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https://en.wikipedia.org/wiki/Regional%20Data%20Exchange%20System
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RDES (the Regional Data Exchange System on food and agricultural statistics in Asia and Pacific countries) is a unitary statistical information system which includes a database on food and agricultural statistics and the web portal of APCAS (Asia and Pacific Commission on Agricultural Statistics) countries under the FAO/Japan cooperative regional project (GCP/RAS/184/JPN). RDES had operated from March 2003 to 2011. The concept of RDES is succeeded to the CountrySTAT project, FAOSTAT.
Database
RDES was designed to contribute to member nations' capacity building and policy analysis through the development of the food and agricultural statistical framework in APCAS countries. It is especially expected to role-play the database on food and agricultural statistics for users, such as policy-makers, decision-makers, researchers, etc.
Time scale
The calendar year is recommended as time scale of the RDES, due to the difference of the crop year in each country.
Definitions
Definitions of the data should be FAO definitions. The unit of the crop production data are production (metric ton), area harvested (hectare), and yield (kilogram per hectare).
Data items
Most data of RDES are crop production and livestock data. Although it depends on the background of food production in each country, the major 19 agricultural products in this region are registered as the basic data items: rice, wheat, maize, cereals, cassava, potatoes, pulses, groundnuts, soybean, seed cotton, sugar cane, tea, cattle, pigs, sheep, goats, chicken, milk total, and hen eggs.
Other data for food security that are required by users, such as other crops and livestock, land area, population, prices, fisheries, etc. are provided by each countries on the basis of its situation.
The database functions of RDES has been strengthened with the CountrySTAT technology. RDES with CountrySTAT was launched on 1 November 2006.
Web Portal
RDES is the gateway to agricultural statistics in APCAS countries. There are pages for each country, which contain not only agricultural statistics but also show the country profile, contact address, and hyperlinks for statistics in each country.
RDES also shows the external hyperlinks to related databases, organizations, and associations for agricultural statistics and food security Information, such as the FAOSTAT, UNSTAT, WFP, etc.
Participating nations
RDES is organizing in cooperation with most APCAS member nations: Bangladesh, Bhutan, Cambodia, the People's Republic of China, Fiji, India, Indonesia, Iran, the Lao People's Democratic Republic, Myanmar, Nepal, Pakistan, Philippines, Sri Lanka, Thailand, and Vietnam. Though some other APCAS countries (Australia, Japan (Donor), Malaysia, the Republic of Korea and the United States of America) are not the participants, but RDES is also organized with their cooperation.
External links
RDES Home Page (closed)
FAO Statistics Home Page
CountrySTAT Web site
FAOSTAT Web site
Agricultural organizations
Agricultur
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https://en.wikipedia.org/wiki/Injective%20metric%20space
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In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces. These properties can be defined in two seemingly different ways: hyperconvexity involves the intersection properties of closed balls in the space, while injectivity involves the isometric embeddings of the space into larger spaces. However it is a theorem of that these two different types of definitions are equivalent.
Hyperconvexity
A metric space is said to be hyperconvex if it is convex and its closed balls have the binary Helly property. That is:
Any two points and can be connected by the isometric image of a line segment of length equal to the distance between the points (i.e. is a path space).
If is any family of closed balls such that each pair of balls in meets, then there exists a point common to all the balls in .
Equivalently, a metric space is hyperconvex if, for any set of points in and radii satisfying for each and , there is a point in that is within distance of each (that is, for all ).
Injectivity
A retraction of a metric space is a function mapping to a subspace of itself, such that
for all we have that ; that is, is the identity function on its image (i.e. it is idempotent), and
for all we have that ; that is, is nonexpansive.
A retract of a space is a subspace of that is an image of a retraction.
A metric space is said to be injective if, whenever is isometric to a subspace of a space , that subspace is a retract of .
Examples
Examples of hyperconvex metric spaces include
The real line
with the ∞ distance
Manhattan distance (L1) in the plane (which is equivalent up to rotation and scaling to the L∞), but not in higher dimensions
The tight span of a metric space
Any complete real tree
– see Metric space aimed at its subspace
Due to the equivalence between hyperconvexity and injectivity, these spaces are all also injective.
Properties
In an injective space, the radius of the minimum ball that contains any set is equal to half the diameter of . This follows since the balls of radius half the diameter, centered at the points of , intersect pairwise and therefore by hyperconvexity have a common intersection; a ball of radius half the diameter centered at a point of this common intersection contains all of . Thus, injective spaces satisfy a particularly strong form of Jung's theorem.
Every injective space is a complete space, and every metric map (or, equivalently, nonexpansive mapping, or short map) on a bounded injective space has a fixed point. A metric space is injective if and only if it is an injective object in the category of metric spaces and metric maps.
Notes
References
Correction (1957), Pacific J. Math. 7: 1729, .
Metric spaces
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https://en.wikipedia.org/wiki/Heinrich%20Suter
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Heinrich Suter (4 January 1848 in Hedingen – 17 March 1922 in Dornach) was a historian of science specializing in Islamic mathematics and astronomy.
Education and career
After graduation from the Industrie Schule at Zürich, Suter studied in Berlin (1869/70) and at ETH Zürich and the University of Zürich. He received in 1871 from the University of Zürich his Promovierung (Ph.D.) with dissertation Geschichte der mathematischen Wissenschaften von den ältesten Zeiten bis Ende des 16. Jahrhunderts. His dissertation was published in 1872 as a book and was subsequently translated into Russian.
In 1874 Suter began teaching as a vicar at the Gymnasium in Schaffhausen, then taught from 1876 to 1886 in Aarau, and finally from 1886 until his retirement in 1916 in Zürich.
Suter in his early forties learned Arabic and acquired some knowledge of Syriac, Persian and Turkish. He studied the history of mathematics and astronomy in the Islamic societies. In Moritz Cantor's "Abhandlungen zur Geschichte der Mathematik“ were published in 1892 Suter's translation of the mathematically related entries in the Kitāb al-Fihrist of Ibn al-Nadim and in 1893 Suter's translation of the mathematical parts of the catalog of the Khedivial Library in Cairo. One of his most important works is his work, commissioned by the Royal Danish Academy of Sciences, on the astronomical tables of Al-Khwarizmi.
In 1904 Suter was an Invited Speaker of the ICM in Heidelberg.
Publications
Books
1871. Geschichte der mathematischen Wissenschaften, Teil 1 : Von den ältesten Zeiten bis Ende des 16. Jahrhunderts. Dissertation, 2. Aufl. 1873. Reprint 1973. Hathi Trust
1875. Geschichte der mathematischen Wissenschaften, Teil II : Vom Anfange des 17. bis gegen Ende des 18. Jahrhunderts. Hathi Trust
1900. Die Mathematiker und Astronomen der Araber und ihre Werke. Abhandl. zur Geschichte der mathematischen Wissenschaften, Heft 10. Reprint 1972 und 1986. bibalex
Articles
1884. Der Tractatus de quadratura Circuli des ALBERTUS DE SAXONIA. ZM. 29, 81.
1886 Ueber diophantische Gleichungen. Z. f. Math. Unterr. 17, 104.
1887. Die Mathematiker auf den Universitäten des Mittelalters. Wiss. Beilage. z. Programm d. Kantonsschule in Zürich.
1889. Die mathematischen und naturphilosophischen Disputationen an der Universität Leipzig, 1512 bis 1526. BM. (2), 3, 17.
1890. Bibliographische Notiz über die math.-hist. Studien in der Schweiz. BM (2), 4, 97.
1892. Das Mathematiker-Verzeichnis im Fihrist des IBN ABI JA QUB AN-NADiM. Abhandl. z. Gesch. d. math. Wissenschaften Heft. 6.
1892. Einiges von NASiR ED-DIN'S EUKLID-Ausgabe. BM. (2), 6, 3.
1893. Zur Geschichte der Trigonometrie. BM. (2), 7, 1.
1893. Der V. Band des Katalogs der arab. Bücher der vicekönigl. Bibliothek in Kairo. ZM. 38, 1. 41. 161.
1893. Zu RUDLOFF und HOCHHEIM, Die Astronomie des GAGMINI. ZDMG 47, 718.
1894. Zur Frage über JOSEPHUS SAPIENS. BM (2), 8, 84.
1895. Die Araber als Vermittler der Wissenschaften in deren Uebergang vom
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https://en.wikipedia.org/wiki/Apartness%20relation
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In constructive mathematics, an apartness relation is a constructive form of inequality, and is often taken to be more basic than equality. It is often written as (⧣ in unicode) to distinguish from the negation of equality (the denial inequality) which is weaker.
Description
An apartness relation is a symmetric irreflexive binary relation with the additional condition that if two elements are apart, then any other element is apart from at least one of them (this last property is often called co-transitivity or comparison).
That is, a binary relation is an apartness relation if it satisfies:
The complement of an apartness relation is an equivalence relation, as the above three conditions become reflexivity, symmetry, and transitivity. If this equivalence relation is in fact equality, then the apartness relation is called tight. That is, is a if it additionally satisfies:
4.
In classical mathematics, it also follows that every apartness relation is the complement of an equivalence relation, and the only tight apartness relation on a given set is the complement of equality. So in that domain, the concept is not useful. In constructive mathematics, however, this is not the case.
The prototypical apartness relation is that of the real numbers: two real numbers are said to be apart if there exists (one can construct) a rational number between them. In other words, real numbers and are apart if there exists a rational number such that or The natural apartness relation of the real numbers is then the disjunction of its natural pseudo-order. The complex numbers, real vector spaces, and indeed any metric space then naturally inherit the apartness relation of the real numbers, even though they do not come equipped with any natural ordering.
If there is no rational number between two real numbers, then the two real numbers are equal. Classically, then, if two real numbers are not equal, one would conclude that there exists a rational number between them. However it does not follow that one can actually construct such a number. Thus to say two real numbers are apart is a stronger statement, constructively, than to say that they are not equal, and while equality of real numbers is definable in terms of their apartness, the apartness of real numbers cannot be defined in terms of their equality. For this reason, in constructive topology especially, the apartness relation over a set is often taken as primitive, and equality is a defined relation.
A set endowed with an apartness relation is known as a constructive setoid. A function where and are constructive setoids is called a morphism for and if
See also
References
Binary relations
Constructivism (mathematics)
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https://en.wikipedia.org/wiki/Orthogonal%20convex%20hull
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In geometry, a set is defined to be orthogonally convex if, for every line that is parallel to one of standard basis vectors, the intersection of with is empty, a point, or a single segment. The term "orthogonal" refers to corresponding Cartesian basis and coordinates in Euclidean space, where different basis vectors are perpendicular, as well as corresponding lines. Unlike ordinary convex sets, an orthogonally convex set is not necessarily connected.
The orthogonal convex hull of a set is the intersection of all connected orthogonally convex supersets of .
These definitions are made by analogy with the classical theory of convexity, in which is convex if, for every line , the intersection of with is empty, a point, or a single segment. Orthogonal convexity restricts the lines for which this property is required to hold, so every convex set is orthogonally convex but not vice versa. For the same reason, the orthogonal convex hull itself is a subset of the convex hull of the same point set. A point belongs to the orthogonal convex hull of if and only if each of the closed axis-aligned orthants having as apex has a nonempty intersection with .
The orthogonal convex hull is also known as the rectilinear convex hull, or, in two dimensions, the - convex hull.
Example
The figure shows a set of 16 points in the plane and the orthogonal convex hull of these points. As can be seen in the figure, the orthogonal convex hull is a polygon with some degenerate edges connecting extreme vertices in each coordinate direction. For a discrete point set such as this one, all orthogonal convex hull edges are horizontal or vertical. In this example, the orthogonal convex hull is connected.
Alternative definitions
In contrast with the classical convexity where there exist several equivalent definitions of the convex hull, definitions of the orthogonal convex hull made by analogy to those of the convex hull result in different geometric objects. So far, researchers have explored the following four definitions of the orthogonal convex hull of a set :
Maximal definition: The definition described in the introduction of this article. It is based on the Maxima of a point set.
Classical definition: The orthogonal convex hull of is the intersection of all orthogonally convex supersets of ; .
Connected definition: The orthogonal convex hull of is the smallest connected orthogonally convex superset of ; .
Functional definition: The orthogonal convex hull of is the intersection of the zero sets of all non-negative orthogonally convex functions that are on ; .
In the figures on the right, the top figure shows a set of six points in the plane. The classical orthogonal convex hull of the point set is the point set itself. From top to bottom, the second to the fourth figures show respectively, the maximal, the connected, and the functional orthogonal convex hull of the point set. As can be seen, the orthogonal convex hull is a polygon with some degenerate "e
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https://en.wikipedia.org/wiki/Welch%27s%20t-test
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In statistics, Welch's t-test, or unequal variances t-test, is a two-sample location test which is used to test the (null) hypothesis that two populations have equal means. It is named for its creator, Bernard Lewis Welch, is an adaptation of Student's t-test, and is more reliable when the two samples have unequal variances and possibly unequal sample sizes. These tests are often referred to as "unpaired" or "independent samples" t-tests, as they are typically applied when the statistical units underlying the two samples being compared are non-overlapping. Given that Welch's t-test has been less popular than Student's t-test and may be less familiar to readers, a more informative name is "Welch's unequal variances t-test" — or "unequal variances t-test" for brevity.
Assumptions
Student's t-test assumes that the sample means being compared for two populations are normally distributed, and that the populations have equal variances. Welch's t-test is designed for unequal population variances, but the assumption of normality is maintained. Welch's t-test is an approximate solution to the Behrens–Fisher problem.
Calculations
Welch's t-test defines the statistic t by the following formula:
where and are the sample mean and its standard error, with denoting the corrected sample standard deviation, and sample size .
Unlike in Student's t-test, the denominator is not based on a pooled variance estimate.
The degrees of freedom associated with this variance estimate is approximated using the Welch–Satterthwaite equation:
This expression can be simplified when :
Here, is the degrees of freedom associated with the i-th variance estimate.
The statistic is approximately from the t-distribution since we have an approximation of the chi-square distribution. This approximation is better done when both and are larger than 5.
Statistical test
Once t and have been computed, these statistics can be used with the t-distribution to test one of two possible null hypotheses:
that the two population means are equal, in which a two-tailed test is applied; or
that one of the population means is greater than or equal to the other, in which a one-tailed test is applied.
The approximate degrees of freedom are real numbers and used as such in statistics-oriented software, whereas they are rounded down to the nearest integer in spreadsheets.
Advantages and limitations
Welch's t-test is more robust than Student's t-test and maintains type I error rates close to nominal for unequal variances and for unequal sample sizes under normality. Furthermore, the power of Welch's t-test comes close to that of Student's t-test, even when the population variances are equal and sample sizes are balanced. Welch's t-test can be generalized to more than 2-samples, which is more robust than one-way analysis of variance (ANOVA).
It is not recommended to pre-test for equal variances and then choose between Student's t-test or Welch's t-test. Rather, Welch's t-test can b
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https://en.wikipedia.org/wiki/Donald%20Richards%20%28statistician%29
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Donald St. P. Richards (born 1955, in Mandeville, Jamaica) is an American statistician conducting research on multivariate statistics, zonal polynomials, distance correlation, total positivity, and hypergeometric functions of matrix argument. He currently serves as a distinguished professor of statistics at the Pennsylvania State University, and is a Fellow of the Institute of Mathematical Statistics and a Fellow of the American Mathematical Society.
Richards obtained his PhD in 1978 at the University of the West Indies, where the statistician Rameshwar D. Gupta was his doctoral advisor.
In 1999, he was elected a Fellow of the Institute of Mathematical Statistics.
In 2012, he was elected a Fellow of the American Mathematical Society.
Personal life
Richards became an American citizen in 1990. He was married to Mercedes Richards, an American Jamaican-born professor of astronomy and astrophysics, until her death in 2016.
References
External links
1955 births
Living people
American statisticians
Jamaican statisticians
Pennsylvania State University faculty
Fellows of the Institute of Mathematical Statistics
Fellows of the American Mathematical Society
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https://en.wikipedia.org/wiki/Proof%20by%20example
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In logic and mathematics, proof by example (sometimes known as inappropriate generalization) is a logical fallacy whereby the validity of a statement is illustrated through one or more examples or cases—rather than a full-fledged proof.
The structure, argument form and formal form of a proof by example generally proceeds as follows:
Structure:
I know that X is such.
Therefore, anything related to X is also such.
Argument form:
I know that x, which is a member of group X, has the property P.
Therefore, all other elements of X must have the property P.
Formal form:
The following example demonstrates why this line of reasoning is a logical fallacy:
I've seen a person shoot someone dead.
Therefore, all people are murderers.
In the common discourse, a proof by example can also be used to describe an attempt to establish a claim using statistically insignificant examples. In which case, the merit of each argument might have to be assessed on an individual basis.
Valid cases of proof by example
In certain circumstances, examples can suffice as logically valid proof.
Proofs of existential statements
In some scenarios, an argument by example may be valid if it leads from a singular premise to an existential conclusion (i.e. proving that a claim is true for at least one case, instead of for all cases). For example:
Socrates is wise.
Therefore, someone is wise.
(or)
I've seen a person steal.
Therefore, (some) people can steal.
These examples outline the informal version of the logical rule known as existential introduction, also known as particularisation or existential generalization:
Existential Introduction
(where denotes the formula formed by substituting all free occurrences of the variable in by .)
Likewise, finding a counterexample disproves (proves the negation of) a universal conclusion. This is used in a proof by contradiction.
Exhaustive proofs
Examples also constitute valid, if inelegant, proof, when it has also been demonstrated that the examples treated cover all possible cases.
In mathematics, proof by example can also be used to refer to attempts to illustrate a claim by proving cases of the claim, with the understanding that these cases contain key ideas which can be generalized into a full-fledged proof.
See also
Affirming the consequent
Anecdotal evidence
Bayesian probability
Counterexample
Hand-waving
Inductive reasoning
Problem of induction
Modus ponens
Proof by construction
Proof by intimidation
References
Further reading
Benjamin Matschke: Valid proofs by example in mathematics (arXiv)
Quantificational fallacies
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https://en.wikipedia.org/wiki/Weyl%27s%20lemma%20%28Laplace%20equation%29
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In mathematics, Weyl's lemma, named after Hermann Weyl, states that every weak solution of Laplace's equation is a smooth solution. This contrasts with the wave equation, for example, which has weak solutions that are not smooth solutions. Weyl's lemma is a special case of elliptic or hypoelliptic regularity.
Statement of the lemma
Let be an open subset of -dimensional Euclidean space , and let denote the usual Laplace operator. Weyl's lemma states that if a locally integrable function is a weak solution of Laplace's equation, in the sense that
for every smooth test function with compact support, then (up to redefinition on a set of measure zero) is smooth and satisfies pointwise in .
This result implies the interior regularity of harmonic functions in , but it does not say anything about their regularity on the boundary .
Idea of the proof
To prove Weyl's lemma, one convolves the function with an appropriate mollifier and shows that the mollification satisfies Laplace's equation, which implies that has the mean value property. Taking the limit as and using the properties of mollifiers, one finds that also has the mean value property, which implies that it is a smooth solution of Laplace's equation. Alternative proofs use the smoothness of the fundamental solution of the Laplacian or suitable a priori elliptic estimates.
Generalization to distributions
More generally, the same result holds for every distributional solution of Laplace's equation: If satisfies for every , then is a regular distribution associated with a smooth solution of Laplace's equation.
Connection with hypoellipticity
Weyl's lemma follows from more general results concerning the regularity properties of elliptic or hypoelliptic operators. A linear partial differential operator with smooth coefficients is hypoelliptic if the singular support of is equal to the singular support of for every distribution . The Laplace operator is hypoelliptic, so if , then the singular support of is empty since the singular support of is empty, meaning that . In fact, since the Laplacian is elliptic, a stronger result is true, and solutions of are real-analytic.
Notes
References
Lemmas in analysis
Partial differential equations
Harmonic functions
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https://en.wikipedia.org/wiki/Polynomial%20matrix
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In mathematics, a polynomial matrix or matrix of polynomials is a matrix whose elements are univariate or multivariate polynomials. Equivalently, a polynomial matrix is a polynomial whose coefficients are matrices.
A univariate polynomial matrix P of degree p is defined as:
where denotes a matrix of constant coefficients, and is non-zero.
An example 3×3 polynomial matrix, degree 2:
We can express this by saying that for a ring R, the rings and
are isomorphic.
Properties
A polynomial matrix over a field with determinant equal to a non-zero element of that field is called unimodular, and has an inverse that is also a polynomial matrix. Note that the only scalar unimodular polynomials are polynomials of degree 0 – nonzero constants, because an inverse of an arbitrary polynomial of higher degree is a rational function.
The roots of a polynomial matrix over the complex numbers are the points in the complex plane where the matrix loses rank.
The determinant of a matrix polynomial with Hermitian positive-definite (semidefinite) coefficients is a polynomial with positive (nonnegative) coefficients.
Note that polynomial matrices are not to be confused with monomial matrices, which are simply matrices with exactly one non-zero entry in each row and column.
If by λ we denote any element of the field over which we constructed the matrix, by I the identity matrix, and we let A be a polynomial matrix, then the matrix λI − A is the characteristic matrix of the matrix A. Its determinant, |λI − A| is the characteristic polynomial of the matrix A.
References
E.V.Krishnamurthy, Error-free Polynomial Matrix computations, Springer Verlag, New York, 1985
Matrices
Polynomials
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https://en.wikipedia.org/wiki/Hamiltonian%20matrix
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In mathematics, a Hamiltonian matrix is a -by- matrix such that is symmetric, where is the skew-symmetric matrix
and is the -by- identity matrix. In other words, is Hamiltonian if and only if where denotes the transpose.
Properties
Suppose that the -by- matrix is written as the block matrix
where , , , and are -by- matrices. Then the condition that be Hamiltonian is equivalent to requiring that the matrices and are symmetric, and that . Another equivalent condition is that is of the form with symmetric.
It follows easily from the definition that the transpose of a Hamiltonian matrix is Hamiltonian. Furthermore, the sum (and any linear combination) of two Hamiltonian matrices is again Hamiltonian, as is their commutator. It follows that the space of all Hamiltonian matrices is a Lie algebra, denoted . The dimension of is . The corresponding Lie group is the symplectic group . This group consists of the symplectic matrices, those matrices which satisfy . Thus, the matrix exponential of a Hamiltonian matrix is symplectic. However the logarithm of a symplectic matrix is not necessarily Hamiltonian because the exponential map from the Lie algebra to the group is not surjective.
The characteristic polynomial of a real Hamiltonian matrix is even. Thus, if a Hamiltonian matrix has as an eigenvalue, then , and are also eigenvalues. It follows that the trace of a Hamiltonian matrix is zero.
The square of a Hamiltonian matrix is skew-Hamiltonian (a matrix is skew-Hamiltonian if ). Conversely, every skew-Hamiltonian matrix arises as the square of a Hamiltonian matrix.
Extension to complex matrices
As for symplectic matrices, the definition for Hamiltonian matrices can be extended to complex matrices in two ways. One possibility is to say that a matrix is Hamiltonian if , as above. Another possibility is to use the condition where the superscript asterisk () denotes the conjugate transpose.
Hamiltonian operators
Let be a vector space, equipped with a symplectic form . A linear map is called a Hamiltonian operator with respect to if the form is symmetric. Equivalently, it should satisfy
Choose a basis in , such that is written as . A linear operator is Hamiltonian with respect to if and only if its matrix in this basis is Hamiltonian.
References
Matrices
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https://en.wikipedia.org/wiki/Cylindric%20algebra
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In mathematics, the notion of cylindric algebra, invented by Alfred Tarski, arises naturally in the algebraization of first-order logic with equality. This is comparable to the role Boolean algebras play for propositional logic. Cylindric algebras are Boolean algebras equipped with additional cylindrification operations that model quantification and equality. They differ from polyadic algebras in that the latter do not model equality.
Definition of a cylindric algebra
A cylindric algebra of dimension (where is any ordinal number) is an algebraic structure such that is a Boolean algebra, a unary operator on for every (called a cylindrification), and a distinguished element of for every and (called a diagonal), such that the following hold:
(C1)
(C2)
(C3)
(C4)
(C5)
(C6) If , then
(C7) If , then
Assuming a presentation of first-order logic without function symbols,
the operator models existential quantification over variable in formula while the operator models the equality of variables and . Hence, reformulated using standard logical notations, the axioms read as
(C1)
(C2)
(C3)
(C4)
(C5)
(C6) If is a variable different from both and , then
(C7) If and are different variables, then
Cylindric set algebras
A cylindric set algebra of dimension is an algebraic structure such that is a field of sets, is given by , and is given by . It necessarily validates the axioms C1–C7 of a cylindric algebra, with instead of , instead of , set complement for complement, empty set as 0, as the unit, and instead of . The set X is called the base.
A representation of a cylindric algebra is an isomorphism from that algebra to a cylindric set algebra. Not every cylindric algebra has a representation as a cylindric set algebra. It is easier to connect the semantics of first-order predicate logic with cylindric set algebra. (For more details, see .)
Generalizations
Cylindric algebras have been generalized to the case of many-sorted logic (Caleiro and Gonçalves 2006), which allows for a better modeling of the duality between first-order formulas and terms.
Relation to monadic Boolean algebra
When and are restricted to being only 0, then becomes , the diagonals can be dropped out, and the following theorem of cylindric algebra (Pinter 1973):
turns into the axiom
of monadic Boolean algebra. The axiom (C4) drops out (becomes a tautology). Thus monadic Boolean algebra can be seen as a restriction of cylindric algebra to the one variable case.
See also
Abstract algebraic logic
Lambda calculus and Combinatory logic—other approaches to modelling quantification and eliminating variables
Hyperdoctrines are a categorical formulation of cylindric algebras
Relation algebras (RA)
Polyadic algebra
Cylindrical algebraic decomposition
Notes
References
Leon Henkin, J. Donald Monk, and Alfred Tarski (1971) Cylindric Algebras, Part I. North-Holland. .
Leon Henkin, J. Donald Monk, and A
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https://en.wikipedia.org/wiki/Conchoid%20of%20D%C3%BCrer
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In geometry, the conchoid of Dürer, also called Dürer's shell curve, is a plane, algebraic curve, named after Albrecht Dürer and introduced in 1525. It is not a true conchoid.
Construction
Suppose two perpendicular lines are given, with intersection point O. For concreteness we may assume that these are the coordinate axes and that O is the origin, that is (0, 0). Let points and move on the axes in such a way that , a constant. On the line , extended as necessary, mark points and at a fixed distance from . The locus of the points and is Dürer's conchoid.
Equation
The equation of the conchoid in Cartesian form is
In parametric form the equation is given by
where the parameter is measured in radians.
Properties
The curve has two components, asymptotic to the lines . Each component is a rational curve. If a > b there is a loop, if a = b there is a cusp at (0,a).
Special cases include:
a = 0: the line y = 0;
b = 0: the line pair together with the circle ;
The envelope of straight lines used in the construction form a parabola (as seen in Durer's original diagram above) and therefore the curve is a point-glissette formed by a line and one of its points sliding respectively against a parabola and one of its tangents.
History
It was first described by the German painter and mathematician Albrecht Dürer (1471–1528) in his book Underweysung der Messung (Instruction in Measurement with Compass and Straightedge p. 38), calling it Ein muschellini (Conchoid or Shell). Dürer only drew one branch of the curve.
See also
Conchoid of de Sluze
List of curves
References
External links
Algebraic curves
Albrecht Dürer
16th-century introductions
1525 in science
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https://en.wikipedia.org/wiki/Newick%20format
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In mathematics, Newick tree format (or Newick notation or New Hampshire tree format) is a way of representing graph-theoretical trees with edge lengths using parentheses and commas. It was adopted by James Archie, William H. E. Day, Joseph Felsenstein, Wayne Maddison, Christopher Meacham, F. James Rohlf, and David Swofford, at two meetings in 1986, the second of which was at Newick's restaurant in Dover, New Hampshire, US. The adopted format is a generalization of the format developed by Meacham in 1984 for the first tree-drawing programs in Felsenstein's PHYLIP package.
Examples
The following tree:
could be represented in Newick format in several ways
((,)); no nodes are named
(A,B,(C,D)); leaf nodes are named
(A,B,(C,D)E)F; all nodes are named
(:0.1,:0.2,(:0.3,:0.4):0.5); all but root node have a distance to parent
(:0.1,:0.2,(:0.3,:0.4):0.5):0.0; all have a distance to parent
(A:0.1,B:0.2,(C:0.3,D:0.4):0.5); distances and leaf names (popular)
(A:0.1,B:0.2,(C:0.3,D:0.4)E:0.5)F; distances and all names
((B:0.2,(C:0.3,D:0.4)E:0.5)F:0.1)A; a tree rooted on a leaf node (rare)
Newick format is typically used for tools like PHYLIP and is a minimal definition for a phylogenetic tree.
Rooted, unrooted, and binary trees
When an unrooted tree is represented in Newick notation, an arbitrary node is chosen as its root. Whether rooted or unrooted, typically a tree's representation is rooted on an internal node and it is rare (but legal) to root a tree on a leaf node.
A rooted binary tree that is rooted on an internal node has exactly two immediate descendant nodes for each internal node.
An unrooted binary tree that is rooted on an arbitrary internal node has exactly three immediate descendant nodes for the root node, and each other internal node has exactly two immediate descendant nodes.
A binary tree rooted from a leaf has at most one immediate descendant node for the root node, and each internal node has exactly two immediate descendant nodes.
Grammar
A grammar for parsing the Newick format (roughly based on ):
The grammar nodes
Tree: The full input Newick Format for a single tree
Subtree: an internal node (and its descendants) or a leaf node
Leaf: a node with no descendants
Internal: a node and its one or more descendants
BranchSet: a set of one or more Branches
Branch: a tree edge and its descendant subtree.
Name: the name of a node
Length: the length of a tree edge.
The grammar rules
Note, "|" separates alternatives.
Tree → Subtree ";"
Subtree → Leaf | Internal
Leaf → Name
Internal → "(" BranchSet ")" Name
BranchSet → Branch | Branch "," BranchSet
Branch → Subtree Length
Name → empty | string
Length → empty | ":" number
Whitespace (spaces, tabs, carriage returns, and linefeeds) within number is prohibited. Whitespace within string is often prohibited. Whitespace elsewhere is ignored. Sometimes the
|
https://en.wikipedia.org/wiki/Missing%20data
|
In statistics, missing data, or missing values, occur when no data value is stored for the variable in an observation. Missing data are a common occurrence and can have a significant effect on the conclusions that can be drawn from the data.
Missing data can occur because of nonresponse: no information is provided for one or more items or for a whole unit ("subject"). Some items are more likely to generate a nonresponse than others: for example items about private subjects such as income. Attrition is a type of missingness that can occur in longitudinal studies—for instance studying development where a measurement is repeated after a certain period of time. Missingness occurs when participants drop out before the test ends and one or more measurements are missing.
Data often are missing in research in economics, sociology, and political science because governments or private entities choose not to, or fail to, report critical statistics, or because the information is not available. Sometimes missing values are caused by the researcher—for example, when data collection is done improperly or mistakes are made in data entry.
These forms of missingness take different types, with different impacts on the validity of conclusions from research: Missing completely at random, missing at random, and missing not at random. Missing data can be handled similarly as censored data.
Types
Understanding the reasons why data are missing is important for handling the remaining data correctly. If values are missing completely at random, the data sample is likely still representative of the population. But if the values are missing systematically, analysis may be biased. For example, in a study of the relation between IQ and income, if participants with an above-average IQ tend to skip the question ‘What is your salary?’, analyses that do not take into account this missing at random (MAR pattern (see below)) may falsely fail to find a positive association between IQ and salary. Because of these problems, methodologists routinely advise researchers to design studies to minimize the occurrence of missing values. Graphical models can be used to describe the missing data mechanism in detail.
Missing completely at random
Values in a data set are missing completely at random (MCAR) if the events that lead to any particular data-item being missing are independent both of observable variables and of unobservable parameters of interest, and occur entirely at random. When data are MCAR, the analysis performed on the data is unbiased; however, data are rarely MCAR.
In the case of MCAR, the missingness of data is unrelated to any study variable: thus, the participants with completely observed data are in effect a random sample of all the participants assigned a particular intervention. With MCAR, the random assignment of treatments is assumed to be preserved, but that is usually an unrealistically strong assumption in practice.
Missing at random
Missing at random (MA
|
https://en.wikipedia.org/wiki/Shift%20matrix
|
In mathematics, a shift matrix is a binary matrix with ones only on the superdiagonal or subdiagonal, and zeroes elsewhere. A shift matrix U with ones on the superdiagonal is an upper shift matrix. The alternative subdiagonal matrix L is unsurprisingly known as a lower shift matrix. The (i,j):th component of U and L are
where is the Kronecker delta symbol.
For example, the 5×5 shift matrices are
Clearly, the transpose of a lower shift matrix is an upper shift matrix and vice versa.
As a linear transformation, a lower shift matrix shifts the components of a column vector one position down, with a zero appearing in the first position. An upper shift matrix shifts the components of a column vector one position up, with a zero appearing in the last position.
Premultiplying a matrix A by a lower shift matrix results in the elements of A being shifted downward by one position, with zeroes appearing in the top row. Postmultiplication by a lower shift matrix results in a shift left.
Similar operations involving an upper shift matrix result in the opposite shift.
Clearly all finite-dimensional shift matrices are nilpotent; an n by n shift matrix S becomes the null matrix when raised to the power of its dimension n.
Shift matrices act on shift spaces. The infinite-dimensional shift matrices are particularly important for the study of ergodic systems. Important examples of infinite-dimensional shifts are the Bernoulli shift, which acts as a shift on Cantor space, and the Gauss map, which acts as a shift on the space of continued fractions (that is, on Baire space.)
Properties
Let L and U be the n by n lower and upper shift matrices, respectively. The following properties hold for both U and L.
Let us therefore only list the properties for U:
det(U) = 0
trace(U) = 0
rank(U) = n − 1
The characteristic polynomials of U is
Un = 0. This follows from the previous property by the Cayley–Hamilton theorem.
The permanent of U is 0.
The following properties show how U and L are related:
If N is any nilpotent matrix, then N is similar to a block diagonal matrix of the form
where each of the blocks S1, S2, ..., Sr is a shift matrix (possibly of different sizes).
Examples
Then,
Clearly there are many possible permutations. For example, is equal to the matrix A shifted up and left along the main diagonal.
See also
Clock and shift matrices
Nilpotent matrix
Subshift of finite type
Notes
References
External links
Shift Matrix - entry in the Matrix Reference Manual
Matrices
Sparse matrices
|
https://en.wikipedia.org/wiki/Ahmed%20bin%20Ateyatalla%20Al%20Khalifa
|
Ahmed bin Ateyatalla Al Khalifa
() has a bachelor’s degree in Maths and Computer Science from University of Salford in Manchester, United Kingdom. Following completion of his education, he worked in the Central Informatics Organisation (CIO), Bahrain, for more than 20 years. During his time at the CIO he managed a number of national projects including Government Data Network project, National Y2K project, National Smartcard project, National GIS Project and National Census project. In addition to him being the executive director for the Bahrain national charter, Census 2001 project, municipal council elections and parliament elections projects. He was made president of CIO in 2004.
Al Khalifa was assigned the seat as Minister of Cabinet Affairs in September 2005. His portfolio included the Civil Service Bureau (CSB), Central Informatics Organisation (CIO), the e-Government Authority (EGA), Telecommunications including Telecommunications Regulatory Authority (TRA), Bahrain Internet Exchange (BIX) and the Bahrain Institute of Public administration (BIPA). He was heavily involved in the National Economic Strategy (NES) 2030 and was the leader for the Portfolio Office covering all programs and projects in his domain.
In 2006, he was responsible for the nationwide scandal revealed in Al Bandar report.
In 2011 Al Khalifa was appointed minister for Follow Up in the Royal Court.
References
Cisco Networkers Bahrain 2010 Keynote Session (6/6)
Cisco Networkers Bahrain 2010 Keynote Session (5/6)
External links
NEW-LOOK CABINET
Shaikh Ahmed bin Ateyatalla Al-Khalifa Appointed Royal Court Minister for Follow-up
Cisco Networkers Bahrain 2010
H.E. Shaikh Ahmed bin Ateyatalla Al Khalifa Officially Opens TRA New Offices
Year of birth missing (living people)
Living people
Ahmed bin Ateyatalla Al Khalifa
Government ministers of Bahrain
|
https://en.wikipedia.org/wiki/The%20Wiggle
|
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The Wiggle is a zig-zagging bicycle route from Market Street to Golden Gate Park in San Francisco, California, that minimizes hilly inclines for bicycle riders. Rising , The Wiggle inclines average 3% and never exceed 6%. The path generally follows the historical route of the long since paved-over Sans Souci Valley watercourse, winding through the Lower Haight neighborhood toward the Panhandle section of Golden Gate Park.
The lower end of the route begins at either end of the Duboce Bikeway in the block of Duboce Avenue just west of Market Street. The elevation is approximately above sea level. It then moves in a zig-zag toward the northwest along Duboce Avenue, Steiner, Waller, Pierce, Haight, Scott, and Fell Streets to the Panhandle Bikeway, above sea level. After climbing 50 more feet, the peak of The Wiggle is reached near Stanyan Street at the peninsular drainage divide, i.e., the dividing point between surface water flowing to the San Francisco Bay on the east side and flowing to the Pacific Ocean on the west.
Bicyclists can travel The Wiggle between major eastern and central neighborhoods (such as Downtown, SoMa, The Mission District, The Castro) and major western neighborhoods (including the Panhandle, Haight-Ashbury, Golden Gate Park, and The Richmond and Sunset Districts).
History
Mint Hill and the hill of Alamo Square on the northeast side are made of underlying serpentine rock, whereas Lone Mountain, Corona Heights, and Buena Vista Hill on the southwest are of the Franciscan chert formation. These hills are the northernmost manifestation of the San Miguel Hills (including Twin Peaks), which themselves comprise the northern tip of the Santa Cruz Mountains.
Over thousands of years, the gentle valley bottom was formed through a process of g
|
https://en.wikipedia.org/wiki/Algebraic%20logic
|
In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables.
What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics for these deductive systems) and connected problems like representation and duality. Well known results like the representation theorem for Boolean algebras and Stone duality fall under the umbrella of classical algebraic logic .
Works in the more recent abstract algebraic logic (AAL) focus on the process of algebraization itself, like classifying various forms of algebraizability using the Leibniz operator .
Calculus of relations
A homogeneous binary relation is found in the power set of for some set X, while a heterogeneous relation is found in the power set of , where . Whether a given relation holds for two individuals is one bit of information, so relations are studied with Boolean arithmetic. Elements of the power set are partially ordered by inclusion, and lattice of these sets becomes an algebra through relative multiplication or composition of relations.
"The basic operations are set-theoretic union, intersection and complementation, the relative multiplication, and conversion."
The conversion refers to the converse relation that always exists, contrary to function theory. A given relation may be represented by a logical matrix; then the converse relation is represented by the transpose matrix. A relation obtained as the composition of two others is then represented by the logical matrix obtained by matrix multiplication using Boolean arithmetic.
Example
An example of calculus of relations arises in erotetics, the theory of questions. In the universe of utterances there are statements S and questions Q. There are two relations and α from Q to S: q α a holds when a is a direct answer to question q. The other relation, q p holds when p is a presupposition of question q. The converse relation T runs from S to Q so that the composition Tα is a homogeneous relation on S. The art of putting the right question to elicit a sufficient answer is recognized in Socratic method dialogue.
Functions
The description of the key binary relation properties has been formulated with the calculus of relations. The univalence property of functions describes a relation that satisfies the formula where is the identity relation on the range of . The injective property corresponds to univalence of , or the formula where this time is the identity on the domain of .
But a univalent relation is only a partial function, while a univalent total relation is a function. The formula for totality is Charles Loewner and Gunther Schmidt use the term mapping for a total, univalent relation.
The facility of complementary relations inspired Augustus De Morgan and Ernst Schröder to introduce equivalences using for
|
https://en.wikipedia.org/wiki/Spread%20of%20a%20matrix
|
In mathematics, and more specifically matrix theory, the spread of a matrix is the largest distance in the complex plane between any two eigenvalues of the matrix.
Definition
Let be a square matrix with eigenvalues . That is, these values are the complex numbers such that there exists a vector on which acts by scalar multiplication:
Then the spread of is the non-negative number
Examples
For the zero matrix and the identity matrix, the spread is zero. The zero matrix has only zero as its eigenvalues, and the identity matrix has only one as its eigenvalues. In both cases, all eigenvalues are equal, so no two eigenvalues can be at nonzero distance from each other.
For a projection, the only eigenvalues are zero and one. A projection matrix therefore has a spread that is either (if all eigenvalues are equal) or (if there are two different eigenvalues).
All eigenvalues of a unitary matrix lie on the unit circle. Therefore, in this case, the spread is at most equal to the diameter of the circle, the number 2.
The spread of a matrix depends only on the spectrum of the matrix (its multiset of eigenvalues). If a second matrix of the same size is invertible, then has the same spectrum as . Therefore, it also has the same spread as .
See also
Field of values
References
Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Dover Publications, 1992, . Chap.III.4.
Linear algebra
Matrix theory
|
https://en.wikipedia.org/wiki/Exchange%20matrix
|
In mathematics, especially linear algebra, the exchange matrices (also called the reversal matrix, backward identity, or standard involutory permutation) are special cases of permutation matrices, where the 1 elements reside on the antidiagonal and all other elements are zero. In other words, they are 'row-reversed' or 'column-reversed' versions of the identity matrix.
Definition
If J is an n × n exchange matrix, then the elements of J are
Properties
Premultiplying a matrix by an exchange matrix flips vertically the positions of the former's rows, i.e.,
Postmultiplying a matrix by an exchange matrix flips horizontally the positions of the former's columns, i.e.,
Exchange matrices are symmetric; that is, JnT = Jn.
For any integer k, Jnk = I if k is even and Jnk = Jn if k is odd. In particular, Jn is an involutory matrix; that is, Jn−1 = Jn.
The trace of Jn is 1 if n is odd and 0 if n is even. In other words, the trace of Jn equals .
The determinant of Jn equals . As a function of n, it has period 4, giving 1, 1, −1, −1 when n is congruent modulo 4 to 0, 1, 2, and 3 respectively.
The characteristic polynomial of Jn is when n is even, and when n is odd.
The adjugate matrix of Jn is .
Relationships
An exchange matrix is the simplest anti-diagonal matrix.
Any matrix A satisfying the condition AJ = JA is said to be centrosymmetric.
Any matrix A satisfying the condition AJ = JAT is said to be persymmetric.
Symmetric matrices A that satisfy the condition AJ = JA are called bisymmetric matrices. Bisymmetric matrices are both centrosymmetric and persymmetric.
See also
Pauli matrices (the first Pauli matrix is a 2 × 2 exchange matrix)
References
Matrices
|
https://en.wikipedia.org/wiki/Lucas%27s%20theorem
|
In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient by a prime number p in terms of the base p expansions of the integers m and n.
Lucas's theorem first appeared in 1878 in papers by Édouard Lucas.
Statement
For non-negative integers m and n and a prime p, the following congruence relation holds:
where
and
are the base p expansions of m and n respectively. This uses the convention that if m < n.
Proofs
There are several ways to prove Lucas's theorem.
Consequences
A binomial coefficient is divisible by a prime p if and only if at least one of the base p digits of n is greater than the corresponding digit of m.
In particular, is odd if and only if the binary digits (bits) in the binary expansion of n are a subset of the bits of m.
Variations and generalizations
Kummer's theorem asserts that the largest integer k such that pk divides the binomial coefficient (or in other words, the valuation of the binomial coefficient with respect to the prime p) is equal to the number of carries that occur when n and m − n are added in the base p.
Generalizations of Lucas's theorem to the case of p being a prime power are given by Davis and Webb (1990) and Granville (1997).
The q-Lucas theorem is a generalization for the q-binomial coefficients, first proved by J. Désarménien.
References
External links
Articles containing proofs
Theorems about prime numbers
|
https://en.wikipedia.org/wiki/Rank%20%28differential%20topology%29
|
In mathematics, the rank of a differentiable map between differentiable manifolds at a point is the rank of the derivative of at . Recall that the derivative of at is a linear map
from the tangent space at p to the tangent space at f(p). As a linear map between vector spaces it has a well-defined rank, which is just the dimension of the image in Tf(p)N:
Constant rank maps
A differentiable map f : M → N is said to have constant rank if the rank of f is the same for all p in M. Constant rank maps have a number of nice properties and are an important concept in differential topology.
Three special cases of constant rank maps occur. A constant rank map f : M → N is
an immersion if rank f = dim M (i.e. the derivative is everywhere injective),
a submersion if rank f = dim N (i.e. the derivative is everywhere surjective),
a local diffeomorphism if rank f = dim M = dim N (i.e. the derivative is everywhere bijective).
The map f itself need not be injective, surjective, or bijective for these conditions to hold, only the behavior of the derivative is important. For example, there are injective maps which are not immersions and immersions which are not injections. However, if f : M → N is a smooth map of constant rank then
if f is injective it is an immersion,
if f is surjective it is a submersion,
if f is bijective it is a diffeomorphism.
Constant rank maps have a nice description in terms of local coordinates. Suppose M and N are smooth manifolds of dimensions m and n respectively, and f : M → N is a smooth map with constant rank k. Then for all p in M there exist coordinates (x1, ..., xm) centered at p and coordinates (y1, ..., yn) centered at f(p) such that f is given by
in these coordinates.
Examples
Maps whose rank is generically maximal, but drops at certain singular points, occur frequently in coordinate systems. For example, in spherical coordinates, the rank of the map from the two angles to a point on the sphere (formally, a map T2 → S2 from the torus to the sphere) is 2 at regular points, but is only 1 at the north and south poles (zenith and nadir).
A subtler example occurs in charts on SO(3), the rotation group. This group occurs widely in engineering, due to 3-dimensional rotations being heavily used in navigation, nautical engineering, and aerospace engineering, among many other uses. Topologically, SO(3) is the real projective space RP3, and it is often desirable to represent rotations by a set of three numbers, known as Euler angles (in numerous variants), both because this is conceptually simple, and because one can build a combination of three gimbals to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus T3 of three angles to the real projective space RP3 of rotations, but this map does not have rank 3 at all points (formally because it cannot be a covering map, as the only (non-trivial) covering space is the hypersphere S3), and the phenomenon of the rank dropping to 2 at ce
|
https://en.wikipedia.org/wiki/Metzler%20matrix
|
In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero):
It is named after the American economist Lloyd Metzler.
Metzler matrices appear in stability analysis of time delayed differential equations and positive linear dynamical systems. Their properties can be derived by applying the properties of nonnegative matrices to matrices of the form M + aI, where M is a Metzler matrix.
Definition and terminology
In mathematics, especially linear algebra, a matrix is called Metzler, quasipositive (or quasi-positive) or essentially nonnegative if all of its elements are non-negative except for those on the main diagonal, which are unconstrained. That is, a Metzler matrix is any matrix A which satisfies
Metzler matrices are also sometimes referred to as -matrices, as a Z-matrix is equivalent to a negated quasipositive matrix.
Properties
The exponential of a Metzler (or quasipositive) matrix is a nonnegative matrix because of the corresponding property for the exponential of a nonnegative matrix. This is natural, once one observes that the generator matrices of continuous-time Markov chains are always Metzler matrices, and that probability distributions are always non-negative.
A Metzler matrix has an eigenvector in the nonnegative orthant because of the corresponding property for nonnegative matrices.
Relevant theorems
Perron–Frobenius theorem
See also
Nonnegative matrices
Delay differential equation
M-matrix
P-matrix
Q-matrix, a specific kind of Metzler matrix
Z-matrix
Hurwitz matrix
Stochastic matrix
Positive systems
Bibliography
Matrices
|
https://en.wikipedia.org/wiki/Lawrence%20D.%20Brown
|
Lawrence David (Larry) Brown (16 December 1940 – 21 February 2018) was Miers Busch Professor and Professor of Statistics at the Wharton School of the University of Pennsylvania in Philadelphia, Pennsylvania. He is known for his groundbreaking work in a broad range of fields including decision theory, recurrence and partial differential equations, nonparametric function estimation, minimax and adaptation theory, and the analysis of census data and call-center data.
Career
Brown was educated at the California Institute of Technology and Cornell University, where he earned his Ph.D. in 1964. He earned numerous honors, including election to the United States National Academy of Sciences, and published widely, beginning with his Ph.D. research, which made major advances in admissibility. He was president of the Institute of Mathematical Statistics in 1992–93. He was elected to the American Academy of Arts and Sciences in 2013.
After having been assistant professor at University of California at Berkeley, associate professor at Cornell Universitywith the latter move entailing a change from a statistics to a mathematics department, allowing him to avoid being drafted for the Vietnam Warand professor at Cornell University and Rutgers University, he was invited to join the Department of Statistics at the Wharton School of the University of Pennsylvania.
Personal life
Brown was born in Los Angeles to parents Louis M. Brown and Hermione Brown. He was married to Linda Zhao, a fellow statistician at the Wharton School.
Honors and awards
Member, American Academy of Arts and Sciences
Member, National Academy of Sciences
Fellow of the American Statistical Association (A.S.A)
Fellow of the Institute of Mathematical Statistics (I.M.S.)
Wald Lecturer, Institute of Mathematical Statistics, August 1985;
Lady Davis Professorship, Hebrew University, 1988
Doctor of Science (Honorary), Purdue University, 1993
Wilks Award of the American Statistical Association, 2002
C.R. and B. Rao Prize, 2007
The Provost's Award for Distinguished Ph.D. Teaching and Mentoring, University of Pennsylvania, 2011
In his honor
Institute of Mathematical Statistics Lawrence D. Brown Ph.D. Student Award
Selected publications
Books
1985. (with Olkin, I., Sacks, J., and Wynn, H.P.) Jack Carl Kiefer Collected Papers, 3 vols., Springer-Verlag, New York.
1986. (with Olkin, I., Sacks, J., and Wynn, H.P.) Jack Carl Kiefer Collected Papers Supplementary Volume, Springer-Verlag, New York.
1986. Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory, Inst. of Math. Statist., Hayward, California.
2005. (with Plewes, T.J., and Gerstein, M.A.) Measuring Research and Development in the United States Economy, National Academies Press.
2010. (with Michael L. Cohen, Daniel L. Cork, and Constance F. Citro) Envisioning the 2020 Census. Panel on the Design of the 2020 Census Program of Evaluations and Experiments, Committee on National Statisti
|
https://en.wikipedia.org/wiki/Signature%20matrix
|
In mathematics, a signature matrix is a diagonal matrix whose diagonal elements are plus or minus 1, that is, any matrix of the form:
Any such matrix is its own inverse, hence is an involutory matrix. It is consequently a square root of the identity matrix. Note however that not all square roots of the identity are signature matrices.
Noting that signature matrices are both symmetric and involutory, they are thus orthogonal. Consequently, any linear transformation corresponding to a signature matrix constitutes an isometry.
Geometrically, signature matrices represent a reflection in each of the axes corresponding to the negated rows or columns.
Properties
If A is a matrix of N*N then:
(Due to the diagonal values being -1 or 1)
The Determinant of A is either 1 or -1 (Due to it being diagonal)
See also
Metric signature
References
Matrices
|
https://en.wikipedia.org/wiki/Epipolar%20geometry
|
Epipolar geometry is the geometry of stereo vision. When two cameras view a 3D scene from two distinct positions, there are a number of geometric relations between the 3D points and their projections onto the 2D images that lead to constraints between the image points. These relations are derived based on the assumption that the cameras can be approximated by the pinhole camera model.
Definitions
The figure below depicts two pinhole cameras looking at point X. In real cameras, the image plane is actually behind the focal center, and produces an image that is symmetric about the focal center of the lens. Here, however, the problem is simplified by placing a virtual image plane in front of the focal center i.e. optical center of each camera lens to produce an image not transformed by the symmetry. OL and OR represent the centers of symmetry of the two cameras lenses. X represents the point of interest in both cameras. Points xL and xR are the projections of point X onto the image planes.
Each camera captures a 2D image of the 3D world. This conversion from 3D to 2D is referred to as a perspective projection and is described by the pinhole camera model. It is common to model this projection operation by rays that emanate from the camera, passing through its focal center. Each emanating ray corresponds to a single point in the image.
Epipole or epipolar point
Since the optical centers of the cameras lenses are distinct, each center projects onto a distinct point into the other camera's image plane. These two image points, denoted by eL and eR, are called epipoles or epipolar points. Both epipoles eL and eR in their respective image planes and both optical centers OL and OR lie on a single 3D line.
Epipolar line
The line OL–X is seen by the left camera as a point because it is directly in line with that camera's lens optical center. However, the right camera sees this line as a line in its image plane. That line (eR–xR) in the right camera is called an epipolar line. Symmetrically, the line OR–X is seen by the right camera as a point and is seen as epipolar line eL–xLby the left camera.
An epipolar line is a function of the position of point X in the 3D space, i.e. as X varies, a set of epipolar lines is generated in both images. Since the 3D line
OL–X passes through the optical center of the lens OL, the corresponding epipolar line in the right image must pass through the epipole eR (and correspondingly for epipolar lines in the left image). All epipolar lines in one image contain the epipolar point of that image. In fact, any line which contains the epipolar point is an epipolar line since it can be derived from some 3D point X.
Epipolar plane
As an alternative visualization, consider the points X, OL & OR that form a plane called the epipolar plane. The epipolar plane intersects each camera's image plane where it forms lines—the epipolar lines. All epipolar planes and epipolar lines intersect the epipole regardless of wh
|
https://en.wikipedia.org/wiki/Segura%20de%20la%20Sierra
|
Segura de la Sierra is a small village in the province of Jaén, (Spain), that belongs to the region of Sierra de Segura in eastern Andalusia.
According to data provided by Spain's national statistics agency, Instituto Nacional de Estadística de España (INE), in 2005 there were 1,771 people living in the town, all them located in the Sierras de Cazorla, Segura y Las Villas Natural Park that includes the following villages:
Cortijos Nuevos
El Ojuelo
Carrasco
La Alberquilla
El Robledo
Rihornos
Trujala
Arroyo frío
Río Madera and Arroyo Canales
Catena
El Tobazo
El Puerto
History
The most important period for Segura de la Sierra was during the Arab occupation, when the town was called Saqura (). The village was conquered in 781 AD by Abul-Asvar who was responsible for building the several walls that surround the town. People were under the rule of the walíes serving the Córdoba kings.
After fighting between the Almohads, the Christians took the control and the king Alfonso VIII donated the village to the military Order of Santiago, many nobles and personalities were born or lived there in those days, including the poet Jorge Manrique.
After it was taken by the Castilian troops, part of its inhabitants resettled in the city of Safi, where they are known to this day by last name Shequri.
With the invasion of Napoleon's troops, the town was set on fire and most of its Archive was destroyed, losing a great part of the history of the village that will never be recovered.
Monuments
Segura de la Sierra was designated in 1972 Conjunto Histórico-Artístico.
The village offers, in essence, the same physiognomy it had in the past, reflected in its silent and beautiful streets.
The most important monument is the Mudéjar Castle, placed on top of the town and surrounded by the ancient walls.
The Fountain of Carlos V decorated with its shield is close to the Church of Nuestra Señora del Collado that has a nice painting of the Descendimiento by Gregorio Hernández and a Romanic sculpture of the Virgen de la Peña.
The old School of the Jesuitas with its plateresca façade was restored and now holds the town Council.
Finally the Arabian baths from the 11th century have also been restored and can now be visited.
Local festivities
The major local celebration is the festivity of the Virgen del Rosario between the 4 and 8 October that mixes religion and culture, those days the town had many people coming from other close villages.
During the day people enjoy the bull fightings in the Arabian square placed down the Castle and the competition of Bolos serranos. The party filled the night with much music and dancings that was typical pasodobles.
References
External links
Portal de la Sierra de Segura
Web site of Segura de la Sierra's Council
Municipalities in the Province of Jaén (Spain)
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https://en.wikipedia.org/wiki/Thomas%20Jones%20%28mathematician%29
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Thomas Jones (23 June 1756 – 18 July 1807) was Head Tutor at Trinity College, Cambridge, for twenty years and an outstanding teacher of mathematics. He is notable as a mentor of Adam Sedgwick.
Biography
Jones was born at Berriew, Montgomeryshire, in Wales.
On completing his studies at Shrewsbury School, Jones was admitted to St John's College, Cambridge, on 28 May 1774, as a 'pensioner' (i.e. a fee-paying student, as opposed to a scholar or sizar). He was believed to be an illegitimate son of Mr Owen Owen, of Tyncoed, and his housekeeper, who afterwards married a Mr Jones, of Traffin, County Kerry, Thomas then being brought up as his son.
On 27 June 1776, Jones migrated from St John's College to Trinity College. He became a scholar in 1777 and obtained his BA in 1779, winning the First Smith's Prize and becoming Senior Wrangler. In 1782, he obtained his MA and became a Fellow of Trinity College in 1781. He became a Junior Dean, 1787–1789 and a Tutor, 1787–1807. He was ordained a deacon at the Peterborough parish on 18 June 1780. Then he was ordained priest, at the Ely parish on 6 June 1784, canon of Fen Ditton, Cambridgeshire, in 1784, and then canon of Swaffham Prior, also 1784. On 11 December 1791, he preached before the university, at Great St Mary's, a sermon against duelling (from Exodus XX. 13), which was prompted by a duel that had lately taken place near Newmarket between Henry Applewhaite and Richard Ryecroft, undergraduates of Pembroke, in which the latter was fatally wounded. Jones died on 18 July 1807, in lodgings in Edgware Road, London. He is buried in the cemetery of Dulwich College. A bust and a memorial tablet are in the ante-chapel of Trinity College.
His academic mentor was John Cranke (1746–1816). His Cambridge tutor was Thomas Postlethwaite.
Notes
References
Dictionary of National Biography, Smith, Elder & Co., 1908–1986, vol. 10, pp. 1055–1056.
J. Wilkes, Encyclopedia Londinensis, Eds. J. Jones and J. Adlard, 1810–1829, vol. 11, pp. 256–258.
J.W. Clark and T.M. Hughes, The Life and Letters of the Reverend Adam Sedgwick, Cambridge University Press: 1890; vol. 1, pp. 73–75.
J. Gascoigne, Cambridge in the Age of Enlightenment, 1989, pp. 226–227, p. 232, p. 234, p. 243.
P. Searby, A History of the University of Cambridge, vol. 3 (1750–1870), ed. C.N.L. Brooke et al., 1997. pp. 309–310.
Oxford Dictionary of National Biography, vol. 30, eds. H. C. G. Matthew and B. Harrison, 2004, p. 645.
1756 births
1807 deaths
Welsh mathematicians
Senior Wranglers
Alumni of St John's College, Cambridge
Alumni of Trinity College, Cambridge
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https://en.wikipedia.org/wiki/South%20Italy
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South Italy ( or just ) is one of the five official statistical regions of Italy used by the National Institute of Statistics (ISTAT), a first level NUTS region and a European Parliament constituency. South Italy encompasses six of the country's 20 regions:
Abruzzo
Apulia
Basilicata
Calabria
Campania
Molise
South Italy is defined only for statistical and electoral purposes. It should not be confused with the Mezzogiorno, or Southern Italy, which refers to the areas of the former Kingdom of the Two Sicilies (once including the southern half of the Italian peninsula and Sicily) with the usual addition of the Western Mediterranean island of Sardinia. The latter and Sicily form a distinct statistical region, called Insular Italy.
Geography
South Italy borders central Italy to the northwest, while it is washed by the Adriatic Sea to the northeast, the Ionian Sea to the southeast and the Tyrrhenian Sea to the southwest.
The territory of south Italy is predominantly hilly and mountainous. The largest plains are the Tavoliere delle Puglie (second largest plain on the Italian peninsula), the Tavoliere salentino, the Campania plain, the Sele plain, the Metaponto plain, the Sibari plain and the Gioia Tauro plain. It is crossed from north to south by the Apennine Mountains, whose highest mountain is the Gran Sasso d'Italia ().
Demography
In 2022, the population resident in south Italy amounts to inhabitants.
Regions
Most populous municipalities
Below is the list of the population residing in 2022 in municipalities with more than inhabitants:
Economy
The gross domestic product (GDP) of the region was 271.1 billion euros in 2018, accounting for 15.4% of Italy's economic output. The GDP per capita adjusted for purchasing power was 19,300 euros, or 64% of the EU27 average in the same year.
See also
National Institute of Statistics (Italy)
NUTS statistical regions of Italy
Italian NUTS level 1 regions:
Northwest Italy
Northeast Italy
Insular Italy
Northern Italy
Central Italy
Southern Italy
References
Geography of Italy
NUTS 1 statistical regions of the European Union
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https://en.wikipedia.org/wiki/Cauchy%20matrix
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In mathematics, a Cauchy matrix, named after Augustin-Louis Cauchy, is an m×n matrix with elements aij in the form
where and are elements of a field , and and are injective sequences (they contain distinct elements).
The Hilbert matrix is a special case of the Cauchy matrix, where
Every submatrix of a Cauchy matrix is itself a Cauchy matrix.
Cauchy determinants
The determinant of a Cauchy matrix is clearly a rational fraction in the parameters and . If the sequences were not injective, the determinant would vanish, and tends to infinity if some tends to . A subset of its zeros and poles are thus known. The fact is that there are no more zeros and poles:
The determinant of a square Cauchy matrix A is known as a Cauchy determinant and can be given explicitly as
(Schechter 1959, eqn 4; Cauchy 1841, p. 154, eqn. 10).
It is always nonzero, and thus all square Cauchy matrices are invertible. The inverse A−1 = B = [bij] is given by
(Schechter 1959, Theorem 1)
where Ai(x) and Bi(x) are the Lagrange polynomials for and , respectively. That is,
with
Generalization
A matrix C is called Cauchy-like if it is of the form
Defining X=diag(xi), Y=diag(yi), one sees that both Cauchy and Cauchy-like matrices satisfy the displacement equation
(with for the Cauchy one). Hence Cauchy-like matrices have a common displacement structure, which can be exploited while working with the matrix. For example, there are known algorithms in literature for
approximate Cauchy matrix-vector multiplication with ops (e.g. the fast multipole method),
(pivoted) LU factorization with ops (GKO algorithm), and thus linear system solving,
approximated or unstable algorithms for linear system solving in .
Here denotes the size of the matrix (one usually deals with square matrices, though all algorithms can be easily generalized to rectangular matrices).
See also
Toeplitz matrix
Fay's trisecant identity
References
TiIo Finck, Georg Heinig, and Karla Rost: "An Inversion Formula and Fast Algorithms for Cauchy-Vandermonde Matrices", Linear Algebra and its Applications, vol.183 (1993), pp.179-191.
Dario Fasino: "Orthogonal Cauchy-like matrices", Numerical Algorithms, vol.92 (2023), pp.619-637. url=https://doi.org/10.1007/s11075-022-01391-y .
Matrices
Determinants
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https://en.wikipedia.org/wiki/Siavash%20Shahshahani
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Siavash Mirshams Shahshahani (Persian: سیاوش میرشمس شهشهانی) (born 1942) is an Iranian mathematician. He is a professor of mathematics and head of Mathematical Sciences Department at Sharif University of Technology. He headed up the IRNIC registry for the .ir ccTLD until his retirement from that position in late 2008. He has also served as a director of APTLD (the Asia Pacific Top Level Domain Association) between 2007 and his retirement from that position in February 2009.
Education
Shahshahani completed his PhD under the supervision of Stephen Smale at the University of California at Berkeley in 1969. He has since devoted a substantial part of his career to mathematical education.
Books
External links
Siavash Shahshahani Homepage at Mathematics Department of Sharif University of Technology
Siavash Shahshahani at ICANNWiki
APTLD homepage
21st-century Iranian mathematicians
Academic staff of Sharif University of Technology
University of California, Berkeley alumni
1942 births
Living people
Iranian Science and Culture Hall of Fame recipients in Mathematics and Physics
20th-century Iranian mathematicians
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https://en.wikipedia.org/wiki/Shear%20matrix
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In mathematics (particularly linear algebra), a shear matrix or transvection is an elementary matrix that represents the addition of a multiple of one row or column to another. Such a matrix may be derived by taking the identity matrix and replacing one of the zero elements with a non-zero value.
The name shear reflects the fact that the matrix represents a shear transformation. Geometrically, such a transformation takes pairs of points in a vector space that are purely axially separated along the axis whose row in the matrix contains the shear element, and effectively replaces those pairs by pairs whose separation is no longer purely axial but has two vector components. Thus, the shear axis is always an eigenvector of .
Definition
A typical shear matrix is of the form
This matrix shears parallel to the axis in the direction of the fourth dimension of the underlying vector space.
A shear parallel to the axis results in and . In matrix form:
Similarly, a shear parallel to the axis has and . In matrix form:
In 3D space this matrix shear the YZ plane into the diagonal plane passing through these 3 points:
The determinant will always be 1, as no matter where the shear element is placed, it will be a member of a skew-diagonal that also contains zero elements (as all skew-diagonals have length at least two) hence its product will remain zero and will not contribute to the determinant. Thus every shear matrix has an inverse, and the inverse is simply a shear matrix with the shear element negated, representing a shear transformation in the opposite direction. In fact, this is part of an easily derived more general result: if is a shear matrix with shear element , then is a shear matrix whose shear element is simply . Hence, raising a shear matrix to a power multiplies its shear factor by .
Properties
If is an shear matrix, then:
has rank and therefore is invertible
1 is the only eigenvalue of , so and
the eigenspace of (associated with the eigenvalue 1) has dimensions.
is defective
is asymmetric
may be made into a block matrix by at most 1 column interchange and 1 row interchange operation
the area, volume, or any higher order interior capacity of a polytope is invariant under the shear transformation of the polytope's vertices.
Composition
Two or more shear transformations can be combined.
If two shear matrices are and
then their composition matrix is
which also has determinant 1, so that area is preserved.
In particular, if , we have
which is a positive definite matrix.
Applications
Shear matrices are often used in computer graphics.
See also
Transformation matrix
Notes
References
Matrices
Linear algebra
Sparse matrices
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https://en.wikipedia.org/wiki/Pattern%20Blocks
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Pattern Blocks are a set of mathematical manipulatives developed in the 1960s. The six shapes are both a play resource and a tool for learning in mathematics, which serve to develop spatial reasoning skills that are fundamental to the learning of mathematics. Among other things, they allow children to see how shapes can be composed and decomposed into other shapes, and introduce children to ideas of tilings. Pattern blocks sets are multiple copies of just six shapes:
Equilateral triangle (Green)
60° rhombus (2 triangles) (Blue) that can be matched with two of the green triangles
30° Narrow rhombus (Beige) with the same side-length as the green triangle
Trapezoid (half hexagon or 3 triangles) (Red) that can be matched with three of the green triangles
Regular Hexagon (6 triangles) (Yellow) that can be matched with six of the green triangles
Square (Orange) with the same side-length as the green triangle
All the angles are multiples of 30° (1/12 of a circle): 30° (1×), 60° (2×), 90° (3×), 120° (4×), and 150° (5×).
Use
The block designed with their for both mathematics and play in mind. The advice given in the 1968 EDC Teacher's Guide is: "Take out the blocks, and play with them yourself. Try out some of your own ideas. Then, when you give the blocks to the children, sit back and watch what they do." The blocks are sufficiently mathematically structured that children’s self-directed play can lead to a variety of mathematical experience. Billy Hargrove and JJ Maybanks identifies a number of frequent features of play which occur:
Composing and Decomposing
Symmetry
Patterns
Three Dimensions
Negative Space
Representational
The EDC Teacher's Guide continues: "Many children start by making abstract designs — both symmetrical and asymmetrical. As play continues these designs may become more and more elegant and complex, or they become simple as the child refines his ideas."
An example of their use is given by Meha Agrawal: "Starting from the center, I would add tier after tier of blocks to build my pattern — it was an iterative process, because if something didn't look aesthetically appealing or fit correctly, it would require peeling off a layer and reevaluating ways to fix it. The best part was the gratification I received when my creation was complete. Though individually boring, collectively these blocks produced an intricate masterpiece that brought art and math, big-picture and detail, simplicity and complexity closer together".
History
Pattern blocks were developed, along with a Teacher's Guide to their use, at the Education Development Center in Newton, Massachusetts as part of the Elementary Science Study (ESS) project. The first Trial Edition of the Teacher's Guide states: "Work on Pattern Blocks was begun by Edward Prenowitz in 1963. He developed most of the ideas for the blocks and their uses and arranged for the first classroom trials. Many ESS staff members tried the materials and suggested additional activities." When Marion Walt
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https://en.wikipedia.org/wiki/Mathematical%20Institute%2C%20University%20of%20Oxford
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The Mathematical Institute is the mathematics department at the University of Oxford in England. It is one of the nine departments of the university's Mathematical, Physical and Life Sciences Division. The institute includes both pure and applied mathematics (Statistics is a separate department) and is one of the largest mathematics departments in the United Kingdom with about 200 academic staff. It was ranked (in a joint submission with Statistics) as the top mathematics department in the UK in the 2021 Research Excellence Framework. Research at the Mathematical Institute covers all branches of mathematical sciences ranging from, for example, algebra, number theory, and geometry to the application of mathematics to a wide range of fields including industry, finance, networks, and the brain. It has more than 850 undergraduates and 550 doctoral or masters students. The institute inhabits a purpose-built building between Somerville College and Green Templeton College on Woodstock Road, next to the Faculty of Philosophy.
History
The earliest forerunner of the Mathematical Institute was the School of Geometry and Arithmetic in the Bodleian Library's main quadrangle. This was completed in 1620.
Notable mathematicians associated with the university include Christopher Wren who, before his notable career as an architect, made contributions in analytical mathematics, astronomy, and mathematical physics; Edmond Halley who published a series of profound papers on astronomy while Savilian Professor of Geometry in the early 18th century; John Wallis, whose innovations include using the symbol for infinity; Charles Dodgson, who made significant contributions to geometry and logic while also achieving fame as a children's author under his pen name Lewis Carroll; and Henry John Stephen Smith, another Savilian Professor of Geometry, whose work in number theory and matrices attracted international recognition to Oxford mathematics. Dodgson jokingly proposed that the university should grant its mathematicians a narrow strip of level ground, reaching "ever so far", so that they could test whether or not parallel lines ever meet.
The building of an institute was originally proposed by G. H. Hardy in 1930. Lectures were normally given in the individual colleges of the university and Hardy proposed a central space where mathematics lectures could be held and where mathematicians could regularly meet. This proposal was too ambitious for the university, who allocated just six rooms for mathematicians in an extension to the Radcliffe Science Library built in 1934. A dedicated Mathematical Institute was built in 1966 and was located at the northern end of St Giles' near the junction with Banbury Road in central north Oxford. The needs of the institute soon outgrew its building, so it also occupied a neighbouring house on St Giles and two annexes: Dartington House on Little Clarendon Street, and the Gibson Building on the site of the Radcliffe Infirmary.
In 2008 the
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https://en.wikipedia.org/wiki/Population%20Reference%20Bureau
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The Population Reference Bureau (PRB) is a private, nonprofit organization specializing in collecting and supplying statistics necessary for research and/or academic purposes focused on the environment, and health and structure of populations. The PRB works in the United States and internationally with a wide range of partners in the government, nonprofit, research, business, and philanthropy sectors.
History
In the early 1930s, the Population Reference Bureau (PRB) shared office space with the Population Association of America, which was created in May 1931 in New York City. This association focuses its work around many aspects, such as reproductive health and fertility, children and families, global health, urbanization, and more.
Funding and partners
The PRB receives support from a number of foundations, non-governmental organizations, and government agencies. Examples of such funding include the Annie E. Casey Foundation, the Johns Hopkins Bloomberg School of Public Health, the United States Census Bureau, and the World Health Organization.
The PRB partners with about 80 other organizations all around the world, in countries like Sudan, Egypt, and Uganda, to name a few. These partners vary in foci and location, ranging from renowned research institutions such as the International Center for Research on Women to public education institutions such as the University of South Florida.
Capabilities
The Population Reference Bureau has many capabilities in providing information to individuals all around the world regarding population, health, and the environment. The organization specializes in the translation of the population demographics and health research, the analysis of the United States and international demographics, social and economic trends, and expanding the platform for general database research.
Services
The Population Reference Bureau offers an annual World Population Data Sheet, which is a chart containing data from 200 countries concerning important demographic and health variables, such as total population, fertility rates, infant mortality rates, HIV/AIDS prevalence, and contraceptive use.
The PRB's online data allows users to search a database of hundreds of demographic, health, economic, and environmental variables for countries and regions all around the world, such as the Middle East, Latin America, and Sub-Saharan Africa. The database provides scholarly articles about an assortment of topics, ranging from noncommunicable diseases and nutrition to the labor force and family planning.
The PRB also publishes a Population Bulletins, information about demographic concepts to help in educating the public on population studies.
Among these, other data and population tools available to the public from the PRB include population bulletins and customizable training and educational materials, presented through visual, written and online publications.
Programs and projects
The method used by the Population Reference Bureau
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https://en.wikipedia.org/wiki/Rellich%E2%80%93Kondrachov%20theorem
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In mathematics, the Rellich–Kondrachov theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Austrian-German mathematician Franz Rellich and the Russian mathematician Vladimir Iosifovich Kondrashov. Rellich proved the L2 theorem and Kondrashov the Lp theorem.
Statement of the theorem
Let Ω ⊆ Rn be an open, bounded Lipschitz domain, and let 1 ≤ p < n. Set
Then the Sobolev space W1,p(Ω; R) is continuously embedded in the Lp space Lp∗(Ω; R) and is compactly embedded in Lq(Ω; R) for every 1 ≤ q < p∗. In symbols,
and
Kondrachov embedding theorem
On a compact manifold with boundary, the Kondrachov embedding theorem states that if and then the Sobolev embedding
is completely continuous (compact).
Consequences
Since an embedding is compact if and only if the inclusion (identity) operator is a compact operator, the Rellich–Kondrachov theorem implies that any uniformly bounded sequence in W1,p(Ω; R) has a subsequence that converges in Lq(Ω; R). Stated in this form, in the past the result was sometimes referred to as the Rellich–Kondrachov selection theorem, since one "selects" a convergent subsequence. (However, today the customary name is "compactness theorem", whereas "selection theorem" has a precise and quite different meaning, referring to set-valued functions).
The Rellich–Kondrachov theorem may be used to prove the Poincaré inequality, which states that for u ∈ W1,p(Ω; R) (where Ω satisfies the same hypotheses as above),
for some constant C depending only on p and the geometry of the domain Ω, where
denotes the mean value of u over Ω.
References
Literature
Kondrachov, V. I., On certain properties of functions in the space L p .Dokl. Akad. Nauk SSSR 48, 563–566 (1945).
Leoni, Giovanni (2009). A First Course in Sobolev Spaces. Graduate Studies in Mathematics. 105. American Mathematical Society. pp. xvi+607. . MR 2527916. Zbl 1180.46001
Theorems in analysis
Sobolev spaces
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https://en.wikipedia.org/wiki/Institute%20of%20Education%20Sciences
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The Institute of Education Sciences (IES) is the independent, non-partisan statistics, research, and evaluation arm of the U.S. Department of Education. IES' stated mission is to provide scientific evidence on which to ground education practice and policy and to share this information in formats that are useful and accessible to educators, parents, policymakers, researchers, and the public. It was created as part of the Education Sciences Reform Act of 2002.
The first director of IES was Grover Whitehurst, who was appointed in November 2002 and served for six years. Mark Schneider is currently the Director of IES.
Divisions
IES is divided into four major research and statistics centers:
National Center for Education Evaluation and Regional Assistance (NCEE)—NCEE conducts large-scale evaluations and provides research-based technical assistance and information about high-quality research to educators and policymakers in a variety of different formats. NCEE's work includes evaluations of education programs and practices supported by federal funds; the Regional Educational Laboratory Program; the Education Resources Information Center (ERIC); the What Works Clearinghouse; and the National Library of Education. Matthew Soldner is the Commissioner of NCEE.
National Center for Education Research (NCER)—NCER supports research to improve student outcomes and education quality in the United States and pursue workable solutions to the challenges faced by educators and the education community. NCER also supports training programs to prepare researchers to conduct high quality, scientific education research. Elizabeth Albro is the Commissioner of NCER.
National Center for Education Statistics (NCES)—NCES is the primary federal entity that collects and analyzes data related to education in the United States and other nations. Among the programs and initiatives that NCES oversees is the National Assessment of Educational Progress. James Lynn Woodworth is the Commissioner of NCES.
National Center for Special Education Research (NCSER)—NCSER sponsors and supports comprehensive research that is designed to expand the knowledge and understanding of infants, toddlers, and children with disabilities, or those who are at risk of developing disabilities. NCSER also supports training programs to prepare researchers to conduct high quality, scientific special education research. Joan E. McLaughlin is the commissioner of NCSER.
National Board for Education Sciences
The National Board for Education Sciences serves as an advisory board for IES and has 15 voting members, who are appointed by the President of the United States. The Board also includes several ex-officio, non-voting members, including the director of IES, the commissioners of the four centers, and representatives of the National Institute of Child Health and Human Development, the U.S. Census Bureau, the U.S. Department of Labor, and the National Science Foundation. The Board advises and consults with
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https://en.wikipedia.org/wiki/Cayley%2C%20Alberta
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Cayley is a hamlet in southern Alberta, Canada within the Foothills County. It is also recognized as a designated place by Statistics Canada.
Cayley is approximately south of Calgary, south of High River and west of Highway 2 on Range Road 290 (former designated as Highway 2A). It is located within Census Division No. 6.
History
The community was named for the Hon. Hugh St. Quentin Cayley, a barrister and the publisher of the Calgary Herald in 1884, who also represented Calgary in the Northwest Territories legislature from 1886 to 1894. The hamlet originally contained at least seven grain elevators; all have been demolished. Cayley is also home to a Hutterite colony and a colony school; in 2001, two Cayley Colony girls were the first students from an Alberta colony school to write provincial diploma exams and graduate from high school.
Incorporation history
Previously incorporated as a village on August 4, 1904, Cayley dissolved to hamlet status on June 1, 1996.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Cayley had a population of 414 living in 166 of its 170 total private dwellings, a change of from its 2016 population of 377. With a land area of , it had a population density of in 2021.
As a designated place in the 2016 Census of Population conducted by Statistics Canada, Cayley had a population of 340 living in 143 of its 143 total private dwellings, a change of from its 2011 population of 265. With a land area of , it had a population density of in 2016.
See also
List of communities in Alberta
List of designated places in Alberta
List of former urban municipalities in Alberta
List of hamlets in Alberta
References
Hamlets in Alberta
Calgary Region
Designated places in Alberta
Foothills County
Former villages in Alberta
Populated places disestablished in 1996
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https://en.wikipedia.org/wiki/Killing%20tensor
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In mathematics, a Killing tensor or Killing tensor field is a generalization of a Killing vector, for symmetric tensor fields instead of just vector fields. It is a concept in pseudo-Riemannian geometry, and is mainly used in the theory of general relativity. Killing tensors satisfy an equation similar to Killing's equation for Killing vectors. Like Killing vectors, every Killing tensor corresponds to a quantity which is conserved along geodesics. However, unlike Killing vectors, which are associated with symmetries (isometries) of a manifold, Killing tensors generally lack such a direct geometric interpretation. Killing tensors are named after Wilhelm Killing.
Definition and properties
In the following definition, parentheses around tensor indices are notation for symmetrization. For example:
Definition
A Killing tensor is a tensor field (of some order m) on a (pseudo)-Riemannian manifold which is symmetric (that is, ) and satisfies:
This equation is a generalization of Killing's equation for Killing vectors:
Properties
Killing vectors are a special case of Killing tensors. Another simple example of a Killing tensor is the metric tensor itself. A linear combination of Killing tensors is a Killing tensor. A symmetric product of Killing tensors is also a Killing tensor; that is, if and are Killing tensors, then is a Killing tensor too.
Every Killing tensor corresponds to a constant of motion on geodesics. More specifically, for every geodesic with tangent vector , the quantity is constant along the geodesic.
Examples
Since Killing tensors are a generalization of Killing vectors, the examples at are also examples of Killing tensors. The following examples focus on Killing tensors not simply obtained from Killing vectors.
FLRW metric
The Friedmann–Lemaître–Robertson–Walker metric, widely used in cosmology, has spacelike Killing vectors corresponding to its spatial symmetries. It also has a Killing tensor
where a is the scale factor, is the t-coordinate basis vector, and the −+++ signature convention is used.
Kerr metric
The Kerr metric, describing a rotating black hole, has two independent Killing vectors. One Killing vector corresponds to the time translation symmetry of the metric, and another corresponds to the axial symmetry about the axis of rotation. In addition, as shown by Walker and Penrose (1970), there is a nontrivial Killing tensor of order 2. The constant of motion corresponding to this Killing tensor is called the Carter constant.
Killing-Yano tensor
An antisymmetric tensor of order p, , is a Killing-Yano tensor :fr:Tenseur de Killing-Yano if it satisfies the equation
.
While also a generalization of the Killing vector, it differs from the usual Killing tensor in that the covariant derivative is only contracted with one tensor index.
See also
Killing form
Killing vector field
Wilhelm Killing
References
Riemannian geometry
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https://en.wikipedia.org/wiki/Nordic%20Mathematical%20Contest
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The Nordic Mathematical Contest (NMC) is a mathematics competition for secondary school students from the five Nordic countries: Denmark, Finland, Iceland, Norway and Sweden. It takes place every year in March or April and serves the double purpose of being a regional secondary school level mathematics competition for the Nordic region and a step in the process of selection of the teams of the participating countries for the International Mathematical Olympiad (IMO) and the regional Baltic Way competition.
Participation
At most twenty participants from each country are appointed by the organisers of the national secondary school level mathematics competitions. They must either be eligible to the IMO or attend a secondary school. (The foreword of ref. renders the eligibility requirements unlike the past and present regulations.)
Problems
The exam consists of four problems to be answered in four hours. Only writing and drawing tools are permitted. For each problem the contestant can get from zero to seven points. The problems are of the IMO type and harder than those of the national secondary school level competitions in mathematics of the Nordic countries but not as hard as those of the IMO. They are chosen by the organising committee of the host country of the year from proposals submitted by the national organising committees.
The official web site of the NMC provides a complete collection in English with solutions of the problems from all the years. It is compiled by Matti Lehtinen. Selected versions of the problems in other Nordic languages are also available at the site
Organisation
The NMC is run in a decentralised manner involving no travel of the contestants nor any other personnel. The contestants write the exam in their own schools on the same day. Thence the papers are sent to a committee in the contestants' country who mark them preliminarily. They are then forwarded with the preliminary marking to a committee in the host country of the year, who coordinate the marking and decide the final result of each contestant. Hosting the NMC rotates among the participating countries in a fixed order. In each country, the NMC is run by the organisers of the country's secondary school level mathematics competition.
History
Start
The early history of the NMC is documented in a series of reports in the journal Normat. According to Åke Samuelsson, the NMC was founded at a meeting of the leaders of the Nordic teams at the 27th IMO in Warsaw 1986. As Denmark did not participate in the IMO before 1991, no team leader from Denmark was there. The founding countries were Finland, Iceland, Norway and Sweden. The first NMC took place on 30 March 1987 hosted by Sweden with 47 contestants from the four participating countries. It is stressed in the report that the organisation was minimal. The 2nd, 3rd and 4th NMCs took place in 1988-90 hosted by Norway, Iceland and Finland, respectively. This established the hosting order for the future.
Rumors of thi
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https://en.wikipedia.org/wiki/Islam%20in%20Myanmar
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Islam is a minority religion in Myanmar, practised by about 2.1% of the population, according to the 2014 Myanmar official statistics.
History
In the early Bagan era (AD 652-660), Arab Muslim merchants landed at ports such as Thaton and Martaban. Arab Muslim ships sailed from Madagascar to China, often going in and out of Burma. Arab travellers visited the Andaman Islands in the Bay of Bengal south of Burma.
The first Muslims had landed in Myanmar (Burma's) Ayeyarwady River delta, on the Tanintharyi coast and in Rakhine in the 9th century, prior to the establishment of the first Burmese empire in 1055 AD by King Anawrahta of Bagan. The sea posts of Burma such as Kyaukpyu, Bassein, Syriam, Martaban, Mergui, etc. are rife with the legendary accounts of early shipwrecks in their neighbourhood: of Kular shipwrecked sailors, traders and soldiers. At first Muslims arrived on the Arakan coast and moved into the upward hinterland to Maungdaw. The time when the Muslims arrived in Burma and in Arakan and Maungdaw is uncertain.
These early Muslim settlements and the propagation of Islam were documented by Arab, Persian, European and Chinese travelers of the 9th century. Burmese Muslims are the descendants of Muslim peoples who settled and intermarried with the local Burmese ethnic groups. Muslims arrived in Burma as traders or settlers, military personnel, and prisoners of war, refugees, and as victims of slavery. However, many early Muslims also as saying goes held positions of status as royal advisers, royal administrators, port authorities, mayors, and traditional medicine men.
The broadminded King Mindon of Mandalay, Burma permitted the Chinese Muslims known as Panthays to build a mosque in the capital, Mandalay. The Panthays of Mandalay requested donations from the Sultan Sulaiman of Yunnan. The Sultan agreed to finance the Mosque and sent his Colonel Mah Too-tu in 1868 to supervise the project. The Mosque, which is still standing, constitutes a historic landmark. It signifies the beginning of the first Panthay Jama'at (Congregation) in Mandalay Ratanabon Naypyidaw.
Persian Muslims arrived in northern Burma on the border with the Chinese region of Yunnan as recorded in the Chronicles of China in 860 AD. Burma's contacts with Islam via Yunnan thus go back to Sai-tien-ch'th (Shamsuddin), State councillor of Yunnan and his family. (1274-1279). His son Na-su-la-ting (Nasiruddin) was the commander of first Mongol invasion of Burma. (1277–78).
Burmese Muslims were sometimes called Pathi, a name believed to be derived from Persian. Many settlements in the southern region near present-day Thailand were noted for the Muslim populations, in which Muslims often outnumbered the local Buddhists. In one record, Pathein was said to be populated with Pathis, and was ruled by three Indian Muslim Kings in the 13th century. Arab merchants also arrived in Martaban, Mergui, and there were Arab settlements in the present Myeik archipelago's mid-western quarter
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https://en.wikipedia.org/wiki/Magic%20circle%20%28disambiguation%29
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A magic circle is a ritually defined space in a number of magical traditions.
Magic circle or Magic Circle may also refer to:
Magic circle (mathematics), an arrangement of natural numbers on circles such that the sum of the numbers on each circle and the sum of numbers on each diameter are identical
Magic circle (social), a concept used in sociology and psychology
Magic circle (virtual worlds), a membrane enclosing virtual worlds
Magic Circle (law firms), a group of leading London law firms
Offshore magic circle, a group of law firms practicing in offshore jurisdictions
The Magic Circle (organisation), a British organisation dedicated to stage magic
Magic Circle (album), a 2005 album by Wizard
The Magic Circle (Waterhouse paintings), two 1886 paintings by John William Waterhouse
The Magic Circle (video game)
Magic Circle Music, a record label founded by Manowar's bassist Joey DeMaio in 2003
Magic Circle Festival, a music festival founded in 2007 by Joey DeMaio, headlined by Manowar
Sala gang or Den Magiska Cirkeln, a 1930s Swedish criminal and occult organization
The Magic Circle, a 1993 novel by Donna Jo Napoli
See also
Circle of Magic, a series of fantasy novels by Tamora Pierce
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https://en.wikipedia.org/wiki/Involutory%20matrix
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In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix A is an involution if and only if A2 = I, where I is the n × n identity matrix. Involutory matrices are all square roots of the identity matrix. This is simply a consequence of the fact that any invertible matrix multiplied by its inverse is the identity.
Examples
The 2 × 2 real matrix is involutory provided that
The Pauli matrices in M(2, C) are involutory:
One of the three classes of elementary matrix is involutory, namely the row-interchange elementary matrix. A special case of another class of elementary matrix, that which represents multiplication of a row or column by −1, is also involutory; it is in fact a trivial example of a signature matrix, all of which are involutory.
Some simple examples of involutory matrices are shown below.
where
I is the 3 × 3 identity matrix (which is trivially involutory);
R is the 3 × 3 identity matrix with a pair of interchanged rows;
S is a signature matrix.
Any block-diagonal matrices constructed from involutory matrices will also be involutory, as a consequence of the linear independence of the blocks.
Symmetry
An involutory matrix which is also symmetric is an orthogonal matrix, and thus represents an isometry (a linear transformation which preserves Euclidean distance). Conversely every orthogonal involutory matrix is symmetric.
As a special case of this, every reflection and 180° rotation matrix is involutory.
Properties
An involution is non-defective, and each eigenvalue equals , so an involution diagonalizes to a signature matrix.
A normal involution is Hermitian (complex) or symmetric (real) and also unitary (complex) or orthogonal (real).
The determinant of an involutory matrix over any field is ±1.
If A is an n × n matrix, then A is involutory if and only if P+ = (I + A)/2 is idempotent. This relation gives a bijection between involutory matrices and idempotent matrices. Similarly, A is involutory if and only if P− = (I − A)/2 is idempotent. These two operators form the symmetric and antisymmetric projections of a vector with respect to the involution A, in the sense that , or . The same construct applies to any involutory function, such as the complex conjugate (real and imaginary parts), transpose (symmetric and antisymmetric matrices), and Hermitian adjoint (Hermitian and skew-Hermitian matrices).
If A is an involutory matrix in M(n, R), which is a matrix algebra over the real numbers, and A is not a scalar multiple of I, then the subalgebra generated by A is isomorphic to the split-complex numbers.
If A and B are two involutory matrices which commute with each other (i.e. AB = BA) then AB is also involutory.
If A is an involutory matrix then every integer power of A is involutory. In fact, An will be equal to A if n is odd and I if n is even.
See also
Affine involution
References
Matrices
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https://en.wikipedia.org/wiki/Essentially%20unique
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In mathematics, the term essentially unique is used to describe a weaker form of uniqueness, where an object satisfying a property is "unique" only in the sense that all objects satisfying the property are equivalent to each other. The notion of essential uniqueness presupposes some form of "sameness", which is often formalized using an equivalence relation.
A related notion is a universal property, where an object is not only essentially unique, but unique up to a unique isomorphism (meaning that it has trivial automorphism group). In general there can be more than one isomorphism between examples of an essentially unique object.
Examples
Set theory
At the most basic level, there is an essentially unique set of any given cardinality, whether one labels the elements or .
In this case, the non-uniqueness of the isomorphism (e.g., match 1 to or 1 to ) is reflected in the symmetric group.
On the other hand, there is an essentially unique ordered set of any given finite cardinality: if one writes and , then the only order-preserving isomorphism is the one which maps 1 to , 2 to , and 3 to .
Number theory
The fundamental theorem of arithmetic establishes that the factorization of any positive integer into prime numbers is essentially unique, i.e., unique up to the ordering of the prime factors.
Group theory
In the context of classification of groups, there is an essentially unique group containing exactly 2 elements. Similarly, there is also an essentially unique group containing exactly 3 elements: the cyclic group of order three. In fact, regardless of how one chooses to write the three elements and denote the group operation, all such groups can be shown to be isomorphic to each other, and hence are "the same".
On the other hand, there does not exist an essentially unique group with exactly 4 elements, as there are in this case two non-isomorphic groups in total: the cyclic group of order 4 and the Klein four group.
Measure theory
There is an essentially unique measure that is translation-invariant, strictly positive and locally finite on the real line. In fact, any such measure must be a constant multiple of Lebesgue measure, specifying that the measure of the unit interval should be 1—before determining the solution uniquely.
Topology
There is an essentially unique two-dimensional, compact, simply connected manifold: the 2-sphere. In this case, it is unique up to homeomorphism.
In the area of topology known as knot theory, there is an analogue of the fundamental theorem of arithmetic: the decomposition of a knot into a sum of prime knots is essentially unique.
Lie theory
A maximal compact subgroup of a semisimple Lie group may not be unique, but is unique up to conjugation.
Category theory
An object that is the limit or colimit over a given diagram is essentially unique, as there is a unique isomorphism to any other limiting/colimiting object.
Coding theory
Given the task of using 24-bit words to store 12 bits of information i
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https://en.wikipedia.org/wiki/William%20A.%20Stein
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William Arthur Stein (born February 21, 1974 in Santa Barbara, California) is a software developer and previously a professor of mathematics at the University of Washington.
He is the lead developer of SageMath and founder of CoCalc. Stein does computational and theoretical research into the problem of computing with modular forms and the Birch and Swinnerton-Dyer conjecture. He is considered "a leading expert in the field of computational arithmetic".
References
External links
1974 births
20th-century American mathematicians
21st-century American mathematicians
Computer programmers
Free software programmers
Living people
Northern Arizona University alumni
Number theorists
University of California, Berkeley alumni
University of Washington faculty
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https://en.wikipedia.org/wiki/List%20of%20chess%20gambits
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This is a list of chess openings that are gambits.
The gambits are organized into sections by the parent chess opening, giving the gambit name, ECO code, and defining moves in algebraic chess notation.
Alekhine's Defence
Alekhine Gambit – B02 – 1.e4 Nf6 2.e5 Nd5 3.c4 Nb6 4.d4 d6 5.Nf3 Bg4 6.Be2 dxe5 7.Nxe5
John Tracy Gambit – B02 – 1.e4 Nf6 2.Nf3
Krejcik Gambit – B02 – 1.e4 Nf6 2.Bc4 Nxe4 3.Bxf7+
Buntin Gambit – B02 – 1.e4 Nf6 2.e5 Nd5 3.e6
Geschev Gambit – B02 – 1.e4 Nf6 2.Nc3 d5 3.exd5 c6
O'Sullivan Gambit – B03 – 1.e4 Nf6 2.e5 Nd5 3.d4 b5
Myers Gambit – B02 – 1.e4 Nf6 2.Nc3 d5 3.d3 dxe4 4.Bg5
Spielmann Gambit – B02 – 1.e4 Nf6 2.Nc3 d5 3.e5 Nfd7 4.e6
Cambridge Gambit – B03 – 1.e4 Nf6 2.e5 Nd5 3.d4 d6 4.c4 Nb6 5.f4 g5
Lasker Simul Gambit – B02 – 1.e4 Nf6 2.e5 Nd5 3.c4 Nb6 4.c5 Nd5 5.Nc3 e6 6.Bc4
Matsukevich Gambit – B02 – 1.e4 Nf6 2.e5 Nd5 3.c4 Nb6 4.c5 Nd5 5.Nc3 Nxc3 6.dxc3 d6 7.Bg5
Mikėnas Gambit – B02 – 1.e4 Nf6 2.e5 Nd5 3.c4 Nb6 4.c5 Nd5 5.Bc4 e6 6.Nc3 d6 7.Nxd5 exd5 8.Bxd5
Benko Gambit
Nescafe Frappe Attack – A57 – 1.d4 Nf6 2.c4 c5 3.d5 b5 4.cxb5 a6 5.Nc3 axb5 6.e4 b4 7.Nb5 d6
Hjoerring Countergambit – A57 – 1.d4 Nf6 2.c4 c5 3.d5 b5 4.e4
Mutkin Countergambit – A57 – 1.d4 Nf6 2.c4 c5 3.d5 b5 4.g4
Bird's Opening
From's Gambit – A02 – 1.f4 e5
Swiss Gambit – A02 – 1.f4 f5 2.e4 fxe4 3.Nc3 Nf6 4.g4
Williams Gambit – A03 – 1.f4 d5 2.e4
Hobbs Gambit – A02 – 1.f4 g5
Dudweiler Gambit – A02 – 1.f4 d5 2.g4
Bahr Gambit – A02 – 1.f4 e5 2.Nc3
Sturm Gambit – A02 – 1.f4 d5 2.c4
Wagner-Zwitersch Gambit – A02 – 1.f4 f5 2.e4 fxe4 3.Nc3 Nf6 4.g4
Hobbs-Zilbermints Gambit – A02 – 1.f4 h6 2.Nf3 g5
Lasker Gambit – A02 – 1.f4 e5 2.fxe5 f6
Platz Gambit – A02 – 1.f4 e5 2.fxe5 Ne7
Schlechter Gambit – A02 – 1.f4 e5 2.fxe5 Nc6
Batavo Gambit – A02 – 1.f4 d5 2.Nf3 c5 3.e4 dxe4
Langheld Gambit – A02 – 1.f4 e5 2.fxe5 d6 3.exd6 Nf6
Siegener Gambit – A02 – 1.f4 e5 2.d4 exd4 3.Nf3 c5 4.c3
Williams-Zilbermints Gambit – A02 – 1.f4 d5 2.e4 dxe4 3.Nc3 Nf6 4.Nge2
Thomas Gambit – A02 – 1.f4 d5 2.b3 Nf6 3.Bb2 d4 4.Nf3 c5 5.e3
Prokofiev Gambit – A03 – 1.f4 d5 2.e4 dxe4 3.d3
Bishop's Opening
Calabrian Countergambit – C23 – 1.e4 e5 2.Bc4 f5
Four Pawns Gambit – C23 – 1.e4 e5 2.Bc4 Bc5 3.b4 Bxb4 4.f4 exf4 5.Nf3 Be7 6.d4 Bh4+ 7.g3 fxg3 8.0-0 gxh2+ 9.Kh1
Lewis Countergambit – C23 – 1.e4 e5 2.Bc4 Bc5 3.c3 d5
McDonnell Double Gambit – C23 – 1.e4 e5 2.Bc4 Bc5 3.b4 Bxb4 4.f4
Petroff Gambit – C23 – 1.e4 e5 2.Bc4 Bc5 3.Nf3 d6 4.c3 Qe7 5.d4
Wing Gambit – C23 – 1.e4 e5 2.Bc4 Bc5 3.b4
Boden–Kieseritzky Gambit – C24 – 1.e4 e5 2.Bc4 Nf6 3.Nf3 Nxe4 4.Nc3
Greco Gambit – C24 – 1.e4 e5 2.Bc4 Nf6 3.f4
Ponziani Gambit – C24 – 1.e4 e5 2.Bc4 Nf6 3.d4
Urusov (Ponziani) Gambit – C24 – 1.e4 e5 2.Bc4 Nf6 3.d4 exd4 4.Nf3
Khan Gambit – C23 – 1.e4 e5 2.Bc4 d5
Lewis Gambit – C23 – 1.e4 e5 2.Bc4 Bc5 3.d4
Stein Gambit – C23 – 1.e4 e5 2.Bc4 Bc5 3.f4
Anderssen Gambit – C23 – 1.e4 e5 2.Bc4 b5 3.Bxb5 c6
Thorold Gambit – C23 – 1.e4 e5 2.Bc4 b5 3.Bxb5 f5
Lopez Gambit – C23 – 1.e4 e5 2.Bc4 Bc5 3.Qe2 Nf
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https://en.wikipedia.org/wiki/Subnormal%20operator
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In mathematics, especially operator theory, subnormal operators are bounded operators on a Hilbert space defined by weakening the requirements for normal operators. Some examples of subnormal operators are isometries and Toeplitz operators with analytic symbols.
Definition
Let H be a Hilbert space. A bounded operator A on H is said to be subnormal if A has a normal extension. In other words, A is subnormal if there exists a Hilbert space K such that H can be embedded in K and there exists a normal operator N of the form
for some bounded operators
Normality, quasinormality, and subnormality
Normal operators
Every normal operator is subnormal by definition, but the converse is not true in general. A simple class of examples can be obtained by weakening the properties of unitary operators. A unitary operator is an isometry with dense range. Consider now an isometry A whose range is not necessarily dense. A concrete example of such is the unilateral shift, which is not normal. But A is subnormal and this can be shown explicitly. Define an operator U on
by
Direct calculation shows that U is unitary, therefore a normal extension of A. The operator U is called the unitary dilation of the isometry A.
Quasinormal operators
An operator A is said to be quasinormal if A commutes with A*A. A normal operator is thus quasinormal; the converse is not true. A counter example is given, as above, by the unilateral shift. Therefore, the family of normal operators is a proper subset of both quasinormal and subnormal operators. A natural question is how are the quasinormal and subnormal operators related.
We will show that a quasinormal operator is necessarily subnormal but not vice versa. Thus the normal operators is a proper subfamily of quasinormal operators, which in turn are contained by the subnormal operators. To argue the claim that a quasinormal operator is subnormal, recall the following property of quasinormal operators:
Fact: A bounded operator A is quasinormal if and only if in its polar decomposition A = UP, the partial isometry U and positive operator P commute.
Given a quasinormal A, the idea is to construct dilations for U and P in a sufficiently nice way so everything commutes. Suppose for the moment that U is an isometry. Let V be the unitary dilation of U,
Define
The operator N = VQ is clearly an extension of A. We show it is a normal extension via direct calculation. Unitarity of V means
On the other hand,
Because UP = PU and P is self adjoint, we have U*P = PU* and DU*P = DU*P. Comparing entries then shows N is normal. This proves quasinormality implies subnormality.
For a counter example that shows the converse is not true, consider again the unilateral shift A. The operator B = A + s for some scalar s remains subnormal. But if B is quasinormal, a straightforward calculation shows that A*A = AA*, which is a contradiction.
Minimal normal extension
Non-uniqueness of normal extensions
Given a subnormal operator A, its normal e
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https://en.wikipedia.org/wiki/2004%20Claxton%20Shield
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Results and statistics for the 2004 Claxton Shield
Ladder
Championship series
23 January 2004 - Semi Final 1 - Western Australia Vs Queensland Rams
24 January 2004 - Semi Final 2 - South Australia Vs New South Wales Patriots
24 January 2004 - Grand Final - New South Wales Patriots Vs Queensland Rams
*Box Score
Award winners
Top Stats
All-Star Team
External links
Official 2004 Claxton Shield Website
Claxton Shield
Claxton Shield
Claxton Shield
January 2004 sports events in Australia
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https://en.wikipedia.org/wiki/Dialectometry
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Dialectometry is the quantitative and computational branch of dialectology, the study of dialect. This sub-field of linguistics studies language variation using the methods of statistics; it arose in the 1970s and 80s as a result of seminal work by J. Séguy and Hans Goebl.
The research concentrates mainly on the regional distribution of dialect similarities, such as cores of dialect and overlapping zones, which can be labelled according to a more or less slight variance of dialect between bordering locations. However, analysis of dialect relationships cannot always be clearly depicted by cladistics, since there are often dialect continuum cases and also examples with elements of convergence, as well as division.
Language atlases serve as an empirical database which document the dialect profile of a large number of locations in detail. Different well-known numerical classification methodologies are used to abstract and visualise a basic pattern from the immense amount of data found in the language atlases.
Not one solid classification can be expected to result from the calculations; rather, different aspects of the basic pattern being searched for can be discovered by using the different methodologies. Principally speaking, there is more interest in the diversity of the taxometric methodologies, the results and the linguistic interpretations which can be made from them.
References
Further reading
Bauer, Roland. 2002-2003. Dolomitenladinische Ähnlichkeitsprofile aus dem Gadertal. Ein Werkstattbericht zur dialektometrischen Analyse des ALD-I, in: Ladinia XXVI-XXVII, 209-250.
Bauer, Roland. 2003. Sguardo dialettometrico su alcune zone di transizione dell'Italia nord-orientale (lombardo vs. trentino vs. veneto), in: Parallela X. Sguardi reciproci. Vicende linguistiche e culturali dell'area italofona e germanofona. Atti del Decimo Incontro italo-austriaco dei linguisti, Bombi, Raffaella / Fusco, Fabiana (Hrsg.), Udine: Forum Editrice, 93-119.
Bauer, Roland. 2004. Dialekte - Dialektmerkmale - dialektale Spannungen. Von "Cliquen", "Störenfrieden" und "Sündenböcken" im Netz des dolomitenladinischen Sprachatlasses ALD-I, in: Ladinia XXVIII, 201-242.
Bauer, Roland. 2005. La classificazione dialettometrica dei basiletti altoitaliani e ladini rappresentati nell'Atlante linguistico del ladino dolomitico e dei dialetti limitrofi (ALD-I), in: Lingue, istituzioni, territori. Riflessioni teoriche, proposte metodologiche ed esperienze di politica linguistica, Guardiano, Cristina et al. (Hrsg.), Roma: Bulzoni, 347-365.
Goebl, Hans. 1982. Dialektometrie; Prinzipien und Methoden des Einsatzes der numerischen Taxonomie im Bereich der Dialektgeographie. Wien: Verlag der Öst. Akademie der Wissenschaften.
Goebl, Hans. 1984. Dialektometrische Studien anhand italoromanischer, rätoromanischer und galloromanischer Sprachmaterialien aus AIS und ALF. Bd.1 (Bd.2 und 3 enthalten Karten und Tabellen). Tübingen: Max Niemeyer.
Goebl, Hans. 1985. Coup d'oeil dialectomét
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https://en.wikipedia.org/wiki/Unit%20tangent%20bundle
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In Riemannian geometry, the unit tangent bundle of a Riemannian manifold (M, g), denoted by T1M, UT(M) or simply UTM, is the unit sphere bundle for the tangent bundle T(M). It is a fiber bundle over M whose fiber at each point is the unit sphere in the tangent bundle:
where Tx(M) denotes the tangent space to M at x. Thus, elements of UT(M) are pairs (x, v), where x is some point of the manifold and v is some tangent direction (of unit length) to the manifold at x. The unit tangent bundle is equipped with a natural projection
which takes each point of the bundle to its base point. The fiber π−1(x) over each point x ∈ M is an (n−1)-sphere Sn−1, where n is the dimension of M. The unit tangent bundle is therefore a sphere bundle over M with fiber Sn−1.
The definition of unit sphere bundle can easily accommodate Finsler manifolds as well. Specifically, if M is a manifold equipped with a Finsler metric F : TM → R, then the unit sphere bundle is the subbundle of the tangent bundle whose fiber at x is the indicatrix of F:
If M is an infinite-dimensional manifold (for example, a Banach, Fréchet or Hilbert manifold), then UT(M) can still be thought of as the unit sphere bundle for the tangent bundle T(M), but the fiber π−1(x) over x is then the infinite-dimensional unit sphere in the tangent space.
Structures
The unit tangent bundle carries a variety of differential geometric structures. The metric on M induces a contact structure on UTM. This is given in terms of a tautological one-form, defined at a point u of UTM (a unit tangent vector of M) by
where is the pushforward along π of the vector v ∈ TuUTM.
Geometrically, this contact structure can be regarded as the distribution of (2n−2)-planes which, at the unit vector u, is the pullback of the orthogonal complement of u in the tangent space of M. This is a contact structure, for the fiber of UTM is obviously an integral manifold (the vertical bundle is everywhere in the kernel of θ), and the remaining tangent directions are filled out by moving up the fiber of UTM. Thus the maximal integral manifold of θ is (an open set of) M itself.
On a Finsler manifold, the contact form is defined by the analogous formula
where gu is the fundamental tensor (the hessian of the Finsler metric). Geometrically, the associated distribution of hyperplanes at the point u ∈ UTxM is the inverse image under π* of the tangent hyperplane to the unit sphere in TxM at u.
The volume form θ∧dθn−1 defines a measure on M, known as the kinematic measure, or Liouville measure, that is invariant under the geodesic flow of M. As a Radon measure, the kinematic measure μ is defined on compactly supported continuous functions ƒ on UTM by
where dV is the volume element on M, and μp is the standard rotationally-invariant Borel measure on the Euclidean sphere UTpM.
The Levi-Civita connection of M gives rise to a splitting of the tangent bundle
into a vertical space V = kerπ* and horizontal space H on which π* is a linear
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https://en.wikipedia.org/wiki/Tonearm%20%28musician%29
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Tonearm is the stage name of Russian-born, New York-based musician Ilia Bis (Илья Бис). Bis grew up in Moscow and later moved to the United States to study mathematics and computer sound analysis. After doing graduate work at the University of Chicago, he decided to pursue music full-time.
Tonearm usually performs as a one-man band, combining electronic processing with singing and playing an electric guitar. He also collaborates closely with video artists, and all shows are accompanied by tightly-scripted live video projections. Despite not having released a full record to date, Tonearm has received considerable critical attention both in the United States and in his native Russia for his strong songwriting and original production. In 2006, he was reportedly working on a debut album.
References
External links
Official website
Tonearm's instrumental music on myspace.com
Radio Liberty article
Moscow Times article
American electronic musicians
American male singer-songwriters
Intelligent dance musicians
Russian emigrants to the United States
University of Chicago alumni
Ableton Live users
Living people
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/M-matrix
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In mathematics, especially linear algebra, an M-matrix is a Z-matrix with eigenvalues whose real parts are nonnegative. The set of non-singular M-matrices are a subset of the class of P-matrices, and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging to the class of positive matrices). The name M-matrix was seemingly originally chosen by Alexander Ostrowski in reference to Hermann Minkowski, who proved that if a Z-matrix has all of its row sums positive, then the determinant of that matrix is positive.
Characterizations
An M-matrix is commonly defined as follows:
Definition: Let be a real Z-matrix. That is, where for all . Then matrix A is also an M-matrix if it can be expressed in the form , where with , for all , where is at least as large as the maximum of the moduli of the eigenvalues of , and is an identity matrix.
For the non-singularity of , according to the Perron–Frobenius theorem, it must be the case that . Also, for a non-singular M-matrix, the diagonal elements of A must be positive. Here we will further characterize only the class of non-singular M-matrices.
Many statements that are equivalent to this definition of non-singular M-matrices are known, and any one of these statements can serve as a starting definition of a non-singular M-matrix. For example, Plemmons lists 40 such equivalences. These characterizations has been categorized by Plemmons in terms of their relations to the properties of: (1) positivity of principal minors, (2) inverse-positivity and splittings,
(3) stability, and (4) semipositivity and diagonal dominance. It makes sense to categorize the properties in this way because the statements within a particular group are related to each other even when matrix is an arbitrary matrix, and not necessarily a Z-matrix. Here we mention a few characterizations from each category.
Equivalences
Below, denotes the element-wise order (not the usual positive semidefinite order on matrices). That is, for any real matrices A, B of size , we write if for all .
Let A be a real Z-matrix, then the following statements are equivalent to A being a non-singular M-matrix:
Positivity of principal minors
All the principal minors of A are positive. That is, the determinant of each submatrix of A obtained by deleting a set, possibly empty, of corresponding rows and columns of A is positive.
is non-singular for each nonnegative diagonal matrix D.
Every real eigenvalue of A is positive.
All the leading principal minors of A are positive.
There exist lower and upper triangular matrices L and U respectively, with positive diagonals, such that .
Inverse-positivity and splittings
A is inverse-positive. That is, exists and .
A is monotone. That is, implies .
A has a convergent regular splitting. That is, A has a representation , where with convergent. That is, .
There exist inverse-positive matrices and with .
Every regular splitting of A is convergent.
Stability
There ex
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https://en.wikipedia.org/wiki/Quantum%20Markov%20chain
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In mathematics, the quantum Markov chain is a reformulation of the ideas of a classical Markov chain, replacing the classical definitions of probability with quantum probability.
Introduction
Very roughly, the theory of a quantum Markov chain resembles that of a measure-many automaton, with some important substitutions: the initial state is to be replaced by a density matrix, and the projection operators are to be replaced by positive operator valued measures.
Formal statement
More precisely, a quantum Markov chain is a pair with a density matrix and a quantum channel such that
is a completely positive trace-preserving map, and a C*-algebra of bounded operators. The pair must obey the quantum Markov condition, that
for all .
See also
Quantum walk
References
Gudder, Stanley. "Quantum Markov chains." Journal of Mathematical Physics 49.7 (2008): 072105.
Exotic probabilities
Quantum information science
Markov models
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https://en.wikipedia.org/wiki/Lund%20v.%20Commonwealth
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Lund v. Commonwealth (Va. 1977) 232 S.E.2d 745 is a Supreme Court of Virginia case involving theft of services.
Facts
Charles Walter Lund was a statistics graduate student at the Virginia Polytechnic Institute and State University. While working on his Ph.D. research in the 1970s, Lund utilized the resources of Virginia Tech's computer lab.
The workings of the lab were complex. The computers were leased from IBM computers and the cost was distributed through various departments that used the computer facilities. Student who wished to use the computers were required to obtain the approval of the department head. Access keys were required to gain access to the lab, and a key was required to use PO boxes used to receive materials printed out on the computers. The student would ask for an item to be printed. The department would print the item and it would be placed in the PO box for retrieval. If the printed projects were too large to fit in the PO box a note would be placed there instead so the student could pick it up at the computer center main window.
Lund was put under surveillance on October 12, 1974, because departments were noticing unauthorized charges being made to their accounts. When asked about his activities on the computers, Lund initially denied any use of the computers. Later he admitted that he had been using it and turned over seven PO box keys to the investigator. Mr. Lund claimed that other students had given him those keys. Upon searching Lund's apartment, a large number of computer cards and print-outs were taken, the estimated value by the university being as much as $26,384.16.
Lund was charged in an indictment with the theft of keys, computer cards, computer printouts and using "without authority computer operation time and services of Computer Center Personnel... with intent to defraud, such property and services having a value of one hundred dollars or more." Lund waived his right to a jury trial and was convicted of grand larceny and sentenced to two years in the state penitentiary. His sentence was suspended and he was placed on probation for five years.
Holding
The Supreme Court of Virginia held that labor and services and the unauthorized use of the University's computer cannot be construed to be subject of larceny. The Court reasoned that labor or services cannot be the subject of the crime of larceny because neither time nor services may be taken or carried away, and that the unauthorized use of the computer could not be the subject of larceny, because "Nowhere in Code §§ 18.1-100[…] do we find the word 'use'."
On the subject of the "stolen" items, the Commonwealth argued that the printouts and the computer cards had as much market value as scrap paper: "The cost of producing the print-outs is not the proper criterion of value for the purpose here. Where there is no market value of an article that has been stolen, the better rule is that its actual value should be proved..."
The judgment and indictment of
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https://en.wikipedia.org/wiki/Ad%C4%B1yaman%20Airport
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Adıyaman Airport is an airport located at Adıyaman, Adıyaman Province, Turkey.
Airlines and destinations
Traffic Statistics
(*)Source: DHMI.gov.tr
The airport was closed for runway works for part of 2011.
References
External links
Airports in Turkey
Buildings and structures in Adıyaman Province
Transport in Adıyaman Province
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https://en.wikipedia.org/wiki/Schur%20orthogonality%20relations
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In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups.
They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3).
Finite groups
Intrinsic statement
The space of complex-valued class functions of a finite group G has a natural inner product:
where means the complex conjugate of the value of on g. With respect to this inner product, the irreducible characters form an orthonormal basis
for the space of class functions, and this yields the orthogonality relation for the rows of the character
table:
For , applying the same inner product to the columns of the character table yields:
where the sum is over all of the irreducible characters of G and the symbol denotes the order of the centralizer of . Note that since and are conjugate iff they are in the same column of the character table, this implies that the columns of the character table are orthogonal.
The orthogonality relations can aid many computations including:
decomposing an unknown character as a linear combination of irreducible characters;
constructing the complete character table when only some of the irreducible characters are known;
finding the orders of the centralizers of representatives of the conjugacy classes of a group; and
finding the order of the group.
Coordinates statement
Let be a matrix element of an irreducible matrix representation
of a finite group of order |G|, i.e. G has |G| elements. Since it can be proven that any matrix representation of any finite group is equivalent to a unitary representation, we assume is unitary:
where is the (finite) dimension of the irreducible representation .
The orthogonality relations, only valid for matrix elements of irreducible representations, are:
Here is the complex conjugate of and the sum is over all elements of G.
The Kronecker delta is unity if the matrices are in the same irreducible representation . If and are non-equivalent
it is zero. The other two Kronecker delta's state that
the row and column indices must be equal ( and ) in order to obtain a non-vanishing result. This theorem is also known as the Great (or Grand) Orthogonality Theorem.
Every group has an identity representation (all group elements mapped onto the real number 1).
This is an irreducible representation. The great orthogonality relations immediately imply that
for and any irreducible representation not equal to the identity representation.
Example of the permutation group on 3 objects
The 3! permutations of three objects form a group of order 6, commonly denoted (symmetric group). This group is isomorphic to the point group , consisting of a threefold rotation axis and three vertical mirror planes. The groups have a 2-dimensional irreducible representation (l = 2). In the case of one usually labels this representation
by
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https://en.wikipedia.org/wiki/Clifford%20torus
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In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the Cartesian product of two circles S and S (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3. To see why R4 is necessary, note that if S and S each exists in its own independent embedding space R and R, the resulting product space will be R4 rather than R3. The historically popular view that the Cartesian product of two circles is an R3 torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis z available to it after the first circle consumes x and y.
Stated another way, a torus embedded in R3 is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in R4. The relationship is similar to that of projecting the edges of a cube onto a sheet of paper. Such a projection creates a lower-dimensional image that accurately captures the connectivity of the cube edges, but also requires the arbitrary selection and removal of one of the three fully symmetric and interchangeable axes of the cube.
If S and S each has a radius of , their Clifford torus product will fit perfectly within the unit 3-sphere S3, which is a 3-dimensional submanifold of R4. When mathematically convenient, the Clifford torus can be viewed as residing inside the complex coordinate space C2, since C2 is topologically equivalent to R4.
The Clifford torus is an example of a square torus, because it is isometric to a square with opposite sides identified. It is further known as a Euclidean 2-torus (the "2" is its topological dimension); figures drawn on it obey Euclidean geometry as if it were flat, whereas the surface of a common "doughnut"-shaped torus is positively curved on the outer rim and negatively curved on the inner. Although having a different geometry than the standard embedding of a torus in three-dimensional Euclidean space, the square torus can also be embedded into three-dimensional space, by the Nash embedding theorem; one possible embedding modifies the standard torus by a fractal set of ripples running in two perpendicular directions along the surface.
Formal definition
The unit circle S1 in R2 can be parameterized by an angle coordinate:
In another copy of R2, take another copy of the unit circle
Then the Clifford torus is
Since each copy of S1 is an embedded submanifold of R2, the Clifford torus is an embedded torus in = R4.
If R4 is given by coordinates (x1, y1, x2, y2), then the Clifford torus is given by
This shows that in R4 the Clifford torus is a submanifold of the unit 3-sphere S3.
It is easy to verify that the Clifford torus is a minimal surface in S3.
Alternative derivation using complex numbers
It is also common to consider the Clifford torus as an embedded torus in C2. In two copies of C, we have the follo
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https://en.wikipedia.org/wiki/Romeu%20%28footballer%29
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Romeu Pereira dos Santos (born 13 February 1985), known as Romeu, is a Brazilian former professional footballer who last played as a defensive midfielder for Greek club Levadiakos.
Career statistics
(correct as of 1 October 2013)
Honours
Club
Fluminense
Copa do Brasil: 2007
External links
CBF
sambafoot
zerozero.pt
1985 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Fluminense FC players
Athlitiki Enosi Larissa F.C. players
Levadiakos F.C. players
Panthrakikos F.C. players
Campeonato Brasileiro Série A players
Super League Greece players
Expatriate men's footballers in Greece
Footballers from Bahia
Men's association football midfielders
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https://en.wikipedia.org/wiki/Sign%20%28mathematics%29
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In mathematics, the sign of a real number is its property of being either positive, negative, or 0.
In some contexts, it makes sense to consider a signed zero (such as floating-point representations of real numbers within computers).
Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it may be considered both positive and negative (having both signs). Whenever not specifically mentioned, this article adheres to the first convention (zero having undefined sign).
In mathematics and physics, the phrase "change of sign" is associated with the generation of the additive inverse (negation, or multiplication by −1) of any object that allows for this construction, and is not restricted to real numbers. It applies among other objects to vectors, matrices, and complex numbers, which are not prescribed to be only either positive, negative, or zero.
The word "sign" is also often used to indicate other binary aspects of mathematical objects that resemble positivity and negativity, such as odd and even (sign of a permutation), sense of orientation or rotation (cw/ccw), one sided limits, and other concepts described in below.
Sign of a number
Numbers from various number systems, like integers, rationals, complex numbers, quaternions, octonions, ... may have multiple attributes, that fix certain properties of a number. A number system that bears the structure of an ordered ring contains a unique number that when added with any number leaves the latter unchanged. This unique number is known as the system's additive identity element. For example, the integers has the structure of an ordered ring. This number is generally denoted as Because of the total order in this ring, there are numbers greater than zero, called the positive numbers. Another property required for a ring to be ordered is that, for each positive number, there exists a unique corresponding number less than whose sum with the original positive number is These numbers less than are called the negative numbers. The numbers in each such pair are their respective additive inverses. This attribute of a number, being exclusively either zero , positive , or negative , is called its sign, and is often encoded to the real numbers , , and , respectively (similar to the way the sign function is defined). Since rational and real numbers are also ordered rings (in fact ordered fields), the sign attribute also applies to these number systems.
When a minus sign is used in between two numbers, it represents the binary operation of subtraction. When a minus sign is written before a single number, it represents the unary operation of yielding the additive inverse (sometimes called negation) of the operand. Abstractly then, the difference of two number is the sum of the minuend with the additive inverse of the subtrahend. While is its own additive inverse (), the additive inverse of a positive number is negative, and t
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https://en.wikipedia.org/wiki/Higher%20spin%20alternating%20sign%20matrix
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In mathematics, a higher spin alternating sign matrix is a generalisation of the alternating sign matrix (ASM), where the columns and rows sum to an integer r (the spin) rather than simply summing to 1 as in the usual alternating sign matrix definition. HSASMs are square matrices whose elements may be integers in the range −r to +r. When traversing any row or column of an ASM or HSASM, the partial sum of its entries must always be non-negative.
High spin ASMs have found application in statistical mechanics and physics, where they have been found to represent symmetry groups in ice crystal formation.
Some typical examples of HSASMs are shown below:
The set of HSASMs is a superset of the ASMs. The extreme points of the convex hull of the set of r-spin HSASMs are themselves integer multiples of the usual ASMs.
References
Matrices
Statistical mechanics
Enumerative combinatorics
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https://en.wikipedia.org/wiki/D%27Alembert%E2%80%93Euler%20condition
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In mathematics and physics, especially the study of mechanics and fluid dynamics, the d'Alembert-Euler condition is a requirement that the streaklines of a flow are irrotational. Let x = x(X,t) be the coordinates of the point x into which X is carried at time t by a (fluid) flow. Let be the second material derivative of x. Then the d'Alembert-Euler condition is:
The d'Alembert-Euler condition is named for Jean le Rond d'Alembert and Leonhard Euler who independently first described its use in the mid-18th century. It is not to be confused with the Cauchy–Riemann conditions.
References
See sections 45–48.
d'Alembert–Euler conditions on the Springer Encyclopedia of Mathematics
Fluid mechanics
Mechanical engineering
Vector calculus
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https://en.wikipedia.org/wiki/Restricted%20sumset
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In additive number theory and combinatorics, a restricted sumset has the form
where are finite nonempty subsets of a field F and is a polynomial over F.
If is a constant non-zero function, for example for any , then is the usual sumset which is denoted by if
When
S is written as which is denoted by if
Note that |S| > 0 if and only if there exist with
Cauchy–Davenport theorem
The Cauchy–Davenport theorem, named after Augustin Louis Cauchy and Harold Davenport, asserts that for any prime p and nonempty subsets A and B of the prime order cyclic group we have the inequality
where , i.e. we're using modular arithmetic. It can be generalised to arbitrary (not necessarily abelian) groups using a Dyson transform. If are subsets of a group , then
where is the size of the smallest nontrivial subgroup of (we set it to if there is no such subgroup).
We may use this to deduce the Erdős–Ginzburg–Ziv theorem: given any sequence of 2n−1 elements in the cyclic group , there are n elements that sum to zero modulo n. (Here n does not need to be prime.)
A direct consequence of the Cauchy-Davenport theorem is: Given any sequence S of p−1 or more nonzero elements, not necessarily distinct, of , every element of can be written as the sum of the elements of some subsequence (possibly empty) of S.
Kneser's theorem generalises this to general abelian groups.
Erdős–Heilbronn conjecture
The Erdős–Heilbronn conjecture posed by Paul Erdős and Hans Heilbronn in 1964 states that if p is a prime and A is a nonempty subset of the field Z/pZ. This was first confirmed by J. A. Dias da Silva and Y. O. Hamidoune in 1994
who showed that
where A is a finite nonempty subset of a field F, and p(F) is a prime p if F is of characteristic p, and p(F) = ∞ if F is of characteristic 0. Various extensions of this result were given by Noga Alon, M. B. Nathanson and I. Ruzsa in 1996, Q. H. Hou and Zhi-Wei Sun in 2002,
and G. Karolyi in 2004.
Combinatorial Nullstellensatz
A powerful tool in the study of lower bounds for cardinalities of various restricted sumsets is the following fundamental principle: the combinatorial Nullstellensatz. Let be a polynomial over a field . Suppose that the coefficient of the monomial in is nonzero and is the total degree of . If are finite subsets of with for , then there are such that .
This tool was rooted in a paper of N. Alon and M. Tarsi in 1989,
and developed by Alon, Nathanson and Ruzsa in 1995–1996,
and reformulated by Alon in 1999.
See also
Polynomial method in combinatorics
References
External links
Augustin-Louis Cauchy
Sumsets
Additive combinatorics
Additive number theory
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https://en.wikipedia.org/wiki/Nested%20sampling%20algorithm
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The nested sampling algorithm is a computational approach to the Bayesian statistics problems of comparing models and generating samples from posterior distributions. It was developed in 2004 by physicist John Skilling.
Background
Bayes' theorem can be applied to a pair of competing models and for data , one of which may be true (though which one is unknown) but which both cannot be true simultaneously. The posterior probability for may be calculated as:
The prior probabilities and are already known, as they are chosen by the researcher ahead of time. However, the remaining Bayes factor is not so easy to evaluate, since in general it requires marginalizing nuisance parameters. Generally, has a set of parameters that can be grouped together and called , and has its own vector of parameters that may be of different dimensionality, but is still termed . The marginalization for is
and likewise for . This integral is often analytically intractable, and in these cases it is necessary to employ a numerical algorithm to find an approximation. The nested sampling algorithm was developed by John Skilling specifically to approximate these marginalization integrals, and it has the added benefit of generating samples from the posterior distribution . It is an alternative to methods from the Bayesian literature such as bridge sampling and defensive importance sampling.
Here is a simple version of the nested sampling algorithm, followed by a description of how it computes the marginal probability density where is or :
Start with points sampled from prior.
for to do % The number of iterations j is chosen by guesswork.
current likelihood values of the points;
Save the point with least likelihood as a sample point with weight .
Update the point with least likelihood with some Markov chain Monte Carlo steps according to the prior, accepting only steps that
keep the likelihood above .
end
return ;
At each iteration, is an estimate of the amount of prior mass covered by the hypervolume in parameter space of all points with likelihood greater than . The weight factor is an estimate of the amount of prior mass that lies between two nested hypersurfaces and . The update step computes the sum over of to numerically approximate the integral
In the limit , this estimator has a positive bias of order which can be removed by using instead of the in the above algorithm.
The idea is to subdivide the range of and estimate, for each interval , how likely it is a priori that a randomly chosen would map to this interval. This can be thought of as a Bayesian's way to numerically implement Lebesgue integration.
Implementations
Example implementations demonstrating the nested sampling algorithm are publicly available for download, written in several programming languages.
Simple examples in C, R, or Python are on John Skilling's website.
A Haskell port of the above simple codes is on Hack
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https://en.wikipedia.org/wiki/GWG
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GWG is a three letter acronym that can stand for:
Girls with guns
Game-winning goal, in sports
Global warming gases (greenhouse gases)
Geometry Wars: Galaxies, a 2007 shoot 'em up video game
Organizations
Goodwill Games (Olympic style games started during the Cold War)
The former Garden Writers' Guild (renamed the Garden Media Guild in late 2007), a British trade association for garden writers, photographers and broadcasters
Great Western Garment Co.
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https://en.wikipedia.org/wiki/Saber%20Mirghorbani
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Seyed Saber Mirghorbani (, born September 17, 1983, in Babolsar, Iran) is an Iranian football player.
Club career
Club career statistics
Assist Goals
International career
Saber Mirghorbani was called up to the Iran national football team in June 2007 for the West Asian Football Federation Championship 2007. He made his debut for Iran in a match vs Palestine.
References
External links
Iranian men's footballers
Iran men's international footballers
Men's association football forwards
Sanat Mes Kerman F.C. players
Sanat Naft Abadan F.C. players
People from Babolsar
1983 births
Living people
PAS Hamedan F.C. players
Fajr Sepasi Shiraz F.C. players
Saipa F.C. players
Persian Gulf Pro League players
Azadegan League players
Footballers from Mazandaran province
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https://en.wikipedia.org/wiki/Fab%C3%A3o%20%28footballer%2C%20born%201976%29
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José Fábio Alves Azevedo (born June 15, 1976 in Vera Cruz), or simply Fabão, is a Brazilian centre back currently playing for Sobradinho.
Club statistics
Honors
Club
FIFA Club World Cup: 2005
Copa Libertadores: 2005
Campeonato Brasileiro Série A: 2006
J. League: 2007
Individual
Campeonato Brasileiro Série A Team of the Year: 2006
References
External links
1976 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Men's association football defenders
Paraná Clube players
Esporte Clube Bahia players
CR Flamengo footballers
Real Betis players
Córdoba CF players
Goiás Esporte Clube players
São Paulo FC players
Kashima Antlers players
Santos FC players
Guarani FC players
Henan F.C. players
Comercial Futebol Clube (Ribeirão Preto) players
Campeonato Brasileiro Série A players
Segunda División players
J1 League players
Copa Libertadores-winning players
Expatriate men's footballers in Spain
Expatriate men's footballers in Japan
Brazilian expatriate sportspeople in Japan
Expatriate men's footballers in China
Brazilian expatriate sportspeople in China
Brazilian expatriate sportspeople in Spain
Chinese Super League players
People from Vera Cruz, Bahia
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https://en.wikipedia.org/wiki/Sequential%20probability%20ratio%20test
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The sequential probability ratio test (SPRT) is a specific sequential hypothesis test, developed by Abraham Wald and later proven to be optimal by Wald and Jacob Wolfowitz. Neyman and Pearson's 1933 result inspired Wald to reformulate it as a sequential analysis problem. The Neyman-Pearson lemma, by contrast, offers a rule of thumb for when all the data is collected (and its likelihood ratio known).
While originally developed for use in quality control studies in the realm of manufacturing, SPRT has been formulated for use in the computerized testing of human examinees as a termination criterion.
Theory
As in classical hypothesis testing, SPRT starts with a pair of hypotheses, say and for the null hypothesis and alternative hypothesis respectively. They must be specified as follows:
The next step is to calculate the cumulative sum of the log-likelihood ratio, , as new data arrive: with , then, for =1,2,...,
The stopping rule is a simple thresholding scheme:
: continue monitoring (critical inequality)
: Accept
: Accept
where and () depend on the desired type I and type II errors, and . They may be chosen as follows:
and
In other words, and must be decided beforehand in order to set the thresholds appropriately. The numerical value will depend on the application. The reason for being only an approximation is that, in the discrete case, the signal may cross the threshold between samples. Thus, depending on the penalty of making an error and the sampling frequency, one might set the thresholds more aggressively. The exact bounds are correct in the continuous case.
Example
A textbook example is parameter estimation of a probability distribution function. Consider the exponential distribution:
The hypotheses are
Then the log-likelihood function (LLF) for one sample is
The cumulative sum of the LLFs for all is
Accordingly, the stopping rule is:
After re-arranging we finally find
The thresholds are simply two parallel lines with slope . Sampling should stop when the sum of the samples makes an excursion outside the continue-sampling region.
Applications
Manufacturing
The test is done on the proportion metric, and tests that a variable p is equal to one of two desired points, p1 or p2. The region between these two points is known as the indifference region (IR). For example, suppose you are performing a quality control study on a factory lot of widgets. Management would like the lot to have 3% or less defective widgets, but 1% or less is the ideal lot that would pass with flying colors. In this example, p1 = 0.01 and p2 = 0.03 and the region between them is the IR because management considers these lots to be marginal and is OK with them being classified either way. Widgets would be sampled one at a time from the lot (sequential analysis) until the test determines, within an acceptable error level, that the lot is ideal or should be rejected.
Testing of human examinees
The SPRT is currently the predominant method of c
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https://en.wikipedia.org/wiki/Predictive%20medicine
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Predictive medicine is a field of medicine that entails predicting the probability of disease and instituting preventive measures in order to either prevent the disease altogether or significantly decrease its impact upon the patient (such as by preventing mortality or limiting morbidity).
While different prediction methodologies exist, such as genomics, proteomics, and cytomics, the most fundamental way to predict future disease is based on genetics. Although proteomics and cytomics allow for the early detection of disease, much of the time those detect biological markers that exist because a disease process has already started. However, comprehensive genetic testing (such as through the use of DNA arrays or full genome sequencing) allows for the estimation of disease risk years to decades before any disease even exists, or even whether a healthy fetus is at higher risk for developing a disease in adolescence or adulthood. Individuals who are more susceptible to disease in the future can be offered lifestyle advice or medication with the aim of preventing the predicted illness.
Current genetic testing guidelines supported by the health care professionals discourage purely predictive genetic testing of minors until they are competent to understand the relevancy of genetic screening so as to allow them to participate in the decision about whether or not it is appropriate for them. Genetic screening of newborns and children in the field of predictive medicine is deemed appropriate if there is a compelling clinical reason to do so, such as the availability of prevention or treatment as a child that would prevent future disease.
The goal
The goal of predictive medicine is to predict the probability of future disease so that health care professionals and the patient themselves can be proactive in instituting lifestyle modifications and increased physician surveillance, such as bi-annual full body skin exams by a dermatologist or internist if their patient is found to have an increased risk of melanoma, an EKG and cardiology examination by a cardiologist if a patient is found to be at increased risk for a cardiac arrhythmia or alternating MRIs or mammograms every six months if a patient is found to be at increased risk for breast cancer. Predictive medicine is intended for both healthy individuals ("predictive health") and for those with diseases ("predictive medicine"), its purpose being to predict susceptibility to a particular disease and to predict progression and treatment response for a given disease.
A number of association studies have been published in scientific literature that show associations between specific genetic variants in a person's genetic code and a specific disease. Association and correlation studies have found that a female individual with a mutation in the BRCA1 gene has a 65% cumulative risk of breast cancer. Additionally, new tests from Genetic Technologies LTD and Phenogen Sciences Inc. comparing non-coding DNA to a
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https://en.wikipedia.org/wiki/Lehmer%20matrix
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In mathematics, particularly matrix theory, the n×n Lehmer matrix (named after Derrick Henry Lehmer) is the constant symmetric matrix defined by
Alternatively, this may be written as
Properties
As can be seen in the examples section, if A is an n×n Lehmer matrix and B is an m×m Lehmer matrix, then A is a submatrix of B whenever m>n. The values of elements diminish toward zero away from the diagonal, where all elements have value 1.
The inverse of a Lehmer matrix is a tridiagonal matrix, where the superdiagonal and subdiagonal have strictly negative entries. Consider again the n×n A and m×m B Lehmer matrices, where m>n. A rather peculiar property of their inverses is that A−1 is nearly a submatrix of B−1, except for the A−1n,n element, which is not equal to B−1n,n.
A Lehmer matrix of order n has trace n.
Examples
The 2×2, 3×3 and 4×4 Lehmer matrices and their inverses are shown below.
See also
Derrick Henry Lehmer
Hilbert matrix
References
M. Newman and J. Todd, The evaluation of matrix inversion programs, Journal of the Society for Industrial and Applied Mathematics, Volume 6, 1958, pages 466-476.
Matrices
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https://en.wikipedia.org/wiki/Renaming
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Renaming may refer to:
Place names
Geographical renaming
Lists of renamed places
Computing
Batch renaming
Great Renaming
Register renaming
Rename (computing)
Rename (relational algebra)
See also
Rename (disambiguation)
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https://en.wikipedia.org/wiki/Completely%20metrizable%20space
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In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (X, T) for which there exists at least one metric d on X such that (X, d) is a complete metric space and d induces the topology T. The term topologically complete space is employed by some authors as a synonym for completely metrizable space, but sometimes also used for other classes of topological spaces, like completely uniformizable spaces or Čech-complete spaces.
Difference between complete metric space and completely metrizable space
The difference between completely metrizable space and complete metric space is in the words there exists at least one metric in the definition of completely metrizable space, which is not the same as there is given a metric (the latter would yield the definition of complete metric space). Once we make the choice of the metric on a completely metrizable space (out of all the complete metrics compatible with the topology), we get a complete metric space. In other words, the category of completely metrizable spaces is a subcategory of that of topological spaces, while the category of complete metric spaces is not (instead, it is a subcategory of the category of metric spaces). Complete metrizability is a topological property while completeness is a property of the metric.
Examples
The space , the open unit interval, is not a complete metric space with its usual metric inherited from , but it is completely metrizable since it is homeomorphic to .
The space of rational numbers with the subspace topology inherited from is metrizable but not completely metrizable.
Properties
A topological space X is completely metrizable if and only if X is metrizable and a Gδ in its Stone–Čech compactification βX.
A subspace of a completely metrizable space is completely metrizable if and only if it is in .
A countable product of nonempty metrizable spaces is completely metrizable in the product topology if and only if each factor is completely metrizable. Hence, a product of nonempty metrizable spaces is completely metrizable if and only if at most countably many factors have more than one point and each factor is completely metrizable.
For every metrizable space there exists a completely metrizable space containing it as a dense subspace, since every metric space has a completion. In general, there are many such completely metrizable spaces, since completions of a topological space with respect to different metrics compatible with its topology can give topologically different completions.
Completely metrizable abelian topological groups
When talking about spaces with more structure than just topology, like topological groups, the natural meaning of the words “completely metrizable” would arguably be the existence of a complete metric that is also compatible with that extra structure, in addition to inducing its topology. For abelian topological groups and topological vector spaces, “compatible with the e
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https://en.wikipedia.org/wiki/Complex%20line
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In mathematics, a complex line is a one-dimensional affine subspace of a vector space over the complex numbers. A common point of confusion is that while a complex line has dimension one over C (hence the term "line"), it has dimension two over the real numbers R, and is topologically equivalent to a real plane, not a real line.
The "complex plane" commonly refers to the graphical representation of the complex line on the real plane, and is thus generally synonymous with the complex line, and not a two-dimensional space over the complex numbers.
See also
Algebraic geometry
Complex vector
Riemann sphere
References
Geometry
Complex analysis
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https://en.wikipedia.org/wiki/Continuity%20theorem
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In mathematics and statistics, the continuity theorem may refer to one of the following results:
the Lévy continuity theorem on random variables;
the Kolmogorov continuity theorem on stochastic processes.
See also
Continuity (disambiguation)
Continuous mapping theorem
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https://en.wikipedia.org/wiki/Maryland%20Academy%20of%20Technology%20and%20Health%20Sciences
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Maryland Academy of Technology and Health Sciences (MATHS) was a public charter school in Baltimore, Maryland, United States that opened in 2006 and closed in 2018. The school was open to all students, did not charge tuition and provided a comprehensive college preparatory education for students in grades 6 through 12. There was an emphasis on preparing students for future careers in biotechnology and health sciences. The school was funded by the state of Maryland and was founded by Rebekah Ghosh in an effort to help the city's faltering graduation rates and better prepare students for solid careers.
References
Public high schools in Maryland
Public middle schools in Maryland
Charter schools in Maryland
Educational institutions established in 2006
2006 establishments in Maryland
Educational institutions disestablished in 2018
2018 disestablishments in Maryland
Public schools in Baltimore
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https://en.wikipedia.org/wiki/Leandro%20%28footballer%2C%20born%201980%29
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Leandro Lessa Azevedo (born 13 August 1980), simply known as Leandro, is a Brazilian former footballer who played as a striker.
Club statistics
Honours
Corinthians
Brazil Cup: 2002
Tournament Rio - São Paulo: 2002
Fluminense
Rio de Janeiro State League: 2005
São Paulo
Brazilian League: 2006, 2007
Grêmio
Rio Grande do Sul State League: 2010
Vasco da Gama
Brazil Cup: 2011
External links
globoesporte.globo.com
CBF
sambafoot
1980 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Botafogo Futebol Clube (SP) players
Sport Club Corinthians Paulista players
FC Lokomotiv Moscow players
Expatriate men's footballers in Russia
Goiás Esporte Clube players
Fluminense FC players
São Paulo FC players
Tokyo Verdy players
Grêmio Foot-Ball Porto Alegrense players
CR Vasco da Gama players
Fortaleza Esporte Clube players
Expatriate men's footballers in Japan
Campeonato Brasileiro Série A players
Russian Premier League players
J1 League players
J2 League players
Men's association football forwards
Footballers from Ribeirão Preto
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https://en.wikipedia.org/wiki/Correlation%20%28projective%20geometry%29
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In projective geometry, a correlation is a transformation of a d-dimensional projective space that maps subspaces of dimension k to subspaces of dimension , reversing inclusion and preserving incidence. Correlations are also called reciprocities or reciprocal transformations.
In two dimensions
In the real projective plane, points and lines are dual to each other. As expressed by Coxeter,
A correlation is a point-to-line and a line-to-point transformation that preserves the relation of incidence in accordance with the principle of duality. Thus it transforms ranges into pencils, pencils into ranges, quadrangles into quadrilaterals, and so on.
Given a line m and P a point not on m, an elementary correlation is obtained as follows: for every Q on m form the line PQ. The inverse correlation starts with the pencil on P: for any line q in this pencil take the point . The composition of two correlations that share the same pencil is a perspectivity.
In three dimensions
In a 3-dimensional projective space a correlation maps a point to a plane. As stated in one textbook:
If κ is such a correlation, every point P is transformed by it into a plane , and conversely, every point P arises from a unique plane π′ by the inverse transformation κ−1.
Three-dimensional correlations also transform lines into lines, so they may be considered to be collineations of the two spaces.
In higher dimensions
In general n-dimensional projective space, a correlation takes a point to a hyperplane. This context was described by Paul Yale:
A correlation of the projective space P(V) is an inclusion-reversing permutation of the proper subspaces of P(V).
He proves a theorem stating that a correlation φ interchanges joins and intersections, and for any projective subspace W of P(V), the dimension of the image of W under φ is , where n is the dimension of the vector space V used to produce the projective space P(V).
Existence of correlations
Correlations can exist only if the space is self-dual. For dimensions 3 and higher, self-duality is easy to test: A coordinatizing skewfield exists and self-duality fails if and only if the skewfield is not isomorphic to its opposite.
Special types of correlations
Polarity
If a correlation φ is an involution (that is, two applications of the correlation equals the identity: for all points P) then it is called a polarity. Polarities of projective spaces lead to polar spaces, which are defined by taking the collection of all subspace which are contained in their image under the polarity.
Natural correlation
There is a natural correlation induced between a projective space P(V) and its dual P(V∗) by the natural pairing between the underlying vector spaces V and its dual V∗, where every subspace W of V∗ is mapped to its orthogonal complement W⊥ in V, defined as
Composing this natural correlation with an isomorphism of projective spaces induced by a semilinear map produces a correlation of P(V) to itself. In this way, every nondegenerate s
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https://en.wikipedia.org/wiki/Multivariate%20t-distribution
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In statistics, the multivariate t-distribution (or multivariate Student distribution) is a multivariate probability distribution. It is a generalization to random vectors of the Student's t-distribution, which is a distribution applicable to univariate random variables. While the case of a random matrix could be treated within this structure, the matrix t-distribution is distinct and makes particular use of the matrix structure.
Definition
One common method of construction of a multivariate t-distribution, for the case of dimensions, is based on the observation that if and are independent and distributed as and (i.e. multivariate normal and chi-squared distributions) respectively, the matrix is a p × p matrix, and is a constant vector then the random variable has the density
and is said to be distributed as a multivariate t-distribution with parameters . Note that is not the covariance matrix since the covariance is given by (for ).
The constructive definition of a multivariate t-distribution simultaneously serves as a sampling algorithm:
Generate and , independently.
Compute .
This formulation gives rise to the hierarchical representation of a multivariate t-distribution as a scale-mixture of normals: where indicates a gamma distribution with density proportional to , and conditionally follows .
In the special case , the distribution is a multivariate Cauchy distribution.
Derivation
There are in fact many candidates for the multivariate generalization of Student's t-distribution. An extensive survey of the field has been given by Kotz and Nadarajah (2004). The essential issue is to define a probability density function of several variables that is the appropriate generalization of the formula for the univariate case. In one dimension (), with and , we have the probability density function
and one approach is to write down a corresponding function of several variables. This is the basic idea of elliptical distribution theory, where one writes down a corresponding function of variables that replaces by a quadratic function of all the . It is clear that this only makes sense when all the marginal distributions have the same degrees of freedom . With , one has a simple choice of multivariate density function
which is the standard but not the only choice.
An important special case is the standard bivariate t-distribution, p = 2:
Note that .
Now, if is the identity matrix, the density is
The difficulty with the standard representation is revealed by this formula, which does not factorize into the product of the marginal one-dimensional distributions. When is diagonal the standard representation can be shown to have zero correlation but the marginal distributions do not agree with statistical independence.
Cumulative distribution function
The definition of the cumulative distribution function (cdf) in one dimension can be extended to multiple dimensions by defining the following probability (here is a real vector):
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https://en.wikipedia.org/wiki/Q-Vandermonde%20identity
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In mathematics, in the field of combinatorics, the q-Vandermonde identity is a q-analogue of the Chu–Vandermonde identity. Using standard notation for q-binomial coefficients, the identity states that
The nonzero contributions to this sum come from values of j such that the q-binomial coefficients on the right side are nonzero, that is,
Other conventions
As is typical for q-analogues, the q-Vandermonde identity can be rewritten in a number of ways. In the conventions common in applications to quantum groups, a different q-binomial coefficient is used. This q-binomial coefficient, which we denote here by , is defined by
In particular, it is the unique shift of the "usual" q-binomial coefficient by a power of q such that the result is symmetric in q and . Using this q-binomial coefficient, the q-Vandermonde identity can be written in the form
Proof
As with the (non-q) Chu–Vandermonde identity, there are several possible proofs of the q-Vandermonde identity. The following proof uses the q-binomial theorem.
One standard proof of the Chu–Vandermonde identity is to expand the product in two different ways. Following Stanley, we can tweak this proof to prove the q-Vandermonde identity, as well. First, observe that the product
can be expanded by the q-binomial theorem as
Less obviously, we can write
and we may expand both subproducts separately using the q-binomial theorem. This yields
Multiplying this latter product out and combining like terms gives
Finally, equating powers of between the two expressions yields the desired result.
This argument may also be phrased in terms of expanding the product in two different ways, where A and B are operators (for example, a pair of matrices) that "q-commute," that is, that satisfy BA = qAB.
Notes
References
Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, , ,
Combinatorics
Q-analogs
Mathematical identities
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https://en.wikipedia.org/wiki/Koszul%20algebra
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In abstract algebra, a Koszul algebra is a graded -algebra over which the ground field has a linear minimal graded free resolution, i.e., there exists an exact sequence:
Here, is the graded algebra with grading shifted up by , i.e. . The exponents refer to the -fold direct sum. Choosing bases for the free modules in the resolution, the chain maps are given by matrices, and the definition requires the matrix entries to be zero or linear forms.
An example of a Koszul algebra is a polynomial ring over a field, for which the Koszul complex is the minimal graded free resolution of the ground field. There are Koszul algebras whose ground fields have infinite minimal graded free resolutions, e.g, .
The concept is named after the French mathematician Jean-Louis Koszul.
See also
Koszul duality
Complete intersection ring
References
.
.
.
.
Algebras
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https://en.wikipedia.org/wiki/Paul%20J.%20Zak
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Paul J. Zak (born 9 February 1962) is an American neuroeconomist.
Background
Zak graduated with degrees in mathematics and economics from San Diego State University before acquiring a PhD in Economics from the University of Pennsylvania. He is professor at Claremont Graduate University in Southern California. He has studied brain imaging, and was among the first to identify the role of oxytocin in mediating trusting behaviors between unacquainted humans. Zak directs the Center for Neuroeconomics Studies at Claremont Graduate University and is a member of the Neurology Department at Loma Linda University Medical Center. He edited Moral Markets: The Critical Role of Values in the Economy (Princeton University Press, 2008). His book, The Moral Molecule was published in 2012 by Dutton. The book summarizes his findings on oxytocin and discusses the role of oxytocin in human experiences and behaviors such as empathy, altruism, and morality.
Zak's research aims to challenge the thought that people generally are driven primarily to act for what they consider their self-interest, and asks how morality may modulate one's interpretation of what constitutes "self-interest" in one's own personal terms. Methodological questions have arisen in regards to Zak's work, however. Other commentators though have called his work "one of the most revealing experiments in the history of economics."
According to The Moral Molecule, Zak's father was an engineer and he takes an engineering approach to neuroscience, seeking to create predictive models of behavior.
His research and ideas have garnered some criticism, particularly from science writer Ed Yong, who points out that oxytocin administration boosts schadenfreude and envy. Oxytocin administration increases the salience of social cues, suggesting that priming effects in these experiments explain their findings. For example, Zak has shown that endogenous oxytocin release eliminates in-group bias indicating that the critiqued effects are due to supraphysiologic doses of oxytocin coupled with antisocial priming.
Neuroscientist Molly Crockett also disputes Zak's claims, referring to studies that show oxytocin increases gloating, bias at the expense of other groups, and in some cases decreasing cooperation; suggesting oxytocin is as much an "immoral molecule" as 'the moral molecule' Paul Zak claims.
Neuromanagement
Zak has coined the term "neuromanagement" to describe how findings in neuroscience can be used to create organizational cultures that are highly engaging for employees and produce high performance for organizations. He has developed a methodology called Ofactor that quantifies organizational culture and identifies how to continuously improve culture to increase trust, joy, and performance. He has used Ofactor to help organizations ranging from nonprofits to startups to Fortune 50 companies change their cultures. His Ofactor research reflects the approach advocated by his late colleague at Claremont Gr
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https://en.wikipedia.org/wiki/Fresia%2C%20Chile
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Fresia () is a city and commune in Llanquihue Province, Los Lagos Region, Chile.
Demographics
According to the 2002 census of the National Statistics Institute, Fresia spans an area of and has 12,804 inhabitants (6,580 men and 6,224 women). Of these, 6,144 (48%) lived in urban areas and 6,660 (52%) in rural areas. The population fell by 1.6% (209 persons) between the 1992 and 2002 censuses.
Administration
As a commune, Fresia is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2008-2012 alcalde is Bernardo Espinoza Villalobos (PS).
Within the electoral divisions of Chile, Fresia is represented in the Chamber of Deputies by Fidel Espinoza (PS) and Carlos Recondo (UDI) as part of the 56th electoral district, together with Puyehue, Río Negro, Purranque, Puerto Octay, Frutillar, Llanquihue, Puerto Varas and Los Muermos. The commune is represented in the Senate by Camilo Escalona Medina (PS) and Carlos Kuschel Silva (RN) as part of the 17th senatorial constituency (Los Lagos Region).
References
External links
Municipality of Fresia
Populated places in Llanquihue Province
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https://en.wikipedia.org/wiki/Householder%20operator
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In linear algebra, the Householder operator is defined as follows. Let be a finite-dimensional inner product space with inner product and unit vector . Then
is defined by
This operator reflects the vector across a plane given by the normal vector .
It is also common to choose a non-unit vector , and normalize it directly in the Householder operator's expression:
Properties
The Householder operator satisfies the following properties:
It is linear; if is a vector space over a field , then
It is self-adjoint.
If , then it is orthogonal; otherwise, if , then it is unitary.
Special cases
Over a real or complex vector space, the Householder operator is also known as the Householder transformation.
References
Numerical linear algebra
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https://en.wikipedia.org/wiki/Integrodifference%20equation
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In mathematics, an integrodifference equation is a recurrence relation on a function space, of the following form:
where is a sequence in the function space and is the domain of those functions. In most applications, for any , is a probability density function on . Note that in the definition above, can be vector valued, in which case each element of has a scalar valued integrodifference equation associated with it. Integrodifference equations are widely used in mathematical biology, especially theoretical ecology, to model the dispersal and growth of populations. In this case, is the population size or density at location at time , describes the local population growth at location and , is the probability of moving from point to point , often referred to as the dispersal kernel. Integrodifference equations are most commonly used to describe univoltine populations, including, but not limited to, many arthropod, and annual plant species. However, multivoltine populations can also be modeled with integrodifference equations, as long as the organism has non-overlapping generations. In this case, is not measured in years, but rather the time increment between broods.
Convolution kernels and invasion speeds
In one spatial dimension, the dispersal kernel often depends only on the distance between the source and the destination, and can be
written as . In this case, some natural conditions on f and k imply that there is a well-defined
spreading speed for waves of invasion generated from compact initial conditions. The wave speed is often calculated
by studying the linearized equation
where .
This can be written as the convolution
Using a moment-generating-function transformation
it has been shown that the critical wave speed
Other types of equations used to model population dynamics through space include reaction–diffusion equations and metapopulation equations. However, diffusion equations do not as easily allow for the inclusion of explicit dispersal patterns and are only biologically accurate for populations with overlapping generations. Metapopulation equations are different from integrodifference equations in the fact that they break the population down into discrete patches rather than a continuous landscape.
References
Mathematical and theoretical biology
Recurrence relations
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https://en.wikipedia.org/wiki/Fr%C3%A9chet%20surface
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In mathematics, a Fréchet surface is an equivalence class of parametrized surfaces in a metric space. In other words, a Fréchet surface is a way of thinking about surfaces independently of how they are "written down" (parametrized). The concept is named after the French mathematician Maurice Fréchet.
Definitions
Let be a compact 2-dimensional manifold, either closed or with boundary, and let be a metric space. A parametrized surface in is a map
that is continuous with respect to the topology on and the metric topology on Let
where the infimum is taken over all homeomorphisms of to itself. Call two parametrized surfaces and in equivalent if and only if
An equivalence class of parametrized surfaces under this notion of equivalence is called a Fréchet surface; each of the parametrized surfaces in this equivalence class is called a parametrization of the Fréchet surface
Properties
Many properties of parametrized surfaces are actually properties of the Fréchet surface, that is, of the whole equivalence class, and not of any particular parametrization.
For example, given two Fréchet surfaces, the value of is independent of the choice of the parametrizations and and is called the Fréchet distance between the Fréchet surfaces.
References
Metric geometry
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https://en.wikipedia.org/wiki/Convergence%20in%20measure
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Convergence in measure is either of two distinct mathematical concepts both of which generalize
the concept of convergence in probability.
Definitions
Let be measurable functions on a measure space . The sequence is said to converge globally in measure to if for every ,
,
and to converge locally in measure to if for every and every with
,
.
On a finite measure space, both notions are equivalent. Otherwise, convergence in measure can refer to either global convergence in measure or local convergence in measure, depending on the author.
Properties
Throughout, f and fn (n N) are measurable functions X → R.
Global convergence in measure implies local convergence in measure. The converse, however, is false; i.e., local convergence in measure is strictly weaker than global convergence in measure, in general.
If, however, or, more generally, if f and all the fn vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears.
If μ is σ-finite and (fn) converges (locally or globally) to f in measure, there is a subsequence converging to f almost everywhere. The assumption of σ-finiteness is not necessary in the case of global convergence in measure.
If μ is σ-finite, (fn) converges to f locally in measure if and only if every subsequence has in turn a subsequence that converges to f almost everywhere.
In particular, if (fn) converges to f almost everywhere, then (fn) converges to f locally in measure. The converse is false.
Fatou's lemma and the monotone convergence theorem hold if almost everywhere convergence is replaced by (local or global) convergence in measure.
If μ is σ-finite, Lebesgue's dominated convergence theorem also holds if almost everywhere convergence is replaced by (local or global) convergence in measure.
If X = [a,b] ⊆ R and μ is Lebesgue measure, there are sequences (gn) of step functions and (hn) of continuous functions converging globally in measure to f.
If f and fn (n ∈ N) are in Lp(μ) for some p > 0 and (fn) converges to f in the p-norm, then (fn) converges to f globally in measure. The converse is false.
If fn converges to f in measure and gn converges to g in measure then fn + gn converges to f + g in measure. Additionally, if the measure space is finite, fngn also converges to fg.
Counterexamples
Let , μ be Lebesgue measure, and f the constant function with value zero.
The sequence converges to f locally in measure, but does not converge to f globally in measure.
The sequence where and (The first five terms of which are ) converges to 0 globally in measure; but for no x does fn(x) converge to zero. Hence (fn) fails to converge to f almost everywhere.
The sequence converges to f almost everywhere and globally in measure, but not in the p-norm for any .
Topology
There is a topology, called the topology of (local) convergence in measure, on the collection of measurable functions from X such that local convergence in measure corre
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https://en.wikipedia.org/wiki/Hermann%20Schubert
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Hermann Cäsar Hannibal Schubert (22 May 1848 – 20 July 1911) was a German mathematician.
Schubert was one of the leading developers of enumerative geometry, which considers those parts of algebraic geometry that involve a finite number of solutions. In 1874, Schubert won a prize for solving a question posed by Zeuthen. Schubert calculus was named after him.
Schubert tutored Adolf Hurwitz at the Realgymnasium Andreanum in Hildesheim, Hanover, and arranged for Hurwitz to study under Felix Klein at University.
See also
Schubert cycle or Schubert variety
Schubert polynomial
Publications
References
Werner Burau and Bodo Renschuch, "Ergänzungen zur Biographie von Hermann Schubert," (Complements to the biography of Hermann Schubert,) Mitt. Math. Ges. Hamb. 13, pp. 63–65 (1993), ISSN 0340-4358.
External links
1848 births
1911 deaths
19th-century German mathematicians
20th-century German mathematicians
Algebraic geometers
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https://en.wikipedia.org/wiki/Copula
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Copula may refer to:
Copula (linguistics), a word used to link subject and predicate
Copula (music), a type of polyphonic texture similar to organum
Copula (probability theory), a function linking marginal variables into a multivariate distribution
Copula (cnidarian), a genus of box jellyfish
Beatmania IIDX 23: Copula, a video game
See also
Copula linguae, an embryonic structure of the tongue
Copulas in signal processing
Copulation (zoology)
Cupola, an architectural term
Cupola furnace, a foundry device
Cupula (disambiguation)
Cupule (disambiguation)
Indo-European copula
Romance copula
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https://en.wikipedia.org/wiki/Golden%E2%80%93Thompson%20inequality
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In physics and mathematics, the Golden–Thompson inequality is a trace inequality between exponentials of symmetric and Hermitian matrices proved independently by and . It has been developed in the context of statistical mechanics, where it has come to have a particular significance.
Statement
The Golden–Thompson inequality states that for (real) symmetric or (complex) Hermitian matrices A and B, the following trace inequality holds:
This inequality is well defined, since the quantities on either side are real numbers. For the expression on right hand side of the inequality, this can be seen by rewriting it as using the cyclic property of the trace.
Motivation
The Golden–Thompson inequality can be viewed as a generalization of a stronger statement for real numbers. If a and b are two real numbers, then the exponential of a+b is the product of the exponential of a with the exponential of b:
If we replace a and b with commuting matrices A and B, then the same inequality holds.
This relationship is not true if A and B do not commute. In fact, proved that if A and B are two Hermitian matrices for which the Golden–Thompson inequality is verified as an equality, then the two matrices commute. The Golden–Thompson inequality shows that, even though and are not equal, they are still related by an inequality.
Generalizations
The Golden–Thompson inequality generalizes to any unitarily invariant norm. If A and B are Hermitian matrices and is a unitarily invariant norm, then
The standard Golden–Thompson inequality is a special case of the above inequality, where the norm is the Schatten norm with . Since and are both positive semidefinite matrices, and .
The inequality has been generalized to three matrices by and furthermore to any arbitrary number of Hermitian matrices by . A naive attempt at generalization does not work: the inequality
is false. For three matrices, the correct generalization takes the following form:
where the operator is the derivative of the matrix logarithm given by .
Note that, if and commute, then , and the inequality for three matrices reduces to the original from Golden and Thompson.
used the Kostant convexity theorem to generalize the Golden–Thompson inequality to all compact Lie groups.
References
External links
Linear algebra
Matrix theory
Inequalities
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https://en.wikipedia.org/wiki/Spectral%20gap
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In mathematics, the spectral gap is the difference between the moduli of the two largest eigenvalues of a matrix or operator; alternately, it is sometimes taken as the smallest non-zero eigenvalue. Various theorems relate this difference to other properties of the system.
See also
Cheeger constant (graph theory)
Cheeger constant (Riemannian geometry)
Eigengap
Spectral gap (physics)
Spectral radius
References
External links
Spectral theory
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https://en.wikipedia.org/wiki/G%C3%A5rding%27s%20inequality
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In mathematics, Gårding's inequality is a result that gives a lower bound for the bilinear form induced by a real linear elliptic partial differential operator. The inequality is named after Lars Gårding.
Statement of the inequality
Let be a bounded, open domain in -dimensional Euclidean space and let denote the Sobolev space of -times weakly differentiable functions with weak derivatives in . Assume that satisfies the -extension property, i.e., that there exists a bounded linear operator such that for all .
Let L be a linear partial differential operator of even order 2k, written in divergence form
and suppose that L is uniformly elliptic, i.e., there exists a constant θ > 0 such that
Finally, suppose that the coefficients Aαβ are bounded, continuous functions on the closure of Ω for |α| = |β| = k and that
Then Gårding's inequality holds: there exist constants C > 0 and G ≥ 0
where
is the bilinear form associated to the operator L.
Application: the Laplace operator and the Poisson problem
Be careful, in this application, Garding's Inequality seems useless here as the final result is a direct consequence of Poincaré's Inequality, or Friedrich Inequality. (See talk on the article).
As a simple example, consider the Laplace operator Δ. More specifically, suppose that one wishes to solve, for f ∈ L2(Ω) the Poisson equation
where Ω is a bounded Lipschitz domain in Rn. The corresponding weak form of the problem is to find u in the Sobolev space H01(Ω) such that
where
The Lax–Milgram lemma ensures that if the bilinear form B is both continuous and elliptic with respect to the norm on H01(Ω), then, for each f ∈ L2(Ω), a unique solution u must exist in H01(Ω). The hypotheses of Gårding's inequality are easy to verify for the Laplace operator Δ, so there exist constants C and G ≥ 0
Applying the Poincaré inequality allows the two terms on the right-hand side to be combined, yielding a new constant K > 0 with
which is precisely the statement that B is elliptic. The continuity of B is even easier to see: simply apply the Cauchy–Schwarz inequality and the fact that the Sobolev norm is controlled by the L2 norm of the gradient.
References
(Theorem 9.17)
Theorems in functional analysis
Inequalities
Partial differential equations
Sobolev spaces
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https://en.wikipedia.org/wiki/Riggatron
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A Riggatron is a magnetic confinement fusion reactor design created by Robert W. Bussard in the late 1970s. It is a tokamak on the basis of its magnetic geometry, but some unconventional engineering choices were made. In particular, Riggatron used copper magnets positioned inside the lithium blanket, which was hoped to lead to much lower construction costs. Originally referred to as the Demountable Tokamak Fusion Core (DTFC), the name was later changed to refer to the Riggs Bank, which funded development along with Bob Guccione, publisher of the adult magazine Penthouse.
Conventional tokamak design
In a conventional tokamak design the confinement magnets are arranged outside a "blanket" of liquid lithium. The lithium serves two purposes, one is to absorb the neutrons from the fusion reactions and produce tritium which is then used to fuel the reactor, and as a secondary role, as shielding to prevent those neutrons from reaching the magnets. Without the lithium blanket the neutrons degrade the magnets quite quickly.
This arrangement has two disadvantages. One is that a magnetic field must be produced not only in the plasma, where it is needed, but also in the blanket, where it is not, significantly raising the construction costs. The other is that the core, where the magnetic coils penetrate the machine along its axis, must be large enough to contain the shielding, which limits the achievable aspect ratio. A higher aspect ratio generally results in better performance.
Riggatron improvement
The Riggatron re-arranged the layout of the conventional design, reducing the role of the lithium to producing tritium only. The magnets were to be directly exposed on the inside of the reactor core, bearing the full neutron flux. This precluded the use of superconducting magnets, and even copper magnets would have to be disposed in as little as 30 days of operation. The Riggatron was laid out to make this core replacement as easy and fast as possible. After removal and replacement, the magnets would then be melted down and reprocessed. Although this process would be costly, the smaller magnetized volume (major radius only ), the larger aspect ratio, and the reduction in complexity by avoiding superconducting magnets was a tradeoff that would, it was hoped, pay off.
Another advantage of the parameters chosen was that ignition appeared to be possible with ohmic heating alone, as opposed to more expensive systems like ion injection normally required. The first proposal, made in the late 1970s, projected that the device would be able to produce about three or four times the power in fusion reactions as it used in powering the heaters and magnets. This represents a fusion energy gain factor (or simply "fusion gain" or Q) of three or four. The project was never completed as Guccioni was unable to secure the $150 million needed to build the full-sized device (much of which would have been for a large homopolar generator).
Fusion research establishment considerati
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https://en.wikipedia.org/wiki/Yoav%20Freund
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Yoav Freund (; born 1961) is an Israeli professor of computer science at the University of California San Diego who mainly works on machine learning, probability theory and related fields and applications.
He is best known for his work on the AdaBoost algorithm, an ensemble learning algorithm which is used to combine many "weak" learning machines to create a more robust one. He and Robert Schapire received the Gödel prize in 2003 for their joint work on AdaBoost.
He is an alumnus of the prestigious Talpiot program of the Israeli army.
Selected works
References
External links
Freund's homepage at UCSD
Living people
American computer scientists
Gödel Prize laureates
University of California, San Diego faculty
University of California, Santa Cruz alumni
1961 births
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https://en.wikipedia.org/wiki/Trygve%20Nygaard
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Trygve Nygaard (born 19 August 1975) is a retired Norwegian footballer.
External links
100% Fotball (Norwegian Premier League statistics)
nifs.no Profile
1975 births
Living people
Sportspeople from Haugesund
Footballers from Rogaland
FK Haugesund players
Viking FK players
Norwegian men's footballers
SK Vard Haugesund players
Eliteserien players
Men's association football midfielders
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https://en.wikipedia.org/wiki/Divided%20power%20structure
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In mathematics, specifically commutative algebra, a divided power structure is a way of making expressions of the form meaningful even when it is not possible to actually divide by .
Definition
Let A be a commutative ring with an ideal I. A divided power structure (or PD-structure, after the French puissances divisées) on I is a collection of maps for n = 0, 1, 2, ... such that:
and for , while for n > 0.
for .
for .
for , where is an integer.
for and , where is an integer.
For convenience of notation, is often written as when it is clear what divided power structure is meant.
The term divided power ideal refers to an ideal with a given divided power structure, and divided power ring refers to a ring with a given ideal with divided power structure.
Homomorphisms of divided power algebras are ring homomorphisms that respects the divided power structure on its source and target.
Examples
The free divided power algebra over on one generator:
If A is an algebra over then every ideal I has a unique divided power structure where Indeed, this is the example which motivates the definition in the first place.
If M is an A-module, let denote the symmetric algebra of M over A. Then its dual has a canonical structure of divided power ring. In fact, it is canonically isomorphic to a natural completion of (see below) if M has finite rank.
Constructions
If A is any ring, there exists a divided power ring
consisting of divided power polynomials in the variables
that is sums of divided power monomials of the form
with . Here the divided power ideal is the set of divided power polynomials with constant coefficient 0.
More generally, if M is an A-module, there is a universal A-algebra, called
with PD ideal
and an A-linear map
(The case of divided power polynomials is the special case in which M is a free module over A of finite rank.)
If I is any ideal of a ring A, there is a universal construction which extends A with divided powers of elements of I to get a divided power envelope of I in A.
Applications
The divided power envelope is a fundamental tool in the theory of PD differential operators and crystalline cohomology, where it is used to overcome technical difficulties which arise in positive characteristic.
The divided power functor is used in the construction of co-Schur functors.
See also
Crystalline cohomology
References
p-adic derived de Rham cohomology - contains excellent material on PD-polynomial rings and PD-envelopes
What's the name for the analogue of divided power algebras for x^i/i - contains useful equivalence to divided power algebras as dual algebras
Commutative algebra
Polynomials
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https://en.wikipedia.org/wiki/Generalizations%20of%20Fibonacci%20numbers
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In mathematics, the Fibonacci numbers form a sequence defined recursively by:
That is, after two starting values, each number is the sum of the two preceding numbers.
The Fibonacci sequence has been studied extensively and generalized in many ways, for example, by starting with other numbers than 0 and 1, by adding more than two numbers to generate the next number, or by adding objects other than numbers.
Extension to negative integers
Using , one can extend the Fibonacci numbers to negative integers. So we get:
... −8, 5, −3, 2, −1, 1, 0, 1, 1, 2, 3, 5, 8, ...
and .
See also Negafibonacci coding.
Extension to all real or complex numbers
There are a number of possible generalizations of the Fibonacci numbers which include the real numbers (and sometimes the complex numbers) in their domain. These each involve the golden ratio , and are based on Binet's formula
The analytic function
has the property that for even integers . Similarly, the analytic function:
satisfies for odd integers .
Finally, putting these together, the analytic function
satisfies for all integers .
Since for all complex numbers , this function also provides an extension of the Fibonacci sequence to the entire complex plane. Hence we can calculate the generalized Fibonacci function of a complex variable, for example,
Vector space
The term Fibonacci sequence is also applied more generally to any function from the integers to a field for which . These functions are precisely those of the form , so the Fibonacci sequences form a vector space with the functions and as a basis.
More generally, the range of may be taken to be any abelian group (regarded as a Z-module). Then the Fibonacci sequences form a 2-dimensional Z-module in the same way.
Similar integer sequences
Fibonacci integer sequences
The 2-dimensional -module of Fibonacci integer sequences consists of all integer sequences satisfying . Expressed in terms of two initial values we have:
where is the golden ratio.
The ratio between two consecutive elements converges to the golden ratio, except in the case of the sequence which is constantly zero and the sequences where the ratio of the two first terms is .
The sequence can be written in the form
in which if and only if . In this form the simplest non-trivial example has , which is the sequence of Lucas numbers:
We have and . The properties include:
Every nontrivial Fibonacci integer sequence appears (possibly after a shift by a finite number of positions) as one of the rows of the Wythoff array. The Fibonacci sequence itself is the first row, and a shift of the Lucas sequence is the second row.
See also Fibonacci integer sequences modulo n.
Lucas sequences
A different generalization of the Fibonacci sequence is the Lucas sequences of the kind defined as follows:
where the normal Fibonacci sequence is the special case of and . Another kind of Lucas sequence begins with , . Such sequences have applications in number theory and primali
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https://en.wikipedia.org/wiki/Peter%20Cameron%20%28mathematician%29
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Peter Jephson Cameron FRSE (born 23 January 1947) is an Australian mathematician who works in group theory, combinatorics, coding theory, and model theory. He is currently half-time Professor of Mathematics at the University of St Andrews, and Emeritus Professor at Queen Mary University of London.
Cameron received a B.Sc. from the University of Queensland and a D.Phil. in 1971 from the University of Oxford as a Rhodes Scholar, with Peter M. Neumann as his supervisor. Subsequently, he was a Junior Research Fellow and later a Tutorial Fellow at Merton College, Oxford, and also lecturer at Bedford College, London.
Work
Cameron specialises in algebra and combinatorics; he has written books about combinatorics, algebra, permutation groups, and logic, and has produced over 350 academic papers. In 1988, he posed the Cameron–Erdős conjecture with Paul Erdős.
Honours and awards
He was awarded the London Mathematical Society's Whitehead Prize in 1979 and is joint winner of the 2003 Euler Medal. In 2008, he was selected as the Forder Lecturer of the LMS and New Zealand Mathematical Society. In 2018 he was elected a Fellow of the Royal Society of Edinburgh.
Books
Notes
References
Short biography
External links
Home page at Queen Mary University of London
Home page at University of St Andrews
Peter Cameron's 60th birthday conference
Theorems by Peter Cameron at Theorem of the Day
Peter Cameron's blog
Academics of Queen Mary University of London
Academics of the University of St Andrews
1947 births
Living people
Australian Rhodes Scholars
Algebraists
Coding theorists
Combinatorialists
Alumni of Balliol College, Oxford
University of Queensland alumni
Whitehead Prize winners
Model theorists
20th-century Australian mathematicians
21st-century Australian mathematicians
Group theorists
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https://en.wikipedia.org/wiki/Compound%20of%20two%20tetrahedra
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In geometry, a compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra.
Stellated octahedron
There is only one uniform polyhedral compound, the stellated octahedron, which has octahedral symmetry, order 48. It has a regular octahedron core, and shares the same 8 vertices with the cube.
If the edge crossings were treated as their own vertices, the compound would have identical surface topology to the rhombic dodecahedron; were face crossings also considered edges of their own the shape would effectively become a nonconvex triakis octahedron.
Lower symmetry constructions
There are lower symmetry variations on this compound, based on lower symmetry forms of the tetrahedron.
A facetting of a rectangular cuboid, creating compounds of two tetragonal or two rhombic disphenoids, with a bipyramid or rhombic fusil cores. This is first in a set of uniform compound of two antiprisms.
A facetting of a trigonal trapezohedron creates a compound of two right triangular pyramids with a triangular antiprism core. This is first in a set of compounds of two pyramids positioned as point reflections of each other.
Other compounds
If two regular tetrahedra are given the same orientation on the 3-fold axis, a different compound is made, with D3h, [3,2] symmetry, order 12.
Other orientations can be chosen as 2 tetrahedra within the compound of five tetrahedra and compound of ten tetrahedra the latter of which can be seen as a hexagrammic pyramid:
See also
Compound of cube and octahedron
Compound of dodecahedron and icosahedron
Compound of small stellated dodecahedron and great dodecahedron
Compound of great stellated dodecahedron and great icosahedron
References
Cundy, H. and Rollett, A. "Five Tetrahedra in a Dodecahedron". §3.10.8 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 139-141, 1989.
External links
Compounds of Polyhedra VRML model:
Polyhedral compounds
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https://en.wikipedia.org/wiki/Compound%20of%20dodecahedron%20and%20icosahedron
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In geometry, this polyhedron can be seen as either a polyhedral stellation or a compound.
As a compound
It can be seen as the compound of an icosahedron and dodecahedron. It is one of four compounds constructed from a Platonic solid or Kepler-Poinsot solid, and its dual.
It has icosahedral symmetry (Ih) and the same vertex arrangement as a rhombic triacontahedron.
This can be seen as the three-dimensional equivalent of the compound of two pentagons ({10/2} "decagram"); this series continues into the fourth dimension as the compound of 120-cell and 600-cell and into higher dimensions as compounds of hyperbolic tilings.
As a stellation
This polyhedron is the first stellation of the icosidodecahedron, and given as Wenninger model index 47.
The stellation facets for construction are:
In popular culture
In the film Tron (1982), the character Bit took this shape when not speaking.
In the cartoon series Steven Universe (2013-2019), Steven's shield bubble, briefly used in the episode Change Your Mind, had this shape.
See also
Compound of two tetrahedra
Compound of cube and octahedron
Compound of small stellated dodecahedron and great dodecahedron
Compound of great stellated dodecahedron and great icosahedron
References
External links
Polyhedral stellation
Polyhedral compounds
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https://en.wikipedia.org/wiki/Costa%27s%20minimal%20surface
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In mathematics, Costa's minimal surface, is an embedded minimal surface discovered in 1982 by the Brazilian mathematician Celso José da Costa. It is also a surface of finite topology, which means that it can be formed by puncturing a compact surface. Topologically, it is a thrice-punctured torus.
Until its discovery, the plane, helicoid and the catenoid were believed to be the only embedded minimal surfaces that could be formed by puncturing a compact surface. The Costa surface evolves from a torus, which is deformed until the planar end becomes catenoidal. Defining these surfaces on rectangular tori of arbitrary dimensions yields the Costa surface. Its discovery triggered research and discovery into several new surfaces and open conjectures in topology.
The Costa surface can be described using the Weierstrass zeta and the Weierstrass elliptic functions.
References
Ph.D. Thesis, IMPA, Rio de Janeiro, Brazil.
Bol. Soc. Bras. Mat. 15, 47–54.
Differential geometry
Minimal surfaces
Articles containing video clips
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