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https://en.wikipedia.org/wiki/Index%20of%20a%20subgroup
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In mathematics, specifically group theory, the index of a subgroup H in a group G is the
number of left cosets of H in G, or equivalently, the number of right cosets of H in G.
The index is denoted or or .
Because G is the disjoint union of the left cosets and because each left coset has the same size as H, the index is related to the orders of the two groups by the formula
(interpret the quantities as cardinal numbers if some of them are infinite).
Thus the index measures the "relative sizes" of G and H.
For example, let be the group of integers under addition, and let be the subgroup consisting of the even integers. Then has two cosets in , namely the set of even integers and the set of odd integers, so the index is 2. More generally, for any positive integer n.
When G is finite, the formula may be written as , and it implies
Lagrange's theorem that divides .
When G is infinite, is a nonzero cardinal number that may be finite or infinite.
For example, , but is infinite.
If N is a normal subgroup of G, then is equal to the order of the quotient group , since the underlying set of is the set of cosets of N in G.
Properties
If H is a subgroup of G and K is a subgroup of H, then
If H and K are subgroups of G, then
with equality if . (If is finite, then equality holds if and only if .)
Equivalently, if H and K are subgroups of G, then
with equality if . (If is finite, then equality holds if and only if .)
If G and H are groups and is a homomorphism, then the index of the kernel of in G is equal to the order of the image:
Let G be a group acting on a set X, and let x ∈ X. Then the cardinality of the orbit of x under G is equal to the index of the stabilizer of x:
This is known as the orbit-stabilizer theorem.
As a special case of the orbit-stabilizer theorem, the number of conjugates of an element is equal to the index of the centralizer of x in G.
Similarly, the number of conjugates of a subgroup H in G is equal to the index of the normalizer of H in G.
If H is a subgroup of G, the index of the normal core of H satisfies the following inequality:
where ! denotes the factorial function; this is discussed further below.
As a corollary, if the index of H in G is 2, or for a finite group the lowest prime p that divides the order of G, then H is normal, as the index of its core must also be p, and thus H equals its core, i.e., it is normal.
Note that a subgroup of lowest prime index may not exist, such as in any simple group of non-prime order, or more generally any perfect group.
Examples
The alternating group has index 2 in the symmetric group and thus is normal.
The special orthogonal group has index 2 in the orthogonal group , and thus is normal.
The free abelian group has three subgroups of index 2, namely
.
More generally, if p is prime then has subgroups of index p, corresponding to the nontrivial homomorphisms .
Similarly, the free group has subgroups of index p.
The infinite dihed
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https://en.wikipedia.org/wiki/John%20H.%20Coates
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John Henry Coates (26 January 1945 – 9 May 2022) was an Australian mathematician who was the Sadleirian Professor of Pure Mathematics at the University of Cambridge in the United Kingdom from 1986 to 2012.
Early life and education
Coates was born the son of J. H. Coates and B. L. Lee on 26 January 1945 and grew up in Possum Brush (near Taree) in New South Wales, Australia. Coates Road in Possum Brush is named after the family farm on which he grew up. Before university he spent a summer working for BHP in Newcastle, New South Wales, though he was not successful in gaining a university scholarship with the company. Coates attended Australian National University on scholarship as one of the first undergraduates, from which he gained a BSc degree. He then moved to France, doing further study at the École Normale Supérieure in Paris, before moving again, this time to England.
Career
In England he did postgraduate research at the University of Cambridge, his doctoral dissertation being on p-adic analogues of Baker's method. In 1969, Coates was appointed assistant professor of mathematics at Harvard University in the United States, before moving again in 1972 to Stanford University where he became an associate professor.
In 1975, he returned to England, where he was made a fellow of Emmanuel College, and took up a lectureship. Here he supervised the PhD of Andrew Wiles, and together they proved a partial case of the Birch and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication.
In 1977, Coates moved back to Australia, becoming a professor at the Australian National University, where he had been an undergraduate. The following year, he moved back to France, taking up a professorship at the University of Paris XI at Orsay. In 1985, he returned to the École Normale Supérieure, this time as professor and director of mathematics.
From 1986 until his death, Coates worked in the Department of Pure Mathematics and Mathematical Statistics (DPMMS) of the University of Cambridge. He was head of DPMMS from 1991 to 1997.
His research interests included Iwasawa theory, number theory and arithmetical algebraic geometry.
He served on the Mathematical Sciences jury for the Infosys Prize in 2009.
Awards and honours
Coates was elected a fellow of the Royal Society of London in 1985, and was President of the London Mathematical Society from 1988 to 1990. The latter organisation awarded him the Senior Whitehead Prize in 1997, for "his fundamental research in number theory and for his many contributions to mathematical life both in the UK and internationally". His nomination for the Royal Society reads:
Personal life
Coates married Julie Turner in 1966, with whom he had three sons. He collected Japanese pottery and porcelain. He died on 9 May 2022.
References
1945 births
2022 deaths
20th-century Australian mathematicians
21st-century Australian mathematicians
Alumni of Trinity College, Cambridge
Australian National University alumni
Academ
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https://en.wikipedia.org/wiki/Parabolic
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Parabolic usually refers to something in a shape of a parabola, but may also refer to a parable.
Parabolic may refer to:
In mathematics:
In elementary mathematics, especially elementary geometry:
Parabolic coordinates
Parabolic cylindrical coordinates
parabolic Möbius transformation
Parabolic geometry (disambiguation)
Parabolic spiral
Parabolic line
In advanced mathematics:
Parabolic cylinder function
Parabolic induction
Parabolic Lie algebra
Parabolic partial differential equation
In physics:
Parabolic trajectory
In technology:
Parabolic antenna
Parabolic microphone
Parabolic reflector
Parabolic trough - a type of solar thermal energy collector
Parabolic flight - a way of achieving weightlessness
Parabolic action, or parabolic bending curve - a term often used to refer to a progressive bending curve in fishing rods.
In commodities and stock markets:
Parabolic SAR - a chart pattern in which prices rise or fall with an increasingly steeper slope
Other
Parabolic dune, a sand formation
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https://en.wikipedia.org/wiki/Bertrand%27s%20postulate
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In number theory, Bertrand's postulate is a theorem stating that for any integer , there always exists at least one prime number with
A less restrictive formulation is: for every , there is always at least one prime such that
Another formulation, where is the -th prime, is: for
This statement was first conjectured in 1845 by Joseph Bertrand (1822–1900). Bertrand himself verified his statement for all integers .
His conjecture was completely proved by Chebyshev (1821–1894) in 1852 and so the postulate is also called the Bertrand–Chebyshev theorem or Chebyshev's theorem. Chebyshev's theorem can also be stated as a relationship with , the prime-counting function (number of primes less than or equal to ):
, for all .
Prime number theorem
The prime number theorem (PNT) implies that the number of primes up to x is roughly x/ln(x), so if we replace x with 2x then we see the number of primes up to 2x is asymptotically twice the number of primes up to x (the terms ln(2x) and ln(x) are asymptotically equivalent). Therefore, the number of primes between n and 2n is roughly n/ln(n) when n is large, and so in particular there are many more primes in this interval than are guaranteed by Bertrand's postulate. So Bertrand's postulate is comparatively weaker than the PNT. But PNT is a deep theorem, while Bertrand's Postulate can be stated more memorably and proved more easily, and also makes precise claims about what happens for small values of n. (In addition, Chebyshev's theorem was proved before the PNT and so has historical interest.)
The similar and still unsolved Legendre's conjecture asks whether for every n ≥ 1, there is a prime p such that n2 < p < (n + 1)2. Again we expect that there will be not just one but many primes between n2 and (n + 1)2, but in this case the PNT doesn't help: the number of primes up to x2 is asymptotic to x2/ln(x2) while the number of primes up to (x + 1)2 is asymptotic to (x + 1)2/ln((x + 1)2), which is asymptotic to the estimate on primes up to x2. So unlike the previous case of x and 2x we don't get a proof of Legendre's conjecture even for all large n. Error estimates on the PNT are not (indeed, cannot be) sufficient to prove the existence of even one prime in this interval.
Generalizations
In 1919, Ramanujan (1887–1920) used properties of the Gamma function to give a simpler proof than Chebyshev's. His short paper included a generalization of the postulate, from which would later arise the concept of Ramanujan primes. Further generalizations of Ramanujan primes have also been discovered; for instance, there is a proof that
with pk the kth prime and Rn the nth Ramanujan prime.
Other generalizations of Bertrand's postulate have been obtained using elementary methods. (In the following, n runs through the set of positive integers.) In 2006, M. El Bachraoui proved that there exists a prime between 2n and 3n. In 1973, Denis Hanson proved that there exists a prime between 3n and 4n. Furthermore, in 2011, A
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https://en.wikipedia.org/wiki/Hypergraph
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In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices.
Formally, a directed hypergraph is a pair , where is a set of elements called nodes, vertices, points, or elements and is a set of pairs of subsets of . Each of these pairs is called an edge or hyperedge; the vertex subset is known as its tail or domain, and as its head or codomain.
The order of a hypergraph is the number of vertices in . The size of the hypergraph is the number of edges in . The order of an edge in a directed hypergraph is : that is, the number of vertices in its tail followed by the number of vertices in its head.
The definition above generalizes from a directed graph to a directed hypergraph by defining the head or tail of each edge as a set of vertices ( or ) rather than as a single vertex. A graph is then the special case where each of these sets contains only one element. Hence any standard graph theoretic concept that is independent of the edge orders will generalize to hypergraph theory.
Under one definition, an undirected hypergraph is a directed hypergraph which has a symmetric edge set: If then . For notational simplicity one can remove the "duplicate" hyperedges since the modifier "undirected" is precisely informing us that they exist: If then where means implicitly in.
While graph edges connect only 2 nodes, hyperedges connect an arbitrary number of nodes. However, it is often desirable to study hypergraphs where all hyperedges have the same cardinality; a k-uniform hypergraph is a hypergraph such that all its hyperedges have size k. (In other words, one such hypergraph is a collection of sets, each such set a hyperedge connecting k nodes.) So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. An undirected hypergraph is also called a set system or a family of sets drawn from the universal set.
Hypergraphs can be viewed as incidence structures. In particular, there is a bipartite "incidence graph" or "Levi graph" corresponding to every hypergraph, and conversely, every bipartite graph can be regarded as the incidence graph of a hypergraph when it is 2-colored and it is indicated which color class corresponds to hypergraph vertices and which to hypergraph edges.
Hypergraphs have many other names. In computational geometry, an undirected hypergraph may sometimes be called a range space and then the hyperedges are called ranges.
In cooperative game theory, hypergraphs are called simple games (voting games); this notion is applied to solve problems in social choice theory. In some literature edges are referred to as hyperlinks or connectors.
The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms.
Applications
Undirected hypergraphs are useful in modelling such things as satisfiability problems, databases, machine learning, and
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https://en.wikipedia.org/wiki/Mathematics%20of%20paper%20folding
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The discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability (whether the model can be flattened without damaging it), and the use of paper folds to solve up-to cubic mathematical equations.
Computational origami is a recent branch of computer science that is concerned with studying algorithms that solve paper-folding problems. The field of computational origami has also grown significantly since its inception in the 1990s with Robert Lang's TreeMaker algorithm to assist in the precise folding of bases. Computational origami results either address origami design or origami foldability. In origami design problems, the goal is to design an object that can be folded out of paper given a specific target configuration. In origami foldability problems, the goal is to fold something using the creases of an initial configuration. Results in origami design problems have been more accessible than in origami foldability problems.
History
In 1893, Indian civil servant T. Sundara Row published Geometric Exercises in Paper Folding which used paper folding to demonstrate proofs of geometrical constructions. This work was inspired by the use of origami in the kindergarten system. Row demonstrated an approximate trisection of angles and implied construction of a cube root was impossible.
In 1922, Harry Houdini published "Houdini's Paper Magic," which described origami techniques that drew informally from mathematical approaches that were later formalized.
In 1936 Margharita P. Beloch showed that use of the 'Beloch fold', later used in the sixth of the Huzita–Hatori axioms, allowed the general cubic equation to be solved using origami.
In 1949, R C Yeates' book "Geometric Methods" described three allowed constructions corresponding to the first, second, and fifth of the Huzita–Hatori axioms.
The Yoshizawa–Randlett system of instruction by diagram was introduced in 1961.
In 1980 was reported a construction which enabled an angle to be trisected. Trisections are impossible under Euclidean rules.
Also in 1980, Kōryō Miura and Masamori Sakamaki demonstrated a novel map-folding technique whereby the folds are made in a prescribed parallelogram pattern, which allows the map to be expandable without any right-angle folds in the conventional manner. Their pattern allows the fold lines to be interdependent, and hence the map can be unpacked in one motion by pulling on its opposite ends, and likewise folded by pushing the two ends together. No unduly complicated series of movements are required, and folded Miura-ori can be packed into a very compact shape. In 1985 Miura reported a method of packaging and deployment of large membranes in outer space, and as early as 2012 this technique had been applied to solar panels on spacecraft.
In 1986, Messer reported a construction by which one could double the cube, which is impossible with Euclidean constructions.
The
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https://en.wikipedia.org/wiki/Degeneracy%20%28mathematics%29
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In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case.
The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the angles and the side lengths of a triangle are supposed to be positive. The limiting cases, where one or several of these inequalities become equalities, are degeneracies. In the case of triangles, one has a degenerate triangle if at least one side length or angle is zero. Equivalently, it becomes a "line segment".
Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line, which makes its dimension one. This is similar to the case of a circle, whose dimension shrinks from two to zero as it degenerates into a point. As another example, the solution set of a system of equations that depends on parameters generally has a fixed cardinality and dimension, but cardinality and/or dimension may be different for some exceptional values, called degenerate cases. In such a degenerate case, the solution set is said to be degenerate.
For some classes of composite objects, the degenerate cases depend on the properties that are specifically studied. In particular, the class of objects may often be defined or characterized by systems of equations. In most scenarios, a given class of objects may be defined by several different systems of equations, and these different systems of equations may lead to different degenerate cases, while characterizing the same non-degenerate cases. This may be the reason for which there is no general definition of degeneracy, despite the fact that the concept is widely used and defined (if needed) in each specific situation.
A degenerate case thus has special features which makes it non-generic, or a special case. However, not all non-generic or special cases are degenerate. For example, right triangles, isosceles triangles and equilateral triangles are non-generic and non-degenerate. In fact, degenerate cases often correspond to singularities, either in the object or in some configuration space. For example, a conic section is degenerate if and only if it has singular points (e.g., point, line, intersecting lines).
In geometry
Conic section
A degenerate conic is a conic section (a second-degree plane curve, defined by a polynomial equation of degree two) that fails to be an irreducible curve.
A point is a degenerate circle, namely one with radius 0.
The line is a degenerate case of a parabola if the parabola resides on a tangent plane. In inversive geometry, a line is a degenerate case of a circle, with infinite radius.
Two parallel lines also form a degenerate par
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https://en.wikipedia.org/wiki/Borwein%27s%20algorithm
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In mathematics, Borwein's algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of . They devised several other algorithms. They published the book Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity.
Ramanujan–Sato series
These two are examples of a Ramanujan–Sato series. The related Chudnovsky algorithm uses a discriminant with class number 1.
Class number 2 (1989)
Start by setting
Then
Each additional term of the partial sum yields approximately 25 digits.
Class number 4 (1993)
Start by setting
Then
Each additional term of the series yields approximately 50 digits.
Iterative algorithms
Quadratic convergence (1984)
Start by setting
Then iterate
Then pk converges quadratically to ; that is, each iteration approximately doubles the number of correct digits. The algorithm is not self-correcting; each iteration must be performed with the desired number of correct digits for 's final result.
Cubic convergence (1991)
Start by setting
Then iterate
Then ak converges cubically to ; that is, each iteration approximately triples the number of correct digits.
Quartic convergence (1985)
Start by setting
Then iterate
Then ak converges quartically against ; that is, each iteration approximately quadruples the number of correct digits. The algorithm is not self-correcting; each iteration must be performed with the desired number of correct digits for 's final result.
One iteration of this algorithm is equivalent to two iterations of the Gauss–Legendre algorithm.
A proof of these algorithms can be found here:
Quintic convergence
Start by setting
where is the golden ratio. Then iterate
Then ak converges quintically to (that is, each iteration approximately quintuples the number of correct digits), and the following condition holds:
Nonic convergence
Start by setting
Then iterate
Then ak converges nonically to ; that is, each iteration approximately multiplies the number of correct digits by nine.
See also
Bailey–Borwein–Plouffe formula
Chudnovsky algorithm
Gauss–Legendre algorithm
Ramanujan–Sato series
References
External links
Pi Formulas from Wolfram MathWorld
Pi algorithms
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https://en.wikipedia.org/wiki/Clopen%20set
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In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open closed, and therefore clopen. As described by topologist James Munkres, unlike a door, "a set can be open, or closed, or both, or neither!" emphasizing that the meaning of "open"/"closed" for is unrelated to their meaning for (and so the open/closed door dichotomy does not transfer to open/closed sets). This contrast to doors gave the class of topological spaces known as "door spaces" their name.
Examples
In any topological space the empty set and the whole space are both clopen.
Now consider the space which consists of the union of the two open intervals and of The topology on is inherited as the subspace topology from the ordinary topology on the real line In the set is clopen, as is the set This is a quite typical example: whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen.
Now let be an infinite set under the discrete metricthat is, two points have distance 1 if they're not the same point, and 0 otherwise. Under the resulting metric space, any singleton set is open; hence any set, being the union of single points, is open. Since any set is open, the complement of any set is open too, and therefore any set is closed. So, all sets in this metric space are clopen.
As a less trivial example, consider the space of all rational numbers with their ordinary topology, and the set of all positive rational numbers whose square is bigger than 2. Using the fact that is not in one can show quite easily that is a clopen subset of ( is a clopen subset of the real line ; it is neither open nor closed in )
Properties
A topological space is connected if and only if the only clopen sets are the empty set and itself.
A set is clopen if and only if its boundary is empty.
Any clopen set is a union of (possibly infinitely many) connected components.
If all connected components of are open (for instance, if has only finitely many components, or if is locally connected), then a set is clopen in if and only if it is a union of connected components.
A topological space is discrete if and only if all of its subsets are clopen.
Using the union and intersection as operations, the clopen subsets of a given topological space form a Boolean algebra. Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.
See also
Notes
References
General topology
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https://en.wikipedia.org/wiki/Yearbook
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A yearbook, also known as an annual, is a type of a book published annually. One use is to record, highlight, and commemorate the past year of a school. The term also refers to a book of statistics or facts published annually. A yearbook often has an overarching theme that is present throughout the entire book.
Many high schools, colleges, elementary and middle schools publish yearbooks; however, many schools are dropping yearbooks or decreasing page counts given social media alternatives to a mass-produced physical photographically-oriented record. From 1995 to 2013, the number of U.S. college yearbooks dropped from roughly 2,400 to 1,000.
History
A marble slab commemorating a class of military cadets in Ancient Athens during the time of the Roman Empire is an early example of this sort of document. Proto-yearbooks in the form of scrapbooks appeared in US East Coast schools towards the end of the 17th century. The first formal modern yearbook was the 1806 Profiles of Part of the Class Graduated at Yale College.
Yearbooks by country
Australia
Yearbooks published by Australian schools follow a consistent structure to their North American counterparts. Australian yearbooks function as an annual magazine for the school body, with a significant focus on objectively reporting the events that occurred during the schooling year. Yearbook staff predominantly consists of only one or two school teachers who serve as editors in chief. Australian school yearbooks are predominantly created on A4 paper size, featuring a softcover style front-and-back cover, typically 250 or 300 g/m2 density. Hardcover-style yearbooks are not as common, although exceptions occur.
This is sold as allowing a higher level of student involvement whilst making the workflow simpler and easier for all involved. Additionally, some schools feature a separate yearbook for students in year 2.
Publishing
Australian school yearbooks are primarily published with technology, with a mix of color, spot color, and black and white pages, depending on the school's budget.
India
India does not have a long history of publishing school yearbooks. However, top Business schools and Engineering colleges publish custom yearbooks. This is typically created by the final-year students of the batch. A yearbook or a memory book would consist of testimonials and common pages such as the Director's address and events, and festivals' picture collages.
Most top schools create school magazines that are shared with each student. Some of the early adopters among school students are starting to create custom yearbooks along the same lines as those created by students from the US or Europe. This trend is likely to pick up with the advent of technology platforms that make it easy for students to create them.
Nigeria
In Nigeria, it is very common to find yearbooks in schools as it is in countries such as the US and Canada, though several schools allocate annual funding and publish yearbooks at the end of the
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https://en.wikipedia.org/wiki/List%20of%20Classical%20Greek%20phrases
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Αα
Ageōmétrētos mēdeìs eisítō.
"Let no one untrained in geometry enter."
Motto over the entrance to Plato's Academy (quoted in Elias' commentary on Aristotle's Categories: Eliae in Porphyrii Isagogen et Aristotelis categorias commentaria, CAG XVIII.1, Berlin 1900, p. 118.13–19).
.
Aeì Libúē phérei ti kainón.
"Libya always bears something new", Aristotle, Historia Animalium.
Compare the Latin proverb ex Africa semper aliquid novi 'from Africa always something new', based on Pliny the Elder.
.
Aeì koloiòs parà koloiôi hizánei.
"A jackdaw is always found near a jackdaw"
Similar to English "birds of a feather flock together."
.
Aeì ho theòs geōmetreî.
"God always geometrizes." — Plato
Plutarch elaborated on this phrase in his essay Πῶς Πλάτων ἔλεγε τὸν θεὸν ἀεί γεωμετρεῖν "What is Plato’s meaning when he says that God always applies geometry". Based on the phrase of Plato, above, a present-day mnemonic for π (pi) was derived:
.
Aeì ho theòs ho mégas geōmetreî tò súmpan.
Always the great God applies geometry to the universe.
.
Aetoû gêras, korydoû neótēs.
"An eagle's old age (is worth) a sparrow's youth."
aièn aristeúein
"Ever to Excel"
Motto of the University of St Andrews (founded 1410), the Edinburgh Academy (founded 1824), and Boston College (founded 1863). The source is the sixth book of Homer's Iliad, (Iliad 6. 208) in a speech Glaucus delivers to Diomedes:
"Hippolocus begat me. I claim to be his son, and he sent me to Troy with strict instructions: Ever to excel, to do better than others, and to bring glory to your forebears, who indeed were very great ... This is my ancestry; this is the blood I am proud to inherit."
.
Aíka.
"If."
Plutarch reports that Philip II of Macedon sent word to the Spartans, saying that "if I should invade Laconia, I shall drive you out" (ἂν ἐμβάλω εἰς τὴν Λακωνικήν, ἀναστάτους ὑμᾶς ποιήσω). The Spartans laconically responded with "if."
.
Anánkāi d' oudè theoì mákhontai.
"Not even the gods fight necessity" — Simonides, 8, 20.
Allà tì êi moi taûta perì drûn ḕ perì pétrēn?
"But why all this about oak or stone?"
English : Why waste time on trivial subjects, or "Why make a mountain out of a mole hill?"
Hesiod, Theogony, 35.
.
Andrôn gàr epiphanôn pâsa gê táphos.
For illustrious men have the whole earth for their tomb. Pericles' Funeral Oration from Thucydides, History of the Peloponnesian War 2.43.3
.
Anerrhíphthō kúbos.
Alea iacta est.
Latin: "The die has been cast"; Greek: "Let the die be cast."
Julius Caesar as reported by Plutarch, when he entered Italy with his army in 49 BC. Translated into Latin by Suetonius as alea iacta est.
.
Ánthrōpos métron.
"Man [is] the measure [of all things]"
Motto of Protagoras (as quoted in Plato's Theaetetus 152a).
hápax legómenon
"Once said"
A word that only occurs once.
apò mēkhanês Theós
Deus ex machina
"God from the machine"
The phrase originates from the way deity figures appeared in ancient Greek theaters, held high up by a machine, to solve a problem in the plot.
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https://en.wikipedia.org/wiki/Fr%C3%A9chet%20space
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In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the metric induced by the norm).
All Banach and Hilbert spaces are Fréchet spaces.
Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically Banach spaces.
A Fréchet space is defined to be a locally convex metrizable topological vector space (TVS) that is complete as a TVS, meaning that every Cauchy sequence in converges to some point in (see footnote for more details).
Important note: Not all authors require that a Fréchet space be locally convex (discussed below).
The topology of every Fréchet space is induced by some translation-invariant complete metric.
Conversely, if the topology of a locally convex space is induced by a translation-invariant complete metric then is a Fréchet space.
Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space" to mean a complete metrizable topological vector space, without the local convexity requirement (such a space is today often called an "F-space").
The local convexity requirement was added later by Nicolas Bourbaki.
It's important to note that a sizable number of authors (e.g. Schaefer) use "F-space" to mean a (locally convex) Fréchet space while others do not require that a "Fréchet space" be locally convex.
Moreover, some authors even use "F-space" and "Fréchet space" interchangeably.
When reading mathematical literature, it is recommended that a reader always check whether the book's or article's definition of "-space" and "Fréchet space" requires local convexity.
Definitions
Fréchet spaces can be defined in two equivalent ways: the first employs a translation-invariant metric, the second a countable family of seminorms.
Invariant metric definition
A topological vector space is a Fréchet space if and only if it satisfies the following three properties:
It is locally convex.
Its topology be induced by a translation-invariant metric, that is, a metric such that for all This means that a subset of is open if and only if for every there exists an such that is a subset of
Some (or equivalently, every) translation-invariant metric on inducing the topology of is complete.
Assuming that the other two conditions are satisfied, this condition is equivalent to being a complete topological vector space, meaning that is a complete uniform space when it is endowed with its canonical uniformity (this canonical uniformity is independent of any metric on and is defined entirely in terms of vector subtraction and 's neighborhoods of the origin; moreover, the uniformity induced by any (topology-defining) translation invariant metric on is identical to this canonical uniformity).
Note there is no natural notion of distance between two p
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https://en.wikipedia.org/wiki/Aggregate%20pattern
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An Aggregate pattern can refer to concepts in either statistics or computer programming. Both uses deal with considering a large case as composed of smaller, simpler, pieces.
Statistics
An aggregate pattern is an important statistical concept in many fields that rely on statistics to predict the behavior of large groups, based on the tendencies of subgroups to consistently behave in a certain way. It is particularly useful in sociology, economics, psychology, and criminology.
Computer programming
In Design Patterns, an aggregate is not a design pattern but rather refers to an object such as a list, vector, or generator which provides an interface for creating iterators. The following example code is in Python.
def fibonacci(n: int):
a, b = 0, 1
count = 0
while count < n:
count += 1
a, b = b, a + b
yield a
for x in fibonacci(10):
print(x)
def fibsum(n: int) -> int:
total = 0
for x in fibonacci(n):
total += x
return total
def fibsum_alt(n: int) -> int:
"""
Alternate implementation. demonstration that Python's built-in function sum()
works with arbitrary iterators.
"""
return sum(fibonacci(n))
myNumbers = [1, 7, 4, 3, 22]
def average(g) -> float:
return float(sum(g)) / len(g) # In Python 3 the cast to float is no longer be necessary
Python hides essentially all of the details using the iterator protocol. Confusingly, Design Patterns uses "aggregate" to refer to the blank in the code for x in ___: which is unrelated to the term "aggregation". Neither of these terms refer to the statistical aggregation of data such as the act of adding up the Fibonacci sequence or taking the average of a list of numbers.
See also
Visitor pattern
Template class
Facade pattern
Type safety
Functional programming
References
Software design patterns
Articles with example Python (programming language) code
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https://en.wikipedia.org/wiki/Automorphic%20number
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In mathematics, an automorphic number (sometimes referred to as a circular number) is a natural number in a given number base whose square "ends" in the same digits as the number itself.
Definition and properties
Given a number base , a natural number with digits is an automorphic number if is a fixed point of the polynomial function over , the ring of integers modulo . As the inverse limit of is , the ring of -adic integers, automorphic numbers are used to find the numerical representations of the fixed points of over .
For example, with , there are four 10-adic fixed points of , the last 10 digits of which are one of these
Thus, the automorphic numbers in base 10 are 0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376, 59918212890625, ... .
A fixed point of is a zero of the function . In the ring of integers modulo , there are zeroes to , where the prime omega function is the number of distinct prime factors in . An element in is a zero of if and only if or for all . Since there are two possible values in , and there are such , there are zeroes of , and thus there are fixed points of . According to Hensel's lemma, if there are zeroes or fixed points of a polynomial function modulo , then there are corresponding zeroes or fixed points of the same function modulo any power of , and this remains true in the inverse limit. Thus, in any given base there are -adic fixed points of .
As 0 is always a zero-divisor, 0 and 1 are always fixed points of , and 0 and 1 are automorphic numbers in every base. These solutions are called trivial automorphic numbers. If is a prime power, then the ring of -adic numbers has no zero-divisors other than 0, so the only fixed points of are 0 and 1. As a result, nontrivial automorphic numbers, those other than 0 and 1, only exist when the base has at least two distinct prime factors.
Automorphic numbers in base b
All -adic numbers are represented in base , using A−Z to represent digit values 10 to 35.
Extensions
Automorphic numbers can be extended to any such polynomial function of degree with b-adic coefficients . These generalised automorphic numbers form a tree.
a-automorphic numbers
An -automorphic number occurs when the polynomial function is
For example, with and , as there are two fixed points for in ( and ), according to Hensel's lemma there are two 10-adic fixed points for ,
so the 2-automorphic numbers in base 10 are 0, 8, 88, 688, 4688...
Trimorphic numbers
A trimorphic number or spherical number occurs when the polynomial function is . All automorphic numbers are trimorphic. The terms circular and spherical were formerly used for the slightly different case of a number whose powers all have the same last digit as the number itself.
For base , the trimorphic numbers are:
0, 1, 4, 5, 6, 9, 24, 2
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https://en.wikipedia.org/wiki/Automorphic
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Automorphic may refer to
Automorphic number, in mathematics
Automorphic form, in mathematics
Automorphic representation, in mathematics
Automorphic L-function, in mathematics
Automorphism, in mathematics
Rock microstructure#Crystal shapes
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https://en.wikipedia.org/wiki/Ackermann%20steering%20geometry
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The Ackermann steering geometry is a geometric arrangement of linkages in the steering of a car or other vehicle designed to solve the problem of wheels on the inside and outside of a turn needing to trace out circles of different radii.
It was invented by the German carriage builder Georg Lankensperger in Munich in 1816, then patented by his agent in England, Rudolph Ackermann (1764–1834) in 1818 for horse-drawn carriages. Erasmus Darwin may have a prior claim as the inventor dating from 1758. He devised his steering system because he was injured when a carriage tipped over.
Advantages
The intention of Ackermann geometry is to avoid the need for tires to slip sideways when following the path around a curve.
The geometrical solution to this is for all wheels to have their axles arranged as radii of circles with a common centre point. As the rear wheels are fixed, this centre point must be on a line extended from the rear axle. Intersecting the axes of the front wheels on this line as well requires that the inside front wheel be turned, when steering, through a greater angle than the outside wheel.
Rather than the preceding "turntable" steering, where both front wheels turned around a common pivot, each wheel gained its own pivot, close to its own hub. While more complex, this arrangement enhances controllability by avoiding large inputs from road surface variations being applied to the end of a long lever arm, as well as greatly reducing the fore-and-aft travel of the steered wheels. A linkage between these hubs pivots the two wheels together, and by careful arrangement of the linkage dimensions the Ackermann geometry could be approximated. This was achieved by making the linkage not a simple parallelogram, but by making the length of the track rod (the moving link between the hubs) shorter than that of the axle, so that the steering arms of the hubs appeared to "toe out". As the steering moved, the wheels turned according to Ackermann, with the inner wheel turning further. If the track rod is placed ahead of the axle, it should instead be longer in comparison, thus preserving this same "toe out".
Design and choice of geometry
A simple approximation to perfect Ackermann steering geometry may be generated by moving the steering pivot points inward so as to lie on a line drawn between the steering kingpins, which is the pivot point, and the centre of the rear axle. The steering pivot points are joined by a rigid bar called the tie rod, which can also be part of the steering mechanism, in the form of a rack and pinion for instance. With perfect Ackermann, at any angle of steering, the centre point of all of the circles traced by all wheels will lie at a common point.
Modern cars do not use pure Ackermann steering, partly because it ignores important dynamic and compliant effects, but the principle is sound for low-speed maneuvers. Some racing cars use reverse Ackermann geometry to compensate for the large difference in slip angle between the
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https://en.wikipedia.org/wiki/Games%20started
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In baseball statistics, games started (denoted by GS) indicates the number of games that a pitcher has started for his team. A pitcher is credited with starting the game if he throws the first pitch to the first opposing batter. If a player is listed in the starting lineup as the team's pitcher, but is replaced before facing an opposing batter, the player is credited with a game pitched but not a game started; there have been instances in major league history in which a starting pitcher was removed before his first pitch due to an injury, perhaps suffered while batting or running the bases during the top half of the first inning.
The all-time leader for games started is Cy Young with 815 over a 22-year career. The players with the most starts in a single season are Pud Galvin and Will White, each with 75 games started.
For position players, games started is also used to denote the number of times their names appear in a team's starting lineup during the season.
The statistic is also used in football.
See also
List of Major League Baseball leaders in games started
List of Major League Baseball leaders in games finished
Complete game
Shutouts in baseball
References
Pitching statistics
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https://en.wikipedia.org/wiki/Lists%20of%20integrals
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Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.
Historical development of integrals
A compilation of a list of integrals (Integraltafeln) and techniques of integral calculus was published by the German mathematician (also spelled Meyer Hirsch) in 1810. These tables were republished in the United Kingdom in 1823. More extensive tables were compiled in 1858 by the Dutch mathematician David Bierens de Haan for his Tables d'intégrales définies, supplemented by Supplément aux tables d'intégrales définies in ca. 1864. A new edition was published in 1867 under the title Nouvelles tables d'intégrales définies.
These tables, which contain mainly integrals of elementary functions, remained in use until the middle of the 20th century. They were then replaced by the much more extensive tables of Gradshteyn and Ryzhik. In Gradshteyn and Ryzhik, integrals originating from the book by Bierens de Haan are denoted by BI.
Not all closed-form expressions have closed-form antiderivatives; this study forms the subject of differential Galois theory, which was initially developed by Joseph Liouville in the 1830s and 1840s, leading to Liouville's theorem which classifies which expressions have closed-form antiderivatives. A simple example of a function without a closed-form antiderivative is , whose antiderivative is (up to constants) the error function.
Since 1968 there is the Risch algorithm for determining indefinite integrals that can be expressed in term of elementary functions, typically using a computer algebra system. Integrals that cannot be expressed using elementary functions can be manipulated symbolically using general functions such as the Meijer G-function.
Lists of integrals
More detail may be found on the following pages for the lists of integrals:
List of integrals of rational functions
List of integrals of irrational functions
List of integrals of trigonometric functions
List of integrals of inverse trigonometric functions
List of integrals of hyperbolic functions
List of integrals of inverse hyperbolic functions
List of integrals of exponential functions
List of integrals of logarithmic functions
List of integrals of Gaussian functions
Gradshteyn, Ryzhik, Geronimus, Tseytlin, Jeffrey, Zwillinger, and Moll's (GR) Table of Integrals, Series, and Products contains a large collection of results. An even larger, multivolume table is the Integrals and Series by Prudnikov, Brychkov, and Marichev (with volumes 1–3 listing integrals and series of elementary and special functions, volume 4–5 are tables of Laplace transforms). More compact collections can be found in e.g. Brychkov, Marichev, Prudnikov's Tables of Indefinite Integrals, or as chapters i
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https://en.wikipedia.org/wiki/James%20Joseph%20Sylvester
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James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership role in American mathematics in the later half of the 19th century as a professor at the Johns Hopkins University and as founder of the American Journal of Mathematics. At his death, he was a professor at Oxford University.
Biography
James Joseph was born in London on 3 September 1814, the son of Abraham Joseph, a Jewish merchant. James later adopted the surname Sylvester when his older brother did so upon emigration to the United States.
At the age of 14, Sylvester was a student of Augustus De Morgan at the University of London. His family withdrew him from the University after he was accused of stabbing a fellow student with a knife. Subsequently, he attended the Liverpool Royal Institution.
Sylvester began his study of mathematics at St John's College, Cambridge in 1831, where his tutor was John Hymers. Although his studies were interrupted for almost two years due to a prolonged illness, he nevertheless ranked second in Cambridge's famous mathematical examination, the tripos, for which he sat in 1837. However, Sylvester was not issued a degree, because graduates at that time were required to state their acceptance of the Thirty-nine Articles of the Church of England, and Sylvester could not do so because he was Jewish. For the same reason, he was unable to compete for a Fellowship or obtain a Smith's prize. In 1838, Sylvester became professor of natural philosophy at University College London and in 1839 a Fellow of the Royal Society of London. In 1841, he was awarded a BA and an MA by Trinity College Dublin. In the same year he moved to the United States to become a professor of mathematics at the University of Virginia, but left after less than four months. A student who had been reading a newspaper in one of Sylvester's lectures insulted him and Sylvester struck him with a sword stick. The student collapsed in shock and Sylvester believed (wrongly) that he had killed him. Sylvester resigned when he felt that the university authorities had not sufficiently disciplined the student. He moved to New York City and began friendships with the Harvard mathematician Benjamin Peirce (father of Charles Sanders Peirce) and the Princeton physicist Joseph Henry. However, he left in November 1843 after being denied appointment as Professor of Mathematics at Columbia College (now University), again for his Judaism, and returned to England.
On his return to England, he was hired in 1844 by the Equity and Law Life Assurance Society for which he developed successful actuarial models and served as de facto CEO, a position that required a law degree. As a result, he studied for the Bar, meeting a fellow British mathematician studying law, Arthur Cayley, with whom he made significant contributions to invariant theory and als
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https://en.wikipedia.org/wiki/Nth%20root
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In mathematics, taking the nth root is an operation involving two numbers, the radicand and the index or degree. Taking the nth root is written as , where is the radicand and n is the index (also sometimes called the degree). This is pronounced as "the nth root of x". The definition then of an nth root of a number x is a number r (the root) which, when raised to the power of the positive integer n, yields x:
A root of degree 2 is called a square root (usually written without the n as just ) and a root of degree 3, a cube root (written ). Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an th root is a root extraction.
For example, 3 is a square root of 9, since 3 = 9, and −3 is also a square root of 9, since (−3) = 9.
Any non-zero number considered as a complex number has different complex th roots, including the real ones (at most two). The th root of 0 is zero for all positive integers , since . In particular, if is even and is a positive real number, one of its th roots is real and positive, one is negative, and the others (when ) are non-real complex numbers; if is even and is a negative real number, none of the th roots is real. If is odd and is real, one th root is real and has the same sign as , while the other () roots are not real. Finally, if is not real, then none of its th roots are real.
Roots of real numbers are usually written using the radical symbol or radix , with denoting the positive square root of if is positive; for higher roots, denotes the real th root if is odd, and the positive nth root if is even and is positive. In the other cases, the symbol is not commonly used as being ambiguous.
When complex th roots are considered, it is often useful to choose one of the roots, called principal root, as a principal value. The common choice is to choose the principal th root of as the th root with the greatest real part, and when there are two (for real and negative), the one with a positive imaginary part. This makes the th root a function that is real and positive for real and positive, and is continuous in the whole complex plane, except for values of that are real and negative.
A difficulty with this choice is that, for a negative real number and an odd index, the principal th root is not the real one. For example, has three cube roots, , and The real cube root is and the principal cube root is
An unresolved root, especially one using the radical symbol, is sometimes referred to as a surd or a radical. Any expression containing a radical, whether it is a square root, a cube root, or a higher root, is called a radical expression, and if it contains no transcendental functions or transcendental numbers it is called an algebraic expression.
The positive root of a number is the inverse operation of Exponentiation with positive integer exponents. Roots can also be defined as special cases of exponentiation, where the exponent is
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https://en.wikipedia.org/wiki/Random%20walk
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In mathematics, a random walk, sometimes known as a drunkard's walk, is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line which starts at 0, and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, or the price of a fluctuating stock and the financial status of a gambler. Random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology, economics, and sociology. The term random walk was first introduced by Karl Pearson in 1905.
Lattice random walk
A popular random walk model is that of a random walk on a regular lattice, where at each step the location jumps to another site according to some probability distribution. In a simple random walk, the location can only jump to neighboring sites of the lattice, forming a lattice path. In a simple symmetric random walk on a locally finite lattice, the probabilities of the location jumping to each one of its immediate neighbors are the same. The best-studied example is the random walk on the d-dimensional integer lattice (sometimes called the hypercubic lattice) .
If the state space is limited to finite dimensions, the random walk model is called a simple bordered symmetric random walk, and the transition probabilities depend on the location of the state because on margin and corner states the movement is limited.
One-dimensional random walk
An elementary example of a random walk is the random walk on the integer number line, , which starts at 0 and at each step moves +1 or −1 with equal probability.
This walk can be illustrated as follows. A marker is placed at zero on the number line, and a fair coin is flipped. If it lands on heads, the marker is moved one unit to the right. If it lands on tails, the marker is moved one unit to the left. After five flips, the marker could now be on -5, -3, -1, 1, 3, 5. With five flips, three heads and two tails, in any order, it will land on 1. There are 10 ways of landing on 1 (by flipping three heads and two tails), 10 ways of landing on −1 (by flipping three tails and two heads), 5 ways of landing on 3 (by flipping four heads and one tail), 5 ways of landing on −3 (by flipping four tails and one head), 1 way of landing on 5 (by flipping five heads), and 1 way of landing on −5 (by flipping five tails). See the figure below for an illustration of the possible outcomes of 5 flips.
To define this walk formally, take independent random variables , where each variable is either 1 or −1, with a 50% probability for either value, and set and The series is called the simple random walk on . This series (the sum of the sequence of −1s and 1s) gives the net distance walked, if each part of th
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https://en.wikipedia.org/wiki/Spearman%27s%20rank%20correlation%20coefficient
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In statistics, Spearman's rank correlation coefficient or Spearman's ρ, named after Charles Spearman and often denoted by the Greek letter (rho) or as , is a nonparametric measure of rank correlation (statistical dependence between the rankings of two variables). It assesses how well the relationship between two variables can be described using a monotonic function.
The Spearman correlation between two variables is equal to the Pearson correlation between the rank values of those two variables; while Pearson's correlation assesses linear relationships, Spearman's correlation assesses monotonic relationships (whether linear or not). If there are no repeated data values, a perfect Spearman correlation of +1 or −1 occurs when each of the variables is a perfect monotone function of the other.
Intuitively, the Spearman correlation between two variables will be high when observations have a similar (or identical for a correlation of 1) rank (i.e. relative position label of the observations within the variable: 1st, 2nd, 3rd, etc.) between the two variables, and low when observations have a dissimilar (or fully opposed for a correlation of −1) rank between the two variables.
Spearman's coefficient is appropriate for both continuous and discrete ordinal variables. Both Spearman's and Kendall's can be formulated as special cases of a more general correlation coefficient.
Definition and calculation
The Spearman correlation coefficient is defined as the Pearson correlation coefficient between the rank variables.
For a sample of size n, the n raw scores are converted to ranks , and is computed as
where
denotes the usual Pearson correlation coefficient, but applied to the rank variables,
is the covariance of the rank variables,
and are the standard deviations of the rank variables.
Only if all n ranks are distinct integers, it can be computed using the popular formula
where
is the difference between the two ranks of each observation,
n is the number of observations.
Consider a bivariate sample with corresponding ranks .
Then the Spearman correlation coefficient of is
where, as usual,
,
,
,
and
,
We shall show that can be expressed purely in terms of
,
provided we assume that there be no ties within each sample.
Under this assumption, we have that can be viewed as random variables
distributed like a uniformly distributed random variable, , on .
Hence
and
,
where
,
,
and thus
.
(These sums can be computed using the formulas for the triangular number and Square pyramidal number,
or basic summation results from discrete mathematics.)
Observe now that
Putting this all together thus yields
Identical values are usually each assigned fractional ranks equal to the average of their positions in the ascending order of the values, which is equivalent to averaging over all possible permutations.
If ties are present in the data set, the simplified formula above yields incorrect results: Only if in both variables all ranks are
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https://en.wikipedia.org/wiki/Unitary%20transformation
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In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Formal definition
More precisely, a unitary transformation is an isomorphism between two inner product spaces (such as Hilbert spaces). In other words, a unitary transformation is a bijective function
between two inner product spaces, and such that
Properties
A unitary transformation is an isometry, as one can see by setting in this formula.
Unitary operator
In the case when and are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator.
Antiunitary transformation
A closely related notion is that of antiunitary transformation, which is a bijective function
between two complex Hilbert spaces such that
for all and in , where the horizontal bar represents the complex conjugate.
See also
Antiunitary
Orthogonal transformation
Time reversal
Unitary group
Unitary operator
Unitary matrix
Wigner's theorem
Unitary transformations in quantum mechanics
Linear algebra
Functional analysis
ru:Унитарное преобразование
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https://en.wikipedia.org/wiki/Elliptic%20geometry
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Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.
The appearance of this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally, including hyperbolic geometry.
Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interior angles of any triangle is always greater than 180°.
Definitions
In elliptic geometry, two lines perpendicular to a given line must intersect. In fact, the perpendiculars on one side all intersect at a single point called the absolute pole of that line. The perpendiculars on the other side also intersect at a point. However, unlike in spherical geometry, the poles on either side are the same. This is because there are no antipodal points in elliptic geometry. For example, this is achieved in the hyperspherical model (described below) by making the "points" in our geometry actually be pairs of opposite points on a sphere. The reason for doing this is that it allows elliptic geometry to satisfy the axiom that there is a unique line passing through any two points.
Every point corresponds to an absolute polar line of which it is the absolute pole. Any point on this polar line forms an absolute conjugate pair with the pole. Such a pair of points is orthogonal, and the distance between them is a quadrant.
The distance between a pair of points is proportional to the angle between their absolute polars.
As explained by H. S. M. Coxeter:
The name "elliptic" is possibly misleading. It does not imply any direct connection with the curve called an ellipse, but only a rather far-fetched analogy. A central conic is called an ellipse or a hyperbola according as it has no asymptote or two asymptotes. Analogously, a non-Euclidean plane is said to be elliptic or hyperbolic according as each of its lines contains no point at infinity or two points at infinity.
Two dimensions
Elliptic plane
The elliptic plane is the real projective plane provided with a metric. Kepler and Desargues used the gnomonic projection to relate a plane σ to points on a hemisphere tangent to it. With O the center of the hemisphere, a point P in σ determines a line OP intersecting the hemisphere, and any line L ⊂ σ determines a plane OL which intersects the hemisphere in half of a great circle. The hemisphere is bounded by a plane through O and parallel to σ. No ordinary line of σ corresponds to this plane; instead a line at infinity is appended to σ. As any line in this extension of σ c
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https://en.wikipedia.org/wiki/Truncated%20binary%20encoding
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Truncated binary encoding is an entropy encoding typically used for uniform probability distributions with a finite alphabet. It is parameterized by an alphabet with total size of number n. It is a slightly more general form of binary encoding when n is not a power of two.
If n is a power of two, then the coded value for 0 ≤ x < n is the simple binary code for x of length log2(n).
Otherwise let k = floor(log2(n)), such that 2k < n < 2k+1
and let u = 2k+1 − n.
Truncated binary encoding assigns the first u symbols codewords of length k and then assigns the remaining n − u symbols the last n − u codewords of length k + 1. Because all the codewords of length k + 1 consist of an unassigned codeword of length k with a "0" or "1" appended, the resulting code is a prefix code.
History
Used since at least 1984, phase-in codes, also known as economy codes,
are also known as truncated binary encoding.
Example with n = 5
For example, for the alphabet {0, 1, 2, 3, 4}, n = 5 and 22 ≤ n < 23, hence k = 2 and u = 23 − 5 = 3. Truncated binary encoding assigns the first u symbols the codewords 00, 01, and 10, all of length 2, then assigns the last n − u symbols the codewords 110 and 111, the last two codewords of length 3.
For example, if n is 5, plain binary encoding and truncated binary encoding allocates the following codewords. Digits shown struck are not transmitted in truncated binary.
It takes 3 bits to encode n using straightforward binary encoding, hence 23 − n = 8 − 5 = 3 are unused.
In numerical terms, to send a value x, where 0 ≤ x < n, and where there are 2k ≤ n < 2k+1 symbols, there are u = 2k+1 − n unused entries when the alphabet size is rounded up to the nearest power of two. The process to encode the number x in truncated binary is: if x is less than u, encode it in k binary bits; if x is greater than or equal to u, encode the value x + u in k + 1 binary bits.
Example with n = 10
Another example, encoding an alphabet of size 10 (between 0 and 9) requires 4 bits, but there are 24 − 10 = 6 unused codes, so input values less than 6 have the first bit discarded, while input values greater than or equal to 6 are offset by 6 to the end of the binary space. (Unused patterns are not shown in this table.)
To decode, read the first k bits. If they encode a value less than u, decoding is complete. Otherwise, read an additional bit and subtract u from the result.
Example with n = 7
Here is a more extreme case: with n = 7 the next power of 2 is 8, so k = 2 and u = 23 − 7 = 1:
This last example demonstrates that a leading zero bit does not always indicate a short code; if u < 2k, some long codes will begin with a zero bit.
Simple algorithm
Generate the truncated binary encoding for a value x, 0 ≤ x < n, where n > 0 is the size of the alphabet containing x. n need not be a power of two.
string TruncatedBinary (int x, int n)
{
// Set k = floor(log2(n)), i.e., k such that 2^k <= n < 2^(k+1).
int k = 0, t = n;
while (t > 1) { k++; t >>= 1; }
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https://en.wikipedia.org/wiki/Subsequence
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In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence is a subsequence of obtained after removal of elements and The relation of one sequence being the subsequence of another is a preorder.
Subsequences can contain consecutive elements which were not consecutive in the original sequence. A subsequence which consists of a consecutive run of elements from the original sequence, such as from is a substring. The substring is a refinement of the subsequence.
The list of all subsequences for the word "apple" would be "a", "ap", "al", "ae", "app", "apl", "ape", "ale", "appl", "appe", "aple", "apple", "p", "pp", "pl", "pe", "ppl", "ppe", "ple", "pple", "l", "le", "e", "" (empty string).
Common subsequence
Given two sequences and a sequence is said to be a common subsequence of and if is a subsequence of both and For example, if
then is said to be a common subsequence of and
This would be the longest common subsequence, since only has length 3, and the common subsequence has length 4. The longest common subsequence of and is
Applications
Subsequences have applications to computer science, especially in the discipline of bioinformatics, where computers are used to compare, analyze, and store DNA, RNA, and protein sequences.
Take two sequences of DNA containing 37 elements, say:
SEQ1 = ACGGTGTCGTGCTATGCTGATGCTGACTTATATGCTA
SEQ2 = CGTTCGGCTATCGTACGTTCTATTCTATGATTTCTAA
The longest common subsequence of sequences 1 and 2 is:
LCS(SEQ1,SEQ2) = CGTTCGGCTATGCTTCTACTTATTCTA
This can be illustrated by highlighting the 27 elements of the longest common subsequence into the initial sequences:
SEQ1 = AGGTGAGGAG
SEQ2 = CTAGTTAGTA
Another way to show this is to align the two sequences, that is, to position elements of the longest common subsequence in a same column (indicated by the vertical bar) and to introduce a special character (here, a dash) for padding of arisen empty subsequences:
SEQ1 = ACGGTGTCGTGCTAT-G--C-TGATGCTGA--CT-T-ATATG-CTA-
| || ||| ||||| | | | | || | || | || | |||
SEQ2 = -C-GT-TCG-GCTATCGTACGT--T-CT-ATTCTATGAT-T-TCTAA
Subsequences are used to determine how similar the two strands of DNA are, using the DNA bases: adenine, guanine, cytosine and thymine.
Theorems
Every infinite sequence of real numbers has an infinite monotone subsequence (This is a lemma used in the proof of the Bolzano–Weierstrass theorem).
Every infinite bounded sequence in has a convergent subsequence (This is the Bolzano–Weierstrass theorem).
For all integers and every finite sequence of length at least contains a monotonically increasing subsequence of length a monotonically decreasing subsequence of length (This is the Erdős–Szekeres theorem).
A metric space is compact if every sequence in has a convergent subsequence whose limit is in .
See also
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https://en.wikipedia.org/wiki/Haken%20manifold
|
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface.
A 3-manifold finitely covered by a Haken manifold is said to be virtually Haken. The Virtually Haken conjecture asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken. This conjecture was proven by Ian Agol.
Haken manifolds were introduced by . proved that Haken manifolds have a hierarchy, where they can be split up into 3-balls along incompressible surfaces. Haken also showed that there was a finite procedure to find an incompressible surface if the 3-manifold had one. gave an algorithm to determine if a 3-manifold was Haken.
Normal surfaces are ubiquitous in the theory of Haken manifolds and their simple and rigid structure leads quite naturally to algorithms.
Haken hierarchy
We will consider only the case of orientable Haken manifolds, as this simplifies the discussion; a regular neighborhood of an orientable surface in an orientable 3-manifold is just a "thickened up" version of the surface, i.e., a trivial I-bundle. So the regular neighborhood is a 3-dimensional submanifold with boundary containing two copies of the surface.
Given an orientable Haken manifold M, by definition it contains an orientable, incompressible surface S. Take the regular neighborhood of S and delete its interior from M, resulting in M' . In effect, we've cut M along the surface S. (This is analogous, in one less dimension, to cutting a surface along a circle or arc.) It is a theorem that any orientable compact manifold with a boundary component that is not a sphere has an infinite first homology group, which implies that it has a properly embedded 2-sided non-separating incompressible surface, and so is again a Haken manifold. Thus, we can pick another incompressible surface in M' , and cut along that. If eventually this sequence of cutting results in a manifold whose pieces (or components) are just 3-balls, we call this sequence a hierarchy.
Applications
The hierarchy makes proving certain kinds of theorems about Haken manifolds a matter of induction. One proves the theorem for 3-balls. Then one proves that if the theorem is true for pieces resulting from a cutting of a Haken manifold, then it is true for that Haken manifold. The key here is that the cutting takes place along a surface that was very "nice", i.e., incompressible. This makes proving the induction step feasible in many cases.
Haken sketched out a proof of an algorithm to check if two Haken manifolds were homeomorphic or not. His outline was filled in by substantive efforts by Friedhelm Waldhausen, Klaus Johannson, Geoffrey Hemion, Sergeĭ Matveev, et al. Since there is an a
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https://en.wikipedia.org/wiki/Logarithmic
|
Logarithmic can refer to:
Logarithm, a transcendental function in mathematics
Logarithmic scale, the use of the logarithmic function to describe measurements
Logarithmic spiral,
Logarithmic growth
Logarithmic distribution, a discrete probability distribution
Natural logarithm
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https://en.wikipedia.org/wiki/Median%20test
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In statistics, Mood's median test is a special case of Pearson's chi-squared test. It is a nonparametric test that tests the null hypothesis that the medians of the populations from which two or more samples are drawn are identical. The data in each sample are assigned to two groups, one consisting of data whose values are higher than the median value in the two groups combined, and the other consisting of data whose values are at the median or below. A Pearson's chi-squared test is then used to determine whether the observed frequencies in each sample differ from expected frequencies derived from a distribution combining the two groups.
Relation to other tests
The test has low power (efficiency) for moderate to large sample sizes. The Wilcoxon–Mann–Whitney U two-sample test or its generalisation for more samples, the Kruskal–Wallis test, can often be considered instead. The relevant aspect of the median test is that it only considers the position of each observation relative to the overall median, whereas the Wilcoxon–Mann–Whitney test takes the ranks of each observation into account. Thus the other mentioned tests are usually more powerful than the median test. Moreover, the median test can only be used for quantitative data.
It is crucial to note, however, that the null hypothesis verified by the Wilcoxon–Mann–Whitney U (and so the Kruskal–Wallis test) is not about medians. The test is sensitive also to differences in scale parameters and symmetry. As a consequence, if the Wilcoxon–Mann–Whitney U test rejects the null hypothesis, one cannot say that the rejection was caused only by the shift in medians. It is easy to prove by simulations, where samples with equal medians, yet different scales and shapes, lead the Wilcoxon–Mann–Whitney U test to fail completely.
However, although the alternative Kruskal-Wallis test does not assume normal distributions, it does assume that the variance is approximately equal across samples. Hence, in situations where that assumption does not hold, the median test is an appropriate test. Moreover, Siegel & Castellan (1988, p. 124) suggest that there is no alternative to the median test when one or more observations are "off the scale."
See also
Sign test – a paired alternative to the median test.
References
Corder, G.W. & Foreman, D.I. (2014). Nonparametric Statistics: A Step-by-Step Approach, Wiley. .
Siegel, S., & Castellan, N. J. Jr. (1988, 2nd ed.). Nonparametric statistics for the behavioral sciences. New York: McGraw–Hill.
Friedlin, B. & Gastwirth, J. L. (2000). Should the median test be retired from general use? The American Statistician, 54, 161–164.
Statistical tests
Nonparametric statistics
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https://en.wikipedia.org/wiki/Markov%20chain%20Monte%20Carlo
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In statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain a sample of the desired distribution by recording states from the chain. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution. Various algorithms exist for constructing chains, including the Metropolis–Hastings algorithm.
Application domains
MCMC methods are primarily used for calculating numerical approximations of multi-dimensional integrals, for example in Bayesian statistics, computational physics, computational biology and computational linguistics.
In Bayesian statistics, the recent development of MCMC methods has made it possible to compute large hierarchical models that require integrations over hundreds to thousands of unknown parameters.
In rare event sampling, they are also used for generating samples that gradually populate the rare failure region.
General explanation
Markov chain Monte Carlo methods create samples from a continuous random variable, with probability density proportional to a known function. These samples can be used to evaluate an integral over that variable, as its expected value or variance.
Practically, an ensemble of chains is generally developed, starting from a set of points arbitrarily chosen and sufficiently distant from each other. These chains are stochastic processes of "walkers" which move around randomly according to an algorithm that looks for places with a reasonably high contribution to the integral to move into next, assigning them higher probabilities.
Random walk Monte Carlo methods are a kind of random simulation or Monte Carlo method. However, whereas the random samples of the integrand used in a conventional Monte Carlo integration are statistically independent, those used in MCMC are autocorrelated. Correlations of samples introduces the need to use the Markov chain central limit theorem when estimating the error of mean values.
These algorithms create Markov chains such that they have an equilibrium distribution which is proportional to the function given.
Reducing correlation
While MCMC methods were created to address multi-dimensional problems better than generic Monte Carlo algorithms, when the number of dimensions rises they too tend to suffer the curse of dimensionality: regions of higher probability tend to stretch and get lost in an increasing volume of space that contributes little to the integral. One way to address this problem could be shortening the steps of the walker, so that it doesn't continuously try to exit the highest probability region, though this way the process would be highly autocorrelated and expensive (i.e. many steps would be required for an accurate result). More sophisticated methods such as Hamiltonian Monte Carlo and the Wang and Landau algorit
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https://en.wikipedia.org/wiki/Rational%20%28disambiguation%29
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Rational may refer to:
Rational number, a number that can be expressed as a ratio of two integers
Rational point of an algebraic variety, a point defined over the rational numbers
Rational function, a function that may be defined as the quotient of two polynomials
Rational fraction, an expression built from the integers and some variables by addition, subtraction, multiplication and division
Rational Software, a software company now owned by IBM
Tenberry Software, formerly Rational Systems, a defunct American software company
Rational AG, a German manufacturer of food processors
RationaL, stage name of Canadian hip-hop artist Matt Brotzel
The Rationals, a former American rock and roll band
Rational, a personality classification in the Keirsey Temperament Sorter
Priestly breastplate, called a 'rational' in older Biblical translations, from the Vulgate name for the breastplate: 'rationale'
See also
Rationality
Rationale (disambiguation)
Rationalism (disambiguation)
Rationalization (disambiguation)
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https://en.wikipedia.org/wiki/Probability-generating%20function
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In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(X = i) in the probability mass function for a random variable X, and to make available the well-developed theory of power series with non-negative coefficients.
Definition
Univariate case
If X is a discrete random variable taking values in the non-negative integers {0,1, ...}, then the probability generating function of X is defined as
where p is the probability mass function of X. Note that the subscripted notations GX and pX are often used to emphasize that these pertain to a particular random variable X, and to its distribution. The power series converges absolutely at least for all complex numbers z with |z| ≤ 1; in many examples the radius of convergence is larger.
Multivariate case
If is a discrete random variable taking values in the d-dimensional non-negative integer lattice {0,1, ...}d, then the probability generating function of X is defined as
where p is the probability mass function of X. The power series converges absolutely at least for all complex vectors with .
Properties
Power series
Probability generating functions obey all the rules of power series with non-negative coefficients. In particular, G(1−) = 1, where G(1−) = limz→1G(z) from below, since the probabilities must sum to one. So the radius of convergence of any probability generating function must be at least 1, by Abel's theorem for power series with non-negative coefficients.
Probabilities and expectations
The following properties allow the derivation of various basic quantities related to X:
The probability mass function of X is recovered by taking derivatives of G,
It follows from Property 1 that if random variables X and Y have probability-generating functions that are equal, , then . That is, if X and Y have identical probability-generating functions, then they have identical distributions.
The normalization of the probability density function can be expressed in terms of the generating function by
The expectation of is given by
More generally, the kth factorial moment, of X is given by
So the variance of X is given by
Finally, the kth raw moment of X is given by
where X is a random variable, is the probability generating function (of X) and is the moment-generating function (of X) .
Functions of independent random variables
Probability generating functions are particularly useful for dealing with functions of independent random variables. For example:
If X1, X2, ..., XN is a sequence of independent (and not necessarily identically distributed) random variables that take on natural-number values, and
where the ai are constant natural numbers, then the probability generating function is given by
For example, if
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https://en.wikipedia.org/wiki/Flatness
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Flatness may refer to:
Flatness (art)
Flatness (cosmology)
Flatness (liquids)
Flatness (manufacturing), a geometrical tolerance required in certain manufacturing situations
Flatness (mathematics)
Flatness (systems theory), a property of nonlinear dynamic systems
Spectral flatness
Flat intonation
Flat module in abstract algebra
See also
Flattening
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https://en.wikipedia.org/wiki/Quotient%20space%20%28topology%29
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In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space.
Intuitively speaking, the points of each equivalence class are or "glued together" for forming a new topological space. For example, identifying the points of a sphere that belong to the same diameter produces the projective plane as a quotient space.
Definition
Let be a topological space, and let be an equivalence relation on The quotient set is the set of equivalence classes of elements of The equivalence class of is denoted
The construction of defines a canonical surjection As discussed below, is a quotient mapping, commonly called the canonical quotient map, or canonical projection map, associated to
The quotient space under is the set equipped with the quotient topology, whose open sets are those subsets whose preimage is open. In other words, is open in the quotient topology on if and only if is open in Similarly, a subset is closed if and only if is closed in
The quotient topology is the final topology on the quotient set, with respect to the map
Quotient map
A map is a quotient map (sometimes called an identification map) if it is surjective and is equipped with the final topology induced by The latter condition admits two more-elementary phrasings: a subset is open (closed) if and only if is open (resp. closed). Every quotient map is continuous but not every continuous map is a quotient map.
Saturated sets
A subset of is called saturated (with respect to ) if it is of the form for some set which is true if and only if
The assignment establishes a one-to-one correspondence (whose inverse is ) between subsets of and saturated subsets of
With this terminology, a surjection is a quotient map if and only if for every subset of is open in if and only if is open in
In particular, open subsets of that are saturated have no impact on whether the function is a quotient map (or, indeed, continuous: a function is continuous if and only if, for every saturated such that is open in the set is open in
Indeed, if is a topology on and is any map then set of all that are saturated subsets of forms a topology on If is also a topological space then is a quotient map (respectively, continuous) if and only if the same is true of
Quotient space of fibers characterization
Given an equivalence relation on denote the equivalence class of a point by and let denote the set of equivalence classes. The map that sends points
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https://en.wikipedia.org/wiki/Abel%27s%20theorem
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In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.
Theorem
Let the Taylor series
be a power series with real coefficients with radius of convergence Suppose that the series
converges.
Then is continuous from the left at that is,
The same theorem holds for complex power series
provided that entirely within a single Stolz sector, that is, a region of the open unit disk where
for some fixed finite . Without this restriction, the limit may fail to exist: for example, the power series
converges to at but is unbounded near any point of the form so the value at is not the limit as tends to 1 in the whole open disk.
Note that is continuous on the real closed interval for by virtue of the uniform convergence of the series on compact subsets of the disk of convergence. Abel's theorem allows us to say more, namely that is continuous on
Stolz sector
The Stolz sector has explicit formulaand is plotted on the right for various values.
The left end of the sector is , and the right end is . On the right end, it becomes a cone with angle , where .
Remarks
As an immediate consequence of this theorem, if is any nonzero complex number for which the series
converges, then it follows that
in which the limit is taken from below.
The theorem can also be generalized to account for sums which diverge to infinity. If
then
However, if the series is only known to be divergent, but for reasons other than diverging to infinity, then the claim of the theorem may fail: take, for example, the power series for
At the series is equal to but
We also remark the theorem holds for radii of convergence other than : let
be a power series with radius of convergence and suppose the series converges at Then is continuous from the left at that is,
Applications
The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (that is, ) approaches from below, even in cases where the radius of convergence, of the power series is equal to and we cannot be sure whether the limit should be finite or not. See for example, the binomial series. Abel's theorem allows us to evaluate many series in closed form. For example, when
we obtain
by integrating the uniformly convergent geometric power series term by term on ; thus the series
converges to by Abel's theorem. Similarly,
converges to
is called the generating function of the sequence Abel's theorem is frequently useful in dealing with generating functions of real-valued and non-negative sequences, such as probability-generating functions. In particular, it is useful in the theory of Galton–Watson processes.
Outline of proof
After subtracting a constant from we may assume that Let Then substituting and performing a simple manipulation of the series (summation by parts) results in
Given pick large en
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https://en.wikipedia.org/wiki/Electron%20density
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Electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typically denoted as either or . The density is determined, through definition, by the normalised -electron wavefunction which itself depends upon variables ( spatial and spin coordinates). Conversely, the density determines the wave function modulo up to a phase factor, providing the formal foundation of density functional theory.
According to quantum mechanics, due to the uncertainty principle on an atomic scale the exact location of an electron cannot be predicted, only the probability of its being at a given position; therefore electrons in atoms and molecules act as if they are "smeared out" in space. For one-electron systems, the electron density at any point is proportional to the square magnitude of the wavefunction.
Definition
The electronic density corresponding to a normalised -electron wavefunction (with and denoting spatial and spin variables respectively) is defined as
where the operator corresponding to the density observable is
Computing as defined above we can simplify the expression as follows.
In words: holding a single electron still in position we sum over all possible arrangements of the other electrons. The factor N arises since all electrons are indistinguishable, and hence all the integrals evaluate to the same value.
In Hartree–Fock and density functional theories, the wave function is typically represented as a single Slater determinant constructed from orbitals, , with corresponding occupations . In these situations, the density simplifies to
General properties
From its definition, the electron density is a non-negative function integrating to the total number of electrons. Further, for a system with kinetic energy T, the density satisfies the inequalities
For finite kinetic energies, the first (stronger) inequality places the square root of the density in the Sobolev space . Together with the normalization and non-negativity this defines a space containing physically acceptable densities as
The second inequality places the density in the L3 space. Together with the normalization property places acceptable densities within the intersection of L1 and L3 – a superset of .
Topology
The ground state electronic density of an atom is conjectured to be a monotonically decaying function of the distance from the nucleus.
Nuclear cusp condition
The electronic density displays cusps at each nucleus in a molecule as a result of the unbounded electron-nucleus Coulomb potential. This behaviour is quantified by the Kato cusp condition formulated in terms of the spherically averaged density, , about any given nucleus as
That is, the radial derivative of the spherically averaged density, evaluated at any nucleus, is equal to twice the density at that nucleus multiplied by the
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https://en.wikipedia.org/wiki/Converge
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Converge may refer to:
Converge (band), American hardcore punk band
Converge (Baptist denomination), American national evangelical Baptist body
Limit (mathematics)
Converge ICT, internet service provider in the Philippines
CONVERGE CFD software, created by Convergent Science
See also
Comverge, a company that provides software, hardware, and services to electric utilities
Convergence (disambiguation)
Convergent (disambiguation)
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https://en.wikipedia.org/wiki/Differintegral
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In fractional calculus, an area of mathematical analysis, the differintegral is a combined differentiation/integration operator. Applied to a function ƒ, the q-differintegral of f, here denoted by
is the fractional derivative (if q > 0) or fractional integral (if q < 0). If q = 0, then the q-th differintegral of a function is the function itself. In the context of fractional integration and differentiation, there are several legitimate definitions of the differintegral.
Standard definitions
The four most common forms are:
The Riemann–Liouville differintegralThis is the simplest and easiest to use, and consequently it is the most often used. It is a generalization of the Cauchy formula for repeated integration to arbitrary order. Here, .
The Grunwald–Letnikov differintegralThe Grunwald–Letnikov differintegral is a direct generalization of the definition of a derivative. It is more difficult to use than the Riemann–Liouville differintegral, but can sometimes be used to solve problems that the Riemann–Liouville cannot.
The Weyl differintegral This is formally similar to the Riemann–Liouville differintegral, but applies to periodic functions, with integral zero over a period.
The Caputo differintegralIn opposite to the Riemann-Liouville differintegral, Caputo derivative of a constant is equal to zero. Moreover, a form of the Laplace transform allows to simply evaluate the initial conditions by computing finite, integer-order derivatives at point .
Definitions via transforms
The definitions of fractional derivatives given by Liouville, Fourier, and Grunwald and Letnikov coincide. They can be represented via Laplace, Fourier transforms or via Newton series expansion.
Recall the continuous Fourier transform, here denoted :
Using the continuous Fourier transform, in Fourier space, differentiation transforms into a multiplication:
So,
which generalizes to
Under the bilateral Laplace transform, here denoted by and defined as , differentiation transforms into a multiplication
Generalizing to arbitrary order and solving for , one obtains
Representation via Newton series is the Newton interpolation over consecutive integer orders:
For fractional derivative definitions described in this section, the following identities hold:
Basic formal properties
Linearity rules
Zero rule
Product rule
In general, composition (or semigroup) rule is a desirable property, but is hard to achieve mathematically and hence is not always completely satisfied by each proposed operator; this forms part of the decision making process on which one to choose:
(ideally)
(in practice)
See also
Fractional-order integrator
References
External links
MathWorld – Fractional calculus
MathWorld – Fractional derivative
Specialized journal: Fractional Calculus and Applied Analysis (1998-2014) and Fractional Calculus and Applied Analysis (from 2015)
Specialized journal: Fractional Differential Equations (FDE)
Specialized journal: Communications in Fractional C
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https://en.wikipedia.org/wiki/Product%20%28category%20theory%29
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In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.
Definition
Product of two objects
Fix a category Let and be objects of A product of and is an object typically denoted equipped with a pair of morphisms satisfying the following universal property:
For every object and every pair of morphisms there exists a unique morphism such that the following diagram commutes:
Whether a product exists may depend on or on and If it does exist, it is unique up to canonical isomorphism, because of the universal property, so one may speak of the product. This has the following meaning: let be another cartesian product, there exists a unique isomorphism such that and .
The morphisms and are called the canonical projections or projection morphisms; the letter (pronounced pi) alliterates with projection. Given and the unique morphism is called the product of morphisms and and is denoted
Product of an arbitrary family
Instead of two objects, we can start with an arbitrary family of objects indexed by a set
Given a family of objects, a product of the family is an object equipped with morphisms satisfying the following universal property:
For every object and every -indexed family of morphisms there exists a unique morphism such that the following diagrams commute for all
The product is denoted If then it is denoted and the product of morphisms is denoted
Equational definition
Alternatively, the product may be defined through equations. So, for example, for the binary product:
Existence of is guaranteed by existence of the operation
Commutativity of the diagrams above is guaranteed by the equality: for all and all
Uniqueness of is guaranteed by the equality: for all
As a limit
The product is a special case of a limit. This may be seen by using a discrete category (a family of objects without any morphisms, other than their identity morphisms) as the diagram required for the definition of the limit. The discrete objects will serve as the index of the components and projections. If we regard this diagram as a functor, it is a functor from the index set considered as a discrete category. The definition of the product then coincides with the definition of the limit, being a cone and projections being the limit (limiting cone).
Universal property
Just as the limit is a special case of the universal construction, so is the product. Starting with the definition given for the universal property of limits, take as the discrete category with two objects, so that is simply the product category The diagonal functor assigns to each object the ordere
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https://en.wikipedia.org/wiki/Markov%27s%20inequality
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In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant. It is named after the Russian mathematician Andrey Markov, although it appeared earlier in the work of Pafnuty Chebyshev (Markov's teacher), and many sources, especially in analysis, refer to it as Chebyshev's inequality (sometimes, calling it the first Chebyshev inequality, while referring to Chebyshev's inequality as the second Chebyshev inequality) or Bienaymé's inequality.
Markov's inequality (and other similar inequalities) relate probabilities to expectations, and provide (frequently loose but still useful) bounds for the cumulative distribution function of a random variable.
Statement
If is a nonnegative random variable and , then the probability
that is at least is at most the expectation of divided by :
Let (where ); then we can rewrite the previous inequality as
In the language of measure theory, Markov's inequality states that if is a measure space, is a measurable extended real-valued function, and , then
This measure-theoretic definition is sometimes referred to as Chebyshev's inequality.
Extended version for nondecreasing functions
If is a nondecreasing nonnegative function, is a (not necessarily nonnegative) random variable, and , then
An immediate corollary, using higher moments of supported on values larger than 0, is
Proofs
We separate the case in which the measure space is a probability space from the more general case because the probability case is more accessible for the general reader.
Intuition
where is larger than or equal to 0 as the random variable is non-negative and is larger than or equal to because the conditional expectation only takes into account of values larger than or equal to which r.v. can take.
Hence intuitively , which directly leads to .
Probability-theoretic proof
Method 1:
From the definition of expectation:
However, X is a non-negative random variable thus,
From this we can derive,
From here, dividing through by allows us to see that
Method 2:
For any event , let be the indicator random variable of , that is, if occurs and otherwise.
Using this notation, we have if the event occurs, and if . Then, given ,
which is clear if we consider the two possible values of . If , then , and so . Otherwise, we have , for which and so .
Since is a monotonically increasing function, taking expectation of both sides of an inequality cannot reverse it. Therefore,
Now, using linearity of expectations, the left side of this inequality is the same as
Thus we have
and since a > 0, we can divide both sides by a.
Measure-theoretic proof
We may assume that the function is non-negative, since only its absolute value enters in the equation. Now, consider the real-valued function s on X given by
Then . By the definition of the Lebesgue integral
and since , both sides can be divided by ,
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https://en.wikipedia.org/wiki/Direct%20proof
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In mathematics and logic, a direct proof is a way of showing the
truth or falsehood of a given statement by a straightforward combination of
established facts, usually axioms, existing lemmas and theorems, without making any further assumptions. In order to directly prove a conditional statement of the form "If p, then q", it suffices to consider the situations in which the statement p is true. Logical deduction is employed to reason from assumptions to conclusion. The type of logic employed is almost invariably first-order logic, employing the quantifiers for all and there exists. Common proof rules used are modus ponens and universal instantiation.
In contrast, an indirect proof may begin with certain hypothetical scenarios and then proceed to eliminate the uncertainties in each of these scenarios until an inescapable conclusion is forced. For example, instead of showing directly p ⇒ q, one proves its contrapositive ~q ⇒ ~p (one assumes ~q and shows that it leads to ~p). Since p ⇒ q and ~q ⇒ ~p are equivalent by the principle of transposition (see law of excluded middle), p ⇒ q is indirectly proved. Proof methods that are not direct include proof by contradiction, including proof by infinite descent. Direct proof methods include proof by exhaustion and proof by induction.
History and etymology
A direct proof is the simplest form of proof there is. The word ‘proof’ comes from the Latin word probare, which means “to test”. The earliest use of proofs was prominent in legal proceedings. A person with authority, such as a nobleman, was said to have probity, which means that the evidence was by his relative authority, which outweighed empirical testimony. In days gone by, mathematics and proof was often intertwined with practical questions – with populations like the Egyptians and the Greeks showing an interest in surveying land. This led to a natural curiosity with regards to geometry and trigonometry – particularly triangles and rectangles. These were the shapes which provided the most questions in terms of practical things, so early geometrical concepts were focused on these shapes, for example, the likes of buildings and pyramids used these shapes in abundance. Another shape which is crucial in the history of direct proof is the circle, which was crucial for the design of arenas and water tanks. This meant that ancient geometry (and Euclidean Geometry) discussed circles.
The earliest form of mathematics was phenomenological. For example, if someone could draw a reasonable picture, or give a convincing description, then that met all the criteria for something to be described as a mathematical “fact”. On occasion, analogical arguments took place, or even by “invoking the gods”. The idea that mathematical statements could be proven had not been developed yet, so these were the earliest forms of the concept of proof, despite not being actual proof at all.
Proof as we know it came about with one specific question: “what is a proof?” Traditional
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https://en.wikipedia.org/wiki/Power%20of%20a%20test
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In statistics, the power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis () when a specific alternative hypothesis () is true. It is commonly denoted by , and represents the chances of a true positive detection conditional on the actual existence of an effect to detect. Statistical power ranges from 0 to 1, and as the power of a test increases, the probability of making a type II error by wrongly failing to reject the null hypothesis decreases.
Notation
This article uses the following notation:
β = probability of a Type II error, known as a "false negative"
1 − β = probability of a "true positive", i.e., correctly rejecting the null hypothesis. "1 − β" is also known as the power of the test.
α = probability of a Type I error, known as a "false positive"
1 − α = probability of a "true negative", i.e., correctly not rejecting the null hypothesis
Description
For a type II error probability of , the corresponding statistical power is 1 − . For example, if experiment E has a statistical power of 0.7, and experiment F has a statistical power of 0.95, then there is a stronger probability that experiment E had a type II error than experiment F. This reduces experiment E's sensitivity to detect significant effects. However, experiment E is consequently more reliable than experiment F due to its lower probability of a type I error. It can be equivalently thought of as the probability of accepting the alternative hypothesis () when it is true – that is, the ability of a test to detect a specific effect, if that specific effect actually exists. Thus,
If is not an equality but rather simply the negation of (so for example with for some unobserved population parameter we have simply ) then power cannot be calculated unless probabilities are known for all possible values of the parameter that violate the null hypothesis. Thus one generally refers to a test's power against a specific alternative hypothesis.
As the power increases, there is a decreasing probability of a type II error, also called the false negative rate () since the power is equal to 1 − . A similar concept is the type I error probability, also referred to as the false positive rate or the level of a test under the null hypothesis.
In the context of binary classification, the power of a test is called its statistical sensitivity, its true positive rate, or its probability of detection.
Power analysis
A related concept is "power analysis". Power analysis can be used to calculate the minimum sample size required so that one can be reasonably likely to detect an effect of a given size. For example: "How many times do I need to toss a coin to conclude it is rigged by a certain amount?" Power analysis can also be used to calculate the minimum effect size that is likely to be detected in a study using a given sample size. In addition, the concept of power is used to make comparisons between different statistical testing procedures: fo
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https://en.wikipedia.org/wiki/Permutation%20matrix
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In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, when used to multiply another matrix, say , results in permuting the rows (when pre-multiplying, to form ) or columns (when post-multiplying, to form ) of the matrix .
Definition
Given a permutation of m elements,
represented in two-line form by
there are two natural ways to associate the permutation with a permutation matrix; namely, starting with the m × m identity matrix, , either permute the columns or permute the rows, according to . Both methods of defining permutation matrices appear in the literature and the properties expressed in one representation can be easily converted to the other representation. This article will primarily deal with just one of these representations and the other will only be mentioned when there is a difference to be aware of.
The m × m permutation matrix P = (pij) obtained by permuting the columns of the identity matrix , that is, for each i, if j = (i) and otherwise, will be referred to as the column representation in this article. Since the entries in row i are all 0 except that a 1 appears in column (i), we may write
where , a standard basis vector, denotes a row vector of length m with 1 in the jth position and 0 in every other position.
For example, the permutation matrix P corresponding to the permutation is
Observe that the jth column of the identity matrix now appears as the (j)th column of P.
The other representation, obtained by permuting the rows of the identity matrix , that is, for each j, pij = 1 if i = (j) and otherwise, will be referred to as the row representation.
Properties
The column representation of a permutation matrix is used throughout this section, except when otherwise indicated.
Multiplying times a column vector g will permute the rows of the vector:
Repeated use of this result shows that if is an appropriately sized matrix, the product, is just a permutation of the rows of . However, observing that
for each shows that the permutation of the rows is given by −1. ( is the transpose of matrix .)
As permutation matrices are orthogonal matrices (that is, ), the inverse matrix exists and can be written as
Multiplying a row vector h times will permute the columns of the vector:
Again, repeated application of this result shows that post-multiplying a matrix by the permutation matrix , that is, , results in permuting the columns of . Notice also that
Given two permutations and of elements, the corresponding permutation matrices and acting on column vectors are composed with
The same matrices acting on row vectors (that is, post-multiplication) compose according to the same rule
To be clear, the above formulas use the prefix notation for permutation composition, that is,
Let be the permutation matrix correspon
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https://en.wikipedia.org/wiki/Reliability%20%28statistics%29
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In statistics and psychometrics, reliability is the overall consistency of a measure. A measure is said to have a high reliability if it produces similar results under consistent conditions:"It is the characteristic of a set of test scores that relates to the amount of random error from the measurement process that might be embedded in the scores. Scores that are highly reliable are precise, reproducible, and consistent from one testing occasion to another. That is, if the testing process were repeated with a group of test takers, essentially the same results would be obtained. Various kinds of reliability coefficients, with values ranging between 0.00 (much error) and 1.00 (no error), are usually used to indicate the amount of error in the scores." For example, measurements of people's height and weight are often extremely reliable.
Types
There are several general classes of reliability estimates:
Inter-rater reliability assesses the degree of agreement between two or more raters in their appraisals. For example, a person gets a stomach ache and different doctors all give the same diagnosis.
Test-retest reliability assesses the degree to which test scores are consistent from one test administration to the next. Measurements are gathered from a single rater who uses the same methods or instruments and the same testing conditions. This includes intra-rater reliability.
Inter-method reliability assesses the degree to which test scores are consistent when there is a variation in the methods or instruments used. This allows inter-rater reliability to be ruled out. When dealing with forms, it may be termed parallel-forms reliability.
Internal consistency reliability, assesses the consistency of results across items within a test.
Difference from validity
Reliability does not imply validity. That is, a reliable measure that is measuring something consistently is not necessarily measuring what you want to be measured. For example, while there are many reliable tests of specific abilities, not all of them would be valid for predicting, say, job performance.
While reliability does not imply validity, reliability does place a limit on the overall validity of a test. A test that is not perfectly reliable cannot be perfectly valid, either as a means of measuring attributes of a person or as a means of predicting scores on a criterion. While a reliable test may provide useful valid information, a test that is not reliable cannot possibly be valid.
For example, if a set of weighing scales consistently measured the weight of an object as 500 grams over the true weight, then the scale would be very reliable, but it would not be valid (as the returned weight is not the true weight). For the scale to be valid, it should return the true weight of an object. This example demonstrates that a perfectly reliable measure is not necessarily valid, but that a valid measure necessarily must be reliable.
General model
In practice, testing measures are never perfectl
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https://en.wikipedia.org/wiki/Validity%20%28statistics%29
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Validity is the main extent to which a concept, conclusion, or measurement is well-founded and likely corresponds accurately to the real world. The word "valid" is derived from the Latin validus, meaning strong. The validity of a measurement tool (for example, a test in education) is the degree to which the tool measures what it claims to measure. Validity is based on the strength of a collection of different types of evidence (e.g. face validity, construct validity, etc.) described in greater detail below.
In psychometrics, validity has a particular application known as test validity: "the degree to which evidence and theory support the interpretations of test scores" ("as entailed by proposed uses of tests").
It is generally accepted that the concept of scientific validity addresses the nature of reality in terms of statistical measures and as such is an epistemological and philosophical issue as well as a question of measurement. The use of the term in logic is narrower, relating to the relationship between the premises and conclusion of an argument. In logic, validity refers to the property of an argument whereby if the premises are true then the truth of the conclusion follows by necessity. The conclusion of an argument is true if the argument is sound, which is to say if the argument is valid and its premises are true. By contrast, "scientific or statistical validity" is not a deductive claim that is necessarily truth preserving, but is an inductive claim that remains true or false in an undecided manner. This is why "scientific or statistical validity" is a claim that is qualified as being either strong or weak in its nature, it is never necessary nor certainly true. This has the effect of making claims of "scientific or statistical validity" open to interpretation as to what, in fact, the facts of the matter mean.
Validity is important because it can help determine what types of tests to use, and help to ensure researchers are using methods that are not only ethical and cost-effective, but also those that truly measure the ideas or constructs in question.
Test validity
Validity (accuracy)
Validity of an assessment is the degree to which it measures what it is supposed to measure. This is not the same as reliability, which is the extent to which a measurement gives results that are very consistent. Within validity, the measurement does not always have to be similar, as it does in reliability. However, just because a measure is reliable, it is not necessarily valid. E.g. a scale that is 5 pounds off is reliable but not valid. A test cannot be valid unless it is reliable. Validity is also dependent on the measurement measuring what it was designed to measure, and not something else instead. Validity (similar to reliability) is a relative concept; validity is not an all-or-nothing idea. There are many different types of validity.
Construct validity
Construct validity refers to the extent to which operationalizations of a construct
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https://en.wikipedia.org/wiki/John%20Wallis
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John Wallis (; ; ) was an English clergyman and mathematician, who is given partial credit for the development of infinitesimal calculus.
Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal court. He is credited with introducing the symbol ∞ to represent the concept of infinity. He similarly used 1/∞ for an infinitesimal. John Wallis was a contemporary of Newton and one of the greatest intellectuals of the early renaissance of mathematics.
Biography
Educational background
Cambridge, M.A., Oxford, D.D.
Grammar School at Tenterden, Kent, 1625–31.
School of Martin Holbeach at Felsted, Essex, 1631–2.
Cambridge University, Emmanuel College, 1632–40; B.A., 1637; M.A., 1640.
D.D. at Oxford in 1654
Family
On 14 March 1645 he married Susanna Glynde ( – 16 March 1687). They had three children:
Anne Blencoe (4 June 1656 – 5 April 1718), married Sir John Blencowe (30 November 1642 – 6 May 1726) in 1675, with issue
John Wallis (26 December 1650 – 14 March 1717), MP for Wallingford 1690–1695, married Elizabeth Harris (d. 1693) on 1 February 1682, with issue: one son and two daughters
Elizabeth Wallis (1658–1703), married William Benson (1649–1691) of Towcester, died with no issue
Life
John Wallis was born in Ashford, Kent. He was the third of five children of Reverend John Wallis and Joanna Chapman. He was initially educated at a school in Ashford but moved to James Movat's school in Tenterden in 1625 following an outbreak of plague. Wallis was first exposed to mathematics in 1631, at Felsted School (then known as Martin Holbeach's school in Felsted); he enjoyed maths, but his study was erratic, since "mathematics, at that time with us, were scarce looked on as academical studies, but rather mechanical" (Scriba 1970). At the school in Felsted, Wallis learned how to speak and write Latin. By this time, he also was proficient in French, Greek, and Hebrew. As it was intended he should be a doctor, he was sent in 1632 to Emmanuel College, Cambridge. While there, he kept an act on the doctrine of the circulation of the blood; that was said to have been the first occasion in Europe on which this theory was publicly maintained in a disputation. His interests, however, centred on mathematics. He received his Bachelor of Arts degree in 1637 and a Master's in 1640, afterwards entering the priesthood. From 1643 to 1649, he served as a nonvoting scribe at the Westminster Assembly. He was elected to a fellowship at Queens' College, Cambridge in 1644, from which he had to resign following his marriage.
Throughout this time, Wallis had been close to the Parliamentarian party, perhaps as a result of his exposure to Holbeach at Felsted School. He rendered them great practical assistance in deciphering Royalist dispatches. The quality of cryptography at that time was mixed; despite the individual successes of mathematicians such as François Viète, the principles underlying cipher design and analysis were very poorly unde
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https://en.wikipedia.org/wiki/Population%20process
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In applied probability, a population process is a Markov chain in which the state of the chain is analogous to the number of individuals in a population (0, 1, 2, etc.), and changes to the state are analogous to the addition or removal of individuals from the population. Typical population processes include birth–death processes and birth, death and catastrophe processes.
Although named by analogy to biological populations from population dynamics, population processes find application in a much wider range of fields than just ecology and other biological sciences. These other applications include telecommunications and queueing theory, chemical kinetics and financial mathematics, and hence the "population" could be of packets in a computer network, of molecules in a chemical reaction, or even of units in a financial index.
Population processes are typically characterized by processes of birth and immigration, and of death, emigration and catastrophe, which correspond to the basic demographic processes and broad environmental effects to which a population is subject. However, population processes are also often equivalent to other processes that may typically be characterised under other paradigms (in the literal sense of "patterns"). Queues, for example, are often characterised by an arrivals process, a service process, and the number of servers. In appropriate circumstances, however, arrivals at a queue are functionally equivalent to births or immigration and the service of waiting "customers" is equivalent to death or emigration.
See also
Moran process
References
Population
Markov models
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https://en.wikipedia.org/wiki/Quadratic
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In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms. Quadratus is Latin for square.
Mathematics
Algebra (elementary and abstract)
Quadratic function (or quadratic polynomial), a polynomial function that contains terms of at most second degree
Complex quadratic polynomials, are particularly interesting for their sometimes chaotic properties under iteration
Quadratic equation, a polynomial equation of degree 2 (reducible to 0 = ax2 + bx + c)
Quadratic formula, calculation to solve a quadratic equation for the independent variable (x)
Quadratic field, an algebraic number field of degree two over the field of rational numbers
Quadratic irrational or "quadratic surd", an irrational number that is a root of a quadratic polynomial
Calculus
Quadratic integral, the integral of the reciprocal of a second-degree polynomial
Statistics and stochastics
Quadratic form (statistics), scalar quantity ε'Λε for an n-dimensional square matrix
Quadratic mean, the square root of the mean of the squares of the data
Quadratic variation, in stochastics, useful for the analysis of Brownian motion and martingales
Number theory
Quadratic reciprocity, a theorem from number theory
Quadratic residue, an integer that is a square modulo n
Quadratic sieve, a modern integer factorization algorithm
Other mathematics
Quadratic convergence, in which the distance to a convergent sequence's limit is squared at each step
Quadratic differential, a form on a Riemann surface that locally looks like the square of an abelian differential
Quadratic form, a homogeneous polynomial of degree two in any number of variables
Quadratic programming, a special type of mathematical optimization problem
Quadratic growth, an asymptotic growth rate proportional to a quadratic function
Periodic points of complex quadratic mappings, a type of graph that can be used to explore stability in control systems
Quadratic bézier curve, a type of bezier curve
Computer science
Quadratic probing, a scheme in computer programming for resolving collisions in hash tables
Quadratic classifier, used in machine learning to separate measurements of two or more classes of objects
Quadratic time, in referring to algorithms with quadratic time complexity
Other
Quadratic (collection), a 1953 collection of science fiction novels by Olaf Stapledon and Murray Leinster
See also
Cubic (disambiguation), relating to a cube or degree 3, as next higher above quadratic
Linear, relating to a line or degree 1, as next lower below quadratic
Quad (disambiguation)
Quadratic transformation (disambiguation)
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https://en.wikipedia.org/wiki/Constructive%20analysis
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In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics.
Introduction
The name of the subject contrasts with classical analysis, which in this context means analysis done according to the more common principles of classical mathematics. However, there are various schools of thought and many different formalizations of constructive analysis. Whether classical or constructive in some fashion, any such framework of analysis axiomatizes the real number line by some means, a collection extending the rationals and with an apartness relation definable from an asymmetric order structure. Center stage takes a positivity predicate, here denoted , which governs an equality-to-zero . The members of the collection are generally just called the real numbers. While this term is thus overloaded in the subject, all the frameworks share a broad common core of results that are also theorems of classical analysis.
Constructive frameworks for its formulation are extensions of Heyting arithmetic by types including , constructive second-order arithmetic, or strong enough topos-, type- or constructive set theories such as , a constructive counter-part of . Of course, a direct axiomatization may be studied as well.
Logical preliminaries
The base logic of constructive analysis is intuitionistic logic, which means that the principle of excluded middle is not automatically assumed for every proposition. If a proposition is provable, this exactly means that the non-existence claim being provable would be absurd, and so the latter cannot also be provable in a consistent theory. The double-negated existence claim is a logically negative statement and implied by, but generally not equivalent to the existence claim itself. Much of the intricacies of constructive analysis can be framed in terms of the weakness of propositions of the logically negative form , which is generally weaker than . In turn, also an implication can generally be not reversed.
While a constructive theory proves fewer theorems than its classical counter-part in its classical presentation, it may exhibit attractive meta-logical properties. For example, if a theory exhibits the disjunction property, then if it proves a disjunction then also or . Already in classical arithmetic, this is violated for the most basic propositions about sequences of numbers - as demonstrated next.
Undecidable predicates
Consider a decidable predicate on the naturals, which in the constructive vernacular means is provable, and let be the characteristic function defined to equal exactly where is true.
A common strategy of formalization of real numbers is in terms of sequences or rationals, . For motivation, consider the sequence , which is monotone with values non-strictly growing between the bounds and .
For the sake of demonstration, defining an extensional equality to the zero sequence , it follows that . Note that the symbol "" is used in seve
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https://en.wikipedia.org/wiki/Peter%20Gustav%20Lejeune%20Dirichlet
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Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a function.
Although his surname is Lejeune Dirichlet, he is commonly referred to by his mononym Dirichlet, in particular for results named after him.
Biography
Early life (1805–1822)
Gustav Lejeune Dirichlet was born on 13 February 1805 in Düren, a town on the left bank of the Rhine which at the time was part of the First French Empire, reverting to Prussia after the Congress of Vienna in 1815. His father Johann Arnold Lejeune Dirichlet was the postmaster, merchant, and city councilor. His paternal grandfather had come to Düren from Richelette (or more likely Richelle), a small community north east of Liège in Belgium, from which his surname "Lejeune Dirichlet" ("", French for "the youth from Richelette") was derived.
Although his family was not wealthy and he was the youngest of seven children, his parents supported his education. They enrolled him in an elementary school and then private school in hope that he would later become a merchant. The young Dirichlet, who showed a strong interest in mathematics before age 12, persuaded his parents to allow him to continue his studies. In 1817 they sent him to the under the care of Peter Joseph Elvenich, a student his family knew. In 1820, Dirichlet moved to the Jesuit Gymnasium in Cologne, where his lessons with Georg Ohm helped widen his knowledge in mathematics. He left the gymnasium a year later with only a certificate, as his inability to speak fluent Latin prevented him from earning the Abitur.
Studies in Paris (1822–1826)
Dirichlet again persuaded his parents to provide further financial support for his studies in mathematics, against their wish for a career in law. As Germany provided little opportunity to study higher mathematics at the time, with only Gauss at the University of Göttingen who was nominally a professor of astronomy and anyway disliked teaching, Dirichlet decided to go to Paris in May 1822. There he attended classes at the Collège de France and at the University of Paris, learning mathematics from Hachette among others, while undertaking private study of Gauss's Disquisitiones Arithmeticae, a book he kept close for his entire life. In 1823 he was recommended to General Maximilien Foy, who hired him as a private tutor to teach his children German, the wage finally allowing Dirichlet to become independent from his parents' financial support.
His first original research, comprising part of a proof of Fermat's Last Theorem for the case , brought him immediate fame, being the first advance in the theorem since Fermat's own proof of the case and Euler's proof for . Adrien-Marie Legendre, one of the referees,
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https://en.wikipedia.org/wiki/Indicator%20function
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In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then if and otherwise, where is a common notation for the indicator function. Other common notations are and
The indicator function of is the Iverson bracket of the property of belonging to ; that is,
For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers.
Definition
The indicator function of a subset of a set is a function
defined as
The Iverson bracket provides the equivalent notation, or to be used instead of
The function is sometimes denoted , , , or even just .
Notation and terminology
The notation is also used to denote the characteristic function in convex analysis, which is defined as if using the reciprocal of the standard definition of the indicator function.
A related concept in statistics is that of a dummy variable. (This must not be confused with "dummy variables" as that term is usually used in mathematics, also called a bound variable.)
The term "characteristic function" has an unrelated meaning in classic probability theory. For this reason, traditional probabilists use the term indicator function for the function defined here almost exclusively, while mathematicians in other fields are more likely to use the term characteristic function to describe the function that indicates membership in a set.
In fuzzy logic and modern many-valued logic, predicates are the characteristic functions of a probability distribution. That is, the strict true/false valuation of the predicate is replaced by a quantity interpreted as the degree of truth.
Basic properties
The indicator or characteristic function of a subset of some set maps elements of to the range .
This mapping is surjective only when is a non-empty proper subset of . If then By a similar argument, if then
If and are two subsets of then
and the indicator function of the complement of i.e. is:
More generally, suppose is a collection of subsets of . For any
is clearly a product of s and s. This product has the value 1 at precisely those that belong to none of the sets and is 0 otherwise. That is
Expanding the product on the left hand side,
where is the cardinality of . This is one form of the principle of inclusion-exclusion.
As suggested by the previous example, the indicator function is a useful notational device in combinatorics. The notation is used in other places as well, for instance in probability theory: if is a probability space with probability measure and is a measurable set, then becomes a random variable whose expected value is equal to the probability of :
This identity is used in a simple proof of Markov's inequality.
In many cases, such as order theory, the inverse of the indicator function may be defined. This is commonly called the ge
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https://en.wikipedia.org/wiki/For%20each
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For each may refer to:
In mathematics, Universal quantification. Also read as: "for all"
In computer science, foreach loop
See also
Each (disambiguation)
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https://en.wikipedia.org/wiki/Hyperbolic%20geometry
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In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R.
(Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.)
The hyperbolic plane is a plane where every point is a saddle point.
Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane.
A modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model.
When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geometry under many different names; Felix Klein finally gave the subject the name hyperbolic geometry to include it in the now rarely used sequence elliptic geometry (spherical geometry), parabolic geometry (Euclidean geometry), and hyperbolic geometry.
In the former Soviet Union, it is commonly called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky.
This page is mainly about the 2-dimensional (planar) hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry. See hyperbolic space for more information on hyperbolic geometry extended to three and more dimensions.
Properties
Relation to Euclidean geometry
Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate.
When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry.
There are two kinds of absolute geometry, Euclidean and hyperbolic.
All theorems of absolute geometry, including the first 28 propositions of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry.
Propositions 27 and 28 of Book One of Euclid's Elements prove the existence of parallel/non-intersecting lines.
This difference also has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry; new concepts need to be introduced.
Further, because of the angle of parallelism, hyperbolic geometry has an absolute scale, a relation between distance and angle measurements.
Lines
Single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points uniquely define a line, and line segments can be infinitely extended.
Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two distinct lines can intersect in no more
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https://en.wikipedia.org/wiki/Zariski%20topology
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In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space.
The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces.
The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of the variety. In the case of an algebraic variety over the complex numbers, the Zariski topology is thus coarser than the usual topology, as every algebraic set is closed for the usual topology.
The generalization of the Zariski topology to the set of prime ideals of a commutative ring follows from Hilbert's Nullstellensatz, that establishes a bijective correspondence between the points of an affine variety defined over an algebraically closed field and the maximal ideals of the ring of its regular functions. This suggests defining the Zariski topology on the set of the maximal ideals of a commutative ring as the topology such that a set of maximal ideals is closed if and only if it is the set of all maximal ideals that contain a given ideal. Another basic idea of Grothendieck's scheme theory is to consider as points, not only the usual points corresponding to maximal ideals, but also all (irreducible) algebraic varieties, which correspond to prime ideals. Thus the Zariski topology on the set of prime ideals (spectrum) of a commutative ring is the topology such that a set of prime ideals is closed if and only if it is the set of all prime ideals that contain a fixed ideal.
Zariski topology of varieties
In classical algebraic geometry (that is, the part of algebraic geometry in which one does not use schemes, which were introduced by Grothendieck around 1960), the Zariski topology is defined on algebraic varieties. The Zariski topology, defined on the points of the variety, is the topology such that the closed sets are the algebraic subsets of the variety. As the most elementary algebraic varieties are affine and projective varieties, it is useful to make this definition more explicit in both cases. We assume that we are working over a fixed, algebraically closed field k (in classical algebraic geometry, k is usually the field of complex numbers).
Affine varieties
First, we define the topology on the affine space formed by the -tuples of elements of . The topology is defined
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https://en.wikipedia.org/wiki/List%20of%20mathematical%20shapes
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Following is a list of some mathematically well-defined shapes.
Algebraic curves
Cubic plane curve
Quartic plane curve
Rational curves
Degree 2
Conic sections
Unit circle
Unit hyperbola
Degree 3
Folium of Descartes
Cissoid of Diocles
Conchoid of de Sluze
Right strophoid
Semicubical parabola
Serpentine curve
Trident curve
Trisectrix of Maclaurin
Tschirnhausen cubic
Witch of Agnesi
Degree 4
Ampersand curve
Bean curve
Bicorn
Bow curve
Bullet-nose curve
Cruciform curve
Deltoid curve
Devil's curve
Hippopede
Kampyle of Eudoxus
Kappa curve
Lemniscate of Booth
Lemniscate of Gerono
Lemniscate of Bernoulli
Limaçon
Cardioid
Limaçon trisectrix
Trifolium curve
Degree 5
Quintic of l'Hospital
Degree 6
Astroid
Atriphtaloid
Nephroid
Quadrifolium
Families of variable degree
Epicycloid
Epispiral
Epitrochoid
Hypocycloid
Lissajous curve
Poinsot's spirals
Rational normal curve
Rose curve
Curves of genus one
Bicuspid curve
Cassini oval
Cassinoide
Cubic curve
Elliptic curve
Watt's curve
Curves with genus greater than one
Butterfly curve
Elkies trinomial curves
Hyperelliptic curve
Klein quartic
Classical modular curve
Bolza surface
Macbeath surface
Curve families with variable genus
Polynomial lemniscate
Fermat curve
Sinusoidal spiral
Superellipse
Hurwitz surface
Transcendental curves
Bowditch curve
Brachistochrone
Butterfly curve
Catenary
Clélies
Cochleoid
Cycloid
Horopter
Isochrone
Isochrone of Huygens (Tautochrone)
Isochrone of Leibniz
Isochrone of Varignon
Lamé curve
Pursuit curve
Rhumb line
Spirals
Archimedean spiral
Cornu spiral
Cotes' spiral
Fermat's spiral
Galileo's spiral
Hyperbolic spiral
Lituus
Logarithmic spiral
Nielsen's spiral
Syntractrix
Tractrix
Trochoid
Piecewise constructions
Bézier curve
Splines
B-spline
Nonuniform rational B-spline
Ogee
Loess curve
Lowess
Polygonal curve
Maurer rose
Reuleaux triangle
Bézier triangle
Curves generated by other curves
Caustic including Catacaustic and Diacaustic
Cissoid
Conchoid
Evolute
Glissette
Inverse curve
Involute
Isoptic including Orthoptic
Orthotomic
Negative pedal curve
Pedal curve
Parallel curve
Radial curve
Roulette
Strophoid
Space curves
Conchospiral
Helix
Tendril perversion (a transition between back-to-back helices)
Hemihelix, a quasi-helical shape characterized by multiple tendril perversions
Seiffert's spiral
Slinky spiral
Twisted cubic
Viviani's curve
Surfaces in 3-space
Plane
Quadric surfaces
Cone
Cylinder
Ellipsoid
Spheroid
Sphere
Hyperboloid
Paraboloid
Bicylinder
Tricylinder
Möbius strip
Torus
Minimal surfaces
Catalan's minimal surface
Costa's minimal surface
Catenoid
Enneper surface
Gyroid
Helicoid
Lidinoid
Riemann's minimal surface
Saddle tower
Scherk surface
Schwarz minimal surface
Triply periodic minimal surface
Non-orientable surfaces
Klein bottle
Real projective plane
Cross-cap
Roman surface
Boy's surface
Quadrics
Sphere
Spheroid
Oblate spheroid
Cone
Ellipsoid
Hyperboloid of one sheet
Hyperboloid of two sheets
Hyperbolic paraboloid (a ruled surface)
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https://en.wikipedia.org/wiki/Difference%20quotient
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In single-variable calculus, the difference quotient is usually the name for the expression
which when taken to the limit as h approaches 0 gives the derivative of the function f. The name of the expression stems from the fact that it is the quotient of the difference of values of the function by the difference of the corresponding values of its argument (the latter is (x + h) - x = h in this case). The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length h). The limit of the difference quotient (i.e., the derivative) is thus the instantaneous rate of change.
By a slight change in notation (and viewpoint), for an interval [a, b], the difference quotient
is called the mean (or average) value of the derivative of f over the interval [a, b]. This name is justified by the mean value theorem, which states that for a differentiable function f, its derivative f′ reaches its mean value at some point in the interval. Geometrically, this difference quotient measures the slope of the secant line passing through the points with coordinates (a, f(a)) and (b, f(b)).
Difference quotients are used as approximations in numerical differentiation, but they have also been subject of criticism in this application.
Difference quotients may also find relevance in applications involving Time discretization, where the width of the time step is used for the value of h.
The difference quotient is sometimes also called the Newton quotient (after Isaac Newton) or Fermat's difference quotient (after Pierre de Fermat).
Overview
The typical notion of the difference quotient discussed above is a particular case of a more general concept. The primary vehicle of calculus and other higher mathematics is the function. Its "input value" is its argument, usually a point ("P") expressible on a graph. The difference between two points, themselves, is known as their Delta (ΔP), as is the difference in their function result, the particular notation being determined by the direction of formation:
Forward difference: ΔF(P) = F(P + ΔP) − F(P);
Central difference: δF(P) = F(P + ½ΔP) − F(P − ½ΔP);
Backward difference: ∇F(P) = F(P) − F(P − ΔP).
The general preference is the forward orientation, as F(P) is the base, to which differences (i.e., "ΔP"s) are added to it. Furthermore,
If |ΔP| is finite (meaning measurable), then ΔF(P) is known as a finite difference, with specific denotations of DP and DF(P);
If |ΔP| is infinitesimal (an infinitely small amount——usually expressed in standard analysis as a limit: ), then ΔF(P) is known as an infinitesimal difference, with specific denotations of dP and dF(P) (in calculus graphing, the point is almost exclusively identified as "x" and F(x) as "y").
The function difference divided by the point difference is known as "difference quotient":
If ΔP is infinitesimal, then the difference quotient is a derivative, otherwise it is a divided difference:
Defining the
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https://en.wikipedia.org/wiki/Secant%20line
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In geometry, a secant is a line that intersects a curve at a minimum of two distinct points.
The word secant comes from the Latin word secare, meaning to cut. In the case of a circle, a secant intersects the circle at exactly two points. A chord is the line segment determined by the two points, that is, the interval on the secant whose ends are the two points.
Circles
A straight line can intersect a circle at zero, one, or two points. A line with intersections at two points is called a secant line, at one point a tangent line and at no points an exterior line. A chord is the line segment that joins two distinct points of a circle. A chord is therefore contained in a unique secant line and each secant line determines a unique chord.
In rigorous modern treatments of plane geometry, results that seem obvious and were assumed (without statement) by Euclid in his treatment, are usually proved.
For example, Theorem (Elementary Circular Continuity): If is a circle and a line that contains a point that is inside and a point that is outside of then is a secant line for .
In some situations phrasing results in terms of secant lines instead of chords can help to unify statements. As an example of this consider the result:
If two secant lines contain chords and in a circle and intersect at a point that is not on the circle, then the line segment lengths satisfy .
If the point lies inside the circle this is Euclid III.35, but if the point is outside the circle the result is not contained in the Elements. However, Robert Simson following Christopher Clavius demonstrated this result, sometimes called the intersecting secants theorem, in their commentaries on Euclid.
Curves
For curves more complicated than simple circles, the possibility that a line that intersects a curve in more than two distinct points arises. Some authors define a secant line to a curve as a line that intersects the curve in two distinct points. This definition leaves open the possibility that the line may have other points of intersection with the curve. When phrased this way the definitions of a secant line for circles and curves are identical and the possibility of additional points of intersection just does not occur for a circle.
Secants and tangents
Secants may be used to approximate the tangent line to a curve, at some point , if it exists. Define a secant to a curve by two points, and , with fixed and variable. As approaches along the curve, if the slope of the secant approaches a limit value, then that limit defines the slope of the tangent line at . The secant lines are the approximations to the tangent line. In calculus, this idea is the geometric definition of the derivative.
A tangent line to a curve at a point may be a secant line to that curve if it intersects the curve in at least one point other than . Another way to look at this is to realize that being a tangent line at a point is a local property, depending only on the curve in the immediate n
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https://en.wikipedia.org/wiki/Projective%20space
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In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines.
This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, point and line are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the following definition, which is more often encountered in modern textbooks.
Using linear algebra, a projective space of dimension is defined as the set of the vector lines (that is, vector subspaces of dimension one) in a vector space of dimension . Equivalently, it is the quotient set of by the equivalence relation "being on the same vector line". As a vector line intersects the unit sphere of in two antipodal points, projective spaces can be equivalently defined as spheres in which antipodal points are identified. A projective space of dimension 1 is a projective line, and a projective space of dimension 2 is a projective plane.
Projective spaces are widely used in geometry, as allowing simpler statements and simpler proofs. For example, in affine geometry, two distinct lines in a plane intersect in at most one point, while, in projective geometry, they intersect in exactly one point. Also, there is only one class of conic sections, which can be distinguished only by their intersections with the line at infinity: two intersection points for hyperbolas; one for the parabola, which is tangent to the line at infinity; and no real intersection point of ellipses.
In topology, and more specifically in manifold theory, projective spaces play a fundamental role, being typical examples of non-orientable manifolds.
Motivation
As outlined above, projective spaces were introduced for formalizing statements like "two coplanar lines intersect in exactly one point, and this point is at infinity if the lines are parallel." Such statements are suggested by the study of perspective, which may be considered as a central projection of the three dimensional space onto a plane (see Pinhole camera model). More precisely, the entrance pupil of a camera or of the eye of an observer is the center of projection, and the image is formed on the projection plane.
Mathematically, the center of projection is a point of the space (the intersection of the axes in the figure); the projection plane (, in blue
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https://en.wikipedia.org/wiki/Negative
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Negative may refer to:
Science and mathematics
Negative number
Negative mass
Negative energy
Negative charge, one of the two types of electric charge
Negative (electrical polarity), in electric circuits
Negative result (disambiguation)
Negative lenses, uses to describe diverging optics
Photography
Negative (photography), an image with inverted luminance or a strip of film with such an image
Original camera negative, the film in a motion picture camera which captures the original image
Paper negative, a negative image printed on paper used to create the final print of a photograph
Linguistics
A negative answer, commonly expressed with the word no
A type of grammatical construction; see affirmative and negative
A double negative is a construction occurring when two forms of grammatical negation are used in the same sentence.
Music
Negative (Finnish band), a Finnish band established in 1997
Negative (Serbian band), a Serbian band established in 1999
The Negatives, a band fronted by Lloyd Cole
Negative (Yōsui Inoue album), 1987
Negative (Negative album), 1999
Negatives (album), a 2004 album by Phantom Planet
The Negatives, a 2014 album by Cruel Hand
Negative (song), a 1998 song by Mansun
Negative (album) by Act Of Denial
Other uses
Negatives (film), a 1968 film
Negative (policy debate) (NEG), the team which negates the resolution in policy debate
Negative feedback, a feedback loop that responds in the opposite direction to a perturbation
Negative liberty is freedom from interference by other people
Negative repetition, the performance of the eccentric phase of weight lifting
Negative sign, the passive or feminine signs of the zodiac in astrology
Negative space, in art, the space around or between elements of the subject
Negative, several distinct concepts within the game of contract bridge, including:
Double negative, see
Herbert negative, see see
Negative double
Negative free bid
Negative inference, see see
Negative response, see see
Photo negative casting, a casting technique in acting by reversing the skin colours of actor and character.
Boys Name Cameron
See also
Mu (negative)
Negation (disambiguation)
Negativity (disambiguation)
Positive (disambiguation)
Double Negative (disambiguation)
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https://en.wikipedia.org/wiki/Positive
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Positive is a property of positivity and may refer to:
Mathematics and science
Positive formula, a logical formula not containing negation
Positive number, a number that is greater than 0
Plus sign, the sign "+" used to indicate a positive number
Positive operator, a type of linear operator in mathematics
Positive result, a result that has been found significant in statistical hypothesis testing
Positive test, a diagnostic test result that indicates some parameter being evaluated was present
Positive charge, one of the two types of electrical charge
Positive (electrical polarity), in electrical circuits
Positive lens, in optics
Positive (photography), a positive image, in which the color and luminance correlates directly with that in the depicted scene
Positive sense, said of an RNA sequence that codes for a protein
Philosophy and humanities
Affirmative (policy debate), the team which affirms the resolution
Negative and positive rights, concerning the moral obligation of a person to do something for/to someone
Positive economics, in economics, about predictions of behavior of economic actors, as opposed to the normative aspect
Positive law, man-made law (statutes) in contrast with natural law (derived from deities or morality)
Positive liberty, the opportunity and ability to act to fulfill one's own potential
Affirmative (linguistics), a property of a non-negated expression (the opposite of negative)
Positive (linguistics), the form of an adjective or adverb on which comparative and superlative are formed with suffixes or the use of more or less
Positive affectivity, the psychological capability to respond positively
Positive psychology, a branch of psychology
Positive statement, in economics and philosophy, a (possibly incorrect) factual statement about what is, as opposed to what should be (a normative statement)
Films and television
Positive (1990 film), a documentary film about AIDS and activism
Positive (2007 film), a short film in Hindi on HIV and AIDS
Positive (2008 film), a 2008 Malayalam language film directed by V. K. Prakash
Positive (TV series), a Filipino drama series
Music
Positive (Peabo Bryson album)
Positive (Tofubeats album)
Positive (EP), the sixth EP by South Korean band Pentagon
"Positive", a 1994 song by Baboon from Face Down in Turpentine
"Positive", a 2011 song by Taio Cruz from TY.O
"(Gotta Be) Positive", a song by Eddy Grant from Reparation
Positive hardcore, a subgenre of hardcore punk
Positive organ
Other uses
Positive sign, in western astrology, the supposedly extroverted personalities of the fire and air signs
See also
HIV-positive people
Negative (disambiguation)
Negative (photography), as opposed to positive images used in such applications as slide projection or photo emulsion stencil-making
Normative
Optimism
Positif (disambiguation)
Positive action (disambiguation)
Positivism (disambiguation)
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https://en.wikipedia.org/wiki/Extrema
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Extrema may refer to:
Extrema (mathematics), maxima and minima values
Extremities (disambiguation)
Extrema, Minas Gerais, town in Brazil
Extrema, Rondônia, town in Brazil
Extrema (band), Antiprotestionarialconstructionaryism
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https://en.wikipedia.org/wiki/Hampshire%20College%20Summer%20Studies%20in%20Mathematics
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The Hampshire College Summer Studies in Mathematics (HCSSiM) is an American residential program for mathematically talented high school students. The program has been conducted each summer since 1971, with the exceptions of 1981 and 1996, and has more than 1500 alumni. Due to the Coronavirus pandemic, the 2020 Summer Studies ran online for a shortened program of four weeks.
The program was created by and is still headed by David Kelly, a professor emeritus of mathematics at Hampshire College.
Background
The program is housed at Hampshire College in Amherst, Massachusetts, and generally runs for six weeks from early July until mid-August. The program itself consists of lectures, study sessions, math workshops (general-knowledge classes), maxi-courses (three-week classes run by the senior staff members), and mini-courses (specialized shorter classes).
On a typical day, students spend four hours in the morning in class, have lunch together with the faculty, and then have several hours to use at their leisure. During this "down time" students and faculty members often host quasis, where they participate in an activity as a small group, such as juggling or making sushi. They return for the "Prime Time Theorem" (an hour-long talk on an interesting piece of mathematics given by a faculty member or a visitor), have dinner, and then spend three hours in a problem solving session. One of the instructors blogged the content of her class.
Many students go on to professional careers in mathematics. An occasional publication has resulted from work done at the program. Well-known alumni of the program include two MacArthur Fellows, Eric Lander and Erik Winfree, as well as Lisa Randall, Dana Randall, and Eugene Volokh. Many alumni return to the campus for a few days around Yellow Pig's Day (July 17) of each year. This observance was formalized for 2006 in "Yellow Pig Math Days," which was conducted in observance of 2006 being the 34th offering of the HCSSiM Program (34 being a multiple of 17).
The Summer Studies has been funded in the past by the American Mathematical Society and the U.S. National Science Foundation.
Notable alumni
Bram Cohen, developer of BitTorrent, co-founder of CodeCon
Matthew Cook, group leader at the Institute for Neuroinformatics at ETH Zurich and computer scientist who proved the Turing universality of Wolfram's Rule 110 cellular automaton
Lenore Cowen, computer scientist and mathematician at Tufts University
Alan Edelman, professor of mathematics at the Massachusetts Institute of Technology, Sloan Fellow
Alan Grayson, former member of the U.S House of Representatives for Florida's 8th and 9th Congressional Districts
Neil Immerman, professor of computer science at the University of Massachusetts Amherst, Guggenheim Fellow
Susan Landau, professor of social science and policy studies at Worcester Polytechnic Institute, Guggenheim Fellow
Eric Lander, professor of biology at MIT and science advisor to Presidents Barack O
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https://en.wikipedia.org/wiki/Class%20field%20theory
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In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.
Hilbert is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin (with the help of Chebotarev's theorem).
One of the major results is: given a number field F, and writing K for the maximal abelian unramified extension of F, the Galois group of K over F is canonically isomorphic to the ideal class group of F. This statement was generalized to the so called Artin reciprocity law; in the idelic language, writing CF for the idele class group of F, and taking L to be any finite abelian extension of F, this law gives a canonical isomorphism
where denotes the idelic norm map from L to F. This isomorphism is named the reciprocity map.
The existence theorem states that the reciprocity map can be used to give a bijection between the set of abelian extensions of F and the set of closed subgroups of finite index of
A standard method for developing global class field theory since the 1930s was to construct local class field theory, which describes abelian extensions of local fields, and then use it to construct global class field theory. This was first done by Emil Artin and Tate using the theory of group cohomology, and in particular by developing the notion of class formations. Later, Neukirch found a proof of the main statements of global class field theory without using cohomological ideas. His method was explicit and algorithmic.
Inside class field theory one can distinguish special class field theory and general class field theory.
Explicit class field theory provides an explicit construction of maximal abelian extensions of a number field in various situations. This portion of the theory consists of Kronecker–Weber theorem, which can be used to construct the abelian extensions of , and the theory of complex multiplication to construct abelian extensions of CM-fields.
There are three main generalizations of class field theory: higher class field theory, the Langlands program (or 'Langlands correspondences'), and anabelian geometry.
Formulation in contemporary language
In modern mathematical language, class field theory (CFT) can be formulated as follows. Consider the maximal abelian extension A of a local or global field K. It is of infinite degree over K; the Galois group G of A over K is an infinite profinite group, so a compact topological group, and it is abelian. The central aims of class field theory are: to describe G in terms of certain appropriate topological objects associated to K, to desc
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https://en.wikipedia.org/wiki/List%20of%20mathematics%20reference%20tables
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See also: List of reference tables
Mathematics
List of mathematical topics
List of statistical topics
List of mathematical functions
List of mathematical theorems
List of mathematical proofs
List of matrices
List of numbers
List of relativistic equations
List of small groups
Mathematical constants
Sporadic group
Table of bases
Table of Clebsch-Gordan coefficients
Table of derivatives
Table of divisors
Table of integrals
Table of mathematical symbols
Table of prime factors
Taylor series
Timeline of mathematics
Trigonometric identities
Truth table
Reference tables
List
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https://en.wikipedia.org/wiki/Number%20%28disambiguation%29
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A number describes quantity and assesses multitude.
Number and numbers may also refer to:
Mathematics and language
Grammatical number, a morphological grammatical category indicating the quantity of referents
Number Forms, a Unicode block containing common fractions and Roman numerals
Nominal number, a label to identify an item uniquely
Number theory, a mathematical discipline
Numbering scheme, a method of assigning numbers to items
Numeral system, a writing system for expressing numbers
Numeral (linguistics), words or phrases that describe a numerical quantity
Number sign, a
Literature
Book of Numbers, part of the Torah; the fourth book of the Bible
Number (magazine), a Japanese sports magazine
Numbers (magazine), a literary magazine published in Cambridge, England
Numbers: The Universal Language, a 1996 illustrated book by Denis Guedj
The Number, a book by Jonny Steinberg
Entertainment
"Numbers" (Lost), an episode of Lost
The Numbers (Lost), the numbers 4, 8, 15, 16, 23, 42 in Lost
Numbers (Nanoha), a group of characters in Magical Girl Lyrical Nanoha StrikerS
Numbers (TV series) (stylised as Numb3rs), an American TV series
The Numbers (website), a website that tracks box office revenue and film sales
Numbers monsters, a set of cards in Yu-Gi-Oh! Zexal
Numbers, a play by Kieron Barry
Numbers, a character in the Dick Tracy franchise
A Number, a 2002 play by the English playwright Caryl Churchill
Numbers, a historic nightclub in the Montrose neighborhood of Houston, Texas
The Number (film), a 2017 South African film
Numbers (South Korean TV series), a 2023 television series
Music
Number (music)
Number opera, an opera consisting of individual musical numbers
Numbers (American band), an American indie rock/electronic group
The Numbers Band, an American rock group
The Numbers (band), an Australian rock group
Numbers (record label), a Scottish record label
Albums
Numbers (Cat Stevens album) (1975)
Numbers (Rufus album) (1979)
Numbers (The Briggs album) (2003)
Numbers (Days Difference album) (2007)
Number(s) (Woe, Is Me album) (2010)
Numbers (MellowHype album) (2012)
Songs
"Numbers" (Kraftwerk song), from the 1981 album Computer World
"Numbers", a song by Soft Cell from their 1983 album The Art of Falling Apart
"Numbers", a song by Basshunter from his 2009 album Bass Generation
"Numbers", a song by Skepta featuring Pharrell Williams from the album Konnichiwa
"The Numbers", a song by Radiohead from their 2016 album A Moon Shaped Pool
"Numbers", a song by Melanie Martinez from her 2020 deluxe album K-12
"Numbers", a song by Weezer from their 2021 album OK Human
People with the name
Ronald Numbers (born 1942), an American professor and historian of science
Other uses
Numbers (spreadsheet), a spreadsheet application developed by Apple as part of its iWork suite
Number (periodicals), a number to indicate a particular issue of a periodical
Number (sports), a number assigned to an athlete
Telephone
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https://en.wikipedia.org/wiki/Exp
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Exp or EXP may stand for:
Exponential function, in mathematics
Expiry date of organic compounds like food or medicines
Experience points, in role-playing games
EXPTIME, a complexity class in computing
Ford EXP, a car manufactured in the 1980s
Exp (band), an Italian group in the 1990s
"EXP" (song), a song by The Jimi Hendrix Experience from the album Axis: Bold as Love
EXP (calculator key), to enter numbers in scientific or engineering notation
See also
Exponential map (disambiguation)
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https://en.wikipedia.org/wiki/Spectral%20method
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Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain "basis functions" (for example, as a Fourier series which is a sum of sinusoids) and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible.
Spectral methods and finite element methods are closely related and built on the same ideas; the main difference between them is that spectral methods use basis functions that are generally nonzero over the whole domain, while finite element methods use basis functions that are nonzero only on small subdomains (compact support). Consequently, spectral methods connect variables globally while finite elements do so locally. Partially for this reason, spectral methods have excellent error properties, with the so-called "exponential convergence" being the fastest possible, when the solution is smooth. However, there are no known three-dimensional single domain spectral shock capturing results (shock waves are not smooth). In the finite element community, a method where the degree of the elements is very high or increases as the grid parameter h increases is sometimes called a spectral element method.
Spectral methods can be used to solve differential equations (PDEs, ODEs, eigenvalue, etc) and optimization problems. When applying spectral methods to time-dependent PDEs, the solution is typically written as a sum of basis functions with time-dependent coefficients; substituting this in the PDE yields a system of ODEs in the coefficients which can be solved using any numerical method for ODEs. Eigenvalue problems for ODEs are similarly converted to matrix eigenvalue problems .
Spectral methods were developed in a long series of papers by Steven Orszag starting in 1969 including, but not limited to, Fourier series methods for periodic geometry problems, polynomial spectral methods for finite and unbounded geometry problems, pseudospectral methods for highly nonlinear problems, and spectral iteration methods for fast solution of steady-state problems. The implementation of the spectral method is normally accomplished either with collocation or a Galerkin or a Tau approach . For very small problems, the spectral method is unique in that solutions may be written out symbolically, yielding a practical alternative to series solutions for differential equations.
Spectral methods can be computationally less expensive and easier to implement than finite element methods; they shine best when high accuracy is sought in simple domains with smooth solutions. However, because of their global nature, the matrices associated with step computation are dense and computational efficiency will quickly suffer when there are many degrees of freedom (with some exceptions, for example if matrix applications can be written as Fourier transforms). For lar
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https://en.wikipedia.org/wiki/Homogeneous%20coordinates
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In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix. They are also used in fundamental elliptic curve cryptography algorithms.
If homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infinity, the number of coordinates required to allow this extension is one more than the dimension of the projective space being considered. For example, two homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates are required to specify a point in the projective plane.
Introduction
The real projective plane can be thought of as the Euclidean plane with additional points added, which are called points at infinity, and are considered to lie on a new line, the line at infinity. There is a point at infinity corresponding to each direction (numerically given by the slope of a line), informally defined as the limit of a point that moves in that direction away from the origin. Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. Given a point on the Euclidean plane, for any non-zero real number Z, the triple is called a set of homogeneous coordinates for the point. By this definition, multiplying the three homogeneous coordinates by a common, non-zero factor gives a new set of homogeneous coordinates for the same point. In particular, is such a system of homogeneous coordinates for the point .
For example, the Cartesian point can be represented in homogeneous coordinates as or . The original Cartesian coordinates are recovered by dividing the first two positions by the third. Thus unlike Cartesian coordinates, a single point can be represented by infinitely many homogeneous coordinates.
The equation of a line through the origin may be written where n and m are not both 0. In parametric form this can be written . Let Z = 1/t, so the coordinates of a point on the line may be written . In homogeneous coordinates this becomes . In the limit, as t approaches infinity, in other words, as the point moves away from the origin, Z approaches 0 and the homogeneous coordinates of the point become . Thus we define as the homogeneous coordinates of the point at infini
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https://en.wikipedia.org/wiki/Green%27s%20theorem
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In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem.
Theorem
Let be a positively oriented, piecewise smooth, simple closed curve in a plane, and let be the region bounded by . If and are functions of defined on an open region containing and have continuous partial derivatives there, then
where the path of integration along is anticlockwise.
In physics, Green's theorem finds many applications. One is solving two-dimensional flow integrals, stating that the sum of fluid outflowing from a volume is equal to the total outflow summed about an enclosing area. In plane geometry, and in particular, area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter.
Proof when D is a simple region
The following is a proof of half of the theorem for the simplified area D, a type I region where C1 and C3 are curves connected by vertical lines (possibly of zero length). A similar proof exists for the other half of the theorem when D is a type II region where C2 and C4 are curves connected by horizontal lines (again, possibly of zero length). Putting these two parts together, the theorem is thus proven for regions of type III (defined as regions which are both type I and type II). The general case can then be deduced from this special case by decomposing D into a set of type III regions.
If it can be shown that
and
are true, then Green's theorem follows immediately for the region D. We can prove () easily for regions of type I, and () for regions of type II. Green's theorem then follows for regions of type III.
Assume region D is a type I region and can thus be characterized, as pictured on the right, by
where g1 and g2 are continuous functions on . Compute the double integral in ():
Now compute the line integral in (). C can be rewritten as the union of four curves: C1, C2, C3, C4.
With C1, use the parametric equations: x = x, y = g1(x), a ≤ x ≤ b. Then
With C3, use the parametric equations: x = x, y = g2(x), a ≤ x ≤ b. Then
The integral over C3 is negated because it goes in the negative direction from b to a, as C is oriented positively (anticlockwise). On C2 and C4, x remains constant, meaning
Therefore,
Combining () with (), we get () for regions of type I. A similar treatment yields () for regions of type II. Putting the two together, we get the result for regions of type III.
Proof for rectifiable Jordan curves
We are going to prove the following
We need the following lemmas whose proofs can be found in:
Now we are in position to prove the theorem:
Proof of Theorem. Let be an arbitrary positive real number. By continuity of , and compactness of , given , there exists such that whenever two points of are less than apart, their images under are less than apart. For this , consider the decomposition giv
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https://en.wikipedia.org/wiki/Erlangen%20program
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In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as Vergleichende Betrachtungen über neuere geometrische Forschungen. It is named after the University Erlangen-Nürnberg, where Klein worked.
By 1872, non-Euclidean geometries had emerged, but without a way to determine their hierarchy and relationships. Klein's method was fundamentally innovative in three ways:
Projective geometry was emphasized as the unifying frame for all other geometries considered by him. In particular, Euclidean geometry was more restrictive than affine geometry, which in turn is more restrictive than projective geometry.
Klein proposed that group theory, a branch of mathematics that uses algebraic methods to abstract the idea of symmetry, was the most useful way of organizing geometrical knowledge; at the time it had already been introduced into the theory of equations in the form of Galois theory.
Klein made much more explicit the idea that each geometrical language had its own, appropriate concepts, thus for example projective geometry rightly talked about conic sections, but not about circles or angles because those notions were not invariant under projective transformations (something familiar in geometrical perspective). The way the multiple languages of geometry then came back together could be explained by the way subgroups of a symmetry group related to each other.
Later, Élie Cartan generalized Klein's homogeneous model spaces to Cartan connections on certain principal bundles, which generalized Riemannian geometry.
The problems of nineteenth century geometry
Since Euclid, geometry had meant the geometry of Euclidean space of two dimensions (plane geometry) or of three dimensions (solid geometry). In the first half of the nineteenth century there had been several developments complicating the picture. Mathematical applications required geometry of four or more dimensions; the close scrutiny of the foundations of the traditional Euclidean geometry had revealed the independence of the parallel postulate from the others, and non-Euclidean geometry had been born. Klein proposed an idea that all these new geometries are just special cases of the projective geometry, as already developed by Poncelet, Möbius, Cayley and others. Klein also strongly suggested to mathematical physicists that even a moderate cultivation of the projective purview might bring substantial benefits to them.
With every geometry, Klein associated an underlying group of symmetries. The hierarchy of geometries is thus mathematically represented as a hierarchy of these groups, and hierarchy of their invariants. For example, lengths, angles and areas are preserved with respect to the Euclidean group of symmetries, while only the incidence structure and the cross-ratio are preserved under the most general projective transformations. A concept of parallelism, which is preserved in
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https://en.wikipedia.org/wiki/Cubic%20function
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In mathematics, a cubic function is a function of the form that is, a polynomial function of degree three. In many texts, the coefficients , , , and are supposed to be real numbers, and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to complex numbers. In other cases, the coefficients may be complex numbers, and the function is a complex function that has the set of the complex numbers as its codomain, even when the domain is restricted to the real numbers.
Setting produces a cubic equation of the form
whose solutions are called roots of the function.
A cubic function with real coefficients has either one or three real roots (which may not be distinct); all odd-degree polynomials with real coefficients have at least one real root.
The graph of a cubic function always has a single inflection point. It may have two critical points, a local minimum and a local maximum. Otherwise, a cubic function is monotonic. The graph of a cubic function is symmetric with respect to its inflection point; that is, it is invariant under a rotation of a half turn around this point. Up to an affine transformation, there are only three possible graphs for cubic functions.
Cubic functions are fundamental for cubic interpolation.
History
Critical and inflection points
The critical points of a cubic function are its stationary points, that is the points where the slope of the function is zero. Thus the critical points of a cubic function defined by
,
occur at values of such that the derivative
of the cubic function is zero.
The solutions of this equation are the -values of the critical points and are given, using the quadratic formula, by
The sign of the expression inside the square root determines the number of critical points. If it is positive, then there are two critical points, one is a local maximum, and the other is a local minimum. If , then there is only one critical point, which is an inflection point. If , then there are no (real) critical points. In the two latter cases, that is, if is nonpositive, the cubic function is strictly monotonic. See the figure for an example of the case .
The inflection point of a function is where that function changes concavity. An inflection point occurs when the second derivative is zero, and the third derivative is nonzero. Thus a cubic function has always a single inflection point, which occurs at
Classification
The graph of a cubic function is a cubic curve, though many cubic curves are not graphs of functions.
Although cubic functions depend on four parameters, their graph can have only very few shapes. In fact, the graph of a cubic function is always similar to the graph of a function of the form
This similarity can be built as the composition of translations parallel to the coordinates axes, a homothecy (uniform scaling), and, possibly, a reflection (mirror image) with respect to the -axis. A further non-uni
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https://en.wikipedia.org/wiki/Projective%20geometry
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In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice-versa.
Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant with respect to projective transformations, as is seen in perspective drawing. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of projective geometry in two dimensions.
While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory of complex projective space, the coordinates used (homogeneous coordinates) being complex numbers. Several major types of more abstract mathematics (including invariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programme resulting in the study of the classical groups) were motivated by projective geometry. It was also a subject with many practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry.
The topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry (the study of projective varieties) and projective differential geometry (the study of differential invariants of the projective transformations).
Overview
Projective geometry is an elementary non-metrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines. That there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes (that always meet), and other linear s
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https://en.wikipedia.org/wiki/Affine%20geometry
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In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle.
As the notion of parallel lines is one of the main properties that is independent of any metric, affine geometry is often considered as the study of parallel lines. Therefore, Playfair's axiom (Given a line and a point not on , there is exactly one line parallel to that passes through .) is fundamental in affine geometry. Comparisons of figures in affine geometry are made with affine transformations, which are mappings that preserve alignment of points and parallelism of lines.
Affine geometry can be developed in two ways that are essentially equivalent.
In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom).
Affine geometry can also be developed on the basis of linear algebra. In this context an affine space is a set of points equipped with a set of transformations (that is bijective mappings), the translations, which forms a vector space (over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; the composition of two translations is their sum in the vector space of the translations.
In more concrete terms, this amounts to having an operation that associates to any ordered pair of points a vector and another operation that allows translation of a point by a vector to give another point; these operations are required to satisfy a number of axioms (notably that two successive translations have the effect of translation by the sum vector). By choosing any point as "origin", the points are in one-to-one correspondence with the vectors, but there is no preferred choice for the origin; thus an affine space may be viewed as obtained from its associated vector space by "forgetting" the origin (zero vector).
The idea of forgetting the metric can be applied in the theory of manifolds. That is developed in the article on the affine connection.
History
In 1748, Leonhard Euler introduced the term affine () in his book (volume 2, chapter XVIII). In 1827, August Möbius wrote on affine geometry in his (chapter 3).
After Felix Klein's Erlangen program, affine geometry was recognized as a generalization of Euclidean geometry.
In 1918, Hermann Weyl referred to affine geometry for his text Space, Time, Matter. He used affine geometry to introduce vector addition and subtraction at the earliest stages of his development of mathematical physics. Later, E. T. Whittaker wrote:
Weyl's geometry is interesting historically as having been the first of the affine geometries to be worked out in detail: it is based on a special type of parallel transport [...using] worldlines of light-signals in four-dimensional space-time. A short element of one of these world-lines may be called a null-vector;
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https://en.wikipedia.org/wiki/Multilinear%20algebra
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Multilinear algebra is the study of functions with multiple vector-valued arguments, which are linear maps with respect to each argument. Concepts such as matrices, vectors, systems of linear equations, higher-dimensional spaces, determinants, inner and outer products, and dual spaces emerge naturally in the mathematics of multilinear functions. Multilinear algebra is a foundational mathematical tool in engineering, machine learning, physics, and mathematics.
Origin
While many theoretical concepts and applications are concerned with single vectors, mathematicians such as Hermann Grassmann considered the structures involving pairs, triplets, and general multi-vectors that generalize vectors. With multiple combinational possibilities, the space of multi-vectors expands to 2n dimensions, where n is the dimension of the relevant vector space. The determinant can be formulated abstractly using the structures of multilinear algebra. Multilinear algebra appears in the study of the mechanical response of materials to stress and strain, involving various moduli of elasticity. The term "tensor" describes elements within the multilinear space due to its added structure. Despite Grassmann's early work in 1844 with his Ausdehnungslehre, which was also republished in 1862, the subject was initially not widely understood, as even ordinary linear algebra posed many challenges at the time.
The concepts of multilinear algebra find applications in certain studies of multivariate calculus and manifolds, particularly in relation to the Jacobian matrix. Infinitesimal differentials encountered in single-variable calculus are transformed into differential forms in multivariate calculus, and their manipulation is carried out using exterior algebra.
Following Grassmann, developments in multilinear algebra were made by Victor Schlegel in 1872 with the publication of the first part of his System der Raumlehre and by Elwin Bruno Christoffel. Notably, significant advancements came through the work of Gregorio Ricci-Curbastro and Tullio Levi-Civita, particularly in the form of absolute differential calculus within multilinear algebra. Marcel Grossmann and Michele Besso introduced this form to Albert Einstein, and in 1915, Einstein's publication on general relativity, explaining the precession of Mercury's perihelion, established multilinear algebra and tensors as important mathematical tools in physics.
In 1958, Nicolas Bourbaki included a chapter on multilinear algebra titled "Algèbre Multilinéaire" in his series Éléments de mathématique, specifically within the book on algebra. The chapter covers topics such as bilinear functions, the tensor product of two modules, and the properties of tensor products.
Topics in multilinear algebra
The field of multilinear algebra has experienced many changes in its presentation over the years. The following pages provide additional information that is central to the topic:
Multivector
Geometric algebra
Clifford algebra
Closed and ex
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https://en.wikipedia.org/wiki/Singular%20%28software%29
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Singular (typeset Singular) is a computer algebra system for polynomial computations with special emphasis on the needs of commutative and non-commutative algebra, algebraic geometry, and singularity theory. Singular has been released under the terms of GNU General Public License. Problems in non-commutative algebra can be tackled with the Singular offspring Plural. Singular is developed under the direction of Wolfram Decker, Gert-Martin Greuel, Gerhard Pfister, and Hans Schönemann, who head Singular's core development team within the Department of Mathematics of the Technische Universität Kaiserslautern.
In the DFG Priority Program 1489, interfaces to GAP, Polymake and Gfan are being developed in order to cover recently established areas of mathematics involving convex and algebraic geometry, such as toric and tropical geometry.
See also
Comparison of computer algebra systems
References
Further reading
External links
Online Manual – PLURAL
Computer algebra system software for Linux
Computer algebra system software for macOS
Computer algebra system software for Windows
Free computer algebra systems
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https://en.wikipedia.org/wiki/Window%20function
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In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually approaching a maximum in the middle, and usually tapering away from the middle. Mathematically, when another function or waveform/data-sequence is "multiplied" by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the "view through the window". Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions.
The reasons for examining segments of a longer function include detection of transient events and time-averaging of frequency spectra. The duration of the segments is determined in each application by requirements like time and frequency resolution. But that method also changes the frequency content of the signal by an effect called spectral leakage. Window functions allow us to distribute the leakage spectrally in different ways, according to the needs of the particular application. There are many choices detailed in this article, but many of the differences are so subtle as to be insignificant in practice.
In typical applications, the window functions used are non-negative, smooth, "bell-shaped" curves. Rectangle, triangle, and other functions can also be used. A more general definition of window functions does not require them to be identically zero outside an interval, as long as the product of the window multiplied by its argument is square integrable, and, more specifically, that the function goes sufficiently rapidly toward zero.
Applications
Window functions are used in spectral analysis/modification/resynthesis, the design of finite impulse response filters, as well as beamforming and antenna design.
Spectral analysis
The Fourier transform of the function is zero, except at frequency ±ω. However, many other functions and waveforms do not have convenient closed-form transforms. Alternatively, one might be interested in their spectral content only during a certain time period.
In either case, the Fourier transform (or a similar transform) can be applied on one or more finite intervals of the waveform. In general, the transform is applied to the product of the waveform and a window function. Any window (including rectangular) affects the spectral estimate computed by this method.
Filter design
Windows are sometimes used in the design of digital filters, in particular to convert an "ideal" impulse response of infinite duration, such as a sinc function, to a finite impulse response (FIR) filter design. That is called the window method.
Statistics and curve fitting
Window functions are sometimes used in the field of sta
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https://en.wikipedia.org/wiki/Matroid
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In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite simple matroid is equivalent to a geometric lattice.
Matroid theory borrows extensively from the terminology of both linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry, topology, combinatorial optimization, network theory and coding theory.
Definition
There are many equivalent ways to define a (finite) matroid.
Independent sets
In terms of independence, a finite matroid is a pair , where is a finite set (called the ground set) and is a family of subsets of (called the independent sets) with the following properties:
(I1) The empty set is independent, i.e., .
(I2) Every subset of an independent set is independent, i.e., for each , if then . This is sometimes called the hereditary property, or the downward-closed property.
(I3) If and are two independent sets (i.e., each set is independent) and has more elements than , then there exists such that is in . This is sometimes called the augmentation property or the independent set exchange property.
The first two properties define a combinatorial structure known as an independence system (or abstract simplicial complex). Actually, assuming (I2), property (I1) is equivalent to the fact that at least one subset of is independent, i.e., .
Bases and circuits
A subset of the ground set that is not independent is called dependent. A maximal independent set—that is, an independent set that becomes dependent upon adding any element of —is called a basis for the matroid. A circuit in a matroid is a minimal dependent subset of —that is, a dependent set whose proper subsets are all independent. The terminology arises because the circuits of graphic matroids are cycles in the corresponding graphs.
The dependent sets, the bases, or the circuits of a matroid characterize the matroid completely: a set is independent if and only if it is not dependent, if and only if it is a subset of a basis, and if and only if it does not contain a circuit. The collections of dependent sets, of bases, and of circuits each have simple properties that may be taken as axioms for a matroid. For instance, one may define a matroid to be a pair , where is a finite set as before and is a collection of subsets of , called "bases", with the following properties:
(B1) is nonempty.
(B2) If and are distinct members of and , then there exists an element such that . This property is called the basis exchange property.
It follows from the basis exchange property that no member of can
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https://en.wikipedia.org/wiki/Improper%20rotation
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In geometry, an improper rotation (also called rotation-reflection, rotoreflection, rotary reflection, or rotoinversion) is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis. Reflection and inversion are each special case of improper rotation. Any improper rotation is an affine transformation and, in cases that keep the coordinate origin fixed, a linear transformation.
It is used as a symmetry operation in the context of geometric symmetry, molecular symmetry and crystallography, where an object that is unchanged by a combination of rotation and reflection is said to have improper rotation symmetry.
Three dimensions
In 3 dimensions, improper rotation is equivalently defined as a combination of rotation about an axis and inversion in a point on the axis. For this reason it is also called a rotoinversion or rotary inversion. The two definitions are equivalent because rotation by an angle θ followed by reflection is the same transformation as rotation by θ + 180° followed by inversion (taking the point of inversion to be in the plane of reflection). In both definitions, the operations commute.
A three-dimensional symmetry that has only one fixed point is necessarily an improper rotation.
An improper rotation of an object thus produces a rotation of its mirror image. The axis is called the rotation-reflection axis. This is called an n-fold improper rotation if the angle of rotation, before or after reflexion, is 360°/n (where n must be even). There are several different systems for naming individual improper rotations:
In the Schoenflies notation the symbol Sn (German, , for mirror), where n must be even, denotes the symmetry group generated by an n-fold improper rotation. For example, the symmetry operation S6 is the combination of a rotation of (360°/6)=60° and a mirror plane reflection. (This should not be confused with the same notation for symmetric groups).
In Hermann–Mauguin notation the symbol is used for an n-fold rotoinversion; i.e., rotation by an angle of rotation of 360°/n with inversion. If n is even it must be divisible by 4. (Note that would be simply a reflection, and is normally denoted "m", for "mirror".) When n is odd this corresponds to a 2n-fold improper rotation (or rotary reflexion).
The Coxeter notation for S2n is [2n+,2+] and , as an index 4 subgroup of [2n,2], , generated as the product of 3 reflections.
The Orbifold notation is n×, order 2n.
Subgroups
The direct subgroup of S2n is Cn, order n, index 2, being the rotoreflection generator applied twice.
For odd n, S2n contains an inversion, denoted Ci or S2. S2n is the direct product: S2n = Cn × S2, if n is odd.
For any n, if odd p is a divisor of n, then S2n/p is a subgroup of S2n, index p. For example S4 is a subgroup of S12, index 3.
As an indirect isometry
In a wider sense, an improper rotation may be defined as any indirect isometry; i.e., an element of E(3)\E+(3): t
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https://en.wikipedia.org/wiki/Scalar%20multiplication
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In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. Scalar multiplication is the multiplication of a vector by a scalar (where the product is a vector), and is to be distinguished from inner product of two vectors (where the product is a scalar).
Definition
In general, if K is a field and V is a vector space over K, then scalar multiplication is a function from K × V to V.
The result of applying this function to k in K and v in V is denoted kv.
Properties
Scalar multiplication obeys the following rules (vector in boldface):
Additivity in the scalar: (c + d)v = cv + dv;
Additivity in the vector: c(v + w) = cv + cw;
Compatibility of product of scalars with scalar multiplication: (cd)v = c(dv);
Multiplying by 1 does not change a vector: 1v = v;
Multiplying by 0 gives the zero vector: 0v = 0;
Multiplying by −1 gives the additive inverse: (−1)v = −v.
Here, + is addition either in the field or in the vector space, as appropriate; and 0 is the additive identity in either.
Juxtaposition indicates either scalar multiplication or the multiplication operation in the field.
Interpretation
Scalar multiplication may be viewed as an external binary operation or as an action of the field on the vector space. A geometric interpretation of scalar multiplication is that it stretches or contracts vectors by a constant factor. As a result, it produces a vector in the same or opposite direction of the original vector but of a different length.
As a special case, V may be taken to be K itself and scalar multiplication may then be taken to be simply the multiplication in the field.
When V is Kn, scalar multiplication is equivalent to multiplication of each component with the scalar, and may be defined as such.
The same idea applies if K is a commutative ring and V is a module over K.
K can even be a rig, but then there is no additive inverse.
If K is not commutative, the distinct operations left scalar multiplication cv and right scalar multiplication vc may be defined.
Scalar multiplication of matrices
The left scalar multiplication of a matrix with a scalar gives another matrix of the same size as . It is denoted by , whose entries of are defined by
explicitly:
Similarly, even though there is no widely-accepted definition, the right scalar multiplication of a matrix with a scalar could be defined to be
explicitly:
When the entries of the matrix and the scalars are from the same commutative field, for example, the real number field or the complex number field, these two multiplications are the same, and can be simply called scalar multiplication. For matrices over a more general field that is not commutative, they may not be equal.
For a real scalar and matrix
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https://en.wikipedia.org/wiki/Bipartite
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Bipartite may refer to:
2 (number)
Bipartite (theology), a philosophical term describing the human duality of body and soul
Bipartite graph, in mathematics, a graph in which the vertices are partitioned into two sets and every edge has an endpoint in each set
Bipartite uterus, a type of uterus found in deer and moose, etc.
Bipartite treaty, a treaty between two parties
See also
Dichotomy
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https://en.wikipedia.org/wiki/Total
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Total may refer to:
Mathematics
Total, the summation of a set of numbers
Total order, a partial order without incomparable pairs
Total relation, which may also mean
connected relation (a binary relation in which any two elements are comparable).
Total function, a partial function that is also a total relation
Business
TotalEnergies, a French petroleum company
Total (cereal), a food brand by General Mills
Total, a brand of strained yogurt made by Fage
Total, a database management system marketed by Cincom Systems
Total Linhas Aéreas - a brazilian airline
Total, a line of dental products by Colgate
Music and culture
Total (group), an American R&B girl group
Total: From Joy Division to New Order, a compilation album
Total (Sebastian album)
Total (Total album)
Total (Teenage Bottlerocket album)
Total (Seigmen album)
Total (Wanessa album)
Total (Belinda Peregrín album)
Total 1, an annual compilation album
Total, the one time recording name of British musician Matthew Bower
Total!, a British videogames magazine
Sports
Total (football club)
See also
La Totale! (film), 1991 French spy comedy film
TOTALe, a software suite related to Rosetta Stone (software)
Total war, a large-scale military conflict
Totaled, the write-off of a damaged vehicle on cost grounds
Totally (disambiguation)
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https://en.wikipedia.org/wiki/Odd
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Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Mathematics
Even and odd numbers, an integer is odd if dividing by two does not yield an integer
Even and odd functions, a function is odd if f(−x) = −f(x) for all x
Even and odd permutations, a permutation of a finite set is odd if it is composed of an odd number of transpositions
Arts and entertainment
Odd Della Robbia, a character in the animated television series Code Lyoko
Odd Thomas (character), a character in a series of novels by Dean Koontz
the protagonist of Odd and the Frost Giants, a book by Neil Gaiman
"Odd", a science fiction short story by John Wyndham in the collection The Seeds of Time
Odd (Shinee album), an album by the South Korean boy band Shinee
"Odd", a song by Loona Odd Eye Circle from Mix & Match
Ships
HNoMS Odd, a Storm-class patrol boat of the Royal Norwegian Navy
, a Norwegian whaler
Other uses
Odd (name), a male name common in Norway
Odd, West Virginia, U.S., an unincorporated community
See also
ODD (disambiguation), an initialism
Odds, from probability theory and gambling
Oddity (disambiguation)
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https://en.wikipedia.org/wiki/Hodge%20conjecture
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In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.
In simple terms, the Hodge conjecture asserts that the basic topological information like the number of holes in certain geometric spaces, complex algebraic varieties, can be understood by studying the possible nice shapes sitting inside those spaces, which look like zero sets of polynomial equations. The latter objects can be studied using algebra and the calculus of analytic functions, and this allows one to indirectly understand the broad shape and structure of often higher-dimensional spaces which can not be otherwise easily visualized.
More specifically, the conjecture states that certain de Rham cohomology classes are algebraic; that is, they are sums of Poincaré duals of the homology classes of subvarieties. It was formulated by the Scottish mathematician William Vallance Douglas Hodge as a result of a work in between 1930 and 1940 to enrich the description of de Rham cohomology to include extra structure that is present in the case of complex algebraic varieties. It received little attention before Hodge presented it in an address during the 1950 International Congress of Mathematicians, held in Cambridge, Massachusetts. The Hodge conjecture is one of the Clay Mathematics Institute's Millennium Prize Problems, with a prize of $1,000,000 US for whoever can prove or disprove the Hodge conjecture.
Motivation
Let X be a compact complex manifold of complex dimension n. Then X is an orientable smooth manifold of real dimension , so its cohomology groups lie in degrees zero through . Assume X is a Kähler manifold, so that there is a decomposition on its cohomology with complex coefficients
where is the subgroup of cohomology classes which are represented by harmonic forms of type . That is, these are the cohomology classes represented by differential forms which, in some choice of local coordinates , can be written as a harmonic function times
Since X is a compact oriented manifold, X has a fundamental class, and so X can be integrated over.
Let Z be a complex submanifold of X of dimension k, and let be the inclusion map. Choose a differential form of type . We can integrate over Z using the pullback function ,
To evaluate this integral, choose a point of Z and call it . The inclusion of Z in X means that we can choose a local basis on X and have . If , then must contain some where pulls back to zero on Z. The same is true for if . Consequently, this integral is zero if .
The Hodge conjecture then (loosely) asks:
Which cohomology classes in come from complex subvarieties Z?
Statement of the Hodge conjecture
Let
We call this the group of Hodge classes of degree 2k on X.
The modern statement of the Hodge conjecture is
Hodge conjecture. Let X be a non-singular complex projective manifold. Then every Hodge cla
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https://en.wikipedia.org/wiki/Rank%20of%20an%20abelian%20group
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In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the rational numbers of dimension rank A. For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup. Torsion-free abelian groups of rank 1 have been completely classified. However, the theory of abelian groups of higher rank is more involved.
The term rank has a different meaning in the context of elementary abelian groups.
Definition
A subset {aα} of an abelian group A is linearly independent (over Z) if the only linear combination of these elements that is equal to zero is trivial: if
where all but finitely many coefficients nα are zero (so that the sum is, in effect, finite), then all coefficients are zero. Any two maximal linearly independent sets in A have the same cardinality, which is called the rank of A.
The rank of an abelian group is analogous to the dimension of a vector space. The main difference with the case of vector space is a presence of torsion. An element of an abelian group A is classified as torsion if its order is finite. The set of all torsion elements is a subgroup, called the torsion subgroup and denoted T(A). A group is called torsion-free if it has no non-trivial torsion elements. The factor-group A/T(A) is the unique maximal torsion-free quotient of A and its rank coincides with the rank of A.
The notion of rank with analogous properties can be defined for modules over any integral domain, the case of abelian groups corresponding to modules over Z. For this, see finitely generated module#Generic rank.
Properties
The rank of an abelian group A coincides with the dimension of the Q-vector space A ⊗ Q. If A is torsion-free then the canonical map A → A ⊗ Q is injective and the rank of A is the minimum dimension of Q-vector space containing A as an abelian subgroup. In particular, any intermediate group Zn < A < Qn has rank n.
Abelian groups of rank 0 are exactly the periodic abelian groups.
The group Q of rational numbers has rank 1. Torsion-free abelian groups of rank 1 are realized as subgroups of Q and there is a satisfactory classification of them up to isomorphism. By contrast, there is no satisfactory classification of torsion-free abelian groups of rank 2.
Rank is additive over short exact sequences: if
is a short exact sequence of abelian groups then rk B = rk A + rk C. This follows from the flatness of Q and the corresponding fact for vector spaces.
Rank is additive over arbitrary direct sums:
where the sum in the right hand side uses cardinal arithmetic.
Groups of higher rank
Abelian groups of rank greater than 1 are sources of interesting examples. For instance, for every cardinal d there exist torsion-free abel
|
https://en.wikipedia.org/wiki/Injective%20cogenerator
|
In category theory, a branch of mathematics, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality. Generators are objects which cover other objects as an approximation, and (dually) cogenerators are objects which envelope other objects as an approximation.
More precisely:
A generator of a category with a zero object is an object G such that for every nonzero object H there exists a nonzero morphism f:G → H.
A cogenerator is an object C such that for every nonzero object H there exists a nonzero morphism f:H → C. (Note the reversed order).
The abelian group case
Assuming one has a category like that of abelian groups, one can in fact form direct sums of copies of G until the morphism
f: Sum(G) →H
is surjective; and one can form direct products of C until the morphism
f:H→ Prod(C)
is injective.
For example, the integers are a generator of the category of abelian groups (since every abelian group is a quotient of a free abelian group). This is the origin of the term generator. The approximation here is normally described as generators and relations.
As an example of a cogenerator in the same category, we have Q/Z, the rationals modulo the integers, which is a divisible abelian group. Given any abelian group A, there is an isomorphic copy of A contained inside the product of |A| copies of Q/Z. This approximation is close to what is called the divisible envelope - the true envelope is subject to a minimality condition.
General theory
Finding a generator of an abelian category allows one to express every object as a quotient of a direct sum of copies of the generator. Finding a cogenerator allows one to express every object as a subobject of a direct product of copies of the cogenerator. One is often interested in projective generators (even finitely generated projective generators, called progenerators) and minimal injective cogenerators. Both examples above have these extra properties.
The cogenerator Q/Z is useful in the study of modules over general rings. If H is a left module over the ring R, one forms the (algebraic) character module H* consisting of all abelian group homomorphisms from H to Q/Z. H* is then a right R-module. Q/Z being a cogenerator says precisely that H* is 0 if and only if H is 0. Even more is true: the * operation takes a homomorphism
f:H → K
to a homomorphism
f*:K* → H*,
and f* is 0 if and only if f is 0. It is thus a faithful contravariant functor from left R-modules to right R-modules.
Every H* is pure-injective (also called algebraically compact). One can often consider a problem after applying the * to simplify matters.
All of this can also be done for continuous modules H: one forms the topological character module of continuous group homomorphisms from H to the circle group R/Z.
In general topology
The Tietze extension theorem can be used to show that an interval is an injective cogenerator in a category of topological spaces subject to separation axioms.
Re
|
https://en.wikipedia.org/wiki/Higher-order%20function
|
In mathematics and computer science, a higher-order function (HOF) is a function that does at least one of the following:
takes one or more functions as arguments (i.e. a procedural parameter, which is a parameter of a procedure that is itself a procedure),
returns a function as its result.
All other functions are first-order functions. In mathematics higher-order functions are also termed operators or functionals. The differential operator in calculus is a common example, since it maps a function to its derivative, also a function. Higher-order functions should not be confused with other uses of the word "functor" throughout mathematics, see Functor (disambiguation).
In the untyped lambda calculus, all functions are higher-order; in a typed lambda calculus, from which most functional programming languages are derived, higher-order functions that take one function as argument are values with types of the form .
General examples
map function, found in many functional programming languages, is one example of a higher-order function. It takes as arguments a function f and a collection of elements, and as the result, returns a new collection with f applied to each element from the collection.
Sorting functions, which take a comparison function as a parameter, allowing the programmer to separate the sorting algorithm from the comparisons of the items being sorted. The C standard function qsort is an example of this.
filter
fold
apply
Function composition
Integration
Callback
Tree traversal
Montague grammar, a semantic theory of natural language, uses higher-order functions
Support in programming languages
Direct support
The examples are not intended to compare and contrast programming languages, but to serve as examples of higher-order function syntax
In the following examples, the higher-order function takes a function, and applies the function to some value twice. If has to be applied several times for the same it preferably should return a function rather than a value. This is in line with the "don't repeat yourself" principle.
APL
twice←{⍺⍺ ⍺⍺ ⍵}
plusthree←{⍵+3}
g←{plusthree twice ⍵}
g 7
13
Or in a tacit manner:
twice←⍣2
plusthree←+∘3
g←plusthree twice
g 7
13
C++
Using in C++11:
#include <iostream>
#include <functional>
auto twice = [](const std::function<int(int)>& f)
{
return [f](int x) {
return f(f(x));
};
};
auto plus_three = [](int i)
{
return i + 3;
};
int main()
{
auto g = twice(plus_three);
std::cout << g(7) << '\n'; // 13
}
Or, with generic lambdas provided by C++14:
#include <iostream>
auto twice = [](const auto& f)
{
return [f](int x) {
return f(f(x));
};
};
auto plus_three = [](int i)
{
return i + 3;
};
int main()
{
auto g = twice(plus_three);
std::cout << g(7) << '\n'; // 13
}
C#
Using just delegates:
using System;
public class Program
{
public static void Main(string
|
https://en.wikipedia.org/wiki/Weil%20conjectures
|
In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
The conjectures concern the generating functions (known as local zeta functions) derived from counting points on algebraic varieties over finite fields. A variety over a finite field with elements has a finite number of rational points (with coordinates in the original field), as well as points with coordinates in any finite extension of the original field. The generating function has coefficients derived from the numbers of points over the extension field with elements.
Weil conjectured that such zeta functions for smooth varieties are rational functions, satisfy a certain functional equation, and have their zeros in restricted places. The last two parts were consciously modelled on the Riemann zeta function, a kind of generating function for prime integers, which obeys a functional equation and (conjecturally) has its zeros restricted by the Riemann hypothesis. The rationality was proved by , the functional equation by , and the analogue of the Riemann hypothesis by .
Background and history
The earliest antecedent of the Weil conjectures is by Carl Friedrich Gauss and appears in section VII of his Disquisitiones Arithmeticae , concerned with roots of unity and Gaussian periods. In article 358, he moves on from the periods that build up towers of quadratic extensions, for the construction of regular polygons; and assumes that is a prime number congruent to 1 modulo 3. Then there is a cyclic cubic field inside the cyclotomic field of th roots of unity, and a normal integral basis of periods for the integers of this field (an instance of the Hilbert–Speiser theorem). Gauss constructs the order-3 periods, corresponding to the cyclic group of non-zero residues modulo under multiplication and its unique subgroup of index three. Gauss lets , , and be its cosets. Taking the periods (sums of roots of unity) corresponding to these cosets applied to , he notes that these periods have a multiplication table that is accessible to calculation. Products are linear combinations of the periods, and he determines the coefficients. He sets, for example, equal to the number of elements of which are in and which, after being increased by one, are also in . He proves that this number and related ones are the coefficients of the products of the periods. To see the relation of these sets to the Weil conjectures, notice that if and are both in , then there exist and in such that and ; consequently, . Therefore is related to the number of solutions to in the finite field . The other coefficients have similar interpretations. Gauss's determination of the coefficients of the products of the periods therefore counts the number of points on these elliptic curves, and as a byproduct he proves the analog of the Ri
|
https://en.wikipedia.org/wiki/Sphenic%20number
|
In number theory, a sphenic number (from , 'wedge') is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers.
Definition
A sphenic number is a product pqr where p, q, and r are three distinct prime numbers. In other words, the sphenic numbers are the square-free 3-almost primes.
Examples
The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes.
The first few sphenic numbers are
30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ...
the largest known sphenic number is
(282,589,933 − 1) × (277,232,917 − 1) × (274,207,281 − 1).
It is the product of the three largest known primes.
Divisors
All sphenic numbers have exactly eight divisors. If we express the sphenic number as , where p, q, and r are distinct primes, then the set of divisors of n will be:
The converse does not hold. For example, 24 is not a sphenic number, but it has exactly eight divisors.
Properties
All sphenic numbers are by definition squarefree, because the prime factors must be distinct.
The Möbius function of any sphenic number is −1.
The cyclotomic polynomials , taken over all sphenic numbers n, may contain arbitrarily large coefficients (for n a product of two primes the coefficients are or 0).
Any multiple of a sphenic number (except by 1) isn't a sphenic number. This is easily provable by the multiplication process at a minimum adding another prime factor, or raising an existing factor to a higher power.
Consecutive sphenic numbers
The first case of two consecutive sphenic integers is 230 = 2×5×23 and 231 = 3×7×11. The first case of three is 1309 = 7×11×17, 1310 = 2×5×131, and 1311 = 3×19×23. There is no case of more than three, because every fourth consecutive positive integer is divisible by 4 = 2×2 and therefore not squarefree.
The numbers 2013 (3×11×61), 2014 (2×19×53), and 2015 (5×13×31) are all sphenic. The next three consecutive sphenic years will be 2665 (5×13×41), 2666 (2×31×43) and 2667 (3×7×127) .
See also
Semiprimes, products of two prime numbers.
Almost prime
References
Integer sequences
Prime numbers
|
https://en.wikipedia.org/wiki/L-function
|
In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An L-series is a Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation. The Riemann zeta function is an example of an L-function, and one important conjecture involving L-functions is the Riemann hypothesis and its generalization.
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way. Because of the Euler product formula there is a deep connection between L-functions and the theory of prime numbers.
The mathematical field that studies L-functions is sometimes called analytic theory of L-functions.
Construction
We distinguish at the outset between the L-series, an infinite series representation (for example the Dirichlet series for the Riemann zeta function), and the L-function, the function in the complex plane that is its analytic continuation. The general constructions start with an L-series, defined first as a Dirichlet series, and then by an expansion as an Euler product indexed by prime numbers. Estimates are required to prove that this converges in some right half-plane of the complex numbers. Then one asks whether the function so defined can be analytically continued to the rest of the complex plane (perhaps with some poles).
It is this (conjectural) meromorphic continuation to the complex plane which is called an L-function. In the classical cases, already, one knows that useful information is contained in the values and behaviour of the L-function at points where the series representation does not converge. The general term L-function here includes many known types of zeta functions. The Selberg class is an attempt to capture the core properties of L-functions in a set of axioms, thus encouraging the study of the properties of the class rather than of individual functions.
Conjectural information
One can list characteristics of known examples of L-functions that one would wish to see generalized:
location of zeros and poles;
functional equation, with respect to some vertical line Re(s) = constant;
interesting values at integers related to quantities from algebraic K-theory.
Detailed work has produced a large body of plausible conjectures, for example about the exact type of functional equation that should apply. Since the Riemann zeta function connects through its values at positive even integers (and negative odd integers) to the Bernoulli numbers, one looks for an appropriate generalisation of that phenomenon. In that case results have been obtained for p-adic L-functions, which describe certain Galois modules.
The sta
|
https://en.wikipedia.org/wiki/Jean%20Leray
|
Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology.
Life and career
He was born in Chantenay-sur-Loire (today part of Nantes). He studied at École Normale Supérieure from 1926 to 1929. He received his Ph.D. in 1933. In 1934 Leray published an important paper that founded the study of weak solutions of the Navier–Stokes equations. In the same year, he and Juliusz Schauder discovered a topological invariant, now called the Leray–Schauder degree, which they applied to prove the existence of solutions for partial differential equations lacking uniqueness.
From 1938 to 1939 he was professor at the University of Nancy. He did not join the Bourbaki group, although he was close with its founders.
His main work in topology was carried out while he was in a prisoner of war camp in Edelbach, Austria from 1940 to 1945. He concealed his expertise on differential equations, fearing that its connections with applied mathematics could lead him to be asked to do war work.
Leray's work of this period proved seminal to the development of spectral sequences and sheaves. These were subsequently developed by many others, each separately becoming an important tool in homological algebra.
He returned to work on partial differential equations from about 1950.
He was professor at the University of Paris from 1945 to 1947, and then at the Collège de France until 1978.
He was awarded the Malaxa Prize (Romania, 1938), the Grand Prix in mathematical sciences (French Academy of Sciences, 1940), the Feltrinelli Prize (Accademia dei Lincei, 1971), the Wolf Prize in Mathematics (Israel, 1979), and the Lomonosov Gold Medal (Moscow, 1988). He was an elected to the American Academy of Arts and Sciences and the American Philosophical Society in 1959 and the United States National Academy of Sciences in 1965.
See also
Leray–Schauder theorem
References
External links
"Jean Leray (1906–1998)", by Armand Borel, Gennadi M. Henkin, and Peter D. Lax, Notices of the American Mathematical Society, vol. 47, no. 3, March 2000.
Jean Leray Short biography
1906 births
1998 deaths
20th-century French mathematicians
Mathematical analysts
Topologists
École Normale Supérieure alumni
Wolf Prize in Mathematics laureates
Members of the French Academy of Sciences
Foreign Members of the Royal Society
Foreign Members of the USSR Academy of Sciences
Foreign Members of the Russian Academy of Sciences
Foreign associates of the National Academy of Sciences
Institute for Advanced Study visiting scholars
PDE theorists
Academic staff of Nancy-Université
French prisoners of war in World War II
Members of the American Philosophical Society
|
https://en.wikipedia.org/wiki/Sheaf%20%28mathematics%29
|
In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts).
The field of mathematics that studies sheaves is called sheaf theory.
Sheaves are understood conceptually as general and abstract objects. Their correct definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets.
There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category. On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of sheaf theory.
Due to their general nature and versatility, sheaves have several applications in topology and especially in algebraic and differential geometry. First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space. In such contexts, several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general cohomology theory, which encompasses also the "usual" topological cohomology theories such as singular cohomology. Especially in algebraic geometry and the theory of complex manifolds, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of D-modules, which provide applications to the theory of differential equations. In addition, generalisations of sheaves to more general settings than topological spaces, such as Grothendieck topology, have provided applications to mathematical logic and to number theory.
Definitions and examples
In many mathematical branches, several structures defined on a topological space (e.g., a differentiable manifold) can be naturally localised or restricted to open subsets : typical examples include continuous real-valued or complex-valued functions, -times differentiable (real-valued or complex-valued) functions, bounded real-valued functions, vector fields, and sections of any vector bundle on the space. The
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https://en.wikipedia.org/wiki/Gaussian%20function
|
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form
and with parametric extension
for arbitrary real constants , and non-zero . It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter is the height of the curve's peak, is the position of the center of the peak, and (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell".
Gaussian functions are often used to represent the probability density function of a normally distributed random variable with expected value and variance . In this case, the Gaussian is of the form
Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform.
Properties
Gaussian functions arise by composing the exponential function with a concave quadratic function:
where
(Note: in ,
not to be confused with )
The Gaussian functions are thus those functions whose logarithm is a concave quadratic function.
The parameter is related to the full width at half maximum (FWHM) of the peak according to
The function may then be expressed in terms of the FWHM, represented by :
Alternatively, the parameter can be interpreted by saying that the two inflection points of the function occur at .
The full width at tenth of maximum (FWTM) for a Gaussian could be of interest and is
Gaussian functions are analytic, and their limit as is 0 (for the above case of ).
Gaussian functions are among those functions that are elementary but lack elementary antiderivatives; the integral of the Gaussian function is the error function:
Nonetheless, their improper integrals over the whole real line can be evaluated exactly, using the Gaussian integral
and one obtains
This integral is 1 if and only if (the normalizing constant), and in this case the Gaussian is the probability density function of a normally distributed random variable with expected value and variance :
These Gaussians are plotted in the accompanying figure.
Gaussian functions centered at zero minimize the Fourier uncertainty principle.
The product of two Gaussian functions is a Gaussian, and the convolution of two Gaussian functions is also a Gaussian, with variance being the sum of the original variances: . The product of two Gaussian probability density functions (PDFs), though, is not in general a Gaussian PDF.
Taking the Fourier transform (unitary, angular-frequency convention) of a Gaussian function with parameters , and yields another Gaussian function, with parameters , and . So in particular the Gaussian functions with and are kept fixed by the Fourier transform (they are eigenfunctions of the Fourie
|
https://en.wikipedia.org/wiki/Trigonometric%20integral
|
In mathematics, trigonometric integrals are a family of integrals involving trigonometric functions.
Sine integral
The different sine integral definitions are
Note that the integrand is the sinc function, and also the zeroth spherical Bessel function.
Since is an even entire function (holomorphic over the entire complex plane), is entire, odd, and the integral in its definition can be taken along any path connecting the endpoints.
By definition, is the antiderivative of whose value is zero at , and is the antiderivative whose value is zero at . Their difference is given by the Dirichlet integral,
In signal processing, the oscillations of the sine integral cause overshoot and ringing artifacts when using the sinc filter, and frequency domain ringing if using a truncated sinc filter as a low-pass filter.
Related is the Gibbs phenomenon: If the sine integral is considered as the convolution of the sinc function with the heaviside step function, this corresponds to truncating the Fourier series, which is the cause of the Gibbs phenomenon.
Cosine integral
The different cosine integral definitions are
where is the Euler–Mascheroni constant. Some texts use instead of .
is the antiderivative of (which vanishes as ). The two definitions are related by
is an even, entire function. For that reason, some texts treat as the primary function, and derive in terms of .
Hyperbolic sine integral
The hyperbolic sine integral is defined as
It is related to the ordinary sine integral by
Hyperbolic cosine integral
The hyperbolic cosine integral is
where is the Euler–Mascheroni constant.
It has the series expansion
Auxiliary functions
Trigonometric integrals can be understood in terms of the so-called "auxiliary functions"
Using these functions, the trigonometric integrals may be re-expressed as
(cf. Abramowitz & Stegun, p. 232)
Nielsen's spiral
The spiral formed by parametric plot of is known as Nielsen's spiral.
The spiral is closely related to the Fresnel integrals and the Euler spiral. Nielsen's spiral has applications in vision processing, road and track construction and other areas.
Expansion
Various expansions can be used for evaluation of trigonometric integrals, depending on the range of the argument.
Asymptotic series (for large argument)
These series are asymptotic and divergent, although can be used for estimates and even precise evaluation at .
Convergent series
These series are convergent at any complex , although for , the series will converge slowly initially, requiring many terms for high precision.
Derivation of series expansion
From the Maclaurin series expansion of sine:
Relation with the exponential integral of imaginary argument
The function
is called the exponential integral. It is closely related to and ,
As each respective function is analytic except for the cut at negative values of the argument, the area of validity of the relation should be extended to (Outside this range, additional terms whi
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https://en.wikipedia.org/wiki/Convex%20function
|
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include a linear function (where is a real number), a quadratic function ( as a nonnegative real number) and an exponential function ( as a nonnegative real number). In simple terms, a convex function refers to a function whose graph is shaped like a cup (or a straight line like a linear function), while a concave function's graph is shaped like a cap .
Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. For instance, a strictly convex function on an open set has no more than one minimum. Even in infinite-dimensional spaces, under suitable additional hypotheses, convex functions continue to satisfy such properties and as a result, they are the most well-understood functionals in the calculus of variations. In probability theory, a convex function applied to the expected value of a random variable is always bounded above by the expected value of the convex function of the random variable. This result, known as Jensen's inequality, can be used to deduce inequalities such as the arithmetic–geometric mean inequality and Hölder's inequality.
Definition
Let be a convex subset of a real vector space and let be a function.
Then is called if and only if any of the following equivalent conditions hold:
For all and all :
The right hand side represents the straight line between and in the graph of as a function of increasing from to or decreasing from to sweeps this line. Similarly, the argument of the function in the left hand side represents the straight line between and in or the -axis of the graph of So, this condition requires that the straight line between any pair of points on the curve of to be above or just meets the graph.
For all and all such that :
The difference of this second condition with respect to the first condition above is that this condition does not include the intersection points (for example, and ) between the straight line passing through a pair of points on the curve of (the straight line is represented by the right hand side of this condition) and the curve of the first condition includes the intersection points as it becomes or at or or In fact, the intersection points do not need to be considered in a condition of convex using because and are always true (so not useful to be a part of a condition).
The second statement characterizing
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https://en.wikipedia.org/wiki/Euler%20integral
|
In mathematics, there are two types of Euler integral:
The Euler integral of the first kind is the beta function
The Euler integral of the second kind is the gamma function
For positive integers and , the two integrals can be expressed in terms of factorials and binomial coefficients:
See also
Leonhard Euler
List of topics named after Leonhard Euler
References
External links and references
Wolfram MathWorld on the Euler Integral
NIST Digital Library of Mathematical Functions dlmf.nist.gov/5.2.1 relation 5.2.1 and dlmf.nist.gov/5.12 relation 5.12.1
Gamma and related functions
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https://en.wikipedia.org/wiki/Beta%20function
|
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
for complex number inputs
such that .
The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its symbol is a Greek capital beta.
Properties
The beta function is symmetric, meaning that
for all inputs and .
A key property of the beta function is its close relationship to the gamma function:
A proof is given below in .
The beta function is also closely related to binomial coefficients. When (or , by symmetry) is a positive integer, it follows from the definition of the gamma function that
Relationship to the gamma function
A simple derivation of the relation can be found in Emil Artin's book The Gamma Function, page 18–19.
To derive this relation, write the product of two factorials as
Changing variables by and , because and , we have that the limits of integrations for are 0 to ∞ and the limits of integration for are 0 to 1. Thus produces
Dividing both sides by gives the desired result.
The stated identity may be seen as a particular case of the identity for the integral of a convolution. Taking
one has:
Derivatives
We have
where denotes the polygamma function.
Approximation
Stirling's approximation gives the asymptotic formula
for large and large .
If on the other hand is large and is fixed, then
Other identities and formulas
The integral defining the beta function may be rewritten in a variety of ways, including the following:
where in the second-to-last identity is any positive real number. One may move from the first integral to the second one by substituting .
The beta function can be written as an infinite sum
(where is the rising factorial)
and as an infinite product
The beta function satisfies several identities analogous to corresponding identities for binomial coefficients, including a version of Pascal's identity
and a simple recurrence on one coordinate:
The positive integer values of the beta function are also the partial derivatives of a 2D function: for all nonnegative integers and ,
where
The Pascal-like identity above implies that this function is a solution to the first-order partial differential equation
For , the beta function may be written in terms of a convolution involving the truncated power function :
Evaluations at particular points may simplify significantly; for example,
and
By taking in this last formula, it follows that .
Generalizing this into a bivariate identity for a product of beta functions leads to:
Euler's integral for the beta function may be converted into an integral over the Pochhammer contour as
This Pochhammer contour integral converges for all values of and and so gives the analytic continuation of the beta function.
Just as the gamma function for integers describe
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https://en.wikipedia.org/wiki/Inverse%20Laplace%20transform
|
In mathematics, the inverse Laplace transform of a function F(s) is the piecewise-continuous and exponentially-restricted real function f(t) which has the property:
where denotes the Laplace transform.
It can be proven that, if a function F(s) has the inverse Laplace transform f(t), then f(t) is uniquely determined (considering functions which differ from each other only on a point set having Lebesgue measure zero as the same). This result was first proven by Mathias Lerch in 1903 and is known as Lerch's theorem.
The Laplace transform and the inverse Laplace transform together have a number of properties that make them useful for analysing linear dynamical systems.
Mellin's inverse formula
An integral formula for the inverse Laplace transform, called the Mellin's inverse formula, the Bromwich integral, or the Fourier–Mellin integral, is given by the line integral:
where the integration is done along the vertical line Re(s) = γ in the complex plane such that γ is greater than the real part of all singularities of F(s) and F(s) is bounded on the line, for example if the contour path is in the region of convergence. If all singularities are in the left half-plane, or F(s) is an entire function , then γ can be set to zero and the above inverse integral formula becomes identical to the inverse Fourier transform.
In practice, computing the complex integral can be done by using the Cauchy residue theorem.
Post's inversion formula
Post's inversion formula for Laplace transforms, named after Emil Post, is a simple-looking but usually impractical formula for evaluating an inverse Laplace transform.
The statement of the formula is as follows: Let f(t) be a continuous function on the interval [0, ∞) of exponential order, i.e.
for some real number b. Then for all s > b, the Laplace transform for f(t) exists and is infinitely differentiable with respect to s. Furthermore, if F(s) is the Laplace transform of f(t), then the inverse Laplace transform of F(s) is given by
for t > 0, where F(k) is the k-th derivative of F with respect to s.
As can be seen from the formula, the need to evaluate derivatives of arbitrarily high orders renders this formula impractical for most purposes.
With the advent of powerful personal computers, the main efforts to use this formula have come from dealing with approximations or asymptotic analysis of the Inverse Laplace transform, using the Grunwald–Letnikov differintegral to evaluate the derivatives.
Post's inversion has attracted interest due to the improvement in computational science and the fact that it is not necessary to know where the poles of F(s) lie, which make it possible to calculate the asymptotic behaviour for big x using inverse Mellin transforms for several arithmetical functions related to the Riemann hypothesis.
Software tools
InverseLaplaceTransform performs symbolic inverse transforms in Mathematica
Numerical Inversion of Laplace Transform with Multiple Precision Using the Comple
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https://en.wikipedia.org/wiki/John%20Tate%20%28mathematician%29
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John Torrence Tate Jr. (March 13, 1925 – October 16, 2019) was an American mathematician distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry, and related areas in algebraic geometry. He was awarded the Abel Prize in 2010.
Biography
Tate was born in Minneapolis, Minnesota. His father, John Tate Sr., was a professor of physics at the University of Minnesota and a longtime editor of Physical Review. His mother, Lois Beatrice Fossler, was a high school English teacher. Tate Jr. received his bachelor's degree in mathematics in 1946 from Harvard University and entered the doctoral program in physics at Princeton University. He later transferred to the mathematics department and received his PhD in mathematics in 1950 after completing a doctoral dissertation titled "Fourier analysis in number fields and Hecke's zeta functions" under the supervision of Emil Artin. Tate taught at Harvard for 36 years before joining the University of Texas in 1990 as a Sid W. Richardson Foundation Regents Chair. He retired from the Texas mathematics department in 2009 and returned to Harvard as a professor emeritus.
Tate died at his home in Lexington, Massachusetts on October 16, 2019, at the age of 94.
Mathematical work
Tate's thesis (1950) on Fourier analysis in number fields has become one of the ingredients for the modern theory of automorphic forms and their L-functions, notably by its use of the adele ring, its self-duality and harmonic analysis on it; independently and a little earlier, Kenkichi Iwasawa obtained a similar theory. Together with his advisor Emil Artin, Tate gave a cohomological treatment of global class field theory using techniques of group cohomology applied to the idele class group and Galois cohomology. This treatment made more transparent some of the algebraic structures in the previous approaches to class field theory, which used central division algebras to compute the Brauer group of a global field.
Subsequently, Tate introduced what are now known as Tate cohomology groups. In the decades following that discovery he extended the reach of Galois cohomology with the Poitou–Tate duality, the Tate–Shafarevich group, and relations with algebraic K-theory. With Jonathan Lubin, he recast local class field theory by the use of formal groups, creating the Lubin–Tate local theory of complex multiplication.
He has also made a number of individual and important contributions to p-adic theory; for example, Tate's invention of rigid analytic spaces can be said to have spawned the entire field of rigid analytic geometry. He found a p-adic analogue of Hodge theory, now called Hodge–Tate theory, which has blossomed into another central technique of modern algebraic number theory. Other innovations of his include the "Tate curve" parametrization for certain p-adic elliptic curves and the p-divisible (Tate–Barsotti) groups.
Many of his results were not immediately published and some of them were written up by
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https://en.wikipedia.org/wiki/Elimination%20theory
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In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating some variables between polynomials of several variables, in order to solve systems of polynomial equations.
Classical elimination theory culminated with the work of Francis Macaulay on multivariate resultants, as described in the chapter on Elimination theory in the first editions (1930) of Bartel van der Waerden's Moderne Algebra. After that, elimination theory was ignored by most algebraic geometers for almost thirty years, until the introduction of new methods for solving polynomial equations, such as Gröbner bases, which were needed for computer algebra.
History and connection to modern theories
The field of elimination theory was motivated by the need of methods for solving systems of polynomial equations.
One of the first results was Bézout's theorem, which bounds the number of solutions (in the case of two polynomials in two variables at Bézout time).
Except for Bézout's theorem, the general approach was to eliminate variables for reducing the problem to a single equation in one variable.
The case of linear equations was completely solved by Gaussian elimination, where the older method of Cramer's rule does not proceed by elimination, and works only when the number of equations equals the number of variables. In the 19th century, this was extended to linear Diophantine equations and abelian group with Hermite normal form and Smith normal form.
Before the 20th century, different types of eliminants were introduced, including resultants, and various kinds of discriminants. In general, these eliminants are also invariant under various changes of variables, and are also fundamental in invariant theory.
All these concepts are effective, in the sense that their definitions include a method of computation. Around 1890, David Hilbert introduced non-effective methods, and this was seen as a revolution, which led most algebraic geometers of the first half of the 20th century to try to "eliminate elimination". Nevertheless Hilbert's Nullstellensatz, may be considered to belong to elimination theory, as it asserts that a system of polynomial equations does not have any solution if and only if one may eliminate all unknowns to obtain the constant equation 1 = 0.
Elimination theory culminated with the work of Leopold Kronecker, and finally Macaulay, who introduced multivariate resultants and U-resultants, providing complete elimination methods for systems of polynomial equations, which are described in the chapter on Elimination theory in the first editions (1930) of van der Waerden's Moderne Algebra.
Later, elimination theory was considered old-fashioned and removed from subsequent editions of Moderne Algebra. It was generally ignored until the introduction of computers, and more specifically of computer algebra, which again made relevant the design of efficient elimination algorithms, rather than merely existence
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https://en.wikipedia.org/wiki/Commutative%20algebra
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Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers ; and p-adic integers.
Commutative algebra is the main technical tool in the local study of schemes.
The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras.
Overview
Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry.
In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic have led to the notion of a valuation ring. The restriction of algebraic field extensions to subrings has led to the notions of integral extensions and integrally closed domains as well as the notion of ramification of an extension of valuation rings.
The notion of localization of a ring (in particular the localization with respect to a prime ideal, the localization consisting in inverting a single element and the total quotient ring) is one of the main differences between commutative algebra and the theory of non-commutative rings. It leads to an important class of commutative rings, the local rings that have only one maximal ideal. The set of the prime ideals of a commutative ring is naturally equipped with a topology, the Zariski topology. All these notions are widely used in algebraic geometry and are the basic technical tools for the definition of scheme theory, a generalization of algebraic geometry introduced by Grothendieck.
Many other notions of commutative algebra are counterparts of geometrical notions occurring in algebraic geometry. This is the case of Krull dimension, primary decomposition, regular rings, Cohen–Macaulay rings, Gorenstein rings and many other notions.
History
The subject, first known as ideal theory, began with Richard Dedekind's work on ideals, itself based on the earlier work of Ernst Kummer and Leopold Kronecker. Later, David Hilbert introduced the term ring to generalize the earlier term number ring. Hilbert introduced a more abstract approach to replace the more concrete and computationally oriented methods grounded in such things as complex analysis and classical invariant theory. In turn, Hilbert strongly influenced Emmy Noether, who recast many earlier results in terms of an ascending chain condition, now known as the Noetherian condition. Another important milestone was the work of Hilbert's student Emanuel Lasker, who introduced primary ideals and proved the first version of the Lasker–Noether theorem.
The main figure responsible for the birth of commutative a
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