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https://en.wikipedia.org/wiki/Fredholm
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Fredholm is a Swedish surname. Notable people with the surname include:
Erik Ivar Fredholm (1866–1927), Swedish mathematician
Fredholm alternative, in mathematics
Fredholm determinant, in mathematics
Fredholm integral equation, in mathematics
Fredholm kernel, in mathematics
Fredholm module, In noncommutative geometry
Fredholm number, in number theory, apparently not in fact studied by Fredholm
Fredholm operator, in mathematics
Fredholm's theorem, in mathematics
Analytic Fredholm theorem, in mathematics
Fredholm theory, in mathematics
Fredholm (crater), a small lunar impact crater
21659 Fredholm (1999 PR3), main-belt asteroid discovered in 1999 by P. G. Comba
(1830–1891), Swedish industrialist
Gert Fredholm (born 1941), Danish film director and screenwriter
Patrik Fredholm (born 1978), Swedish footballer
Swedish-language surnames
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https://en.wikipedia.org/wiki/Theorem%20of%20the%20cube
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In mathematics, the theorem of the cube is a condition for a line bundle over a product of three complete varieties to be trivial. It was a principle discovered, in the context of linear equivalence, by the Italian school of algebraic geometry. The final version of the theorem of the cube was first published by , who credited it to André Weil. A discussion of the history has been given by . A treatment by means of sheaf cohomology, and description in terms of the Picard functor, was given
by .
Statement
The theorem states that for any complete varieties U, V and W over an algebraically closed field, and given points u, v and w on them, any invertible sheaf L which has a trivial restriction to each of U× V × {w}, U× {v} × W, and {u} × V × W, is itself trivial. (Mumford p. 55; the result there is slightly stronger, in that one of the varieties need not be complete and can be replaced by a connected scheme.)
Special cases
On a ringed space X, an invertible sheaf L is trivial if isomorphic to OX, as an OX-module. If the base X is a complex manifold, then an invertible sheaf is (the sheaf of sections of) a holomorphic line bundle, and trivial means holomorphically equivalent to a trivial bundle, not just topologically equivalent.
Restatement using biextensions
Weil's result has been restated in terms of biextensions, a concept now generally used in the duality theory of abelian varieties.
Theorem of the square
The theorem of the square is a corollary (also due to Weil) applying to an abelian variety A. One version of it states that the function φL taking x∈A to TL⊗L−1 is a group homomorphism from A to Pic(A) (where T is translation by x on line bundles).
References
Notes
Abelian varieties
Algebraic varieties
Theorems in geometry
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https://en.wikipedia.org/wiki/Absolute%20irreducibility
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In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field. For example, is absolutely irreducible, but while is irreducible over the integers and the reals, it is reducible over the complex numbers as and thus not absolutely irreducible.
More generally, a polynomial defined over a field K is absolutely irreducible if it is irreducible over every algebraic extension of K, and an affine algebraic set defined by equations with coefficients in a field K is absolutely irreducible if it is not the union of two algebraic sets defined by equations in an algebraically closed extension of K. In other words, an absolutely irreducible algebraic set is a synonym of an algebraic variety, which emphasizes that the coefficients of the defining equations may not belong to an algebraically closed field.
Absolutely irreducible is also applied, with the same meaning, to linear representations of algebraic groups.
In all cases, being absolutely irreducible is the same as being irreducible over the algebraic closure of the ground field.
Examples
A univariate polynomial of degree greater than or equal to 2 is never absolutely irreducible, due to the fundamental theorem of algebra.
The irreducible two-dimensional representation of the symmetric group S3 of order 6, originally defined over the field of rational numbers, is absolutely irreducible.
The representation of the circle group by rotations in the plane is irreducible (over the field of real numbers), but is not absolutely irreducible. After extending the field to complex numbers, it splits into two irreducible components. This is to be expected, since the circle group is commutative and it is known that all irreducible representations of commutative groups over an algebraically closed field are one-dimensional.
The real algebraic variety defined by the equation
is absolutely irreducible. It is the ordinary circle over the reals and remains an irreducible conic section over the field of complex numbers. Absolute irreducibility more generally holds over any field not of characteristic two. In characteristic two, the equation is equivalent to (x + y −1)2 = 0. Hence it defines the double line x + y =1, which is a non-reduced scheme.
The algebraic variety given by the equation
is not absolutely irreducible. Indeed, the left hand side can be factored as
where is a square root of −1.
Therefore, this algebraic variety consists of two lines intersecting at the origin and is not absolutely irreducible. This holds either already over the ground field, if −1 is a square, or over the quadratic extension obtained by adjoining i.
References
Algebraic geometry
Representation theory
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https://en.wikipedia.org/wiki/Dual%20abelian%20variety
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In mathematics, a dual abelian variety can be defined from an abelian variety A, defined over a field K.
Definition
To an abelian variety A over a field k, one associates a dual abelian variety Av (over the same field), which is the solution to the following moduli problem. A family of degree 0 line bundles parametrized by a k-variety T is defined to be a line bundle L on
A×T such that
for all , the restriction of L to A×{t} is a degree 0 line bundle,
the restriction of L to {0}×T is a trivial line bundle (here 0 is the identity of A).
Then there is a variety Av and a line bundle ,, called the Poincaré bundle, which is a family of degree 0 line bundles parametrized by Av in the sense of the above definition. Moreover, this family is universal, that is, to any family L parametrized by T is associated a unique morphism f: T → Av so that L is isomorphic to the pullback of P along the morphism 1A×f: A×T → A×Av. Applying this to the case when T is a point, we see that the points of Av correspond to line bundles of degree 0 on A, so there is a natural group operation on Av given by tensor product of line bundles, which makes it into an abelian variety.
In the language of representable functors one can state the above result as follows. The contravariant functor, which associates to each k-variety T the set of families of degree 0 line bundles parametrised by T and to each k-morphism f: T → T the mapping induced by the pullback with f, is representable. The universal element representing this functor is the pair (Av, P).
This association is a duality in the sense that there is a natural isomorphism between the double dual Avv and A (defined via the Poincaré bundle) and that it is contravariant functorial, i.e. it associates to all morphisms f: A → B dual morphisms fv: Bv → Av in a compatible way. The n-torsion of an abelian variety and the n-torsion of its dual are dual to each other when n is coprime to the characteristic of the base. In general - for all n - the n-torsion group schemes of dual abelian varieties are Cartier duals of each other. This generalizes the Weil pairing for elliptic curves.
History
The theory was first put into a good form when K was the field of complex numbers. In that case there is a general form of duality between the Albanese variety of a complete variety V, and its Picard variety; this was realised, for definitions in terms of complex tori, as soon as André Weil had given a general definition of Albanese variety. For an abelian variety A, the Albanese variety is A itself, so the dual should be Pic0(A), the connected component of the identity element of what in contemporary terminology is the Picard scheme.
For the case of the Jacobian variety J of a compact Riemann surface C, the choice of a principal polarization of J gives rise to an identification of J with its own Picard variety. This in a sense is just a consequence of Abel's theorem. For general abelian varieties, still over the complex numbers, A is in t
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https://en.wikipedia.org/wiki/Tonelli%E2%80%93Hobson%20test
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In mathematics, the Tonelli–Hobson test gives sufficient criteria for a function ƒ on R2 to be an integrable function. It is often used to establish that Fubini's theorem may be applied to ƒ. It is named for Leonida Tonelli and E. W. Hobson.
More precisely, the Tonelli–Hobson test states that if ƒ is a real-valued measurable function on R2, and either of the two iterated integrals
or
is finite, then ƒ is Lebesgue-integrable on R2.
Integral calculus
Theorems in analysis
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https://en.wikipedia.org/wiki/Matrix%20pencil
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In linear algebra, if are complex matrices for some nonnegative integer , and (the zero matrix), then the matrix pencil of degree is the matrix-valued function defined on the complex numbers
A particular case is a linear matrix pencil with where and are complex (or real) matrices. We denote it briefly with the notation .
A pencil is called regular if there is at least one value of such that . We call eigenvalues of a matrix pencil all complex numbers for which (see eigenvalue for comparison). The set of the eigenvalues is called the spectrum of the pencil and is written .
Moreover, the pencil is said to have one or more eigenvalues at infinity if has one or more 0 eigenvalues.
Applications
Matrix pencils play an important role in numerical linear algebra. The problem of finding the eigenvalues of a pencil is called the generalized eigenvalue problem. The most popular algorithm for this task is the QZ algorithm, which is an implicit version of the QR algorithm to solve the associated eigenvalue problem without forming explicitly the matrix (which could be impossible or ill-conditioned if is singular or near-singular)
Pencil generated by commuting matrices
If , then the pencil generated by and :
consists only of matrices similar to a diagonal matrix, or
has no matrices in it similar to a diagonal matrix, or
has exactly one matrix in it similar to a diagonal matrix.
See also
Generalized eigenvalue problem
Generalized pencil-of-function method
Nonlinear eigenproblem
Quadratic eigenvalue problem
Generalized Rayleigh quotient
Notes
References
Linear algebra
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https://en.wikipedia.org/wiki/Hitting%20time
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In the study of stochastic processes in mathematics, a hitting time (or first hit time) is the first time at which a given process "hits" a given subset of the state space. Exit times and return times are also examples of hitting times.
Definitions
Let be an ordered index set such as the natural numbers, the non-negative real numbers, , or a subset of these; elements can be thought of as "times". Given a probability space and a measurable state space , let be a stochastic process, and let be a measurable subset of the state space . Then the first hit time is the random variable defined by
The first exit time (from ) is defined to be the first hit time for , the complement of in . Confusingly, this is also often denoted by .
The first return time is defined to be the first hit time for the singleton set which is usually a given deterministic element of the state space, such as the origin of the coordinate system.
Examples
Any stopping time is a hitting time for a properly chosen process and target set. This follows from the converse of the Début theorem (Fischer, 2013).
Let denote standard Brownian motion on the real line starting at the origin. Then the hitting time satisfies the measurability requirements to be a stopping time for every Borel measurable set
For as above, let () denote the first exit time for the interval , i.e. the first hit time for Then the expected value and variance of satisfy
For as above, the time of hitting a single point (different from the starting point 0) has the Lévy distribution.
Début theorem
The hitting time of a set is also known as the début of . The Début theorem says that the hitting time of a measurable set , for a progressively measurable process, is a stopping time. Progressively measurable processes include, in particular, all right and left-continuous adapted processes.
The proof that the début is measurable is rather involved and involves properties of analytic sets. The theorem requires the underlying probability space to be complete or, at least, universally complete.
The converse of the Début theorem states that every stopping time defined with respect to a filtration over a real-valued time index can be represented by a hitting time. In particular, for essentially any such stopping time there exists an adapted, non-increasing process with càdlàg (RCLL) paths that takes the values 0 and 1 only, such that the hitting time of the set by this process is the considered stopping time. The proof is very simple.
See also
Stopping time
References
Stochastic processes
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https://en.wikipedia.org/wiki/Poincar%C3%A9%20residue
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In mathematics, the Poincaré residue is a generalization, to several complex variables and complex manifold theory, of the residue at a pole of complex function theory. It is just one of a number of such possible extensions.
Given a hypersurface defined by a degree polynomial and a rational -form on with a pole of order on , then we can construct a cohomology class . If we recover the classical residue construction.
Historical construction
When Poincaré first introduced residues he was studying period integrals of the form for where was a rational differential form with poles along a divisor . He was able to make the reduction of this integral to an integral of the form for where , sending to the boundary of a solid -tube around on the smooth locus of the divisor. Ifon an affine chart where is irreducible of degree and (so there is no poles on the line at infinity page 150). Then, he gave a formula for computing this residue aswhich are both cohomologous forms.
Construction
Preliminary definition
Given the setup in the introduction, let be the space of meromorphic -forms on which have poles of order up to . Notice that the standard differential sends
Define
as the rational de-Rham cohomology groups. They form a filtrationcorresponding to the Hodge filtration.
Definition of residue
Consider an -cycle . We take a tube around (which is locally isomorphic to ) that lies within the complement of . Since this is an -cycle, we can integrate a rational -form and get a number. If we write this as
then we get a linear transformation on the homology classes. Homology/cohomology duality implies that this is a cohomology class
which we call the residue. Notice if we restrict to the case , this is just the standard residue from complex analysis (although we extend our meromorphic -form to all of . This definition can be summarized as the map
Algorithm for computing this class
There is a simple recursive method for computing the residues which reduces to the classical case of . Recall that the residue of a -form
If we consider a chart containing where it is the vanishing locus of , we can write a meromorphic -form with pole on as
Then we can write it out as
This shows that the two cohomology classes
are equal. We have thus reduced the order of the pole hence we can use recursion to get a pole of order and define the residue of as
Example
For example, consider the curve defined by the polynomial
Then, we can apply the previous algorithm to compute the residue of
Since
and
we have that
This implies that
See also
Grothendieck residue
Leray residue
Bott residue
Sheaf of logarithmic differential forms
normal crossing singularity
Adjunction formula#Poincare residue
Hodge structure
Jacobian ideal
References
Introductory
Poincaré and algebraic geometry
Infinitesimal variations of Hodge structure and the global Torelli problem - Page 7 contains general computation formula using Cech cohomology
Higher Dimensiona
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https://en.wikipedia.org/wiki/Casson%20invariant
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In 3-dimensional topology, a part of the mathematical field of geometric topology, the Casson invariant is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson.
Kevin Walker (1992) found an extension to rational homology 3-spheres, called the Casson–Walker invariant, and Christine Lescop (1995) extended the invariant to all closed oriented 3-manifolds.
Definition
A Casson invariant is a surjective map
λ from oriented integral homology 3-spheres to Z satisfying the following properties:
λ(S3) = 0.
Let Σ be an integral homology 3-sphere. Then for any knot K and for any integer n, the difference
is independent of n. Here denotes Dehn surgery on Σ by K.
For any boundary link K ∪ L in Σ the following expression is zero:
The Casson invariant is unique (with respect to the above properties) up to an overall multiplicative constant.
Properties
If K is the trefoil then
.
The Casson invariant is 1 (or −1) for the Poincaré homology sphere.
The Casson invariant changes sign if the orientation of M is reversed.
The Rokhlin invariant of M is equal to the Casson invariant mod 2.
The Casson invariant is additive with respect to connected summing of homology 3-spheres.
The Casson invariant is a sort of Euler characteristic for Floer homology.
For any integer n
where is the coefficient of in the Alexander–Conway polynomial , and is congruent (mod 2) to the Arf invariant of K.
The Casson invariant is the degree 1 part of the Le–Murakami–Ohtsuki invariant.
The Casson invariant for the Seifert manifold is given by the formula:
where
The Casson invariant as a count of representations
Informally speaking, the Casson invariant counts half the number of conjugacy classes of representations of the fundamental group of a homology 3-sphere M into the group SU(2). This can be made precise as follows.
The representation space of a compact oriented 3-manifold M is defined as where denotes the space of irreducible SU(2) representations of . For a Heegaard splitting of , the Casson invariant equals times the algebraic intersection of with .
Generalizations
Rational homology 3-spheres
Kevin Walker found an extension of the Casson invariant to rational homology 3-spheres. A Casson-Walker invariant is a surjective map λCW from oriented rational homology 3-spheres to Q satisfying the following properties:
1. λ(S3) = 0.
2. For every 1-component Dehn surgery presentation (K, μ) of an oriented rational homology sphere M′ in an oriented rational homology sphere M:
where:
m is an oriented meridian of a knot K and μ is the characteristic curve of the surgery.
ν is a generator the kernel of the natural map H1(∂N(K), Z) → H1(M−K, Z).
is the intersection form on the tubular neighbourhood of the knot, N(K).
Δ is the Alexander polynomial normalized so that the action of t corresponds to an action of the generator of in the infinite cyclic cover of M−K, and is symmetric and evaluates to 1 at 1.
where x, y are generato
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https://en.wikipedia.org/wiki/Don%20Orlich
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Don Orlich is professor emeritus of the Science Mathematics Engineering Education Center at Washington State University. He has published more than 100 professional papers, co-authored more than 30 monographs and books, and is the senior co-author of “Teaching Strategies: A guide to Effective Teaching,” published by Houghton Mifflin in 2004.
He has conducted an independent study of Washington's WASL standards based assessment, concluding, “The WASL is a disaster” Orlich has concluded that the fifth grade science WASL exceeds the intellectual level of the majority of fifth graders, the seventh grade math WASL is more like a ninth grade test. Learning goals for the seventh grade is almost identical to many 10th grade goals.
He has authored a soon to be released book titled “School Reform and the Great American Brain Robbery,”. He analyzed areas of the WASL using criteria from developmental psychology and the Scales of the National Assessment of Education Progress (NAEP). Orlich has found areas of the Grade Level Expectations (GLEs), hence the WASL test, to be developmentally inappropriate. He has won a national award from the Association for Supervision and Curriculum Development for a critical analysis he wrote on the fourth-grade WASL, although the OSPI disagrees with the analysis.
Notes
External links
Orlich on the WASL
Education reform
Year of birth missing (living people)
Living people
Washington State University faculty
Place of birth missing (living people)
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https://en.wikipedia.org/wiki/Kurtosis%20risk
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In statistics and decision theory, kurtosis risk is the risk that results when a statistical model assumes the normal distribution, but is applied to observations that have a tendency to occasionally be much farther (in terms of number of standard deviations) from the average than is expected for a normal distribution.
Overview
Kurtosis risk applies to any kurtosis-related quantitative model that assumes the normal distribution for certain of its independent variables when the latter may in fact have kurtosis much greater than does the normal distribution. Kurtosis risk is commonly referred to as "fat tail" risk. The "fat tail" metaphor explicitly describes the situation of having more observations at either extreme than the tails of the normal distribution would suggest; therefore, the tails are "fatter".
Ignoring kurtosis risk will cause any model to understate the risk of variables with high kurtosis. For instance, Long-Term Capital Management, a hedge fund cofounded by Myron Scholes, ignored kurtosis risk to its detriment. After four successful years, this hedge fund had to be bailed out by major investment banks in the late 1990s because it understated the kurtosis of many financial securities underlying the fund's own trading positions.
Research by Mandelbrot
Benoit Mandelbrot, a French mathematician, extensively researched this issue. He felt that the extensive reliance on the normal distribution for much of the body of modern finance and investment theory is a serious flaw of any related models including the Black–Scholes option model developed by Myron Scholes and Fischer Black, and the capital asset pricing model developed by William F. Sharpe. Mandelbrot explained his views and alternative finance theory in his book: The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward published on September 18, 1997.
See also
Kurtosis
Skewness risk
Stochastic volatility
Holy grail distribution
Taleb distribution
The Black Swan: The Impact of the Highly Improbable by Nassim Nicholas Taleb
Notes
References
Premaratne, G., Bera, A. K. (2000). Modeling Asymmetry and Excess Kurtosis in Stock Return Data. Office of Research Working Paper Number 00-0123, University of Illinois
Normal distribution
Investment
Risk analysis
Mathematical finance
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https://en.wikipedia.org/wiki/Mediation%20%28statistics%29
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In statistics, a mediation model seeks to identify and explain the mechanism or process that underlies an observed relationship between an independent variable and a dependent variable via the inclusion of a third hypothetical variable, known as a mediator variable (also a mediating variable, intermediary variable, or intervening variable). Rather than a direct causal relationship between the independent variable and the dependent variable, a mediation model proposes that the independent variable influences the mediator variable, which in turn influences the dependent variable. Thus, the mediator variable serves to clarify the nature of the relationship between the independent and dependent variables.
Mediation analyses are employed to understand a known relationship by exploring the underlying mechanism or process by which one variable influences another variable through a mediator variable. In particular, mediation analysis can contribute to better understanding the relationship between an independent variable and a dependent variable when these variables do not have an obvious direct connection.
Baron and Kenny's (1986) steps for mediation analysis
Baron and Kenny (1986) laid out several requirements that must be met to form a true mediation relationship. They are outlined below using a real-world example. See the diagram above for a visual representation of the overall mediating relationship to be explained. Note: Hayes (2009) critiqued Baron and Kenny's mediation steps approach, and as of 2019, David A. Kenny on his website stated that mediation can exist in the absence of a 'significant' total effect, and therefore step 1 below may not be needed. This situation is sometimes referred to as "inconsistent mediation". Later publications by Hayes also questioned the concepts of full or partial mediation and advocated for these terms, along with the classical mediation steps approach outlined below, to be abandoned.
Step 1
Regress the dependent variable on the independent variable to confirm that the independent variable is a significant predictor of the dependent variable.
Independent variable dependent variable
β11 is significant
Step 2
Regress the mediator on the independent variable to confirm that the independent variable is a significant predictor of the mediator. If the mediator is not associated with the independent variable, then it couldn’t possibly mediate anything.
Independent variable mediator
β21 is significant
Step 3
Regress the dependent variable on both the mediator and independent variable to confirm that a) the mediator is a significant predictor of the dependent variable, and b) the strength of the coefficient of the previously significant independent variable in Step #1 is now greatly reduced, if not rendered nonsignificant.
β32 is significant
β31 should be smaller in absolute value than the original effect for the independent variable (β11 above)
Example
The following example, drawn from Howell (20
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https://en.wikipedia.org/wiki/Mikhail%20Goussarov
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Mikhail Goussarov (March 8, 1958, Leningrad – June 25, 1999, Tel Aviv) was a Soviet mathematician who worked in low-dimensional topology. He and Victor Vassiliev independently discovered finite type invariants of knots and links. He drowned at the age of 41 in an accident in Tel Aviv, Israel.
See also
Memorial page maintained by Dror Bar-Natan.
1958 births
1999 deaths
20th-century Russian mathematicians
Deaths by drowning
Accidental deaths in Israel
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https://en.wikipedia.org/wiki/Christopher%20Knight%20%28author%29
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Christopher Knight is an author who has written several books dealing with pseudoscientific conspiracy theories such as 366-degree geometry and the origins of Freemasonry.
In an interview about the book Who Built the Moon?: 2005 Knight stated that the moon is an artificial construction probably built by humans with a message in "base ten arithmetic so it looks as though it is directed to a ten digit species that is living on Earth right now - which seems to mean humans." He believes that it was created to make life on Earth possible, including humans, and that the most likely builders were humans of the future using time travel.
Books
Co-authored with Robert Lomas:
The Hiram Key. 1996, Century.
The Second Messiah. 1997, Century.
The Holy Grail (Mysteries of the Ancient World). 1997, Weidenfeld and Nicolson.
Uriel's Machine. 1999, Century.
The Book Of Hiram. 2003, Century.
Co-authored with Alan Butler
Civilization One. 1999, Watkins Publishing.
Who Built the Moon?. 2005, Watkins Publishing.
Solomon's Power Brokers. 2007, Watkins Publishing.
Before the Pyramids. 2009, Watkins Publishing.
The Hiram Key Revisited. 2010, Watkins Publishing.
See also
Archaeoastronomy
References
British writers
Pseudohistorians
Living people
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/Jeremiah%20Farrell
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Jeremiah (Jerry) Farrell (December 12, 1937, in Hastings, Nebraska – July 4, 2022, in Indianapolis, Indiana) was an American professor emeritus of mathematics at Butler University in Indiana. He was well known for having constructed Will Shortz's favorite puzzle, the famous 1996 "Election Day" crossword in The New York Times. He also wrote puzzles for many other books and newspapers, such as Scott Kim's puzzle column for Discover magazine.
Biography
Farrell was born in Hastings, Nebraska, the oldest of three children to Belle Einsphar and Paul Farrell, a third-generation railroad man. Farrell himself worked for one summer on the railroad, as a "grinder", one who planes down the railroad tracks so they stay smooth. He attended Hastings High School, graduating in 1955, and then the University of Nebraska, graduating in 1963 with degrees in mathematics, chemistry, and physics. He later obtained a master's degree in mathematics, and in 1966 was hired by Butler University, where he worked for the next 40 years, teaching nearly every subject in the mathematics department. He officially retired in 1994 but continued to teach.
He was best known for constructing many crossword puzzles for The New York Times, starting in the 1970s for editor Margaret Farrar, and then continuing to design new puzzles after Shortz took over. In 1996, he designed his most famous puzzle, the "Election Day" crossword. One of the words had the clue "lead story tomorrow", with a 14-letter answer. The puzzle had two correct solutions: "Bob Dole elected" and "Clinton elected", and all the crossing words were designed such that they could be one of two different words, to make either answer work. Shortz called it an "amazing" feat and his favorite puzzle.
With his wife Karen, Farrell helped organize the biannual Gathering for Gardner conferences, which started in 1993 as an invitation-only event for people connected with Martin Gardner.
In 2006 Farrell and his wife took over from A. Ross Eckler, Jr. as editors and publishers of the quarterly publication Word Ways: the Journal of Recreational Linguistics, established in 1968.
Contributed works
Zen and the Art of Magic Squares
A.K. Peters publications (where he is called a "mathemagician")
Discover magazine
References
NYT "Election Day" crossword
New York Sun, "A Washington Square Park Puzzle Is Solved", May 19–21, 2006
Indianapolis Star, May 25, 2006, "Butler Prof Figures It Out"
Butler University faculty
20th-century American mathematicians
21st-century American mathematicians
Crossword creators
Puzzle designers
People from Hastings, Nebraska
Word Ways people
Hastings Senior High School (Nebraska) alumni
1937 births
2022 deaths
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https://en.wikipedia.org/wiki/Centrosymmetric%20matrix
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In mathematics, especially in linear algebra and matrix theory, a centrosymmetric matrix is a matrix which is symmetric about its center. More precisely, an n×n matrix A = [Ai,j] is centrosymmetric when its entries satisfy
Ai,j = An−i + 1,n−j + 1 for i, j ∊{1, ..., n}.
If J denotes the n×n exchange matrix with 1 on the antidiagonal and 0 elsewhere (that is, Ji,n + 1 − i = 1; Ji,j = 0 if j ≠ n +1− i), then a matrix A is centrosymmetric if and only if AJ = JA.
Examples
All 2×2 centrosymmetric matrices have the form
All 3×3 centrosymmetric matrices have the form
Symmetric Toeplitz matrices are centrosymmetric.
Algebraic structure and properties
If A and B are centrosymmetric matrices over a field F, then so are A + B and cA for any c in F. Moreover, the matrix product AB is centrosymmetric, since JAB = AJB = ABJ. Since the identity matrix is also centrosymmetric, it follows that the set of n×n centrosymmetric matrices over F is a subalgebra of the associative algebra of all n×n matrices.
If A is a centrosymmetric matrix with an m-dimensional eigenbasis, then its m eigenvectors can each be chosen so that they satisfy either x = Jx or x = −Jx where J is the exchange matrix.
If A is a centrosymmetric matrix with distinct eigenvalues, then the matrices that commute with A must be centrosymmetric.
The maximum number of unique elements in a m × m centrosymmetric matrix is .
Related structures
An n×n matrix A is said to be skew-centrosymmetric if its entries satisfy Ai,j = −An−i+1,n−j+1 for i, j ∊ {1, ..., n}. Equivalently, A is skew-centrosymmetric if AJ = −JA, where J is the exchange matrix defined above.
The centrosymmetric relation AJ = JA lends itself to a natural generalization, where J is replaced with an involutory matrix K (i.e., K2 = I) or, more generally, a matrix K satisfying Km = I for an integer m > 1. The inverse problem for the commutation relation of identifying all involutory K that commute with a fixed matrix A has also been studied.
Symmetric centrosymmetric matrices are sometimes called bisymmetric matrices. When the ground field is the field of real numbers, it has been shown that bisymmetric matrices are precisely those symmetric matrices whose eigenvalues remain the same aside from possible sign changes following pre- or post-multiplication by the exchange matrix. A similar result holds for Hermitian centrosymmetric and skew-centrosymmetric matrices.
References
Further reading
External links
Centrosymmetric matrix on MathWorld.
Linear algebra
Matrices
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https://en.wikipedia.org/wiki/Contributions%20of%20Leonhard%20Euler%20to%20mathematics
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The Swiss mathematician Leonhard Euler (1707–1783) is among the most prolific and successful mathematicians in the history of the field. His seminal work had a profound impact in numerous areas of mathematics and he is widely credited for introducing and popularizing modern notation and terminology.
Mathematical notation
Euler introduced much of the mathematical notation in use today, such as the notation f(x) to describe a function and the modern notation for the trigonometric functions. He was the first to use the letter e for the base of the natural logarithm, now also known as Euler's number. The use of the Greek letter to denote the ratio of a circle's circumference to its diameter was also popularized by Euler (although it did not originate with him). He is also credited for inventing the notation i to denote .
Complex analysis
Euler made important contributions to complex analysis. He introduced scientific notation. He discovered what is now known as Euler's formula, that for any real number , the complex exponential function satisfies
This has been called "the most remarkable formula in mathematics" by Richard Feynman. Euler's identity is a special case of this:
This identity is particularly remarkable as it involves e, , i, 1, and 0, arguably the five most important constants in mathematics.
Analysis
The development of calculus was at the forefront of 18th-century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Understanding the infinite was the major focus of Euler's research. While some of Euler's proofs may not have been acceptable under modern standards of rigor, his ideas were responsible for many great advances. First of all, Euler introduced the concept of a function, and introduced the use of the exponential function and logarithms in analytic proofs.
Euler frequently used the logarithmic functions as a tool in analysis problems, and discovered new ways by which they could be used. He discovered ways to express various logarithmic functions in terms of power series, and successfully defined logarithms for complex and negative numbers, thus greatly expanding the scope where logarithms could be applied in mathematics. Most researchers in the field long held the view that for any positive real since by using the additivity property of logarithms . In a 1747 letter to Jean Le Rond d'Alembert, Euler defined the natural logarithm of −1 as , a pure imaginary.
Euler is well known in analysis for his frequent use and development of power series: that is, the expression of functions as sums of infinitely many terms, such as
Notably, Euler discovered the power series expansions for e and the inverse tangent function
His use of power series enabled him to solve the famous Basel problem in 1735:
In addition, Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quarti
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https://en.wikipedia.org/wiki/Kolmogorov%20continuity%20theorem
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In mathematics, the Kolmogorov continuity theorem is a theorem that guarantees that a stochastic process that satisfies certain constraints on the moments of its increments will be continuous (or, more precisely, have a "continuous version"). It is credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov.
Statement
Let be some complete metric space, and let be a stochastic process. Suppose that for all times , there exist positive constants such that
for all . Then there exists a modification of that is a continuous process, i.e. a process such that
is sample-continuous;
for every time ,
Furthermore, the paths of are locally -Hölder-continuous for every .
Example
In the case of Brownian motion on , the choice of constants , , will work in the Kolmogorov continuity theorem. Moreover, for any positive integer , the constants , will work, for some positive value of that depends on and .
See also
Kolmogorov extension theorem
References
p. 51
Theorems regarding stochastic processes
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https://en.wikipedia.org/wiki/Mathland
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MathLand was one of several elementary mathematics curricula that were designed around the 1989 NCTM standards. It was developed and published by Creative Publications and was initially adopted by the U.S. state of California and schools run by the US Department of Defense by the mid 1990s. Unlike curricula such as Investigations in Numbers, Data, and Space, by 2007 Mathland was no longer offered by the publisher, and has since been dropped by many early adopters. Its demise may have been, at least in part, a result of intense scrutiny by critics (see below).
Adoption
Mathland was among the math curricula rated as "promising" by an Education Department panel, although subsequently 200 mathematicians and scientists, including four Nobel Prize recipients and two winners of the Fields Medal, published a letter in the Washington Post deploring the findings of that panel. MathLand was adopted in many California school districts as its material most closely fit the legal mandate of the 1992 California Framework. That framework has since been discredited and abandoned as misguided and replaced by a newer standard based on traditional mathematics. It bears noting that the process by which the framework was replaced itself came under serious scrutiny.
Concept
Mathland focuses on "attention to conceptual understanding, communication, reasoning and problem solving." Children meet in small groups and invent their own ways to add, subtract, multiply and divide, which spares young learners from "teacher-imposed rules." In the spirit of not chaining instruction to fixed content, MathLand does away with textbooks.
A textbook as well as other practice books were available to reinforce concepts taught in the lesson.
Standard Arithmetic
MathLand does not teach standard arithmetic algorithms, including carrying and borrowing. Such methods familiar to adults are absent from the curriculum, and so would need to be supplemented if desired. The standard method for multi-digit multiplication is not presented until 6th grade, and then only as an example of how it is error-prone. Instead a Russian peasants' algorithm for calculating 13 x 18 = 234 is favored. By cutting and pasting various strips of paper, it can be solved by simply using 3 divisions, 3 multiplications, a cancellation, and an addition of three numbers.
Sixth graders are asked to solve following problem:
"I just checked out a library book that is 1,344 pages long! The book is due in 3 weeks. How many pages will I need to read a day to finish the book in time?"
Long division is not used to divide 1,344 by 21. Instead, the curriculum guide explains that "division in MathLand is not a separate operation to master, but rather a combination of successive approximations, multiplication, adding up and subtracting back, all held together with the students' own number sense."
Criticisms
Debra J. Saunders of the San Francisco Chronicle calls Mathland a math curriculum that prefers not to give lessons with "
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https://en.wikipedia.org/wiki/Omnitruncation
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In geometry, an omnitruncation of a convex polytope is a simple polytope of the same dimension, having a vertex for each flag of the original polytope and a facet for each face of any dimension of the original polytope. Omnitruncation is the dual operation to barycentric subdivision. Because the barycentric subdivision of any polytope can be realized as another polytope, the same is true for the omnitruncation of any polytope.
When omnitruncation is applied to a regular polytope (or honeycomb) it can be described geometrically as a Wythoff construction that creates a maximum number of facets. It is represented in a Coxeter–Dynkin diagram with all nodes ringed.
It is a shortcut term which has a different meaning in progressively-higher-dimensional polytopes:
Uniform polytope truncation operators
For regular polygons: An ordinary truncation, .
Coxeter-Dynkin diagram
For uniform polyhedra (3-polytopes): A cantitruncation, . (Application of both cantellation and truncation operations)
Coxeter-Dynkin diagram:
For uniform polychora: A runcicantitruncation, . (Application of runcination, cantellation, and truncation operations)
Coxeter-Dynkin diagram: , ,
For uniform polytera (5-polytopes): A steriruncicantitruncation, t0,1,2,3,4{p,q,r,s}. . (Application of sterication, runcination, cantellation, and truncation operations)
Coxeter-Dynkin diagram: , ,
For uniform n-polytopes: .
See also
Expansion (geometry)
Omnitruncated polyhedron
References
Further reading
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, (pp. 145–154 Chapter 8: Truncation, p 210 Expansion)
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
External links
Polyhedra
Uniform polyhedra
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https://en.wikipedia.org/wiki/Oswald%20Veblen%20Prize%20in%20Geometry
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The Oswald Veblen Prize in Geometry is an award granted by the American Mathematical Society for notable research in geometry or topology. It was funded in 1961 in memory of Oswald Veblen and first issued in 1964. The Veblen Prize is now worth US$5000, and is awarded every three years.
The first seven prize winners were awarded for works in topology. James Harris Simons and William Thurston were the first ones to receive it for works in geometry (for some distinctions, see geometry and topology). As of 2020, there have been thirty-four prize recipients.
List of recipients
1964 Christos Papakyriakopoulos
1964 Raoul Bott
1966 Stephen Smale
1966 Morton Brown and Barry Mazur
1971 Robion Kirby
1971 Dennis Sullivan
1976 William Thurston
1976 James Harris Simons
1981 Mikhail Gromov for:
Manifolds of negative curvature. Journal of Differential Geometry 13 (1978), no. 2, 223–230.
Almost flat manifolds. Journal of Differential Geometry 13 (1978), no. 2, 231–241.
Curvature, diameter and Betti numbers. Comment. Math. Helv. 56 (1981), no. 2, 179–195.
Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73.
Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99
1981 Shing-Tung Yau for:
On the regularity of the solution of the n-dimensional Minkowski problem. Comm. Pure Appl. Math. 29 (1976), no. 5, 495–516. (with Shiu-Yuen Cheng)
On the regularity of the Monge-Ampère equation . Comm. Pure Appl. Math. 30 (1977), no. 1, 41–68. (with Shiu-Yuen Cheng)
Calabi's conjecture and some new results in algebraic geometry. Proc. Natl. Acad. Sci. U.S.A. 74 (1977), no. 5, 1798–1799.
On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411.
On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys. 65 (1979), no. 1, 45–76. (with Richard Schoen)
Topology of three-dimensional manifolds and the embedding problems in minimal surface theory. Ann. of Math. (2) 112 (1980), no. 3, 441–484. (with William Meeks)
1986 Michael Freedman for:
The topology of four-dimensional manifolds. Journal of Differential Geometry 17 (1982), no. 3, 357–453.
1991 Andrew Casson for:
his work on the topology of low dimensional manifolds and specifically for the discovery of an integer valued invariant of homology three spheres whose reduction mod(2) is the invariant of Rohlin.
1991 Clifford Taubes for:
Self-dual Yang-Mills connections on non-self-dual 4-manifolds. Journal of Differential Geometry 17 (1982), no. 1, 139–170.
Gauge theory on asymptotically periodic 4-manifolds. J. Differential Geom. 25 (1987), no. 3, 363–430.
Casson's invariant and gauge theory. J. Differential Geom. 31 (1990), no. 2, 547–599.
1996 Richard S. Hamilton for:
The formation of singularities in the Ricci flow. Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7–136, Int. Press, Cambridge, MA, 1995.
Four-manifol
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https://en.wikipedia.org/wiki/Hyman%20Bass
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Hyman Bass (; born October 5, 1932) is an American mathematician, known for work in algebra and in mathematics education. From 1959 to 1998 he was Professor in the Mathematics Department at Columbia University. He is currently the Samuel Eilenberg Distinguished University Professor of Mathematics and Professor of Mathematics Education at the University of Michigan.
Life
Born to a Jewish family in Houston, Texas, he earned his B.A. in 1955 from Princeton University and his Ph.D. in 1959 from the University of Chicago. His thesis, titled Global dimensions of rings, was written under the supervision of Irving Kaplansky.
He has held visiting appointments at the Institute for Advanced Study in Princeton, New Jersey, Institut des Hautes Études Scientifiques and École Normale Supérieure (Paris), Tata Institute of Fundamental Research (Bombay), University of Cambridge, University of California, Berkeley, University of Rome, IMPA (Rio), National Autonomous University of Mexico, Mittag-Leffler Institute (Stockholm), and the University of Utah. He was president of the American Mathematical Society.
Bass formerly chaired the Mathematical Sciences Education Board (1992–2000) at the National Academy of Sciences, and the Committee on Education of the American Mathematical Society. He was the President of ICMI from 1999 to 2006. Since 1996 he has been collaborating with Deborah Ball and her research group at the University of Michigan on the mathematical knowledge and resources entailed in the teaching of mathematics at the elementary level. He has worked to build bridges between diverse professional communities and stakeholders involved in mathematics education.
Work
His research interests have been in algebraic K-theory, commutative algebra and algebraic geometry, algebraic groups, geometric methods in group theory, and ζ functions on finite simple graphs.
Awards and recognitions
Bass was elected as a member of the National Academy of Sciences in 1982. In 1983, he was elected a Fellow of the American Academy of Arts and Sciences. In 2002 he was elected a fellow of The World Academy of Sciences. He is a 2006 National Medal of Science laureate. In 2009 he was elected a member of the National Academy of Education. In 2012 he became a fellow of the American Mathematical Society. He was awarded the Mary P. Dolciani Award in 2013.
See also
Bass number
Bass–Serre theory
Bass–Quillen conjecture
References
External links
Directory page at University of Michigan
Author profile in the database zbMATH
1932 births
20th-century American Jews
Algebraists
Columbia University faculty
Fellows of the American Academy of Arts and Sciences
Fellows of the American Mathematical Society
Living people
Mathematics educators
Members of the United States National Academy of Sciences
National Medal of Science laureates
Institute for Advanced Study visiting scholars
Nicolas Bourbaki
Presidents of the American Mathematical Society
Academics from Houston
Princeton University al
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https://en.wikipedia.org/wiki/Inverse%20image%20functor
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In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map , the inverse image functor is a functor from the category of sheaves on Y to the category of sheaves on X. The direct image functor is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features.
Definition
Suppose we are given a sheaf on and that we want to transport to using a continuous map .
We will call the result the inverse image or pullback sheaf . If we try to imitate the direct image by setting
for each open set of , we immediately run into a problem: is not necessarily open. The best we could do is to approximate it by open sets, and even then we will get a presheaf and not a sheaf. Consequently, we define to be the sheaf associated to the presheaf:
(Here is an open subset of and the colimit runs over all open subsets of containing .)
For example, if is just the inclusion of a point of , then is just the stalk of at this point.
The restriction maps, as well as the functoriality of the inverse image follows from the universal property of direct limits.
When dealing with morphisms of locally ringed spaces, for example schemes in algebraic geometry, one often works with sheaves of -modules, where is the structure sheaf of . Then the functor is inappropriate, because in general it does not even give sheaves of -modules. In order to remedy this, one defines in this situation for a sheaf of -modules its inverse image by
.
Properties
While is more complicated to define than , the stalks are easier to compute: given a point , one has .
is an exact functor, as can be seen by the above calculation of the stalks.
is (in general) only right exact. If is exact, f is called flat.
is the left adjoint of the direct image functor . This implies that there are natural unit and counit morphisms and . These morphisms yield a natural adjunction correspondence:
.
However, the morphisms and are almost never isomorphisms.
For example, if denotes the inclusion of a closed subset, the stalk of at a point is canonically isomorphic to if is in and otherwise. A similar adjunction holds for the case of sheaves of modules, replacing by .
References
. See section II.4.
Algebraic geometry
Sheaf theory
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https://en.wikipedia.org/wiki/Star%20polyhedron
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In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality.
There are two general kinds of star polyhedron:
Polyhedra which self-intersect in a repetitive way.
Concave polyhedra of a particular kind which alternate convex and concave or saddle vertices in a repetitive way. Mathematically these figures are examples of star domains.
Mathematical studies of star polyhedra are usually concerned with regular, uniform polyhedra, or the duals of the uniform polyhedra. All these stars are of the self-intersecting kind.
Self-intersecting star polyhedra
Regular star polyhedra
The regular star polyhedra are self-intersecting polyhedra. They may either have self-intersecting faces, or self-intersecting vertex figures.
There are four regular star polyhedra, known as the Kepler–Poinsot polyhedra. The Schläfli symbol {p,q} implies faces with p sides, and vertex figures with q sides. Two of them have pentagrammic {5/2} faces and two have pentagrammic vertex figures.
These images show each form with a single face colored yellow to show the visible portion of that face.
There are also an infinite number of regular star dihedra and hosohedra {2,p/q} and {p/q,2} for any star polygon {p/q}. While degenerate in Euclidean space, they can be realised spherically in nondegenerate form.
Uniform and uniform dual star polyhedra
There are many uniform star polyhedra including two infinite series, of prisms and of antiprisms, and their duals.
The uniform and dual uniform star polyhedra are also self-intersecting polyhedra. They may either have self-intersecting faces, or self-intersecting vertex figures or both.
The uniform star polyhedra have regular faces or regular star polygon faces. The dual uniform star polyhedra have regular faces or regular star polygon vertex figures.
Stellations and facettings
Beyond the forms above, there are unlimited classes of self-intersecting (star) polyhedra.
Two important classes are the stellations of convex polyhedra and their duals, the facettings of the dual polyhedra.
For example, the complete stellation of the icosahedron (illustrated) can be interpreted as a self-intersecting polyhedron composed of 20 identical faces, each a (9/4) wound polygon. Below is an illustration of this polyhedron with one face drawn in yellow.
Star polytopes
A similarly self-intersecting polytope in any number of dimensions is called a star polytope.
A regular polytope {p,q,r,...,s,t} is a star polytope if either its facet {p,q,...s} or its vertex figure {q,r,...,s,t} is a star polytope.
In four dimensions, the 10 regular star polychora are called the Schläfli–Hess polychora. Analogous to the regular star polyhedra, these 10 are all composed of facets which are either one of the five regular Platonic solids or one of the four regular star Kepler–Poinsot polyhedra.
For example, the great grand stellated 120-cell, projected orthogonally into 3-space, looks like
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https://en.wikipedia.org/wiki/Converse
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Converse may refer to:
Mathematics and logic
Converse (logic), the result of reversing the two parts of a definite or implicational statement
Converse implication, the converse of a material implication
Converse nonimplication, a logical connective which is the negation of the converse implication
Converse (semantics), pairs of words that refer to a relationship from opposite points of view
Converse accident, a logical fallacy that can occur in a statistical syllogism when an exception to a generalization is wrongly excluded
Converse relation or inverse relation, in mathematics the relation that occurs when switching the order of the elements in a binary relation
Places in the United States
Converse, Blackford County, Indiana
Converse, Indiana
Converse, Louisiana
Converse, Missouri
Converse, South Carolina
Converse, Texas
Converse County, Wyoming
Converse Basin, a grove of giant sequoia trees located in the Sequoia National Forest in the Sierra Nevada in eastern California
Vessels
USS Converse (DD-291), U.S. Navy destroyer
USS Converse (DD-509), U.S. Navy destroyer
Other uses
Converse (surname), various people with the surname
Converse (lifestyle wear), an American shoe and clothing company
Converses or Chuck Taylor All-Stars, canvas and rubber shoes produced by the company
Converse College, a women's college in Spartanburg, South Carolina
Conversation, a form of communication between people following rules of etiquette
Converse technique, a standard method in ear reconstruction
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https://en.wikipedia.org/wiki/Abel%27s%20test
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In mathematics, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Henrik Abel. There are two slightly different versions of Abel's test – one is used with series of real numbers, and the other is used with power series in complex analysis. Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions dependent on parameters.
Abel's test in real analysis
Suppose the following statements are true:
is a convergent series,
{bn} is a monotone sequence, and
{bn} is bounded.
Then is also convergent.
It is important to understand that this test is mainly pertinent and
useful in the context of non absolutely convergent series .
For absolutely convergent series, this theorem, albeit true, is almost self evident.
This theorem can be proved directly using summation by parts.
Abel's test in complex analysis
A closely related convergence test, also known as Abel's test, can often be used to establish the convergence of a power series on the boundary of its circle of convergence. Specifically, Abel's test states that if a sequence of positive real numbers is decreasing monotonically (or at least that for all n greater than some natural number m, we have ) with
then the power series
converges everywhere on the closed unit circle, except when z = 1. Abel's test cannot be applied when z = 1, so convergence at that single point must be investigated separately. Notice that Abel's test implies in particular that the radius of convergence is at least 1. It can also be applied to a power series with radius of convergence R ≠ 1 by a simple change of variables ζ = z/R. Notice that Abel's test is a generalization of the Leibniz Criterion by taking z = −1.
Proof of Abel's test: Suppose that z is a point on the unit circle, z ≠ 1. For each , we define
By multiplying this function by (1 − z), we obtain
The first summand is constant, the second converges uniformly to zero (since by assumption the sequence converges to zero). It only remains to show that the series converges. We will show this by showing that it even converges absolutely:
where the last sum is a converging telescoping sum. The absolute value vanished because the sequence is decreasing by assumption.
Hence, the sequence converges (even uniformly) on the closed unit disc. If , we may divide by (1 − z) and obtain the result.
Another way to obtain the result is to apply the Dirichlet's test. Indeed, for holds , hence the assumptions of the Dirichlet's test are fulfilled.
Abel's uniform convergence test
Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions or an improper integration of functions dependent on parameters. It is related to Abel's test for the convergence of an ordinary series of real numbers, and the proof relies on the same technique of summation by parts.
The test is as follows. Le
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https://en.wikipedia.org/wiki/Ar%C5%ABnas%20Matelis
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Arūnas Matelis (born 9 April 1961, in Kaunas) is a Lithuanian documentary film director. From 1979 till 1983 Arūnas Matelis studied Mathematics at Vilnius University and later in 1989 graduated from the Lithuanian Music Academy. In 1992, he established one of the first independent film production companies in Lithuania, "Nominum". In 2006 Matelis became a full member of European Film Academy with the right to vote.
Filmography
Pelesos milžinai (1989)
Baltijos kelias (1989)
Dešimt minučių prieš Ikaro skrydį (1991)
Autoportretas (1993)
Iš dar nebaigtų Jeruzalės pasakų (1996)
Pirmasis atsisveikinimas su Rojum (1998)
Priverstinės emigracijos dienoraštis (1999)
Skrydis per Lietuvą arba 510 sekundžių tylos (Flight over Lithuania or 510 seconds of silence) (2000)
Sekmadienis. Evangelija pagal liftininką Albertą (2003)
Prieš parskrendant į žemę (Before Flying Back to Earth) (2005)
Wonderful Losers: A Different World (2017)
Awards
Matelis is one of the recipients of the Lithuanian National Prize of 2005.
"Prieš parskrendant į žemę", the first feature-length documentary by Matelis about children hospitalized with leukemia, is the most highly acclaimed Lithuanian film and is considered one of the best European documentary films of 2005, awarded in numerous festivals:
Best documentary in Directors Guild of America Awards 2006
Best Lithuanian Film 2005 by Lithuanian Filmmakers Union
Silver Wolf in International Documentary Film Festival Amsterdam (IDFA), 2005
Golden Dove in International Leipzig Festival for Documentary and Animated Film, 2005
Main Prize in Festival Documenta Madrid, 2005
"Spirit Award for Documentary" in Brooklyn International Film Festival, 2006
Grand Prix in Pärnu International Film Festival, 2006
Special Jury Mention in Silverdocs Festival, 2006
Veliki pečat international competition award in ZagrebDox Film Festival, 2006
The film is nominated for the European Film Academy best European documentary award of 2005.
Further reading
Arūnas Matelis: Films Emerge from Sensations in: Lithuanian Cinema: Special Edition for Lithuanian Film Days in Poland 2015, Auksė Kancerevičiūtė [ed.]. Vilnius: Lithuanian Film Centre, 2015. .
References
External links
Personal Website
Lithuanian film directors
Lithuanian documentary film directors
Living people
1961 births
Vilnius University alumni
Lithuanian Academy of Music and Theatre alumni
Directors Guild of America Award winners
Film people from Kaunas
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https://en.wikipedia.org/wiki/Soviet%20Union%20men%27s%20national%20handball%20team
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The USSR national handball team was the national handball team of the Soviet Union.
World Championships Record
Summer Olympics Record
Player statistics
Most appearances
100+
Top scorers
250+
National teams of the former Soviet republics
See also
Soviet Union women's national handball team
Russia men's national handball team
Russia women's national handball team
External links
Former national handball teams
National sports teams of the Soviet Union
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https://en.wikipedia.org/wiki/Zappa%E2%80%93Sz%C3%A9p%20product
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In mathematics, especially group theory, the Zappa–Szép product (also known as the Zappa–Rédei–Szép product, general product, knit product, exact factorization or bicrossed product) describes a way in which a group can be constructed from two subgroups. It is a generalization of the direct and semidirect products. It is named after Guido Zappa (1940) and Jenő Szép (1950) although it was independently studied by others including B.H. Neumann (1935), G.A. Miller (1935), and J.A. de Séguier (1904).
Internal Zappa–Szép products
Let G be a group with identity element e, and let H and K be subgroups of G. The following statements are equivalent:
G = HK and H ∩ K = {e}
For each g in G, there exists a unique h in H and a unique k in K such that g = hk.
If either (and hence both) of these statements hold, then G is said to be an internal Zappa–Szép product of H and K.
Examples
Let G = GL(n,C), the general linear group of invertible n × n matrices over the complex numbers. For each matrix A in G, the QR decomposition asserts that there exists a unique unitary matrix Q and a unique upper triangular matrix R with positive real entries on the main diagonal such that A = QR. Thus G is a Zappa–Szép product of the unitary group U(n) and the group (say) K of upper triangular matrices with positive diagonal entries.
One of the most important examples of this is Philip Hall's 1937 theorem on the existence of Sylow systems for soluble groups. This shows that every soluble group is a Zappa–Szép product of a Hall p'-subgroup and a Sylow p-subgroup, and in fact that the group is a (multiple factor) Zappa–Szép product of a certain set of representatives of its Sylow subgroups.
In 1935, George Miller showed that any non-regular transitive permutation group with a regular subgroup is a Zappa–Szép product of the regular subgroup and a point stabilizer. He gives PSL(2,11) and the alternating group of degree 5 as examples, and of course every alternating group of prime degree is an example. This same paper gives a number of examples of groups which cannot be realized as Zappa–Szép products of proper subgroups, such as the quaternion group and the alternating group of degree 6.
External Zappa–Szép products
As with the direct and semidirect products, there is an external version of the Zappa–Szép product for groups which are not known a priori to be subgroups of a given group. To motivate this, let G = HK be an internal Zappa–Szép product of subgroups H and K of the group G. For each k in K and each h in H, there exist α(k, h) in H and β(k, h) in K such that kh = α(k, h) β(k, h). This defines mappings α : K × H → H and β : K × H → K which turn out to have the following properties:
α(e, h) = h and β(k, e) = k for all h in H and k in K.
α(k1k2, h) = α(k1, α(k2, h))
β(k, h1h2) = β(β(k, h1), h2)
α(k, h1h2) = α(k, h1) α(β(k, h1), h2)
β(k1k2, h) = β(k1, α(k2, h)) β(k2, h)
for all h1, h2 in H, k1, k2 in K. From these, it follows that
For each k in K, the mapp
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https://en.wikipedia.org/wiki/Feodor%20Deahna
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Heinrich Wilhelm Feodor Deahna (8 July 1815 – 8 January 1844) was a German mathematician.
He is known for providing proof of what is now known as Frobenius theorem in differential topology, which he published in Crelle's Journal in 1840.
Deahna was born near Bayreuth on July 8, 1815, and was a student at the University of Göttingen in 1834. In 1843 he became an assistant mathematics teacher at the Fulda Gymnasium, but he died soon afterwards in Fulda, on January 8, 1844.
Selected works
Deahna, F. "Über die Bedingungen der Integrabilitat ....", J. Reine Angew. Math. 20 (1840) 340-350.
References
1815 births
1844 deaths
19th-century German mathematicians
Differential geometers
Mathematicians from the Kingdom of Bavaria
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https://en.wikipedia.org/wiki/Gysin%20homomorphism
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In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere bundle. The Gysin sequence is a useful tool for calculating the cohomology rings given the Euler class of the sphere bundle and vice versa. It was introduced by , and is generalized by the Serre spectral sequence.
Definition
Consider a fiber-oriented sphere bundle with total space E, base space M, fiber Sk and projection map
:
Any such bundle defines a degree k + 1 cohomology class e called the Euler class of the bundle.
De Rham cohomology
Discussion of the sequence is clearest with de Rham cohomology. There cohomology classes are represented by differential forms, so that e can be represented by a (k + 1)-form.
The projection map induces a map in cohomology called its pullback
In the case of a fiber bundle, one can also define a pushforward map
which acts by fiberwise integration of differential forms on the oriented sphere – note that this map goes "the wrong way": it is a covariant map between objects associated with a contravariant functor.
Gysin proved that the following is a long exact sequence
where is the wedge product of a differential form with the Euler class e.
Integral cohomology
The Gysin sequence is a long exact sequence not only for the de Rham cohomology of differential forms, but also for cohomology with integral coefficients. In the integral case one needs to replace the wedge product with the Euler class with the cup product, and the pushforward map no longer corresponds to integration.
Gysin homomorphism in algebraic geometry
Let i: X → Y be a (closed) regular embedding of codimension d, Y → Y a morphism and i: X = X ×Y Y → Y the induced map. Let N be the pullback of the normal bundle of i to X. Then the refined Gysin homomorphism i! refers to the composition
where
σ is the specialization homomorphism; which sends a k-dimensional subvariety V to the normal cone to the intersection of V and X in V. The result lies in N through .
The second map is the (usual) Gysin homomorphism induced by the zero-section embedding .
The homomorphism i! encodes intersection product in intersection theory in that one either shows the intersection product of X and V to be given by the formula or takes this formula as a definition.
Example: Given a vector bundle E, let s: X → E be a section of E. Then, when s is a regular section, is the class of the zero-locus of s, where [X] is the fundamental class of X.
See also
Logarithmic form
Wang sequence
Notes
Sources
Algebraic topology
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https://en.wikipedia.org/wiki/John%20Ockendon
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Professor John Richard Ockendon FRS (born c. 1940) is an applied mathematician noted especially for his contribution to fluid dynamics and novel applications of mathematics to real world problems. He is a professor at the University of Oxford and an Emeritus Fellow at St Catherine's College, Oxford, the first director of the Oxford Centre for Collaborative Applied Mathematics (OCCAM) and a former director of the Smith Institute for Industrial Mathematics and System Engineering.
Education
Ockendon was educated at the University of Oxford where he was awarded a Doctor of Philosophy degree in 1965 for research on fluid dynamics supervised by Alan B Tayler.
Research and career
His initial fluid mechanics interests included hypersonic aerodynamics, creeping flow, sloshing and channel flows and leading to flows in porous media, ship hydrodynamics and models for flow separation.
He moved on to free and moving boundary problems. He pioneered the study of diffusion-controlled moving boundary problems in the 1970s his involvement centring on models for phase changes and elastic contact problems all built around the paradigm of the Hele-Shaw free boundary problem. Other industrial collaboration has led to new ideas for lens design, fibre manufacture, extensional and surface-tension- driven flows and glass manufacture, fluidised-bed models, semiconductor device modelling and a range of other problems in mechanics and heat and mass transfer, especially scattering and ray theory, nonlinear wave propagation, nonlinear oscillations, nonlinear diffusion and impact in solids and liquids.
His efforts to promote mathematical collaboration with industry led him to organise annual meetings of the Study Groups with Industry from 1972 to 1989.
Awards and honours
He was elected Fellow of the Royal Society in 1999, and awarded the Gold Medal of the Institute of Mathematics and its Applications in 2006.
Personal life
Ockendon is married to coauthor and colleague Hilary Ockendon.
In Who's Who he lists his recreations as mathematical modelling, bird watching, Hornby-Dublo model trains and old sports cars.
References
Fellows of the Royal Society
Living people
1940s births
20th-century British mathematicians
21st-century British mathematicians
Fellows of the Society for Industrial and Applied Mathematics
Fellows of St Catherine's College, Oxford
Fluid dynamicists
Alumni of the University of Oxford
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https://en.wikipedia.org/wiki/Structure%20tensor
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In mathematics, the structure tensor, also referred to as the second-moment matrix, is a matrix derived from the gradient of a function. It describes the distribution of the gradient in a specified neighborhood around a point and makes the information invariant respect the observing coordinates. The structure tensor is often used in image processing and computer vision.
The 2D structure tensor
Continuous version
For a function of two variables , the structure tensor is the 2×2 matrix
where and are the partial derivatives of with respect to x and y; the integrals range over the plane ; and w is some fixed "window function" (such as a Gaussian blur), a distribution on two variables. Note that the matrix is itself a function of .
The formula above can be written also as , where is the matrix-valued function defined by
If the gradient of is viewed as a 2×1 (single-column) matrix, where denotes transpose operation, turning a row vector to a column vector, the matrix can be written as the matrix product or tensor or outer product . Note however that the structure tensor cannot be factored in this way in general except if is a Dirac delta function.
Discrete version
In image processing and other similar applications, the function is usually given as a discrete array of samples , where p is a pair of integer indices. The 2D structure tensor at a given pixel is usually taken to be the discrete sum
Here the summation index r ranges over a finite set of index pairs (the "window", typically for some m), and w[r] is a fixed "window weight" that depends on r, such that the sum of all weights is 1. The values are the partial derivatives sampled at pixel p; which, for instance, may be estimated from by by finite difference formulas.
The formula of the structure tensor can be written also as , where is the matrix-valued array such that
Interpretation
The importance of the 2D structure tensor stems from the fact eigenvalues (which can be ordered so that ) and the corresponding eigenvectors summarize the distribution of the gradient of within the window defined by centered at .
Namely, if , then (or ) is the direction that is maximally aligned with the gradient within the window.
In particular, if then the gradient is always a multiple of (positive, negative or zero); this is the case if and only if within the window varies along the direction but is constant along . This condition of eigenvalues is also called linear symmetry condition because then the iso-curves of consist in parallel lines, i.e there exists a one dimensional function which can generate the two dimensional function as for some constant vector and the coordinates .
If , on the other hand, the gradient in the window has no predominant direction; which happens, for instance, when the image has rotational symmetry within that window. This condition of eigenvalues is also called balanced body, or directional equilibrium condition because it holds when all g
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https://en.wikipedia.org/wiki/Ruth%20Aaronson%20Bari
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Ruth Aaronson Bari (November 17, 1917 – August 25, 2005) was an American mathematician known for her work in graph theory and algebraic homomorphisms. She was a professor at George Washington University, beginning in 1966.
Career
The daughter of Polish-Jewish immigrants to the United States, Ruth Aaronson was born November 17, 1917, and grew up in Brooklyn, New York.
She attended Brooklyn College, earning her bachelor's degree in mathematics in 1939. She earned her Master of Arts degree at Johns Hopkins University in 1943, but had originally enrolled in the doctoral program. When the university suggested that women in the graduate program should give up their fellowships so that men returning from World War II could study, Bari acceded. After marrying Arthur Bari, she spent the next two decades devoted to their family. They had three daughters together.
She returned to Johns Hopkins for graduate work, and completed her dissertation on "absolute reducibility of maps of at most 19 regions" in 1966 at the age of 47. Bari's dissertation explored chromatic polynomials and the Birkhoff–Lewis conjecture. She determined that, "because of the fact that all other cubic maps with fewer than 20 regions contain at least one absolutely reducible configuration, it follows that the Birkhoff-Lewis conjecture holds for all maps with fewer than 20 regions." Her Ph.D. advisor was Daniel Clark Lewis, Jr. and her thesis was titled, Absolute Reducibility of Maps of at Most 19 Regions.
After she received her degree, mathematician William Tutte invited Bari to spend two weeks lecturing on her work in Canada at the University of Waterloo. Bari's work in the areas of graph theory and homomorphisms—and especially chromatic polynomials—has been recognized as influential.
In 1976, two professors relied on computer work to solve the perennial problem of Bari's dissertation, involving the four-color conjecture. When her daughter Martha asked her if she felt cheated by the technological solution, Bari replied, "I’m just grateful that it was solved within my lifetime and that I had the privilege to witness it."
During her teaching career, Bari participated in a class-action lawsuit against George Washington University which protested inequalities in promotion and pay for female faculty members. The protests were successful. Notable students of Bari include Carol Crawford, Steven Kahn, and Lee Lawrence.
Bari retired at the legally mandated age of 70 in 1988 with the distinction of professor emeritus.
Community and personal life
Bari was active in the Washington, DC community. In the early 1970s, Bari used a grant from the National Science Foundation to start a master's degree program in teaching mathematics. She felt that math teachers in DC public schools were not as well prepared as they needed to be.
Her three daughters became influential in their fields. Judi Bari (1949–1997) was a leading labor and environmental activist and feminist, who lived and worked in Northern
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https://en.wikipedia.org/wiki/Dorothy%20Lewis%20Bernstein
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Dorothy Lewis Bernstein (April 11, 1914 – February 5, 1988) was an American mathematician known for her work in applied mathematics, statistics, computer programming, and her research on the Laplace transform. She was the first woman to be elected president of the Mathematics Association of America.
Early life
Bernstein was born in Chicago, the daughter of Jewish Russian immigrants Jacob and Tille Lewis Bernstein. While her parents had no formal education, they encouraged all of their children to seek education; all five earned either a PhD or MD.
Education
Bernstein attended North Division High School (Milwaukee) in Milwaukee, Wisconsin. In 1930 she attended the University of Wisconsin, where she held a University Scholarship (1933–1934) and was elected to Phi Beta Kappa. In 1934 she graduated with both a B.A degree, summa cum laude, and a M.A. Degree in Mathematics. She did her master's thesis research on finding complex roots of polynomials by an extension of Newton's method. In 1935 she attended Brown University, where she became a member of the scientific society Sigma Xi. She received her Ph.D. in mathematics from Brown in 1939, while simultaneously holding a teaching position at Mount Holyoke College. Her dissertation was entitled "The Double Laplace Integral" and was published in the Duke Mathematical Journal.
Career
From 1943 to 1959 Bernstein taught at the University of Rochester, where she worked on existence theorems for partial differential equations. Her work was motivated by non-linear problems that were just being tackled by high-speed digital computers. In 1950, Princeton University Press published her book, Existence Theorems in Partial Differential Equations.
She spent 1959–1979 as a professor of mathematics at Goucher College, where she was chairman of the mathematics department for most of that time (1960–70, 1974–79).
She professed that she was particularly interested combining pure and applied mathematics in the undergraduate curriculum. Due in great part to Bernstein's ability to get grants from the National Science Foundation, Goucher College was the first women's university to use computers in mathematics instruction, beginning in 1961. She also developed an internship program for Goucher mathematics students to obtain meaningful employment experience. In 1972 Bernstein cofounded the Maryland Association for Educational Uses of Computers, and was interested in incorporating computers into secondary school mathematics.
Bernstein was very active in the Mathematical Association of America, where she was on the board of governors from 1965 to 1968. She served as the vice president in 1972–73, and later became the first female president of the MAA in 1979–80.
Women in mathematics
She noted that attitudes and opportunities for women changed drastically after World War II, which she attributed to two causes. First, that women demonstrated they could handle the jobs formerly held by men, and second that the
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https://en.wikipedia.org/wiki/Lutz%E2%80%93Kelker%20bias
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The Lutz–Kelker bias is a supposed systematic bias that results from the assumption that the probability of a star being at distance increases with the square of the distance which is equivalent to the assumption that the distribution of stars in space is uniform. In particular, it causes measured parallaxes to stars to be larger than their actual values. The bias towards measuring larger parallaxes in turn results in an underestimate of distance and therefore an underestimate on the object's luminosity.
For a given parallax measurement with an accompanying uncertainty, both stars closer and farther may, because of uncertainty in measurement, appear at the given parallax. Assuming uniform stellar distribution in space, the probability density of the true parallax per unit range of parallax will be proportional to (where is the true parallax), and therefore, there will be more stars in the volume shells at farther distance. As a result of this dependence, more stars will have their true parallax smaller than the observed parallax. Thus, the measured parallax will be systematically biased towards a value larger than the true parallax. This causes inferred luminosities and distances to be too small, which poses an apparent problem to astronomers trying to measure distance. The existence (or otherwise) of this bias and the necessity of correcting for it has become relevant in astronomy with the precision parallax measurements made by the Hipparcos satellite and more recently with the high-precision data releases of the Gaia mission.
The correction method due to Lutz and Kelker placed a bound on the true parallax of stars. This is not valid because true parallax (as distinct from measured parallax) cannot be known. Integrating over all true parallaxes (all space) assumes that stars are equally visible at all distances, and leads to divergent integrals yielding an invalid calculation. Consequently, the Lutz-Kelker correction should not be used. In general, other corrections for systematic bias are required, depending on the selection criteria of the stars under consideration.
The scope of effects of the bias are also discussed in the context of the current higher-precision measurements and the choice of stellar sample where the original stellar distribution assumptions are not valid. These differences result in the original discussion of effects to be largely overestimated and highly dependent on the choice of stellar sample. It also remains possible that relations to other forms of statistical bias such as the Malmquist bias may have a counter-effect on the Lutz–Kelker bias for at least some samples.
Mathematical Description
Original Description
The Distribution Function
Mathematically, the Lutz-Kelker Bias originates from the dependence of the number density on the observed parallax that is translated into the conditional probability of parallax measurements. Assuming a Gaussian distribution of the observed parallax about the true parallax
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https://en.wikipedia.org/wiki/Kanti%20Mardia
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Kantilal Vardichand "Kanti" Mardia (born 1935) is an Indian-British statistician specialising in directional statistics, multivariate analysis, geostatistics, statistical bioinformatics and statistical shape analysis. He was born in Sirohi, Rajasthan, India in a Jain family and now resides and works in Leeds. He is known for his series of tests of multivariate normality based measures of multivariate skewness and kurtosis as well as work on the statistical measures of shape.
Life and career
Mardia was educated at the Ismail Yusuf College at the University of Bombay (BSc 1955, MSc in statistics 1957), the University of Poona (MSc in pure mathematics 1961), the University of Rajasthan (PhD 1965) and the University of Newcastle Upon Tyne (PhD 1967, DSc 1973). He held academic positions at the Institute of Science, Mumbai and the University of Hull.
Mardia was appointed professor of applied statistics and head of the Department of Statistics in the School of Mathematics at the University of Leeds in 1973. He retired in 2000 with the title emeritus professor and is currently senior research professor of applied statistics at Leeds. He is also a long-term visiting professor at the University of Oxford, from March 2013, and the Indian Institute of Management Ahmedabad (IIMA), from 2008.
He was instrumental in founding the Centre of Medical Imaging Research (CoMIR) in the University of Leeds, where he held the position of joint director. He was the driving force behind the exchange programs between Leeds and other scholarly centres such as the University of Granada, Spain, and the Indian Statistical Institute, Calcutta. He has written several scholarly books and edited conference proceedings and other special volumes.
In 1973, Mardia founded the University of Leeds Annual Statistics Research Workshops (LASR) which have run for most years and he has edited all the proceedings. These workshops attract an international audience and focus on applied statistical topics especially those involving shape and images, and more recently, bioinformatics.
In 2003, he was awarded the Guy Medal in Silver by the Royal Statistical Society.
In 2013, he was awarded the Wilks Memorial Award by the American Statistical Association.
In 2019, he was awarded the Lifetime Achievement Award by the International Indian Statistical Association.
In 2020, he was awarded the Mahatma Gandhi Medal of Honour by the NRI Institute.
In 2021, he received the OneJAIN Life Achievement Award from the Jain All-Party Parliamentary Group.
The 40th Fisher Memorial Lecture was given by Professor Mardia on 18 November 2022 at the Oxford Mathematical Institute.
He is a practicing Jain and strict vegetarian. His 1990 book The Scientific Foundations of Jainism introduced the Four Noble Truths of Jains. He is the founding and current chairman of the Yorkshire Jain Foundation.
Mardia was appointed Officer of the Order of the British Empire (OBE) in the 2023 New Year Honours for services to s
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https://en.wikipedia.org/wiki/Andrei%20Suslin
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Andrei Suslin (, sometimes transliterated Souslin) was a Russian mathematician who contributed to algebraic K-theory and its connections with algebraic geometry. He was a Trustee Chair and Professor of mathematics at Northwestern University.
He was born on 27 December 1950 in St. Petersburg, Russia. As a youth, he was an "all Leningrad" gymnast. He received his PhD from Leningrad University in 1974; his thesis was titled Projective modules over polynomial rings.
In 1976 he and Daniel Quillen independently proved Serre's conjecture about the triviality of algebraic vector bundles on affine space.
In 1982 he and Alexander Merkurjev proved the Merkurjev–Suslin theorem on the norm residue homomorphism in Milnor K2-theory, with applications to the Brauer group.
Suslin was an invited speaker at the International Congress of Mathematicians in 1978 and 1994, and he gave a plenary invited address at the Congress in 1986. He was awarded the Frank Nelson Cole Prize in Algebra in 2000 by the American Mathematical Society for his work on motivic cohomology.
In 2010 special issues of Journal of K-theory
and of Documenta Mathematica
have been published in honour of his 60th birthday.
He died on 10 July 2018.
References
External links
Anfrei Suslin, faculty profile, Department of Mathematics, Northwestern University
1950 births
2018 deaths
Northwestern University faculty
20th-century Russian mathematicians
21st-century Russian mathematicians
Algebraic geometers
Institute for Advanced Study visiting scholars
International Mathematical Olympiad participants
Mathematicians from Saint Petersburg
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https://en.wikipedia.org/wiki/Wick%20product
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In probability theory, the Wick product is a particular way of defining an adjusted product of a set of random variables. In the lowest order product the adjustment corresponds to subtracting off the mean value, to leave a result whose mean is zero. For the higher order products the adjustment involves subtracting off lower order (ordinary) products of the random variables, in a symmetric way, again leaving a result whose mean is zero. The Wick product is a polynomial function of the random variables, their expected values, and expected values of their products.
The definition of the Wick product immediately leads to the Wick power of a single random variable and this allows analogues of other functions of random variables to be defined on the basis of replacing the ordinary powers in a power-series expansions by the Wick powers. The Wick powers of commonly-seen random variables can be expressed in terms of special functions such as Bernoulli polynomials or Hermite polynomials.
The Wick product is named after physicist Gian-Carlo Wick, cf. Wick's theorem.
Definition
Assume that X1, ..., Xk are random variables with finite moments. The Wick product
is a sort of product defined recursively as follows:
(i.e. the empty product—the product of no random variables at all—is 1). For k ≥ 1, we impose the requirement
where means that Xi is absent, together with the constraint that the average is zero,
Equivalently, the Wick product can be defined by writing the monomial as a "Wick polynomial":
,
where denotes the Wick product if . This is easily seen to satisfy the inductive definition.
Examples
It follows that
Another notational convention
In the notation conventional among physicists, the Wick product is often denoted thus:
and the angle-bracket notation
is used to denote the expected value of the random variable X.
Wick powers
The nth Wick power of a random variable X is the Wick product
with n factors.
The sequence of polynomials Pn such that
form an Appell sequence, i.e. they satisfy the identity
for n = 0, 1, 2, ... and P0(x) is a nonzero constant.
For example, it can be shown that if X is uniformly distributed on the interval [0, 1], then
where Bn is the nth-degree Bernoulli polynomial. Similarly, if X is normally distributed with variance 1, then
where Hn is the nth Hermite polynomial.
Binomial theorem
Wick exponential
References
Wick Product Springer Encyclopedia of Mathematics
Florin Avram and Murad Taqqu, (1987) "Noncentral Limit Theorems and Appell Polynomials", Annals of Probability, volume 15, number 2, pages 767—775, 1987.
Hida, T. and Ikeda, N. (1967) "Analysis on Hilbert space with reproducing kernel arising from multiple Wiener integral". Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66). Vol. II: Contributions to Probability Theory, Part 1 pp. 117–143 Univ. California Press
Wick, G. C. (1950) "The evaluation of the collision matrix". Physical Rev. 80 (2), 268–272.
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https://en.wikipedia.org/wiki/Henry%20Tamburin
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Henry Tamburin (born 1944) is a gambling author with a background in mathematics and a doctorate in chemistry. He is best known for his book Blackjack: Take the Money and Run which explains basic blackjack strategy, managing a bankroll, side bets and advanced tactics like card counting.
Tamburin is also well known for his prowess as a blackjack player and frequently teaches courses in blackjack across the United States. He has published 700 articles on various casino games from craps to video poker in publications like The Gambler Magazine, Gaming South Magazine, Strictly Slots and Casino Player Magazine. Henry Tamburin never considered himself to be a real blackjack expert though all critics ascribe such a status to him. He had been working as a manager of chemical company (an international one) for 30 years and always loved his work; after being retired he began to devote more time to the game of blackjack and became interested in video poker too.
Tamburin also appeared in a televised blackjack tournament entitled the Ultimate Blackjack Tour, which aired on CBS. He is currently editor and publisher of the Blackjack Insider Newsletter and runs his own website called Smart Gaming.
References
External links
Official site
American blackjack players
American gambling writers
American male non-fiction writers
Living people
1944 births
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https://en.wikipedia.org/wiki/Hall%27s%20conjecture
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In mathematics, Hall's conjecture is an open question, , on the differences between perfect squares and perfect cubes. It asserts that a perfect square y2 and a perfect cube x3 that are not equal must lie a substantial distance apart. This question arose from consideration of the Mordell equation in the theory of integer points on elliptic curves.
The original version of Hall's conjecture, formulated by Marshall Hall, Jr. in 1970, says that there is a positive constant C such that for any integers x and y for which y2 ≠ x3,
Hall suggested that perhaps C could be taken as 1/5, which was consistent with all the data known at the time the conjecture was proposed. Danilov showed in 1982 that the exponent 1/2 on the right side (that is, the use of |x|1/2) cannot be replaced by any higher power: for no δ > 0 is there a constant C such that |y2 - x3| > C|x|1/2 + δ whenever y2 ≠ x3.
In 1965, Davenport proved an analogue of the above conjecture in the case of polynomials:
if f(t) and g(t) are nonzero polynomials over C such that
g(t)3 ≠ f(t)2 in C[t], then
The weak form of Hall's conjecture, stated by Stark and Trotter around 1980, replaces the square root on the right side of the inequality by any exponent less than 1/2: for any ε > 0, there is some constant c(ε) depending on ε such that for any integers x and y for which y2 ≠ x3,
The original, strong, form of the conjecture with exponent 1/2 has never been disproved, although it is no longer believed to be true and the term Hall's conjecture now generally means the version with the ε in it. For example, in 1998, Noam Elkies found the example
4478849284284020423079182 - 58538865167812233 = -1641843,
for which compatibility with Hall's conjecture would require C to be less than .0214 ≈ 1/50, so roughly 10 times smaller than the original choice of 1/5 that Hall suggested.
The weak form of Hall's conjecture would follow from the ABC conjecture. A generalization to other perfect powers is Pillai's conjecture, though it is also known that Pillai's conjecture would be true if Hall's conjecture held for any specific 0 < ε < 1/2.
The table below displays the known cases with . Note that y can be computed as the
nearest integer to x3/2.
References
Elkies, N.D. "Rational points near curves and small nonzero | 'x3 - y2'| via lattice reduction", http://arxiv.org/abs/math/0005139
Danilov, L.V., "The Diophantine equation 'x3 - y2 ' ' = k ' and Hall's conjecture", 'Math. Notes Acad. Sci. USSR' 32(1982), 617-618.
Gebel, J., Pethö, A., and Zimmer, H.G.: "On Mordell's equation", 'Compositio Math.' 110(1998), 335-367.
I. Jiménez Calvo, J. Herranz and G. Sáez Moreno, "A new algorithm to search for small nonzero |'x3 - y2'| values", 'Math. Comp.' 78 (2009), pp. 2435-2444.
S. Aanderaa, L. Kristiansen and H. K. Ruud, "Search for good examples of Hall's conjecture", 'Math. Comp.' 87 (2018), 2903-2914.
External links
a page on the problem by Noam Elkies
Conjectures
Unsolved problems in number theo
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https://en.wikipedia.org/wiki/Norwegian%20statistics%20by%20ethnic%20group
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The following selected statistics about ethnic groups living in Norway have been extracted from the results of the Norwegian census.
Average income for couples with children
Listed in Norwegian kroner
Home ownership percentage
Percentages that have been given penalty from Norwegian court
Percentage of people under 67 unable to earn for a living
Higher Education Ages from 19-24
Employees by immigrant groups
listed in percentage for all ages
Number of reported sick
listed in percentage days of valid sick report given from doctor
References
External links
http://ssb.no/emner/02/aktuell_befolkning/200002/ab20002.pdf
http://www.ssb.no/emner/02/sa_innvand/sa66/
http://www.ssb.no/emner/00/02/notat_200466/notat_200466.pdf
Demographics of Norway
Population statistics
Norway-related lists
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https://en.wikipedia.org/wiki/Urban%20cluster%20%28France%29
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In France, a pôle urbain (English: urban cluster) is a statistical area defined by INSEE (France's national statistics office) for the measurement of contiguously built-up areas. It shares the same definition as an unité urbaine ("urban unit"), except that a pôle urbain is not contained within the couronne ("commuter belt") of any other; in other words, a pôle urbain is an urban area that is a core of demographic growth.
See also
urban area
unité urbaine
aire urbaine
References
Urban planning in France
Human habitats
INSEE concepts
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https://en.wikipedia.org/wiki/Exponent%20%28disambiguation%29
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Exponentiation is a mathematical operation.
Exponent may also refer to:
Mathematics
List of exponential topics
Exponential function, a function of a certain form
Matrix exponential, a matrix function on square matrices
The least common multiple of a periodic group
Statistics
Exponential distribution, a probability distribution
Exponential family, a parametric set of probability distributions of a certain form
Exponential growth, a specific way that a quantity may increase over time
Exponential decay, decreasing quantity at a rate proportional to current value
Linguistics
Exponent (linguistics), the expression of one or more grammatical properties by sound.
Music
The Exponents, a New Zealand rock group
Publications
Purdue Exponent, a student newspaper of Purdue University
Woman's Exponent, a publication of The Church of Jesus Christ of Latter-day Saints
Exponent II, a quarterly periodical for Latter-day Saint women
The Exponent (Montana State University), a student newspaper of Montana State University – Bozeman
The Brooklyn Exponent, a weekly newspaper serving communities in Michigan
The Jewish Exponent, a weekly community newspaper in Philadelphia, Pennsylvania
Companies
Exponent (consulting firm), an American engineering and scientific consulting firm
Other uses
Currency exponent, used in ISO 4271
Exponent CMS, an enterprise software framework and content management system
Exponent (podcast), podcast co-hosted by Ben Thompson
See also
Exponential (disambiguation)
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https://en.wikipedia.org/wiki/Robert%20Boyer
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Robert Boyer may refer to:
Robert S. Boyer, professor of computer science, mathematics, and philosophy
See List of Charles Whitman's victims for Robert Hamilton Boyer, professor killed at The University of Texas in 1966
Robert Boyer (artist) (1948–2004), Canadian artist of aboriginal heritage
Robert Boyer (chemist) (1909–1989), chemist employed by Henry Ford
Robert James Boyer (1913–2005), former politician in Ontario, Canada
Bob Boyer (wrestler), retired Canadian professional wrestler
See also
Robert Boyers (1876–1949), American football coach
Robert Bowyer (1758–1834), British painter and publisher
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https://en.wikipedia.org/wiki/Wedderburn%27s%20little%20theorem
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In mathematics, Wedderburn's little theorem states that every finite division ring is a field. In other words, for finite rings, there is no distinction between domains, division rings and fields.
The Artin–Zorn theorem generalizes the theorem to alternative rings: every finite alternative division ring is a field.
History
The original proof was given by Joseph Wedderburn in 1905, who went on to prove it two other ways. Another proof was given by Leonard Eugene Dickson shortly after Wedderburn's original proof, and Dickson acknowledged Wedderburn's priority. However, as noted in , Wedderburn's first proof was incorrect – it had a gap – and his subsequent proofs appeared only after he had read Dickson's correct proof. On this basis, Parshall argues that Dickson should be credited with the first correct proof.
A simplified version of the proof was later given by Ernst Witt. Witt's proof is sketched below. Alternatively, the theorem is a consequence of the Skolem–Noether theorem by the following argument. Let be a finite division algebra with center . Let and denote the cardinality of . Every maximal subfield of has elements; so they are isomorphic and thus are conjugate by Skolem–Noether. But a finite group (the multiplicative group of in our case) cannot be a union of conjugates of a proper subgroup; hence, .
A later "group-theoretic" proof was given by Ted Kaczynski in 1964. This proof, Kaczynski's first published piece of mathematical writing, was a short, two-page note which also acknowledged the earlier historical proofs.
Relationship to the Brauer group of a finite field
The theorem is essentially equivalent to saying that the Brauer group of a finite field is trivial. In fact, this characterization immediately yields a proof of the theorem as follows: let k be a finite field. Since the Herbrand quotient vanishes by finiteness, coincides with , which in turn vanishes by Hilbert 90.
Proof
Let A be a finite domain. For each nonzero x in A, the two maps
are injective by the cancellation property, and thus, surjective by counting. It follows from the elementary group theory that the nonzero elements of form a group under multiplication. Thus, is a skew-field.
To prove that every finite skew-field is a field, we use strong induction on the size of the skew-field. Thus, let be a skew-field, and assume that all skew-fields that are proper subsets of are fields. Since the center of is a field, is a vector space over with finite dimension . Our objective is then to show . If is the order of , then has order . Note that because contains the distinct elements and , . For each in that is not in the center, the centralizer of is clearly a skew-field and thus a field, by the induction hypothesis, and because can be viewed as a vector space over and can be viewed as a vector space over , we have that has order where divides and is less than . Viewing , , and the as groups under multiplication, we can write
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https://en.wikipedia.org/wiki/Symbolic
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Symbolic may refer to:
Symbol, something that represents an idea, a process, or a physical entity
Mathematics, logic, and computing
Symbolic computation, a scientific area concerned with computing with mathematical formulas
Symbolic dynamics, a method for modeling dynamical systems by a discrete space consisting of infinite sequences of abstract symbols
Symbolic execution, the analysis of computer programs by tracking symbolic rather than actual values
Symbolic link, a special type of file in a computer memory storage system
Symbolic logic, the use of symbols for logical operations in logic and mathematics
Music
Symbolic (Death album), a 1995 album by the band Death
Symbolic (Voodoo Glow Skulls album), a 2000 album by the band Voodoo Glow Skulls
Social sciences
Symbolic anthropology, the study of cultural symbols and how those symbols can be interpreted to better understand a particular society
Symbolic capital, the resources available to an individual on the basis of honor, prestige or recognition in sociology and anthropology
Symbolic interaction, a system of interaction in sociology
Symbolic system, a structured system of symbols in anthropology, sociology and psychology
The Symbolic or Symbolic Order, Jacques Lacan's attempt to contrast with The Imaginary and The Real in psychoanalysis
See also
Symbol (disambiguation)
Symbolism (disambiguation)
Symbolic representation (disambiguation)
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https://en.wikipedia.org/wiki/Commutation%20matrix
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In mathematics, especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of its transpose. Specifically, the commutation matrix K(m,n) is the nm × mn matrix which, for any m × n matrix A, transforms vec(A) into vec(AT):
K(m,n) vec(A) = vec(AT) .
Here vec(A) is the mn × 1 column vector obtain by stacking the columns of A on top of one another:
where A = [Ai,j]. In other words, vec(A) is the vector obtained by vectorizing A in column-major order. Similarly, vec(AT) is the vector obtaining by vectorizing A in row-major order.
In the context of quantum information theory, the commutation matrix is sometimes referred to as the swap matrix or swap operator
Properties
The commutation matrix is a special type of permutation matrix, and is therefore orthogonal. In particular, K(m,n) is equal to , where is the permutation over for which
Replacing A with AT in the definition of the commutation matrix shows that Therefore in the special case of m = n the commutation matrix is an involution and symmetric.
The main use of the commutation matrix, and the source of its name, is to commute the Kronecker product: for every m × n matrix A and every r × q matrix B,
This property is often used in developing the higher order statistics of Wishart covariance matrices.
The case of n=q=1 for the above equation states that for any column vectors v,w of sizes m,r respectively,
This property is the reason that this matrix is referred to as the "swap operator" in the context of quantum information theory.
Two explicit forms for the commutation matrix are as follows: if er,j denotes the j-th canonical vector of dimension r (i.e. the vector with 1 in the j-th coordinate and 0 elsewhere) then
The commutation matrix may be expressed as the following block matrix:
Where the p,q entry of n x m block-matrix Ki,j is given by
For example,
Code
For both square and rectangular matrices of m rows and n columns, the commutation matrix can be generated by the code below.
Python
import numpy as np
def comm_mat(m, n):
# determine permutation applied by K
w = np.arange(m * n).reshape((m, n), order="F").T.ravel(order="F")
# apply this permutation to the rows (i.e. to each column) of identity matrix and return result
return np.eye(m * n)[w, :]
Alternatively, a version without imports:
# Kronecker delta
def delta(i, j):
return int(i == j)
def comm_mat(m, n):
# determine permutation applied by K
v = [m * j + i for i in range(m) for j in range(n)]
# apply this permutation to the rows (i.e. to each column) of identity matrix
I = [[delta(i, j) for j in range(m * n)] for i in range(m * n)]
return [I[i] for i in v]
MATLAB
function P = com_mat(m, n)
% determine permutation applied by K
A = reshape(1:m*n, m, n);
v = reshape(A', 1, []);
% apply this permutation to the rows (i.e. to each column) of identity matrix
P = eye(m*n);
P =
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https://en.wikipedia.org/wiki/List%20of%20Torquay%20United%20F.C.%20records%20and%20statistics
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Torquay United Football Club is an English professional association football club based in Torquay, Devon. This list details the major honours and achievements won by Torquay United as well as records set by the club, the players and the managers.
Honours and achievements
Torquay United won their first honour in 1909 as champions of the Torquay & District League before winning the Plymouth & District League (when they were known as Torquay Town) in 1912. After merging with Babbacombe and reverting to the name of Torquay United, the club won the Southern League Western Section in 1927 as well as finishing runners-up in the Western League during the same season.
Since being elected into the Football League in 1927, Torquay United have yet to progress any further than the third tier of English football. United's highest ever League finish was when they were runners-up to Alf Ramsey's Ipswich Town in the Third Division South in 1957. After the Football League was expanded into four nationwide divisions in 1958, Torquay have spent most of their existence in the bottom section. However, United have won automatic promotion to the third division on three occasions (1960, 1966 and 2004) and promoted via the playoffs just once (in 1991). The 1991 Division Four play-off final was the second time Torquay United had played at Wembley Stadium having been runners-up to Bolton Wanderers in the 1989 Sherpa Van Trophy Final. Torquay would make a third appearance at Wembley in 1998 but were beaten again by Colchester United in the Division Three play-off final. Despite being relegated to the Football Conference in 2007, Torquay found themselves at the new Wembley Stadium the following year for the 2008 FA Trophy Final. Unfortunately, Torquay were destined to be the losing side again, being beaten this time by Ebbsfleet United. However, the club's fifth appearance at Wembley for the 2009 Conference National play-off final resulted in a victory against Cambridge United and ensured the club's return to the Football League after a two-year absence.
Torquay United have reached the fourth round of the FA Cup on seven occasions and the third round of the League Cup on four occasions. The club also progressed to the final of the short-lived Third Division South Cup in 1934 before losing to local rivals Exeter City. Unfortunately, despite also reaching the final again in 1939, the match was never played due to the outbreak of World War II. On a local level, Torquay have won the Devon Senior Cup twice (once under the name of Torquay Town) and the Devon Professional Bowl (and its successor the Devon St Luke's Bowl) on fifteen separate occasions.
Leagues
League One / Division Three / Third Division South (level 3):
Runners-up (1): 1956–57
League Two / Division Three / Division Four (level 4):
Promotion (3): 1959–60, 1965–66, 2003–04
Play-off winners (1): 1990–91
Play-off runners-up (3): 1987–88, 1997–98, 2010–11
Conference National (level 5):
Play-off winners
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https://en.wikipedia.org/wiki/Duplication%20and%20elimination%20matrices
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In mathematics, especially in linear algebra and matrix theory, the duplication matrix and the elimination matrix are linear transformations used for transforming half-vectorizations of matrices into vectorizations or (respectively) vice versa.
Duplication matrix
The duplication matrix is the unique matrix which, for any symmetric matrix , transforms into :
.
For the symmetric matrix , this transformation reads
The explicit formula for calculating the duplication matrix for a matrix is:
Where:
is a unit vector of order having the value in the position and 0 elsewhere;
is a matrix with 1 in position and and zero elsewhere
Here is a C++ function using Armadillo (C++ library):
arma::mat duplication_matrix(const int &n) {
arma::mat out((n*(n+1))/2, n*n, arma::fill::zeros);
for (int j = 0; j < n; ++j) {
for (int i = j; i < n; ++i) {
arma::vec u((n*(n+1))/2, arma::fill::zeros);
u(j*n+i-((j+1)*j)/2) = 1.0;
arma::mat T(n,n, arma::fill::zeros);
T(i,j) = 1.0;
T(j,i) = 1.0;
out += u * arma::trans(arma::vectorise(T));
}
}
return out.t();
}
Elimination matrix
An elimination matrix is a matrix which, for any matrix , transforms into :
.
By the explicit (constructive) definition given by , the by elimination matrix is given by
where is a unit vector whose -th element is one and zeros elsewhere, and .
Here is a C++ function using Armadillo (C++ library):
arma::mat elimination_matrix(const int &n) {
arma::mat out((n*(n+1))/2, n*n, arma::fill::zeros);
for (int j = 0; j < n; ++j) {
arma::rowvec e_j(n, arma::fill::zeros);
e_j(j) = 1.0;
for (int i = j; i < n; ++i) {
arma::vec u((n*(n+1))/2, arma::fill::zeros);
u(j*n+i-((j+1)*j)/2) = 1.0;
arma::rowvec e_i(n, arma::fill::zeros);
e_i(i) = 1.0;
out += arma::kron(u, arma::kron(e_j, e_i));
}
}
return out;
}
For the matrix , one choice for this transformation is given by
.
Notes
References
.
Jan R. Magnus and Heinz Neudecker (1988), Matrix Differential Calculus with Applications in Statistics and Econometrics, Wiley. .
Jan R. Magnus (1988), Linear Structures, Oxford University Press.
Matrices
de:Eliminationsmatrix
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https://en.wikipedia.org/wiki/Vectorization%20%28mathematics%29
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In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a vector. Specifically, the vectorization of a matrix A, denoted vec(A), is the column vector obtained by stacking the columns of the matrix A on top of one another:
Here, represents the element in the i-th row and j-th column of A, and the superscript denotes the transpose. Vectorization expresses, through coordinates, the isomorphism between these (i.e., of matrices and vectors) as vector spaces.
For example, for the 2×2 matrix , the vectorization is .
The connection between the vectorization of A and the vectorization of its transpose is given by the commutation matrix.
Compatibility with Kronecker products
The vectorization is frequently used together with the Kronecker product to express matrix multiplication as a linear transformation on matrices. In particular,
for matrices A, B, and C of dimensions k×l, l×m, and m×n. For example, if (the adjoint endomorphism of the Lie algebra of all n×n matrices with complex entries), then , where is the n×n identity matrix.
There are two other useful formulations:
More generally, it has been shown that vectorization is a self-adjunction in the monoidal closed structure of any category of matrices.
Compatibility with Hadamard products
Vectorization is an algebra homomorphism from the space of matrices with the Hadamard (entrywise) product to Cn2 with its Hadamard product:
Compatibility with inner products
Vectorization is a unitary transformation from the space of n×n matrices with the Frobenius (or Hilbert–Schmidt) inner product to Cn2:
where the superscript † denotes the conjugate transpose.
Vectorization as a linear sum
The matrix vectorization operation can be written in terms of a linear sum. Let X be an matrix that we want to vectorize, and let ei be the i-th canonical basis vector for the n-dimensional space, that is . Let Bi be a block matrix defined as follows:
Bi consists of n block matrices of size , stacked column-wise, and all these matrices are all-zero except for the i-th one, which is a identity matrix Im.
Then the vectorized version of X can be expressed as follows:
Multiplication of X by ei extracts the i-th column, while multiplication by Bi puts it into the desired position in the final vector.
Alternatively, the linear sum can be expressed using the Kronecker product:
Half-vectorization
For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the entries on and below the main diagonal. For such matrices, the half-vectorization is sometimes more useful than the vectorization. The half-vectorization, vech(A), of a symmetric matrix A is the column vector obtained by vectorizing only the lower triangular part of A:
For example, for the 2×2 matrix , the
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https://en.wikipedia.org/wiki/Subdivisions%20of%20Brazil
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Brazil is divided into several types and levels of subdivisions.
Regions
Since 1942, the Brazilian Institute of Geography and Statistics has divided Brazil into five geographic regions. On 23 November 1970, the regions of Brazil were adjusted slightly to the definition that is still in use today. The division into regions is merely academic and statistical, as the regions do not enjoy any political autonomy.
North Region (Região Norte)
Northeast Region (Região Nordeste)
Central-West Region (Região Centro-Oeste)
Southeast Region (Região Sudeste)
South Region (Região Sul)
States
Brazil is divided into 27 federative units: 26 states and 1 federal district (Distrito Federal).
(AC)
(AL)
(AP)
(AM)
(BA)
(CE)
(ES)
(GO)
(MA)
(MT)
(MS)
(MG)
(PA)
(PB)
(PR)
(PE)
(PI)
(RJ)
(RN)
(RS)
(RO)
(RR)
(SC)
(SP)
(SE)
(TO)
(DF)
Municipalities
The lowest level of political division of Brazil are the municipalities, which also enjoy political and economical autonomy. There are over 5500 municipalities in Brazil, comprising almost the entirety of the country's territory. The only exceptions are the Federal District (not divided into municipalities, but into 33 administrative regions, without any political autonomy) and the archipelago of Fernando de Noronha, which consists in a state district.
Statistical Areas
For statistical purposes, Brazilian states and the Federal District are divided into "Intermediate Geographic Regions" (), which themselves are divided into smaller "Immediate Geographic Regions" (Regiões Geográficas Imediatas) which correspond to a metropolitan area. From 1989-2017, they were grouped into mesoregions and microregions.
See also
ISO 3166-2:BR
Proposed states and territories of Brazil
Former subdivisions of Brazil
Indigenous territories
References
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https://en.wikipedia.org/wiki/Euler%E2%80%93Maruyama%20method
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In Itô calculus, the Euler–Maruyama method (also called the Euler method) is a method for the approximate numerical solution of a stochastic differential equation (SDE). It is an extension of the Euler method for ordinary differential equations to stochastic differential equations. It is named after Leonhard Euler and Gisiro Maruyama. Unfortunately, the same generalization cannot be done for any arbitrary deterministic method.
Consider the stochastic differential equation (see Itô calculus)
with initial condition X0 = x0, where Wt stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time [0, T]. Then the Euler–Maruyama approximation to the true solution X is the Markov chain Y defined as follows:
partition the interval [0, T] into N equal subintervals of width :
set Y0 = x0
recursively define Yn for 0 ≤ n ≤ N-1 by
where
The random variables ΔWn are independent and identically distributed normal random variables with expected value zero and variance .
Example
Numerical simulation
An area that has benefited significantly from SDE is biology or more precisely mathematical biology. Here the number of publications on the use of stochastic model grew, as most of the models are nonlinear, demanding numerical schemes.
The graphic depicts a stochastic differential equation being solved using the Euler Scheme. The deterministic counterpart is shown as well.
Computer implementation
The following Python code implements the Euler–Maruyama method and uses it to solve the Ornstein–Uhlenbeck process defined by
The random numbers for are generated using the NumPy mathematics package.
# -*- coding: utf-8 -*-
import numpy as np
import matplotlib.pyplot as plt
class Model:
"""Stochastic model constants."""
THETA = 0.7
MU = 1.5
SIGMA = 0.06
def mu(y: float, _t: float) -> float:
"""Implement the Ornstein–Uhlenbeck mu."""
return Model.THETA * (Model.MU - y)
def sigma(_y: float, _t: float) -> float:
"""Implement the Ornstein–Uhlenbeck sigma."""
return Model.SIGMA
def dW(delta_t: float) -> float:
"""Sample a random number at each call."""
return np.random.normal(loc=0.0, scale=np.sqrt(delta_t))
def run_simulation():
""" Return the result of one full simulation."""
T_INIT = 3
T_END = 7
N = 1000 # Compute at 1000 grid points
DT = float(T_END - T_INIT) / N
TS = np.arange(T_INIT, T_END + DT, DT)
assert TS.size == N + 1
Y_INIT = 0
ys = np.zeros(TS.size)
ys[0] = Y_INIT
for i in range(1, TS.size):
t = T_INIT + (i - 1) * DT
y = ys[i - 1]
ys[i] = y + mu(y, t) * DT + sigma(y, t) * dW(DT)
return TS, ys
def plot_simulations(num_sims: int):
""" Plot several simulations in one image."""
for _ in range(num_sims):
plt.plot(*run_simulation())
plt.xlabel("time")
plt.ylabel("y")
plt.show()
if __name__ == "__main__":
NUM_SIMS = 5
plot_simulations(NUM_SIMS)
The following is s
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https://en.wikipedia.org/wiki/Milstein%20method
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In mathematics, the Milstein method is a technique for the approximate numerical solution of a stochastic differential equation. It is named after Grigori N. Milstein who first published it in 1974.
Description
Consider the autonomous Itō stochastic differential equation:
with initial condition , where stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time . Then the Milstein approximation to the true solution is the Markov chain defined as follows:
partition the interval into equal subintervals of width :
set
recursively define for by: where denotes the derivative of with respect to and: are independent and identically distributed normal random variables with expected value zero and variance . Then will approximate for , and increasing will yield a better approximation.
Note that when , i.e. the diffusion term does not depend on , this method is equivalent to the Euler–Maruyama method.
The Milstein scheme has both weak and strong order of convergence, , which is superior to the Euler–Maruyama method, which in turn has the same weak order of convergence, , but inferior strong order of convergence, .
Intuitive derivation
For this derivation, we will only look at geometric Brownian motion (GBM), the stochastic differential equation of which is given by:
with real constants and . Using Itō's lemma we get:
Thus, the solution to the GBM SDE is:
where
See numerical solution is presented above for three different trajectories.
Computer implementation
The following Python code implements the Milstein method and uses it to solve the SDE describing the Geometric Brownian Motion defined by
# -*- coding: utf-8 -*-
# Milstein Method
import numpy as np
import matplotlib.pyplot as plt
num_sims = 1 # One Example
# One Second and thousand grid points
t_init, t_end = 0, 1
N = 1000 # Compute 1000 grid points
dt = float(t_end - t_init) / N
## Initial Conditions
y_init = 1
μ, σ = 3, 1
# dw Random process
def dW(Δt):
"""Random sample normal distribution"""
return np.random.normal(loc=0.0, scale=np.sqrt(Δt))
# vectors to fill
ts = np.arange(t_init, t_end + dt, dt)
ys = np.zeros(N + 1)
ys[0] = y_init
# Loop
for _ in range(num_sims):
for i in range(1, ts.size):
t = (i - 1) * dt
y = ys[i - 1]
# Milstein method
dw_ = dW(dt)
ys[i] = y + μ * dt * y + σ * y * dw_ + 0.5 * σ**2 * y * (dw_**2 - dt)
plt.plot(ts, ys)
# Plot
plt.xlabel("time (s)")
plt.grid()
h = plt.ylabel("y")
h.set_rotation(0)
plt.show()
See also
Euler–Maruyama method
References
Further reading
Numerical differential equations
Stochastic differential equations
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https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta%20method%20%28SDE%29
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In mathematics of stochastic systems, the Runge–Kutta method is a technique for the approximate numerical solution of a stochastic differential equation. It is a generalisation of the Runge–Kutta method for ordinary differential equations to stochastic differential equations (SDEs). Importantly, the method does not involve knowing derivatives of the coefficient functions in the SDEs.
Most basic scheme
Consider the Itō diffusion satisfying the following Itō stochastic differential equation
with initial condition , where stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time . Then the basic Runge–Kutta approximation to the true solution is the Markov chain defined as follows:
partition the interval into subintervals of width :
set ;
recursively compute for by where and
The random variables are independent and identically distributed normal random variables with expected value zero and variance .
This scheme has strong order 1, meaning that the approximation error of the actual solution at a fixed time scales with the time step . It has also weak order 1, meaning that the error on the statistics of the solution scales with the time step . See the references for complete and exact statements.
The functions and can be time-varying without any complication. The method can be generalized to the case of several coupled equations; the principle is the same but the equations become longer.
Variation of the Improved Euler is flexible
A newer Runge—Kutta scheme also of strong order 1 straightforwardly reduces to the improved Euler scheme for deterministic ODEs.
Consider the vector stochastic process that satisfies the general Ito SDE
where drift and volatility are sufficiently smooth functions of their arguments.
Given time step , and given the value estimate by for time via
where for normal random ;
and where , each alternative chosen with probability .
The above describes only one time step.
Repeat this time step times in order to integrate the SDE from time to .
The scheme integrates Stratonovich SDEs to provided one sets throughout (instead of choosing ).
Higher order Runge-Kutta schemes
Higher-order schemes also exist, but become increasingly complex.
Rößler developed many schemes for Ito SDEs,
whereas Komori developed schemes for Stratonovich SDEs. Rackauckas extended these schemes to allow for adaptive-time stepping via Rejection Sampling with Memory (RSwM), resulting in orders of magnitude efficiency increases in practical biological models, along with coefficient optimization for improved stability.
References
Numerical differential equations
Stochastic differential equations
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https://en.wikipedia.org/wiki/List%20of%20Queens%20Park%20Rangers%20F.C.%20records%20and%20statistics
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This article lists records and statistics relating to the English football club Queens Park Rangers.
Team records
Record Football League win: 9–2 v Tranmere Rovers, Football League Division Three (3 December 1960)
Record Winning margin: 8-0 v Merthyr Tydfil, Football League Division Three South (9 March 1929)
Queen's Park Rangers 10 -0 Verona stars . Pre season 2015
Player Records
Record appearances: Tony Ingham (548, 1950–1963)
Record goalscorer: George Goddard (172, 1926–1934)
Most international caps whilst at QPR: Alan McDonald (52 for Northern Ireland)
Most consecutive games played: Mike Keen 263 between December 1963 and September 1968.
Other club records
Most played Football League clubs
This table lists the teams that QPR has met on most occasions in the English Football League / Premier League, and is correct as at 20 March 2017.
Transfers
Highest transfer fees paid
Highest transfer fees received
References
Records
English football club statistics
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https://en.wikipedia.org/wiki/Absolutely%20integrable%20function
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In mathematics, an absolutely integrable function is a function whose absolute value is integrable, meaning that the integral of the absolute value over the whole domain is finite.
For a real-valued function, since
where
both and must be finite. In Lebesgue integration, this is exactly the requirement for any measurable function f to be considered integrable, with the integral then equaling , so that in fact "absolutely integrable" means the same thing as "Lebesgue integrable" for measurable functions.
The same thing goes for a complex-valued function. Let us define
where and are the real and imaginary parts of . Then
so
This shows that the sum of the four integrals (in the middle) is finite if and only if the integral of the absolute value is finite, and the function is Lebesgue integrable only if all the four integrals are finite. So having a finite integral of the absolute value is equivalent to the conditions for the function to be "Lebesgue integrable".
External links
Integral calculus
References
Tao, Terence, Analysis 2, 3rd ed., Texts and Readings in Mathematics, Hindustan Book Agency, New Delhi.
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https://en.wikipedia.org/wiki/Aspherical%20space
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In topology, a branch of mathematics, an aspherical space is a topological space with all homotopy groups equal to 0 when .
If one works with CW complexes, one can reformulate this condition: an aspherical CW complex is a CW complex whose universal cover is contractible. Indeed, contractibility of a universal cover is the same, by Whitehead's theorem, as asphericality of it. And it is an application of the exact sequence of a fibration that higher homotopy groups of a space and its universal cover are same. (By the same argument, if E is a path-connected space and is any covering map, then E is aspherical if and only if B is aspherical.)
Each aspherical space X is, by definition, an Eilenberg–MacLane space of type , where is the fundamental group of X. Also directly from the definition, an aspherical space is a classifying space for its fundamental group (considered to be a topological group when endowed with the discrete topology).
Examples
Using the second of above definitions we easily see that all orientable compact surfaces of genus greater than 0 are aspherical (as they have either the Euclidean plane or the hyperbolic plane as a universal cover).
It follows that all non-orientable surfaces, except the real projective plane, are aspherical as well, as they can be covered by an orientable surface of genus 1 or higher.
Similarly, a product of any number of circles is aspherical. As is any complete, Riemannian flat manifold.
Any hyperbolic 3-manifold is, by definition, covered by the hyperbolic 3-space H3, hence aspherical. As is any n-manifold whose universal covering space is hyperbolic n-space Hn.
Let X = G/K be a Riemannian symmetric space of negative type, and Γ be a lattice in G that acts freely on X. Then the locally symmetric space is aspherical.
The Bruhat–Tits building of a simple algebraic group over a field with a discrete valuation is aspherical.
The complement of a knot in S3 is aspherical, by the sphere theorem
Metric spaces with nonpositive curvature in the sense of Aleksandr D. Aleksandrov (locally CAT(0) spaces) are aspherical. In the case of Riemannian manifolds, this follows from the Cartan–Hadamard theorem, which has been generalized to geodesic metric spaces by Mikhail Gromov and Hans Werner Ballmann. This class of aspherical spaces subsumes all the previously given examples.
Any nilmanifold is aspherical.
Symplectically aspherical manifolds
In the context of symplectic manifolds, the meaning of "aspherical" is a little bit different. Specifically, we say that a symplectic manifold (M,ω) is symplectically aspherical if and only if
for every continuous mapping
where denotes the first Chern class of an almost complex structure which is compatible with ω.
By Stokes' theorem, we see that symplectic manifolds which are aspherical are also symplectically aspherical manifolds. However, there do exist symplectically aspherical manifolds which are not aspherical spaces.
Some references drop the requirement
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https://en.wikipedia.org/wiki/Differential%20calculus%20over%20commutative%20algebras
|
In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this are:
The whole topological information of a smooth manifold is encoded in the algebraic properties of its -algebra of smooth functions as in the Banach–Stone theorem.
Vector bundles over correspond to projective finitely generated modules over via the functor which associates to a vector bundle its module of sections.
Vector fields on are naturally identified with derivations of the algebra .
More generally, a linear differential operator of order k, sending sections of a vector bundle to sections of another bundle is seen to be an -linear map between the associated modules, such that for any elements :
where the bracket is defined as the commutator
Denoting the set of th order linear differential operators from an -module to an -module with we obtain a bi-functor with values in the category of -modules. Other natural concepts of calculus such as jet spaces, differential forms are then obtained as representing objects of the functors and related functors.
Seen from this point of view calculus may in fact be understood as the theory of these functors and their representing objects.
Replacing the real numbers with any commutative ring, and the algebra with any commutative algebra the above said remains meaningful, hence differential calculus can be developed for arbitrary commutative algebras. Many of these concepts are widely used in algebraic geometry, differential geometry and secondary calculus. Moreover, the theory generalizes naturally to the setting of graded commutative algebra, allowing for a natural foundation of calculus on supermanifolds, graded manifolds and associated concepts like the Berezin integral.
See also
References
J. Nestruev, Smooth Manifolds and Observables, Graduate Texts in Mathematics 220, Springer, 2002.
I. S. Krasil'shchik, "Lectures on Linear Differential Operators over Commutative Algebras". Eprint DIPS-01/99.
I. S. Krasil'shchik, A. M. Vinogradov (eds) "Algebraic Aspects of Differential Calculus", Acta Appl. Math. 49 (1997), Eprints: DIPS-01/96, DIPS-02/96, DIPS-03/96, DIPS-04/96, DIPS-05/96, DIPS-06/96, DIPS-07/96, DIPS-08/96.
I. S. Krasil'shchik, A. M. Verbovetsky, "Homological Methods in Equations of Mathematical Physics", Open Ed. and Sciences, Opava (Czech Rep.), 1998; Eprint arXiv:math/9808130v2.
G. Sardanashvily, Lectures on Differential Geometry of Modules and Rings, Lambert Academic Publishing, 2012; Eprint arXiv:0910.1515 [math-ph] 137 pages.
A. M. Vinogradov, "The Logic Algebra for the Theory of Linear Differential Operators", Dokl. Akad. Nauk SSSR, 295(5) (1972) 1025-1028; English transl. in Soviet Math. Dokl. 13(4) (1972), 1058-1062.
A. M. Vinogradov, "Some new homological systems associated with differential calcu
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https://en.wikipedia.org/wiki/Ptolemy%27s%20inequality
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In Euclidean geometry, Ptolemy's inequality relates the six distances determined by four points in the plane or in a higher-dimensional space. It states that, for any four points , , , and , the following inequality holds:
It is named after the Greek astronomer and mathematician Ptolemy.
The four points can be ordered in any of three distinct ways (counting reversals as not distinct) to form three different quadrilaterals, for each of which the sum of the products of opposite sides is at least as large as the product of the diagonals. Thus, the three product terms in the inequality can be additively permuted to put any one of them on the right side of the inequality, so the three products of opposite sides or of diagonals of any one of the quadrilaterals must obey the triangle inequality.
As a special case, Ptolemy's theorem states that the inequality becomes an equality when the four points lie in cyclic order on a circle.
The other case of equality occurs when the four points are collinear in order. The inequality does not generalize from Euclidean spaces to arbitrary metric spaces. The spaces where it remains valid are called the Ptolemaic spaces; they include the inner product spaces, Hadamard spaces, and shortest path distances on Ptolemaic graphs.
Assumptions and derivation
Ptolemy's inequality is often stated for a special case, in which the four points are the vertices of a convex quadrilateral, given in cyclic order. However, the theorem applies more generally to any four points; it is not required that the quadrilateral they form be convex, simple, or even planar.
For points in the plane, Ptolemy's inequality can be derived from the triangle inequality by an inversion centered at one of the four points. Alternatively, it can be derived by interpreting the four points as complex numbers, using the complex number identity:
to construct a triangle whose side lengths are the products of sides of the given quadrilateral, and applying the triangle inequality to this triangle. One can also view the points as belonging to the complex projective line, express the inequality in the form that the absolute values of two cross-ratios of the points sum to at least one, and deduce this from the fact that the cross-ratios themselves add to exactly one.
A proof of the inequality for points in three-dimensional space can be reduced to the planar case, by observing that for any non-planar quadrilateral, it is possible to rotate one of the points around the diagonal until the quadrilateral becomes planar, increasing the other diagonal's length and keeping the other five distances constant. In spaces of higher dimension than three, any four points lie in a three-dimensional subspace, and the same three-dimensional proof can be used.
Four concyclic points
For four points in order around a circle, Ptolemy's inequality becomes an equality, known as Ptolemy's theorem:
In the inversion-based proof of Ptolemy's inequality, transforming four co-circular
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https://en.wikipedia.org/wiki/Factorization%20system
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In mathematics, it can be shown that every function can be written as the composite of a surjective function followed by an injective function. Factorization systems are a generalization of this situation in category theory.
Definition
A factorization system (E, M) for a category C consists of two classes of morphisms E and M of C such that:
E and M both contain all isomorphisms of C and are closed under composition.
Every morphism f of C can be factored as for some morphisms and .
The factorization is functorial: if and are two morphisms such that for some morphisms and , then there exists a unique morphism making the following diagram commute:
Remark: is a morphism from to in the arrow category.
Orthogonality
Two morphisms and are said to be orthogonal, denoted , if for every pair of morphisms and such that there is a unique morphism such that the diagram
commutes. This notion can be extended to define the orthogonals of sets of morphisms by
and
Since in a factorization system contains all the isomorphisms, the condition (3) of the definition is equivalent to
(3') and
Proof: In the previous diagram (3), take (identity on the appropriate object) and .
Equivalent definition
The pair of classes of morphisms of C is a factorization system if and only if it satisfies the following conditions:
Every morphism f of C can be factored as with and
and
Weak factorization systems
Suppose e and m are two morphisms in a category C. Then e has the left lifting property with respect to m (respectively m has the right lifting property with respect to e) when for every pair of morphisms u and v such that ve = mu there is a morphism w such that the following diagram commutes. The difference with orthogonality is that w is not necessarily unique.
A weak factorization system (E, M) for a category C consists of two classes of morphisms E and M of C such that:
The class E is exactly the class of morphisms having the left lifting property with respect to each morphism in M.
The class M is exactly the class of morphisms having the right lifting property with respect to each morphism in E.
Every morphism f of C can be factored as for some morphisms and .
This notion leads to a succinct definition of model categories: a model category is a pair consisting of a category C and classes of (so-called) weak equivalences W, fibrations F and cofibrations C so that
C has all limits and colimits,
is a weak factorization system, and
is a weak factorization system.
A model category is a complete and cocomplete category equipped with a model structure. A map is called a trivial fibration if it belongs to and it is called a trivial cofibration if it belongs to An object is called fibrant if the morphism to the terminal object is a fibration, and it is called cofibrant if the morphism from the initial object is a cofibration.
References
External links
Category theory
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https://en.wikipedia.org/wiki/Generalized%20dihedral%20group
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In mathematics, the generalized dihedral groups are a family of groups with algebraic structures similar to that of the dihedral groups. They include the finite dihedral groups, the infinite dihedral group, and the orthogonal group O(2). Dihedral groups play an important role in group theory, geometry, and chemistry.
Definition
For any abelian group H, the generalized dihedral group of H, written Dih(H), is the semidirect product of H and Z2, with Z2 acting on H by inverting elements. I.e., with φ(0) the identity and φ(1) inversion.
Thus we get:
(h1, 0) * (h2, t2) = (h1 + h2, t2)
(h1, 1) * (h2, t2) = (h1 − h2, 1 + t2)
for all h1, h2 in H and t2 in Z2.
(Writing Z2 multiplicatively, we have (h1, t1) * (h2, t2) = (h1 + t1h2, t1t2) .)
Note that (h, 0) * (0,1) = (h,1), i.e. first the inversion and then the operation in H. Also (0, 1) * (h, t) = (−h, 1 + t); indeed (0,1) inverts h, and toggles t between "normal" (0) and "inverted" (1) (this combined operation is its own inverse).
The subgroup of Dih(H) of elements (h, 0) is a normal subgroup of index 2, isomorphic to H, while the elements (h, 1) are all their own inverse.
The conjugacy classes are:
the sets {(h,0 ), (−h,0 )}
the sets {(h + k + k, 1) | k in H }
Thus for every subgroup M of H, the corresponding set of elements (m,0) is also a normal subgroup. We have:
Dih(H) / M = Dih ( H / M )
Examples
Dihn = Dih(Zn) (the dihedral groups)
For even n there are two sets {(h + k + k, 1) | k in H }, and each generates a normal subgroup of type Dihn / 2. As subgroups of the isometry group of the set of vertices of a regular n-gon they are different: the reflections in one subgroup all have two fixed points, while none in the other subgroup has (the rotations of both are the same). However, they are isomorphic as abstract groups.
For odd n there is only one set {(h + k + k, 1) | k in H }
Dih∞ = Dih(Z) (the infinite dihedral group); there are two sets {(h + k + k, 1) | k in H }, and each generates a normal subgroup of type Dih∞. As subgroups of the isometry group of Z they are different: the reflections in one subgroup all have a fixed point, the mirrors are at the integers, while none in the other subgroup has, the mirrors are in between (the translations of both are the same: by even numbers). However, they are isomorphic as abstract groups.
Dih(S1), or orthogonal group O(2,R), or O(2): the isometry group of a circle, or equivalently, the group of isometries in 2D that keep the origin fixed. The rotations form the circle group S1, or equivalently SO(2,R), also written SO(2), and R/Z ; it is also the multiplicative group of complex numbers of absolute value 1. In the latter case one of the reflections (generating the others) is complex conjugation. There are no proper normal subgroups with reflections. The discrete normal subgroups are cyclic groups of order n for all positive integers n. The quotient groups are isomorphic with the same group Dih(S1).
Dih(Rn ): the group of isometries of Rn consis
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https://en.wikipedia.org/wiki/Divorce%20demography
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Estimates of annual divorces by country
The following are the countries with the most annual divorces according to the United Nations in 2009.
Divorce statistics by country/region (per 1,000 population / year)
Metrics / statistics
Crude divorce rate
This is divorces per 1,000 population per year. For example, if a city has 10,000 people living in it, and 30 couples divorce in one year, then the crude divorce rate for that year is 3 divorces per 1,000 residents.
The crude divorce rate can give a general overview of marriage in an area, but it does not take people who cannot marry into account. For example, it would include young children, who are clearly not of marriageable age in its sample. In a place with large numbers of children or single adults, the crude divorce rate can seem low. In a place with few children and single adults, the crude divorce rate can seem high.
Refined divorce rate
This measures the number of divorces per 1,000 women married to men, so that all unmarried persons are left out of the calculation. For example, if that same city of 10,000 people has 3,000 married women, and 30 couples divorce in one year, then the refined divorce rate is 10 divorces per 1,000 married women.
Divorce-to-marriage ratio
This compares the number of divorces in a given year to the number of marriages in that same year (the ratio of the crude divorce rate to the crude marriage rate). For example, if there are 500 divorces and 1,000 marriages in a given year in a given area, the ratio would be one divorce for every two marriages, e.g. a ratio of 0.50 (50%).
However, this measurement compares two unlike populations – those who can marry and those who can divorce. Say there exists a community with 100,000 married couples, and very few people capable of marriage, for reasons such as age. If 1,000 people obtain divorces and 1,000 people get married in the same year, the ratio is one divorce for every marriage, which may lead people to think that the community's relationships are extremely unstable, despite the number of married people not changing. This is also true in reverse: a community with very many people of marriageable age may have 10,000 marriages and 1,000 divorces, leading people to believe that it has very stable relationships.
Furthermore, these two rates are not directly comparable since the marriage rate only examines the current year, while the divorce rate examines the outcomes of marriages for many previous years. This does not equate to the proportion of marriages in a given single-year cohort that will ultimately end in divorce. In any given year, underlying rates may change, and this can affect the ratio. For example, during an economic downturn, some couples might postpone a divorce because they can't afford to live separately. These individual choices could seem to temporarily improve the divorce-to-marriage ratio.
References
External links
Authorship: United States CDC National Center for Health Stat
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https://en.wikipedia.org/wiki/Peter%20M.%20Neumann
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Peter Michael Neumann OBE (28 December 1940 – 18 December 2020) was a British mathematician. His fields of interest included the history of mathematics and Galois theory.
Biography
Born in December 1940, Neumann was a son of the German-born mathematicians Bernhard Neumann and Hanna Neumann. He gained a BA degree from The Queen's College, Oxford in 1963, and a DPhil degree from the University of Oxford in 1966. Neumann was a Tutorial Fellow at the Queen's College, Oxford, and a lecturer at the University of Oxford. His research work was in the field of group theory. In 1987, Neumann won the Lester R. Ford Award of the Mathematical Association of America for his review of Harold Edwards' book Galois Theory.
He was the first Chairman of the United Kingdom Mathematics Trust, from October 1996 to April 2004, succeeded by Bernard Silverman.
Neumann showed in 1997 that Alhazen's problem (reflecting a light ray off a spherical mirror to hit a target) cannot be solved with a straightedge and compass construction. Although the solution is a straightforward application of Galois theory it settles the constructibility of one of the last remaining geometric construction problems posed in antiquity.
In 2003, the London Mathematical Society awarded him the Senior Whitehead Prize. He was appointed Officer of the Order of the British Empire (OBE) in the 2008 New Year Honours.
After retiring in 2008, he became an Emeritus Fellow at the Queen's College.
Neumann's work in the history of mathematics includes his 2011 publication The Mathematical Writings of Evariste Galois, an English language book on the work of French mathematician Évariste Galois (1811–1832). Neumann was a long-standing supporter of the British Society for the History of Mathematics, whose Neumann Prize is named in his honour.
Neumann was the president of the Mathematical Association from 2015 to 2016.
Neumann died from COVID-19 on 18 December 2020, ten days before his 80th birthday, during the COVID-19 pandemic in the United Kingdom.
Personal life
Neumann married Sylvia Bull in 1962. She was a fellow mathematics undergraduate at Oxford, where they met.
References
External links
Home page at the Mathematical Institute, Oxford
Home page at The Queen's College, Oxford
1940 births
2020 deaths
Deaths from the COVID-19 pandemic in England
Alumni of The Queen's College, Oxford
20th-century English mathematicians
21st-century English mathematicians
Group theorists
British historians of mathematics
Fellows of The Queen's College, Oxford
Officers of the Order of the British Empire
David Crighton medalists
English male writers
English people of German descent
English people of German-Jewish descent
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https://en.wikipedia.org/wiki/Dagger%20category
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In category theory, a branch of mathematics, a dagger category (also called involutive category or category with involution) is a category equipped with a certain structure called dagger or involution. The name dagger category was coined by Peter Selinger.
Formal definition
A dagger category is a category equipped with an involutive contravariant endofunctor which is the identity on objects.
In detail, this means that:
for all morphisms , there exist its adjoint
for all morphisms ,
for all objects ,
for all and ,
Note that in the previous definition, the term "adjoint" is used in a way analogous to (and inspired by) the linear-algebraic sense, not in the category-theoretic sense.
Some sources define a category with involution to be a dagger category with the additional property that its set of morphisms is partially ordered and that the order of morphisms is compatible with the composition of morphisms, that is implies for morphisms , , whenever their sources and targets are compatible.
Examples
The category Rel of sets and relations possesses a dagger structure: for a given relation in Rel, the relation is the relational converse of . In this example, a self-adjoint morphism is a symmetric relation.
The category Cob of cobordisms is a dagger compact category, in particular it possesses a dagger structure.
The category Hilb of Hilbert spaces also possesses a dagger structure: Given a bounded linear map , the map is just its adjoint in the usual sense.
Any monoid with involution is a dagger category with only one object. In fact, every endomorphism hom-set in a dagger category is not simply a monoid, but a monoid with involution, because of the dagger.
A discrete category is trivially a dagger category.
A groupoid (and as trivial corollary, a group) also has a dagger structure with the adjoint of a morphism being its inverse. In this case, all morphisms are unitary (definition below).
Remarkable morphisms
In a dagger category , a morphism is called
unitary if
self-adjoint if
The latter is only possible for an endomorphism . The terms unitary and self-adjoint in the previous definition are taken from the category of Hilbert spaces, where the morphisms satisfying those properties are then unitary and self-adjoint in the usual sense.
See also
*-algebra
Dagger symmetric monoidal category
Dagger compact category
References
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https://en.wikipedia.org/wiki/Addison-Wesley%20Secondary%20Math%3A%20An%20Integrated%20Approach%3A%20Focus%20on%20Algebra
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Focus on Algebra was the widely cited 812-page-long algebra textbook which contained significant content outside the traditional field of mathematics. The real-life context, intended to make mathematics more relevant, included chili recipes, ancient myths, and photographs of famous people.
Although it was a widely used textbook, it made headlines when it was dubbed "rainforest algebra" by critics.
Senate Testimony
Senator Robert Byrd, Democrat from West Virginia, joined critics of reform mathematics on the floor of the senate by dubbing Addison-Wesley Secondary Math: An Integrated Approach: Focus on Algebra the "Texas rainforest algebra book". It had received an "F" grade on a report card produced by Mathematically Correct, a back-to-basics group, who claimed that it had no algebraic content on the first hundred pages.
Structure
Each of the 10 Chapters was composed of two or three themes, or "Superlessons," each of which connected the algebraic content to another discipline. Each Superlesson began with an opening page with discussion questions relating to the theme. Critics of the programs cited these questions as evidence of the lack of math in the books.
Examples of questions cited:
What other kinds of pollution besides air pollution might threaten our planet? [page 163, in the introduction to 3-1 Functional Relationships]
Each year the Oilfield Chili Appreciation Society holds a chili cook-off. . . . 1. The chili cook-off raises money for charity. Describe some ways the organizers could raise money in the cook- off. 2. What is the hottest kind of pepper that you have eaten? People who have tasted them agree that cayenne peppers are hotter than pimento peppers. How would you set up a hotness scale for peppers? . . . . [page 217, in the introduction to 4-1 Solving Linear Equations]
What role should zoos play in today's society? . . . . [page 233, in the introduction to 4-2 Other Techniques for Solving Linear Equations]
Exercises within lessons that related to the theme were also criticized. One example:
Creative Writing The zoo sponsors a creative writing contest for high school students. The topic for the essay this year is "Why should we save an endangered species? The prize winners will split a $9000 scholarship. The prize for first place is 3 times that of third place. The prize for second place is $1500 more than that for third place. a. How much money will be awarded for each place? b. If you were in charge of awarding the prizes, would you have awarded the same amounts for the places. Why or why not? c. Suppose you are a judge. What would you use as criteria for judging the essay? [page 253]
See also
Mathematically Correct
Traditional mathematics
NCTM standards
References
Rain-Forest Algebra and MTV Geometry Marianne M. Jennings
Mathematics textbooks
Education reform
Mathematics education reform
Addison-Wesley books
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https://en.wikipedia.org/wiki/Dagger%20compact%20category
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In category theory, a branch of mathematics, dagger compact categories (or dagger compact closed categories) first appeared in 1989 in the work of Sergio Doplicher and John E. Roberts on the reconstruction of compact topological groups from their category of finite-dimensional continuous unitary representations (that is, Tannakian categories). They also appeared in the work of John Baez and James Dolan as an instance of semistrict k-tuply monoidal n-categories, which describe general topological quantum field theories, for n = 1 and k = 3. They are a fundamental structure in Samson Abramsky and Bob Coecke's categorical quantum mechanics.
Overview
Dagger compact categories can be used to express and verify some fundamental quantum information protocols, namely: teleportation, logic gate teleportation and entanglement swapping, and standard notions such as unitarity, inner-product, trace, Choi–Jamiolkowsky duality, complete positivity, Bell states and many other notions are captured by the language of dagger compact categories. All this follows from the completeness theorem, below. Categorical quantum mechanics takes dagger compact categories as a background structure relative to which other quantum mechanical notions like quantum observables and complementarity thereof can be abstractly defined. This forms the basis for a high-level approach to quantum information processing.
Formal definition
A dagger compact category is a dagger symmetric monoidal category which is also compact closed, together with a relation to tie together the dagger structure to the compact structure. Specifically, the dagger is used to connect the unit to the counit, so that, for all in , the following diagram commutes:
To summarize all of these points:
A category is closed if it has an internal hom functor; that is, if the hom-set of morphisms between two objects of the category is an object of the category itself (rather than of Set).
A category is monoidal if it is equipped with an associative bifunctor that is associative, natural and has left and right identities obeying certain coherence conditions.
A monoidal category is symmetric monoidal, if, for every pair A, B of objects in C, there is an isomorphism that is natural in both A and B, and, again, obeys certain coherence conditions (see symmetric monoidal category for details).
A monoidal category is compact closed, if every object has a dual object . Categories with dual objects are equipped with two morphisms, the unit and the counit , which satisfy certain coherence or yanking conditions.
A category is a dagger category if it is equipped with an involutive functor that is the identity on objects, but maps morphisms to their adjoints.
A monoidal category is dagger symmetric if it is a dagger category and is symmetric, and has coherence conditions that make the various functors natural.
A dagger compact category is then a category that is each of the above, and, in addition, has a condition t
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https://en.wikipedia.org/wiki/Algebraically%20closed%20group
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In group theory, a group is algebraically closed if any finite set of equations and inequations that are applicable to have a solution in without needing a group extension. This notion will be made precise later in the article in .
Informal discussion
Suppose we wished to find an element of a group satisfying the conditions (equations and inequations):
Then it is easy to see that this is impossible because the first two equations imply . In this case we say the set of conditions are inconsistent with . (In fact this set of conditions are inconsistent with any group whatsoever.)
Now suppose is the group with the multiplication table to the right.
Then the conditions:
have a solution in , namely .
However the conditions:
Do not have a solution in , as can easily be checked.
However if we extend the group to the group with the adjacent multiplication table:
Then the conditions have two solutions, namely and .
Thus there are three possibilities regarding such conditions:
They may be inconsistent with and have no solution in any extension of .
They may have a solution in .
They may have no solution in but nevertheless have a solution in some extension of .
It is reasonable to ask whether there are any groups such that whenever a set of conditions like these have a solution at all, they have a solution in itself? The answer turns out to be "yes", and we call such groups algebraically closed groups.
Formal definition
We first need some preliminary ideas.
If is a group and is the free group on countably many generators, then by a finite set of equations and inequations with coefficients in we mean a pair of subsets and of the free product of and .
This formalizes the notion of a set of equations and inequations consisting of variables and elements of . The set represents equations like:
The set represents inequations like
By a solution in to this finite set of equations and inequations, we mean a homomorphism , such that for all and for all , where is the unique homomorphism that equals on and is the identity on .
This formalizes the idea of substituting elements of for the variables to get true identities and inidentities. In the example the substitutions and yield:
We say the finite set of equations and inequations is consistent with if we can solve them in a "bigger" group . More formally:
The equations and inequations are consistent with if there is a group and an embedding such that the finite set of equations and inequations and has a solution in , where is the unique homomorphism that equals on and is the identity on .
Now we formally define the group to be algebraically closed if every finite set of equations and inequations that has coefficients in and is consistent with has a solution in .
Known Results
It is difficult to give concrete examples of algebraically closed groups as the following results indicate:
Every countable group can be embedded in a countable alg
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https://en.wikipedia.org/wiki/Unit%20distance%20graph
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In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points whenever the distance between them is exactly one. To distinguish these graphs from a broader definition that allows some non-adjacent pairs of vertices to be at distance one, they may also be called strict unit distance graphs or faithful unit distance graphs. As a hereditary family of graphs, they can be characterized by forbidden induced subgraphs. The unit distance graphs include the cactus graphs, the matchstick graphs and penny graphs, and the hypercube graphs. The generalized Petersen graphs are non-strict unit distance graphs.
An unsolved problem of Paul Erdős asks how many edges a unit distance graph on vertices can have. The best known lower bound is slightly above linear in —far from the upper bound, proportional to . The number of colors required to color unit distance graphs is also unknown (the Hadwiger–Nelson problem): some unit distance graphs require five colors, and every unit distance graph can be colored with seven colors. For every algebraic number there is a unit distance graph with two vertices that must be that distance apart. According to the Beckman–Quarles theorem, the only plane transformations that preserve all unit distance graphs are the isometries.
It is possible to construct a unit distance graph efficiently, given its points. Finding all unit distances has applications in pattern matching, where it can be a first step in finding congruent copies of larger patterns. However, determining whether a given graph can be represented as a unit distance graph is NP-hard, and more specifically complete for the existential theory of the reals.
Definition
The unit distance graph for a set of points in the plane is the undirected graph having those points as its vertices, with an edge between two vertices whenever their Euclidean distance is exactly one. An abstract graph is said to be a unit distance graph if it is possible to find distinct locations in the plane for its vertices, so that its edges have unit length and so that all non-adjacent pairs of vertices have non-unit distances. When this is possible, the abstract graph is isomorphic to the unit distance graph of the chosen locations. Alternatively, some sources use a broader definition, allowing non-adjacent pairs of vertices to be at unit distance. The resulting graphs are the subgraphs of the unit distance graphs (as defined here). Where the terminology may be ambiguous, the graphs in which non-edges must be a non-unit distance apart may be called strict unit distance graphs or faithful unit distance graphs. The subgraphs of unit distance graphs are equivalently the graphs that can be drawn in the plane using only one edge length. For brevity, this article refers to these as "non-strict unit distance graphs".
Unit distance graphs should not be confused with unit disk graphs, which connect pair
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https://en.wikipedia.org/wiki/F-ratio
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F-ratio or f-ratio may refer to:
The F-ratio used in statistics, which relates the variances of independent samples; see F-distribution
f-ratio (oceanography), which relates recycled and total primary production in the surface ocean
f-number, f-ratio, or focal ratio, the ratio of the focal length of an optical system to the diameter of its entrance pupil
See also
F-number (disambiguation)
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https://en.wikipedia.org/wiki/Instancing
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Instancing may refer to:
Geometry instancing, a technique used in realtime rendering
Dungeon instancing, a technique used in online games to provide individual players or groups of players with their own instance of some sort of content at the same time
See also
Instantiation (disambiguation)
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https://en.wikipedia.org/wiki/Fisher%27s%20method
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In statistics, Fisher's method, also known as Fisher's combined probability test, is a technique for data fusion or "meta-analysis" (analysis of analyses). It was developed by and named for Ronald Fisher. In its basic form, it is used to combine the results from several independence tests bearing upon the same overall hypothesis (H0).
Application to independent test statistics
Fisher's method combines extreme value probabilities from each test, commonly known as "p-values", into one test statistic (X2) using the formula
where pi is the p-value for the ith hypothesis test. When the p-values tend to be small, the test statistic X2 will be large, which suggests that the null hypotheses are not true for every test.
When all the null hypotheses are true, and the pi (or their corresponding test statistics) are independent, X2 has a chi-squared distribution with 2k degrees of freedom, where k is the number of tests being combined. This fact can be used to determine the p-value for X2.
The distribution of X2 is a chi-squared distribution for the following reason; under the null hypothesis for test i, the p-value pi follows a uniform distribution on the interval [0,1]. The negative logarithm of a uniformly distributed value follows an exponential distribution. Scaling a value that follows an exponential distribution by a factor of two yields a quantity that follows a chi-squared distribution with two degrees of freedom. Finally, the sum of k independent chi-squared values, each with two degrees of freedom, follows a chi-squared distribution with 2k degrees of freedom.
Limitations of independence assumption
Dependence among statistical tests is generally positive, which means that the p-value of X2 is too small (anti-conservative) if the dependency is not taken into account. Thus, if Fisher's method for independent tests is applied in a dependent setting, and the p-value is not small enough to reject the null hypothesis, then that conclusion will continue to hold even if the dependence is not properly accounted for. However, if positive dependence is not accounted for, and the meta-analysis p-value is found to be small, the evidence against the null hypothesis is generally overstated. The mean false discovery rate, , reduced for k independent or positively correlated tests, may suffice to control alpha for useful comparison to an over-small p-value from Fisher's X2.
Extension to dependent test statistics
In cases where the tests are not independent, the null distribution of X2 is more complicated. A common strategy is to approximate the null distribution with a scaled random variable. Different approaches may be used depending on whether or not the covariance between the different p-values is known.
Brown's method can be used to combine dependent p-values whose underlying test statistics have a multivariate normal distribution with a known covariance matrix. Kost's method extends Brown's to allow one to combine p-values when the covar
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https://en.wikipedia.org/wiki/Topological%20censorship
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The topological censorship theorem (if valid) states that general relativity does not allow an observer to probe the topology of spacetime: any topological structure collapses too quickly to allow light to traverse it. More precisely, in a globally hyperbolic, asymptotically flat spacetime satisfying the null energy condition, every causal curve from past null infinity to future null infinity is fixed-endpoint homotopic to a curve in a topologically trivial neighbourhood of infinity.
A 2013 paper by Sergey Krasnikov claims that the topological censorship theorem was not proven in the original article because of a gap in the proof.
References
Lorentzian manifolds
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https://en.wikipedia.org/wiki/Nascimento%20%28surname%29
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Nascimento or do Nascimento (, meaning birth) is a common Portuguese surname that refers to the birth of Jesus Christ.
It may also originate from the Dutch surname Nassau.
Statistics
In 2004, about 0.39% of the Portuguese population bore the surname Nascimento.
According to Forebears.io, Nascimento is the 571st most common surname in the world and is most prevalent in Brazil.
Notable people with this surname
Abdias do Nascimento (1914–2011), an Afro-Brazilian scholar, artist, and politician
Alexandra do Nascimento (born 1981), a Brazilian handball player
Alexandre do Nascimento (born 1925), Roman Catholic cardinal and archbishop in Angola
Andrêsa do Nascimento (1859–1927), also known as Preta Fernanda
Eduardo Nascimento (1943–2019), Angolan singer
Emanuel Nascimento (born 1970), a Brazilian freestyle swimmer
Fabíula Nascimento (born 1978), a Brazilian actress
Francisco José do Nascimento (1839–1914), Afro-Brazilian abolitionist
Francisco Manoel de Nascimento (1734–1819), the Portuguese poet known as Filinto Elysio
Lopo do Nascimento (born 1942), an Angolan politician
Luiz Gonzaga do Nascimento (1912–1989), a Brazilian singer, songwriter, musician and poet
Milton Nascimento (born 1942), a Brazilian singer-songwriter
Norton Nascimento (1962–2007), a Brazilian actor
Rodrigo Nascimento (born 1992), a Brazilian mixed martial artist
Yazaldes Nascimento (born 1986), a Portuguese athlete
Sandro Barbosa do Nascimento (1978–2000), a notorious Brazilian criminal
Tuany Nascimento (20th and 21st century), Brazilian ballet dancer and dance teacher
Tasha Nascimento and Tracie Nascimento (born 1995), Brazilian rapper duo
Brazilian footballers
Aldair Santos do Nascimento (born 1965), the former Brazilian football defender
André Luiz Silva do Nascimento (born 1980), the Brazilian footballer
Evandro Silva do Nascimento (born 1987), the Brazilian footballer
Fábio do Nascimento Silva (born 1983), Brazilian footballer
Gabriel dos Santos Nascimento (born 1983), Brazilian footballer
Gi Santos - Giovanna dos Santos Nascimento (born 1997), Brazilian women's football player
Jeovânio Rocha do Nascimento (born 1977), Brazilian defensive midfielder
Robert Kenedy Nunes Nascimento (born 1996), the Brazilian footballer
Leonardo Nascimento de Araújo (born 1969), Brazilian footballer
Luizão — Luiz Carlos Nascimento Júnior (born 1987), Brazilian footballer
Matheus Leite Nascimento (born 1983), the Brazilian footballer
Moacir Barbosa Nascimento (1921–2000), the former Brazilian football goalkeeper
Nasa - Marcos Antonio García Nascimento (born 1979), Brazilian footballer
Paulo Sérgio Silvestre do Nascimento (born 1969), the former Brazilian footballer
Pelé — Edson Arantes do Nascimento (1940–2022), the Brazilian footballer
Edinho — Edson Cholbi Nascimento (born 1970), the former Brazilian football goalkeeper and son of Pelé
Rafael da Silva Nascimento (born 1984), Brazilian footballer
Ramires Santos do Nascimento (born 1987), the Brazilian
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https://en.wikipedia.org/wiki/Absolute%20presentation%20of%20a%20group
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In mathematics, an absolute presentation is one method of defining a group.
Recall that to define a group by means of a presentation, one specifies a set of generators so that every element of the group can be written as a product of some of these generators, and a set of relations among those generators. In symbols:
Informally is the group generated by the set such that for all . But here there is a tacit assumption that is the "freest" such group as clearly the relations are satisfied in any homomorphic image of . One way of being able to eliminate this tacit assumption is by specifying that certain words in should not be equal to That is we specify a set , called the set of irrelations, such that for all
Formal definition
To define an absolute presentation of a group one specifies a set of generators and sets and of relations and irrelations among those
generators. We then say has absolute presentation
provided that:
has presentation
Given any homomorphism such that the irrelations are satisfied in , is isomorphic to .
A more algebraic, but equivalent, way of stating condition 2 is:
2a. If is a non-trivial normal subgroup of then
Remark: The concept of an absolute presentation has been fruitful in fields such as algebraically closed groups and the Grigorchuk topology.
In the literature, in a context where absolute presentations are being discussed, a presentation (in the usual sense of the word) is sometimes referred to as a relative presentation, which is an instance of a retronym.
Example
The cyclic group of order 8 has the presentation
But, up to isomorphism there are three more groups that "satisfy" the relation namely:
and
However none of these satisfy the irrelation . So an absolute presentation for the cyclic group of order 8 is:
It is part of the definition of an absolute presentation that the irrelations are not satisfied in any proper homomorphic image of the group. Therefore:
Is not an absolute presentation for the cyclic group of order 8 because the irrelation is satisfied in the cyclic group of order 4.
Background
The notion of an absolute presentation arises from Bernhard Neumann's study of the isomorphism problem for algebraically closed groups.
A common strategy for considering whether two groups and are isomorphic is to consider whether a presentation for one might be transformed into a presentation for the other. However algebraically closed groups are neither finitely generated nor recursively presented and so it is impossible to compare their presentations. Neumann considered the following alternative strategy:
Suppose we know that a group with finite presentation can be embedded in the algebraically closed group then given another algebraically closed group , we can ask "Can be embedded in ?"
It soon becomes apparent that a presentation for a group does not contain enough information to make this decision for while there may be a homomorphism , this homomorphism need
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https://en.wikipedia.org/wiki/Truncated%20distribution
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In statistics, a truncated distribution is a conditional distribution that results from restricting the domain of some other probability distribution. Truncated distributions arise in practical statistics in cases where the ability to record, or even to know about, occurrences is limited to values which lie above or below a given threshold or within a specified range. For example, if the dates of birth of children in a school are examined, these would typically be subject to truncation relative to those of all children in the area given that the school accepts only children in a given age range on a specific date. There would be no information about how many children in the locality had dates of birth before or after the school's cutoff dates if only a direct approach to the school were used to obtain information.
Where sampling is such as to retain knowledge of items that fall outside the required range, without recording the actual values, this is known as censoring, as opposed to the truncation here.
Definition
The following discussion is in terms of a random variable having a continuous distribution although the same ideas apply to discrete distributions. Similarly, the discussion assumes that truncation is to a semi-open interval y ∈ (a,b] but other possibilities can be handled straightforwardly.
Suppose we have a random variable, that is distributed according to some probability density function, , with cumulative distribution function both of which have infinite support. Suppose we wish to know the probability density of the random variable after restricting the support to be between two constants so that the support, . That is to say, suppose we wish to know how is distributed given .
where for all and everywhere else. That is, where is the indicator function. Note that the denominator in the truncated distribution is constant with respect to the .
Notice that in fact is a density:
.
Truncated distributions need not have parts removed from the top and bottom. A truncated distribution where just the bottom of the distribution has been removed is as follows:
where for all and everywhere else, and is the cumulative distribution function.
A truncated distribution where the top of the distribution has been removed is as follows:
where for all and everywhere else, and is the cumulative distribution function.
Expectation of truncated random variable
Suppose we wish to find the expected value of a random variable distributed according to the density and a cumulative distribution of given that the random variable, , is greater than some known value . The expectation of a truncated random variable is thus:
where again is for all and everywhere else.
Letting and be the lower and upper limits respectively of support for the original density function (which we assume is continuous), properties of , where is some continuous function with a continuous derivative, include:
and
Provided that the lim
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https://en.wikipedia.org/wiki/Spurious
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Spurious may refer to:
Spurious relationship in statistics
Spurious emission or spurious tone in radio engineering
Spurious key in cryptography
Spurious interrupt in computing
Spurious wakeup in computing
Spurious, a 2011 novel by Lars Iyer
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https://en.wikipedia.org/wiki/Borel%20conjecture
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In mathematics, specifically geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. It is a rigidity conjecture, asserting that a weak, algebraic notion of equivalence (namely, homotopy equivalence) should imply a stronger, topological notion (namely, homeomorphism).
Precise formulation of the conjecture
Let and be closed and aspherical topological manifolds, and let
be a homotopy equivalence. The Borel conjecture states that the map is homotopic to a homeomorphism. Since aspherical manifolds with isomorphic fundamental groups are homotopy equivalent, the Borel conjecture implies that aspherical closed manifolds are determined, up to homeomorphism, by their fundamental groups.
This conjecture is false if topological manifolds and homeomorphisms are replaced by smooth manifolds and diffeomorphisms; counterexamples can be constructed by taking a connected sum with an exotic sphere.
The origin of the conjecture
In a May 1953 letter to Jean-Pierre Serre, Armand Borel raised the question whether two aspherical manifolds with isomorphic fundamental groups are homeomorphic. A positive answer to the question "Is every homotopy equivalence between closed aspherical manifolds homotopic to a homeomorphism?" is referred to as the "so-called Borel Conjecture" in a 1986 paper of Jonathan Rosenberg.
Motivation for the conjecture
A basic question is the following: if two closed manifolds are homotopy equivalent, are they homeomorphic? This is not true in general: there are homotopy equivalent lens spaces which are not homeomorphic.
Nevertheless, there are classes of manifolds for which homotopy equivalences between them can be homotoped to homeomorphisms. For instance, the Mostow rigidity theorem states that a homotopy equivalence between closed hyperbolic manifolds is homotopic to an isometry—in particular, to a homeomorphism. The Borel conjecture is a topological reformulation of Mostow rigidity, weakening the hypothesis from hyperbolic manifolds to aspherical manifolds, and similarly weakening the conclusion from an isometry to a homeomorphism.
Relationship to other conjectures
The Borel conjecture implies the Novikov conjecture for the special case in which the reference map is a homotopy equivalence.
The Poincaré conjecture asserts that a closed manifold homotopy equivalent to , the 3-sphere, is homeomorphic to . This is not a special case of the Borel conjecture, because is not aspherical. Nevertheless, the Borel conjecture for the 3-torus implies the Poincaré conjecture for .
References
F. Thomas Farrell, The Borel conjecture. Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001), 225–298, ICTP Lect. Notes, 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2002.
Matthias Kreck, and Wolfgang Lück, The Novikov conjecture. Geometry and algebra. Oberwolfach Seminars, 33. Birkhäuser Verlag, Basel, 2005.
Geo
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https://en.wikipedia.org/wiki/Los%20Angeles%20Rams%20statistics
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This page details statistics about the Los Angeles Rams American football franchise, formerly the St. Louis Rams and the Cleveland Rams.
Franchise firsts
First NFL game – A 28–0 loss to the Detroit Lions, 9/10/37.
First NFL win – A 21–3 victory over the Philadelphia Eagles, 9/17/37.
First winning season – 1945 (9–1).
First championship season – 1945.
First player drafted – Johnny Drake, 1937.
First Ram elected to the Hall of Fame – QB Bob Waterfield, 1965.
First to pass 400 yards in a game – Jim Hardy, 406 yards vs. Chicago Cardinals, 10/31/48.
First to rush 200 yards in a game – Dan Towler, 205 yards vs. the Baltimore Colts, 11/22/53.
First 1,000-yard rusher in a season – Dick Bass, 1,033 yards (1962).
First Super Bowl appearance – A 31–19 loss to the Pittsburgh Steelers in Super Bowl XIV, 1/20/80.
Wins/losses in a season
Most games won in a season (regular season): 14, 2001
Most games won in a season (including postseason): 16, 1999, 2001
Most games lost in a season: 15, 2009
Individual records
Appearances
Most seasons in a Rams uniform – 20, Jackie Slater, (1976–1995).
Most games played in a Rams uniform – 259, Jackie Slater, (1976–1995).
Most consecutive games played in a Rams uniform – 201, Jack Youngblood, (1971–1984).
Most Pro Bowls – 14, Merlin Olsen, (1962–1975).
Game
Points – 24, eleven times, last time by Todd Gurley, vs Seattle Seahawks, 12/17/17
Touchdowns – 4, eleven times, last time by Todd Gurley, vs Seattle Seahawks, 12/17/17
Rushing yards – 247, Willie Ellison, vs Green Bay Packers, 12/05/71
Rushing touchdowns – 4, Marshall Faulk, vs Minnesota Vikings, 12/10/00
Passing yards – 554, Norm Van Brocklin, vs New York Yanks, 28 September 1951
Passing touchdowns – 5, 10 times, last time by Jared Goff, vs Minnesota Vikings, 09/27/18
Receptions – 18, Tom Fears, vs Green Bay Packers, 12/03/50
Receiving yards – 336, Willie "Flipper" Anderson, vs New Orleans Saints, 11/26/89
Receiving touchdowns – 4, four times, last time by Isaac Bruce, vs San Francisco 49ers, 10/10/99
Total yards – 336 (336 receiving), Willie "Flipper" Anderson, vs New Orleans Saints, 11/26/89
Interceptions – 3, many times, last time by Keith Lyle, vs Atlanta Falcons, 12/15/96
Sacks – 5, Gary Jeter, vs Los Angeles Raiders, 09/18/88
Field goals – 7, Greg Zuerlein, at Dallas Cowboys, 10/01/17
Punts – 12, two times, last time by Rusty Jackson, vs San Francisco 49ers, 11/21/76
Punting average yards – 56.4, Johnny Hekker, vs Philadelphia Eagles, 12/10/17
Kickoff returns – 8, Tony Horne, vs Kansas City Chiefs, 10/22/00
Kickoff return yards – 267, Tony Horne, vs Kansas City Chiefs, 10/22/00
Punt returns – 7, nine times, last time by Pharoh Cooper, vs Seattle Seahawks, 12/17/17
Punt return yards – 207, LeRoy Irvin, vs Atlanta Falcons, 11/14/81
Season
Points – 163 Jeff Wilkins (2003)
Touchdowns – 26 (18-run, 8-pass) Marshall Faulk (2000)
Rushing yards – 2,105 Eric Dickerson (1984)
Rushing touchdowns – 18, two times, last time by Marshall Faulk (2000)
Passer rating – 109.2 Ku
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https://en.wikipedia.org/wiki/Wide%20chord
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Wide chord fan refers to the fan blades on a modern turbofan jet engine having a ducted fan with a specific blade geometry - In layman's terms, they would be described as having wider blades than other jet engines. The technology was pioneered by Geoff Wilde at Rolls-Royce in the 1970s.
Overview
The main fan on a jet engine consists of a number of aerofoils mounted at the rotational center on the fan disk, and as the engine core rotates the fan blades accelerate an air mass and create the force to move forward which gives thrust (in accordance with Newton's Third Law).
In theory the larger the fan diameter (the line from the tip of one fanblade to its opposite member) the greater the thrust. In practical applications, fan size is limited by the weight, the space available around the aircraft and by the increased drag (resistance) generated by the larger frontal area.
In the race to achieve better fuel economy, more thrust and less weight & noise from jet engines, designers have refined the blade design and materials to extract more thrust for any given fan disk area. One significant improvement is to make blade chords wider and, more recently, alter the blade geometry to give it a scimitar-like shape. Further refinements include making the blades from a light material such as titanium and to manufacture them with a hollow cross-section.
Modern jet engines such as the Rolls-Royce Trent 900 and the Engine Alliance GP7000, which both power the Airbus A380, are examples of engines with wide-chord fans.
Key design considerations
A wide chord fan has fewer, wider blades compared to the narrower blades on earlier technology fans. The blades are often hollow and made from titanium. The wide-chord fan blade was designed and developed at Rolls-Royce Barnoldswick in Lancashire. The manufacturing process uses superplastic forming, diffusion-bonded technology to achieve a light weight, strong design.
References
External links
Jet engines
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https://en.wikipedia.org/wiki/2005%20World%20Series%20of%20Poker%20results
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This list of 2005 World Series of Poker (WSOP) results includes statistics, final table results and payouts.
Results
Event 1: $500 Casino Employee's No Limit Hold'em
June 2, 2005
This event kicked off the 2005 WSOP. It was a $500 buy-in no limit Texas hold 'em tournament reserved for casino employees that work in Nevada.
Number of buy-ins: 662
Total Prize Pool: $297,900
Number of Payouts: 63
Event 2: $1,500 No-Limit Texas Hold'em
June 3, 2005
Number of buy-ins: 2,305
Total Prize Pool: $3,180,900
Number of Payouts: 200
Event 3: $1,500 Pot-Limit Hold'em
June 4, 2005
Number of buy-ins: 1,071
Total Prize Pool: $1,477,980
Number of Payouts: 100
Event 4: $1,500 Limit Hold 'em
June 5, 2005
Number of buy-ins: 1,049
Total Prize Pool: $1,447,620
Number of Payouts: 100
Event 5: $1,500 Omaha High-Low 8/OB
June 6, 2005
Number of buy-ins: 699
Total Prize Pool: $964,620
Number of Payouts: 63
Event 6: $2,500 No-Limit Hold'em (Six-Handed)
June 7, 2005
Number of buy-ins: 548
Total Prize Pool: $1,260,400
Number of Payouts: 66
Event 7: $1,000 No-limit Hold'em w/Rebuys
June 8, 2005
Number of buy-ins: 826
Number of rebuys: 1,495
Total Prize Pool: $2,201,630
Number of Payouts: 72
Event 8: $1,500 Seven Card Stud
June 9, 2005
Number of buy-ins: 472
Total Prize Pool: $651,360
Number of Payouts: 40
Event 9: $2,000 No Limit Hold'em
June 10, 2005
Number of buy-ins: 1403
Total Prize Pool: $2,581,520
Number of Payouts: 140
Event 10: $2,000 Limit Hold'em
June 11, 2005
Number of buy-ins: 569
Total Prize Pool: $1,046,940
Number of Payouts: 54
Event 11: $2,000 Pot Limit Hold'em
June 12, 2005
Number of buy-ins: 540
Total Prize Pool: $993,600
Number of Payouts: 45
Event 12: $2,000 Pot Limit Omaha w/Rebuys
June 13, 2005
Number of buy-ins: 212
Number of rebuys: 395
Total Prize Pool: $1,156,350
Number of Payouts: 18
Event 13: $5,000 No-Limit Hold'em
June 14, 2005
Number of buy-ins: 466
Total Prize Pool: $2,190,200
Number of Payouts: 45
Event 14: $1000 Seven Card Stud High-Low 8/OB
June 15, 2005
Number of buy-ins: 595
Total Prize Pool: $541,450
Number of Payouts: 48
Event 15: $1,500 Limit Hold'em Shootout
June 16, 2005
Number of buy-ins: 450
Total Prize Pool: $621,000
Number of Payouts: 45
Event 16: $1,500 No Limit Hold'em Shootout
June 17, 2005
Number of buy-ins: 780
Total Prize Pool: $1,076,400
Number of Payouts: 78
Event 17: $2,500 Limit Hold'em
June 18, 2005
Number of buy-ins: 373
Total Prize Pool: $857,900
Number of Payouts: 36
Event 18: $1,500 Pot Limit Omaha
June 19, 2005
Number of buy-ins: 291
Total Prize Pool: $404,710
Number of Payouts: 27
Event 19: $2,000 Seven Card Stud High-Low 8/OB
June 19, 2005
Number of buy-ins: 279
Total Prize Pool: $513,360
Number of Payouts: 24
Event 20: $5,000 Pot-Limit Hold'em
June 20, 2005
Number of buy-ins: 239
Total Prize Pool: $1,123,300
Number of Payouts: 18
Event 21: $2500 Omaha High-Low 8/OB
June 21, 2005
Number of
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https://en.wikipedia.org/wiki/5-cube
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In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.
It is represented by Schläfli symbol {4,3,3,3} or {4,33}, constructed as 3 tesseracts, {4,3,3}, around each cubic ridge. It can be called a penteract, a portmanteau of the Greek word , for 'five' (dimensions), and the word tesseract (the 4-cube). It can also be called a regular deca-5-tope or decateron, being a 5-dimensional polytope constructed from 10 regular facets.
Related polytopes
It is a part of an infinite hypercube family. The dual of a 5-cube is the 5-orthoplex, of the infinite family of orthoplexes.
Applying an alternation operation, deleting alternating vertices of the 5-cube, creates another uniform 5-polytope, called a 5-demicube, which is also part of an infinite family called the demihypercubes.
The 5-cube can be seen as an order-3 tesseractic honeycomb on a 4-sphere. It is related to the Euclidean 4-space (order-4) tesseractic honeycomb and paracompact hyperbolic honeycomb order-5 tesseractic honeycomb.
As a configuration
This configuration matrix represents the 5-cube. The rows and columns correspond to vertices, edges, faces, cells, and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
Cartesian coordinates
The Cartesian coordinates of the vertices of a 5-cube centered at the origin and having edge length 2 are
(±1,±1,±1,±1,±1),
while this 5-cube's interior consists of all points (x0, x1, x2, x3, x4) with -1 < xi < 1 for all i.
Images
n-cube Coxeter plane projections in the Bk Coxeter groups project into k-cube graphs, with power of two vertices overlapping in the projective graphs.
Projection
The 5-cube can be projected down to 3 dimensions with a rhombic icosahedron envelope. There are 22 exterior vertices, and 10 interior vertices. The 10 interior vertices have the convex hull of a pentagonal antiprism. The 80 edges project into 40 external edges and 40 internal ones. The 40 cubes project into golden rhombohedra which can be used to dissect the rhombic icosahedron. The projection vectors are u = {1, φ, 0, -1, φ}, v = {φ, 0, 1, φ, 0}, w = {0, 1, φ, 0, -1}, where φ is the golden ratio, .
It is also possible to project penteracts into three-dimensional space, similarly to projecting a cube into two-dimensional space.
Symmetry
The 5-cube has Coxeter group symmetry B5, abstract structure , order 3840, containing 25 hyperplanes of reflection. The Schläfli symbol for the 5-cube, {4,3,3,3}, matches the Coxeter notation symmetry [4,3,3,3].
Prisms
All hypercubes have lower symmetry forms constructed as prisms. The 5-cube has 7 prismatic forms from the lowest 5-orthotope, { }5, and upwards as orthogonal edges are constrained to be of equal length. The vertices in a prism are equal to the product of the vertices in the elemen
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https://en.wikipedia.org/wiki/Primary%20pseudoperfect%20number
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In mathematics, and particularly in number theory, N is a primary pseudoperfect number if it satisfies the Egyptian fraction equation
where the sum is over only the prime divisors of N.
Properties
Equivalently, N is a primary pseudoperfect number if it satisfies
Except for the primary pseudoperfect number N = 2, this expression gives a representation for N as the sum of distinct divisors of N. Therefore, each primary pseudoperfect number N (except N = 2) is also pseudoperfect.
The eight known primary pseudoperfect numbers are
2, 6, 42, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086 .
The first four of these numbers are one less than the corresponding numbers in Sylvester's sequence, but then the two sequences diverge.
It is unknown whether there are infinitely many primary pseudoperfect numbers, or whether there are any odd primary pseudoperfect numbers.
The prime factors of primary pseudoperfect numbers sometimes may provide solutions to Znám's problem, in which all elements of the solution set are prime. For instance, the prime factors of the primary pseudoperfect number 47058 form the solution set {2,3,11,23,31} to Znám's problem. However, the smaller primary pseudoperfect numbers 2, 6, 42, and 1806 do not correspond to solutions to Znám's problem in this way, as their sets of prime factors violate the requirement that no number in the set can equal one plus the product of the other numbers. Anne (1998) observes that there is exactly one solution set of this type that has k primes in it, for each k ≤ 8, and conjectures that the same is true for larger k.
If a primary pseudoperfect number N is one less than a prime number, then N × (N + 1) is also primary pseudoperfect. For instance, 47058 is primary pseudoperfect, and 47059 is prime, so 47058 × 47059 = 2214502422 is also primary pseudoperfect.
History
Primary pseudoperfect numbers were first investigated and named by Butske, Jaje, and Mayernik (2000). Using computational search techniques, they proved the remarkable result that for each positive integer r up to 8, there exists exactly one primary pseudoperfect number with precisely r (distinct) prime factors, namely, the rth known primary pseudoperfect number. Those with 2 ≤ r ≤ 8, when reduced modulo 288, form the arithmetic progression 6, 42, 78, 114, 150, 186, 222, as was observed by Sondow and MacMillan (2017).
See also
Giuga number
References
.
.
.
External links
Integer sequences
Egyptian fractions
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https://en.wikipedia.org/wiki/Cahen%27s%20constant
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In mathematics, Cahen's constant is defined as the value of an infinite series of unit fractions with alternating signs:
Here denotes Sylvester's sequence, which is defined recursively by
Combining these fractions in pairs leads to an alternative expansion of Cahen's constant as a series of positive unit fractions formed from the terms in even positions of Sylvester's sequence. This series for Cahen's constant forms its greedy Egyptian expansion:
This constant is named after (also known for the Cahen–Mellin integral), who was the first to introduce it and prove its irrationality.
Continued fraction expansion
The majority of naturally occurring mathematical constants have no known simple patterns in their continued fraction expansions. Nevertheless, the complete continued fraction expansion of Cahen's constant is known: it is
where the sequence of coefficients
is defined by the recurrence relation
All the partial quotients of this expansion are squares of integers. Davison and Shallit made use of the continued fraction expansion to prove that is transcendental.
Alternatively, one may express the partial quotients in the continued fraction expansion of Cahen's constant through the terms of Sylvester's sequence: To see this, we prove by induction on that . Indeed, we have , and if holds for some , then
where we used the recursion for in the first step respectively the recursion for in the final step. As a consequence, holds for every , from which it is easy to conclude that
.
Best approximation order
Cahen's constant has best approximation order . That means, there exist constants such that the inequality
has infinitely many solutions , while the inequality has at most finitely many solutions .
This implies (but is not equivalent to) the fact that has irrationality measure 3, which was first observed by .
To give a proof, denote by the sequence of convergents to Cahen's constant (that means, ).
But now it follows from and the recursion for that
for every . As a consequence, the limits
and
(recall that ) both exist by basic properties of infinite products, which is due to the absolute convergence of . Numerically, one can check that . Thus the well-known inequality
yields
and
for all sufficiently large . Therefore has best approximation order 3 (with ), where we use that any solution to
is necessarily a convergent to Cahen's constant.
Notes
References
External links
Mathematical constants
Real transcendental numbers
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https://en.wikipedia.org/wiki/Bureau%20of%20Justice%20Statistics
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The Bureau of Justice Statistics (UJC) of the U.S. Department of Justice is the principal federal agency responsible for measuring crime, criminal victimization, criminal offenders, victims of crime, correlates of crime, and the operation of criminal and civil justice systems at the federal, state, tribal, and local levels. Established on December 27, 1979, BJS collects, analyzes, and publishes data relating to crime in the United States. The agency publishes data regarding statistics gathered from the roughly fifty-thousand agencies, offices, courts, and institutions that together comprise the U.S. justice system.
The mission of BJS is "To collect, analyze, publish, and disseminate information on crime, criminal offenders, victims of crime, and the operation of justice systems at all levels of government."
BJS, along with the National Institute of Justice (NIJ), Bureau of Justice Assistance (BJA), Office of Juvenile Justice and Delinquency Prevention (OJJDP), Office for Victims of Crime (OVC), and other program offices, comprise the Office of Justice Programs (OJP) branch of the Department of Justice.
Programs
The BJS conducts the Annual Survey of Jails of a sample of about 950 U.S. jails, and a periodic Census of Jails covering all U.S. jails.
Data from these programs was used to show that local jails in the U.S. had a sharp decline in inmates from February to May, 2020 of perhaps 185,000 inmates, more than 20% of the inmate population, in response to the danger of covid-19 on a crowded incarcerated population. Many inmates were given an "expedited release".
See also
Uniform Crime Reports (FBI)
Data.gov
USAFacts
BJS Directors
In 2005, the Bush administration replaced BJS Director Lawrence Greenfeld after he refused to remove certain racial statistics from a report, despite having published similar statistics in 2001. The following two references provide analysis and initial reporting, respectively.
Josephf M. Bessette
Eric Lichtblau
More recent directors have included Jeffrey H. Anderson, Jeffrey Sedgwick, Michael Sinclair, John Jay Professor James P. Lynch, and former Deputy Director William Sabol.
Until 2012 the position of the BJS Director required a Senate approval, but since 2012 the post only requires the President's appointment. Alex Piquero is the current BJS Director.
References
External links
most recent reports
Research, Evaluation, and Statistics account on USAspending.gov
Government agencies established in 1979
Crime statistics
National statistical services
Statistics
Statistical organizations in the United States
1979 establishments in Washington, D.C.
Federal Statistical System of the United States
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https://en.wikipedia.org/wiki/Category%20of%20finite-dimensional%20Hilbert%20spaces
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In mathematics, the category FdHilb has all finite-dimensional Hilbert spaces for objects and the linear transformations between them as morphisms. Whereas the theory described by the normal category of Hilbert spaces, Hilb, is ordinary quantum mechanics, the corresponding theory on finite dimensional Hilbert spaces is called fdQM.
Properties
This category
is monoidal,
possesses finite biproducts, and
is dagger compact.
According to a theorem of Selinger, the category of finite-dimensional Hilbert spaces is complete in the dagger compact category. Many ideas from Hilbert spaces, such as the no-cloning theorem, hold in general for dagger compact categories. See that article for additional details.
References
Monoidal categories
Dagger categories
Hilbert spaces
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https://en.wikipedia.org/wiki/Uniform%20topology
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In mathematics, the uniform topology on a space may mean:
In functional analysis, it sometimes refers to a polar topology on a topological vector space.
In general topology, it is the topology carried by a uniform space.
In real analysis, it is the topology of uniform convergence.
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https://en.wikipedia.org/wiki/Uniformly%20Cauchy%20sequence
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In mathematics, a sequence of functions from a set S to a metric space M is said to be uniformly Cauchy if:
For all , there exists such that for all : whenever .
Another way of saying this is that as , where the uniform distance between two functions is defined by
Convergence criteria
A sequence of functions {fn} from S to M is pointwise Cauchy if, for each x ∈ S, the sequence {fn(x)} is a Cauchy sequence in M. This is a weaker condition than being uniformly Cauchy.
In general a sequence can be pointwise Cauchy and not pointwise convergent, or it can be uniformly Cauchy and not uniformly convergent. Nevertheless, if the metric space M is complete, then any pointwise Cauchy sequence converges pointwise to a function from S to M. Similarly, any uniformly Cauchy sequence will tend uniformly to such a function.
The uniform Cauchy property is frequently used when the S is not just a set, but a topological space, and M is a complete metric space. The following theorem holds:
Let S be a topological space and M a complete metric space. Then any uniformly Cauchy sequence of continuous functions fn : S → M tends uniformly to a unique continuous function f : S → M.
Generalization to uniform spaces
A sequence of functions from a set S to a uniform space U is said to be uniformly Cauchy if:
For all and for any entourage , there exists such that whenever .
See also
Modes of convergence (annotated index)
Functional analysis
Convergence (mathematics)
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https://en.wikipedia.org/wiki/5-demicube
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In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.
It was discovered by Thorold Gosset. Since it was the only semiregular 5-polytope (made of more than one type of regular facets), he called it a 5-ic semi-regular. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM5 for a 5-dimensional half measure polytope.
Coxeter named this polytope as 121 from its Coxeter diagram, which has branches of length 2, 1 and 1 with a ringed node on one of the short branches, and Schläfli symbol or {3,32,1}.
It exists in the k21 polytope family as 121 with the Gosset polytopes: 221, 321, and 421.
The graph formed by the vertices and edges of the demipenteract is sometimes called the Clebsch graph, though that name sometimes refers to the folded cube graph of order five instead.
Cartesian coordinates
Cartesian coordinates for the vertices of a demipenteract centered at the origin and edge length 2 are alternate halves of the penteract:
(±1,±1,±1,±1,±1)
with an odd number of plus signs.
As a configuration
This configuration matrix represents the 5-demicube. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-demicube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.
* = The number of elements (diagonal values) can be computed by the symmetry order D5 divided by the symmetry order of the subgroup with selected mirrors removed.
Projected images
Images
Related polytopes
It is a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
There are 23 Uniform 5-polytopes (uniform 5-polytopes) that can be constructed from the D5 symmetry of the demipenteract, 8 of which are unique to this family, and 15 are shared within the penteractic family.
The 5-demicube is third in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes (5-simplices and 5-orthoplexes in the case of the 5-demicube). In Coxeter's notation the 5-demicube is given the symbol 121.
References
T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
H.S.M. Coxeter:
Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, , p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polyto
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https://en.wikipedia.org/wiki/Facet%20%28geometry%29
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In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself. More specifically:
In three-dimensional geometry, a facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a face. To facet a polyhedron is to find and join such facets to form the faces of a new polyhedron; this is the reciprocal process to stellation and may also be applied to higher-dimensional polytopes.
In polyhedral combinatorics and in the general theory of polytopes, a face that has dimension n − 1 (an (n − 1)-face or hyperface) is also called a facet.
A facet of a simplicial complex is a maximal simplex, that is a simplex that is not a face of another simplex of the complex. For (boundary complexes of) simplicial polytopes this coincides with the meaning from polyhedral combinatorics.
References
External links
Polyhedra
Polyhedral combinatorics
Polytopes
Broad-concept articles
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https://en.wikipedia.org/wiki/Grace%20Wahba
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Grace Goldsmith Wahba (born August 3, 1934) is an American statistician and retired I. J. Schoenberg-Hilldale Professor of Statistics at the University of Wisconsin–Madison. She is a pioneer in methods for smoothing noisy data. Best known for the development of generalized cross-validation and "Wahba's problem", she has developed methods with applications in demographic studies, machine learning, DNA microarrays, risk modeling, medical imaging, and climate prediction.
Biography
Wahba had an interest in science from an early age, when she was in junior high she was given a chemistry set. At this time she was also interested in becoming an engineer.
Wahba studied at Cornell University for her undergraduate degree. When she was there women were severely restricted in their privileges, for example she was required to live in a dorm and had a curfew. She received her bachelor's degree from Cornell University in 1956 and a master's degree from the University of Maryland, College Park in 1962. She worked in industry for several years before receiving her doctorate from Stanford University in 1966 and settling in Madison in 1967.
She is the author of Spline Models for Observational Data. She retired in August 2018 from the University of Wisconsin-Madison.
Her life and career are discussed in a 2020 interview.
Honors and awards
Wahba was elected to the American Academy of Arts and Sciences in 1997 and to the National Academy of Sciences in 2000. She is also a fellow of several academic societies including the American Association for the Advancement of Science, the American Statistical Association, and the Institute of Mathematical Statistics.
Over the years she has received a selection of notable awards in the statistics community:
R. A. Fisher Lectureship, COPSS, August 2014
Gottfried E. Noether Senior Researcher Award, Joint Statistics Meetings, August 2009
Committee of Presidents of Statistical Societies Elizabeth Scott Award, 1996
First Emanuel and Carol Parzen Prize for Statistical Innovation, 1994
She received honorary Doctor of Science degrees from the University of Chicago in 2007 and The Ohio State University in 2022.
The Institute of Mathematical Statistics announced the IMS Grace Wahba Award and Lecture in 2021.
References
External links
Grace Wahba's University of Wisconsin website Home page
1934 births
Living people
American women statisticians
Fellows of the American Statistical Association
Members of the United States National Academy of Sciences
20th-century American mathematicians
21st-century American mathematicians
Fellows of the Society for Industrial and Applied Mathematics
University of Wisconsin–Madison faculty
Cornell University alumni
University of Maryland, College Park alumni
Stanford University alumni
Bayesian statisticians
Fellows of the American Academy of Arts and Sciences
Fellows of the American Association for the Advancement of Science
Fellows of the Institute of Mathematical Statistics
Machine learn
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https://en.wikipedia.org/wiki/Optimal%20stopping
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In mathematics, the theory of optimal stopping or early stopping is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance (related to the pricing of American options). A key example of an optimal stopping problem is the secretary problem. Optimal stopping problems can often be written in the form of a Bellman equation, and are therefore often solved using dynamic programming.
Definition
Discrete time case
Stopping rule problems are associated with two objects:
A sequence of random variables , whose joint distribution is something assumed to be known
A sequence of 'reward' functions which depend on the observed values of the random variables in 1:
Given those objects, the problem is as follows:
You are observing the sequence of random variables, and at each step , you can choose to either stop observing or continue
If you stop observing at step , you will receive reward
You want to choose a stopping rule to maximize your expected reward (or equivalently, minimize your expected loss)
Continuous time case
Consider a gain process defined on a filtered probability space and assume that is adapted to the filtration. The optimal stopping problem is to find the stopping time which maximizes the expected gain
where is called the value function. Here can take value .
A more specific formulation is as follows. We consider an adapted strong Markov process defined on a filtered probability space where denotes the probability measure where the stochastic process starts at . Given continuous functions , and , the optimal stopping problem is
This is sometimes called the MLS (which stand for Mayer, Lagrange, and supremum, respectively) formulation.
Solution methods
There are generally two approaches to solving optimal stopping problems. When the underlying process (or the gain process) is described by its unconditional finite-dimensional distributions, the appropriate solution technique is the martingale approach, so called because it uses martingale theory, the most important concept being the Snell envelope. In the discrete time case, if the planning horizon is finite, the problem can also be easily solved by dynamic programming.
When the underlying process is determined by a family of (conditional) transition functions leading to a Markov family of transition probabilities, powerful analytical tools provided by the theory of Markov processes can often be utilized and this approach is referred to as the Markov method. The solution is usually obtained by solving the associated free-boundary problems (Stefan problems).
A jump diffusion result
Let be a Lévy diffusion in given by the SDE
where is an -dimensional Brownian motion, is an -dimensional compensated Poisson random measure, , , and are given functions such that a unique solut
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https://en.wikipedia.org/wiki/Verdier%20duality
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In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact topological spaces of
Alexander Grothendieck's theory of
Poincaré duality in étale cohomology
for schemes in algebraic geometry. It is thus (together with the said étale theory and for example Grothendieck's coherent duality) one instance of Grothendieck's six operations formalism.
Verdier duality generalises the classical Poincaré duality of manifolds in two directions: it applies to continuous maps from one space to another (reducing to the classical case for the unique map from a manifold to a one-point space), and it applies to spaces that fail to be manifolds due to the presence of singularities. It is commonly encountered when studying constructible or perverse sheaves.
Verdier duality
Verdier duality states that (subject to suitable finiteness conditions discussed below)
certain derived image functors for sheaves are actually adjoint functors. There are two versions.
Global Verdier duality states that for a continuous map of locally compact Hausdorff spaces, the derived functor of the direct image with compact (or proper) supports has a right adjoint in the derived category of
sheaves, in other words, for (complexes of) sheaves (of abelian groups) on and on we have
Local Verdier duality states that
in the derived category of sheaves on Y.
It is important to note that the distinction between the global and local versions is that the former relates morphisms between
complexes of sheaves in the derived categories, whereas the latter relates internal Hom-complexes and so can be evaluated locally. Taking global sections of both sides in the local statement gives the global Verdier duality.
These results hold subject to the compactly supported direct image functor having finite cohomological dimension.
This is the case if the there is a bound such that the compactly supported cohomology
vanishes for all fibres (where )
and . This holds if all the fibres are at most -dimensional manifolds or more generally at most -dimensional CW-complexes.
The discussion above is about derived categories of sheaves of abelian groups. It is instead possible to consider a ring
and (derived categories of) sheaves of -modules; the case above corresponds to
.
The dualizing complex on is defined to be
where p is the map from to a point. Part of what makes Verdier duality interesting in the singular setting is that when is not a manifold (a graph or singular algebraic variety for example) then the dualizing complex is not quasi-isomorphic to a sheaf concentrated in a single degree. From this perspective the derived category is necessary in the study of singular spaces.
If is a finite-dimensional locally compact space, and the bounded derived category of sheaves of abelian groups over , then the Verdier dual is a contravariant fun
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https://en.wikipedia.org/wiki/Demographics%20of%20Belgrade
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Belgrade is the capital and largest city of Serbia.
Ethnicity
Source: Bureau of Statistics of Republic of Serbia, Census 2011
Religion
Source: Bureau of Statistics of Republic of Serbia, Census 2011
References
Geography of Belgrade
Belgrade
Belgrade
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https://en.wikipedia.org/wiki/Hirzebruch%20signature%20theorem
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In differential topology, an area of mathematics, the Hirzebruch signature theorem (sometimes called the Hirzebruch index theorem)
is Friedrich Hirzebruch's 1954 result expressing the signature
of a smooth closed oriented manifold by a linear combination of Pontryagin numbers called the
L-genus.
It was used in the proof of the Hirzebruch–Riemann–Roch theorem.
Statement of the theorem
The L-genus is the genus for the multiplicative sequence of polynomials
associated to the characteristic power series
The first two of the resulting L-polynomials are:
(for further L-polynomials see or ).
By taking for the the Pontryagin classes of the tangent bundle of a 4n dimensional smooth closed oriented
manifold M one obtains the L-classes of M.
Hirzebruch showed that the n-th L-class of M evaluated on the fundamental class of M, , is equal to , the signature of M
(i.e. the signature of the intersection form on the 2nth cohomology group of M):
Sketch of proof of the signature theorem
René Thom had earlier proved that the signature was given by some linear combination of Pontryagin numbers, and Hirzebruch found the exact formula for this linear combination
by introducing the notion of the genus of a multiplicative sequence.
Since the rational oriented cobordism ring is equal to
the polynomial algebra generated by the oriented cobordism classes
of the even dimensional complex projective spaces,
it is enough to verify that
for all i.
Generalizations
The signature theorem is a special case of the Atiyah–Singer index theorem for
the signature operator.
The analytic index of the signature operator equals the signature of the manifold, and its topological index is the L-genus of the manifold.
By the Atiyah–Singer index theorem these are equal.
References
Sources
F. Hirzebruch, The Signature Theorem. Reminiscences and recreation. Prospects in Mathematics, Annals of Mathematical Studies, Band 70, 1971, S. 3–31.
Theorems in algebraic topology
Theorems in differential topology
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https://en.wikipedia.org/wiki/Bruss
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Bruss may refer to:
Brös an alternate spelling of a preparation of cheese and grappa
Franz Thomas Bruss, a Belgian-German professor of mathematics at the Université Libre de Bruxelles
Logan Bruss (born 1999), American football player
Robert Bruss, a real estate attorney and syndicated columnist known as "the Dear Abby of real estate"
Arthur Atkinson aka Arthur "Artie" 'Bruss' Atkinson, an English former professional rugby league footballer
Surnames from given names
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https://en.wikipedia.org/wiki/Theta%20characteristic
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In mathematics, a theta characteristic of a non-singular algebraic curve C is a divisor class Θ such that 2Θ is the canonical class. In terms of holomorphic line bundles L on a connected compact Riemann surface, it is therefore L such that L2 is the canonical bundle, here also equivalently the holomorphic cotangent bundle. In terms of algebraic geometry, the equivalent definition is as an invertible sheaf, which squares to the sheaf of differentials of the first kind. Theta characteristics were introduced by
History and genus 1
The importance of this concept was realised first in the analytic theory of theta functions, and geometrically in the theory of bitangents. In the analytic theory, there are four fundamental theta functions in the theory of Jacobian elliptic functions. Their labels are in effect the theta characteristics of an elliptic curve. For that case, the canonical class is trivial (zero in the divisor class group) and so the theta characteristics of an elliptic curve E over the complex numbers are seen to be in 1-1 correspondence with the four points P on E with 2P = 0; this is counting of the solutions is clear from the group structure, a product of two circle groups, when E is treated as a complex torus.
Higher genus
For C of genus 0 there is one such divisor class, namely the class of -P, where P is any point on the curve. In case of higher genus g, assuming the field over which C is defined does not have characteristic 2, the theta characteristics can be counted as
22g
in number if the base field is algebraically closed.
This comes about because the solutions of the equation on the divisor class level will form a single coset of the solutions of
2D = 0.
In other words, with K the canonical class and Θ any given solution of
2Θ = K,
any other solution will be of form
Θ + D.
This reduces counting the theta characteristics to finding the 2-rank of the Jacobian variety J(C) of C. In the complex case, again, the result follows since J(C) is a complex torus of dimension 2g. Over a general field, see the theory explained at Hasse-Witt matrix for the counting of the p-rank of an abelian variety. The answer is the same, provided the characteristic of the field is not 2.
A theta characteristic Θ will be called even or odd depending on the dimension of its space of global sections . It turns out that on C there are even and odd theta characteristics.
Classical theory
Classically the theta characteristics were divided into these two kinds, odd and even, according to the value of the Arf invariant of a certain quadratic form Q with values mod 2. Thus in case of g = 3 and a plane quartic curve, there were 28 of one type, and the remaining 36 of the other; this is basic in the question of counting bitangents, as it corresponds to the 28 bitangents of a quartic. The geometric construction of Q as an intersection form is with modern tools possible algebraically. In fact the Weil pairing applies, in its abelian variety form.
Trip
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https://en.wikipedia.org/wiki/Configuration%20%28geometry%29
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In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points.
Although certain specific configurations had been studied earlier (for instance by Thomas Kirkman in 1849), the formal study of configurations was first introduced by Theodor Reye in 1876, in the second edition of his book Geometrie der Lage, in the context of a discussion of Desargues' theorem. Ernst Steinitz wrote his dissertation on the subject in 1894, and they were popularized by Hilbert and Cohn-Vossen's 1932 book Anschauliche Geometrie, reprinted in English as .
Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the Euclidean or projective planes (these are said to be realizable in that geometry), or as a type of abstract incidence geometry. In the latter case they are closely related to regular hypergraphs and biregular bipartite graphs, but with some additional restrictions: every two points of the incidence structure can be associated with at most one line, and every two lines can be associated with at most one point. That is, the girth of the corresponding bipartite graph (the Levi graph of the configuration) must be at least six.
Notation
A configuration in the plane is denoted by (), where is the number of points, the number of lines, the number of lines per point, and the number of points per line. These numbers necessarily satisfy the equation
as this product is the number of point-line incidences (flags).
Configurations having the same symbol, say (), need not be isomorphic as incidence structures. For instance, there exist three different (93 93) configurations: the Pappus configuration and two less notable configurations.
In some configurations, and consequently, . These are called symmetric or balanced configurations and the notation is often condensed to avoid repetition. For example, (93 93) abbreviates to (93).
Examples
Notable projective configurations include the following:
(11), the simplest possible configuration, consisting of a point incident to a line. Often excluded as being trivial.
(32), the triangle. Each of its three sides meets two of its three vertices, and vice versa. More generally any polygon of sides forms a configuration of type ()
(43 62) and (62 43), the complete quadrangle and complete quadrilateral respectively.
(73), the Fano plane. This configuration exists as an abstract incidence geometry, but cannot be constructed in the Euclidean plane.
(83), the Möbius–Kantor configuration. This configuration describes two quadrilaterals that are simultaneously inscribed and circumscribed in each other. It cannot be constructed in Euclidean plane geometry but the equations defining it have nontrivial solutions in complex numbers.
(93), the Pappus configuration.
(94 123), th
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https://en.wikipedia.org/wiki/Schlegel%20diagram
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In geometry, a Schlegel diagram is a projection of a polytope from into through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in that, together with the original facet, is combinatorially equivalent to the original polytope. The diagram is named for Victor Schlegel, who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes. In dimension 3, a Schlegel diagram is a projection of a polyhedron into a plane figure; in dimension 4, it is a projection of a 4-polytope to 3-space. As such, Schlegel diagrams are commonly used as a means of visualizing four-dimensional polytopes.
Construction
The most elementary Schlegel diagram, that of a polyhedron, was described by Duncan Sommerville as follows:
A very useful method of representing a convex polyhedron is by plane projection. If it is projected from any external point, since each ray cuts it twice, it will be represented by a polygonal area divided twice over into polygons. It is always possible by suitable choice of the centre of projection to make the projection of one face completely contain the projections of all the other faces. This is called a Schlegel diagram of the polyhedron. The Schlegel diagram completely represents the morphology of the polyhedron. It is sometimes convenient to project the polyhedron from a vertex; this vertex is projected to infinity and does not appear in the diagram, the edges through it are represented by lines drawn outwards.
Sommerville also considers the case of a simplex in four dimensions: "The Schlegel diagram of simplex in S4 is a tetrahedron divided into four tetrahedra." More generally, a polytope in n-dimensions has a Schlegel diagram constructed by a perspective projection viewed from a point outside of the polytope, above the center of a facet. All vertices and edges of the polytope are projected onto a hyperplane of that facet. If the polytope is convex, a point near the facet will exist which maps the facet outside, and all other facets inside, so no edges need to cross in the projection.
Examples
See also
Net (polyhedron) – A different approach for visualization by lowering the dimension of a polytope is to build a net, disconnecting facets, and unfolding until the facets can exist on a single hyperplane. This maintains the geometric scale and shape, but makes the topological connections harder to see.
References
Further reading
Victor Schlegel (1883) Theorie der homogen zusammengesetzten Raumgebilde, Nova Acta, Ksl. Leop.-Carol. Deutsche Akademie der Naturforscher, Band XLIV, Nr. 4, Druck von E. Blochmann & Sohn in Dresden.
Victor Schlegel (1886) Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper, Waren.
Coxeter, H.S.M.; Regular Polytopes, (Methuen and Co., 1948). (p. 242)
Regular Polytopes, (3rd edition, 1973), Dover edition,
.
External links
George W. Hart: 4D Polytope Projection Models by 3D Printing
Nrich maths –
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https://en.wikipedia.org/wiki/H.%20F.%20Baker
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Henry Frederick Baker FRS FRSE (3 July 1866 – 17 March 1956) was a British mathematician, working mainly in algebraic geometry, but also remembered for contributions to partial differential equations (related to what would become known as solitons), and Lie groups.
Early life
He was born in Cambridge the son of Henry Baker, a butler, and Sarah Ann Britham.
Education
He was educated at The Perse School before winning a scholarship to St John's College, Cambridge in October 1884. Baker graduated as Senior Wrangler in 1887, bracketed with 3 others.
Career
Baker was elected Fellow of St John's in 1888 where he remained for 68 years.
In June, 1898 he was elected a Fellow of the Royal Society. In 1911, he gave the presidential address to the London Mathematical Society.
Baker was one of the mathematicians (along with E. W. Hobson) to whom Srinivasa Ramanujan wrote before G. H. Hardy but his papers were returned without comment.
In January 1914 he was appointed Lowndean Professor of Astronomy.
Gordon Welchman recalled that in the 1930s before the war Dennis Babbage and he were members of a group of geometers known as Professor Baker's "Tea Party", who met once a week to discuss the areas of research in which we were all interested.
He married twice. Firstly in 1893 to Lilly Isabella Hamfield Klopp, who died in 1903, then he remarried in 1913, to Muriel Irene Woodyard.
He died in Cambridge and is buried at the Parish of the Ascension Burial Ground, with his second wife Muriel (1885 - 1956).
See also
Baker–Campbell–Hausdorff formula
Cremona–Richmond configuration
Eleven-point conic
Publications
Abel's theorem and the allied theory, including the theory of the theta functions (Cambridge: The University Press, 1897)
An introduction to the theory of multiply periodic functions (Cambridge: The University Press, 1907)
1943 An Introduction to Plane Geometry
References
1866 births
1956 deaths
19th-century British mathematicians
20th-century British mathematicians
Alumni of St John's College, Cambridge
Fellows of St John's College, Cambridge
Senior Wranglers
Fellows of the Royal Society
Lowndean Professors of Astronomy and Geometry
PDE theorists
De Morgan Medallists
People educated at The Perse School
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