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https://en.wikipedia.org/wiki/Wassily%20Hoeffding
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Wassily Hoeffding (June 12, 1914 – February 28, 1991) was a Finnish statistician and probabilist. Hoeffding was one of the founders of nonparametric statistics, in which Hoeffding contributed the idea and basic results on U-statistics.
In probability theory, Hoeffding's inequality provides an upper bound on the probability for the sum of random variables to deviate from its expected value.
Personal life
Hoeffding was born in Mustamäki, Finland, (Gorkovskoye, Russia since 1940), although his place of birth is registered as St. Petersburg on his birth certificate. His father was an economist and a disciple of Peter Struve, the Russian social scientist and public figure. His paternal grandparents were Danish and his father's uncle was the Danish philosopher Harald Høffding. His mother, née Wedensky, had studied medicine. Both grandfathers had been engineers. In 1918 the family left Tsarskoye Selo for Ukraine and, after traveling through scenes of civil war, finally left Russia for Denmark in 1920, where Wassily entered school.
In 1924 the family settled in Berlin. Hoeffding obtained his PhD in 1940 at the University of Berlin. He migrated with his mother to the United States in 1946. His younger brother, Oleg, became a military historian in the United States.
Hoeffding's ashes were buried in a small cemetery on land owned by George E. Nicholson, Jr.'s family in Chatham County, NC about 11 miles south of Chapel Hill, NC.
Work
In 1948, he introduced the concept of U-statistics.
See the collected works of Wassily Hoeffding.
Writings
Masstabinvariante Korrelationstheorie, 1940
On the distribution of the rank correlation coefficient t when the variates are not independent in Biometrika, 1947
A class of statistics with asymptotically normal distribution, 1948
A nonparametric test for independence, 1948
The central limit theorem for dependent random variables (with Herbert Robbins), 1948
"Optimum" nonparametric tests, 1951
A combinatorial central limit theorem, 1951
The large-sample power of test based on permutations of observations, 1952
On the distribution of the expected values of the order statistics, 1953
The efficiency of tests (with J. R. Rosenblatt), 1955
On the distribution of the number of successes in independent trials, 1956
Distinguishability of sets of distributions. (The case of independent and identically distributed random variables.), (with Jacob Wolfowitz), 1958
Lower bounds for the expected sample size and the average risk of a sequential procedure, 1960
Probability inequalities for sums of bounded random variables, 1963
See also
Hoeffding's bounds
Hoeffding's C1 statistic
Hoeffding's decomposition
Hoeffding's independence test
Hoeffding's inequality
Hoeffding's lemma
Hoeffding–Blum–Kiefer–Rosenblatt process
Terry–Hoeffding test
References
External links
1914 births
1991 deaths
People from Vyborg District
People from Viipuri Province (Grand Duchy of Finland)
American people of Danish descent
Pr
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https://en.wikipedia.org/wiki/Dynkin%20system
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A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set satisfying a set of axioms weaker than those of -algebra. Dynkin systems are sometimes referred to as -systems (Dynkin himself used this term) or d-system. These set families have applications in measure theory and probability.
A major application of -systems is the - theorem, see below.
Definition
Let be a nonempty set, and let be a collection of subsets of (that is, is a subset of the power set of ). Then is a Dynkin system if
is closed under complements of subsets in supersets: if and then
is closed under countable increasing unions: if is an increasing sequence of sets in then
It is easy to check that any Dynkin system satisfies:
is closed under complements in : if then
Taking shows that
is closed under countable unions of pairwise disjoint sets: if is a sequence of pairwise disjoint sets in (meaning that for all ) then
To be clear, this property also holds for finite sequences of pairwise disjoint sets (by letting for all ).
Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynkin class.
For this reason, a small group of authors have adopted conditions 4-6 to define a Dynkin system as they are easier to verify.
An important fact is that any Dynkin system that is also a -system (that is, closed under finite intersections) is a -algebra. This can be verified by noting that conditions 2 and 3 together with closure under finite intersections imply closure under finite unions, which in turn implies closure under countable unions.
Given any collection of subsets of there exists a unique Dynkin system denoted which is minimal with respect to containing That is, if is any Dynkin system containing then is called the
For instance,
For another example, let and ; then
Sierpiński–Dynkin's π-λ theorem
Sierpiński-Dynkin's - theorem:
If is a -system and is a Dynkin system with then
In other words, the -algebra generated by is contained in Thus a Dynkin system contains a -system if and only if it contains the -algebra generated by that -system.
One application of Sierpiński-Dynkin's - theorem is the uniqueness of a measure that evaluates the length of an interval (known as the Lebesgue measure):
Let be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let be another measure on satisfying and let be the family of sets such that Let and observe that is closed under finite intersections, that and that is the -algebra generated by It may be shown that satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin's - Theorem it follows that in fact includes all of , which is equivalent to showing that the Lebesgue measure is unique on .
Application to probability distributions
See also
Notes
Proofs
References
Families of sets
Lemmas
Probability theory
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https://en.wikipedia.org/wiki/Provable%20prime
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In number theory, a provable prime is an integer that has been calculated to be prime using a primality-proving algorithm. Boot-strapping techniques using Pocklington primality test are the most common ways to generate provable primes for cryptography.
Contrast with probable prime, which is likely (but not certain) to be prime, based on the output of a probabilistic primality test.
In principle, every prime number can be proved to be prime in polynomial time by using the AKS primality test. Other methods which guarantee that their result is prime, but which do not work for all primes, are useful for the random generation of provable primes.
Provable primes have also been generated on embedded devices.
See also
Probable prime
Primality test
References
Primality tests
Prime numbers
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https://en.wikipedia.org/wiki/Finite%20character
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In mathematics, a family of sets is of finite character if for each , belongs to if and only if every finite subset of belongs to . That is,
For each , every finite subset of belongs to .
If every finite subset of a given set belongs to , then belongs to .
Properties
A family of sets of finite character enjoys the following properties:
For each , every (finite or infinite) subset of belongs to .
Every nonempty family of finite character has a maximal element with respect to inclusion (Tukey's lemma): In , partially ordered by inclusion, the union of every chain of elements of also belongs to , therefore, by Zorn's lemma, contains at least one maximal element.
Example
Let be a vector space, and let be the family of linearly independent subsets of . Then is a family of finite character (because a subset is linearly dependent if and only if has a finite subset which is linearly dependent).
Therefore, in every vector space, there exists a maximal family of linearly independent elements. As a maximal family is a vector basis, every vector space has a (possibly infinite) vector basis.
See also
Hereditarily finite set
References
Families of sets
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https://en.wikipedia.org/wiki/End
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End, END, Ending, or ENDS may refer to:
End
Mathematics
End (category theory)
End (topology)
End (graph theory)
End (group theory) (a subcase of the previous)
End (endomorphism)
Sports and games
End (gridiron football)
End, a division of play in the sports of curling, target archery and pétanque
End (dominoes), one of the halves of the face of a domino tile
Entertainment
End (band) an American hardcore punk supergroup formed in 2017
End key on a modern computer keyboard
End Records, a record label
"End", a song by The Cure from Wish
"Ends" (song) a 1998 song by Everlast, off the album Whitey Ford Sings the Blues
End (album), by Explosions in the Sky
Other uses
End, in weaving, a single thread of the warp
Ends (short story collection) (1988 book) anthology of Gordon R. Dickson stories
END
European Nuclear Disarmament
Endoglin, a glycoprotein
Equivalent narcotic depth, a concept used in underwater diving
Environmental noise directive
Ending
Ending (linguistics), a linguistic morpheme
Alternate ending
End of a part of a baseball game
Chess endgame
Ending credits
Post-credits scene
False ending
Happy ending
Multiple endings
Twist ending
Endings (film), a 2012 film
The Ending (Song), a 2012 song by Ellie Goulding off the album Halcyon
This Ending (band) Swedish extreme metal band
A repeat sign, in music theory
"Endings", a Series E episode of the television series QI (2007)
ENDS
ENDS, electronic nicotine delivery system
See also
The End (disambiguation)
Telos (philosophy), a goal or final state
Conclude (disambiguation)
End of the world (disambiguation)
Finale (disambiguation)
Front end (disambiguation)
Terminate (disambiguation)
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https://en.wikipedia.org/wiki/Erwin%20Kreyszig
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Erwin Otto Kreyszig (January 6, 1922 in Pirna, Germany – December 12, 2008) was a German Canadian applied mathematician and the Professor of Mathematics at Carleton University in Ottawa, Ontario, Canada. He was a pioneer in the field of applied mathematics: non-wave replicating linear systems. He was also a distinguished author, having written the textbook Advanced Engineering Mathematics, the leading textbook for civil, mechanical, electrical, and chemical engineering undergraduate engineering mathematics.
Kreyszig received his PhD degree in 1949 at the University of Darmstadt under the supervision of Alwin Walther. He then continued his research activities at the universities of Tübingen and Münster. Prior to joining Carleton University in 1984, he held positions at Stanford University (1954/55), the University of Ottawa (1955/56), Ohio State University (1956–60, professor 1957) and he completed his habilitation at the University of Mainz. In 1960 he became professor at the Technical University of Graz and organized the Graz 1964 Mathematical Congress. He worked at the University of Düsseldorf (1967–71) and at the University of Karlsruhe (1971–73). From 1973 through 1984 he worked at the University of Windsor and since 1984 he had been at Carleton University. He was awarded the title of Distinguished Research Professor in 1991 in recognition of a research career during which he published 176 papers in refereed journals, and 37 in refereed conference proceedings.
Kreyszig was also an administrator, developing a Computer Centre at the University of Graz, and at the Mathematics Institute at the University of Düsseldorf. In 1964, he took a leave of absence from Graz to initiate a doctoral program in mathematics at Texas A&M University.
Kreyszig authored 14 books, including Advanced Engineering Mathematics, which was published in its 10th edition in 2011. He supervised 104 master's and 22 doctoral students as well as 12 postdoctoral researchers. Together with his son he founded the Erwin and Herbert Kreyszig Scholarship which has funded graduate students since 2001.
Books
Statistische Methoden und ihre Anwendungen, Vandenhoeck & Ruprecht, Göttingen, 1965.
Introduction to Differential Geometry and Riemannian Geometry (English Translation), University of Toronto Press, 1968.
(with Kracht, Manfred): Methods of Complex Analysis in Partial Differential Equations with Applications, Wiley, 1988, .
Introductory Functional Analysis with Applications, Wiley, 1989, .
Differentialgeometrie. Leipzig 1957; engl. Differential Geometry, Dover, 1991, .
Advanced Engineering Mathematics, Wiley, (First edition 1962; ninth edition 2006, ; tenth edition (posthumous) 2011, ).
Literature
Manfred W. Kracht: In Honor of Professor Erwin Kreyszig on the Occasion of His Seventieth Birthday, Complex Variables 18, pp. 1–2, (1992)
Obituary by Martin Muldoon
External links
Canadian mathematicians
20th-century German mathematicians
21st-century German mathematicians
Academic
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https://en.wikipedia.org/wiki/Paul%20Bernays
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Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a Swiss mathematician who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator of David Hilbert.
Biography
Bernays was born into a distinguished German-Jewish family of scholars and businessmen. His great-grandfather, Isaac ben Jacob Bernays, served as chief rabbi of Hamburg from 1821 to 1849.
Bernays spent his childhood in Berlin, and attended the Köllner Gymnasium, 1895–1907. At the University of Berlin, he studied mathematics under Issai Schur, Edmund Landau, Ferdinand Georg Frobenius, and Friedrich Schottky; philosophy under Alois Riehl, Carl Stumpf and Ernst Cassirer; and physics under Max Planck. At the University of Göttingen, he studied mathematics under David Hilbert, Edmund Landau, Hermann Weyl, and Felix Klein; physics under Voigt and Max Born; and philosophy under Leonard Nelson.
In 1912, the University of Berlin awarded him a Ph.D. in mathematics for a thesis, supervised by Landau, on the analytic number theory of binary quadratic forms. That same year, the University of Zurich awarded him habilitation for a thesis on complex analysis and Picard's theorem. The examiner was Ernst Zermelo. Bernays was Privatdozent at the University of Zurich, 1912–17, where he came to know George Pólya. His collected communications with Kurt Gödel span many decades.
Starting in 1917, David Hilbert employed Bernays to assist him with his investigations of the foundation of arithmetic. Bernays also lectured on other areas of mathematics at the University of Göttingen. In 1918, that university awarded him a second habilitation for a thesis on the axiomatics of the propositional calculus of Principia Mathematica.
In 1922, Göttingen appointed Bernays extraordinary professor without tenure. His most successful student there was Gerhard Gentzen. After Nazi Germany enacted the Law for the Restoration of the Professional Civil Service in 1933, the university fired Bernays because of his Jewish ancestry.
After working privately for Hilbert for six months, Bernays and his family moved to Switzerland, whose nationality he had inherited from his father, and where the ETH Zurich employed him on occasion. He also visited the University of Pennsylvania and was a visiting scholar at the Institute for Advanced Study in 1935–36 and again in 1959–60.
Mathematical work
Bernays's collaboration with Hilbert culminated in the two volume work, Grundlagen der Mathematik (English: Foundations of Mathematics) published in 1934 and 1939, which is discussed in Sieg and Ravaglia (2005). A proof in this work that a sufficiently strong consistent theory cannot contain its own reference functor is known as the Hilbert–Bernays paradox.
In seven papers, published between 1937 and 1954 in the Journal of Symbolic Logic (republished in Müller 1976), Bernays set out an axiomatic set theory whose starting point was a related
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https://en.wikipedia.org/wiki/Schur%27s%20Inequality
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In mathematics, Schur's inequality, named after Issai Schur,
establishes that for all non-negative real numbers
x, y, z, and t>0,
with equality if and only if x = y = z or two of them are equal and the other is zero. When t is an even positive integer, the inequality holds for all real numbers x, y and z.
When , the following well-known special case can be derived:
Proof
Since the inequality is symmetric in we may assume without loss of generality that . Then the inequality
clearly holds, since every term on the left-hand side of the inequality is non-negative. This rearranges to Schur's inequality.
Extensions
A generalization of Schur's inequality is the following:
Suppose a,b,c are positive real numbers. If the triples (a,b,c) and (x,y,z) are similarly sorted, then the following inequality holds:
In 2007, Romanian mathematician Valentin Vornicu showed that a yet further generalized form of Schur's inequality holds:
Consider , where , and either or . Let , and let be either convex or monotonic. Then,
The standard form of Schur's is the case of this inequality where x = a, y = b, z = c, k = 1, ƒ(m) = mr.
Another possible extension states that if the non-negative real numbers with and the positive real number t are such that x + v ≥ y + z then
Notes
Inequalities
Articles containing proofs
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https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison%20formula
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In mathematics, in particular linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of the sum of an invertible matrix and the outer product, , of vectors and . The Sherman–Morrison formula is a special case of the Woodbury formula. Though named after Sherman and Morrison, it appeared already in earlier publications.
Statement
Suppose is an invertible square matrix and are column vectors. Then is invertible iff . In this case,
Here, is the outer product of two vectors and . The general form shown here is the one published by Bartlett.
Proof
() To prove that the backward direction is invertible with inverse given as above) is true, we verify the properties of the inverse. A matrix (in this case the right-hand side of the Sherman–Morrison formula) is the inverse of a matrix (in this case ) if and only if .
We first verify that the right hand side () satisfies .
To end the proof of this direction, we need to show that in a similar way as above:
(In fact, the last step can be avoided since for square matrices and , is equivalent to .)
() Reciprocally, if , then via the matrix determinant lemma, , so is not invertible.
Application
If the inverse of is already known, the formula provides a numerically cheap way to compute the inverse of corrected by the matrix (depending on the point of view, the correction may be seen as a perturbation or as a rank-1 update). The computation is relatively cheap because the inverse of does not have to be computed from scratch (which in general is expensive), but can be computed by correcting (or perturbing) .
Using unit columns (columns from the identity matrix) for or , individual columns or rows of may be manipulated and a correspondingly updated inverse computed relatively cheaply in this way. In the general case, where is a -by- matrix and and are arbitrary vectors of dimension , the whole matrix is updated and the computation takes scalar multiplications. If is a unit column, the computation takes only scalar multiplications. The same goes if is a unit column. If both and are unit columns, the computation takes only scalar multiplications.
This formula also has application in theoretical physics. Namely, in quantum field theory, one uses this formula to calculate the propagator of a spin-1 field. The inverse propagator (as it appears in the Lagrangian) has the form . One uses the Sherman–Morrison formula to calculate the inverse (satisfying certain time-ordering boundary conditions) of the inverse propagator—or simply the (Feynman) propagator—which is needed to perform any perturbative calculation involving the spin-1 field.
One of the issues with the formula is that little is known about its numerical stability. There are no published results concerning its error bounds. Anecdotal evidence suggests that the Woodbury matrix identity (a general case of the Sherman–Morrison formula) may diverge even for
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https://en.wikipedia.org/wiki/Lagrange%27s%20identity
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In algebra, Lagrange's identity, named after Joseph Louis Lagrange, is:
which applies to any two sets {a1, a2, ..., an} and {b1, b2, ..., bn} of real or complex numbers (or more generally, elements of a commutative ring). This identity is a generalisation of the Brahmagupta–Fibonacci identity and a special form of the Binet–Cauchy identity.
In a more compact vector notation, Lagrange's identity is expressed as:
where a and b are n-dimensional vectors with components that are real numbers. The extension to complex numbers requires the interpretation of the dot product as an inner product or Hermitian dot product. Explicitly, for complex numbers, Lagrange's identity can be written in the form:
involving the absolute value.
Since the right-hand side of the identity is clearly non-negative, it implies Cauchy's inequality in the finite-dimensional real coordinate space Rn and its complex counterpart Cn.
Geometrically, the identity asserts that the square of the volume of the parallelepiped spanned by a set of vectors is the Gram determinant of the vectors.
Lagrange's identity and exterior algebra
In terms of the wedge product, Lagrange's identity can be written
Hence, it can be seen as a formula which gives the length of the wedge product of two vectors, which is the area of the parallelogram they define, in terms of the dot products of the two vectors, as
Lagrange's identity and vector calculus
In three dimensions, Lagrange's identity asserts that if a and b are vectors in R3 with lengths |a| and |b|, then Lagrange's identity can be written in terms of the cross product and dot product:
Using the definition of angle based upon the dot product (see also Cauchy–Schwarz inequality), the left-hand side is
where is the angle formed by the vectors a and b. The area of a parallelogram with sides and and angle is known in elementary geometry to be
so the left-hand side of Lagrange's identity is the squared area of the parallelogram. The cross product appearing on the right-hand side is defined by
which is a vector whose components are equal in magnitude to the areas of the projections of the parallelogram onto the yz, zx, and xy planes, respectively.
Seven dimensions
For a and b as vectors in R7, Lagrange's identity takes on the same form as in the case of R3
However, the cross product in 7 dimensions does not share all the properties of the cross product in 3 dimensions. For example, the direction of a × b in 7-dimensions may be the same as c × d even though c and d are linearly independent of a and b. Also the seven-dimensional cross product is not compatible with the Jacobi identity.
Quaternions
A quaternion p is defined as the sum of a scalar t and a vector v:
The product of two quaternions and is defined by
The quaternionic conjugate of q is defined by
and the norm squared is
The multiplicativity of the norm in the quaternion algebra provides, for quaternions p and q:
The quaternions p and q are called imaginary if their
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https://en.wikipedia.org/wiki/Curved%20space
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Curved space often refers to a spatial geometry which is not "flat", where a flat space has zero curvature, as described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Curved spaces play an essential role in general relativity, where gravity is often visualized as curved space. The Friedmann–Lemaître–Robertson–Walker metric is a curved metric which forms the current foundation for the description of the expansion of space and shape of the universe.
Simple two-dimensional example
A very familiar example of a curved space is the surface of a sphere. While to our familiar outlook the sphere looks three-dimensional, if an object is constrained to lie on the surface, it only has two dimensions that it can move in. The surface of a sphere can be completely described by two dimensions since no matter how rough the surface may appear to be, it is still only a surface, which is the two-dimensional outside border of a volume. Even the surface of the Earth, which is fractal in complexity, is still only a two-dimensional boundary along the outside of a volume.
Embedding
One of the defining characteristics of a curved space is its departure from the Pythagorean theorem. In a curved space
.
The Pythagorean relationship can often be restored by describing the space with an extra dimension.
Suppose we have a non-euclidean three-dimensional space with coordinates . Because it is not flat
.
But if we now describe the three-dimensional space with four dimensions () we can choose coordinates such that
.
Note that the coordinate is not the same as the coordinate .
For the choice of the 4D coordinates to be valid descriptors of the original 3D space it must have the same number of degrees of freedom. Since four coordinates have four degrees of freedom it must have a constraint placed on it. We can choose a constraint such that Pythagorean theorem holds in the new 4D space. That is
.
The constant can be positive or negative. For convenience we can choose the constant to be
where now is positive and .
We can now use this constraint to eliminate the artificial fourth coordinate . The differential of the constraining equation is
leading to .
Plugging into the original equation gives
.
This form is usually not particularly appealing and so a coordinate transform is often applied: , , . With this coordinate transformation
.
Without embedding
The geometry of a n-dimensional space can also be described with Riemannian geometry. An isotropic and homogeneous space can be described by the metric:
.
This reduces to Euclidean space when . But a space can be said to be "flat" when the Weyl tensor has all zero components. In three dimensions this condition is met when the Ricci tensor () is equal to the metric times the Ricci scalar (, not to be confused with the R of the previous section). That is . Calculation of these components from the metric gives that
wher
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https://en.wikipedia.org/wiki/Leibniz%20integral%20rule
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In calculus, the Leibniz integral rule for differentiation under the integral sign states that for an integral of the form
where and the integrands are functions dependent on the derivative of this integral is expressible as
where the partial derivative indicates that inside the integral, only the variation of with is considered in taking the derivative. It is named after Gottfried Leibniz.
In the special case where the functions and are constants and with values that do not depend on this simplifies to:
If is constant and , which is another common situation (for example, in the proof of Cauchy's repeated integration formula), the Leibniz integral rule becomes:
This important result may, under certain conditions, be used to interchange the integral and partial differential operators, and is particularly useful in the differentiation of integral transforms. An example of such is the moment generating function in probability theory, a variation of the Laplace transform, which can be differentiated to generate the moments of a random variable. Whether Leibniz's integral rule applies is essentially a question about the interchange of limits.
General form: differentiation under the integral sign
The right hand side may also be written using Lagrange's notation as:
Stronger versions of the theorem only require that the partial derivative exist almost everywhere, and not that it be continuous. This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just the particular case of the above formula where is constant, and does not depend on
If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation:
where is the partial derivative with respect to and is the integral operator with respect to over a fixed interval. That is, it is related to the symmetry of second derivatives, but involving integrals as well as derivatives. This case is also known as the Leibniz integral rule.
The following three basic theorems on the interchange of limits are essentially equivalent:
the interchange of a derivative and an integral (differentiation under the integral sign; i.e., Leibniz integral rule);
the change of order of partial derivatives;
the change of order of integration (integration under the integral sign; i.e., Fubini's theorem).
Three-dimensional, time-dependent case
A Leibniz integral rule for a two dimensional surface moving in three dimensional space is
where:
is a vector field at the spatial position at time ,
is a surface bounded by the closed curve ,
is a vector element of the surface ,
is a vector element of the curve ,
is the velocity of movement of the region ,
is the vector divergence,
is the vector cross product,
The double integrals are surface integrals over the surface , and the line integral is over the bounding curve .
Higher dimensions
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https://en.wikipedia.org/wiki/Grundz%C3%BCge%20der%20Mengenlehre
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(German for "Basics of Set Theory") is a book on set theory written by Felix Hausdorff.
First published in April 1914, was the first comprehensive introduction to set theory. Besides the systematic treatment of known results in set theory, the book also contains chapters on measure theory and topology, which were then still considered parts of set theory. Hausdorff presented and developed original material which was later to become the basis for those areas. In 1927 Hausdorff published an extensively revised second edition under the title Mengenlehre (German for "Set Theory"), with many of the topics of the first edition omitted. In 1935 there was a third German edition, which in 1957 was translated by John R. Aumann et al. into English under the title Set Theory.
Chelsea Publishing Company reprinted the German 1914 edition in New York City in German in 1944, 1949, 1965, 1978 and 1991 but never issued an English translation of this first edition (or the 1927 second edition) to date. When the American Mathematical Society took over and set up AMS Chelsea Publishing it published editions in 2005 and 2021.
References
.
Reprinted by Chelsea Publishing Company in 1944, 1949 and 1965 .
Republished by Dover Publications, New York, N. Y., 1944
Republished by AMS-Chelsea 2005.
. Extended edition of a chapter in The Princeton Companion to Mathematics.
1914 non-fiction books
Mathematics books
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https://en.wikipedia.org/wiki/Maria%20Simon%20%28actress%29
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Maria Simon (born 6 February 1976) is a German actress.
Family and background
Simon's German father originally hailed from Leipzig and studied mathematics in Leningrad. There he met Simon's Russian-Jewish mother, Olga, who studied electronics and originally hailed from Kazakhstan. The couple married while studying.
Maria Simon is the younger sister of actress Susanna Simon, who was born on 23 July 1968, in Almaty, Kazakhstan. Maria was born and brought up in the former East Germany, but moved to New York City in 1990 to live with her father, a computer expert with the United Nations, and her sister Dalena Simon. She also has a sister named Alyssa.
Simon has four children, the first from a former relationship with the actor Devid Striesow, and three with her ex-husband, the actor Bernd Michael Lade.
Education
After finishing school she moved back to the newly reunited Germany to study acting at the Academy of Performing Arts Ernst Busch in Berlin where she received her diploma in 1999.
Roles and awards
She won the award for Best Actress for her role in the film Zornige Küsse at the 22nd Moscow International Film Festival in 2000. Simon was nominated as the best supporting actress in the 2003 German Film Awards, and was named European Shooting Star (i.e., best newcomer) at the 2004 Berlinale. In the same year she played Polly in Bertolt Brechts Dreigroschenoper at the Maxim Gorki Theater in Berlin. Her TV movie Kleine Schwester was nominated for the Adolf Grimme Awards in 2005.
Filmography
Angry Kisses (1999), as Lea
(2001), as Johanna
Erste Ehe (2002), as Dorit (USA title: Portrait of a Married Couple)
My Daughter's Tears (2002), as Stefanie (USA title: Against All Evidence, German title: Meine Tochter ist keine Mörderin)
Good Bye, Lenin! (2003), as Ariane Kerner
Distant Lights (2003), as Sonja
Luther (2003), as Hanna
TV Work
Jenny Berlin: Tod am Meer (2000, TV series episode), as Tanja Schulz
Mord im Swingerclub (2000, TV film), as Susanna Bach
HeliCops – Einsatz über Berlin: Fehlgeleitet (2001, TV series episode), as Biene Virchow
Balko: Der Schweinemann (2001, TV series episode), as Marischka
Verbotene Küsse (2001, TV film), as Andrea
Jonathans Liebe (2001, TV film), as Nina Buchwald
Tatort: Verrat (2002, TV series episode), as Lisa Mattern
Alarm für Cobra 11 - Die Autobahnpolizei: Die Clique (2002, TV series episode), as Laura Friedrich
Tatort: Reise ins Nichts (2002, TV series episode), as Sabine Hallmeier
Spurlos – Ein Baby verschwindet (2003, TV film), as Andrea Bär
Fast perfekt verlobt (2003, TV film), as Nika Kreschninski
K3 – Kripo Hamburg: Auf dünnem Eis (2003, TV series episode), as Kathrin Leutgeb
Spur & Partner (2003, TV series), as Frau Stolz/Hausmädchen/Eva Hermann
Carola Stern's Double Life (2004, TV film), as Carola Stern
Kleine Schwester (2004, TV film), as Katrin Rubakow
Tatort: Feuertaufe (2005, TV series episode), as Sabine Gerber
Tatort: Minenspiel (2005, TV series episode), as Hannah Siems
Die Pathologin (2006, T
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https://en.wikipedia.org/wiki/Orthogonal%20trajectory
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In mathematics, an orthogonal trajectory is a curve which intersects any curve of a given pencil of (planar) curves orthogonally.
For example, the orthogonal trajectories of a pencil of concentric circles are the lines through their common center (see diagram).
Suitable methods for the determination of orthogonal trajectories are provided by solving differential equations. The standard method establishes a first order ordinary differential equation and solves it by separation of variables. Both steps may be difficult or even impossible. In such cases one has to apply numerical methods.
Orthogonal trajectories are used in mathematics, for example as curved coordinate systems (i.e. elliptic coordinates) and appear in physics as electric fields and their equipotential curves.
If the trajectory intersects the given curves by an arbitrary (but fixed) angle, one gets an isogonal trajectory.
Determination of the orthogonal trajectory
In cartesian coordinates
Generally, one assumes that the pencil of curves is given implicitly by an equation
(0) 1. example 2. example
where is the parameter of the pencil. If the pencil is given explicitly by an equation , one can change the representation into an implicit one: . For the considerations below, it is supposed that all necessary derivatives do exist.
Step 1.
Differentiating implicitly for yields
(1) in 1. example 2. example
Step 2.
Now it is assumed that equation (0) can be solved for parameter , which can thus be eliminated from equation (1). One gets the differential equation of first order
(2) in 1. example 2. example
which is fulfilled by the given pencil of curves.
Step 3.
Because the slope of the orthogonal trajectory at a point is the negative multiplicative inverse of the slope of the given curve at this point, the orthogonal trajectory satisfies the differential equation of first order
(3) in 1. example 2. example
Step 4.
This differential equation can (hopefully) be solved by a suitable method.
For both examples separation of variables is suitable. The solutions are:
in example 1, the lines and
in example 2, the ellipses
In polar coordinates
If the pencil of curves is represented implicitly in polar coordinates by
(0p)
one determines, alike the cartesian case, the parameter free differential equation
(1p)
(2p)
of the pencil. The differential equation of the orthogonal trajectories is then (see Redheffer & Port p. 65, Heuser, p. 120)
(3p)
Example: Cardioids:
(0p) (in diagram: blue)
(1p)
Elimination of yields the differential equation of the given pencil:
(2p)
Hence the differential equation of the orthogonal trajectories is:
(3p)
After solving this differential equation by separation of variables one gets
which describes the pencil of cardioids (red in diagram), symmetric to the given pencil.
Isogonal trajectory
A curve, which intersects any curve of a given pencil of (planar) curves by a fixed angle is called isogonal trajectory.
Between the slope o
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https://en.wikipedia.org/wiki/Mirimanoff%27s%20congruence
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In number theory, a branch of mathematics, a Mirimanoff's congruence is one of a collection of expressions in modular arithmetic which, if they hold, entail the truth of Fermat's Last Theorem. Since the theorem has now been proven, these are now of mainly historical significance, though the Mirimanoff polynomials are interesting in their own right. The theorem is due to Dmitry Mirimanoff.
Definition
The nth Mirimanoff polynomial for the prime p is
In terms of these polynomials, if t is one of the six values {-X/Y, -Y/X, -X/Z, -Z/X, -Y/Z, -Z/Y} where Xp+Yp+Zp=0 is a solution to Fermat's Last Theorem, then
φp-1(t) ≡ 0 (mod p)
φp-2(t)φ2(t) ≡ 0 (mod p)
φp-3(t)φ3(t) ≡ 0 (mod p)
...
φ(p+1)/2(t)φ(p-1)/2(t) ≡ 0 (mod p)
Other congruences
Mirimanoff also proved the following:
If an odd prime p does not divide one of the numerators of the Bernoulli numbers Bp-3, Bp-5, Bp-7 or Bp-9, then the first case of Fermat's Last Theorem, where p does not divide X, Y or Z in the equation Xp+Yp+Zp=0, holds.
If the first case of Fermat's Last Theorem fails for the prime p, then 3p-1 ≡ 1 (mod p2). A prime number with this property is sometimes called a Mirimanoff prime, in analogy to a Wieferich prime which is a prime such that 2p-1 ≡ 1 (mod p2). The existence of primes satisfying such congruences was recognized long before their implications for the first case of Fermat's Last Theorem became apparent; but while the discovery of the first Wieferich prime came after these theoretical developments and was prompted by them, the first instance of a Mirimanoff prime is so small that it was already known before Mirimanoff formulated the connection to FLT in 1910, which fact may explain the reluctance of some writers to use the name. So early as his 1895 paper (p. 298), Mirimanoff alludes to a rather complicated test for the primes now known by his name, deriving from a formula published by Sylvester in 1861, which is of little computational value but great theoretical interest. This test was considerably simplified by Lerch (1905), p. 476, who showed that in general, for p > 3,
so that a prime possesses the Mirimanoff property if it divides the expression within the curly braces. The condition was further refined in an important paper by Emma Lehmer (1938), in which she considered the intriguing and still unanswered question of whether it is possible for a number to satisfy the congruences of Wieferich and Mirimanoff simultaneously. To date, the only known Mirimanoff primes are 11 and 1006003 . The discovery of the second of these appears to be due to K.E. Kloss (1965).
References
K.E. Kloss, "Some Number-Theoretic Calculations," Journal of Research of the National Bureau of Standards—B. Mathematics and Mathematical Physics 69 (1965), pp. 335–336.
Emma Lehmer, "On Congruences involving Bernoulli Numbers and the Quotients of Fermat and Wilson," Annals of Mathematics 39 (1938), pp. 350–360.
M. Lerch, "Zur Theorie des Fermatschen Quotienten…," Mathematische Annalen 60
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https://en.wikipedia.org/wiki/Volodymyr%20Korolyuk
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Volodymyr Semenovych Korolyuk (, 19 August 1925 – 4 April 2020) was a Soviet and Ukrainian mathematician who made significant contributions to probability theory and its applications, academician of the National Academy of Sciences of Ukraine (1976).
Korolyuk was born in Kyiv in August 1925. Between 1949 and 2005 Volodymyr Korolyuk published over 300 papers and 22 monographs. He died in Kyiv in April 2020 at the age of 94.
Awards and honors
Volodymyr Korolyuk has been awarded a number of scientific prizes.
Krylov Prize of the National Academy of Sciences of Ukraine, 1976
State Prize of the Ukrainian Soviet Socialist Republic, 1978
Glushkov Prize of the National Academy of Sciences of Ukraine, 1988
Bogolyubov Prize of the National Academy of Sciences of Ukraine, 1995
Ostrogradsky Medal, 2002
State Prize of Ukraine, 2003
References
Biography at the website of the Kyiv Mathematical Society (in Ukrainian)
Yu. A. Mitropolskiy, A. V. Skorokhod, D. V. Gusak, Vladimir Semenovich Korolyuk (in honor of 60th anniversary), Ukrainian Math. Journal, 37, No 4, 1985, pp 488–489 (in Russian)
1925 births
2020 deaths
20th-century Ukrainian mathematicians
Scientists from Kyiv
Laureates of the State Prize of Ukraine in Science and Technology
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https://en.wikipedia.org/wiki/Narrow%20class%20group
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In algebraic number theory, the narrow class group of a number field K is a refinement of the class group of K that takes into account some information about embeddings of K into the field of real numbers.
Formal definition
Suppose that K is a finite extension of Q. Recall that the ordinary class group of K is defined as the quotient
where IK is the group of fractional ideals of K, and PK is the subgroup of principal fractional ideals of K, that is, ideals of the form aOK where a is an element of K.
The narrow class group is defined to be the quotient
where now PK+ is the group of totally positive principal fractional ideals of K; that is, ideals of the form aOK where a is an element of K such that σ(a) is positive for every embedding
Uses
The narrow class group features prominently in the theory of representing integers by quadratic forms. An example is the following result (Fröhlich and Taylor, Chapter V, Theorem 1.25).
Theorem. Suppose that where d is a square-free integer, and that the narrow class group of K is trivial. Suppose that
is a basis for the ring of integers of K. Define a quadratic form
,
where NK/Q is the norm. Then a prime number p is of the form
for some integers x and y if and only if either
or
or
where dK is the discriminant of K, and
denotes the Legendre symbol.
Examples
For example, one can prove that the quadratic fields Q(), Q(), Q() all have trivial narrow class group. Then, by choosing appropriate bases for the integers of each of these fields, the above theorem implies the following:
A prime p is of the form p = x2 + y 2 for integers x and y if and only if
(This is known as Fermat's theorem on sums of two squares.)
A prime p is of the form p = x2 − 2y 2 for integers x and y if and only if
A prime p is of the form p = x2 − xy + y 2 for integers x and y if and only if
(cf. Eisenstein prime)
An example that illustrates the difference between the narrow class group and the usual class group is the case of Q(). This has trivial class group, but its narrow class group has order 2. Because the class group is trivial, the following statement is true:
A prime p or its inverse −p is of the form ± p = x2 − 6y 2 for integers x and y if and only if
However, this statement is false if we focus only on p and not −p (and is in fact even false for p = 2), because the narrow class group is nontrivial. The statement that classifies the positive p is the following:
A prime p is of the form p = x2 − 6y 2 for integers x and y if and only if p = 3 or
(Whereas the first statement allows primes , the second only allows primes .)
See also
Class group
Quadratic form
References
A. Fröhlich and M. J. Taylor, Algebraic Number Theory (p. 180), Cambridge University Press, 1991.
Algebraic number theory
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https://en.wikipedia.org/wiki/Polymatroid
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In mathematics, a polymatroid is a polytope associated with a submodular function. The notion was introduced by Jack Edmonds in 1970. It is also described as the multiset analogue of the matroid.
Definition
Let be a finite set and a non-decreasing submodular function, that is, for each we have , and for each we have . We define the polymatroid associated to to be the following polytope:
.
When we allow the entries of to be negative we denote this polytope by , and call it the extended polymatroid associated to .
An equivalent definition
Let be a finite set. If then we denote by the sum of the entries of , and write whenever for every (notice that this gives an order to ). A polymatroid on the ground set is a nonempty compact subset in , the set of independent vectors, such that:
We have that if , then for every :
If with , then there is a vector such that .
This definition is equivalent to the one described before, where is the function defined by for every .
Relation to matroids
To every matroid on the ground set we can associate the set , where is the set of independent sets of and we denote by the characteristic vector of : for every
By taking the convex hull of we get a polymatroid. It is associated to the rank function of . The conditions of the second definition reflect the axioms for the independent sets of a matroid.
Relation to generalized permutahedra
Because generalized permutahedra can be constructed from submodular functions, and every generalized permutahedron has an associated submodular function, we have that there should be a correspondence between generalized permutahedra and polymatroids. In fact every polymatroid is a generalized permutahedron that has been translated to have a vertex in the origin. This result suggests that the combinatorial information of polymatroids is shared with generalized permutahedra.
Properties
is nonempty if and only if and that is nonempty if and only if .
Given any extended polymatroid there is a unique submodular function such that and .
Contrapolymatroids
For a supermodular f one analogously may define the contrapolymatroid
This analogously generalizes the dominant of the spanning set polytope of matroids.
Discrete polymatroids
When we only focus on the lattice points of our polymatroids we get what is called, discrete polymatroids. Formally speaking, the definition of a discrete polymatroid goes exactly as the one for polymatroids except for where the vectors will live in, instead of they will live in . This combinatorial object is of great interest because of their relationship to monomial ideals.
References
Footnotes
Additional reading
Matroid theory
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https://en.wikipedia.org/wiki/Harmonic%20number%20%28disambiguation%29
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In number theory, the harmonic numbers are the sums of the inverses of integers, forming the harmonic series. Harmonic number may also refer to:
Harmonic, a periodic wave with a frequency that is an integral multiple of the frequency of another wave
Harmonic divisor numbers, also called Ore numbers or Ore's harmonic numbers, positive integers whose divisors have an integral harmonic mean
3-smooth numbers, numbers whose only prime factors are 2 and 3
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https://en.wikipedia.org/wiki/Ponta%20Grossa
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Ponta Grossa () is a municipality in the state of Paraná, southern Brazil. The estimated population is 355,336 according to official data from the Brazilian Institute of Geography and Statistics and it is the 4th most populous city in Paraná (76th in Brazil). It is also the largest city close to Greater Curitiba region, so within a radius of 186 miles (300 km) of Ponta Grossa.
It is also known as Princesa dos Campos (in English: Princess of the Fields) and Capital Cívica do Paraná (in English: Civic Capital of Paraná). The city is connected to the Caminho das Tropas (in English: Path of the Troops), being one of the network of routes used by drovers (tropeiros) in the middle of a high hill inside a grassy vegetation. The city is considered of average size, located around a central hill, while most of its growth occurred in the second half of the twentieth century with the weakening of the primary economy.
Ponta Grossa is one of the largest tourist destinations in the Paraná, especially because of the area of natural beauty, Vila Velha State Park which is located within the limits of the municipality. The cup of Vila Velha refers to its location in the collective imagination. The München Fest, a party dedicated to German culture and also known as the Festa Nacional do Chopp Escuro (in English: Dark Chopp National Party), is the biggest event in Paraná and usually lasts a week between November and December.
In this city, the industrial sector is fundamental (supported by agriculture). The city hosts the largest concentration of industry in the interior of Paraná. Agroindustry, lumber and metalworking are the major industries. The result is reflected in national GDP with the contribution from this city within the interior of Brazil, being only below Foz do Iguaçu. Municipal GDP increased over the state and national average between 2013 and 2019, this was also seen in the number of registered companies and employees.
Etymology
The place where it is located has a toponymia related to a hill seen long distance during trips to the Campos Gerais. The name would have originated from a high hill that stood out before the whole pastures landscape by its prominent height and the capo of bush that covers it. The tropeiros to refer to its location said that they were near Ponta Grossa. But other stories have the same idea, like that of the foreman when he tells the farmer the place chosen to establish his farm, "there at Ponta Grossa". Or even that the name had seen later when the owner ceded the lands for settlement.
Ponta Grossa was founded on the farm of Miguel da Rocha Ferreira Carvalhaes who chose it as favorable agricultural land. The farm still exists towards Castro. In 1871 the city came to be called Pitangui, but the following year it resumed its original name. Sometimes it is the target of malicious humor due to what its name can send like in Portuguese. One way or another describes the characteristics of the vegetation and the regional topog
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https://en.wikipedia.org/wiki/Picard%20horn
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A Picard horn, also called the Picard topology or Picard model, is one of the oldest known hyperbolic 3-manifolds, first described by Émile Picard in 1884. The manifold is the quotient of the upper half-plane model of hyperbolic 3-space by the projective special linear group, . It was proposed as a model for the
shape of the universe in 2004. The term "horn" is due to pseudosphere models of hyperbolic space.
Geometry and topology
A modern description, in terms of fundamental domain and identifications, can be found in section 3.2, page 63 of Grunewald and Huntebrinker, along with the first 80 eigenvalues of the Laplacian, tabulated on page 72, where is a fundamental domain of the Picard space.
Cosmology
The term was coined in 2004 by Ralf Aurich, Sven Lustig, Frank Steiner, and Holger Then in their paper Hyperbolic Universes with a Horned Topology and the CMB Anisotropy.
The model was chosen in an attempt to describe the microwave background radiation apparent in the universe, and has finite volume and useful spectral characteristics (the first several eigenvalues of the Laplacian are computed and in good accord with observation). In this model one end of the figure curves finitely into the bell of the horn. The curve along any side of horn is considered to be a negative curve. The other end extends to infinity.
See also
Gabriel's Horn
References
3-manifolds
Hyperbolic geometry
Physical cosmology
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https://en.wikipedia.org/wiki/Inductive%20set
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Bourbaki also defines an inductive set to be a partially ordered set that satisfies the hypothesis of Zorn's lemma when nonempty.
In descriptive set theory, an inductive set of real numbers (or more generally, an inductive subset of a Polish space) is one that can be defined as the least fixed point of a monotone operation definable by a positive Σ1n formula, for some natural number n, together with a real parameter.
The inductive sets form a boldface pointclass; that is, they are closed under continuous preimages. In the Wadge hierarchy, they lie above the projective sets and below the sets in L(R). Assuming sufficient determinacy, the class of inductive sets has the scale property and thus the prewellordering property.
The term having a number of different meanings.
According to:
Russell's definition, an inductive set is a nonempty partially ordered set in which every element has a successor. An example is the set of natural numbers N, where 0 is the first element, and the others are produced by adding 1 successively.
Roitman considers the same construction in a more abstract form: the elements are sets, 0 is replaced by the empty set, and the successor of every element y is the set y union {y}. In particular, every inductive set contains a sequence of the form.
For many other authors (e.g., Bourbaki), an inductive set is a partially ordered set in which every totally ordered subset has an upper bound, i.e., it is a set fulfilling the assumption of Zorn's lemma.
References
Descriptive set theory
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https://en.wikipedia.org/wiki/Chaos%20game
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In mathematics, the term chaos game originally referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it. The fractal is created by iteratively creating a sequence of points, starting with the initial random point, in which each point in the sequence is a given fraction of the distance between the previous point and one of the vertices of the polygon; the vertex is chosen at random in each iteration. Repeating this iterative process a large number of times, selecting the vertex at random on each iteration, and throwing out the first few points in the sequence, will often (but not always) produce a fractal shape. Using a regular triangle and the factor 1/2 will result in the Sierpinski triangle, while creating the proper arrangement with four points and a factor 1/2 will create a display of a "Sierpinski Tetrahedron", the three-dimensional analogue of the Sierpinski triangle. As the number of points is increased to a number N, the arrangement forms a corresponding (N-1)-dimensional Sierpinski Simplex.
The term has been generalized to refer to a method of generating the attractor, or the fixed point, of any iterated function system (IFS). Starting with any point x0, successive iterations are formed as xk+1 = fr(xk), where fr is a member of the given IFS randomly selected for each iteration. The iterations converge to the fixed point of the IFS. Whenever x0 belongs to the attractor of the IFS, all iterations xk stay inside the attractor and, with probability 1, form a dense set in the latter.
The "chaos game" method plots points in random order all over the attractor. This is in contrast to other methods of drawing fractals, which test each pixel on the screen to see whether it belongs to the fractal. The general shape of a fractal can be plotted quickly with the "chaos game" method, but it may be difficult to plot some areas of the fractal in detail.
With the aid of the "chaos game" a new fractal can be made and while making the new fractal some parameters can be obtained. These parameters are useful for applications of fractal theory such as classification and identification. The new fractal is self-similar to the original in some important features such as fractal dimension.
Restricted chaos game
If the chaos game is run with a square, no fractal appears and the interior of the square fills evenly with points. However, if restrictions are placed on the choice of vertices, fractals will appear in the square. For example, if the current vertex cannot be chosen in the next iteration, this fractal appears:
If the current vertex cannot be one place away (anti-clockwise) from the previously chosen vertex, this fractal appears:
If the point is prevented from landing on a particular region of the square, the shape of that region will be reproduced as a fractal in other and apparently unrestricted parts of the square.
Jumps other than 1/2
When the length of the jump towards a vertex or another p
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https://en.wikipedia.org/wiki/Integrable%20system
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In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals that its motion is confined to a submanifold
of much smaller dimensionality than that of its phase space.
Three features are often referred to as characterizing integrable systems:
the existence of a maximal set of conserved quantities (the usual defining property of complete integrability)
the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability)
the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability)
Integrable systems may be seen as very different in qualitative character from more generic dynamical systems,
which are more typically chaotic systems. The latter generally have no conserved quantities, and are asymptotically intractable, since an arbitrarily small perturbation in initial conditions may lead to arbitrarily large deviations in their trajectories over a sufficiently large time.
Many systems studied in physics are completely integrable, in particular, in the Hamiltonian sense, the key example being multi-dimensional harmonic oscillators. Another standard example is planetary motion about either one fixed center (e.g., the sun) or two. Other elementary examples include the motion of a rigid body about its center of mass (the Euler top) and the motion of an axially symmetric rigid body about a point in its axis of symmetry (the Lagrange top).
In the late 1960's, it was realized that there are completely integrable systems in physics having an infinite number of degrees of freedom, such as some models of shallow water waves (Korteweg–de Vries equation), the Kerr effect in optical fibres, described by the nonlinear Schrödinger equation, and certain integrable many-body systems, such as the Toda lattice. The modern theory of integrable systems was revived with the numerical discovery of solitons by Martin Kruskal and Norman Zabusky in 1965, which led to the inverse scattering transform method in 1967.
In the special case of Hamiltonian systems, if there are enough independent Poisson commuting first integrals for the flow parameters to be able to serve as a coordinate system on the invariant level sets (the leaves of the Lagrangian foliation), and if the flows are complete and the energy level set is compact, this implies the Liouville-Arnold theorem; i.e., the existence of action-angle variables. General dynamical systems have no such conserved quantities; in the case of autonomous Hamiltonian systems, the energy is generally the only one, and on the energy level sets, the flows are typically chaotic.
A key ingredient in characterizing integrable systems is the Frobenius theorem, which states that a system is
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https://en.wikipedia.org/wiki/Fubini%E2%80%93Study%20metric
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In mathematics, the Fubini–Study metric (IPA: /fubini-ʃtuːdi/) is a Kähler metric on projective Hilbert space, that is, on a complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.
A Hermitian form in (the vector space) Cn+1 defines a unitary subgroup U(n+1) in GL(n+1,C). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(n+1) action; thus it is homogeneous. Equipped with a Fubini–Study metric, CPn is a symmetric space. The particular normalization on the metric depends on the application. In Riemannian geometry, one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the (2n+1)-sphere. In algebraic geometry, one uses a normalization making CPn a Hodge manifold.
Construction
The Fubini–Study metric arises naturally in the quotient space construction of complex projective space.
Specifically, one may define CPn to be the space consisting of all complex lines in Cn+1, i.e., the quotient of Cn+1\{0} by the equivalence relation relating all complex multiples of each point together. This agrees with the quotient by the diagonal group action of the multiplicative group C* = C \ {0}:
This quotient realizes Cn+1\{0} as a complex line bundle over the base space CPn. (In fact this is the so-called tautological bundle over CPn.) A point of CPn is thus identified with an equivalence class of (n+1)-tuples [Z0,...,Zn] modulo nonzero complex rescaling; the Zi are called homogeneous coordinates of the point.
Furthermore, one may realize this quotient mapping in two steps: since multiplication by a nonzero complex scalar z = R eiθ can be uniquely thought of as the composition of a dilation by the modulus R followed by a counterclockwise rotation about the origin by an angle , the quotient mapping Cn+1 → CPn splits into two pieces.
where step (a) is a quotient by the dilation Z ~ RZ for R ∈ R+, the multiplicative group of positive real numbers, and step (b) is a quotient by the rotations Z ~ eiθZ.
The result of the quotient in (a) is the real hypersphere S2n+1 defined by the equation |Z|2 = |Z0|2 + ... + |Zn|2 = 1. The quotient in (b) realizes CPn = S2n+1/S1, where S1 represents the group of rotations. This quotient is realized explicitly by the famous Hopf fibration S1 → S2n+1 → CPn, the fibers of which are among the great circles of .
As a metric quotient
When a quotient is taken of a Riemannian manifold (or metric space in general), care must be taken to ensure that the quotient space is endowed with a metric that is well-defined. For instance, if a group G acts on a Riemannian manifold (X,g), then in order for the orbit space X/G to possess an induced metric, must be constant along G-orbits in the sense that for any element h ∈ G and pair of vector fields we must have g(Xh,Yh) = g(X,Y).
The standard Hermitian metric on Cn+1 is given in the standard basis by
whose
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https://en.wikipedia.org/wiki/Fr%C3%B6licher%20spectral%20sequence
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In mathematics, the Frölicher spectral sequence (often misspelled as Fröhlicher) is a tool in the theory of complex manifolds, for expressing the potential failure of the results of cohomology theory that are valid in general only for Kähler manifolds. It was introduced by . A spectral sequence is set up, the degeneration of which would give the results of Hodge theory and Dolbeault's theorem.
See also
Hodge–de Rham spectral sequence
References
Complex manifolds
Spectral sequences
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https://en.wikipedia.org/wiki/Compactly%20supported%20homology
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In mathematics, a homology theory in algebraic topology is compactly supported if, in every degree n, the relative homology group Hn(X, A) of every pair of spaces
(X, A)
is naturally isomorphic to the direct limit of the nth relative homology groups of pairs (Y, B), where Y varies over compact subspaces of X and B varies over compact subspaces of A.
Singular homology is compactly supported, since each singular chain is a finite sum of simplices, which are compactly supported. Strong homology is not compactly supported.
If one has defined a homology theory over compact pairs, it is possible to extend it into a compactly supported homology theory in the wider category of Hausdorff pairs (X, A) with A closed in X, by defining that the homology of a Hausdorff pair (X, A) is the direct limit over pairs (Y, B), where Y, B are compact, Y is a subset of X, and B is a subset of A.
References
Homology theory
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https://en.wikipedia.org/wiki/Duration%20calculus
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Duration calculus (DC) is an interval logic for real-time systems. It was originally developed by Zhou Chaochen with the help of Anders P. Ravn and C. A. R. Hoare on the European ESPRIT Basic Research Action (BRA) ProCoS project on Provably Correct Systems.
Duration calculus is mainly useful at the requirements level of the software development process for real-time systems. Some tools are available (e.g., DCVALID, IDLVALID, etc.). Subsets of duration calculus have been studied (e.g., using discrete time rather than continuous time). Duration calculus is especially espoused by UNU-IIST in Macau and the Tata Institute of Fundamental Research in Mumbai, which are major centres of excellence for the approach.
See also
Interval temporal logic (ITL)
Temporal logic
Temporal logic of actions (TLA)
Modal logic
References
External links
Duration Calculus — Virtual Library entry
1991 introductions
Formal specification languages
Temporal logic
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https://en.wikipedia.org/wiki/Happy%20ending%20problem
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In mathematics, the "happy ending problem" (so named by Paul Erdős because it led to the marriage of George Szekeres and Esther Klein) is the following statement:
This was one of the original results that led to the development of Ramsey theory.
The happy ending theorem can be proven by a simple case analysis: if four or more points are vertices of the convex hull, any four such points can be chosen. If on the other hand, the convex hull has the form of a triangle with two points inside it, the two inner points and one of the triangle sides can be chosen. See for an illustrated explanation of this proof, and for a more detailed survey of the problem.
The Erdős–Szekeres conjecture states precisely a more general relationship between the number of points in a general-position point set and its largest subset forming a convex polygon, namely that the smallest number of points for which any general position arrangement contains a convex subset of points is . It remains unproven, but less precise bounds are known.
Larger polygons
proved the following generalisation:
The proof appeared in the same paper that proves the Erdős–Szekeres theorem on monotonic subsequences in sequences of numbers.
Let denote the minimum for which any set of points in general position must contain a convex N-gon. It is known that
, trivially.
.
. A set of eight points with no convex pentagon is shown in the illustration, demonstrating that ; the more difficult part of the proof is to show that every set of nine points in general position contains the vertices of a convex pentagon.
.
The value of is unknown for all . By the result of , is known to be finite for all finite .
On the basis of the known values of for N = 3, 4 and 5, Erdős and Szekeres conjectured in their original paper that
They proved later, by constructing explicit examples, that
.
In 2016 Andrew Suk showed that for
Suk actually proves, for N sufficiently large,
A 2020 preprint by Andreas F. Holmsen, Hossein Nassajian Mojarrad, János Pach and Gábor Tardos claims an improvement on Suk:
Empty convex polygons
There is also the question of whether any sufficiently large set of points in general position has an "empty" convex quadrilateral, pentagon, etc.,
that is, one that contains no other input point. The original solution to the happy ending problem can be adapted to show that any five points in general position have an empty convex quadrilateral, as shown in the illustration, and any ten points in general position have an empty convex pentagon. However, there exist arbitrarily large sets of points in general position that contain no empty convex heptagon.
For a long time the question of the existence of empty hexagons remained open, but and proved that every sufficiently large point set in general position contains a convex empty hexagon. More specifically, Gerken showed that the number of points needed is no more than f(9) for the same function f defined above, while Nicolás
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https://en.wikipedia.org/wiki/Zhou%20Chaochen
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Zhou Chaochen (; born 1 November 1937) is a Chinese computer scientist.
Zhou was born in Nanhui, Shanghai, China. He studied as an undergraduate at the Department of Mathematics and Mechanics, Peking University (1954–1958) and as a postgraduate at the Institute of Computing Technology, Chinese Academy of Sciences (CAS) (1963–1967).
He worked at Peking University and CAS until his visit to the Oxford University Computing Laboratory (now the Oxford University Department of Computer Science) (1989–1992). During this time, he was the prime investigator of the duration calculus, an interval logic for real-time systems as part of the European ESPRIT ProCoS project on Provably Correct Systems.
During the periods 1990–1992 and 1995–1996, Zhou Chaochen was visiting professor at the Department of Computer Science, Technical University of Denmark, Lyngby, on the invitation of Professor Dines Bjørner. He was Principal Research Fellow (1992–1997) and later Director of UNU-IIST in Macau (1997–2002), until his retirement, when he returned to Beijing.
In 2007, Zhou and Dines Bjørner, the first Director of UNU-IIST, were honoured on the occasion of their 70th birthdays. Zhou is a member of the Chinese Academy of Sciences.
Books
Zhou, Chaochen and Hansen, Michael R., Duration Calculus: A Formal Approach to Real-Time Systems. Springer-Verlag, Monographs in Theoretical Computer Science, An EATCS Series, 2003. .
References
External links
Institute of Software, Chinese Academy of Sciences (ISCAS) information
1937 births
Living people
Chinese computer scientists
Formal methods people
Members of the Chinese Academy of Sciences
Members of the Department of Computer Science, University of Oxford
Peking University alumni
Academic staff of Peking University
Scientists from Shanghai
Academic staff of the Technical University of Denmark
Academic staff of United Nations University
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https://en.wikipedia.org/wiki/Homogeneity%20and%20heterogeneity%20%28statistics%29
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In statistics, homogeneity and its opposite, heterogeneity, arise in describing the properties of a dataset, or several datasets. They relate to the validity of the often convenient assumption that the statistical properties of any one part of an overall dataset are the same as any other part. In meta-analysis, which combines the data from several studies, homogeneity measures the differences or similarities between the several studies (see also Study heterogeneity).
Homogeneity can be studied to several degrees of complexity. For example, considerations of homoscedasticity examine how much the variability of data-values changes throughout a dataset. However, questions of homogeneity apply to all aspects of the statistical distributions, including the location parameter. Thus, a more detailed study would examine changes to the whole of the marginal distribution. An intermediate-level study might move from looking at the variability to studying changes in the skewness. In addition to these, questions of homogeneity apply also to the joint distributions.
The concept of homogeneity can be applied in many different ways and, for certain types of statistical analysis, it is used to look for further properties that might need to be treated as varying within a dataset once some initial types of non-homogeneity have been dealt with.
Of variance
Examples
Regression
Differences in the typical values across the dataset might initially be dealt with by constructing a regression model using certain explanatory variables to relate variations in the typical value to known quantities. There should then be a later stage of analysis to examine whether the errors in the predictions from the regression behave in the same way across the dataset. Thus the question becomes one of the homogeneity of the distribution of the residuals, as the explanatory variables change. See regression analysis.
Time series
The initial stages in the analysis of a time series may involve plotting values against time to examine homogeneity of the series in various ways: stability across time as opposed to a trend; stability of local fluctuations over time.
Combining information across sites
In hydrology, data-series across a number of sites composed of annual values of the within-year annual maximum river-flow are analysed. A common model is that the distributions of these values are the same for all sites apart from a simple scaling factor, so that the location and scale are linked in a simple way. There can then be questions of examining the homogeneity across sites of the distribution of the scaled values.
Combining information sources
In meteorology, weather datasets are acquired over many years of record and, as part of this, measurements at certain stations may cease occasionally while, at around the same time, measurements may start at nearby locations. There are then questions as to whether, if the records are combined to form a single longer set of records, those records c
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https://en.wikipedia.org/wiki/Bernoulli%20scheme
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In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical systems. Many important dynamical systems (such as Axiom A systems) exhibit a repellor that is the product of the Cantor set and a smooth manifold, and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift. This is essentially the Markov partition. The term shift is in reference to the shift operator, which may be used to study Bernoulli schemes. The Ornstein isomorphism theorem shows that Bernoulli shifts are isomorphic when their entropy is equal.
Definition
A Bernoulli scheme is a discrete-time stochastic process where each independent random variable may take on one of N distinct possible values, with the outcome i occurring with probability , with i = 1, ..., N, and
The sample space is usually denoted as
as a shorthand for
The associated measure is called the Bernoulli measure
The σ-algebra on X is the product sigma algebra; that is, it is the (countable) direct product of the σ-algebras of the finite set {1, ..., N}. Thus, the triplet
is a measure space. A basis of is the cylinder sets. Given a cylinder set , its measure is
The equivalent expression, using the notation of probability theory, is
for the random variables
The Bernoulli scheme, as any stochastic process, may be viewed as a dynamical system by endowing it with the shift operator T where
Since the outcomes are independent, the shift preserves the measure, and thus T is a measure-preserving transformation. The quadruplet
is a measure-preserving dynamical system, and is called a Bernoulli scheme or a Bernoulli shift. It is often denoted by
The N = 2 Bernoulli scheme is called a Bernoulli process. The Bernoulli shift can be understood as a special case of the Markov shift, where all entries in the adjacency matrix are one, the corresponding graph thus being a clique.
Matches and metrics
The Hamming distance provides a natural metric on a Bernoulli scheme. Another important metric is the so-called metric, defined via a supremum over string matches.
Let and be two strings of symbols. A match is a sequence M of pairs of indexes into the string, i.e. pairs such that understood to be totally ordered. That is, each individual subsequence and are ordered: and likewise
The -distance between and is
where the supremum is being taken over all matches between and . This satisfies the triangle inequality only when and so is not quite a true metric; despite this, it is commonly called a "distance" in the literature.
Generalizations
Most of the properties of the Bernoulli scheme follow from the countable direct product, rather than from the finite base space. Thus, one may take the base space to be any standard probability space , and define the Bernoulli scheme as
This works because the countable
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https://en.wikipedia.org/wiki/MathSciNet
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MathSciNet is a searchable online bibliographic database created by the American Mathematical Society in 1996. It contains all of the contents of the journal Mathematical Reviews (MR) since 1940 along with an extensive author database, links to other MR entries, citations, full journal entries, and links to original articles. It contains almost 3.6 million items and over 2.3 million links to original articles.
Along with its parent publication Mathematical Reviews, MathSciNet has become an essential tool for researchers in the mathematical sciences. Access to the database is by subscription only and is not generally available to individual researchers who are not affiliated with a larger subscribing institution.
For the first 40 years of its existence, traditional typesetting was used to produce the Mathematical Reviews journal. Starting in 1980 bibliographic information and the reviews themselves were produced in both print and electronic form. This formed the basis of the first purely electronic version called MathFile launched in 1982. Further enhancements were added over the next 18 years and the current version known as MathSciNet went online in 1996.
Unlike most other abstracting databases, MathSciNet takes care to uniquely identify authors. Its author search allows the user to find publications associated with a given author record, even if multiple authors have exactly the same name or if the same person publishes under multiple names or name variants. Mathematical Reviews personnel will sometimes even contact authors to ensure that MathSciNet has correctly attributed their papers.
MathSciNet co-develops the Mathematics Subject Classification taxonomy with zbMATH.
Scope
MathSciNet contains information on over 3 million articles and over eight hundred thousand authors indexed from 1800 mathematical journals, many of them abstracted "cover-to-cover". A portion of those journals (about 450 in 2012) are designated as "Reference List Journals"; for MathSciNet entries of papers from these journals original reference lists are included.
In addition, reviews or bibliographical information on selected articles is included from many engineering, computer science and other applied journals abstracted by MathSciNet. The selection is done by the editors of Mathematical Reviews. The editors accept suggestions to cover additional journals, but do not reconsider missing articles for inclusion.
See also
All-Russian Mathematical Portal
Zentralblatt MATH
List of academic databases and search engines
References
External links
Official website
Bibliographic databases and indexes
Mathematical databases
Bibliographic databases in computer science
Publications established in 1980
Scholarly search services
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https://en.wikipedia.org/wiki/Motor%20variable
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In mathematics, a function of a motor variable is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers. William Kingdon Clifford coined the term motor for a kinematic operator in his "Preliminary Sketch of Biquaternions" (1873). He used split-complex numbers for scalars in his split-biquaternions. Motor variable is used here in place of split-complex variable for euphony and tradition.
For example,
Functions of a motor variable provide a context to extend real analysis and provide compact representation of mappings of the plane. However, the theory falls well short of function theory on the ordinary complex plane. Nevertheless, some of the aspects of conventional complex analysis have an interpretation given with motor variables, and more generally in hypercomplex analysis.
Elementary functions
Let D = , the split-complex plane. The following exemplar functions f have domain and range in D:
The action of a hyperbolic versor is combined with translation to produce the affine transformation
. When c = 0, the function is equivalent to a squeeze mapping.
The squaring function has no analogy in ordinary complex arithmetic. Let
and note that
The result is that the four quadrants are mapped into one, the identity component:
.
Note that forms the unit hyperbola . Thus, the
reciprocation
involves the hyperbola as curve of reference as opposed to the circle in C.
Linear fractional transformations
Using the concept of a projective line over a ring, the projective line P(D) is formed. The construction uses homogeneous coordinates with split-complex number components. The projective line P(D) is transformed by linear fractional transformations:
sometimes written
provided cz + d is a unit in D.
Elementary linear fractional transformations include
hyperbolic rotations
translations and
the inversion
Each of these has an inverse, and compositions fill out a group of linear fractional transformations. The motor variable is characterized by hyperbolic angle in its polar coordinates, and this angle is preserved by motor variable linear fractional transformations just as circular angle is preserved by the Möbius transformations of the ordinary complex plane. Transformations preserving angles are called conformal, so linear fractional transformations are conformal maps.
Transformations bounding regions can be compared: For example, on the ordinary complex plane, the Cayley transform carries the upper half-plane to the unit disk, thus bounding it. A mapping of the identity component U1 of D into a rectangle provides a comparable bounding action:
where T = {z = x + jy : |y| < x < 1 or |y| < 2 – x when 1 ≤ x <2}.
To realize the linear fractional transformations as bijections on the projective line a compactification of D is used. See the section given below.
Exp, log, and square root
The exponential function carries the whole plane D into U1:
.
Thus when
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https://en.wikipedia.org/wiki/Two-element%20Boolean%20algebra
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In mathematics and abstract algebra, the two-element Boolean algebra is the Boolean algebra whose underlying set (or universe or carrier) B is the Boolean domain. The elements of the Boolean domain are 1 and 0 by convention, so that B = {0, 1}. Paul Halmos's name for this algebra "2" has some following in the literature, and will be employed here.
Definition
B is a partially ordered set and the elements of B are also its bounds.
An operation of arity n is a mapping from Bn to B. Boolean algebra consists of two binary operations and unary complementation. The binary operations have been named and notated in various ways. Here they are called 'sum' and 'product', and notated by infix '+' and '∙', respectively. Sum and product commute and associate, as in the usual algebra of real numbers. As for the order of operations, brackets are decisive if present. Otherwise '∙' precedes '+'. Hence is parsed as and not as . Complementation is denoted by writing an overbar over its argument. The numerical analog of the complement of is . In the language of universal algebra, a Boolean algebra is a ∙ algebra of type .
Either one-to-one correspondence between {0,1} and {True,False} yields classical bivalent logic in equational form, with complementation read as NOT. If 1 is read as True, '+' is read as OR, and '∙' as AND, and vice versa if 1 is read as False. These two operations define a commutative semiring, known as the Boolean semiring.
Some basic identities
2 can be seen as grounded in the following trivial "Boolean" arithmetic:
Note that:
'+' and '∙' work exactly as in numerical arithmetic, except that 1+1=1. '+' and '∙' are derived by analogy from numerical arithmetic; simply set any nonzero number to 1.
Swapping 0 and 1, and '+' and '∙' preserves truth; this is the essence of the duality pervading all Boolean algebras.
This Boolean arithmetic suffices to verify any equation of 2, including the axioms, by examining every possible assignment of 0s and 1s to each variable (see decision procedure).
The following equations may now be verified:
Each of '+' and '∙' distributes over the other:
That '∙' distributes over '+' agrees with elementary algebra, but not '+' over '∙'. For this and other reasons, a sum of products (leading to a NAND synthesis) is more commonly employed than a product of sums (leading to a NOR synthesis).
Each of '+' and '∙' can be defined in terms of the other and complementation:
We only need one binary operation, and concatenation suffices to denote it. Hence concatenation and overbar suffice to notate 2. This notation is also that of Quine's Boolean term schemata. Letting (X) denote the complement of X and "()" denote either 0 or 1 yields the syntax of the primary algebra of G. Spencer-Brown's Laws of Form.
A basis for 2 is a set of equations, called axioms, from which all of the above equations (and more) can be derived. There are many known bases for all Boolean algebras and hence for 2. An elegant basis notated us
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https://en.wikipedia.org/wiki/Japanese%20theorem%20for%20cyclic%20polygons
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In geometry, the Japanese theorem states that no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant.
Conversely, if the sum of inradii is independent of the triangulation, then the polygon is cyclic. The Japanese theorem follows from Carnot's theorem; it is a Sangaku problem.
Proof
This theorem can be proven by first proving a special case: no matter how one triangulates a cyclic quadrilateral, the sum of inradii of triangles is constant.
After proving the quadrilateral case, the general case of the cyclic polygon theorem is an immediate corollary. The quadrilateral rule can be applied to quadrilateral components of a general partition of a cyclic polygon, and repeated application of the rule, which "flips" one diagonal, will generate all the possible partitions from any given partition, with each "flip" preserving the sum of the inradii.
The quadrilateral case follows from a simple extension of the Japanese theorem for cyclic quadrilaterals, which shows that a rectangle is formed by the two pairs of incenters corresponding to the two possible triangulations of the quadrilateral. The steps of this theorem require nothing beyond basic constructive Euclidean geometry.
With the additional construction of a parallelogram having sides parallel to the diagonals, and tangent to the corners of the rectangle of incenters, the quadrilateral case of the cyclic polygon theorem can be proved in a few steps. The equality of the sums of the radii of the two pairs is equivalent to the condition that the constructed parallelogram be a rhombus, and this is easily shown in the construction.
Another proof of the quadrilateral case is available due to Wilfred Reyes (2002). In the proof, both the Japanese theorem for cyclic quadrilaterals and the quadrilateral case of the cyclic polygon theorem are proven as a consequence of Thébault's problem III.
See also
Carnot's theorem, which is used in a proof of the theorem above
Equal incircles theorem
Tangent lines to circles
Notes
References
Claudi Alsina, Roger B. Nelsen: Icons of Mathematics: An Exploration of Twenty Key Images. MAA, 2011, , pp. 121-125
Wilfred Reyes: An Application of Thebault’s Theorem. Forum Geometricorum, Volume 2, 2002, pp. 183–185
External links
Mangho Ahuja, Wataru Uegaki, Kayo Matsushita: In Search of the Japanese Theorem
Japanese theorem at Mathworld
Japanese Theorem interactive demonstration at the C.a.R. website
Wataru Uegaki: "Japanese Theoremの起源と歴史" (On the Origin and History of the Japanese Theorem) http://hdl.handle.net/10076/4917
Euclidean plane geometry
Japanese mathematics
Theorems about triangles and circles
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https://en.wikipedia.org/wiki/MLD
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MLD may refer to:
Medicine
Manual lymphatic drainage
Metachromatic leukodystrophy, a rare neurometabolic genetic condition
Science and technology
Mean log deviation in statistics and econometrics
Mixed layer depth in hydrography
Multicast Listener Discovery, in computer networking
Million liter per day, in environmental engineering
Other
ICAO airline designator of Air Moldova
Maldives, ITU country code
Maniac Latin Disciples, a street gang
Marine Luchtvaart Dienst, the Dutch Naval Aviation Service
Mutually locally derivable, a mathematical property of aperiodic tile sets
EU Money Laundering Directive
Miluo East railway station, China Railway pinyin code MLD
Monolingual learner's dictionary, type of dictionary designed to meet the reference needs of people learning a foreign language.
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https://en.wikipedia.org/wiki/Tannaka%E2%80%93Krein%20duality
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In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups that are compact but noncommutative. The theory is named after Tadao Tannaka and Mark Grigorievich Krein. In contrast to the case of commutative groups considered by Lev Pontryagin, the notion dual to a noncommutative compact group is not a group, but a category of representations Π(G) with some additional structure, formed by the finite-dimensional representations of G.
Duality theorems of Tannaka and Krein describe the converse passage from the category Π(G) back to the group G, allowing one to recover the group from its category of representations. Moreover, they in effect completely characterize all categories that can arise from a group in this fashion. Alexander Grothendieck later showed that by a similar process, Tannaka duality can be extended to the case of algebraic groups via Tannakian formalism. Meanwhile, the original theory of Tannaka and Krein continued to be developed and refined by mathematical physicists. A generalization of Tannaka–Krein theory provides the natural framework for studying representations of quantum groups, and is currently being extended to quantum supergroups, quantum groupoids and their dual Hopf algebroids.
The idea of Tannaka–Krein duality: category of representations of a group
In Pontryagin duality theory for locally compact commutative groups, the dual object to a group G is its character group which consists of its one-dimensional unitary representations. If we allow the group G to be noncommutative, the most direct analogue of the character group is the set of equivalence classes of irreducible unitary representations of G. The analogue of the product of characters is the tensor product of representations. However, irreducible representations of G in general fail to form a group, or even a monoid, because a tensor product of irreducible representations is not necessarily irreducible. It turns out that one needs to consider the set of all finite-dimensional representations, and treat it as a monoidal category, where the product is the usual tensor product of representations, and the dual object is given by the operation of the contragredient representation.
A representation of the category is a monoidal natural transformation from the identity functor to itself. In other words, it is a non-zero function that associates with any an endomorphism of the space of T and satisfies the conditions of compatibility with tensor products, , and with arbitrary intertwining operators , namely, . The collection of all representations of the category can be endowed with multiplication and topology, in which convergence is defined pointwise, i.e., a sequence converges to some if and only if converges to for all . It can be shown that the set
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https://en.wikipedia.org/wiki/Norman%20Lloyd%20Johnson
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Norman Lloyd Johnson (9 January 1917, Ilford, Essex, England – 18 November 2004, Chapel Hill, North Carolina, United States) was a professor of statistics and author or editor of several standard reference works in statistics and probability theory.
Education
Johnson attended Ilford County High School, and went on to University College London, where he obtained a B.Sc. in mathematics 1936 and a B.Sc. and M.Sc. in statistics in 1937 and 1938.
Career
On qualification in 1938, Johnson was appointed Assistant Lecturer in the Department of Statistics at UCL. During World War II, he served under his former Professor Egon Pearson as an Experimental Officer with the Ordnance Board. He returned to the Statistics Department at UCL in 1945 and stayed there until 1962, as Assistant Lecturer, Lecturer and then Reader. In 1948 he was awarded a Ph.D. in Statistics for his work on the Johnson system of frequency curves. In 1949 he became a Fellow of the Institute of Actuaries.
Two visiting appointments in the USA, at the University of North Carolina at Chapel Hill (UNC) in 1952–1953 and at Case Institute of Technology in Cleveland, Ohio, in 1960–1961, led to his permanent appointment as Professor in the Department of Statistics at UNC in 1962. He was Chairman 1971–1976 and officially retired in 1982, but continued to be active in scholarship and research as Professor Emeritus almost until his death. UNC named a distinguished endowed chair in his honour. He expressed a wish to retire completely and return to live in Ilford, but never managed it.
Publications
He wrote, together with Samuel Kotz, a standard reference series, Distributions in Statistics. This series has been described as of "virtually Biblical authority", a comment that he (a devout Christian) firmly rejected. He was editor-in-chief of the 10-volume Encyclopedia of Statistical Sciences, widely regarded as one of the most important reference works in statistical methodology. He also wrote several textbooks and about 180 papers. His book "Survival Models" was co-authored with his wife Regina Elandt Johnson, herself a professor of biostatistics.
Honours
He was honoured in numerous ways, including the Wilks Award of the American Statistical Association, the Shewhart Medal of the American Society for Quality Control and a D.Sc. degree from UCL.
References
Campbell B. Read (2004) A Conversation with Norman L. Johnson, Statistical Science, 19, 544–560. Project Euclid
External links
Norman L. Johnson
1917 births
2004 deaths
People educated at Ilford County High School
Academics of University College London
Alumni of University College London
English statisticians
20th-century English mathematicians
People from Ilford
Fellows of the American Statistical Association
Mathematical statisticians
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https://en.wikipedia.org/wiki/Bapoo%20Mama
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Bapoo Burjorji 'B.B.' Mama (April 8, 1924 in Bombay – March 18, 1995 in Bombay) was a cricket statistician.
Bapoo Mama was a major figure in Indian cricket statistics in the second half of the twentieth century. In the 1970s and eighties, he contributed columns like Follow 'em with BBM, Figures are Fun, Factfile and Down the Memory Lane to prominent periodicals like the Sportstar, Sportsweek, Times of India and Pakistan Cricket International. He was also a regular contributor to the Indian sections in the Wisden. He served as the official statistician of the national channel Doordarshan from 1973 to 1988.
Mama was educated in Bombay but moved to Panchgani in Maharashtra in 1948 for reasons of health. He lived most of the rest of his life there. He died following a short illness of intestinal and lung complications.
References
Obituary in Indian Cricket 1996
Cricket historians and writers
Cricket scorers
Parsi people
1995 deaths
1924 births
Cricket statisticians
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https://en.wikipedia.org/wiki/Recursive%20Bayesian%20estimation
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In probability theory, statistics, and machine learning, recursive Bayesian estimation, also known as a Bayes filter, is a general probabilistic approach for estimating an unknown probability density function (PDF) recursively over time using incoming measurements and a mathematical process model. The process relies heavily upon mathematical concepts and models that are theorized within a study of prior and posterior probabilities known as Bayesian statistics.
In robotics
A Bayes filter is an algorithm used in computer science for calculating the probabilities of multiple beliefs to allow a robot to infer its position and orientation. Essentially, Bayes filters allow robots to continuously update their most likely position within a coordinate system, based on the most recently acquired sensor data. This is a recursive algorithm. It consists of two parts: prediction and innovation. If the variables are normally distributed and the transitions are linear, the Bayes filter becomes equal to the Kalman filter.
In a simple example, a robot moving throughout a grid may have several different sensors that provide it with information about its surroundings. The robot may start out with certainty that it is at position (0,0). However, as it moves farther and farther from its original position, the robot has continuously less certainty about its position; using a Bayes filter, a probability can be assigned to the robot's belief about its current position, and that probability can be continuously updated from additional sensor information.
Model
The measurements are the manifestations of a hidden Markov model (HMM), which means the true state is assumed to be an unobserved Markov process. The following picture presents a Bayesian network of a HMM.
Because of the Markov assumption, the probability of the current true state given the immediately previous one is conditionally independent of the other earlier states.
Similarly, the measurement at the k-th timestep is dependent only upon the current state, so is conditionally independent of all other states given the current state.
Using these assumptions the probability distribution over all states of the HMM can be written simply as:
However, when using the Kalman filter to estimate the state x, the probability distribution of interest is associated with the current states conditioned on the measurements up to the current timestep. (This is achieved by marginalising out the previous states and dividing by the probability of the measurement set.)
This leads to the predict and update steps of the Kalman filter written probabilistically. The probability distribution associated with the predicted state is the sum (integral) of the products of the probability distribution associated with the transition from the (k - 1)-th timestep to the k-th and the probability distribution associated with the previous state, over all possible .
The probability distribution of update is proportional to the product o
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https://en.wikipedia.org/wiki/Octant
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Octant may refer to:
Octant (solid geometry), one of the eight divisions of 3-dimensional space by orthogonal coordinate planes
Octant of a sphere, a spherical triangle with three right angles
Octant (plane geometry), one eighth of a full circle
Octant (instrument) for celestial navigation
Octans, a constellation also called The Octant
Octant (band), from Seattle, Washington
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https://en.wikipedia.org/wiki/Simple%20linear%20regression
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In statistics, simple linear regression is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable (conventionally, the x and y coordinates in a Cartesian coordinate system) and finds a linear function (a non-vertical straight line) that, as accurately as possible, predicts the dependent variable values as a function of the independent variable.
The adjective simple refers to the fact that the outcome variable is related to a single predictor.
It is common to make the additional stipulation that the ordinary least squares (OLS) method should be used: the accuracy of each predicted value is measured by its squared residual (vertical distance between the point of the data set and the fitted line), and the goal is to make the sum of these squared deviations as small as possible. Other regression methods that can be used in place of ordinary least squares include least absolute deviations (minimizing the sum of absolute values of residuals) and the Theil–Sen estimator (which chooses a line whose slope is the median of the slopes determined by pairs of sample points). Deming regression (total least squares) also finds a line that fits a set of two-dimensional sample points, but (unlike ordinary least squares, least absolute deviations, and median slope regression) it is not really an instance of simple linear regression, because it does not separate the coordinates into one dependent and one independent variable and could potentially return a vertical line as its fit.
The remainder of the article assumes an ordinary least squares regression.
In this case, the slope of the fitted line is equal to the correlation between and corrected by the ratio of standard deviations of these variables. The intercept of the fitted line is such that the line passes through the center of mass of the data points.
Fitting the regression line
Consider the model function
which describes a line with slope and -intercept . In general such a relationship may not hold exactly for the largely unobserved population of values of the independent and dependent variables; we call the unobserved deviations from the above equation the errors. Suppose we observe data pairs and call them }. We can describe the underlying relationship between and involving this error term by
This relationship between the true (but unobserved) underlying parameters and and the data points is called a linear regression model.
The goal is to find estimated values and for the parameters and which would provide the "best" fit in some sense for the data points. As mentioned in the introduction, in this article the "best" fit will be understood as in the least-squares approach: a line that minimizes the sum of squared residuals (see also Errors and residuals) (differences between actual and predicted values of the dependent variable y), each of which is given by, for any
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https://en.wikipedia.org/wiki/Chevalley%20scheme
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A Chevalley scheme in algebraic geometry was a precursor notion of scheme theory.
Let X be a separated integral noetherian scheme, R its function field. If we denote by the set of subrings of R, where x runs through X (when , we denote by ), verifies the following three properties
For each , R is the field of fractions of M.
There is a finite set of noetherian subrings of R so that and that, for each pair of indices i,j, the subring of R generated by is an -algebra of finite type.
If in are such that the maximal ideal of M is contained in that of N, then M=N.
Originally, Chevalley also supposed that R was an extension of finite type of a field K and that the 's were algebras of finite type over a field too (this simplifies the second condition above).
Bibliography
Online
Scheme theory
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https://en.wikipedia.org/wiki/Pierpont%20prime
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In number theory, a Pierpont prime is a prime number of the form
for some nonnegative integers and . That is, they are the prime numbers for which is 3-smooth. They are named after the mathematician James Pierpont, who used them to characterize the regular polygons that can be constructed using conic sections. The same characterization applies to polygons that can be constructed using ruler, compass, and angle trisector, or using paper folding.
Except for 2 and the Fermat primes, every Pierpont prime must be 1 modulo 6. The first few Pierpont primes are:
It has been conjectured that there are infinitely many Pierpont primes, but this remains unproven.
Distribution
A Pierpont prime with is of the form , and is therefore a Fermat prime (unless ). If is positive then must also be positive (because would be an even number greater than 2 and therefore not prime), and therefore the non-Fermat Pierpont primes all have the form , when is a positive integer (except for 2, when ).
Empirically, the Pierpont primes do not seem to be particularly rare or sparsely distributed; there are 42 Pierpont primes less than 106, 65 less than 109, 157 less than 1020, and 795 less than 10100. There are few restrictions from algebraic factorisations on the Pierpont primes, so there are no requirements like the Mersenne prime condition that the exponent must be prime. Thus, it is expected that among -digit numbers of the correct form , the fraction of these that are prime should be proportional to , a similar proportion as the proportion of prime numbers among all -digit numbers.
As there are numbers of the correct form in this range, there should be Pierpont primes.
Andrew M. Gleason made this reasoning explicit, conjecturing there are infinitely many Pierpont primes, and more specifically that there should be approximately Pierpont primes up to . According to Gleason's conjecture there are Pierpont primes smaller than N, as opposed to the smaller conjectural number of Mersenne primes in that range.
Primality testing
When , is a Proth number and thus its primality can be tested by Proth's theorem. On the other hand, when alternative primality tests for are possible based on the factorization of as a small even number multiplied by a large power of 3.
Pierpont primes found as factors of Fermat numbers
As part of the ongoing worldwide search for factors of Fermat numbers, some Pierpont primes have been announced as factors. The following table gives values of m, k, and n such that
The left-hand side is a Fermat number; the right-hand side is a Pierpont prime.
, the largest known Pierpont prime is 81 × 220498148 + 1 (6,170,560 decimal digits), whose primality was discovered in June 2023.
Polygon construction
In the mathematics of paper folding, the Huzita axioms define six of the seven types of fold possible. It has been shown that these folds are sufficient to allow the construction of the points that solve any cubic equation.
It follows tha
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https://en.wikipedia.org/wiki/Determinacy
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Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "determinacy" is the property of a game whereby such a strategy exists. Determinacy was introduced by Gale and Stewart in 1950, under the name "determinateness".
The games studied in set theory are usually Gale–Stewart games—two-player games of perfect information in which the players make an infinite sequence of moves and there are no draws. The field of game theory studies more general kinds of games, including games with draws such as tic-tac-toe, chess, or infinite chess, or games with imperfect information such as poker.
Basic notions
Games
The first sort of game we shall consider is the two-player game of perfect information of length ω, in which the players play natural numbers. These games are often called Gale–Stewart games.
In this sort of game there are two players, often named I and II, who take turns playing natural numbers, with I going first. They play "forever"; that is, their plays are indexed by the natural numbers. When they're finished, a predetermined condition decides which player won. This condition need not be specified by any definable rule; it may simply be an arbitrary (infinitely long) lookup table saying who has won given a particular sequence of plays.
More formally, consider a subset A of Baire space; recall that the latter consists of all ω-sequences of natural numbers. Then in the game GA,
I plays a natural number a0, then II plays a1, then I plays a2, and so on. Then I wins the game if and only if
and otherwise II wins. A is then called the payoff set of GA.
It is assumed that each player can see all moves preceding each of his moves, and also knows the winning condition.
Strategies
Informally, a strategy for a player is a way of playing in which his plays are entirely determined by the foregoing plays. Again, such a "way" does not have to be capable of being captured by any explicable "rule", but may simply be a lookup table.
More formally, a strategy for player I (for a game in the sense of the preceding subsection) is a function that accepts as an argument any finite sequence of natural numbers, of even length, and returns a natural number. If σ is such a strategy and <a0,...,a2n-1>
is a sequence of plays, then σ(<a0,...,a2n-1>) is the next play I will make, if I is following the strategy σ. Strategies for II are just the same, substituting "odd" for "even".
Note that we have said nothing, as yet, about whether a strategy is in any way good. A strategy might direct a player to make aggressively bad moves, and it would still be a strategy. In fact it is not necessary even to know the winning condition for a game, to know what strategies exist for the game.
Winning strategies
A strategy is winning if the player following it must necessaril
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https://en.wikipedia.org/wiki/Bulgarian%20solitaire
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In mathematics and game theory, Bulgarian solitaire is a card game that was introduced by Martin Gardner.
In the game, a pack of cards is divided into several piles. Then for each pile, remove one card; collect the removed cards together to form a new pile (piles of zero size are ignored).
If is a triangular number (that is, for some ), then it is known that Bulgarian solitaire will reach a stable configuration in which the sizes of the piles are . This state is reached in moves or fewer. If is not triangular, no stable configuration exists and a limit cycle is reached.
Random Bulgarian solitaire
In random Bulgarian solitaire or stochastic Bulgarian solitaire a pack of cards is divided into several piles. Then for each pile, either leave it intact or, with a fixed probability , remove one card; collect the removed cards together to form a new pile (piles of zero size are ignored). This is a finite irreducible Markov chain.
In 2004, Brazilian probabilist of Russian origin Serguei Popov showed that stochastic Bulgarian solitaire spends "most" of its time in a "roughly" triangular distribution.
References
20th-century card games
Combinatorial game theory
Year of introduction missing
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https://en.wikipedia.org/wiki/RSSSF
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The Rec.Sport.Soccer Statistics Foundation (RSSSF) is an international organization dedicated to collecting statistics about association football. The foundation aims to build an exhaustive archive of football-related information from around the world.
History
This enterprise, according to its founders, was created in January 1994 by three regulars of the Rec.Sport.Soccer (RSS) Usenet newsgroup: Lars Aarhus, Kent Hedlundh, and Karel Stokkermans. It was originally known as the "North European Rec.Sport.Soccer Statistics Foundation", but the geographical reference was dropped as its membership from other regions grew.
The RSSSF has members and contributors from all around the world and has spawned seven spin-off projects to more closely follow the leagues of that project's home country. The spin-off projects are dedicated to Albania, Brazil, Denmark, Norway, Romania, Uruguay, Venezuela and Egypt. In November 2002, the Polish service 90minut.pl became the official branch of RSSSF Poland.
Reception
RSSSF's database has been described as the "very best" for football data.
Rec.Sport.Soccer Player of the Year
Since 1992 a vote for the Best Footballer in the World among the readers of the rec.sport.soccer newsgroup. It was held yearly until 2005, when it was discontinued. The voting works as follows: each voter chooses five players, at most two of the same nationality, in order; these obtain five to one points. The nationality restriction was dropped for the 2003 vote, in which voting was restricted to 173 pre-selected players.
Wins by player
Wins by country
Wins by club
References
Bibliography
Archived 23 January 2014
External links
Association football websites
Usenet
Organizations established in 1994
Internet properties established in 1994
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https://en.wikipedia.org/wiki/Graham%20Higman
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Graham Higman FRS (19 January 1917 – 8 April 2008) was a prominent English mathematician known for his contributions to group theory.
Biography
Higman was born in Louth, Lincolnshire, and attended Sutton High School, Plymouth, winning a scholarship to Balliol College, Oxford. In 1939 he co-founded The Invariant Society, the student mathematics society, and earned his DPhil from the University of Oxford in 1941. His thesis, The units of group-rings, was written under the direction of J. H. C. Whitehead.
From 1960 to 1984 he was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford.
Higman was awarded the Senior Berwick Prize in 1962 and the De Morgan Medal of the London Mathematical Society in 1974. He was the founder of the Journal of Algebra and its editor from 1964 to 1984. Higman had 51 D.Phil. students, including Jonathan Lazare Alperin, Rosemary A. Bailey, Marston Conder, John Mackintosh Howie, and Peter M. Neumann.
He was also a local preacher in the Oxford Circuit of the Methodist Church. During the Second World War he was a conscientious objector, working at the Meteorological Office in Northern Ireland and Gibraltar.
He died in Oxford.
Publications
Graham Higman (1966) Odd characterisations of finite simple groups, U. of Michigan Press
*
Graham Higman and Elizabeth Scott (1988), Existentially closed groups, LMS Monographs, Clarendon Press, Oxford
See also
Higman–Sims group, named after Donald G. Higman, but studied also by Graham Higman.
Higman's embedding theorem
Feit-Higman theorem
Higman group
Higman's lemma
HNN extension
Hall–Higman theorem
Notes
References
Death notice, Oxford University Gazette, 17 April 2008
External links
1917 births
2008 deaths
20th-century English mathematicians
21st-century English mathematicians
Group theorists
Alumni of Balliol College, Oxford
Fellows of Magdalen College, Oxford
Fellows of the Royal Society
People from Louth, Lincolnshire
English conscientious objectors
British Methodists
Waynflete Professors of Pure Mathematics
Presidents of the London Mathematical Society
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https://en.wikipedia.org/wiki/Noncentral%20chi-squared%20distribution
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In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power analysis of statistical tests in which the null distribution is (perhaps asymptotically) a chi-squared distribution; important examples of such tests are the likelihood-ratio tests.
Definitions
Background
Let be k independent, normally distributed random variables with means and unit variances. Then the random variable
is distributed according to the noncentral chi-squared distribution. It has two parameters: which specifies the number of degrees of freedom (i.e. the number of ), and which is related to the mean of the random variables by:
is sometimes called the noncentrality parameter. Note that some references define in other ways, such as half of the above sum, or its square root.
This distribution arises in multivariate statistics as a derivative of the multivariate normal distribution. While the central chi-squared distribution is the squared norm of a random vector with distribution (i.e., the squared distance from the origin to a point taken at random from that distribution), the non-central is the squared norm of a random vector with distribution. Here is a zero vector of length k, and is the identity matrix of size k.
Density
The probability density function (pdf) is given by
where is distributed as chi-squared with degrees of freedom.
From this representation, the noncentral chi-squared distribution is seen to be a Poisson-weighted mixture of central chi-squared distributions. Suppose that a random variable J has a Poisson distribution with mean , and the conditional distribution of Z given J = i is chi-squared with k + 2i degrees of freedom. Then the unconditional distribution of Z is non-central chi-squared with k degrees of freedom, and non-centrality parameter .
Alternatively, the pdf can be written as
where is a modified Bessel function of the first kind given by
Using the relation between Bessel functions and hypergeometric functions, the pdf can also be written as:
Siegel (1979) discusses the case k = 0 specifically (zero degrees of freedom), in which case the distribution has a discrete component at zero.
Derivation of the pdf
The derivation of the probability density function is most easily done by performing the following steps:
Since have unit variances, their joint distribution is spherically symmetric, up to a location shift.
The spherical symmetry then implies that the distribution of depends on the means only through the squared length, . Without loss of generality, we can therefore take and .
Now derive the density of (i.e. the k = 1 case). Simple transformation of random variables shows that
where is the standard normal density.
Expand the cosh term in a Taylor series. This gives the Poisson-weighted mixture representation of the densi
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https://en.wikipedia.org/wiki/Pierre%20Samuel
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Pierre Samuel (12 September 1921 – 23 August 2009) was a French mathematician, known for his work in commutative algebra and its applications to algebraic geometry. The two-volume work Commutative Algebra that he wrote with Oscar Zariski is a classic. Other books of his covered projective geometry and algebraic number theory.
Early life and education
Samuel studied at the Lycée Janson-de-Sailly in Paris before attending the École Normale Supérieure where he studied for his Agrégé de mathematique. He received his Master of Arts and then a Ph.D. from Princeton University in 1947, under the supervision of Oscar Zariski, with a thesis "Ultrafilters and Compactification of Uniform Spaces".
Career
Samuel ran a Paris seminar during the 1960s, and became Professeur émérite at the Université Paris-Sud (Orsay). His lectures on unique factorization domains published by the Tata Institute of Fundamental Research played a significant role in computing the Picard group of a Zariski surface via the work of Jeffrey Lang and collaborators. The method was inspired by earlier work of Nathan Jacobson and Pierre Cartier another outstanding member of the Bourbaki group. Nicholas Katz related this to the concept of p-curvature of a connection introduced by Alexander Grothendieck.
He was a member of the Bourbaki group, and filmed some of their meetings. A French television documentary on Bourbaki broadcast some of this footage in 2000.
Samuel was also active in issues of social justice, including concerns about environmental degradation (where he was influenced by Grothendieck), and arms control. He died in Paris in August 2009.
His doctoral students include Lucien Szpiro and Daniel Lazard.
Awards and honors
In 1958 he was an invited speaker (Relations d'équivalence en géométrie algébrique) at the ICM in Edinburgh. In 1969 he won the Lester R. Ford Award.
Works
with Oscar Zariski:
with Oscar Zariski:
Anneaux factoriels, Publicaçoes da Sociedade de Matematica de São Paulo, 1962
Écologie: détente ou cycle infernal, Union générale d'éditions, Collection 10-18, 1973
Amazones, guerrières et gaillardes, éditions Complexe & Presses universitaires de Grenoble, 1975
Le nucléaire en question, 1980
Géométrie projective, Presses universitaires de France, 1986
Colloque en l'honneur de Pierre Samuel, Mém. Société mathématique de France (1989)
References
Further reading
Colloque en l'honneur de Pierre Samuel, Mem. Math. Soc. Fr. (1989)
Tangente Magazine
External links
1921 births
2009 deaths
Academic staff of Paris-Sud University
20th-century French mathematicians
Algebraists
Algebraic geometers
École Normale Supérieure alumni
Nicolas Bourbaki
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https://en.wikipedia.org/wiki/Reeb%20foliation
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In mathematics, the Reeb foliation is a particular foliation of the 3-sphere, introduced by the French mathematician Georges Reeb (1920–1993).
It is based on dividing the sphere into two solid tori, along a 2-torus: see Clifford torus. Each of the solid tori is then foliated internally, in codimension 1, and the dividing torus surface forms one more leaf.
By Novikov's compact leaf theorem, every smooth foliation of the 3-sphere includes a compact torus leaf, bounding a solid torus foliated in the same way.
Illustrations
References
External links
Foliations
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https://en.wikipedia.org/wiki/Puiseux%20series
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In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series
is a Puiseux series in the indeterminate . Puiseux series were first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850.
The definition of a Puiseux series includes that the denominators of the exponents must be bounded. So, by reducing exponents to a common denominator , a Puiseux series becomes a Laurent series in an th root of the indeterminate. For example, the example above is a Laurent series in Because a complex number has th roots, a convergent Puiseux series typically defines functions in a neighborhood of .
Puiseux's theorem, sometimes also called the Newton–Puiseux theorem, asserts that, given a polynomial equation with complex coefficients, its solutions in , viewed as functions of , may be expanded as Puiseux series in that are convergent in some neighbourhood of . In other words, every branch of an algebraic curve may be locally described by a Puiseux series in (or in when considering branches above a neighborhood of ).
Using modern terminology, Puiseux's theorem asserts that the set of Puiseux series over an algebraically closed field of characteristic 0 is itself an algebraically closed field, called the field of Puiseux series. It is the algebraic closure of the field of formal Laurent series, which itself is the field of fractions of the ring of formal power series.
Definition
If is a field (such as the complex numbers), a Puiseux series with coefficients in is an expression of the form
where is a positive integer and is an integer. In other words, Puiseux series differ from Laurent series in that they allow for fractional exponents of the indeterminate, as long as these fractional exponents have bounded denominator (here n). Just as with Laurent series, Puiseux series allow for negative exponents of the indeterminate as long as these negative exponents are bounded below (here by ). Addition and multiplication are as expected: for example,
and
One might define them by first "upgrading" the denominator of the exponents to some common denominator and then performing the operation in the corresponding field of formal Laurent series of .
The Puiseux series with coefficients in form a field, which is the union
of fields of formal Laurent series in (considered as an indeterminate).
This yields an alternative definition of the field of Puiseux series in terms of a direct limit. For every positive integer , let be an indeterminate (meant to represent ), and be the field of formal Laurent series in If divides , the mapping induces a field homomorphism and these homomorphisms form a direct system that has the field of Puiseux series as a direct limit. The fact that every field homomorphism is injective shows that this direct limit can be identified with the above union, and that the two definitions are equivalent (up t
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https://en.wikipedia.org/wiki/Bitangent
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In geometry, a bitangent to a curve is a line that touches in two distinct points and and that has the same direction as at these points. That is, is a tangent line at and at .
Bitangents of algebraic curves
In general, an algebraic curve will have infinitely many secant lines, but only finitely many bitangents.
Bézout's theorem implies that an algebraic plane curve with a bitangent must have degree at least 4. The case of the 28 bitangents of a quartic was a celebrated piece of geometry of the nineteenth century, a relationship being shown to the 27 lines on the cubic surface.
Bitangents of polygons
The four bitangents of two disjoint convex polygons may be found efficiently by an algorithm based on binary search in which one maintains a binary search pointer into the lists of edges of each polygon and moves one of the pointers left or right at each steps depending on where the tangent lines to the edges at the two pointers cross each other. This bitangent calculation is a key subroutine in data structures for maintaining convex hulls dynamically . describe an algorithm for efficiently listing all bitangent line segments that do not cross any of the other curves in a system of multiple disjoint convex curves, using a technique based on pseudotriangulation.
Bitangents may be used to speed up the visibility graph approach to solving the Euclidean shortest path problem: the shortest path among a collection of polygonal obstacles may only enter or leave the boundary of an obstacle along one of its bitangents, so the shortest path can be found by applying Dijkstra's algorithm to a subgraph of the visibility graph formed by the visibility edges that lie on bitangent lines .
Related concepts
A bitangent differs from a secant line in that a secant line may cross the curve at the two points it intersects it. One can also consider bitangents that are not lines; for instance, the symmetry set of a curve is the locus of centers of circles that are tangent to the curve in two points.
Bitangents to pairs of circles figure prominently in Jakob Steiner's 1826 construction of the Malfatti circles, in the belt problem of calculating the length of a belt connecting two pulleys, in Casey's theorem characterizing sets of four circles with a common tangent circle, and in Monge's theorem on the collinearity of intersection points of certain bitangents.
References
.
.
.
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Differential geometry
Algebraic curves
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https://en.wikipedia.org/wiki/Birational%20invariant
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In algebraic geometry, a birational invariant is a property that is preserved under birational equivalence.
Formal definition
A birational invariant is a quantity or object that is well-defined on a birational equivalence class of algebraic varieties. In other words, it depends only on the function field of the variety.
Examples
The first example is given by the grounding work of Riemann himself: in his thesis, he shows that one can define a Riemann surface to each algebraic curve; every Riemann surface comes from an algebraic curve, well defined up to birational equivalence and two birational equivalent curves give the same surface. Therefore, the Riemann surface, or more simply its Geometric genus is a birational invariant.
A more complicated example is given by Hodge theory: in the case of an algebraic surface, the Hodge numbers h0,1 and h0,2 of a non-singular projective complex surface are birational invariants. The Hodge number h1,1 is not, since the process of blowing up a point to a curve on the surface can augment it.
References
.
Birational geometry
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https://en.wikipedia.org/wiki/Complete%20quadrangle
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In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six lines connecting the six pairs of points. Dually, a complete quadrilateral is a system of four lines, no three of which pass through the same point, and the six points of intersection of these lines. The complete quadrangle was called a tetrastigm by , and the complete quadrilateral was called a tetragram; those terms are occasionally still used.
Diagonals
The six lines of a complete quadrangle meet in pairs to form three additional points called the diagonal points of the quadrangle. Similarly, among the six points of a complete quadrilateral there are three pairs of points that are not already connected by lines; the line segments connecting these pairs are called diagonals. For points and lines in the Euclidean plane, the diagonal points cannot lie on a single line, and the diagonals cannot have a single point of triple crossing. Due to the discovery of the Fano plane, a finite geometry in which the diagonal points of a complete quadrangle are collinear, some authors have augmented the axioms of projective geometry with Fano's axiom that the diagonal points are not collinear, while others have been less restrictive.
A set of contracted expressions for the parts of a complete quadrangle were introduced by G. B. Halsted: He calls the vertices of the quadrangle dots, and the diagonal points he calls codots. The lines of the projective space are called straights, and in the quadrangle they are called connectors. The "diagonal lines" of Coxeter are called opposite connectors by Halsted. Opposite connectors cross at a codot. The configuration of the complete quadrangle is a tetrastim.
Projective properties
As systems of points and lines in which all points belong to the same number of lines and all lines contain the same number of points, the complete quadrangle and the complete quadrilateral both form projective configurations; in the notation of projective configurations, the complete quadrangle is written as (4362) and the complete quadrilateral is written (6243), where the numbers in this notation refer to the numbers of points, lines per point, lines, and points per line of the configuration.
The projective dual of a complete quadrangle is a complete quadrilateral, and vice versa. For any two complete quadrangles, or any two complete quadrilaterals, there is a unique projective transformation taking one of the two configurations into the other.
Karl von Staudt reformed mathematical foundations in 1847 with the complete quadrangle when he noted that a "harmonic property" could be based on concomitants of the quadrangle: When each pair of opposite sides of the quadrangle intersect on a line, then the diagonals intersect the line at projective harmonic conjugate positions. The four points on the line
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https://en.wikipedia.org/wiki/Topological%20divisor%20of%20zero
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In mathematics, an element of a Banach algebra is called a topological divisor of zero if there exists a sequence of elements of such that
The sequence converges to the zero element, but
The sequence does not converge to the zero element.
If such a sequence exists, then one may assume that for all .
If is not commutative, then is called a "left" topological divisor of zero, and one may define "right" topological divisors of zero similarly.
Examples
If has a unit element, then the invertible elements of form an open subset of , while the non-invertible elements are the complementary closed subset. Any point on the boundary between these two sets is both a left and right topological divisor of zero.
In particular, any quasinilpotent element is a topological divisor of zero (e.g. the Volterra operator).
An operator on a Banach space , which is injective, not surjective, but whose image is dense in , is a left topological divisor of zero.
Generalization
The notion of a topological divisor of zero may be generalized to any topological algebra. If the algebra in question is not first-countable, one must substitute nets for the sequences used in the definition.
Topological algebra
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https://en.wikipedia.org/wiki/Interval%20exchange%20transformation
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In mathematics, an interval exchange transformation is a kind of dynamical system that generalises circle rotation. The phase space consists of the unit interval, and the transformation acts by cutting the interval into several subintervals, and then permuting these subintervals. They arise naturally in the study of polygonal billiards and in area-preserving flows.
Formal definition
Let and let be a permutation on . Consider a vector of positive real numbers (the widths of the subintervals), satisfying
Define a map called the interval exchange transformation associated with the pair as follows. For let
Then for , define
if lies in the subinterval . Thus acts on each subinterval of the form by a translation, and it rearranges these subintervals so that the subinterval at position is moved to position .
Properties
Any interval exchange transformation is a bijection of to itself that preserves the Lebesgue measure. It is continuous except at a finite number of points.
The inverse of the interval exchange transformation is again an interval exchange transformation. In fact, it is the transformation where for all .
If and (in cycle notation), and if we join up the ends of the interval to make a circle, then is just a circle rotation. The Weyl equidistribution theorem then asserts that if the length is irrational, then is uniquely ergodic. Roughly speaking, this means that the orbits of points of are uniformly evenly distributed. On the other hand, if is rational then each point of the interval is periodic, and the period is the denominator of (written in lowest terms).
If , and provided satisfies certain non-degeneracy conditions (namely there is no integer such that ), a deep theorem which was a conjecture of M.Keane and due independently to William A. Veech and to Howard Masur asserts that for almost all choices of in the unit simplex the interval exchange transformation is again uniquely ergodic. However, for there also exist choices of so that is ergodic but not uniquely ergodic. Even in these cases, the number of ergodic invariant measures of is finite, and is at most .
Interval maps have a topological entropy of zero.
Odometers
The dyadic odometer can be understood as an interval exchange transformation of a countable number of intervals. The dyadic odometer is most easily written as the transformation
defined on the Cantor space The standard mapping from Cantor space into the unit interval is given by
This mapping is a measure-preserving homomorphism from the Cantor set to the unit interval, in that it maps the standard Bernoulli measure on the Cantor set to the Lebesgue measure on the unit interval. A visualization of the odometer and its first three iterates appear on the right.
Higher dimensions
Two and higher-dimensional generalizations include polygon exchanges, polyhedral exchanges and piecewise isometries.
See also
Odometer
Notes
References
Artur Avila and Giovanni Forni, We
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https://en.wikipedia.org/wiki/Edwin%20Ray%20Guthrie
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Edwin Ray Guthrie (; January 9, 1886 – April 23, 1969) was a behavioral psychologist who began his career as a mathematics teacher and philosopher. But, he became a psychologist at the age of 33. He spent most of his career at the University of Washington, where he became full professor and then emeritus professor in psychology.
Guthrie is best known for his theory that all learning is based on a stimulus–response association. This was variously described as one trial theory, non-reinforcement, and contiguity learning. The theory was:
"A combination of stimuli which has accompanied a movement will on its recurrence tend to be followed by that movement".
One word that his coworkers and students used to describe Guthrie and his theories was "simple", referring to how he described complex ideas in simple terms. Some critics have considered his teaching style defective, with one claiming that "...many reviews of Guthrie in the literature have mistaken incompleteness for simplicity".
Early life and education
Guthrie was born in Lincoln, Nebraska, to a father who owned a store selling pianos and bicycles, and a mother who was a school teacher. He remarked that his theories got an early start when he and a friend read Darwin's Origin of Species and The Expression of the Emotions in Man and Animals while they were both in eighth grade. Guthrie graduated at the age of 17 after writing a senior thesis that argued: "that both science and religion, being dependent on words, and words being symbols dependent for their meanings on the experience of their users and auditors, would have no chance at expressing Absolute Truth". Guthrie received the title of lay reader in his local Episcopal Church while pursuing a philosophy degree from the University of Nebraska. This university he credited with helping him pursue his varied interests because "the university had none of the present requirements of required courses and set curricula ... This freedom made possible the inclusion of courses in both Latin and Greek which had been begun in high school; mathematics through calculus."
Psychology interest
While Guthrie was going to graduate school he was the only student in a seminary taught by Wilhelm Wundt’s protégé Harry Kirke Wolfe, where they debated the philosophy of science. Guthrie later characterized the classes that he took for his degree as philosophy courses that "took much interest in issues that would now be recognized as psychological". His focus upon a theoretical approach to psychology as opposed to an experimental research approach can be found in his account of his single experimental psychology course which he described as "a research course under Bolton devoted a winter to observations with an aesthesiometer on the limen of twoness, and served to quench [my] interest in psychophysics, which was the chief preoccupation of psychological laboratories then".
His professional psychology career did not start in full until he met Stevenson Smith, who
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https://en.wikipedia.org/wiki/AUCTeX
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AUCTeX is an extensible package for writing and formatting TeX files in Emacs and XEmacs.
AUCTeX provides syntax highlighting, smart indentation and formatting, previews of mathematics and other elements directly in the editing buffer, smart folding of syntactical elements, macro and environment completion. It also supports the self-documenting .dtx format from the LaTeX project and, to a limited extent, ConTeXt and plain TeX.
AUCTeX, originating from the ‘tex-mode.el’ package of Emacs 16, was created by students from Aalborg University Center (now Aalborg University), hence the name AUCTeX. Lars Peter Fischer wrote the first functions to insert font macros and Danish characters back in 1986. Per Abrahamsen wrote the functions to insert environments and sections, and to indent the text, as well as the outline minor mode in 1987. Kresten Krab Thorup wrote the buffer handling and debugging functions, the macro completion, and much more, including much improved indentation and text formatting functions, and made the first public release of AUCTeX in 1991.
AUCTeX is distributed under the GNU General Public License.
See also
RefTeX
Comparison of TeX editors
References
External links
Official homepage
GNU Project software
Emacs
Free TeX editors
Free software programmed in Lisp
Aalborg University
TeX software for macOS
TeX software for Windows
Linux TeX software
TeX editors
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https://en.wikipedia.org/wiki/Positive%20energy%20theorem
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The positive energy theorem (also known as the positive mass theorem) refers to a collection of foundational results in general relativity and differential geometry. Its standard form, broadly speaking, asserts that the gravitational energy of an isolated system is nonnegative, and can only be zero when the system has no gravitating objects. Although these statements are often thought of as being primarily physical in nature, they can be formalized as mathematical theorems which can be proven using techniques of differential geometry, partial differential equations, and geometric measure theory.
Richard Schoen and Shing-Tung Yau, in 1979 and 1981, were the first to give proofs of the positive mass theorem. Edward Witten, in 1982, gave the outlines of an alternative proof, which were later filled in rigorously by mathematicians. Witten and Yau were awarded the Fields medal in mathematics in part for their work on this topic.
An imprecise formulation of the Schoen-Yau / Witten positive energy theorem states the following:
The meaning of these terms is discussed below. There are alternative and non-equivalent formulations for different notions of energy-momentum and for different classes of initial data sets. Not all of these formulations have been rigorously proven, and it is currently an open problem whether the above formulation holds for initial data sets of arbitrary dimension.
Historical overview
The original proof of the theorem for ADM mass was provided by Richard Schoen and Shing-Tung Yau in 1979 using variational methods and minimal surfaces. Edward Witten gave another proof in 1981 based on the use of spinors, inspired by positive energy theorems in the context of supergravity. An extension of the theorem for the Bondi mass was given by Ludvigsen and James Vickers, Gary Horowitz and Malcolm Perry, and Schoen and Yau.
Gary Gibbons, Stephen Hawking, Horowitz and Perry proved extensions of the theorem to asymptotically anti-de Sitter spacetimes and to Einstein–Maxwell theory. The mass of an asymptotically anti-de Sitter spacetime is non-negative and only equal to zero for anti-de Sitter spacetime. In Einstein–Maxwell theory, for a spacetime with electric charge and magnetic charge , the mass of the spacetime satisfies (in Gaussian units)
with equality for the Majumdar–Papapetrou extremal black hole solutions.
Initial data sets
An initial data set consists of a Riemannian manifold and a symmetric 2-tensor field on . One says that an initial data set :
is time-symmetric if is zero
is maximal if
satisfies the dominant energy condition if
where denotes the scalar curvature of .
Note that a time-symmetric initial data set satisfies the dominant energy condition if and only if the scalar curvature of is nonnegative. One says that a Lorentzian manifold is a development of an initial data set if there is a (necessarily spacelike) hypersurface embedding of into , together with a continuous unit normal vector field, such tha
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https://en.wikipedia.org/wiki/Artin%20reciprocity%20law
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The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.
Statement
Let be a Galois extension of global fields and stand for the idèle class group
of . One of the statements of the Artin reciprocity law is that there is a canonical isomorphism called the global symbol map
where denotes the abelianization of a group. The map is defined by assembling the maps called the local Artin symbol, the local reciprocity map or the norm residue symbol
for different places of . More precisely, is given by the local maps on the -component of an idèle class. The maps are isomorphisms. This is the content of the local reciprocity law, a main theorem of local class field theory.
Proof
A cohomological proof of the global reciprocity law can be achieved by first establishing that
constitutes a class formation in the sense of Artin and Tate. Then one proves that
where denote the Tate cohomology groups. Working out the cohomology groups establishes that is an isomorphism.
Significance
Artin's reciprocity law implies a description of the abelianization of the absolute Galois group of a global field K which is based on the Hasse local–global principle and the use of the Frobenius elements. Together with the Takagi existence theorem, it is used to describe the abelian extensions of K in terms of the arithmetic of K and to understand the behavior of the nonarchimedean places in them. Therefore, the Artin reciprocity law can be interpreted as one of the main theorems of global class field theory. It can be used to prove that Artin L-functions are meromorphic, and also to prove the Chebotarev density theorem.
Two years after the publication of his general reciprocity law in 1927, Artin rediscovered the transfer homomorphism of I. Schur and used the reciprocity law to translate the principalization problem for ideal classes of algebraic number fields into the group theoretic task of determining the kernels of transfers of finite non-abelian groups.
Finite extensions of global fields
The definition of the Artin map for a finite abelian extension L/K of global fields (such as a finite abelian extension of ) has a concrete description in terms of prime ideals and Frobenius elements.
If is a prime of K then the decomposition groups of primes above are equal in Gal(L/K) since the latter group is abelian. If is unramified in L, then the decomposition group is canonically isomorphic to the Galois group of the extension of residue fields over . There is therefore a canonically defin
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https://en.wikipedia.org/wiki/Rational%20sieve
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In mathematics, the rational sieve is a general algorithm for factoring integers into prime factors. It is a special case of the general number field sieve. While it is less efficient than the general algorithm, it is conceptually simpler. It serves as a helpful first step in understanding how the general number field sieve works.
Method
Suppose we are trying to factor the composite number n. We choose a bound B, and identify the factor base (which we will call P), the set of all primes less than or equal to B. Next, we search for positive integers z such that both z and z+n are B-smooth — i.e. all of their prime factors are in P. We can therefore write, for suitable exponents ,
and likewise, for suitable , we have
.
But and are congruent modulo , and so each such integer z that we find yields a multiplicative relation (mod n) among the elements of P, i.e.
(where the ai and bi are nonnegative integers.)
When we have generated enough of these relations (it's generally sufficient that the number of relations be a few more than the size of P), we can use the methods of linear algebra to multiply together these various relations in such a way that the exponents of the primes are all even. This will give us a congruence of squares of the form a2≡b2 (mod n), which can be turned into a factorization of n = gcd(a-b,n)×gcd(a+b,n). This factorization might turn out to be trivial (i.e. n=n×1), in which case we have to try again with a different combination of relations; but with luck we will get a nontrivial pair of factors of n, and the algorithm will terminate.
Example
We will factor the integer n = 187 using the rational sieve. We'll arbitrarily try the value B=7, giving the factor base P = {2,3,5,7}. The first step is to test n for divisibility by each of the members of P; clearly if n is divisible by one of these primes, then we are finished already. However, 187 is not divisible by 2, 3, 5, or 7. Next, we search for suitable values of z; the first few are 2, 5, 9, and 56. The four suitable values of z give four multiplicative relations (mod 187):
There are now several essentially different ways to combine these and end up with even exponents. For example,
()×(): After multiplying these and canceling out the common factor of 7 (which we can do since 7, being a member of P, has already been determined to be coprime with n), this reduces to 24 ≡ 38 (mod n), or 42 ≡ 812 (mod n). The resulting factorization is 187 = gcd(81-4,187) × gcd(81+4,187) = 11×17.
Alternatively, equation () is in the proper form already:
(): This says 32 ≡ 142 (mod n), which gives the factorization 187 = gcd(14-3,187) × gcd(14+3,187) = 11×17.
Limitations of the algorithm
The rational sieve, like the general number field sieve, cannot factor numbers of the form pm, where p is a prime and m is an integer. This is not a huge problem, though—such numbers are statistically rare, and moreover there is a simple and fast process to check whether a given number is of this
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https://en.wikipedia.org/wiki/Peetre%27s%20inequality
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In mathematics, Peetre's inequality, named after Jaak Peetre, says that for any real number and any vectors and in the following inequality holds:
The inequality was proved by J. Peetre in 1959 and has founds applications in functional analysis and Sobolev spaces.
See also
References
.
.
.
External links
Planetmath.org: Peetre's inequality
Functional analysis
Inequalities
Linear algebra
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https://en.wikipedia.org/wiki/Rolling%20ball%20argument
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In topology, quantum mechanics and geometrodynamics, rolling-ball arguments are used to describe how the perceived geometry and connectedness of a surface can be scale-dependent.
If a researcher probes the shape of an intricately curved surface by rolling a ball across it, then features that are continually curved but whose curvature radius is smaller than the ball radius may appear in the ball's description of the geometry as abrupt points, barriers and singularities.
Scale-dependent topology
If the surface being probed contains connections whose scale is smaller than the ball diameter, then these connections may not appear in the ball's map. If the surface contains a wormhole whose throat narrows to slightly less than the ball's diameter, the ball may be able to enter and explore each wormhole mouth, but will not be able to pass through the throat, and will produce a map in which the narrowing mouth walls each terminate in a sharp geometrical spike.
The smooth and multiply connected surface will be mapped by the physics of a "large" particle as being singly connected and including geometrical singularities.
Topology change without topology change
If the surface being explored is flexible or elastic, the way the ball is used may affect the reported topology. If the ball is forced into a wormhole mouth that is slightly too small, and the ball and/or throat distorts to allow the ball through, then in the ball's description of the surface, a "new" wormhole connection has suddenly appeared and disappeared again, and the connectivity of the surface has fluctuated unexpectedly.
In this case, no real geometry-change occurs in the deduced shape of the underlying metric – the process identified and "caught" a wormhole candidate (getting the ball wedged in the throat), then modified the curvature of the metric over time, forcing the throat to inflate to dimensions that allowed it to be traversed.
Quantum foam
In John Wheeler's geometrodynamic description of quantum mechanics, the small-scale structure of spacetime is described as a quantum foam whose connectivities are not obvious part in large-scale physics, but whose behaviours become more apparent as we probe the surface at progressively smaller scales.
In wormhole theory, the idea of this "quantum foam" is sometimes invoked as a possible way of achieving large-scale wormholes without geometry change – instead of creating a wormhole from scratch, it may be theoretically possible to pluck an existing wormhole connection from the quantum foam and inflate it to a useful size.
See also
Fractals
Wormholes
John Wheeler
Pregeometry
Wormhole theory
Quantum gravity
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https://en.wikipedia.org/wiki/Quadratic%20differential
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In mathematics, a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle. If the section is holomorphic, then the quadratic differential is said to be holomorphic. The vector space of holomorphic quadratic differentials on a Riemann surface has a natural interpretation as the cotangent space to the Riemann moduli space, or Teichmüller space.
Local form
Each quadratic differential on a domain in the complex plane may be written as , where is the complex variable, and is a complex-valued function on .
Such a "local" quadratic differential is holomorphic if and only if is holomorphic. Given a chart for a general Riemann surface and a quadratic differential on , the pull-back defines a quadratic differential on a domain in the complex plane.
Relation to abelian differentials
If is an abelian differential on a Riemann surface, then is a quadratic differential.
Singular Euclidean structure
A holomorphic quadratic differential determines a Riemannian metric on the complement of its zeroes. If is defined on a domain in the complex plane, and , then the associated Riemannian metric is , where . Since is holomorphic, the curvature of this metric is zero. Thus, a holomorphic quadratic differential defines a flat metric on the complement of the set of such that .
References
Kurt Strebel, Quadratic differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 5. Springer-Verlag, Berlin, 1984. xii + 184 pp. .
Y. Imayoshi and M. Taniguchi, M. An introduction to Teichmüller spaces. Translated and revised from the Japanese version by the authors. Springer-Verlag, Tokyo, 1992. xiv + 279 pp. .
Frederick P. Gardiner, Teichmüller Theory and Quadratic Differentials. Wiley-Interscience, New York, 1987. xvii + 236 pp. .
Complex manifolds
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https://en.wikipedia.org/wiki/Patos%20de%20Minas
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Patos de Minas is a municipality in the state of Minas Gerais in Brazil.
Geography
According to the modern (2017) geographic classification by Brazil's National Institute of Geography and Statistics (IBGE), the city is the main municipality in the Intermediate Geographic Region of Patos de Minas.
History
The name is derived from the ranch known as farmhouse Os Patos, owned by the original settlers, which had the name because of the large number of wild ducks found in the region. In 1826 the first settlers, Antônio da Silva Guerra and his wife, Luísa Correia de Andrade, donated lands for the new settlement, called Vila de Santo Antônio dos Patos. In 1892 Patos de Minas gained city status with the name of Patos, which was lengthened in 1944 to Patos de Minas.
High standard of living
Patos de Minas occupies a privileged position in the ranking of cities in the state and country in socio-economic development. A study in Veja magazine in 2001 placed Patos in fifth place among five thousand cities with the greatest socio-economic development between 1970 and 1996.
Rare among Brazilian cities of this size, around 99% of all the streets of the city are paved and have public lighting and almost all of the inhabitants (97%) receive treated water. Sewage treatment reached 98% of the urban population. City government site
Municipal Human Development Index: 0.813 (2000)
State ranking: 19 out of 853 municipalities as of 2000
National ranking: 318 out of 5,138 municipalities as of 2000
Literacy rate: 92%
Life expectancy: 74 (average of males and females)
The highest ranking municipality in Minas Gerais in 2000 was Poços de Caldas with 0.841, while the lowest was Setubinha with 0.568. Nationally the highest was São Caetano do Sul in São Paulo with 0.919, while the lowest was Setubinha. In more recent statistics (considering 5,507 municipalities) Manari in the state of Pernambuco has the lowest rating in the country—0,467—putting it in last place.
Economy
The main sources of income are agriculture, services, light industry, and livestock raising. Dairy products, jerked beef, and pork by-products are processed in the city. In 2005 the GDP was R$1,217 billion, with 750 million generated by services, 189 million by industry, and 154 million by agriculture.
Patos is known nationally for its corn production. The "Princess of the Alto Paranaíba" as it is called, has become the national capital of corn, a fact that has attracted investors from all over the country to set up industries in agro-industry, garments, beverages, and packaging. It is also one of the most important seed producing centers in the country.
Corn is so important in the life of the town that every May Patos puts on a festival, the biggest in the state, called Fenamilho, the Festa Nacional do Milho (National Corn Festival), in which there are rodeos and performances by country and western groups.
Main crops in 2006
Coffee: 5,700 ha.
Cotton: 310 ha.
Potatoes: 130 ha.
Sugarcane: 170
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https://en.wikipedia.org/wiki/Gilbert%20Strang
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William Gilbert Strang (born November 27, 1934) is an American mathematician known for his contributions to finite element theory, the calculus of variations, wavelet analysis and linear algebra. He has made many contributions to mathematics education, including publishing mathematics textbooks. Strang was the MathWorks Professor of Mathematics at the Massachusetts Institute of Technology. He taught Linear Algebra, Computational Science, and Engineering, Learning from Data, and his lectures are freely available through MIT OpenCourseWare.
Biography
Strang was born in Chicago in 1934. His parents William and Mary Catherine Strang migrated to the USA from Scotland. He and his sister Vivian grew up in Washington DC and Cincinnati, and went to high school at Principia in St. Louis.
Strang graduated from MIT in 1955 with a Bachelor of Science in mathematics. He then received a Rhodes Scholarship to University of Oxford, where he received his B.A. and M.A. from Balliol College in 1957.
Strang earned his Ph.D. from University of California, Los Angeles in 1959 as a National Science Foundation Fellow, under the supervision of Peter K. Henrici. His dissertation was titled "Difference Methods for Mixed Boundary Value Problems".
While at Oxford, Strang met his future wife Jillian Shannon, and they married in 1958. Following his Ph.D. at UCLA, they have lived in Wellesley, Massachusetts for almost all of his 62 years on the MIT faculty. The Strangs have three sons David, John, and Robert and describe themselves as a very close-knit family. He retired on May 15, 2023 after giving his final Linear Algebra and Learning from Data lecture at MIT.
Strang's teaching has focused on linear algebra which has helped the subject become essential for students of many majors. His linear algebra video lectures are popular on YouTube and MIT OpenCourseware. Strang founded Wellesley-Cambridge Press to publish Introduction to Linear Algebra (now in 6th edition) and ten other books.
University Positions
Following his PhD studies, from 1959 to 1961, Strang was a C. L. E. Moore instructor at M.I.T. in the Mathematics department. From 1961-1962 he was a NATO Postdoctoral Fellow at Oxford University. From 1962 until 2023, Strang was a mathematics professor at MIT.
He has received Honorary Titles and Fellowships from the following institutes:
Alfred P. Sloan Fellow (1966–1967)
Honorary Professor, Xi'an Jiaotong University, China (1980)
Honorary Fellow, Balliol College, Oxford University (1999)
Honorary Member, Irish Mathematical Society (2002)
Fellow of the Society for Industrial and Applied Mathematics (2009)
Doctor Honoris Causa, University of Toulouse (2010)
Fellow of the American Mathematical Society (2012)
Doctor Honoris Causa, Aalborg University (2013)
Awards
Rhodes Scholar (1955)
National Science Foundation Graduate Research Fellowship (1957)
Chauvenet Prize, Mathematical Association of America (1977)
American Academy of Arts and Sciences (1985)
Aw
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https://en.wikipedia.org/wiki/Cylinder%20set
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In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra.
General definition
Given a collection of sets, consider the Cartesian product of all sets in the collection. The canonical projection corresponding to some is the function that maps every element of the product to its component. A cylinder set is a preimage of a canonical projection or finite intersection of such preimages. Explicitly, it is a set of the form,
for any choice of , finite sequence of sets and subsets for . Here denotes the component of .
Then, when all sets in are topological spaces, the product topology is generated by cylinder sets corresponding to the components' open sets. That is cylinders of the form where for each , is open in . In the same manner, in case of measurable spaces, the cylinder σ-algebra is the one which is generated by cylinder sets corresponding to the components' measurable sets.
The restriction that the cylinder set be the intersection of a finite number of open cylinders is important; allowing infinite intersections generally results in a finer topology. In the latter case, the resulting topology is the box topology; cylinder sets are never Hilbert cubes.
Cylinder sets in products of discrete sets
Let be a finite set, containing n objects or letters. The collection of all bi-infinite strings in these letters is denoted by
The natural topology on is the discrete topology. Basic open sets in the discrete topology consist of individual letters; thus, the open cylinders of the product topology on are
The intersections of a finite number of open cylinders are the cylinder sets
Cylinder sets are clopen sets. As elements of the topology, cylinder sets are by definition open sets. The complement of an open set is a closed set, but the complement of a cylinder set is a union of cylinders, and so cylinder sets are also closed, and are thus clopen.
Definition for vector spaces
Given a finite or infinite-dimensional vector space over a field K (such as the real or complex numbers), the cylinder sets may be defined as
where is a Borel set in , and each is a linear functional on ; that is, , the algebraic dual space to . When dealing with topological vector spaces, the definition is made instead for elements , the continuous dual space. That is, the functionals are taken to be continuous linear functionals.
Applications
Cylinder sets are often used to define a topology on sets that are subsets of and occur frequently in the study of symbolic dynamics; see, for example, subshift of finite type. Cylinder sets are often used to define a measure, using the Kolmogorov extension theorem; for example, the measure of a cylinder set of length m might be given by or by .
Cylinder sets may be used to define a metric on the space: for example, one says that two strings are ε-close if a fraction 1−ε of the letters in the strings match.
Since stri
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https://en.wikipedia.org/wiki/Steven%20Kerckhoff
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Steven Paul Kerckhoff (born 1952) is a professor of mathematics at Stanford University, who works on hyperbolic 3-manifolds and Teichmüller spaces.
He received his Ph.D. in mathematics from Princeton University in 1978, under the direction of William Thurston. Among his most famous results is his resolution of the Nielsen realization problem, a 1932 conjecture by Jakob Nielsen. Along with William J. Floyd, he wrote large parts of Thurston's influential Princeton lecture notes, and he is well known for his work (some of which is joint with Craig Hodgson) in exploring and clarifying Thurston's hyperbolic Dehn surgery.
Kerckhoff is one of four academics from Stanford University, along with Gunnar Carlsson, Ralph Cohen, and R. James Milgram, who were instrumental in developing the controversial California Mathematics Academic Content Standards for the State Board of Education.
Selected publications
Kerckhoff, Steven P.; Thurston, William P., Noncontinuity of the action of the modular group at Bers' boundary of Teichmüller space. Inventiones Mathematicae 100 (1990), no. 1, 25–47.
Kerckhoff, Steven; Masur, Howard; Smillie, John, Ergodicity of billiard flows and quadratic differentials. Annals of Mathematics (2) 124 (1986), no. 2, 293–311.
Cooper, Daryl; Hodgson, Craig D.; Kerckhoff, Steven P. Three-dimensional orbifolds and cone-manifolds. With a postface by Sadayoshi Kojima. MSJ Memoirs, 5. Mathematical Society of Japan, Tokyo, 2000. x+170 pp.
Hodgson, Craig D.; Kerckhoff, Steven P., Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery. Journal of Differential Geometry 48 (1998), no. 1, 1–59.
References
Big Business, Race, and Gender in Mathematics Reform, by Steven Krantz ().
Excerpt from this book.
Letter from Kerckhoff, Wayne Bishop, Jane Friedman, and Yat-Sun Poon, to California State Curriculum Commission and California State Board of Education, dated November 21, 2000
External links
Topologists
Geometers
Princeton University alumni
Stanford University Department of Mathematics faculty
1952 births
Living people
20th-century American mathematicians
21st-century American mathematicians
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https://en.wikipedia.org/wiki/Commensurability
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Two concepts or things are commensurable if they are measurable or comparable by a common standard.
Commensurability most commonly refers to commensurability (mathematics). It may also refer to:
Commensurability (astronomy), whether two orbital periods are mathematically commensurate.
Commensurability (crystal structure), whether periodic material properties repeat over a distance that is mathematically commensurate with the length of the unit cell.
Commensurability (economics), whether economic value can always be measured by money
Commensurability (ethics), the commensurability of values in ethics
Commensurability (group theory), when two groups have a subgroup of finite index in common
Commensurability (philosophy of science)
Commensurability (physics), a concept in dimensional analysis that concerns conversion of units of measurement
Apples and oranges, common idiom related to incommensurability
it:Incommensurabilità
simple:Incommensurability
sv:Inkommensurabilitet
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https://en.wikipedia.org/wiki/Omega%20network
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An Omega network is a network configuration often used in parallel computing architectures. It is an indirect topology that relies on the perfect shuffle interconnection algorithm.
Connection architecture
An 8x8 Omega network is a multistage interconnection network, meaning that processing elements (PEs) are connected using multiple stages of switches. Inputs and outputs are given addresses as shown in the figure. The outputs from each stage are connected to the inputs of the next stage using a perfect shuffle connection system. This means that the connections at each stage represent the movement of a deck of cards divided into 2 equal decks and then shuffled together, with each card from one deck alternating with the corresponding card from the other deck. In terms of binary representation of the PEs, each stage of the perfect shuffle can be thought of as a cyclic logical left shift; each bit in the address is shifted once to the left, with the most significant bit moving to the least significant bit.
At each stage, adjacent pairs of inputs are connected to a simple exchange element, which can be set either straight (pass inputs directly through to outputs) or crossed (send top input to bottom output, and vice versa). For N processing element, an Omega network contains N/2 switches at each stage, and log2N stages. The manner in which these switches are set determines the connection paths available in the network at any given time. Two such methods are destination-tag routing and XOR-tag routing, discussed in detail below.
The Omega Network is highly blocking, though one path can always be made from any input to any output in a free network.
Destination-tag routing
In destination-tag routing, switch settings are determined solely by the message destination. The most significant bit of the destination address is used to select the output of the switch in the first stage; if the most significant bit is 0, the upper output is selected, and if it is 1, the lower output is selected. The next-most significant bit of the destination address is used to select the output of the switch in the next stage, and so on until the final output has been selected.
For example, if a message's destination is PE 001, the switch settings are: upper, upper, lower. If a message's destination is PE 101, the switch settings are: lower, upper, lower. These switch settings hold regardless of the PE sending the message.
XOR-tag routing
In XOR-tag routing, switch settings are based on (source PE) XOR (destination PE). This XOR-tag contains 1s in the bit positions that must be swapped and 0s in the bit positions that both source and destination have in common. The most significant bit of the XOR-tag is used to select the setting of the switch in the first stage; if the most significant bit is 0, the switch is set to pass-through, and if it is 1, the switch is crossed. The next-most significant bit of the tag is used to set the switch in the next stage, and so on until the
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https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Woods%20number
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In number theory, a positive integer is said to be an Erdős–Woods number if it has the following property:
there exists a positive integer such that in the sequence of consecutive integers, each of the elements has a non-trivial common factor with one of the endpoints. In other words, is an Erdős–Woods number if there exists a positive integer such that for each integer between and , at least one of the greatest common divisors or is greater than .
Examples
The first Erdős–Woods numbers are
16, 22, 34, 36, 46, 56, 64, 66, 70, 76, 78, 86, 88, 92, 94, 96, 100, 106, 112, 116 … .
History
Investigation of such numbers stemmed from the following prior conjecture by Paul Erdős:
There exists a positive integer such that every integer is uniquely determined by the list of prime divisors of .
Alan R. Woods investigated this question for his 1981 thesis. Woods conjectured that whenever , the interval always includes a number coprime to both endpoints. It was only later that he found the first counterexample, , with . The existence of this counterexample shows that 16 is an Erdős–Woods number.
proved that there are infinitely many Erdős–Woods numbers, and showed that the set of Erdős–Woods numbers is recursive.
References
External links
Woods number
Integer sequences
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https://en.wikipedia.org/wiki/MSU%20Faculty%20of%20Computational%20Mathematics%20and%20Cybernetics
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MSU Faculty of Computational Mathematics and Cybernetics (CMC) (), founded in 1970 by Andrey Tikhonov, is a part of Moscow State University.
Education
CMC is a Russian research and training center in the fields of applied mathematics, computing and software development . Education at CMC combines theoretical studies, practical exercises, and research.
Main 12 Master's programs:
Mathematical physics
Mathematical modeling
Computational diagnostics
Numerical methods
Theory of probability and mathematical statistics
Operations research and systems analysis
Optimization and optimal control
Mathematical cybernetics
Software for computers and computer systems
Networks software
System programming
Decision making in Economics and Finance
History
A group of professors and scholars from Department of Physics and Department of Mechanics and Mathematics led by Andrey Tikhonov founded CMC in 1970. The three departments are still closely connected.
The faculty houses the 33,072-processor Lomonosov supercomputer in Moscow. The system was designed by T-Platforms, and used Xeon 2.93 GHz processors, Nvidia 2070 GPUs, and an Infiniband interconnect.
Following companies work with CS MSU: Intel, Microsoft, Sun Microsystems, Borland, Software AG, Siemens, IBM/Lotus, Samsung, HP.
Following the school's support for the 2022 Russian invasion of Ukraine, Intel and AMD, the largest chip manufacturers in the world, whose processors are used in the Moscow State University supercomputer, as well as NVIDIA, reacted by suspending deliveries of their processors to Russia.
Deans
The deans of the faculty:
Andrey Tikhonov (1970–1990)
Dmitrij Kostomarov (1991–1999)
Evgeny Moiseev (1999-2019)
Igor Sokolov (since March 2019)
Structure
Departments
The faculty consists of 19 Academic departments:
Scientific laboratories
The faculty includes 18 research laboratories:
Laboratory of Mathematical Physics
Laboratory of Computational Electrodynamics
Laboratory of Heat and Mass Transfer Processes Simulation
Laboratory of Inverse Problems
Laboratory of Mathematical Methods of Image Processing
Laboratory of Mathematical Modeling in Physics
Laboratory of Difference Methods
Open Laboratory of Information Technologies
Laboratory of Statistical Analysis
Laboratory of Mathematical Problems of Computer Security
Laboratory of Computational Practice and Information Systems
The Computer Systems Laboratory
Laboratory of Information Systems Security
Computer Graphics and Multimedia Laboratory
Laboratory of Programming Technologies
Laboratory of Ternary Informatics
Research Laboratory of Computational Modeling Tools
Laboratory of Industrial Mathematics
Рrofessors
Faculty staff consists of more than 550 professors and research scientists. The list of scientists that worked in the Faculty of Computational Mathematics and Cybernetics includes:
Lev Pontryagin, the founder and the first chair of the department of Optimal Control.
Sergey Yablonsky, the founder an
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https://en.wikipedia.org/wiki/Faithful%20representation
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In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group on a vector space is a linear representation in which different elements of are represented by distinct linear mappings .
In more abstract language, this means that the group homomorphism is injective (or one-to-one).
Caveat
While representations of over a field are de facto the same as -modules (with denoting the group algebra of the group ), a faithful representation of is not necessarily a faithful module for the group algebra. In fact each faithful -module is a faithful representation of , but the converse does not hold. Consider for example the natural representation of the symmetric group in dimensions by permutation matrices, which is certainly faithful. Here the order of the group is while the matrices form a vector space of dimension . As soon as is at least 4, dimension counting means that some linear dependence must occur between permutation matrices (since ); this relation means that the module for the group algebra is not faithful.
Properties
A representation of a finite group over an algebraically closed field of characteristic zero is faithful (as a representation) if and only if every irreducible representation of occurs as a subrepresentation of (the -th symmetric power of the representation ) for a sufficiently high . Also, is faithful (as a representation) if and only if every irreducible representation of occurs as a subrepresentation of
(the -th tensor power of the representation ) for a sufficiently high .
References
Representation theory
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https://en.wikipedia.org/wiki/William%20Newton-Smith
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William Herbert Newton-Smith (May 25, 1943 – April 8, 2023) was a Canadian philosopher of science.
Biography
Newton-Smith's undergraduate degree from Queen's University was in Mathematics and Philosophy, in 1966. He took an MA from Cornell University in Philosophy, in 1968, and a DPhil in philosophy from Balliol College, Oxford, in 1974. His working life before retirement was mainly as a Fellow of Balliol.
Newton-Smith's 1980 book The Structure of Time is on the philosophy of time.
Newton-Smith led Central European University from its foundation in 1991 until Alfred Stepan was elected rector in 1993. In the 1980s he led a small team of British philosophers, including Kathy Wilkes and Roger Scruton, who travelled to Czechoslovakia to give unauthorized philosophy lectures.
Newton-Smith had two daughters with his first wife Dorris Heffron. His daughter Rain Newton-Smith is an economist who became the Director General of the Confederation of British Industry (CBI) in April 2023.
In 2003, Newton-Smith and his second wife Nancy Durham were the first to grow lavender on a field scale in Wales. They became the sole distillers of lavender oil in Wales. Their company, Welsh Lavender Ltd, produces face and body creams.
Newton-Smith died of throat cancer on April 8, 2023, at the age of 79.
Works
The Structure of Time (1980)
The Rationality of Science (1981)
Logic (1984)
Modelling the Mind (1990) editor with K. V. Wilkes
Popper in China (1992) editor with J. Tianji
Chapter 1 - Popper, ciência e racionalidade. In Karl Popper: Filosofia e problemas (1997), organized by Anthony O'Hear, translated to Portuguese by Luiz Paulo Rouanet. Editora Unesp. Cambridge University Press.
The Companion to the Philosophy of Science (2000)
References
External links
Welsh Lavender
1943 births
2023 deaths
Central European University
Cornell University alumni
Fellows of Balliol College, Oxford
Philosophers of science
Philosophers of time
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https://en.wikipedia.org/wiki/Gravitational%20instanton
|
In mathematical physics and differential geometry, a gravitational instanton is a four-dimensional complete Riemannian manifold satisfying the vacuum Einstein equations. They are so named because they are analogues in quantum theories of gravity of instantons in Yang–Mills theory. In accordance with this analogy with self-dual Yang–Mills instantons, gravitational instantons are usually assumed to look like four dimensional Euclidean space at large distances, and to have a self-dual Riemann tensor. Mathematically, this means that they are asymptotically locally Euclidean (or perhaps asymptotically locally flat) hyperkähler 4-manifolds, and in this sense, they are special examples of Einstein manifolds. From a physical point of view, a gravitational instanton is a non-singular solution of the vacuum Einstein equations with positive-definite, as opposed to Lorentzian, metric.
There are many possible generalizations of the original conception of a gravitational instanton: for example one can allow gravitational instantons to have a nonzero cosmological constant or a Riemann tensor which is not self-dual. One can also relax the boundary condition that the metric is asymptotically Euclidean.
There are many methods for constructing gravitational instantons, including the Gibbons–Hawking Ansatz, twistor theory, and the hyperkähler quotient construction.
Introduction
Gravitational instantons are interesting, as they offer insights into the quantization of gravity. For example, positive definite asymptotically locally Euclidean metrics are needed as they obey the positive-action conjecture; actions that are unbounded below create divergence in the quantum path integral.
A four-dimensional Kähler–Einstein manifold has a self-dual Riemann tensor.
Equivalently, a self-dual gravitational instanton is a four-dimensional complete hyperkähler manifold.
Gravitational instantons are analogous to self-dual Yang–Mills instantons.
Several distinctions can be made with respect to the structure of the Riemann curvature tensor, pertaining to flatness and self-duality. These include:
Einstein (non-zero cosmological constant)
Ricci flatness (vanishing Ricci tensor)
Conformal flatness (vanishing Weyl tensor)
Self-duality
Anti-self-duality
Conformally self-dual
Conformally anti-self-dual
Taxonomy
By specifying the 'boundary conditions', i.e. the asymptotics of the metric 'at infinity' on a noncompact Riemannian manifold, gravitational instantons are divided into a few classes, such as asymptotically locally Euclidean spaces (ALE spaces), asymptotically locally flat spaces (ALF spaces).
They can be further characterized by whether the Riemann tensor is self-dual, whether the Weyl tensor is self-dual, or neither; whether or not they are Kahler manifolds; and various characteristic classes, such as Euler characteristic, the Hirzebruch signature (Pontryagin class), the Rarita–Schwinger index (spin-3/2 index), or generally the Chern class. The ability to suppor
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https://en.wikipedia.org/wiki/Conformal%20Killing%20vector%20field
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In conformal geometry, a conformal Killing vector field on a manifold of dimension n with (pseudo) Riemannian metric (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field whose (locally defined) flow defines conformal transformations, that is, preserve up to scale and preserve the conformal structure. Several equivalent formulations, called the conformal Killing equation, exist in terms of the Lie derivative of the flow e.g.
for some function on the manifold. For there are a finite number of solutions, specifying the conformal symmetry of that space, but in two dimensions, there is an infinity of solutions. The name Killing refers to Wilhelm Killing, who first investigated Killing vector fields.
Densitized metric tensor and Conformal Killing vectors
A vector field is a Killing vector field if and only if its flow preserves the metric tensor (strictly speaking for each compact subsets of the manifold, the flow need only be defined for finite time). Formulated mathematically, is Killing if and only if it satisfies
where is the Lie derivative.
More generally, define a w-Killing vector field as a vector field whose (local) flow preserves the densitized metric , where is the volume density defined by (i.e. locally ) and is its weight. Note that a Killing vector field preserves and so automatically also satisfies this more general equation. Also note that is the unique weight that makes the combination invariant under scaling of the metric. Therefore, in this case, the condition depends only on the conformal structure.
Now is a w-Killing vector field if and only if
Since this is equivalent to
Taking traces of both sides, we conclude . Hence for , necessarily and a w-Killing vector field is just a normal Killing vector field whose flow preserves the metric. However, for , the flow of has to only preserve the conformal structure and is, by definition, a conformal Killing vector field.
Equivalent formulations
The following are equivalent
is a conformal Killing vector field,
The (locally defined) flow of preserves the conformal structure,
for some function
The discussion above proves the equivalence of all but the seemingly more general last form.
However, the last two forms are also equivalent: taking traces shows that necessarily .
The last form makes it clear that any Killing vector is also a conformal Killing vector, with
The conformal Killing equation
Using that where is the Levi Civita derivative of (aka covariant derivative), and is the dual 1 form of (aka associated covariant vector aka vector with lowered indices), and is projection on the symmetric part, one can write the conformal Killing equation in abstract index notation as
Another index notation to write the conformal Killing equations is
Examples
Flat space
In -dimensional flat space, that is Euclidean space or pseudo-Euclidean space, there exist globally flat coordinates in which we have a c
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https://en.wikipedia.org/wiki/Coordinate-measuring%20machine
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A coordinate measuring machine (CMM) is a device that measures the geometry of physical objects by sensing discrete points on the surface of the object with a probe. Various types of probes are used in CMMs, the most common being mechanical and laser sensors, though optical and white light sensor do exist. Depending on the machine, the probe position may be manually controlled by an operator or it may be computer controlled. CMMs typically specify a probe's position in terms of its displacement from a reference position in a three-dimensional Cartesian coordinate system (i.e., with XYZ axes). In addition to moving the probe along the X, Y, and Z axes, many machines also allow the probe angle to be controlled to allow measurement of surfaces that would otherwise be unreachable.
Description
The typical 3D "bridge" CMM allows probe movement along three axes, X, Y and Z, which are orthogonal to each other in a three-dimensional Cartesian coordinate system. Each axis has a sensor that monitors the position of the probe on that axis, with typical accuracy in the order of microns. When the probe contacts (or otherwise detects) a particular location on the object, the machine samples the axis position sensors, thus measuring the location of one point on the object's surface, as well as the 3-dimensional vector of the measurement taken. This process is repeated as necessary, moving the probe each time, to produce a "point cloud" which describes the surface areas of interest. The points can be measured either manually by an operator or automatically via Direct Computer Control (DCC) or automatically using scripted programs; thus, an automated CMM is a specialized form of industrial robot.
A common use of CMMs is in manufacturing and assembly processes to test a part or assembly against the design intent. The measured points can be used to verify the distance between features. They can also be used to construct geometric features such as cylinders and planes etc. for GD&T such as roundness, flatness and perpendicularity can be assessed.
Technical facts
Parts
Coordinate-measuring machines include three main components:
The main structure includes three axes of motion. The material used to construct the moving frame has varied over the years. Granite and steel were used in the early CMM's. Today all the major CMM manufacturers build frames from Granite, aluminum alloy or some derivative and also use ceramic to increase the stiffness of the Z axis for scanning applications. Few CMM builders today still manufacture granite frame CMM due to market requirement for improved metrology dynamics and increasing trend to install CMM outside of the quality lab. The increasing trend towards scanning also requires the CMM Z axis to be stiffer and new materials have been introduced such as black granite, ceramic and silicon carbide.
Probing system
Data collection and reduction system — typically includes a machine controller, desktop computer and application softwa
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https://en.wikipedia.org/wiki/Flow%20%28mathematics%29
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In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow is a group action of the real numbers on a set.
The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups. Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, the mean curvature flow, and Anosov flows. Flows may also be defined for systems of random variables and stochastic processes, and occur in the study of ergodic dynamical systems. The most celebrated of these is perhaps the Bernoulli flow.
Formal definition
A flow on a set is a group action of the additive group of real numbers on . More explicitly, a flow is a mapping
such that, for all and all real numbers and ,
It is customary to write instead of , so that the equations above can be expressed as (the identity function) and (group law). Then, for all the mapping is a bijection with inverse This follows from the above definition, and the real parameter may be taken as a generalized functional power, as in function iteration.
Flows are usually required to be compatible with structures furnished on the set . In particular, if is equipped with a topology, then is usually required to be continuous. If is equipped with a differentiable structure, then is usually required to be differentiable. In these cases the flow forms a one-parameter group of homeomorphisms and diffeomorphisms, respectively.
In certain situations one might also consider s, which are defined only in some subset
called the of . This is often the case with the flows of vector fields.
Alternative notations
It is very common in many fields, including engineering, physics and the study of differential equations, to use a notation that makes the flow implicit. Thus, is written for and one might say that the variable depends on the time and the initial condition . Examples are given below.
In the case of a flow of a vector field on a smooth manifold , the flow is often denoted in such a way that its generator is made explicit. For example,
Orbits
Given in , the set is called the orbit of under . Informally, it may be regarded as the trajectory of a particle that was initially positioned at . If the flow is generated by a vector field, then its orbits are the images of its integral curves.
Examples
Algebraic equation
Let be a time-dependent trajectory which is a bijective function. Then a flow can be defined by
Autonomous systems of ordinary differential equations
Let be a (time-independent) vector field
and the solution of the initial value problem
Then is the flow of the vector field . It is a well-defined local fl
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https://en.wikipedia.org/wiki/Ernst%20Witt
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Ernst Witt (26 June 1911 – 3 July 1991) was a German mathematician, one of the leading algebraists of his time.
Biography
Witt was born on the island of Alsen, then a part of the German Empire. Shortly after his birth, his parents moved the family to China to work as missionaries, and he did not return to Europe until he was nine.
After his schooling, Witt went to the University of Freiburg and the University of Göttingen. He joined the NSDAP (Nazi Party) and was an active party member. Witt was awarded a Ph.D. at the University of Göttingen in 1934 with a thesis titled: "Riemann-Roch theorem and zeta-Function in hypercomplexes" (Riemann-Rochscher Satz und Zeta-Funktion im Hyperkomplexen) that was supervised by Gustav Herglotz with Emmy Noether suggesting the topic for the doctorate. He qualified to become a lecturer and gave guest lectures in Göttingen and Hamburg. He became associated with the team led by Helmut Hasse who led his habilitation. In June 1936, he gave his habilitation lecture.
During World War II he joined a group of five mathematicians, recruited by Wilhelm Fenner, and which included Georg Aumann, Alexander Aigner, Oswald Teichmüller, Johann Friedrich Schultze and their leader professor Wolfgang Franz, to form the backbone of the new mathematical research department in the late 1930s, which would eventually be called: Section IVc of Cipher Department of the High Command of the Wehrmacht (abbr. OKW/Chi).
From 1937 until 1979, he taught at the University of Hamburg. He died in Hamburg in 1991, shortly after his 80th birthday.
Work
Witt's work has been highly influential. His invention of the Witt vectors clarifies and generalizes the structure of the p-adic numbers. It has become fundamental to p-adic Hodge theory.
Witt was the founder of the theory of quadratic forms over an arbitrary field. He proved several of the key results, including the Witt cancellation theorem. He defined the Witt ring of all quadratic forms over a field, now a central object in the theory.
The Poincaré–Birkhoff–Witt theorem is basic to the study of Lie algebras. In algebraic geometry, the Hasse–Witt matrix of an algebraic curve over a finite field determines the cyclic étale coverings of degree p of a curve in characteristic p.
In the 1970s, Witt claimed that in 1940 he had discovered what would eventually be named the "Leech lattice" many years before John Leech discovered it in 1965, but Witt did not publish his discovery and the details of exactly what he did are unclear.
See also
Leech lattice
Verschiebung operator
Wedderburn's little theorem
List of things named after Ernst Witt
References
Bibliography
External links
20th-century German mathematicians
Algebraists
1911 births
1991 deaths
German cryptographers
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https://en.wikipedia.org/wiki/Vector%20flow
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In mathematics, the vector flow refers to a set of closely related concepts of the flow determined by a vector field. These appear in a number of different contexts, including differential topology, Riemannian geometry and Lie group theory. These related concepts are explored in a spectrum of articles:
exponential map (Riemannian geometry)
matrix exponential
exponential function
infinitesimal generator (→ Lie group)
integral curve (→ vector field)
one-parameter subgroup
flow (geometry)
geodesic flow
Hamiltonian flow
Ricci flow
Anosov flow
injectivity radius (→ glossary)
Vector flow in differential topology
Relevant concepts: (flow, infinitesimal generator, integral curve, complete vector field)
Let V be a smooth vector field on a smooth manifold M. There is a unique maximal flow D → M whose infinitesimal generator is V. Here D ⊆ R × M is the flow domain. For each p ∈ M the map Dp → M is the unique maximal integral curve of V starting at p.
A global flow is one whose flow domain is all of R × M. Global flows define smooth actions of R on M. A vector field is complete if it generates a global flow. Every smooth vector field on a compact manifold without boundary is complete.
Vector flow in Riemannian geometry
Relevant concepts: (geodesic, exponential map, injectivity radius)
The exponential map
exp : TpM → M
is defined as exp(X) = γ(1) where γ : I → M is the unique geodesic passing through p at 0 and whose tangent vector at 0 is X. Here I is the maximal open interval of R for which the geodesic is defined.
Let M be a pseudo-Riemannian manifold (or any manifold with an affine connection) and let p be a point in M. Then for every V in TpM there exists a unique geodesic γ : I → M for which γ(0) = p and Let Dp be the subset of TpM for which 1 lies in I.
Vector flow in Lie group theory
Relevant concepts: (exponential map, infinitesimal generator, one-parameter group)
Every left-invariant vector field on a Lie group is complete. The integral curve starting at the identity is a one-parameter subgroup of G. There are one-to-one correspondences
{one-parameter subgroups of G} ⇔ {left-invariant vector fields on G} ⇔ g = TeG.
Let G be a Lie group and g its Lie algebra. The exponential map is a map exp : g → G given by exp(X) = γ(1) where γ is the integral curve starting at the identity in G generated by X.
The exponential map is smooth.
For a fixed X, the map t exp(tX) is the one-parameter subgroup of G generated by X.
The exponential map restricts to a diffeomorphism from some neighborhood of 0 in g to a neighborhood of e in G.
The image of the exponential map always lies in the connected component of the identity in G.
See also
Geodesic (mathematics)
Differential topology
Lie groups
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https://en.wikipedia.org/wiki/Unconditional%20convergence
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In mathematics, specifically functional analysis, a series is unconditionally convergent if all reorderings of the series converge to the same value. In contrast, a series is conditionally convergent if it converges but different orderings do not all converge to that same value. Unconditional convergence is equivalent to absolute convergence in finite-dimensional vector spaces, but is a weaker property in infinite dimensions.
Definition
Let be a topological vector space. Let be an index set and for all
The series is called unconditionally convergent to if
the indexing set is countable, and
for every permutation (bijection) of the following relation holds:
Alternative definition
Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence with the series
converges.
If is a Banach space, every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general. Indeed, if is an infinite-dimensional Banach space, then by Dvoretzky–Rogers theorem there always exists an unconditionally convergent series in this space that is not absolutely convergent. However when by the Riemann series theorem, the series is unconditionally convergent if and only if it is absolutely convergent.
See also
References
Ch. Heil: A Basis Theory Primer
Convergence (mathematics)
Mathematical analysis
Mathematical series
Summability theory
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https://en.wikipedia.org/wiki/Brocard%27s%20conjecture
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In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between (pn)2 and (pn+1)2, where pn is the nth prime number, for every n ≥ 2. The conjecture is named after Henri Brocard. It is widely believed that this conjecture is true. However, it remains unproven as of 2022.
The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ... .
Legendre's conjecture that there is a prime between consecutive integer squares directly implies that there are at least two primes between prime squares for pn ≥ 3 since pn+1 − pn ≥ 2.
See also
Prime-counting function
Notes
Conjectures about prime numbers
Unsolved problems in number theory
Squares in number theory
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https://en.wikipedia.org/wiki/One-to-many
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One-to-many may refer to:
Fat link, a one-to-many link in hypertext
Multivalued function, a one-to-many function in mathematics
One-to-many (data model), a type of relationship and cardinality in systems analysis
Point-to-multipoint communication, communication which has a one-to-many relationship
See also
Cardinality (data modeling)
Multicast
One Too Many (data modeling)
One-to-one (disambiguation)
Point-to-point (disambiguation)
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https://en.wikipedia.org/wiki/Crystal%20Ball%20function
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The Crystal Ball function, named after the Crystal Ball Collaboration (hence the capitalized initial letters), is a probability density function commonly used to model various lossy processes in high-energy physics. It consists of a Gaussian core portion and a power-law low-end tail, below a certain threshold. The function itself and its first derivative are both continuous.
The Crystal Ball function is given by:
where
,
,
,
,
.
(Skwarnicki 1986) is a normalization factor and , , and are parameters which are fitted with the data. erf is the error function.
External links
J. E. Gaiser, Appendix-F Charmonium Spectroscopy from Radiative Decays of the J/Psi and Psi-Prime, Ph.D. Thesis, SLAC-R-255 (1982). (This is a 205-page document in .pdf form – the function is defined on p. 178.)
M. J. Oreglia, A Study of the Reactions psi prime --> gamma gamma psi, Ph.D. Thesis, SLAC-R-236 (1980), Appendix D.
T. Skwarnicki, A study of the radiative CASCADE transitions between the Upsilon-Prime and Upsilon resonances, Ph.D Thesis, DESY F31-86-02(1986), Appendix E.
Functions and mappings
Continuous distributions
Experimental particle physics
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https://en.wikipedia.org/wiki/Relative%20risk
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The relative risk (RR) or risk ratio is the ratio of the probability of an outcome in an exposed group to the probability of an outcome in an unexposed group. Together with risk difference and odds ratio, relative risk measures the association between the exposure and the outcome.
Statistical use and meaning
Relative risk is used in the statistical analysis of the data of ecological, cohort, medical and intervention studies, to estimate the strength of the association between exposures (treatments or risk factors) and outcomes. Mathematically, it is the incidence rate of the outcome in the exposed group, , divided by the rate of the unexposed group, . As such, it is used to compare the risk of an adverse outcome when receiving a medical treatment versus no treatment (or placebo), or for environmental risk factors. For example, in a study examining the effect of the drug apixaban on the occurrence of thromboembolism, 8.8% of placebo-treated patients experienced the disease, but only 1.7% of patients treated with the drug did, so the relative risk is .19 (1.7/8.8): patients receiving apixaban had 19% the disease risk of patients receiving the placebo. In this case, apixaban is a protective factor rather than a risk factor, because it reduces the risk of disease.
Assuming the causal effect between the exposure and the outcome, values of relative risk can be interpreted as follows:
RR = 1 means that exposure does not affect the outcome
RR < 1 means that the risk of the outcome is decreased by the exposure, which is a "protective factor"
RR > 1 means that the risk of the outcome is increased by the exposure, which is a "risk factor"
As always, correlation does not mean causation; the causation could be reversed, or they could both be caused by a common confounding variable. The relative risk of having cancer when in the hospital versus at home, for example, would be greater than 1, but that is because having cancer causes people to go to the hospital.
Usage in reporting
Relative risk is commonly used to present the results of randomized controlled trials. This can be problematic if the relative risk is presented without the absolute measures, such as absolute risk, or risk difference. In cases where the base rate of the outcome is low, large or small values of relative risk may not translate to significant effects, and the importance of the effects to the public health can be overestimated. Equivalently, in cases where the base rate of the outcome is high, values of the relative risk close to 1 may still result in a significant effect, and their effects can be underestimated. Thus, presentation of both absolute and relative measures is recommended.
Inference
Relative risk can be estimated from a 2×2 contingency table:
The point estimate of the relative risk is
The sampling distribution of the is closer to normal than the distribution of RR, with standard error
The confidence interval for the is then
where is the standard score for t
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https://en.wikipedia.org/wiki/Champernowne
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Champernowne may refer to:
Arthur Champernowne (disambiguation), multiple people
D. G. Champernowne (1912-2000), English economist and mathematician
Champernowne constant, in mathematics
Champernowne distribution, in statistics
Joan Champernowne (died 1553), lady-in-waiting at the court of Henry VIII of England
Katherine Champernowne, maiden name of Kat Ashley, governess and friend of Elizabeth I of England
Clyst Champernowne, ancient name of Clyst St George, a village in East Devon, England
See also
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https://en.wikipedia.org/wiki/Adjustment
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Adjustment may refer to:
Adjustment (law), with several meanings
Adjustment (psychology), the process of balancing conflicting needs
Adjustment of observations, in mathematics, a method of solving an overdetermined system of equations
Calibration, in metrology
Spinal adjustment, in chiropractic practice
In statistics, compensation for confounding variables
See also
Setting (disambiguation)
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https://en.wikipedia.org/wiki/Autonomous%20category
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In mathematics, an autonomous category is a monoidal category where dual objects exist.
Definition
A left (resp. right) autonomous category is a monoidal category where every object has a left (resp. right) dual. An autonomous category is a monoidal category where every object has both a left and a right dual. Rigid category is a synonym for autonomous category.
In a symmetric monoidal category, the existence of left duals is equivalent to the existence of right duals, categories of this kind are called (symmetric) compact closed categories.
In categorial grammars, categories which are both left and right rigid are often called pregroups, and are employed in Lambek calculus, a non-symmetric extension of linear logic.
The concepts of *-autonomous category and autonomous category are directly related, specifically, every autonomous category is *-autonomous. A *-autonomous category may be described as a linearly distributive category with (left and right) negations; such categories have two monoidal products linked with a sort of distributive law. In the case where the two monoidal products coincide and the distributivities are taken from the associativity isomorphism of the single monoidal structure, one obtains autonomous categories.
Notes and references
Sources
Monoidal categories
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https://en.wikipedia.org/wiki/Georgia%20Academy%20of%20Arts%2C%20Mathematics%2C%20Engineering%20and%20Science
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The Georgia Academy of Arts, Mathematics, Engineering and Sciences, (formerly known as GAMES), is a dual-enrollment early college entrance program created in 1997 and facilitated by the University System of Georgia in the United States. Typically, juniors in high school who meet the base requirements of GPA and SAT/ACT scores may apply and be admitted to the two-year program which is located at the Cochran, Georgia campus of Middle Georgia State University, although rising seniors and exceptional sophomores may also apply. Students at the Georgia Academy receive college-level education with specialization in the fields of the arts, mathematics, engineering and science. Academy students take a full college course load and can participate in activities such as the Honors Program, Undergraduate Research, and collegiate clubs such as Science Club, Anime Club, Dungeons & Dragons, the PSYCH-KNIGHTS, Model African Union, Mock Mediation and Math Competition.
Students live in residence halls located on the Middle Georgia State University campus in Cochran, interact with faculty, and are given similar status to traditional students within the university. When students complete the program, they are awarded associate's degrees as well as high school diplomas from their former high schools, and can enter a four-year college or university with junior standing.
More than 700 students have been admitted to The Georgia Academy since its inception in 1997, and The Academy counts two Gates Millennium Scholars (2009 & 2014) among its many very successful alumni.
After the Academy
Following completion of an associate degree from Middle Georgia State University and receiving a high school diploma, Academy alumni have gone on to attend schools such as:
Agnes Scott College
Auburn University
Brenau University
Brown University
California Institute of Technology
Carnegie Mellon University
Clemson University
College of Charleston
Cornell University
Duke University
Emory University
Fisk University
Florida A&M University
George Washington University
Furman University
Georgetown University
Georgia Institute of Technology
Georgia State University
Howard University
Illinois State University
Iowa State University
Johns Hopkins University
Loyola University
Massachusetts Institute of Technology
Middle Georgia State University
New York University
Oglethorpe University
Pennsylvania State University
Purdue University
Rochester Institute of Technology
Samford University
Tulane University
University of Alabama
University of California
University of Chicago
University of Colorado
University of Florida
University of Georgia
University of Miami
University of Oklahoma Honors College
University of Pennsylvania
University of South Carolina
University of Southern California
University of Tennessee
University of Virginia
U.S. Naval Academy
Vanderbilt University
Washington University
Washington & Lee University
Yale University
The Georgia
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https://en.wikipedia.org/wiki/Extensional%20context
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In any of several fields of study that treat the use of signs — for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language — an extensional context (or transparent context) is a syntactic environment in which a sub-sentential expression e can be replaced by an expression with the same extension and without affecting the truth-value of the sentence as a whole. Extensional contexts are contrasted with opaque contexts where truth-preserving substitutions are not possible.
Take the case of Clark Kent, who is secretly Superman. Suppose that Lois Lane fell out of a window and Superman caught her. Thus the sentence "Superman caught Lois Lane" is true. Because this sentence is an extensional context, the sentence "Clark Kent caught Lois Lane" is also true. Anybody that Superman caught, Clark Kent caught.
In opposition to extensional contexts are intensional contexts (which can involve modal operators and modal logic), where terms cannot be substituted without potentially compromising the truth-value. Suppose that Lois Lane believes that Clark Kent will investigate a news story with her. Thus, the sentence "Lois Lane believes that Clark Kent will investigate a news story with her" is true. However, the statement, "Lois Lane believes that Superman will investigate a news story with her," is false. This is because 'believes' typically induces an intensional context. Lois Lane doesn't believe that Superman is Clark Kent and the propositional attitude "believe" induces an intensional context, so the substitution alters the meaning of the original sentence.
See also
De dicto and de re
Extension (semantics)
Extensional definition
Extensionalism
Intensional logic
Opaque context
Propositional attitude
W.V. Quine
Further reading
Francis Watanabe Dauer, Critical Thinking: An Introduction to Reasoning, Oxford University Press, 1989, p. 392.
Philosophy of language
Logic
Semantics
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https://en.wikipedia.org/wiki/Pacific%20Journal%20of%20Mathematics
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The Pacific Journal of Mathematics is a mathematics research journal supported by several universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisation, and the University of California, Berkeley.
It was founded in 1951 by František Wolf and Edwin F. Beckenbach and has been published continuously since, with five two-issue volumes per year and 12 issues per year. Full-text PDF versions of all journal articles are available on-line via the journal's website with a subscription.
The journal is incorporated as a 501(c)(3) organization.
References
Mathematics journals
Academic journals established in 1951
Mathematical Sciences Publishers academic journals
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https://en.wikipedia.org/wiki/Translation%20plane
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In mathematics, a translation plane is a projective plane which admits a certain group of symmetries (described below). Along with the Hughes planes and the Figueroa planes, translation planes are among the most well-studied of the known non-Desarguesian planes, and the vast majority of known non-Desarguesian planes are either translation planes, or can be obtained from a translation plane via successive iterations of dualization and/or derivation.
In a projective plane, let represent a point, and represent a line. A central collineation with center and axis is a collineation fixing every point on and every line through . It is called an elation if is on , otherwise it is called a homology. The central collineations with center and axis form a group. A line in a projective plane is a translation line if the group of all elations with axis acts transitively on the points of the affine plane obtained by removing from the plane , (the affine derivative of ). A projective plane with a translation line is called a translation plane.
The affine plane obtained by removing the translation line is called an affine translation plane. While it is often easier to work with projective planes, in this context several authors use the term translation plane to mean affine translation plane.
Algebraic construction with coordinates
Every projective plane can be coordinatized by at least one planar ternary ring. For translation planes, it is always possible to coordinatize with a quasifield. However, some quasifields satisfy additional algebraic properties, and the corresponding planar ternary rings coordinatize translation planes which admit additional symmetries. Some of these special classes are:
Nearfield planes - coordinatized by nearfields.
Semifield planes - coordinatized by semifields, semifield planes have the property that their dual is also a translation plane.
Moufang planes - coordinatized by alternative division rings, Moufang planes are exactly those translation planes that have at least two translation lines. Every finite Moufang plane is Desarguesian and every Desarguesian plane is a Moufang plane, but there are infinite Moufang planes that are not Desarguesian (such as the Cayley plane).
Given a quasifield with operations + (addition) and (multiplication), one can define a planar ternary ring to create coordinates for a translation plane. However, it is more typical to create an affine plane directly from the quasifield by defining the points as pairs where and are elements of the quasifield, and the lines are the sets of points satisfying an equation of the form , as and vary over the elements of the quasifield, together with the sets of points satisfying an equation of the form , as varies over the elements of the quasifield.
Geometric construction with spreads (Bruck/Bose)
Translation planes are related to spreads of odd-dimensional projective spaces by the Bruck-Bose construction. A spread of , where is an i
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https://en.wikipedia.org/wiki/Le%20Cam%27s%20theorem
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In probability theory, Le Cam's theorem, named after Lucien Le Cam (1924 – 2000), states the following.
Suppose:
are independent random variables, each with a Bernoulli distribution (i.e., equal to either 0 or 1), not necessarily identically distributed.
(i.e. follows a Poisson binomial distribution)
Then
In other words, the sum has approximately a Poisson distribution and the above inequality bounds the approximation error in terms of the total variation distance.
By setting pi = λn/n, we see that this generalizes the usual Poisson limit theorem.
When is large a better bound is possible: , where represents the operator.
It is also possible to weaken the independence requirement.
References
External links
Probability theorems
Probabilistic inequalities
Statistical inequalities
Theorems in statistics
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https://en.wikipedia.org/wiki/Normal%20scheme
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In algebraic geometry, an algebraic variety or scheme X is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety X (understood to be irreducible) is normal if and only if the ring O(X) of regular functions on X is an integrally closed domain. A variety X over a field is normal if and only if every finite birational morphism from any variety Y to X is an isomorphism.
Normal varieties were introduced by .
Geometric and algebraic interpretations of normality
A morphism of varieties is finite if the inverse image of every point is finite and the morphism is proper. A morphism of varieties
is birational if it restricts to an isomorphism between dense open subsets. So, for example, the cuspidal cubic curve X in the affine plane A2 defined by x2 = y3 is not normal, because there is a finite birational morphism A1 → X
(namely, t maps to (t3, t2)) which is not an isomorphism. By contrast, the affine line A1 is normal: it cannot be simplified any further by finite birational morphisms.
A normal complex variety X has the property, when viewed as a stratified space using the classical topology, that every link is connected. Equivalently, every complex point x has arbitrarily small neighborhoods U such that U minus
the singular set of X is connected. For example, it follows that the nodal cubic curve X in the figure, defined by x2 = y2(y + 1), is not normal. This also follows from the definition of normality, since there is a finite birational morphism from A1 to X which is not an isomorphism; it sends two points of A1 to the same point in X.
More generally, a scheme X is normal if each of its local rings
OX,x
is an integrally closed domain. That is, each of these rings is an integral domain R, and every ring S with R ⊆ S ⊆ Frac(R) such that S is finitely generated as an R-module is equal to R. (Here Frac(R) denotes the field of fractions of R.) This is a direct translation, in terms of local rings, of the geometric condition that every finite birational morphism to X is an isomorphism.
An older notion is that a subvariety X of projective space is linearly normal if the linear system giving the embedding is complete. Equivalently, X ⊆ Pn is not the linear projection of an embedding X ⊆ Pn+1 (unless X is contained
in a hyperplane Pn). This is the meaning of "normal" in the phrases rational normal curve and rational normal scroll.
Every regular scheme is normal. Conversely, showed that every normal variety is regular outside a subset of codimension at least 2, and a similar result is true for schemes. So, for example, every normal curve is regular.
The normalization
Any reduced scheme X has a unique normalization: a normal scheme Y with an integral birational morphism Y → X. (For X a variety over a field, the morphism Y → X is finite, which is stronger than "integral".) The normalization of a scheme of dimension 1 is regular, and the normalization of a scheme of dimension
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https://en.wikipedia.org/wiki/Rational%20singularity
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In mathematics, more particularly in the field of algebraic geometry, a scheme has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a proper birational map
from a regular scheme such that the higher direct images of applied to are trivial. That is,
for .
If there is one such resolution, then it follows that all resolutions share this property, since any two resolutions of singularities can be dominated by a third.
For surfaces, rational singularities were defined by .
Formulations
Alternately, one can say that has rational singularities if and only if the natural map in the derived category
is a quasi-isomorphism. Notice that this includes the statement that and hence the assumption that is normal.
There are related notions in positive and mixed characteristic of
pseudo-rational
and
F-rational
Rational singularities are in particular Cohen-Macaulay, normal and Du Bois. They need not be Gorenstein or even Q-Gorenstein.
Log terminal singularities are rational.
Examples
An example of a rational singularity is the singular point of the quadric cone
Artin showed that
the rational double points of algebraic surfaces are the Du Val singularities.
See also
Elliptic singularity
References
Algebraic surfaces
Singularity theory
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https://en.wikipedia.org/wiki/Pentadiagonal%20matrix
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In linear algebra, a pentadiagonal matrix is a special case of band matrices.
Its only nonzero entries are on the main diagonal and the first two upper and two lower diagonals.
So, it is of the form.
It follows that a pentadiagonal matrix has at most nonzero entries, where n is the size of the matrix. Hence, pentadiagonal matrices are sparse, making them useful in numerical analysis.
See also
Tridiagonal matrix
Heptadiagonal matrix
Sparse matrices
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https://en.wikipedia.org/wiki/Lightface%20analytic%20game
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In descriptive set theory, a lightface analytic game is a game whose payoff set A is a subset of Baire space; that is, there is a tree T on which is a computable subset of , such that A is the projection of the set of all branches of T.
The determinacy of all lightface analytic games is equivalent to the existence of 0#.
Effective descriptive set theory
Determinacy
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