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https://en.wikipedia.org/wiki/API-Calculus
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API Calculus is a program that solves calculus problems using operating systems within a device that solves calculus problems. In 1989, the PI- Calculus was created by Robin Milner and was very successful throughout the years. The PI Calculus is an extension of the process algebra CCS, a tool that has algebraic languages that are specific to processing and formulating statements. The PI Calculus provides a formal theory for modeling systems and reasoning about their behaviors. In the PI Calculus there are two specific variables such as name and processes. But it was not until 2002 when Shahram Rahimi decided to create an upgraded version of the PI- Calculus and call it the API Calculus. Milner claimed the detailed characteristics of the API Calculus to be its "Communication Ability, Capacity for Cooperation, Capacity for Reasoning and Learning, Adaptive Behavior and Trustworthiness." The main purpose of creating this mobile advancement is to better network and communicate with other operators while completing a task. Unfortunately, the API Calculus is not perfect and has faced a problem with its security system. The language has seven features that was created within the device that the PI Calculus does not have. Since this program is so advanced by the way the software was created and the different abilities that are offered in the program, it is required to be converted to other programming languages so it can be used on various devices and other computing languages. Although the API Calculus is currently being used by various other programming languages, modifications are still being done since the security on the API Calculus is causing problems to users.
What Does It Do?
The API Calculus is the main demonstration for modeling migration, intelligence, natural grouping and security in agent-based systems. This calculus programming language is usually used in various other program languages such as Java. In Java, a famous programming language used by various corporations such as IBM, TCS, and Google, the API Calculus is commonly used to solve equations and programs involving calculus.
Features
The API Calculus has a wide variety of features those similar to the PI Calculus but has new and improved features such as:
accepts processes to be passed over communication links
natural grouping of mobile processes is addressed
features calculus dictionary
includes milieu - a level of abstraction that is between a single mobile agents (combination of computer software and data that is able to transfer from one computer to another independently and still able to work on the most recent computer that data was transferred to) and the device as a whole. It is a very restricted environment that involves zero or many agents or other milieus that work closely together to solve computer based problems.
ability of grouping together hosts ( a physical node - connection point - or software program ) and processes ( computer program that is runni
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https://en.wikipedia.org/wiki/List%20of%20formulas%20in%20Riemannian%20geometry
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This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise.
Christoffel symbols, covariant derivative
In a smooth coordinate chart, the Christoffel symbols of the first kind are given by
and the Christoffel symbols of the second kind by
Here is the inverse matrix to the metric tensor . In other words,
and thus
is the dimension of the manifold.
Christoffel symbols satisfy the symmetry relations
or, respectively, ,
the second of which is equivalent to the torsion-freeness of the Levi-Civita connection.
The contracting relations on the Christoffel symbols are given by
and
where |g| is the absolute value of the determinant of the metric tensor . These are useful when dealing with divergences and Laplacians (see below).
The covariant derivative of a vector field with components is given by:
and similarly the covariant derivative of a -tensor field with components is given by:
For a -tensor field with components this becomes
and likewise for tensors with more indices.
The covariant derivative of a function (scalar) is just its usual differential:
Because the Levi-Civita connection is metric-compatible, the covariant derivatives of metrics vanish,
as well as the covariant derivatives of the metric's determinant (and volume element)
The geodesic starting at the origin with initial speed has Taylor expansion in the chart:
Curvature tensors
Definitions
(3,1) Riemann curvature tensor
(3,1) Riemann curvature tensor
Ricci curvature
Scalar curvature
Traceless Ricci tensor
(4,0) Riemann curvature tensor
(4,0) Weyl tensor
Einstein tensor
Identities
Basic symmetries
The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero:
The Ricci tensor, the Einstein tensor, and the traceless Ricci tensor are symmetric 2-tensors:
First Bianchi identity
Second Bianchi identity
Contracted second Bianchi identity
Twice-contracted second Bianchi identity
Equivalently:
Ricci identity
If is a vector field then
which is just the definition of the Riemann tensor. If is a one-form then
More generally, if is a (0,k)-tensor field then
Remarks
A classical result says that if and only if is locally conformally flat, i.e. if and only if can be covered by smooth coordinate charts relative to which the metric tensor is of the form for some function on the chart.
Gradient, divergence, Laplace–Beltrami operator
The gradient of a function is obtained by raising the index of the differential , whose components are given by:
The divergence of a vector field with components is
The Laplace–Beltrami operator acting on a function is given by the divergence of the gradient:
The divergence of an antisymmetric tensor field of type simplifies to
The Hessian of a map is given by
Kulkarni–Nomizu product
The Kulka
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https://en.wikipedia.org/wiki/Calculus%20on%20manifolds
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Calculus on manifolds may refer to:
Calculus on Manifolds, an undergraduate real analysis and differential geometry textbook by Michael Spivak
The generalization of differential and Integral calculus to differentiable manifolds. For this, see Calculus on Euclidean space#Calculus on manifolds.
See also
Differential geometry
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https://en.wikipedia.org/wiki/Extremally%20disconnected%20space
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In mathematics, an extremally disconnected space is a topological space in which the closure of every open set is open. (The term "extremally disconnected" is correct, even though the word "extremally" does not appear in most dictionaries, and is sometimes mistaken by spellcheckers for the homophone extremely disconnected.)
An extremally disconnected space that is also compact and Hausdorff is sometimes called a Stonean space. This is not the same as a Stone space, which is a totally disconnected compact Hausdorff space. Every Stonean space is a Stone space, but not vice versa. In the duality between Stone spaces and Boolean algebras, the Stonean spaces correspond to the complete Boolean algebras.
An extremally disconnected first-countable collectionwise Hausdorff space must be discrete. In particular, for metric spaces, the property of being extremally disconnected (the closure of every open set is open) is equivalent to the property of being discrete (every set is open).
Examples
Every discrete space is extremally disconnected. Every indiscrete space is both extremally disconnected and connected.
The Stone–Čech compactification of a discrete space is extremally disconnected.
The spectrum of an abelian von Neumann algebra is extremally disconnected.
Any commutative AW*-algebra is isomorphic to where is extremally disconnected, compact and Hausdorff.
Any infinite space with the cofinite topology is both extremally disconnected and connected. More generally, every hyperconnected space is extremally disconnected.
The space on three points with base provides a finite example of a space that is both extremally disconnected and connected. Another example is given by the sierpinski space, since it is finite, connected, and hyperconnected.
Equivalent characterizations
A theorem due to says that the projective objects of the category of compact Hausdorff spaces are exactly the extremally disconnected compact Hausdorff spaces. A simplified proof of this fact is given by .
A compact Hausdorff space is extremally disconnected if and only if it is a retract of the Stone–Čech compactification of a discrete space.
Applications
proves the Riesz–Markov–Kakutani representation theorem by reducing it to the case of extremally disconnected spaces, in which case the representation theorem can be proved by elementary means.
See also
Totally disconnected space
References
Properties of topological spaces
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https://en.wikipedia.org/wiki/Paranormal%20space
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In mathematics, in the realm of topology, a paranormal space is a topological space in which every countable discrete collection of closed sets has a locally finite open expansion.
See also
– a topological space in which every two disjoint closed sets have disjoint open neighborhoods
– a topological space in which every open cover admits an open locally finite refinement
References
Properties of topological spaces
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https://en.wikipedia.org/wiki/Rational%20representation
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In mathematics, in the representation theory of algebraic groups, a linear representation of an algebraic group is said to be rational if, viewed as a map from the group to the general linear group, it is a rational map of algebraic varieties.
Finite direct sums and products of rational representations are rational.
A rational module is a module that can be expressed as a sum (not necessarily direct) of rational representations.
References
Springer Online Reference Works: Rational Representation
Representation theory of algebraic groups
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https://en.wikipedia.org/wiki/Observable%20subgroup
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In mathematics, in the representation theory of algebraic groups, an observable subgroup is an algebraic subgroup of a linear algebraic group whose every finite-dimensional rational representation arises as the restriction to the subgroup of a finite-dimensional rational representation of the whole group.
An equivalent formulation, in case the base field is closed, is that K is an observable subgroup of G if and only if the quotient variety G/K is a quasi-affine variety.
Some basic facts about observable subgroups:
Every normal algebraic subgroup of an algebraic group is observable.
Every observable subgroup of an observable subgroup is observable.
External links
Extensions of Representations of algebraic linear groups
Representation theory of algebraic groups
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https://en.wikipedia.org/wiki/Ambient%20construction
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In conformal geometry, the ambient construction refers to a construction of Charles Fefferman and Robin Graham for which a conformal manifold of dimension n is realized (ambiently) as the boundary of a certain Poincaré manifold, or alternatively as the celestial sphere of a certain pseudo-Riemannian manifold.
The ambient construction is canonical in the sense that it is performed only using the conformal class of the metric: it is conformally invariant. However, the construction only works asymptotically, up to a certain order of approximation. There is, in general, an obstruction to continuing this extension past the critical order. The obstruction itself is of tensorial character, and is known as the (conformal) obstruction tensor. It is, along with the Weyl tensor, one of the two primitive invariants in conformal differential geometry.
Aside from the obstruction tensor, the ambient construction can be used to define a class of conformally invariant differential operators known as the GJMS operators.
A related construction is the tractor bundle.
Overview
The model flat geometry for the ambient construction is the future null cone in Minkowski space, with the origin deleted. The celestial sphere at infinity is the conformal manifold M, and the null rays in the cone determine a line bundle over M. Moreover, the null cone carries a metric which degenerates in the direction of the generators of the cone.
The ambient construction in this flat model space then asks: if one is provided with such a line bundle, along with its degenerate metric, to what extent is it possible to extend the metric off the null cone in a canonical way, thus recovering the ambient Minkowski space? In formal terms, the degenerate metric supplies a Dirichlet boundary condition for the extension problem and, as it happens, the natural condition is for the extended metric to be Ricci flat (because of the normalization of the normal conformal connection.)
The ambient construction generalizes this to the case when M is conformally curved, first by constructing a natural null line bundle N with a degenerate metric, and then solving the associated Dirichlet problem on N × (-1,1).
Details
This section provides an overview of the construction, first of the null line bundle, and then of its ambient extension.
The null line bundle
Suppose that M is a conformal manifold, and that [g] denotes the conformal metric defined on M. Let π : N → M denote the tautological subbundle of T*M ⊗ T*M defined by all representatives of the conformal metric. In terms of a fixed background metric g0, N consists of all positive multiples ω2g0 of the metric. There is a natural action of R+ on N, given by
Moreover, the total space of N carries a tautological degenerate metric, for if p is a point of the fibre of π : N → M corresponding to the conformal representative gp, then let
This metric degenerates along the vertical directions. Furthermore, it is homogeneous of degree 2 under the R
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https://en.wikipedia.org/wiki/Bach%20tensor
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In differential geometry and general relativity, the Bach tensor is a trace-free tensor of rank 2 which is conformally invariant in dimension . Before 1968, it was the only known conformally invariant tensor that is algebraically independent of the Weyl tensor. In abstract indices the Bach tensor is given by
where is the Weyl tensor, and the Schouten tensor given in terms of the Ricci tensor and scalar curvature by
See also
Cotton tensor
Obstruction tensor
References
Further reading
Arthur L. Besse, Einstein Manifolds. Springer-Verlag, 2007. See Ch.4, §H "Quadratic Functionals".
Demetrios Christodoulou, Mathematical Problems of General Relativity I. European Mathematical Society, 2008. Ch.4 §2 "Sketch of the proof of the global stability of Minkowski spacetime".
Yvonne Choquet-Bruhat, General Relativity and the Einstein Equations. Oxford University Press, 2011. See Ch.XV §5 "Christodoulou-Klainerman theorem" which notes the Bach tensor is the "dual of the Coton tensor which vanishes for conformally flat metrics".
Thomas W. Baumgarte, Stuart L. Shapiro, Numerical Relativity: Solving Einstein's Equations on the Computer. Cambridge University Press, 2010. See Ch.3.
Tensors
Tensors in general relativity
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https://en.wikipedia.org/wiki/Rothe%E2%80%93Hagen%20identity
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In mathematics, the Rothe–Hagen identity is a mathematical identity valid for all complex numbers () except where its denominators vanish:
It is a generalization of Vandermonde's identity, and is named after Heinrich August Rothe and Johann Georg Hagen.
References
.
. See especially pp. 89–91.
. As cited by .
.
. As cited by .
Factorial and binomial topics
Mathematical identities
Complex analysis
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https://en.wikipedia.org/wiki/Notre-Dame-du-Mont-Carmel%2C%20Quebec
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Notre-Dame-du-Mont-Carmel (Parish municipality)
in the Mauricie region of the province of Quebec in Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Notre-Dame-du-Mont-Carmel had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Government
The mayor is the municipality's highest elected official. Officially, mayoral elections in Notre-Dame-du-Mont-Carmel are on a non-partisan basis. The following list may be incomplete.
Photos
Related article
La Gabelle Generating Station
References
External links
https://web.archive.org/web/20110529035007/http://mont-carmel.org/Default.aspx?idPage=1
Parish municipalities in Quebec
Incorporated places in Mauricie
Les Chenaux Regional County Municipality
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https://en.wikipedia.org/wiki/Grosshans%20subgroup
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In mathematics, in the representation theory of algebraic groups, a Grosshans subgroup, named after Frank Grosshans, is an algebraic subgroup of an algebraic group that is an observable subgroup for which the ring of functions on the quotient variety is finitely generated.
References
External links
Invariants of Unipotent subgroups
Representation theory of algebraic groups
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https://en.wikipedia.org/wiki/Chebyshev%20function
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In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function or is given by
where denotes the natural logarithm, with the sum extending over all prime numbers that are less than or equal to .
The second Chebyshev function is defined similarly, with the sum extending over all prime powers not exceeding
where is the von Mangoldt function. The Chebyshev functions, especially the second one , are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, (see the exact formula below.) Both Chebyshev functions are asymptotic to , a statement equivalent to the prime number theorem.
Tchebycheff function, Chebyshev utility function, or weighted Tchebycheff scalarizing function is used when one has several functions to be minimized and one wants to "scalarize" them to a single function:
By minimizing this function for different values of , one obtains every point on a Pareto front, even in the nonconvex parts. Often the functions to be minimized are not but for some scalars . Then
All three functions are named in honour of Pafnuty Chebyshev.
Relationships
The second Chebyshev function can be seen to be related to the first by writing it as
where is the unique integer such that and . The values of are given in . A more direct relationship is given by
Note that this last sum has only a finite number of non-vanishing terms, as
The second Chebyshev function is the logarithm of the least common multiple of the integers from 1 to .
Values of for the integer variable are given at .
Relationships between and
Source:
The following theorem relates the two quotients and .
Theorem: For , we have
Note: This inequality implies that
In other words, if one of the or tends to a limit then so does the other, and the two limits are equal.
Proof: Since , we find that
But from the definition of we have the trivial inequality
so
Lastly, divide by to obtain the inequality in the theorem.
Asymptotics and bounds
The following bounds are known for the Chebyshev functions: (in these formulas is the th prime number; , , etc.)
Furthermore, under the Riemann hypothesis,
for any .
Upper bounds exist for both and such that
for any .
An explanation of the constant 1.03883 is given at .
The exact formula
In 1895, Hans Carl Friedrich von Mangoldt proved an explicit expression for as a sum over the nontrivial zeros of the Riemann zeta function:
(The numerical value of is .) Here runs over the nontrivial zeros of the zeta function, and is the same as , except that at its jump discontinuities (the prime powers) it takes the value halfway between the values to the left and the right:
From the Taylor series for the logarithm, the last term in the explicit formula can be understood as a summation of over the trivial zeros of the zeta function, , i.e.
Similarly,
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https://en.wikipedia.org/wiki/Implicational%20propositional%20calculus
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In mathematical logic, the implicational propositional calculus is a version of classical propositional calculus which uses only one connective, called implication or conditional. In formulas, this binary operation is indicated by "implies", "if ..., then ...", "→", "", etc..
Functional (in)completeness
Implication alone is not functionally complete as a logical operator because one cannot form all other two-valued truth functions from it.
For example, the two-place truth function that always returns false is not definable from → and arbitrary sentence variables: any formula constructed from → and propositional variables must receive the value true when all of its variables are evaluated to true.
It follows that {→} is not functionally complete.
However, if one adds a nullary connective ⊥ for falsity, then one can define all other truth functions. Formulas over the resulting set of connectives {→, ⊥} are called f-implicational. If P and Q are propositions, then:
¬P is equivalent to P → ⊥
P ∧ Q is equivalent to (P → (Q → ⊥)) → ⊥
P ∨ Q is equivalent to (P → Q) → Q
P ↔ Q is equivalent to ((P → Q) → ((Q → P) → ⊥)) → ⊥
Since the above operators are known to be functionally complete, it follows that any truth function can be expressed in terms of → and ⊥.
Axiom system
The following statements are considered tautologies (irreducible and intuitively true, by definition).
Axiom schema 1 is P → (Q → P).
Axiom schema 2 is (P → (Q → R)) → ((P → Q) → (P → R)).
Axiom schema 3 (Peirce's law) is ((P → Q) → P) → P.
The one non-nullary rule of inference (modus ponens) is: from P and P → Q infer Q.
Where in each case, P, Q, and R may be replaced by any formulas which contain only "→" as a connective. If Γ is a set of formulas and A a formula, then means that A is derivable using the axioms and rules above and formulas from Γ as additional hypotheses.
Łukasiewicz (1948) found an axiom system for the implicational calculus, which replaces the schemas 1–3 above with a single schema
((P → Q) → R) → ((R → P) → (S → P)).
He also argued that there is no shorter axiom system.
Basic properties of derivation
Since all axioms and rules of the calculus are schemata, derivation is closed under substitution:
If then
where σ is any substitution (of formulas using only implication).
The implicational propositional calculus also satisfies the deduction theorem:
If , then
As explained in the deduction theorem article, this holds for any axiomatic extension of the system containing axiom schemas 1 and 2 above and modus ponens.
Completeness
The implicational propositional calculus is semantically complete with respect to the usual two-valued semantics of classical propositional logic. That is, if Γ is a set of implicational formulas, and A is an implicational formula entailed by Γ, then .
Proof
A proof of the completeness theorem is outlined below. First, using the compactness theorem and the deduction theorem, we may reduce the completeness theorem to its special ca
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https://en.wikipedia.org/wiki/Jacobi%20rotation
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In numerical linear algebra, a Jacobi rotation is a rotation, Qkℓ, of a 2-dimensional linear subspace of an n-dimensional inner product space, chosen to zero a symmetric pair of off-diagonal entries of an n×n real symmetric matrix, A, when applied as a similarity transformation:
It is the core operation in the Jacobi eigenvalue algorithm, which is numerically stable and well-suited to implementation on parallel processors .
Only rows k and ℓ and columns k and ℓ of A will be affected, and that A will remain symmetric. Also, an explicit matrix for Qkℓ is rarely computed; instead, auxiliary values are computed and A is updated in an efficient and numerically stable way. However, for reference, we may write the matrix as
That is, Qkℓ is an identity matrix except for four entries, two on the diagonal (qkk and qℓℓ, both equal to c) and two symmetrically placed off the diagonal (qkℓ and qℓk, equal to s and −s, respectively). Here c = cos θ and s = sin θ for some angle θ; but to apply the rotation, the angle itself is not required. Using Kronecker delta notation, the matrix entries can be written:
Suppose h is an index other than k or ℓ (which must themselves be distinct). Then the similarity update produces, algebraically:
Numerically stable computation
To determine the quantities needed for the update, we must solve the off-diagonal equation for zero . This implies that:
Set β to half of this quantity:
If akℓ is zero we can stop without performing an update, thus we never divide by zero. Let t be tan θ. Then with a few trigonometric identities we reduce the equation to:
For stability we choose the solution:
From this we may obtain c and s as:
Although we now could use the algebraic update equations given previously, it may be preferable to rewrite them. Let:
so that ρ = tan(θ/2). Then the revised update equations are:
As previously remarked, we need never explicitly compute the rotation angle θ. In fact, we can reproduce the symmetric update determined by Qkℓ by retaining only the three values k, ℓ, and t, with t set to zero for a null rotation.
See also
Givens rotation
Householder transformation
References
Numerical linear algebra
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https://en.wikipedia.org/wiki/Inner%20measure
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In mathematics, in particular in measure theory, an inner measure is a function on the power set of a given set, with values in the extended real numbers, satisfying some technical conditions. Intuitively, the inner measure of a set is a lower bound of the size of that set.
Definition
An inner measure is a set function
defined on all subsets of a set that satisfies the following conditions:
Null empty set: The empty set has zero inner measure (see also: measure zero); that is,
Superadditive: For any disjoint sets and
Limits of decreasing towers: For any sequence of sets such that for each and
Infinity must be approached: If for a set then for every positive real number there exists some such that
The inner measure induced by a measure
Let be a σ-algebra over a set and be a measure on
Then the inner measure induced by is defined by
Essentially gives a lower bound of the size of any set by ensuring it is at least as big as the -measure of any of its -measurable subsets. Even though the set function is usually not a measure, shares the following properties with measures:
is non-negative,
If then
Measure completion
Induced inner measures are often used in combination with outer measures to extend a measure to a larger σ-algebra. If is a finite measure defined on a σ-algebra over and and are corresponding induced outer and inner measures, then the sets such that form a σ-algebra with .
The set function defined by
for all is a measure on known as the completion of
See also
References
Halmos, Paul R., Measure Theory, D. Van Nostrand Company, Inc., 1950, pp. 58.
A. N. Kolmogorov & S. V. Fomin, translated by Richard A. Silverman, Introductory Real Analysis, Dover Publications, New York, 1970, (Chapter 7)
Measures (measure theory)
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https://en.wikipedia.org/wiki/Mathematics%20and%20Science%20High%20School%20at%20Clover%20Hill
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The Chesterfield County Mathematics and Science High School at Clover Hill is a magnet school in Midlothian, Virginia. The school, which is on the campus of Clover Hill High School, opened in September 1994. The school is a member of the National Consortium for Specialized Secondary Schools of Mathematics, Science, and Technology (NCSSSMST). It was known as the Renaissance Program early in its history.
References
External links
School Website
Public high schools in Virginia
Educational institutions established in 1994
Schools in Chesterfield County, Virginia
Magnet schools in Virginia
1994 establishments in Virginia
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https://en.wikipedia.org/wiki/Tractor%20bundle
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In conformal geometry, the tractor bundle is a particular vector bundle constructed on a conformal manifold whose fibres form an effective representation of the conformal group (see associated bundle).
The term tractor is a portmanteau of "Tracy Thomas" and "twistor", the bundle having been introduced first by T. Y. Thomas as an alternative formulation of the Cartan conformal connection, and later rediscovered within the formalism of local twistors and generalized to projective connections by Michael Eastwood et al. in
References
Differential geometry
Conformal geometry
Vector bundles
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https://en.wikipedia.org/wiki/John%20L.%20Kelley
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John L. Kelley (December 6, 1916, Kansas – November 26, 1999, Berkeley, California) was an American mathematician at the University of California, Berkeley, who worked in general topology and functional analysis.
Kelley's 1955 text, General Topology, which eventually appeared in three editions and several translations, is a classic and widely cited graduate-level introduction to topology. An appendix sets out a new approach to axiomatic set theory, now called Morse–Kelley set theory, that builds on Von Neumann–Bernays–Gödel set theory. He introduced the first definition of a subnet.
After earning B.A. (1936) and M.A. (1937) degrees from the University of California, Los Angeles, he went to the University of Virginia, where he obtained his Ph.D. in 1940. Gordon Whyburn, a student of Robert Lee Moore, supervised his thesis, entitled A Study of Hyperspaces. He taught at the University of Notre Dame until the outbreak of World War II. From 1942 to 1945, he did mathematics (mainly exterior ballistics, including ballistics for the atomic bomb) for the war effort at the Aberdeen Proving Ground, where his work unit included his future Berkeley colleagues Anthony Morse and Charles Morrey. After teaching at the University of Chicago, 1946–47, Kelley spent the rest of his career at Berkeley, from which he retired in 1985. He chaired the Mathematics Department at Berkeley 1957–60 and 1975–80. He held visiting appointments at Cambridge University and the Indian Institute of Technology in Kanpur, India. An Indian mathematician, Vashishtha Narayan Singh, was among those mentored by Kelley.
In 1950, Kelley was one of 29 tenured Berkeley faculty (3 of whom were members of the Mathematics Department) dismissed for refusing to sign a McCarthy-era loyalty oath mandated by the UC Board of Regents. When asked why he refused to swear that he was loyal to his country, he replied, "For the same reason that I would refuse to swear, under duress, that I loved my mother." He then taught at Tulane University and the University of Kansas. He returned to Berkeley in 1953, after the California Supreme Court declared the oath unconstitutional and directed UC Berkeley to rehire the dismissed academics. He was later an outspoken opponent of the Vietnam War.
Kelley's interest in teaching extended well beyond the higher reaches of mathematics. In 1960, he took a leave of absence to serve as the National Teacher on NBC's Continental Classroom television program. He was an active member of the School Mathematics Study Group (SMSG), which played an important role in designing and promulgating the "new math" of that era. In 1964, he led his department to introduce a new major called Mathematics for Teachers, and later taught one of its core courses. These endeavors culminated in the text Kelley and Richert (1970). In 1977–78, he was a member of the U.S. Commission on Mathematical Instruction.
His doctoral students include Vashishtha Narayan Singh, James Michael Gardner Fell, Isaac
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https://en.wikipedia.org/wiki/Conformal%20connection
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In conformal differential geometry, a conformal connection is a Cartan connection on an n-dimensional manifold M arising as a deformation of the Klein geometry given by the celestial n-sphere, viewed as the homogeneous space
O+(n+1,1)/P
where P is the stabilizer of a fixed null line through the origin in Rn+2, in the orthochronous Lorentz group O+(n+1,1) in n+2 dimensions.
Normal Cartan connection
Any manifold equipped with a conformal structure has a canonical conformal connection called the normal Cartan connection.
Formal definition
A conformal connection on an n-manifold M is a Cartan geometry modelled on the conformal sphere, where the latter is viewed as a homogeneous space for O+(n+1,1). In other words, it is an O+(n+1,1)-bundle equipped with
a O+(n+1,1)-connection (the Cartan connection)
a reduction of structure group to the stabilizer of a point in the conformal sphere (a null line in Rn+1,1)
such that the solder form induced by these data is an isomorphism.
References
E. Cartan, "Les espaces à connexion conforme", Ann. Soc. Polon. Math., 2 (1923): 171–221.
K. Ogiue, "Theory of conformal connections" Kodai Math. Sem. Reports, 19 (1967): 193–224.
Le, Anbo. "Cartan connections for CR manifolds." manuscripta mathematica 122.2 (2007): 245–264.
External links
Conformal geometry
Connection (mathematics)
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https://en.wikipedia.org/wiki/Quasitopological%20space
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In mathematics, a quasi-topology on a set X is a function that associates to every compact Hausdorff space C a collection of mappings from C to X satisfying certain natural conditions. A set with a quasi-topology is called a quasitopological space.
They were introduced by Spanier, who showed that there is a natural quasi-topology on the space of continuous maps from one space to another.
References
.
Topology
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https://en.wikipedia.org/wiki/Riesz%20mean
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In mathematics, the Riesz mean is a certain mean of the terms in a series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesàro mean. The Riesz mean should not be confused with the Bochner–Riesz mean or the Strong–Riesz mean.
Definition
Given a series , the Riesz mean of the series is defined by
Sometimes, a generalized Riesz mean is defined as
Here, the are a sequence with and with as . Other than this, the are taken as arbitrary.
Riesz means are often used to explore the summability of sequences; typical summability theorems discuss the case of for some sequence . Typically, a sequence is summable when the limit exists, or the limit exists, although the precise summability theorems in question often impose additional conditions.
Special cases
Let for all . Then
Here, one must take ; is the Gamma function and is the Riemann zeta function. The power series
can be shown to be convergent for . Note that the integral is of the form of an inverse Mellin transform.
Another interesting case connected with number theory arises by taking where is the Von Mangoldt function. Then
Again, one must take c > 1. The sum over ρ is the sum over the zeroes of the Riemann zeta function, and
is convergent for λ > 1.
The integrals that occur here are similar to the Nörlund–Rice integral; very roughly, they can be connected to that integral via Perron's formula.
References
M. Riesz, Comptes Rendus, 12 June 1911
Means
Summability methods
Zeta and L-functions
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https://en.wikipedia.org/wiki/Joseph%20Bernstein
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Joseph Bernstein (sometimes spelled I. N. Bernshtein; ; ; born 18 April 1945) is a Soviet-born Israeli mathematician working at Tel Aviv University. He works in algebraic geometry, representation theory, and number theory.
Biography
Bernstein received his Ph.D. in 1972 under Israel Gelfand at Moscow State University. In 1981, he emigrated to the United States due to growing antisemitism in the Soviet Union.
Bernstein was a professor at Harvard during 1983-1993.
He was a visiting scholar at the Institute for Advanced Study in 1985-86 and again in 1997-98. In 1993, he moved to Israel to take a professorship at Tel Aviv University (emeritus since 2014).
Awards and honors
Bernstein received a gold medal at the 1962 International Mathematical Olympiad. He was elected to the Israel Academy of Sciences and Humanities in 2002 and was elected to the United States National Academy of Sciences in 2004. In 2004, Bernstein was awarded the Israel Prize for mathematics. In 1998, he was an Invited Speaker of the International Congress of Mathematicians in Berlin. In 2012, he became a fellow of the American Mathematical Society.
Publications
Publication list
Some pdf files of papers by Bernstein including Algebraic theory of D-modules and his notes on Meromorphic continuation of Eisenstein series
See also
Bernstein–Sato polynomial
Bernstein–Gelfand–Gelfand resolution
References
External links
Bernstein's home page
1945 births
Living people
20th-century Russian mathematicians
21st-century Russian mathematicians
Israeli mathematicians
Russian emigrants to Israel
Russian Jews
Israeli Jews
Jewish scientists
Algebraic geometers
Group theorists
Number theorists
Israel Prize in mathematics recipients
EMET Prize recipients in the Exact Sciences
Moscow State University alumni
Academic staff of Tel Aviv University
Fellows of the American Mathematical Society
Members of the Israel Academy of Sciences and Humanities
Members of the United States National Academy of Sciences
Soviet Jews
Institute for Advanced Study visiting scholars
International Mathematical Olympiad participants
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https://en.wikipedia.org/wiki/Center%20of%20curvature
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In geometry, the center of curvature of a curve is found at a point that is at a distance from the curve equal to the radius of curvature lying on the normal vector. It is the point at infinity if the curvature is zero. The osculating circle to the curve is centered at the centre of curvature. Cauchy defined the center of curvature C as the intersection point of two infinitely close normal lines to the curve. The locus of centers of curvature for each point on the curve comprise the evolute of the curve. This term is generally used in physics regarding the study of lenses and mirrors (see radius of curvature (optics)).
It can also be defined as the spherical distance between the point at which all the rays falling on a lens or mirror either seems to converge to (in the case of convex lenses and concave mirrors) or diverge from (in the case of concave lenses or convex mirrors) and the lens/mirror itself.
See also
Curvature
Differential geometry of curves
References
Bibliography
Curves
Differential geometry
Curvature
Concepts in physics
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https://en.wikipedia.org/wiki/Holonomic%20function
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In mathematics, and more specifically in analysis, a holonomic function is a smooth function of several variables that is a solution of a system of linear homogeneous differential equations with polynomial coefficients and satisfies a suitable dimension condition in terms of D-modules theory. More precisely, a holonomic function is an element of a holonomic module of smooth functions. Holonomic functions can also be described as differentiably finite functions, also known as D-finite functions. When a power series in the variables is the Taylor expansion of a holonomic function, the sequence of its coefficients, in one or several indices, is also called holonomic. Holonomic sequences are also called P-recursive sequences: they are defined recursively by multivariate recurrences satisfied by the whole sequence and by suitable specializations of it. The situation simplifies in the univariate case: any univariate sequence that satisfies a linear homogeneous recurrence relation with polynomial coefficients, or equivalently a linear homogeneous difference equation with polynomial coefficients, is holonomic.
Holonomic functions and sequences in one variable
Definitions
Let be a field of characteristic 0 (for example, or ).
A function is called D-finite (or holonomic) if there exist polynomials such that
holds for all x. This can also be written as where
and is the differential operator that maps to . is called an annihilating operator of f (the annihilating operators of form an ideal in the ring , called the annihilator of ). The quantity r is called the order of the annihilating operator. By extension, the holonomic function f is said to be of order r when an annihilating operator of such order exists.
A sequence is called P-recursive (or holonomic) if there exist polynomials such that
holds for all n. This can also be written as where
and the shift operator that maps to . is called an annihilating operator of c (the annihilating operators of form an ideal in the ring , called the annihilator of ). The quantity r is called the order of the annihilating operator. By extension, the holonomic sequence c is said to be of order r when an annihilating operator of such order exists.
Holonomic functions are precisely the generating functions of holonomic sequences: if is holonomic, then the coefficients in the power series expansion
form a holonomic sequence. Conversely, for a given holonomic sequence , the function defined by the above sum is holonomic (this is true in the sense of formal power series, even if the sum has a zero radius of convergence).
Closure properties
Holonomic functions (or sequences) satisfy several closure properties. In particular, holonomic functions (or sequences) form a ring. They are not closed under division, however, and therefore do not form a field.
If and are holonomic functions, then the following functions are also holonomic:
, where and are constants
(the Cauchy product of the s
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https://en.wikipedia.org/wiki/Antiparallelogram
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In geometry, an antiparallelogram is a type of self-crossing quadrilateral. Like a parallelogram, an antiparallelogram has two opposite pairs of equal-length sides, but these pairs of sides are not in general parallel. Instead, each pair of sides is antiparallel with respect to the other, with sides in the longer pair crossing each other as in a scissors mechanism. Whereas a parallelogram's opposite angles are equal and oriented the same way, an antiparallelogram's are equal but oppositely oriented. Antiparallelograms are also called contraparallelograms or crossed parallelograms.
Antiparallelograms occur as the vertex figures of certain nonconvex uniform polyhedra. In the theory of four-bar linkages, the linkages with the form of an antiparallelogram are also called butterfly linkages or bow-tie linkages, and are used in the design of non-circular gears. In celestial mechanics, they occur in certain families of solutions to the 4-body problem.
Every antiparallelogram has an axis of symmetry, with all four vertices on a circle. It can be formed from an isosceles trapezoid by adding the two diagonals and removing two parallel sides. The signed area of every antiparallelogram is zero.
Geometric properties
An antiparallelogram is a special case of a crossed quadrilateral, with two pairs of equal-length edges. In general, crossed quadrilaterals can have unequal edges. A special form of the antiparallelogram is a crossed rectangle, in which two opposite edges are parallel. Every antiparallelogram is a cyclic quadrilateral, meaning that its four vertices all lie on a single circle. Additionally, the four extended sides of any antiparallelogram are the bitangents of two circles, making antiparallelograms closely related to the tangential quadrilaterals, ex-tangential quadrilaterals, and kites (which are both tangential and ex-tangential).
Every antiparallelogram has an axis of symmetry through its crossing point. Because of this symmetry, it has two pairs of equal angles and two pairs of equal sides. The four midpoints of its sides lie on a line perpendicular to the axis of symmetry; that is, for this kind of quadrilateral, the Varignon parallelogram is a degenerate quadrilateral of area zero, consisting of four collinear points. The convex hull of an antiparallelogram is an isosceles trapezoid, and every antiparallelogram may be formed from an isosceles trapezoid (or its special cases, the rectangles and squares) by replacing two parallel sides by the two diagonals of the trapezoid.
Because an antiparallelogram forms two congruent triangular regions of the plane, but loops around those two regions in opposite directions, its signed area is the difference between the regions' areas and is therefore zero. The polygon's unsigned area (the total area it surrounds) is the sum, rather than the difference, of these areas. For an antiparallelogram with two parallel diagonals of lengths and , separated by height , this sum is . It follows from applying t
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https://en.wikipedia.org/wiki/Acyclic%20object
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In mathematics, in the field of homological algebra, given an abelian category
having enough injectives and an additive (covariant) functor
,
an acyclic object with respect to , or simply an -acyclic object, is an object in such that
for all ,
where are the right derived functors of
.
References
Homological algebra
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https://en.wikipedia.org/wiki/Nodec%20space
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In topology and related areas of mathematics, a topological space is a nodec space if every nowhere dense subset of is closed. This concept was introduced and studied by .
References
.
General topology
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https://en.wikipedia.org/wiki/James%20Earl%20Baumgartner
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James Earl Baumgartner (March 23, 1943 – December 28, 2011) was an American mathematician who worked in set theory, mathematical logic and foundations, and topology.
Baumgartner was born in Wichita, Kansas, began his undergraduate study at the California Institute of Technology in 1960, then transferred to the University of California, Berkeley, from which he received his PhD in 1970 from for a dissertation titled Results and Independence Proofs in Combinatorial Set Theory. His advisor was Robert Vaught. He became a professor at Dartmouth College in 1969, and spent his entire career there.
One of Baumgartner's results is the consistency of the statement that any two -dense sets of reals are order isomorphic (a set of reals is -dense if it has exactly points in every open interval). With András Hajnal he proved the Baumgartner–Hajnal theorem, which states that the partition relation holds for and . He died in 2011 of a heart attack at his home in Hanover, New Hampshire.
The mathematical context in which Baumgartner worked spans Suslin's problem, Ramsey theory, uncountable order types, disjoint refinements, almost disjoint families, cardinal arithmetics, filters, ideals, and partition relations, iterated forcing and Axiom A, proper forcing and the proper forcing axiom, chromatic number of graphs, a thin very-tall superatomic Boolean algebra, closed unbounded sets, and partition relations.
See also
Baumgartner's axiom
Selected publications
Baumgartner, James E., A new class of order types, Annals of Mathematical Logic, 9:187–222, 1976
Baumgartner, James E., Ineffability properties of cardinals I, Infinite and Finite Sets, Keszthely (Hungary) 1973, volume 10 of Colloquia Mathematica Societatis János Bolyai, pages 109–130. North-Holland, 1975
Baumgartner, James E.; Harrington, Leo; Kleinberg, Eugene, Adding a closed unbounded set, Journal of Symbolic Logic, 41(2):481–482, 1976
Baumgartner, James E., Ineffability properties of cardinals II, Robert E. Butts and Jaakko Hintikka, editors, Logic, Foundations of Mathematics and Computability Theory, pages 87–106. Reidel, 1977
Baumgartner, James E.; Galvin, Fred, Generalized Erdős cardinals and 0#, Annals of Mathematical Logic 15, 289–313, 1978
Baumgartner, James E.; Erdős, Paul; Galvin, Fred; Larson, J., Colorful partitions of cardinal numbers, Can. J. Math. 31, 524–541, 1979
Baumgartner, James E.; Erdős, Paul; Higgs, D., Cross-cuts in the power set of an infinite set, Order 1, 139–145, 1984
Baumgartner, James E. (Editor), Axiomatic Set Theory (Contemporary Mathematics, Volume 31), 1990
References
1943 births
20th-century American mathematicians
21st-century American mathematicians
American logicians
Set theorists
Mathematical logicians
University of California, Berkeley alumni
Dartmouth College faculty
2011 deaths
People from Wichita, Kansas
Mathematicians from Kansas
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https://en.wikipedia.org/wiki/Robbins%20algebra
|
In abstract algebra, a Robbins algebra is an algebra containing a single binary operation, usually denoted by , and a single unary operation usually denoted by satisfying the following axioms:
For all elements a, b, and c:
Associativity:
Commutativity:
Robbins equation:
For many years, it was conjectured, but unproven, that all Robbins algebras are Boolean algebras. This was proved in 1996, so the term "Robbins algebra" is now simply a synonym for "Boolean algebra".
History
In 1933, Edward Huntington proposed a new set of axioms for Boolean algebras, consisting of (1) and (2) above, plus:
Huntington's equation:
From these axioms, Huntington derived the usual axioms of Boolean algebra.
Very soon thereafter, Herbert Robbins posed the Robbins conjecture, namely that the Huntington equation could be replaced with what came to be called the Robbins equation, and the result would still be Boolean algebra. would interpret Boolean join and Boolean complement. Boolean meet and the constants 0 and 1 are easily defined from the Robbins algebra primitives. Pending verification of the conjecture, the system of Robbins was called "Robbins algebra."
Verifying the Robbins conjecture required proving Huntington's equation, or some other axiomatization of a Boolean algebra, as theorems of a Robbins algebra. Huntington, Robbins, Alfred Tarski, and others worked on the problem, but failed to find a proof or counterexample.
William McCune proved the conjecture in 1996, using the automated theorem prover EQP. For a complete proof of the Robbins conjecture in one consistent notation and following McCune closely, see Mann (2003). Dahn (1998) simplified McCune's machine proof.
See also
Algebraic structure
Minimal axioms for Boolean algebra
References
Dahn, B. I. (1998) Abstract to "Robbins Algebras Are Boolean: A Revision of McCune's Computer-Generated Solution of Robbins Problem," Journal of Algebra 208(2): 526–32.
Mann, Allen (2003) "A Complete Proof of the Robbins Conjecture."
William McCune, "Robbins Algebras Are Boolean," With links to proofs and other papers.
Boolean algebra
Formal methods
Computer-assisted proofs
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https://en.wikipedia.org/wiki/Representation%20rigid%20group
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In mathematics, in the representation theory of groups, a group is said to be representation rigid if for every , it has only finitely many isomorphism classes of complex irreducible representations of dimension .
External links
The proalgebraic completion of rigid groups
Properties of groups
Representation theory of groups
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https://en.wikipedia.org/wiki/Capable%20group
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In mathematics, in the realm of group theory, a group is said to be capable if it occurs as the inner automorphism group of some group. These groups were first studied by Reinhold Baer, who showed that a finite abelian group is capable if and only if it is a product of cyclic groups of orders n1, ..., nk where ni divides ni +1 and nk −1 = nk.
References
External links
Bounds on the index of the center in capable groups
Properties of groups
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https://en.wikipedia.org/wiki/Evelyn%20Boyd%20Granville
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Evelyn Boyd Granville (May 1, 1924 – June 27, 2023) was the second African-American woman to receive a Ph.D. in mathematics from an American university; she earned it in 1949 from Yale University. She graduated from Smith College in 1945. She performed pioneering work in the field of computing.
Education
Evelyn Boyd was born in Washington, D.C.; her father worked odd jobs due to the Great Depression but separated from her mother when Boyd was young. Boyd and her older sister were raised by her mother and aunt, who both worked at the Bureau of Engraving and Printing. She was valedictorian at Dunbar High School, which at that time was a segregated but academically competitive school for black students in Washington.
With financial support from her aunt and a small partial scholarship from Phi Delta Kappa, Boyd entered Smith College in the fall of 1941. She majored in mathematics and physics, but also took a keen interest in astronomy. She was elected to Phi Beta Kappa and to Sigma Xi and graduated summa cum laude in 1945. Encouraged by a graduate scholarship from the Smith Student Aid Society of Smith College, she applied to graduate programs in mathematics and was accepted by both Yale University and the University of Michigan; she chose Yale because of the financial aid they offered. There she studied functional analysis under the supervision of Einar Hille, finishing her doctorate in 1949. Her dissertation was "On Laguerre Series in the Complex Domain".
Career
Following graduate school, Boyd went to New York University Institute for Mathematics and performed research and teaching there. After, in 1950, she took a teaching position at Fisk University, a college for black students in Nashville, Tennessee (more prestigious postings being unavailable to black women). Two of her students there, Vivienne Malone-Mayes and Etta Zuber Falconer, went on to earn doctorates in mathematics of their own. But by 1952 she left academia and returned to Washington with a position at the Diamond Ordnance Fuze Laboratories. In January 1956, she moved to IBM as a computer programmer; when IBM received a NASA contract, she moved to Vanguard Computing Center in Washington, D.C.
Boyd moved from Washington to New York City in 1957. In 1960, after marrying Reverend G. Mansfield Collins, Boyd moved to Los Angeles. There she worked for the U.S. Space Technology Laboratories, which became the North American Aviation Space and Information Systems Division in 1962. She worked on various projects for the Apollo program, including celestial mechanics, trajectory computation, and "digital computer techniques".
Forced to move because of a restructuring at IBM, she took a position at California State University, Los Angeles in 1967 as a full professor of mathematics.
After retiring from CSULA in 1984 she taught at Texas College in Tyler, Texas for four years, and then in 1990 joined the faculty of the University of Texas at Tyler as the Sam A. Lindsey Professor of mathematic
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https://en.wikipedia.org/wiki/Heegner%20point
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In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. They were defined by Bryan Birch and named after Kurt Heegner, who used similar ideas to prove Gauss's conjecture on imaginary quadratic fields of class number one.
Gross–Zagier theorem
The Gross–Zagier theorem describes the height of Heegner points in terms of a derivative of the L-function of the elliptic curve at the point s = 1. In particular if the elliptic curve has (analytic) rank 1, then the Heegner points can be used to construct a rational point on the curve of infinite order (so the Mordell–Weil group has rank at least 1). More generally, showed that Heegner points could be used to construct rational points on the curve for each positive integer n, and the heights of these points were the coefficients of a modular form of weight 3/2. Shou-Wu Zhang generalized the Gross–Zagier theorem from elliptic curves to the case of modular abelian varieties (, ).
Birch and Swinnerton-Dyer conjecture
Kolyvagin later used Heegner points to construct Euler systems, and used this to prove much of the Birch–Swinnerton-Dyer conjecture for rank 1 elliptic curves. Brown proved the Birch–Swinnerton-Dyer conjecture for most rank 1 elliptic curves over global fields of positive characteristic .
Computation
Heegner points can be used to compute very large rational points on rank 1 elliptic curves (see for a survey) that could not be found by naive methods. Implementations of the algorithm are available in Magma, PARI/GP, and Sage.
References
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Algebraic number theory
Elliptic curves
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https://en.wikipedia.org/wiki/Rigid%20analytic%20space
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In mathematics, a rigid analytic space is an analogue of a complex analytic space over a nonarchimedean field. Such spaces were introduced by John Tate in 1962, as an outgrowth of his work on uniformizing p-adic elliptic curves with bad reduction using the multiplicative group. In contrast to the classical theory of p-adic analytic manifolds, rigid analytic spaces admit meaningful notions of analytic continuation and connectedness.
Definitions
The basic rigid analytic object is the n-dimensional unit polydisc, whose ring of functions is the Tate algebra , made of power series in n variables whose coefficients approach zero in some complete nonarchimedean field k. The Tate algebra is the completion of the polynomial ring in n variables under the Gauss norm (taking the supremum of coefficients), and the polydisc plays a role analogous to that of affine n-space in algebraic geometry. Points on the polydisc are defined to be maximal ideals in the Tate algebra, and if k is algebraically closed, these correspond to points in whose coordinates have norm at most one.
An affinoid algebra is a k-Banach algebra that is isomorphic to a quotient of the Tate algebra by an ideal. An affinoid is then the subset of the unit polydisc on which the elements of this ideal vanish, i.e., it is the set of maximal ideals containing the ideal in question. The topology on affinoids is subtle, using notions of affinoid subdomains (which satisfy a universality property with respect to maps of affinoid algebras) and admissible open sets (which satisfy a finiteness condition for covers by affinoid subdomains). In fact, the admissible opens in an affinoid do not in general endow it with the structure of a topological space, but they do form a Grothendieck topology (called the G-topology), and this allows one to define good notions of sheaves and gluing of spaces.
A rigid analytic space over k is a pair describing a locally ringed G-topologized space with a sheaf of k-algebras, such that there is a covering by open subspaces isomorphic to affinoids. This is analogous to the notion of manifolds being coverable by open subsets isomorphic to euclidean space, or schemes being coverable by affines. Schemes over k can be analytified functorially, much like varieties over the complex numbers can be viewed as complex analytic spaces, and there is an analogous formal GAGA theorem. The analytification functor respects finite limits.
Other formulations
Around 1970, Michel Raynaud provided an interpretation of certain rigid analytic spaces as formal models, i.e., as generic fibers of formal schemes over the valuation ring R of k. In particular, he showed that the category of quasi-compact quasi-separated rigid spaces over k is equivalent to the localization of the category of quasi-compact admissible formal schemes over R with respect to admissible formal blow-ups. Here, a formal scheme is admissible if it is coverable by formal spectra of topologically finitely presented
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https://en.wikipedia.org/wiki/List%20of%20urban%20areas%20in%20Denmark%20by%20population
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This is a list of urban areas in Denmark by population. For a list on cities in Denmark please see List of cities in Denmark by population.
The population is measured by Statistics Denmark for urban areas (Danish: byområder or bymæssige områder), which is defined as a contiguous built-up area with a maximum distance of 200 m between houses, unless further distance is caused by public areas, cemeteries or similar reasons. Furthermore, to obtain by-status, the area must have at least 200 inhabitants. Some urban areas in Denmark have witnessed conurbation and grown together.
See also
Largest metropolitan areas in the Nordic countries
List of metropolitan areas in Sweden
List of urban areas in Sweden by population
List of urban areas in Norway by population
List of urban areas in the Nordic countries
List of urban areas in Finland by population
List of cities in Iceland
World's largest cities
Notes
References
Denmark
Denmark
Denmark
Urban
ca:Llista de ciutats de Dinamarca
da:Danmarks største byer
eo:Listo de urboj de Danio
fr:Villes du Danemark
kk:Дания қалаларының тізімі
lv:Dānijas pilsētu uzskaitījums
nl:Lijst van grote Deense steden
no:Liste over danske byer
pl:Miasta Danii
ru:Города Дании
sv:Lista över städer i Danmark efter storlek
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https://en.wikipedia.org/wiki/Bill%20Chen
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William Chen (born 1970 in Williamsburg, Virginia) is an American quantitative analyst, poker player, software designer, and badminton player.
Biography
Chen holds a Ph.D. in mathematics (1999) from the University of California, Berkeley. He was an undergraduate at Washington University in St. Louis triple-majoring in Physics, Math, and Computer Science, and was also a research intern in Washington University's Computer Science SURA Program where he co-wrote a technical report inventing an Argument Game. He heads the Statistical arbitrage department at Susquehanna International Group.
Poker career
At the 2006 World Series of Poker Chen won two events, a $3,000 limit Texas hold 'em event with a prize of $343,618, and a $2,500 no limit hold 'em short-handed event with a prize of $442,511. Prior to these events Chen's largest tournament win was for $41,600 at a no limit hold 'em event at the Bicycle Casino's Legends of Poker in 2000.
Chen has been a longtime participant in the rec.gambling.poker newsgroup and its B.A.R.G.E offshoot. He has also been a member of Team PokerStars.
With Jerrod Ankenman, Chen coauthored The Mathematics of Poker, an introduction to quantitative techniques and game theory as applied to poker.
In February 2009, he appeared on Poker After Dark's "Brilliant Minds" week, finishing in 5th place after his lost to Jimmy Warren's after Chen pushed all-in on a flop of .
As of 2017, his total live tournament winnings exceed $1,900,000. His 38 cashes at the WSOP account for over $1,725,000 of those winnings.
World Series of Poker bracelets
Bibliography
References
20th-century American mathematicians
21st-century American mathematicians
American poker players
American people of Chinese descent
American computer programmers
UC Berkeley College of Letters and Science alumni
Washington University in St. Louis alumni
Washington University physicists
Living people
World Series of Poker bracelet winners
1970 births
Auburn High School (Alabama) alumni
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https://en.wikipedia.org/wiki/Truncatable%20prime
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In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading ("left") digit is successively removed, then all resulting numbers are prime. For example, 9137, since 9137, 137, 37 and 7 are all prime. Decimal representation is often assumed and always used in this article.
A right-truncatable prime is a prime which remains prime when the last ("right") digit is successively removed. 7393 is an example of a right-truncatable prime, since 7393, 739, 73, and 7 are all prime.
A left-and-right-truncatable prime is a prime which remains prime if the leading ("left") and last ("right") digits are simultaneously successively removed down to a one- or two-digit prime. 1825711 is an example of a left-and-right-truncatable prime, since 1825711, 82571, 257, and 5 are all prime.
In base 10, there are exactly 4260 left-truncatable primes, 83 right-truncatable primes, and 920,720,315 left-and-right-truncatable primes.
History
An author named Leslie E. Card in early volumes of the Journal of Recreational Mathematics (which started its run in 1968) considered a topic close to that of right-truncatable primes, calling sequences that by adding digits to the right in sequence to an initial number not necessarily prime snowball primes.
Discussion of the topic dates to at least November 1969 issue of Mathematics Magazine, where truncatable primes were called prime primes by two co-authors (Murray Berg and John E. Walstrom).
Decimal truncatable primes
There are 4260 left-truncatable primes:
2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, 197, 223, 283, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683, 743, 773, 797, 823, 853, 883, 937, 947, 953, 967, 983, 997, ...
The largest is the 24-digit 357686312646216567629137.
There are 83 right-truncatable primes. The complete list:
2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, 719, 733, 739, 797, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797, 5939, 7193, 7331, 7333, 7393, 23333, 23339, 23399, 23993, 29399, 31193, 31379, 37337, 37339, 37397, 59393, 59399, 71933, 73331, 73939, 233993, 239933, 293999, 373379, 373393, 593933, 593993, 719333, 739391, 739393, 739397, 739399, 2339933, 2399333, 2939999, 3733799, 5939333, 7393913, 7393931, 7393933, 23399339, 29399999, 37337999, 59393339, 73939133
The largest is the 8-digit 73939133. All primes above 5 end with digit 1, 3, 7 or 9, so a right-truncatable prime can only contain those digits after the leading digit.
There are 920,720,315 left-and-right-truncatable primes:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 127, 131, 137, 139, 151, 157, 173, 179, 223, 227, 229, 233, 239, 251, 257, 271, 277, 331, 337, 353, 359, 373, 379, 421, 431, 433, 439, 457, 479, 521, 523, 557, 571, 577, 631, 653, 659, 673, 677, 727, 733, 739, 751,
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https://en.wikipedia.org/wiki/Sandvika%2C%20Innlandet
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Sandvika is a village in Stange Municipality in Innlandet county, Norway. The village is located along the lake Mjøsa, just across a bay from the city of Hamar. Statistics Norway considers this to be part of the Bekkelaget urban area, so its statistics are not tracked.
References
Stange
Villages in Innlandet
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https://en.wikipedia.org/wiki/Mordell%E2%80%93Weil%20group
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In arithmetic geometry, the Mordell–Weil group is an abelian group associated to any abelian variety defined over a number field , it is an arithmetic invariant of the Abelian variety. It is simply the group of -points of , so is the Mordell–Weil grouppg 207. The main structure theorem about this group is the Mordell–Weil theorem which shows this group is in fact a finitely-generated abelian group. Moreover, there are many conjectures related to this group, such as the Birch and Swinnerton-Dyer conjecture which relates the rank of to the zero of the associated L-function at a special point.
Examples
Constructing explicit examples of the Mordell–Weil group of an abelian variety is a non-trivial process which is not always guaranteed to be successful, so we instead specialize to the case of a specific elliptic curve . Let be defined by the Weierstrass equationover the rational numbers. It has discriminant (and this polynomial can be used to define a global model ). It can be foundthrough the following procedure. First, we find some obvious torsion points by plugging in some numbers, which areIn addition, after trying some smaller pairs of integers, we find is a point which is not obviously torsion. One useful result for finding the torsion part of is that the torsion of prime to , for having good reduction to , denoted injects into , soWe check at two primes and calculate the cardinality of the setsnote that because both primes only contain a factor of , we have found all the torsion points. In addition, we know the point has infinite order because otherwise there would be a prime factor shared by both cardinalities, so the rank is at least . Now, computing the rank is a more arduous process consisting of calculating the group where using some long exact sequences from homological algebra and the Kummer map.
Theorems concerning special cases
There are many theorems in the literature about the structure of the Mordell–Weil groups of abelian varieties of specific dimension, over specific fields, or having some other special property.
Abelian varieties over the rational function field k(t)
For a hyperelliptic curve and an abelian variety defined over a fixed field , we denote the the twist of (the pullback of to the function field ) by a 1-cocylefor Galois cohomology of the field extension associated to the covering map . Note which follows from the map being hyperelliptic. More explicitly, this 1-cocyle is given as a map of groupswhich using universal properties is the same as giving two maps , hence we can write it as a mapwhere is the inclusion map and is sent to negative . This can be used to define the twisted abelian variety defined over using general theory of algebraic geometrypg 5. In particular, from universal properties of this construction, is an abelian variety over which is isomorphic to after base-change to .
Theorem
For the setup given above, there is an isomorphism of abelian groupswhere is the Jacob
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https://en.wikipedia.org/wiki/Kat%C4%9Btov%E2%80%93Tong%20insertion%20theorem
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The Katětov–Tong insertion theorem is a theorem of point-set topology proved independently by Miroslav Katětov and Hing Tong in the 1950s. The theorem states the following:
Let be a normal topological space and let be functions with g upper semicontinuous, h lower semicontinuous and . Then there exists a continuous function with
This theorem has a number of applications and is the first of many classical insertion theorems. In particular it implies the Tietze extension theorem and consequently Urysohn's lemma, and so the conclusion of the theorem is equivalent to normality.
References
General topology
Theorems in topology
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https://en.wikipedia.org/wiki/Conical%20function
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In mathematics, conical functions or Mehler functions are functions which can be expressed in terms of Legendre functions of the first and second kind,
and
The functions were introduced by Gustav Ferdinand Mehler, in 1868, when expanding in series the distance of a point on the axis of a cone to a point located on the surface of the cone. Mehler used the notation to represent these functions. He obtained integral representation and series of functions representations for them. He also established an addition theorem
for the conical functions. Carl Neumann obtained an expansion of the functions in terms
of the Legendre polynomials in 1881. Leonhardt introduced for the conical functions the equivalent of the spherical harmonics in 1882.
External links
G. F. Mehler "Ueber die Vertheilung der statischen Elektricität in einem von zwei Kugelkalotten begrenzten Körper" Journal für die reine und angewandte Mathematik 68, 134 (1868).
G. F. Mehler "Ueber eine mit den Kugel- und Cylinderfunctionen verwandte Function und ihre Anwendung in der Theorie der Elektricitätsvertheilung" Mathematische Annalen 18 p. 161 (1881).
C. Neumann "Ueber die Mehler'schen Kegelfunctionen und deren Anwendung auf elektrostatische Probleme" Mathematische Annalen 18 p. 195 (1881).
G. Leonhardt " Integraleigenschaften der adjungirten Kegelfunctionen" Mathematische Annalen 19 p. 578 (1882).
Milton Abramowitz and Irene Stegun (Eds.) Handbook of Mathematical Functions (Dover, 1972) p. 337
A. Gil, J. Segura, N. M. Temme "Computing the conical function $P^{\mu}_{-1/2+i\tau}(x)$" SIAM J. Sci. Comput. 31(3), 1716–1741 (2009).
Tiwari, U. N.; Pandey, J. N. The Mehler-Fock transform of distributions. Rocky Mountain J. Math. 10 (1980), no. 2, 401–408.
Special functions
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https://en.wikipedia.org/wiki/%C3%98stby%2C%20Innlandet
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Østby is a village in Trysil municipality, Innlandet county, Norway. The population of the village in 2003 was 205, but since 2004 it has not been considered an urban settlement by Statistics Norway, and its data is therefore no longer tracked separately. (Locals say that about 200 people live there).
In Østby, there is a community house, but no longer a supermarket, an abandoned school, Østby Church and a hotel, the Kjølen Hotell, which is well known as a training location for cross country events: in the area surrounding Østby are several cross country tracks of which some are illuminated. Every year the Trysil Skimaraton (42195) takes place in Østby, the starting point of which is the hotel. The Norwegian National Road 25 runs through the village, about east of the border with Sweden.
References
Trysil
Villages in Innlandet
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https://en.wikipedia.org/wiki/Konglungen
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Konglungen is a village in Asker municipality, Norway.
Its population in 1999 was 208, but since 2001 it is not considered an urban settlement by Statistics Norway, and its data is therefore not registered.
References
Villages in Viken (county)
Villages in Akershus
Villages in Asker
Villages in Northern Asker
Asker
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https://en.wikipedia.org/wiki/Bisection%20bandwidth
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In computer networking, if the network is bisected into two equal-sized partitions, the bisection bandwidth of a network topology is the bandwidth available between the two partitions. Bisection should be done in such a way that the bandwidth between two partitions is minimum. Bisection bandwidth gives the true bandwidth available in the entire system. Bisection bandwidth accounts for the bottleneck bandwidth of the entire network. Therefore bisection bandwidth represents bandwidth characteristics of the network better than any other metric.
Bisection bandwidth calculations
For a linear array with n nodes bisection bandwidth is one link bandwidth. For linear array only one link needs to be broken to bisect the network into two partitions.
For ring topology with n nodes two links should be broken to bisect the network, so bisection bandwidth becomes bandwidth of two links.
For tree topology with n nodes can be bisected at the root by breaking one link, so bisection bandwidth is one link bandwidth.
For Mesh topology with n nodes, links should be broken to bisect the network, so bisection bandwidth is bandwidth of links.
For Hyper-cube topology with n nodes, n/2 links should be broken to bisect the network, so bisection bandwidth is bandwidth of n/2 links.
Significance of bisection bandwidth
Theoretical support for the importance of this measure of network performance was developed in the PhD research of Clark Thomborson (formerly Clark Thompson). Thomborson proved that important algorithms for sorting, Fast Fourier transformation, and matrix-matrix multiplication become communication-limited—as opposed to CPU-limited or memory-limited—on computers with insufficient bisection bandwidth. F. Thomson Leighton's PhD research tightened Thomborson's loose bound on the bisection bandwidth of a computationally-important variant of the De Bruijn graph known as the shuffle-exchange network. Based on Bill Dally's analysis of latency, average-case throughput, and hot-spot throughput of m-ary n-cube networks for various m, it can be observed that low-dimensional networks, in comparison to high-dimensional networks (e.g., binary n-cubes) with the same bisection bandwidth (e.g., tori), have reduced latency and higher hot-spot throughput.
References
Information theory
Network management
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https://en.wikipedia.org/wiki/Edison%20Academy%20Magnet%20School
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The Edison Academy Magnet School (formerly known as the Middlesex County Academy for Science, Mathematics and Engineering Technologies) is a four-year career academy and college preparatory magnet public high school located on the campus of the Middlesex County College in Edison, in Middlesex County, New Jersey, United States, serving students in ninth through twelfth grades as part of the Middlesex County Magnet Schools. The school serves students from all over Middlesex County who are eligible to apply to their program of choice while in eighth grade.
As of the 2021–22 school year, the school had an enrollment of 172 students and 11.5 classroom teachers (on an FTE basis), for a student–teacher ratio of 15.0:1. There were 5 students (2.9% of enrollment) eligible for free lunch and 1 (0.6% of students) eligible for reduced-cost lunch.
Awards, recognition and rankings
In September 2013, the academy was one of 15 schools in New Jersey to be recognized by the United States Department of Education as part of the National Blue Ribbon Schools Program, an award called the "most prestigious honor in the United States' education system" and which Education Secretary Arne Duncan described as schools that "represent examples of educational excellence".
Schooldigger.com ranked the school as one of 16 schools tied for first out of 381 public high schools statewide in its 2011 rankings (unchanged from the 2010 ranking) which were based on the combined percentage of students classified as proficient or above proficient on the language arts literacy (100.0%) and mathematics (100.0%) components of the High School Proficiency Assessment (HSPA).
In its listing of "America's Best High Schools 2016", the school was ranked 10th out of 500 best high schools in the country; it was ranked third among all high schools in New Jersey.
The Academy offers students the opportunity to apply and participate in the National Honor Society and Spanish National Honor Society.
History
Founded in 2000, the Academy inaugurated a freshman class of 40 students from all over the county and had its first graduating class in 2004. During the 2010–2011 school year, The Academy underwent the process of incorporating its first AP class into the curriculum, AP English Literature and Composition, running concurrent to the existing senior English class, British Literature. The first exam was administered that year, and the first full year curriculum will begin for the senior class of 2012. The Academy accepts 42 freshman students each year through a competitive admissions process.
Education
The major subjects of studies for four years are the following:
English: World Literature Honors, American Literature I/II Honors, and AP English Literature and Composition
Mathematics: Geometry Honors, Algebra II Honors, Precalculus Honors, AP Calculus AB, and AP Calculus BC
Lab Sciences: Biology Honors, Chemistry Honors, Physics Honors, and AP Environmental Science
Social Science: World History Honor
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https://en.wikipedia.org/wiki/Outline%20of%20algebraic%20structures
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In mathematics, there are many types of algebraic structures which are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms.
Another branch of mathematics known as universal algebra studies algebraic structures in general. From the universal algebra viewpoint, most structures can be divided into varieties and quasivarieties depending on the axioms used. Some axiomatic formal systems that are neither varieties nor quasivarieties, called nonvarieties, are sometimes included among the algebraic structures by tradition.
Concrete examples of each structure will be found in the articles listed.
Algebraic structures are so numerous today that this article will inevitably be incomplete. In addition to this, there are sometimes multiple names for the same structure, and sometimes one name will be defined by disagreeing axioms by different authors. Most structures appearing on this page will be common ones which most authors agree on. Other web lists of algebraic structures, organized more or less alphabetically, include Jipsen and PlanetMath. These lists mention many structures not included below, and may present more information about some structures than is presented here.
Study of algebraic structures
Algebraic structures appear in most branches of mathematics, and one can encounter them in many different ways.
Beginning study: In American universities, groups, vector spaces and fields are generally the first structures encountered in subjects such as linear algebra. They are usually introduced as sets with certain axioms.
Advanced study:
Abstract algebra studies properties of specific algebraic structures.
Universal algebra studies algebraic structures abstractly, rather than specific types of structures.
Varieties
Category theory studies interrelationships between different structures, algebraic and non-algebraic. To study a non-algebraic object, it is often useful to use category theory to relate the object to an algebraic structure.
Example: The fundamental group of a topological space gives information about the topological space.
Types of algebraic structures
In full generality, an algebraic structure may use any number of sets and any number of axioms in its definition. The most commonly studied structures, however, usually involve only one or two sets and one or two binary operations. The structures below are organized by how many sets are involved, and how many binary operations are used. Increased indentation is meant to indicate a more exotic structure, and the least indented levels are the most basic.
One set with no binary operations
Set: a degenerate algebraic structure S having no operations.
Pointed set: S has one or mo
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https://en.wikipedia.org/wiki/Band%20%28algebra%29
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In mathematics, a band (also called idempotent semigroup) is a semigroup in which every element is idempotent (in other words equal to its own square). Bands were first studied and named by .
The lattice of varieties of bands was described independently in the early 1970s by Biryukov, Fennemore and Gerhard. Semilattices, left-zero bands, right-zero bands, rectangular bands, normal bands, left-regular bands, right-regular bands and regular bands are specific subclasses of bands that lie near the bottom of this lattice and which are of particular interest; they are briefly described below.
Varieties of bands
A class of bands forms a variety if it is closed under formation of subsemigroups, homomorphic images and direct product. Each variety of bands can be defined by a single defining identity.
Semilattices
Semilattices are exactly commutative bands; that is, they are the bands satisfying the equation
for all and .
Bands induce a preorder that may be defined as if . Requiring commutativity implies that this preorder becomes a (semilattice) partial order.
Zero bands
A left-zero band is a band satisfying the equation
,
whence its Cayley table has constant rows.
Symmetrically, a right-zero band is one satisfying
,
so that the Cayley table has constant columns.
Rectangular bands
A rectangular band is a band that satisfies
for all , or equivalently,
for all ,
In any semigroup the first identity is sufficient to characterize a Nowhere commutative semigroup.
Nowhere commutative semigroup implies the first identity.
In any flexible magma so every element commutes with its square. So in any Nowhere commutative semigroup every element is idempotent thus any Nowhere commutative semigroup is in fact a Nowhere commutative band.
Thus in any Nowhere commutative semigroup
So commutes with and thus - the first characteristic identity.
In a any semigroup the first identity implies idempotence since so so idempotent (a band). Then
nowhere commutative since a band So in a band
In any semigroup the first identity also implies the second because .
The idempotents of a rectangular semigroup form a sub band that is a rectangular band but a rectangular semigroup may have elements that are not idempotent. In a band the second identity obviously implies the first but that requires idempotence. There exist semigroups that satisfy the second identity but are not bands and do not satisfy the first.
There is a complete classification of rectangular bands. Given arbitrary sets and one can define a magma operation on by setting
This operation is associative because for any three pairs , , we have
and likewise
These two magma identities
and
are together equivalent to the second characteristic identity above.
The two together also imply associativity . Any magma that satisfies these two rectangular identities and idempotence is therefore a rectangular band. So any magma that satisfies both the characteristic identiti
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https://en.wikipedia.org/wiki/Zerosumfree%20monoid
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In abstract algebra, an additive monoid is said to be zerosumfree, conical, centerless or positive if nonzero elements do not sum to zero. Formally:
This means that the only way zero can be expressed as a sum is as .
References
Semigroup theory
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https://en.wikipedia.org/wiki/Maha%20Ganapathi%20Mahammaya%20Temple
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The Shirali Maha Ganapathi Mahammaya Temple is the Kuladevata Temple (family temple) of the Goud Saraswat Brahmin community. The temple deity is a Kuladev of the Kamaths, Bhats, Puraniks, Prabhus, Joishys, Mallyas, Kudvas and Nayak families from the Goud Saraswat Brahmin community. The temple is located at Shirali in the Uttar Kannada district of Karnataka state. It is a five-minute drive from either Bhatkal or Murdeshwar. The Temple was built about 400 years ago. It was renovated in 1904.
These families are referred to as the Kulavis of the temple. The temple was established by devotees who migrated from Goa about 400–500 years ago. The presiding deities are Shri Mahaganapati (Vinayaka) and Shri Mahamaya (Shantadurga).
The deities were moved from Ella in Goa to nearby Golthi and Naveli on Divar Island during the Muslim destruction of Goa. Golthi / Goltim and Naveli / Navelim are located in Divar Island, Tiswadi Taluka of Goa. Shri Gomanteshwar and his affiliated deities still remain in Ella in Brahmapur. The older temple has been destroyed. On account of the hostile religious policies pursued by the Portuguese rulers around 1560, the devotees left Goltim and Navelim after the temple destruction. Unable to take with them the idols, they invoked the ‘saanidhya’ or the presence of the deities in the silver trunk of Lord Ganesha and the mask of goddess Mahamaya. When they reached Bhatkal they were unable to construct a temple immediately and kept these two symbols in a shop belonging to a devotee. Later on they constructed a temple in Shirali, a few miles north of Bhatkal, where it stands to this day. The deities are also called Pete Vinayaka and Shantadurga as they are located in a "pete", which means a town in Kannada. The temple has a unique darshan seva called, "mali".
According to the Archaeological Survey of Goa, The idol of Mahaganapati and Mahamaya (also referred as Durgadevi and Shantadurga) were located in Ella - Tiswadi Goa along with Shri Gomanteshwar and its affiliates. During the Muslim invasion in Goa (13th century) the temple in Ella was destroyed and the idols were transferred to Navelim and Goltim. Until the early 16th century, the deities were worshiped on the island, following which they were evicted by the overzealous Portuguese missionaries. The devotees had no choice but to transfer the idols to Khandepar and from there to its final destination Khandola. During the zealous Portuguese missionary acts, many devotees fled Goa and entered Karnataka. Traces of which, can still be found today. Along the way to Karnataka, families that settled in Karwar established a temple in Asnoti. For those who settled in Ankola, the sacred coconut of Shantadurga/ Durgadevi/ Mahamaya found no place at homes and was kept at the local temple to worship. Many years later, the local deity came to be known as Shantadurga. Some families went far ahead from northern Karnataka to the south coast of Bhatkal, where they felt secure and thus, establish
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https://en.wikipedia.org/wiki/Grothendieck%20connection
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In algebraic geometry and synthetic differential geometry, a Grothendieck connection is a way of viewing connections in terms of descent data from infinitesimal neighbourhoods of the diagonal.
Introduction and motivation
The Grothendieck connection is a generalization of the Gauss–Manin connection constructed in a manner analogous to that in which the Ehresmann connection generalizes the Koszul connection. The construction itself must satisfy a requirement of geometric invariance, which may be regarded as the analog of covariance for a wider class of structures including the schemes of algebraic geometry. Thus the connection in a certain sense must live in a natural sheaf on a Grothendieck topology. In this section, we discuss how to describe an Ehresmann connection in sheaf-theoretic terms as a Grothendieck connection.
Let be a manifold and a surjective submersion, so that is a manifold fibred over Let be the first-order jet bundle of sections of This may be regarded as a bundle over or a bundle over the total space of With the latter interpretation, an Ehresmann connection is a section of the bundle (over ) The problem is thus to obtain an intrinsic description of the sheaf of sections of this vector bundle.
Grothendieck's solution is to consider the diagonal embedding The sheaf of ideals of in consists of functions on which vanish along the diagonal. Much of the infinitesimal geometry of can be realized in terms of For instance, is the sheaf of sections of the cotangent bundle. One may define a first-order infinitesimal neighborhood of in to be the subscheme corresponding to the sheaf of ideals (See below for a coordinate description.)
There are a pair of projections given by projection the respective factors of the Cartesian product, which restrict to give projections One may now form the pullback of the fibre space along one or the other of or In general, there is no canonical way to identify and with each other. A Grothendieck connection is a specified isomorphism between these two spaces. One may proceed to define curvature and p-curvature of a connection in the same language.
See also
References
Osserman, B., "Connections, curvature, and p-curvature", preprint.
Katz, N., "Nilpotent connections and the monodromy theorem", IHES Publ. Math. 39 (1970) 175–232.
Connection (mathematics)
Algebraic geometry
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https://en.wikipedia.org/wiki/Journal%20of%20Business%20%26%20Economic%20Statistics
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The Journal of Business & Economic Statistics is a quarterly peer-reviewed academic journal published by the American Statistical Association. The journal covers a broad range of applied problems in business and economic statistics, including forecasting, seasonal adjustment, applied demand and cost analysis, applied econometric modeling, empirical finance, analysis of survey and longitudinal data related to business and economic problems, the impact of discrimination on wages and productivity, the returns to education and training, the effects of unionization, and applications of stochastic control theory to business and economic problems.
See also
List of scholarly journals in economics
References
External links
Journal of Business & Economic Statistics
American Statistical Association academic journals
Statistics journals
Econometrics journals
Academic journals established in 1983
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https://en.wikipedia.org/wiki/Technometrics
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Technometrics is a journal of statistics for the physical, chemical, and engineering sciences, published quarterly since 1959 by the American Society for Quality and the American Statistical Association.
Statement of purpose
The purpose of Technometrics is to contribute to the development and use of statistical methods in physical, chemical, and engineering sciences as well as information sciences and technology. This vision includes developments on the interface of statistics and computer science such as data mining, machine learning, large databases, and so on. The journal places a premium on clear communication among statisticians and practitioners of these sciences and an emphasis on the application of statistical concepts and methods to problems that occur in these fields. The journal will publish papers describing new statistical techniques, papers illustrating innovative application of known statistical methods, expository papers on particular statistical methods, and papers dealing with the philosophy and problems of applying statistical methods, when such papers are consistent with the journal's objective. Every article shall include adequate justification of the application of the technique, preferably by means of an actual application to a problem in the physical, chemical, engineering or information sciences. All papers must contain a short, clear summary of contents and conclusions. Mathematical derivations not essential to the flow of the text should be placed in an appendix or a supplementary file. Brief descriptions of problems requiring solution and short technical notes that clearly pertain to the journal's purpose will also be considered for publication. Concise letters to the editor will be published when they are considered timely and appropriate.
Editor
V. Roshan Joseph, Georgia Institute of Technology
Book Reviews Editor
S. Ejaz Ahmed, Brock University, Canada
Former Editors
J. Stuart Hunter (1959–1963)
Fred C. Leone (1964–1968)
Harry Smith, Jr. (1969–1971)
Donald A. Gardiner (1972–1974)
William H. Lawton (1975–1977)
John W. Wilkinson (1978–1980
Robert G. Easterling (1981–1983)
Jerald F. Lawless (1984–1986)
William Q. Meeker, Jr. (1987–1989)
Vijayan Nair (1990–1992)
Stephen B. Vardeman (1993–1995)
Max Morris (1996–1998)
Karen Kafadar (1999–2001)
William I. Notz (2002–2004)
Randy R. Sitter (2005–2007)
David M. Steinberg (2008–2010)
Hugh A. Chipman (2011–2013)
Peihua Qiu (2014–2016)
Daniel Apley (2017-2019)
References
External links
Official website
Home page on the American Statistical Association website
American Statistical Association academic journals
Statistics journals
Computational statistics journals
Academic journals established in 1959
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https://en.wikipedia.org/wiki/Miroslav%20Kat%C4%9Btov
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Miroslav Katětov (; March 17, 1918, Chembar, Russia – December 15, 1995) was a Czech mathematician, chess master, and psychologist. His research interests in mathematics included topology and functional analysis. He was an author of the Katětov–Tong insertion theorem. From 1953 to 1957 he was rector of Charles University in Prague.
External links
Biography
1918 births
1995 deaths
People from Penza Oblast
Czechoslovak mathematicians
Topologists
Czech chess players
Czech psychologists
Charles University alumni
Rectors of Charles University
Czech expatriates in Russia
20th-century chess players
20th-century psychologists
Soviet emigrants to Czechoslovakia
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https://en.wikipedia.org/wiki/Sm%C3%A5land%2C%20Inder%C3%B8y
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Breivika or Breidvik (Statistics Norway calls it Småland) is a village in the municipality of Inderøy in Trøndelag county, Norway. It is located along the Trondheimsfjord in the northern part of the Inderøya peninsula, about northwest of the village of Gangstadhaugen.
The village has a population (2018) of 295 and a population density of .
References
Villages in Trøndelag
Inderøy
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https://en.wikipedia.org/wiki/Forbregd/Lein
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Forbregd and Lein are two small adjoining villages in the municipality of Verdal in Trøndelag county, Norway. Statistics Norway classifies the urban area as Forbregd/Lein. The village area is located about northeast of the town of Verdalsøra and about northwest of Stiklestad, along the southern shore of the lake Leksdalsvatnet. The village has a population (2018) of 849 and a population density of .
References
Verdal
Villages in Trøndelag
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https://en.wikipedia.org/wiki/Sylvan%20Learning
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Sylvan Learning, Inc. (formerly Sylvan Learning Corporation) consists of franchised and corporate supplemental learning centers which provide personalized instruction in reading, writing, mathematics, study skills, homework support, and test preparation for college entrance and state exams. Some centers also offer STEM courses in robotics and coding. Sylvan provides personalized learning programs and primarily serves students in primary and secondary education.
History
Sylvan Learning began in Portland, Oregon in 1979 at the Sylvan Hill Medical Center Building. It was founded by former school teacher W. Berry Fowler, who had also worked with the educational company The Reading Game. By 1983, Sylvan had dozens of franchises and moved its headquarters to Bellevue, WA. In 1986, having over 500 franchises, Sylvan went public on the NASDAQ exchange and used funds to develop corporate learning centers in key cities. By July 1987, KinderCare, then based in Montgomery, AL, owned the majority of stock and moved the company to Alabama. Most of the staff did not relocate.
In 1991 the company was taken over by R. Christopher Hoehn-Saric and Douglas L. Becker. In 1997 the company had an annual revenues of $246 million, and in addition to tutoring centers, Sylvan had expanded to offer teacher training, computerized testing, distance learning, and other services.
In 2003, Sylvan Learning was purchased by Apollo Management from Sylvan Learning Systems Inc., its parent company. (Sylvan Learning Systems Inc. shifted focus to post-secondary education, and to reflect that change was renamed Laureate Education in 2004.) In 2016, John McAuliffe was named as Chief Executive Officer.
Over 28 franchised centers located in multiple states closed between 2008 and 2012, some very suddenly. Sylvan worked with franchisees to open more centers and planned to open 200 non-franchise locations. Most Sylvan locations utilize a digital learning system called SylvanSync, which provides lessons on tablets with instructors teaching lessons, which become more independent if the student progresses. Sylvan employs standardized assessments to place and monitor students and to evaluate instructors.
See also
Storefront school
References
Bibliography
External links
Companies based in Baltimore
Education companies established in 1979
Education companies of the United States
Learning programs
Test preparation companies
1979 establishments in Oregon
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https://en.wikipedia.org/wiki/Vanishing%20cycle
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In mathematics, vanishing cycles are studied in singularity theory and other parts of algebraic geometry. They are those homology cycles of a smooth fiber in a family which vanish in the singular fiber.
For example, in a map from a connected complex surface to the complex projective line, a generic fiber is a smooth Riemann surface of some fixed genus g and, generically, there will be isolated points in the target whose preimages are nodal curves. If one considers an isolated critical value and a small loop around it, in each fiber, one can find a smooth loop such that the singular fiber can be obtained by pinching that loop to a point. The loop in the smooth fibers gives an element of the first homology group of a surface, and the monodromy of the critical value is defined to be the monodromy of the first homology of the fibers as the loop is traversed, i.e. an invertible map of the first homology of a (real) surface of genus g.
A classical result is the Picard–Lefschetz formula, detailing how the monodromy round the singular fiber acts on the vanishing cycles, by a shear mapping.
The classical, geometric theory of Solomon Lefschetz was recast in purely algebraic terms, in SGA7. This was for the requirements of its application in the context of l-adic cohomology; and eventual application to the Weil conjectures. There the definition uses derived categories, and looks very different. It involves a functor, the nearby cycle functor, with a definition by means of the higher direct image and pullbacks. The vanishing cycle functor then sits in a distinguished triangle with the nearby cycle functor and a more elementary functor. This formulation has been of continuing influence, in particular in D-module theory.
See also
Thom–Sebastiani Theorem
References
Dimca, Alexandru; Singularities and Topology of Hypersurfaces.
Section 3 of Peters, C.A.M. and J.H.M. Steenbrink: Infinitesimal variations of Hodge structure and the generic Torelli problem for projective hypersurfaces, in : Classification of Algebraic Manifolds, K. Ueno ed., Progress inMath. 39, Birkhauser 1983.
For the étale cohomology version, see the chapter on monodromy in
, see especially Pierre Deligne, Le formalisme des cycles évanescents, SGA7 XIII and XIV.
External links
Vanishing Cycle in the Encyclopedia of Mathematics
Algebraic topology
Topological methods of algebraic geometry
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https://en.wikipedia.org/wiki/Steinberg%20representation
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In mathematics, the Steinberg representation, or Steinberg module or Steinberg character, denoted by St, is a particular linear representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-dimensional sign representation ε of a Coxeter or Weyl group that takes all reflections to –1.
For groups over finite fields, these representations were introduced by , first for the general linear groups, then for classical groups, and then for all Chevalley groups, with a construction that immediately generalized to the other groups of Lie type that were discovered soon after by Steinberg, Suzuki and Ree.
Over a finite field of characteristic p, the Steinberg representation has degree equal to the largest power of p dividing the order of the group.
The Steinberg representation is the Alvis–Curtis dual of the trivial 1-dimensional representation.
, , and defined analogous Steinberg representations (sometimes called special representations) for algebraic groups over local fields. For the general linear group GL(2), the dimension of the Jacquet module of a special representation is always one.
The Steinberg representation of a finite group
The character value of St on an element g equals, up to sign, the order of a Sylow subgroup of the centralizer of g if g has order prime to p, and is zero if the order of g is divisible by p.
The Steinberg representation is equal to an alternating sum over all parabolic subgroups containing a Borel subgroup, of the representation induced from the identity representation of the parabolic subgroup.
The Steinberg representation is both regular and unipotent, and is the only irreducible regular unipotent representation (for the given prime p).
The Steinberg representation is used in the proof of Haboush's theorem (the Mumford conjecture).
Most finite simple groups have exactly one Steinberg representation. A few have more than one because they are
groups of Lie type in more than one way. For symmetric groups (and other Coxeter groups) the sign representation is analogous to the Steinberg representation. Some of the sporadic simple groups act as doubly transitive permutation groups so have a BN-pair for which one can define a Steinberg representation, but for most of the sporadic groups there is no known analogue of it.
The Steinberg representation of a p-adic group
, , and introduced Steinberg representations for algebraic groups over local fields. showed that the different ways of defining Steinberg representations are equivalent.
and showed how to realize the Steinberg representation in the cohomology group H(X) of the Bruhat–Tits building of the group.
References
Finite Groups of Lie Type: Conjugacy Classes and Complex Characters (Wiley Classics Library) by Roger W. Carter, John Wiley & Sons Inc; New Ed edition (August 1993)
R. Steinberg, Collected Papers, Amer. Math. Soc. (1997) pp. 580–586
Representation theory of algebraic gro
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https://en.wikipedia.org/wiki/James%20Robert%20Brown
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James Robert Brown (born 1949) is a Canadian philosopher of science. He is an emeritus professor of philosophy at the University of Toronto. In the philosophy of mathematics, he has advocated mathematical Platonism, visual reasoning, and in the philosophy of science he has defended scientific realism mostly against anti-realist views associated with social constructivism. He has also argued for the socialization of medical research (especially pharmaceutical research). He is largely known for his work on thought experiments.
Elected: Academy of Sciences Leopoldina (Deutsche Akademie der Naturforscher Leopoldina –
Nationale Akademie der Wissenschaften) 2004, Royal Society of Canada 2007, Académie Internationale de Philosophie des Sciences 2010
Brown was born in Montreal, Quebec. He is married to the philosopher Kathleen Okruhlik.
Books
1989 The Rational and the Social (Routledge 1989)
1991 The Laboratory of the Mind: Thought Experiments in the Natural Sciences (Routledge 1991, second edition 2010)
1994 Smoke and Mirrors: How Science Reflects Reality (Routledge 1994)
1999 Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures (Routledge 1999, second edition 2008)
2001 Who Rules in Science? An Opinionated Guide to the Wars (Harvard 2001)
2012 Platonism, Naturalism, and Mathematical Knowledge (Routledge 2012)
2017 On Foundations of Seismology: Bringing Idealizations Down to Earth (with M. Slawinski)
Books edited include:
2012 Thought Experiments in Philosophy, Science, and the Arts (ed. with M. Frappier and L. Meynell ) (Routledge 2012)
2018 The Routledge Companion to Thought Experiments (ed. with M. Stuart and Y. Fehige), (Routledge 2018)
References
External links
James Brown’s Homepage
"Plato's Heaven: A User's Guide - A conversation with James Robert Brown", Ideas Roadshow, 2013
20th-century Canadian philosophers
Academics from Montreal
Canadian philosophers
Fellows of the Royal Society of Canada
Living people
Members of the German National Academy of Sciences Leopoldina
Moral realists
Philosophers of mathematics
Philosophers of science
University of Guelph alumni
Academic staff of the University of Toronto
1949 births
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https://en.wikipedia.org/wiki/Miles%20Reid
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Miles Anthony Reid FRS (born 30 January 1948) is a mathematician who works in algebraic geometry.
Education
Reid studied the Cambridge Mathematical Tripos at Trinity College, Cambridge and obtained his Ph.D. in 1973 under the supervision of Peter Swinnerton-Dyer and Pierre Deligne.
Career
Reid was a research fellow of Christ's College, Cambridge from 1973 to 1978. He became a lecturer at the University of Warwick in 1978 and was appointed professor there in 1992. He has written two well known books: Undergraduate Algebraic Geometry and Undergraduate Commutative Algebra.
Awards and honours
Reid was elected a Fellow of the Royal Society in 2002. In the same year, he participated as an Invited Speaker in the
International Congress of Mathematicians in Beijing. Reid was awarded the Senior Berwick Prize in 2006 for his paper with Alessio Corti and Aleksandr Pukhlikov, "Fano 3-fold hypersurfaces", which made a big advance in the study of 3-dimensional algebraic varieties. In 2023 he was awarded the Sylvester Medal of the Royal Society.
Personal life
Reid speaks Japanese and Russian and has given lectures in Japanese.
Bibliography
His most famous book is
Undergraduate Algebraic Geometry, Cambridge University Press 1988 ()
Other books
Undergraduate commutative algebra, Cambridge University Press 1995,
with Balazs Szendroi: Geometry and topology, Cambridge University Press 2007
His most famous translation is the two volume work by Shafarevich
Basic Algebraic Geometry 1 ()
Basic Algebraic Geometry 2 ()
References
1948 births
Living people
People from Hoddesdon
20th-century English mathematicians
21st-century English mathematicians
Algebraic geometers
Fellows of Christ's College, Cambridge
Fellows of the Royal Society
Academics of the University of Warwick
Alumni of Trinity College, Cambridge
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https://en.wikipedia.org/wiki/Yuktibh%C4%81%E1%B9%A3%C4%81
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Yuktibhāṣā (), also known as Gaṇita-yukti-bhāṣā and (English: Compendium of Astronomical Rationale), is a major treatise on mathematics and astronomy, written by the Indian astronomer Jyesthadeva of the Kerala school of mathematics around 1530. The treatise, written in Malayalam, is a consolidation of the discoveries by Madhava of Sangamagrama, Nilakantha Somayaji, Parameshvara, Jyeshtadeva, Achyuta Pisharati, and other astronomer-mathematicians of the Kerala school. It also exists in a Sanskrit version, with unclear author and date, composed as a rough translation of the Malayalam original.
The work contains proofs and derivations of the theorems that it presents. Modern historians used to assert, based on the works of Indian mathematics that first became available, that early Indian scholars in astronomy and computation lacked in proofs, but demonstrates otherwise.
Some of its important topics include the infinite series expansions of functions; power series, including of π and π/4; trigonometric series of sine, cosine, and arctangent; Taylor series, including second and third order approximations of sine and cosine; radii, diameters and circumferences.
mainly gives rationale for the results in Nilakantha's Tantra Samgraha. It is considered an early text to give some ideas of calculus like Taylor and infinity series, predating Newton and Leibniz by two centuries. The treatise was largely unnoticed outside India, as it was written in the local language of Malayalam. In modern times, due to wider international cooperation in mathematics, the wider world has taken notice of the work. For example, both Oxford University and the Royal Society of Great Britain have given attribution to pioneering mathematical theorems of Indian origin that predate their Western counterparts.
Contents
contains most of the developments of the earlier Kerala school, particularly Madhava and Nilakantha. The text is divided into two parts – the former deals with mathematical analysis and the latter with astronomy. Beyond this, the continuous text does not have any further division into subjects or topics, so published editions divide the work into chapters based on editorial judgment.
Mathematics
This subjects treated in the mathematics part of the can be divided into seven chapters:
parikarma: logistics (the eight mathematical operations)
daśapraśna: ten problems involving logistics
bhinnagaṇita: arithmetic of fractions
trairāśika: rule of three
kuṭṭakāra: pulverisation (linear indeterminate equations)
paridhi-vyāsa: relation between circumference and diameter: infinite series and approximations for the ratio of the circumference and diameter of a circle
jyānayana: derivation of Rsines: infinite series and approximations for sines.
The first four chapters of the contain elementary mathematics, such as division, the Pythagorean theorem, square roots, etc. Novel ideas are not discussed until the sixth chapter on circumference of a circle. contains a
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https://en.wikipedia.org/wiki/Natural%20filtration
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In the theory of stochastic processes in mathematics and statistics, the generated filtration or natural filtration associated to a stochastic process is a filtration associated to the process which records its "past behaviour" at each time. It is in a sense the simplest filtration available for studying the given process: all information concerning the process, and only that information, is available in the natural filtration.
More formally, let (Ω, F, P) be a probability space; let (I, ≤) be a totally ordered index set; let (S, Σ) be a measurable space; let X : I × Ω → S be a stochastic process. Then the natural filtration of F with respect to X is defined to be the filtration F•X = (FiX)i∈I given by
i.e., the smallest σ-algebra on Ω that contains all pre-images of Σ-measurable subsets of S for "times" j up to i.
In many examples, the index set I is the natural numbers N (possibly including 0) or an interval [0, T] or [0, +∞); the state space S is often the real line R or Euclidean space Rn.
Any stochastic process X is an adapted process with respect to its natural filtration.
References
See also
Filtration (mathematics)
Stochastic processes
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https://en.wikipedia.org/wiki/John%20Nelder
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John Ashworth Nelder (8 October 1924 – 7 August 2010) was a British statistician known for his contributions to experimental design, analysis of variance, computational statistics, and statistical theory.
Contributions
Nelder's work was influential in statistics. While leading research at Rothamsted Experimental Station, Nelder developed and supervised the updating of the statistical software packages GLIM and GenStat: Both packages are flexible high-level programming languages that allow statisticians to formulate linear models concisely. GLIM influenced later environments for statistical computing such as S-PLUS and R. Both GLIM and GenStat have powerful facilities for the analysis of variance for block experiments, an area where Nelder made many contributions.
In statistical theory, Nelder proposed the generalized linear model together with Robert Wedderburn. Nelder and Wedderburn formulated generalized linear models as a way of unifying various other statistical models, including linear regression, logistic regression and Poisson regression. They proposed an iteratively reweighted least squares method for maximum likelihood estimation of the model parameters.
In statistical inference, Nelder (along with George Barnard and A. W. F. Edwards) emphasized the importance of the likelihood in data analysis, promoting this "likelihood approach" as an alternative to frequentist and Bayesian statistics.
In response-surface optimization, Nelder and Roger Mead proposed the Nelder–Mead simplex heuristic, widely used in engineering and statistics.
Biography
Born in Brushford, near Dulverton, Somerset, Nelder was educated at Blundell's School and Sidney Sussex College, Cambridge, where he read Mathematics.
Nelder's appointments included Head of the Statistics Section at the National Vegetable Research Station, Wellesbourne, from 1951 to 1968 and head of the Statistics Department at Rothamsted Experimental Station from 1968 to 1984. During his time at Wellesbourne he spent a year (1965–1966) at the Waite Institute in Adelaide, South Australia, where he worked with Graham Wilkinson on Genstat. He held an appointment as Visiting Professor at Imperial College London from 1972 onwards.
He was responsible, with Max Nicholson and James Ferguson-Lees, for debunking the Hastings Rarities – sightings of a series of rare birds, preserved by a taxidermist and provided with bogus histories.
Nelder died on 7 August 2010 in Luton and Dunstable Hospital, taken there after a fall at home, which was incidental to the cause of death.
Awards and distinctions
Nelder was elected a Fellow of the Royal Society in 1976 and received the Royal Statistical Society's Guy Medal in Gold in 2005. He was also the recipient of the inaugural Karl Pearson Prize of the International Statistical Institute, with Peter McCullagh, "for their monograph Generalized Linear Models (1983)".
As tribute on his eightieth birthday, a festschrift Methods and Models in Statistics: In Honour of Pro
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https://en.wikipedia.org/wiki/Cyril%20Offord
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Albert Cyril Offord FRS FRSE (9 June 1906 – 4 June 2000) was a British mathematician. He was the first professor of mathematics at the London School of Economics.
Life
He was born in London on 9 June 1906 the eldest child of Albert Edwin Offord, a master printer, and his wife Hester Louise, a former opera singer. The family were Plymouth Brethren. He was educated at Hackney Downs Grammar School. He then studied Mathematics at University College, London. He then went to St John's College, Cambridge as a postgraduate, working with Prof John Edensor Littlewood.
He received two Ph.D.s in mathematics: the first from the University of London (under Bosanquet) in 1932, the second from Cambridge (under Hardy) in 1936.
In 1940 he left Cambridge to lecture at University College, Bangor. In 1942 he moved to King's College, Newcastle-upon-Tyne (later being named the University of Newcastle). He was created Professor of Mathematics in 1945.
In 1946 he was elected a Fellow of the Royal Society of Edinburgh. His proposers were Sir Edmund Whittaker, John William Heslop-Harrison, Alexander Aitken and Alfred Dennis Hobson. He was elected a Fellow of the Royal Society of London in 1952.
In 1948 he left Newcastle to become Professor of Mathematics at Birkbeck College in London replacing Prof Dienes. He left in 1966 to take up a new chair at London School of Economics. He retired in 1973 then becoming a senior research fellow at Imperial College, London.
He died in Oxford on 4 June 2000.
Family
In 1945 he married Margaret Yvonne Pickard (generally known as Rita), an English teacher. They had one daughter, Margaret Offord (born 1949).
See also
Littlewood–Offord problem
References
External links
Royal Society basic cv
Royal Society citation
Royal Society: photograph
1906 births
2000 deaths
20th-century British mathematicians
People educated at Hackney Downs School
Fellows of the Royal Society
Fellows of the Royal Society of Edinburgh
Alumni of St John's College, Cambridge
Academics of Birkbeck, University of London
Academics of the London School of Economics
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https://en.wikipedia.org/wiki/Constance%20Reid
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Constance Bowman Reid (January 3, 1918 – October 14, 2010)
was the author of several biographies of mathematicians and popular books about mathematics. She received several awards for mathematical exposition. She was not a mathematician but came from a mathematical family—one of her sisters was Julia Robinson, and her brother-in-law was Raphael M. Robinson.
Background and education
Reid was born in St. Louis, Missouri, the daughter of Ralph Bowers Bowman and Helen (Hall) Bowman. One of her younger sisters was the mathematician Julia Robinson. The family moved to Arizona and then to San Diego when the girls were a few years old.
In 1950 she married a law student, Neil D. Reid, with whom she had two children, Julia and Stewart.
Reid received a Bachelor of Arts degree from San Diego State University in 1938 and a Master of Education degree from University of California, Berkeley in 1949. She worked as a teacher of English and journalism at San Diego High School from 1939 to 1950, and as a free-lance writer since then. She has said, "I always wanted to be a writer, but it took me a while to find my subject."
Works
Reid's first published work was a memoir of her work in a World War II bomber factory, Slacks and Calluses, published in 1944. She also published a short story.
Her first mathematical publication was an article on perfect numbers for Scientific American. Reid remarked in an interview that some readers objected to her as an author: "But the readers (maybe, just one reader, I have forgotten now) objected that articles in Scientific American should be written by authorities in their fields and not by housewives!"
The Scientific American article led to an invitation from Robert L. Crowell of the Thomas Y. Crowell Co. publishing house to write "a little book on numbers" that became From Zero to Infinity. Two more popular math books for Crowell followed: Introduction to Higher Mathematics for the General Reader in 1959 and A Long Way from Euclid in 1963.
After writing these books she felt she had run out of ideas, and her sister Julia Robinson suggested that she should update Eric Temple Bell's collection of mathematical biographies, Men of Mathematics.
After travelling to Göttingen to absorb some mathematical culture, Reid decided instead to write a full-length biography of David Hilbert, who she considered the greatest mathematician of the first half of the twentieth century. Julia encouraged her in this project, and the biography was published in 1970 as Hilbert. The Hilbert biography was a success among mathematicians, and her next book was a biography of another Göttingen figure, Richard Courant, published in 1976 as Courant in Göttingen and New York. Her next book, published in 1982, was a biography of the mathematical statistician Jerzy Neyman, who like Courant had emigrated to the United States and built a new career there.
An attempt to write a biography of Eric Temple Bell proved unexpectedly difficult, as he had been very se
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https://en.wikipedia.org/wiki/Schottky%20group
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In mathematics, a Schottky group is a special sort of Kleinian group, first studied by .
Definition
Fix some point p on the Riemann sphere. Each Jordan curve not passing through p divides the Riemann sphere into two pieces, and we call the piece containing p the "exterior" of the curve, and the other piece its "interior". Suppose there are 2g disjoint Jordan curves A1, B1,..., Ag, Bg in the Riemann sphere with disjoint interiors. If there are Möbius transformations Ti taking the outside of Ai onto the inside of Bi, then the group generated by these transformations is a Kleinian group. A Schottky group is any Kleinian group that can be constructed like this.
Properties
By work of , a finitely generated Kleinian group is Schottky if and only if it is finitely generated, free, has nonempty domain of discontinuity, and all non-trivial elements are loxodromic.
A fundamental domain for the action of a Schottky group G on its regular points Ω(G) in the Riemann sphere is given by the exterior of the Jordan curves defining it. The corresponding quotient space Ω(G)/G is given by joining up the Jordan curves in pairs, so is a compact Riemann surface of genus g. This is the boundary of the 3-manifold given by taking the quotient (H∪Ω(G))/G of 3-dimensional hyperbolic H space plus the regular set Ω(G) by the Schottky group G, which is a handlebody of genus g. Conversely any compact Riemann surface of genus g can be obtained from some Schottky group of genus g.
Classical and non-classical Schottky groups
A Schottky group is called classical if all the disjoint Jordan curves corresponding to some set of generators can be chosen to be circles. gave an indirect and non-constructive proof of the existence of non-classical Schottky groups, and gave an explicit example of one. It has been shown by that all finitely generated classical Schottky groups have limit sets of Hausdorff dimension bounded above strictly by a universal constant less than 2. Conversely, has proved that there exists a universal lower bound on the Hausdorff dimension of limit sets of all non-classical Schottky groups.
Limit sets of Schottky groups
The limit set of a Schottky group, the complement of Ω(G), always has Lebesgue measure zero, but can have positive d-dimensional Hausdorff measure for d < 2. It is perfect and nowhere dense with positive logarithmic capacity.
The statement on Lebesgue measures follows for classical Schottky groups from the existence of the Poincaré series
Poincaré showed that the series | ci |−4 is summable over the non-identity elements of the group. In fact taking a closed disk in the interior of the fundamental domain, its images under different group elements are disjoint and contained in a fixed disk about 0. So the sums of the areas is finite. By the changes of variables formula, the area is greater than a constant times | ci |−4.
A similar argument implies that the limit set has Lebesgue measure zero. For it is contained in the complement o
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https://en.wikipedia.org/wiki/Win%20probability%20added
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Win probability added (WPA) is a sport statistic which attempts to measure a player's contribution to a win by figuring the factor by which each specific play made by that player has altered the outcome of a game. It is used for baseball and American football.
Explanation
Some form of win probability has been around for about 40 years; however, until computer use became widespread, win probability added was often difficult to derive, or imprecise. With the aid of Retrosheet, however, win probability added has become substantially easier to calculate. The win probability for a specific situation in baseball (including the inning, number of outs, men on base, and score) is obtained by first finding all the teams that have encountered this situation. Then the winning percentage of these teams in these situations is found. This probability figure is then adjusted for home-field advantage. Thus win probability added is the difference between the win probability when the player came to bat and the win probability when the play ended.
Win probability and win shares
Some people confuse win probability added with win shares, since both are baseball statistics that attempt to measure a player's win contribution. However, they are quite different. In win shares, a player with 0 win shares has contributed nothing to his team; in win probability added, a player with 0 win probability added points is average. Also, win shares would give the same amount of credit to a player if he hit a lead-off solo home run as if he hit a walk-off solo home run; WPA, however, would give vastly more credit to the player who hit the walk-off homer.
Baseball
MLB postseason
In Game 6 of the 2011 World Series, St. Louis Cardinals' third-baseman David Freese posted the best WPA in Major League Baseball postseason history, with a 0.969, which was 0.099 better than the now-second-best WPA of .870, posted by the Los Angeles Dodgers' Kirk Gibson in Game 1 of the 1988 World Series. The third- and fourth-best WPAs are .854 (by the San Diego Padres' Steve Garvey in Game 4 of the 1984 National League Championship Series) and 0.832 (by the Cardinals' Lance Berkman in Game 6 of the 2011 World Series).
References
Baseball statistics
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https://en.wikipedia.org/wiki/1924%20Kohat%20riots
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The 1924 Kohat riots happened in the Kohat town of the North-West Frontier Province, British India in 1924. In three days (9–11 September) of rioting, official statistics peg the number of casualties among Hindus and Sikhs at more than thrice that of Muslims; almost the entire Hindu population had to be evacuated to Rawalpindi.
Background
In the 1921 census, Kohat had a population of about 8,000 Hindus and Sikhs, and 19,000 Muslims. Much of the bureaucracy was composed of Muslims. The Hindus were economically dominant: income tax records of the same year note indicate they paid four times as much tax as Muslims. Prior to the twentieth century, says Patrick McGinn, relations between the two religious communities were peaceful and exhibited multiple instances of cooperation.
However, with the turn of the century, waves of religiopolitical consciousness swept Kohat. With the Indian Nationalist Movement being extensively Hindu-ised, subcontinental Muslims sought out faith-based avenues for political aspirations. After the disintegration of the Khilafat Movement, local Ulema had rebranded themselves as defenders of Islam. Aggressive efforts by Arya Samaj stirred communal tensions in no insignificant manner either — in 1907, Provincial Commissioner H. Deane held the Arya Samaj to be primarily responsible for the sudden rise in religious antagonism. A minor riot broke out in 1909.
In June 1924, the son of Sardar Makan Singh eloped with a Muslim girl and the affair was communalised. In early August, the editor of Guru Ghantal—a Hindu Newspaper based in Lahore and having a large audience in Kohat—was prosecuted for publishing inflammatory articles attacking Islam. This was the culmination of a months-long partisan tirade in the press where Hindus claimed to be speaking out against forced conversions and Muslims against dishonour of their religion. The same month, Muslims lodged protests against the proposed construction of a bathing ghat for Hindu women in the vicinity of a Muslim neighborhood; the government adjudicated the dispute in favour of Hindus on 2 September. Overall, in the opinion of Patrick McGinn, the dominant discourse in the town on the eve of the riot smacked of competitive communalism.
Riot
Prelude
The immediate trigger of the dispute was a defaming poem written about Krishna during Janmastami and the retaliation by Jiwan Das—secretary of the local branch of Sanatan Dharm Sabha—publishing a pamphlet of bhajans, titled Krishan Sandesh, on the occasion of Krishna Janmashtami, 22 August. One particular bhajan, allegedly written by a poet from Jammu, gave calls to evict all Muslims to Arabia and to construct a Vishnu Temple at Kaaba. The poem was brought to broader public attention on the occasion of a Muslim funeral, on 1 September. Enraged Muslims held meetings in mosques against the "gross defamation" of Islam, and fanatical preachers from outside arrived in Kohat. With tensions escalating rapidly, the Dharm Sabha argued but to li
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https://en.wikipedia.org/wiki/Schwartz%E2%80%93Bruhat%20function
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In mathematics, a Schwartz–Bruhat function, named after Laurent Schwartz and François Bruhat, is a complex valued function on a locally compact abelian group, such as the adeles, that generalizes a Schwartz function on a real vector space. A tempered distribution is defined as a continuous linear functional on the space of Schwartz–Bruhat functions.
Definitions
On a real vector space , the Schwartz–Bruhat functions are just the usual Schwartz functions (all derivatives rapidly decreasing) and form the space .
On a torus, the Schwartz–Bruhat functions are the smooth functions.
On a sum of copies of the integers, the Schwartz–Bruhat functions are the rapidly decreasing functions.
On an elementary group (i.e., an abelian locally compact group that is a product of copies of the reals, the integers, the circle group, and finite groups), the Schwartz–Bruhat functions are the smooth functions all of whose derivatives are rapidly decreasing.
On a general locally compact abelian group , let be a compactly generated subgroup, and a compact subgroup of such that is elementary. Then the pullback of a Schwartz–Bruhat function on is a Schwartz–Bruhat function on , and all Schwartz–Bruhat functions on are obtained like this for suitable and . (The space of Schwartz–Bruhat functions on is endowed with the inductive limit topology.)
On a non-archimedean local field , a Schwartz–Bruhat function is a locally constant function of compact support.
In particular, on the ring of adeles over a global field , the Schwartz–Bruhat functions are finite linear combinations of the products over each place of , where each is a Schwartz–Bruhat function on a local field and is the characteristic function on the ring of integers for all but finitely many . (For the archimedean places of , the are just the usual Schwartz functions on , while for the non-archimedean places the are the Schwartz–Bruhat functions of non-archimedean local fields.)
The space of Schwartz–Bruhat functions on the adeles is defined to be the restricted tensor product of Schwartz–Bruhat spaces of local fields, where is a finite set of places of . The elements of this space are of the form , where for all and for all but finitely many . For each we can write , which is finite and thus is well defined.
Examples
Every Schwartz–Bruhat function can be written as , where each , , and . This can be seen by observing that being a local field implies that by definition has compact support, i.e., has a finite subcover. Since every open set in can be expressed as a disjoint union of open balls of the form (for some and ) we have
. The function must also be locally constant, so for some . (As for evaluated at zero, is always included as a term.)
On the rational adeles all functions in the Schwartz–Bruhat space are finite linear combinations of over all rational primes , where , , and for all but finitely many . The sets and are the field of p-adic numbers and ring of p-
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https://en.wikipedia.org/wiki/Sergio%20Gadea
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Sergio Gadea Panisello (born 30 December 1984 in Puçol, Valencian Community) is a Spanish motorcycle road racer. He started to run professionally in 2003.
Career statistics
Grand Prix motorcycle racing
By season
By year
(Races in bold indicate pole position, races in italics indicate fastest lap of the race)
Superbike World Championship
Races by year
Supersport World Championship
Races by year
(key)
External links
1984 births
Living people
People from Puçol
Sportspeople from the Province of Valencia
Spanish motorcycle racers
125cc World Championship riders
Moto2 World Championship riders
Superbike World Championship riders
Supersport World Championship riders
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https://en.wikipedia.org/wiki/Henry%20Thomas%20Herbert%20Piaggio
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Henry Thomas Herbert Piaggio (2 June 1884–26 June 1967) was an English mathematician. Educated at the City of London School and St John's College, Cambridge, he was appointed lecturer in mathematics at the University of Nottingham in 1908 and then the first Professor of Mathematics in 1919. He was the author of "An Elementary Treatise on Differential Equations and their Applications".-
References
External links
. (MacTutor version of Three Sadleirian Professors)
new members - Margate Civic Society ("The Old and New Meet at the Rendezvous"), Winter 2007, Issue No. 345 Henry's father Francis ("Frank") Piaggio briefly operated a dancing academy in the Marine Palace, which he leased from 1895. The Marine Palace was built in 1884 and destroyed in the Great Storm of 1897, which devastated Margate.
1884 births
1967 deaths
Academics of the University of Nottingham
Mathematicians from London
Alumni of St John's College, Cambridge
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https://en.wikipedia.org/wiki/Uwe%20Storch
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Uwe Storch (born 12 July 1940, Leopoldshall– Lanzarote, 17 September 2017) was a German mathematician. His field of research was
commutative algebra and analytic and algebraic geometry, in particular derivations, divisor class group, resultants.
Storch studied mathematics, physics and mathematical logic in
Münster and in
Heidelberg. He got his PhD 1966 under the supervision of Heinrich Behnke with a thesis on almost (or Q) factorial rings.
1972 Habilitation in Bochum, 1974 professor in Osnabrück and since 1981 professor for algebra and geometry in Bochum. 2005 Emeritation. Uwe Storch is married and has four sons.
Theorem of Eisenbud–Evans–Storch
The Theorem of Eisenbud-Evans-Storch states that
every algebraic variety in n-dimensional affine space
is given geometrically (i.e. up to radical) by n polynomials.
Selected publications
Günther Scheja and Uwe Storch, Lehrbuch der Algebra, 2 volumes, Stuttgart 1980 (1st edition was in 3 volumes), 1988.
Uwe Storch and Hartmut Wiebe, Lehrbuch der Mathematik, 4 volumes.
External links
1940 births
2017 deaths
20th-century German mathematicians
21st-century German mathematicians
Algebraists
People from Staßfurt
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https://en.wikipedia.org/wiki/Ramanujan%27s%20sum
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In number theory, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula
where (a, q) = 1 means that a only takes on values coprime to q.
Srinivasa Ramanujan mentioned the sums in a 1918 paper. In addition to the expansions discussed in this article, Ramanujan's sums are used in the proof of Vinogradov's theorem that every sufficiently large odd number is the sum of three primes.
Notation
For integers a and b, is read "a divides b" and means that there is an integer c such that Similarly, is read "a does not divide b". The summation symbol
means that d goes through all the positive divisors of m, e.g.
is the greatest common divisor,
is Euler's totient function,
is the Möbius function, and
is the Riemann zeta function.
Formulas for cq(n)
Trigonometry
These formulas come from the definition, Euler's formula and elementary trigonometric identities.
and so on (, , , ,.., ,...). cq(n) is always an integer.
Kluyver
Let Then is a root of the equation . Each of its powers,
is also a root. Therefore, since there are q of them, they are all of the roots. The numbers where 1 ≤ n ≤ q are called the q-th roots of unity. is called a primitive q-th root of unity because the smallest value of n that makes is q. The other primitive q-th roots of unity are the numbers where (a, q) = 1. Therefore, there are φ(q) primitive q-th roots of unity.
Thus, the Ramanujan sum cq(n) is the sum of the n-th powers of the primitive q-th roots of unity.
It is a fact that the powers of are precisely the primitive roots for all the divisors of q.
Example. Let q = 12. Then
and are the primitive twelfth roots of unity,
and are the primitive sixth roots of unity,
and are the primitive fourth roots of unity,
and are the primitive third roots of unity,
is the primitive second root of unity, and
is the primitive first root of unity.
Therefore, if
is the sum of the n-th powers of all the roots, primitive and imprimitive,
and by Möbius inversion,
It follows from the identity xq − 1 = (x − 1)(xq−1 + xq−2 + ... + x + 1) that
and this leads to the formula
published by Kluyver in 1906.
This shows that cq(n) is always an integer. Compare it with the formula
von Sterneck
It is easily shown from the definition that cq(n) is multiplicative when considered as a function of q for a fixed value of n: i.e.
From the definition (or Kluyver's formula) it is straightforward to prove that, if p is a prime number,
and if pk is a prime power where k > 1,
This result and the multiplicative property can be used to prove
This is called von Sterneck's arithmetic function. The equivalence of it and Ramanujan's sum is due to Hölder.
Other properties of cq(n)
For all positive integers q,
For a fixed value of q the absolute value of the sequence is bounded by φ(q), and for a fixed value of n the absolute value of the sequence is bounded by n.
If q > 1
Let m1, m2 > 0, m = lcm(m1, m2). T
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https://en.wikipedia.org/wiki/Robert%20Alexander%20Rankin
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Robert Alexander Rankin FRSE FRSAMD (27 October 1915 – 27 January 2001) was a Scottish mathematician who worked in analytic number theory.
Life
Rankin was born in Garlieston in Wigtownshire the son of Rev Oliver Rankin (1885–1954), minister of Sorbie and his wife, Olivia Theresa Shaw. His father took the name Oliver Shaw Rankin on marriage and became Professor of Old Testament Language, Literature and Theology in the University of Edinburgh.
Rankin was educated at Fettes College then studied mathematics at Clare College, Cambridge, graduating in 1937. At Cambridge he was particularly influenced by J.E. Littlewood and A.E. Ingham.
Rankin was elected a Fellow of Clare College in 1939, but his career was interrupted by the Second World War, during which he worked first for the Ministry of Supply then on rocketry research at Fort Halstead. In 1945 he returned to Cambridge as an assistant lecturer, and then moved to the University of Birmingham in 1951 as Mason professor of mathematics. In 1954 he became Professor of Mathematics, Glasgow University, retiring in 1982.
In 1954 he was elected a Fellow of the Royal Society of Edinburgh. His proposers were William M. Smart, Robert Garry, James Norman Davidson and Robert Pollock Gillespie. He served as Vice President 1960 to 1963 and won the Society's Keith Prize for the period 1961–63.
Rankin had a continuing interest in Srinivasa Ramanujan, working initially with G.H. Hardy on Ramanujan's unpublished notes. His research interests lay in the distribution of prime numbers and in modular forms. In 1939 he developed what is now known as the Rankin–Selberg method. In 1977 Cambridge University Press published Rankin's Modular Forms and Functions. In his review, Marvin Knopp wrote:
For, as much as any recent exposition of modular functions, this book succeeds in getting near the research frontier, and in some instances even reaches it – no small feat in this theory. Only someone of Rankin's stature as a research mathematician and experience in the classroom could aspire to such an accomplishment in a self-contained work – beginning with first principles.
In 1987 Rankin received the Senior Whitehead Prize from the London Mathematical Society.
Rankin died in Glasgow on 27 January 2001.
Family
In 1942 he married Mary Ferrier Llewellyn.
See also
Rankin–Cohen bracket
Books
An introduction to mathematical analysis, Pergamon Press 1963; Dover 2007.
The modular group and its subgroups, Madras, Ramanujan Institute, 1969.
Modular forms and functions, Cambridge University Press 1977
References
External links
1915 births
2001 deaths
People educated at Fettes College
Alumni of Clare College, Cambridge
Fellows of Clare College, Cambridge
20th-century Scottish mathematicians
Number theorists
Fellows of the Royal Society of Edinburgh
People from Dumfries and Galloway
Academics of the University of Glasgow
Academics of the University of Birmingham
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https://en.wikipedia.org/wiki/Row%20equivalence
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In linear algebra, two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. Alternatively, two m × n matrices are row equivalent if and only if they have the same row space. The concept is most commonly applied to matrices that represent systems of linear equations, in which case two matrices of the same size are row equivalent if and only if the corresponding homogeneous systems have the same set of solutions, or equivalently the matrices have the same null space.
Because elementary row operations are reversible, row equivalence is an equivalence relation. It is commonly denoted by a tilde (~).
There is a similar notion of column equivalence, defined by elementary column operations; two matrices are column equivalent if and only if their transpose matrices are row equivalent. Two rectangular matrices that can be converted into one another allowing both elementary row and column operations are called simply equivalent.
Elementary row operations
An elementary row operation is any one of the following moves:
Swap: Swap two rows of a matrix.
Scale: Multiply a row of a matrix by a nonzero constant.
Pivot: Add a multiple of one row of a matrix to another row.
Two matrices A and B are row equivalent if it is possible to transform A into B by a sequence of elementary row operations.
Row space
The row space of a matrix is the set of all possible linear combinations of its row vectors. If the rows of the matrix represent a system of linear equations, then the row space consists of all linear equations that can be deduced algebraically from those in the system. Two m × n matrices are row equivalent if and only if they have the same row space.
For example, the matrices
are row equivalent, the row space being all vectors of the form . The corresponding systems of homogeneous equations convey the same information:
In particular, both of these systems imply every equation of the form
Equivalence of the definitions
The fact that two matrices are row equivalent if and only if they have the same row space is an important theorem in linear algebra. The proof is based on the following observations:
Elementary row operations do not affect the row space of a matrix. In particular, any two row equivalent matrices have the same row space.
Any matrix can be reduced by elementary row operations to a matrix in reduced row echelon form.
Two matrices in reduced row echelon form have the same row space if and only if they are equal.
This line of reasoning also proves that every matrix is row equivalent to a unique matrix with reduced row echelon form.
Additional properties
Because the null space of a matrix is the orthogonal complement of the row space, two matrices are row equivalent if and only if they have the same null space.
The rank of a matrix is equal to the dimension of the row space, so row equivalent matrices must have the same rank. This is equal to the number of pivots in the reduc
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https://en.wikipedia.org/wiki/Elementary%20matrix
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In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group when is a field. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations.
Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form. They are also used in Gauss–Jordan elimination to further reduce the matrix to reduced row echelon form.
Elementary row operations
There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations):
Row switching A row within the matrix can be switched with another row.
Row multiplication Each element in a row can be multiplied by a non-zero constant. It is also known as scaling a row.
Row addition A row can be replaced by the sum of that row and a multiple of another row.
If is an elementary matrix, as described below, to apply the elementary row operation to a matrix , one multiplies by the elementary matrix on the left, . The elementary matrix for any row operation is obtained by executing the operation on the identity matrix. This fact can be understood as an instance of the Yoneda lemma applied to the category of matrices.
Row-switching transformations
The first type of row operation on a matrix switches all matrix elements on row with their counterparts on a different row . The corresponding elementary matrix is obtained by swapping row and row of the identity matrix.
So is the matrix produced by exchanging row and row of .
Coefficient wise, the matrix is defined by :
Properties
The inverse of this matrix is itself:
Since the determinant of the identity matrix is unity, It follows that for any square matrix (of the correct size), we have
For theoretical considerations, the row-switching transformation can be obtained from row-addition and row-multiplication transformations introduced below because
Row-multiplying transformations
The next type of row operation on a matrix multiplies all elements on row by where is a non-zero scalar (usually a real number). The corresponding elementary matrix is a diagonal matrix, with diagonal entries 1 everywhere except in the th position, where it is .
So is the matrix produced from by multiplying row by .
Coefficient wise, the matrix is defined by :
Properties
The inverse of this matrix is given by
The matrix and its inverse are diagonal matrices.
Therefore for a square matrix (of the correct size), we have
Row-addition transformations
The final type of row operation on a matrix adds row multiplied by a scalar to row . The corresponding elementary matrix is the identity matrix but with an in the position.
So is the matrix produced from by adding times row to row .
And is the matrix produce
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https://en.wikipedia.org/wiki/Elasticity%20of%20a%20function
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In mathematics, the elasticity or point elasticity of a positive differentiable function f of a positive variable (positive input, positive output) at point a is defined as
or equivalently
It is thus the ratio of the relative (percentage) change in the function's output with respect to the relative change in its input , for infinitesimal changes from a point . Equivalently, it is the ratio of the infinitesimal change of the logarithm of a function with respect to the infinitesimal change of the logarithm of the argument. Generalisations to multi-input-multi-output cases also exist in the literature.
The elasticity of a function is a constant if and only if the function has the form for a constant .
The elasticity at a point is the limit of the arc elasticity between two points as the separation between those two points approaches zero.
The concept of elasticity is widely used in economics and Metabolic Control Analysis; see elasticity (economics) and Elasticity coefficient respectively for details.
Rules
Rules for finding the elasticity of products and quotients are simpler than those for derivatives. Let f, g be differentiable. Then
The derivative can be expressed in terms of elasticity as
Let a and b be constants. Then
,
.
Estimating point elasticities
In economics, the price elasticity of demand refers to the elasticity of a demand function Q(P), and can be expressed as (dQ/dP)/(Q(P)/P) or the ratio of the value of the marginal function (dQ/dP) to the value of the average function (Q(P)/P). This relationship provides an easy way of determining whether a demand curve is elastic or inelastic at a particular point. First, suppose one follows the usual convention in mathematics of plotting the independent variable (P) horizontally and the dependent variable (Q) vertically. Then the slope of a line tangent to the curve at that point is the value of the marginal function at that point. The slope of a ray drawn from the origin through the point is the value of the average function. If the absolute value of the slope of the tangent is greater than the slope of the ray then the function is elastic at the point; if the slope of the secant is greater than the absolute value of the slope of the tangent then the curve is inelastic at the point. If the tangent line is extended to the horizontal axis the problem is simply a matter of comparing angles created by the lines and the horizontal axis. If the marginal angle is greater than the average angle then the function is elastic at the point; if the marginal angle is less than the average angle then the function is inelastic at that point. If, however, one follows the convention adopted by economists and plots the independent variable P on the vertical axis and the dependent variable Q on the horizontal axis, then the opposite rules would apply.
The same graphical procedure can also be applied to a supply function or other functions.
Semi-elasticity
A semi-elasticity (or semielasticity)
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https://en.wikipedia.org/wiki/SMP%20%28computer%20algebra%20system%29
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Symbolic Manipulation Program, usually called SMP, was a computer algebra system designed by Chris A. Cole and Stephen Wolfram at Caltech circa 1979. It was initially developed in the Caltech physics department with contributions from Geoffrey C. Fox, Jeffrey M. Greif, Eric D. Mjolsness, Larry J. Romans, Timothy Shaw, and Anthony E. Terrano.
SMP was first sold commercially in 1981, by the Computer Mathematics Corporation of Los Angeles, which later became part of Inference Corporation. Inference further developed the program and marketed it commercially from 1983 to 1988, but it was not a commercial success, and Inference became pessimistic about the market for symbolic math programs, and so abandoned SMP to concentrate on expert systems.
SMP was influenced by the earlier computer algebra systems Macsyma (of which Wolfram was a user) and Schoonschip (whose code Wolfram studied).
SMP follows a rule-based approach, giving it a "consistent, pattern-directed language". Unlike Macsyma and Reduce, it was written in C.
During the 1980s, it was one of the generally available general-purpose computer algebra systems, along with Reduce, Macsyma, and Scratchpad, and later muMATH and Maple. It was often used for teaching college calculus.
The design of SMP's interactive language and its "map" commands influenced the design of the 1984 version of Scratchpad.
Criticism
SMP has been criticized for various characteristics, notably its use of floating-point numbers instead of exact rational numbers, which can lead to incorrect results, and makes polynomial greatest common divisor calculations problematic. Many other problems in early versions of the system were purportedly fixed in later versions.
References
Additional sources
Chris A. Cole, Stephen Wolfram, "SMP: A Symbolic Manipulation Program", Proceedings of the fourth ACM symposium on Symbolic and algebraic computation (SIGSAM), Snowbird, Utah, 1981. full text
Stephen Wolfram with Chris A. Cole, SMP: A Symbolic Manipulation Program, Reference Manual, California Institute of Technology, 1981; Inference Corporation, 1983. full text
Stephen Wolfram, "Symbolic Mathematical Computation", Communications of the ACM, April 1985 (Volume 28, Issue 4). Despite the general-sounding title the focus is on an introduction to SMP. Online version of this article
J.M. Greif, "The SMP Pattern-Matcher" in B.F. Caviness (editor), Proceedings of EUROCAL 1985, volume 2, pgs. 303-314, Springer-Verlag Lecture Notes in Computer Science, no. 204, A discussion, with examples, of the capabilities, tasks, and design philosophy of the pattern-matcher.
SMP's manual "SMP Handbook"
Stephen Wolfram's blog post on the history of SMP's creation
Computer algebra systems
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https://en.wikipedia.org/wiki/SIGSAM
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SIGSAM is the ACM Special Interest Group on Symbolic and Algebraic Manipulation. It publishes the ACM Communications in Computer Algebra and often sponsors the International Symposium on Symbolic and Algebraic Computation (ISSAC).
External links
ACM Official SIGSAM web site
ISSAC 2009, Seoul, Korea
ISSAC 2008, ("RISC Linz"), Hagenberg, Austria
ISSAC 2007, Waterloo, Ontario
ISSAC 2006, Genoa
ISSAC 2005, Beijing
ISSAC 2004, Santander, Cantabria
ISSAC 2003, Philadelphia
ISSAC 2002, Lille
ISSAC 2001, London, Ontario
ISSAC 2000, St. Andrews
ISSAC 1999, Vancouver
ISSAC 1998, Rostock
ISSAC 1997, Maui
Association for Computing Machinery Special Interest Groups
Computer algebra systems
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https://en.wikipedia.org/wiki/Wrack
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Wrack may refer to:
wrack (mathematics), a concept in knot theory
wrack (seaweed), several species of seaweed
Wrack, a novel by James Bradley (Australian writer)
Charlie Wrack (1899–1979), English footballer
Darren Wrack (born 1976), English footballer
Matt Wrack (born 1962), British firefighter and trade unionist
Wrack, the leading broodmare sire in North America in 1935
Wrack (video game), A first person shooter video game made by Final Boss Entertainment
Wrack zone, covered in high water, uncovered at low water.
See also
Rack (disambiguation)
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https://en.wikipedia.org/wiki/Weil%27s%20criterion
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In mathematics, Weil's criterion is a criterion of André Weil for the Generalized Riemann hypothesis to be true. It takes the form of an equivalent statement, to the effect that a certain generalized function is positive definite.
Weil's idea was formulated first in a 1952 paper. It is based on the explicit formulae of prime number theory, as they apply to Dirichlet L-functions, and other more general global L-functions. A single statement thus combines statements on the complex zeroes of all Dirichlet L-functions.
Weil returned to this idea in a 1972 paper, showing how the formulation extended to a larger class of L-functions (Artin-Hecke L-functions); and to the global function field case. Here the inclusion of Artin L-functions, in particular, implicates Artin's conjecture; so that the criterion involves a Generalized Riemann Hypothesis plus Artin Conjecture.
The case of function fields, of curves over finite fields, is one in which the analogue of the Riemann Hypothesis is known, by Weil's classical work begun in 1940; and Weil also proved the analogue of the Artin Conjecture. Therefore, in that setting, the criterion can be used to show the corresponding statement of positive-definiteness does hold.
References
A. Weil, "Sur les 'formules explicites' de la théorie des nombres premiers", Comm. Lund (vol. dédié a Marcel Riesz) (1952) 252–265; Collected Papers II
A. Weil, "Sur les formules explicites de la théorie des nombres, Izvestia Akad. Nauk S.S.S.R., Ser. Math. 36 (1972) 3-18; Collected Papers III, 249-264
Zeta and L-functions
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https://en.wikipedia.org/wiki/NNO
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NNO may stand for:
Nuveen North Carolina Dividend Advantage Municipal Fund 2 (stock symbol: NNO)
Natural number object, in category theory, a subfield of mathematics
National Night Out, a crime prevention activity in the United States
Nynorsk, ISO 639-2 and ISO 639-3 language codes
Nitrous oxide
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https://en.wikipedia.org/wiki/BICOM
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BICOM may refer to:
Britain Israel Communications and Research Centre
Brunel Institute of Computational Mathematics
Bioresonance therapy, pseudoscientific medical practice
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https://en.wikipedia.org/wiki/Digital%20topology
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Digital topology deals with properties and features of two-dimensional (2D) or three-dimensional (3D) digital images
that correspond to topological properties (e.g., connectedness) or topological features (e.g., boundaries) of objects.
Concepts and results of digital topology are used to specify and justify important (low-level) image analysis algorithms,
including algorithms for thinning, border or surface tracing, counting of components or tunnels, or region-filling.
History
Digital topology was first studied in the late 1960s by the computer image analysis researcher Azriel Rosenfeld (1931–2004), whose publications on the subject played a major role in establishing and developing the field. The term "digital topology" was itself invented by Rosenfeld, who used it in a 1973 publication for the first time.
A related work called the grid cell topology, which could be considered as a link to classic combinatorial topology, appeared in the book of Pavel Alexandrov and Heinz Hopf, Topologie I (1935). Rosenfeld et al. proposed digital connectivity such as 4-connectivity and 8-connectivity in two dimensions as well as 6-connectivity and 26-connectivity in three dimensions. The labeling method for inferring a connected component was studied in the 1970s. Theodosios Pavlidis (1982) suggested the use of graph-theoretic algorithms such as the depth-first search method for finding connected components. Vladimir A. Kovalevsky (1989) extended the Alexandrov–Hopf 2D grid cell topology to three and higher dimensions. He also proposed (2008) a more general axiomatic theory of locally finite topological spaces and abstract cell complexes formerly suggested by Ernst Steinitz (1908). It is the Alexandrov topology. The book from 2008 contains new definitions of topological balls and spheres independent of a metric and numerous applications to digital image analysis.
In the early 1980s, digital surfaces were studied. David Morgenthaler and Rosenfeld (1981) gave a mathematical definition of surfaces in three-dimensional digital space. This definition contains a total of nine types of digital surfaces. The digital manifold was studied in the 1990s. A recursive definition of the digital k-manifold was proposed intuitively by Chen and Zhang in 1993. Many applications were found in image processing and computer vision.
Basic results
A basic (early) result in digital topology says that 2D binary images require the alternative use of 4- or 8-adjacency or "pixel connectivity" (for "object" or "non-object"
pixels) to ensure the basic topological duality of separation and connectedness. This alternative use corresponds to open or closed
sets in the 2D grid cell topology, and the result generalizes to 3D: the alternative use of 6- or 26-adjacency corresponds
to open or closed sets in the 3D grid cell topology. Grid cell topology also applies to multilevel (e.g., color) 2D or 3D images,
for example based on a total order of possible image values and applying a 'maxim
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https://en.wikipedia.org/wiki/Grid%20cell%20topology
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The grid cell topology is studied in digital topology as part of the theoretical basis for (low-level) algorithms in computer image analysis or computer graphics.
The elements of the n-dimensional grid cell topology (n ≥ 1) are all n-dimensional grid cubes and their k-dimensional faces ( for 0 ≤ k ≤ n−1); between these a partial order A ≤ B is defined if A is a subset of B (and thus also dim(A) ≤ dim(B)). The grid cell topology is the Alexandrov topology (open sets are up-sets) with respect to this partial order. (See also poset topology.)
Alexandrov and Hopf first introduced the grid cell topology, for the two-dimensional case, within an exercise in their text Topologie I (1935).
A recursive method to obtain n-dimensional grid cells and an intuitive definition for
grid cell manifolds can be found in Chen, 2004. It is related to digital manifolds.
See also
Pixel connectivity
References
Digital Geometry: Geometric Methods for Digital Image Analysis, by Reinhard Klette and Azriel Rosenfeld, Morgan Kaufmann Pub, May 2004, (The Morgan Kaufmann Series in Computer Graphics)
Topologie I, by Paul Alexandroff and Heinz Hopf, Springer, Berlin, 1935, xiii + 636 pp.
Digital topology
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https://en.wikipedia.org/wiki/Poset%20topology
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In mathematics, the poset topology associated to a poset (S, ≤) is the Alexandrov topology (open sets are upper sets) on the poset of finite chains of (S, ≤), ordered by inclusion.
Let V be a set of vertices. An abstract simplicial complex Δ is a set of finite sets of vertices, known as faces , such that
Given a simplicial complex Δ as above, we define a (point set) topology on Δ by declaring a subset be closed if and only if Γ is a simplicial complex, i.e.
This is the Alexandrov topology on the poset of faces of Δ.
The order complex associated to a poset (S, ≤) has the set S as vertices, and the finite chains of (S, ≤) as faces. The poset topology associated to a poset (S, ≤) is then the Alexandrov topology on the order complex associated to (S, ≤).
See also
Topological combinatorics
References
Poset Topology: Tools and Applications Michelle L. Wachs, lecture notes IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004)
General topology
Order theory
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https://en.wikipedia.org/wiki/3/4
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3/4 or ¾ may refer to:
The fraction (mathematics) three quarters () equal to 0.75
Arts and media
3/4 (film), a 2017 Bulgarian film
time, a form of triple metre in music
3/4 profile, in portraits
3/4 perspective, in video games
Other uses
″ videocassette, better known as the U-matic format
March 4 (month-day date notation)
3 April (day-month date notation)
3rd Battalion 4th Marines, a unit in the United States Marine Corps
Three fourths, alternative name for Capri pants
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https://en.wikipedia.org/wiki/Lindley%27s%20paradox
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Lindley's paradox is a counterintuitive situation in statistics in which the Bayesian and frequentist approaches to a hypothesis testing problem give different results for certain choices of the prior distribution. The problem of the disagreement between the two approaches was discussed in Harold Jeffreys' 1939 textbook; it became known as Lindley's paradox after Dennis Lindley called the disagreement a paradox in a 1957 paper.
Although referred to as a paradox, the differing results from the Bayesian and frequentist approaches can be explained as using them to answer fundamentally different questions, rather than actual disagreement between the two methods.
Nevertheless, for a large class of priors the differences between the frequentist and Bayesian approach are caused by keeping the significance level fixed: as even Lindley recognized, "the theory does not justify the practice of keeping the significance level fixed'' and even "some computations by Prof. Pearson in the discussion to that paper emphasized how the significance level would have to change with the sample size, if the losses and prior probabilities were kept fixed.'' In fact, if the critical value increases with the sample size suitably fast, then the disagreement between the frequentist and Bayesian approaches becomes negligible as the sample size increases.
Description of the paradox
The result of some experiment has two possible explanations, hypotheses and , and some prior distribution representing uncertainty as to which hypothesis is more accurate before taking into account .
Lindley's paradox occurs when
The result is "significant" by a frequentist test of , indicating sufficient evidence to reject , say, at the 5% level, and
The posterior probability of given is high, indicating strong evidence that is in better agreement with than .
These results can occur at the same time when is very specific, more diffuse, and the prior distribution does not strongly favor one or the other, as seen below.
Numerical example
The following numerical example illustrates Lindley's paradox. In a certain city 49,581 boys and 48,870 girls have been born over a certain time period. The observed proportion of male births is thus 49,581/98,451 ≈ 0.5036. We assume the fraction of male births is a binomial variable with parameter . We are interested in testing whether is 0.5 or some other value. That is, our null hypothesis is and the alternative is .
Frequentist approach
The frequentist approach to testing is to compute a p-value, the probability of observing a fraction of boys at least as large as assuming is true. Because the number of births is very large, we can use a normal approximation for the fraction of male births , with and , to compute
We would have been equally surprised if we had seen 49,581 female births, i.e. , so a frequentist would usually perform a two-sided test, for which the p-value would be . In both cases, the p-value is lower than the sign
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https://en.wikipedia.org/wiki/Pentachord
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A pentachord in music theory may be either of two things. In pitch-class set theory, a pentachord is defined as any five pitch classes, regarded as an unordered collection . In other contexts, a pentachord may be any consecutive five-note section of a diatonic scale . A pentad is a five-note chord .
Under the latter definition, a diatonic scale comprises five non-transpositionally equivalent pentachords rather than seven because the Ionian and Mixolydian pentachords and the Dorian and Aeolian pentachords are intervallically identical (CDEFG=GABCD; DEFGA=ABCDE).
The name "pentachord" was also given to a musical instrument, now in disuse, built to the specifications of Sir Edward Walpole. It was demonstrated by Karl Friedrich Abel at his first public concert in London, on 5 April 1759, when it was described as "newly invented" . In the dedication to Walpole of his cello sonatas op. 3, the cellist/composer James Cervetto praised the pentachord, declaring: "I know not a more fit Instrument to Accompany the Voice" . Performances on the instrument are documented as late as 1783, after which it seems to have fallen out of use. It appears to have been similar to a five-string violoncello .
References
Further reading
Chords
Pentatonic scales
Simultaneities (music)
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https://en.wikipedia.org/wiki/Jean-Claude%20Dunyach
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Jean-Claude Dunyach (born 1957) is a French science fiction writer.
Overview
Dunyach has a Ph.D. in applied mathematics and supercomputing from Paul Sabatier University. He works for Airbus in Toulouse in southwestern France.
Dunyach has been writing science fiction since the beginning of the 1980s and has already published nine novels and ten collections of short stories, garnering the French Science-Fiction award in 1983 and the Prix Rosny-Aîné Awards in 1992, as well as the Grand Prix de l’Imaginaire and the Prix Ozone in 1997.
His short story Déchiffrer la Trame (Unravelling the Thread) won both the Prix de l’Imaginaire and the Rosny Award in 1998, and was voted Best Story of the Year by the readers of the magazine Interzone.
His novel, Etoiles Mourantes (Dying Stars), written in collaboration with the French author Ayerdhal, won the prestigious Eiffel Tower Award in 1999 as well as the Prix Ozone.
Dunyach's works have been translated into English, Bulgarian, Croatian, Danish, Hungarian, German, Italian, Russian and Spanish.
Dunyach also writes lyrics for several French singers, which served as an inspiration for one of his novels about a rock and roll singer touring in Antarctica with a zombie philharmonic orchestra...
Bibliography
Autoportrait (Self-Portrait) (collection) (Présence du Futur No. 415, Denoël, Paris, 1986)
Le Temple de Chair (Le Jeu des Sabliers, Tome 1) (The Temple of Flesh (The Game of the Hourglass, Vol. 1)) (Anticipation No. 1592, Fleuve Noir, Paris, 1987)
Le Temple d’Os (Le Jeu des Sabliers, Tome 2) (The Temple of Bones (The Game of the Hourglass, Vol. 2)) (Anticipation No. 1609, Fleuve Noir, Paris, 1988)
Nivôse (Étoiles Mortes, Tome1) (Nivose (Dead Stars, Vol. 1)) (Anticipation No.1837, Fleuve Noir, Paris, 1991)
Aigue-Marine (Étoiles Mortes, Tome 2) (Aigue-Marine (Dead Stars, Vol. 2)) (Anticipation No.1838, Fleuve Noir, Paris, 1991)
Voleurs de Silence (Étoiles Mortes, Tome 3) (Thieves of Silence (Dead Stars, Vol. 3) (Anticipation No. 1858, Fleuve Noir, Paris, 1992)
Roll Over, Amundsen (Anticipation No. 1912, Fleuve Noir, Paris, 1993)
La Guerre des Cercles (The War of the Circles) (Anticipation No. 1963, Fleuve Noir, Paris, 1995)
Étoiles Mourantes (Dying Stars) (with Ayerdhal) (J’ai Lu Millénaire, Paris, 1999)
La Station de l’Agnelle (Station of the Lamb) (collection) (L’Atalante, Nantes, 2000)
Dix Jours Sans Voir la Mer (Ten Days Without Looking at the Sea) (collection) (L’Atalante, Nantes, 2000)
Étoiles Mortes (Dead Stars) (J’ai Lu, Paris, 2000)
Déchiffrer la Trame (Unravelling the Thread) (collection) (L’Atalante, Nantes, 2001)
Le Jeu des Sabliers (The Game of the Hourglass) (ISF, Paris, 2003)
Les Nageurs de Sable (The Sand Swimmers) (collection) (L’Atalante, Nantes, 2003)
Le temps, en s'évaporant... (Time, as it evaporates...) (collection) (L’Atalante, Nantes, 2005)
Séparations (Separations) (collection) (L’Atalante, Nantes, 2007)
Les harmoniques célestes (Celestial harmonics) (collection) (
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https://en.wikipedia.org/wiki/Sextic%20equation
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In algebra, a sextic (or hexic) polynomial is a polynomial of degree six.
A sextic equation is a polynomial equation of degree six—that is, an equation whose left hand side is a sextic polynomial and whose right hand side is zero. More precisely, it has the form:
where and the coefficients may be integers, rational numbers, real numbers, complex numbers or, more generally, members of any field.
A sextic function is a function defined by a sextic polynomial. Because they have an even degree, sextic functions appear similar to quartic functions when graphed, except they may possess an additional local maximum and local minimum each. The derivative of a sextic function is a quintic function.
Since a sextic function is defined by a polynomial with even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If the leading coefficient is positive, then the function increases to positive infinity at both sides and thus the function has a global minimum. Likewise, if is negative, the sextic function decreases to negative infinity and has a global maximum.
Solvable sextics
Some sixth degree equations, such as , can be solved by factorizing into radicals, but other sextics cannot. Évariste Galois developed techniques for determining whether a given equation could be solved by radicals which gave rise to the field of Galois theory.
It follows from Galois theory that a sextic equation is solvable in terms of radicals if and only if its Galois group is contained either in the group of order 48 which stabilizes a partition of the set of the roots into three subsets of two roots or in the group of order 72 which stabilizes a partition of the set of the roots into two subsets of three roots.
There are formulas to test either case, and, if the equation is solvable, compute the roots in term of radicals.
The general sextic equation can be solved by the two-variable Kampé de Fériet function. A more restricted class of sextics can be solved by the one-variable generalised hypergeometric function using Felix Klein's approach to solving the quintic equation.
Examples
Watt's curve, which arose in the context of early work on the steam engine, is a sextic in two variables.
One method of solving the cubic equation involves transforming variables to obtain a sextic equation having terms only of degrees 6, 3, and 0, which can be solved as a quadratic equation in the cube of the variable.
Etymology
The describer "sextic" comes from the Latin stem for 6 or 6th ("sex-t-"), and the Greek suffix meaning "pertaining to" ("-ic"). The much less common "hexic" uses Greek for both its stem (hex- 6) and its suffix (-ik-). In both cases, the prefix refers to the degree of the function. Often, these type of functions will simply be referred to as "6th degree functions".
See also
Cayley's sextic
Cubic function
Septic equation
References
Equations
Galois theory
Polynomials
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https://en.wikipedia.org/wiki/Harmonic%20%28disambiguation%29
|
Harmonic usually refers to the frequency components of a time-varying signal, such as a musical note.
Mathematics, science and engineering
Harmonic (mathematics), a number of concepts in mathematics
Harmonic analysis, representing signals by superposition of basic waves
Harmonic oscillator, a concept in classical mechanics
Simple harmonic motion, a concept in classical mechanics
Harmonic distortion, a measurement of signal distortion
Harmonics (electrical power)
Harmonic series (mathematics), a divergent infinite series
Harmonic tremor, a rhythmic earthquake which may indicate volcanic activity
Music
String harmonic, a string instrument playing technique
Artificial harmonic, a string instrument playing technique
Enharmonic, a "spelling" issue in music
Harmonic series (music), the series of overtones (or partials) present in a musical note, or the vibrational modes of a string or an air column
Scale of harmonics, a musical scale based on harmonic nodes of a string
The Harmonics, a rock a cappella group from Stanford University
Harmony, the musical use of simultaneous pitches, or chords
Inharmonicity, the degree of overtones' departure from integral multiples of the fundamental frequency
Overtone, any resonant frequency higher than the fundamental frequency
Other uses
Harmonic (color), a relationship between three colors
Harmonic Convergence, a New Age astrological term
"Harmonics", the twelfth movement of Mike Oldfield's Tubular Bells 2003 album
Harmonic Inc., a video infrastructure product company, headquartered in San Jose, California
See also
Harmonix, a video game development company
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https://en.wikipedia.org/wiki/Mathematical%20operators%20and%20symbols%20in%20Unicode
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The Unicode Standard encodes almost all standard characters used in mathematics.
Unicode Technical Report #25 provides comprehensive information about the character repertoire, their properties, and guidelines for implementation.
Mathematical operators and symbols are in multiple Unicode blocks. Some of these blocks are dedicated to, or primarily contain, mathematical characters while others are a mix of mathematical and non-mathematical characters. This article covers all Unicode characters with a derived property of "Math".
Dedicated blocks
Mathematical Operators block
The Mathematical Operators block (U+2200–U+22FF) contains characters for mathematical, logical, and set notation.
Supplemental Mathematical Operators block
The Supplemental Mathematical Operators block (U+2A00–U+2AFF) contains various mathematical symbols, including N-ary operators, summations and integrals, intersections and unions, logical and relational operators, and subset/superset relations.
Mathematical Alphanumeric Symbols block
The Mathematical Alphanumeric Symbols block (U+1D400–U+1D7FF) contains Latin and Greek letters and decimal digits that enable mathematicians to denote different notions with different letter styles. The reserved code points (the "holes") in the alphabetic ranges up to U+1D551 duplicate characters in the Letterlike Symbols block.
Letterlike Symbols block
The Letterlike Symbols block (U+2100–U+214F) includes variables. Most alphabetic math symbols are in the Mathematical Alphanumeric Symbols block shown above.
The math subset of this block is U+2102, U+2107, U+210A–U+2113, U+2115, U+2118–U+211D, U+2124, U+2128–U+2129, U+212C–U+212D, U+212F–U+2131, U+2133–U+2138, U+213C–U+2149, and U+214B.
Miscellaneous Mathematical Symbols-A block
The Miscellaneous Mathematical Symbols-A block (U+27C0–U+27EF) contains characters for mathematical, logical, and database notation.
Miscellaneous Mathematical Symbols-B block
The Miscellaneous Mathematical Symbols-B block (U+2980–U+29FF) contains miscellaneous mathematical symbols, including brackets, angles, and circle symbols.
Miscellaneous Technical block
The Miscellaneous Technical block (U+2300–U+23FF) includes braces and operators.
The math subset of this block is U+2308–U+230B, U+2320–U+2321, U+237C, U+239B–U+23B5, 23B7, U+23D0, and U+23DC–U+23E2.
Geometric Shapes block
The Geometric Shapes block (U+25A0–U+25FF) contains geometric shape symbols.
The math subset of this block is U+25A0–25A1, U+25AE–25B7, U+25BC–25C1, U+25C6–25C7, U+25CA–25CB, U+25CF–25D3, U+25E2, U+25E4, U+25E7–25EC, and U+25F8–25FF.
Arrows block
The Arrows block (U+2190–U+21FF) contains line, curve, and semicircle arrows and arrow-like operators.
The math subset of this block is U+2190–U+21A7, U+21A9–U+21AE, U+21B0–U+21B1, U+21B6–U+21B7, U+21BC–U+21DB, U+21DD, U+21E4–U+21E5, U+21F4–U+21FF.
Supplemental Arrows-A block
The Supplemental Arrows-A block (U+27F0–U+27FF) contains arrows and arrow-like operators.
Supplemental
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https://en.wikipedia.org/wiki/Hilbert%20symbol
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In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from K× × K× to the group of nth roots of unity in a local field K such as the fields of reals or p-adic numbers. It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields.
The Hilbert symbol has been generalized to higher local fields.
Quadratic Hilbert symbol
Over a local field K whose multiplicative group of non-zero elements is K×,
the quadratic Hilbert symbol is the function (–, –) from K× × K× to {−1,1} defined by
Equivalently, if and only if is equal to the norm of an element of the quadratic extension page 110.
Properties
The following three properties follow directly from the definition, by choosing suitable solutions of the diophantine equation above:
If a is a square, then (a, b) = 1 for all b.
For all a,b in K×, (a, b) = (b, a).
For any a in K× such that a−1 is also in K×, we have (a, 1−a) = 1.
The (bi)multiplicativity, i.e.,
(a, b1b2) = (a, b1)·(a, b2)
for any a, b1 and b2 in K× is, however, more difficult to prove, and requires the development of local class field theory.
The third property shows that the Hilbert symbol is an example of a Steinberg symbol and thus factors over the second Milnor K-group , which is by definition
K× ⊗ K× / (a ⊗ (1−a), a ∈ K× \ {1})
By the first property it even factors over . This is the first step towards the Milnor conjecture.
Interpretation as an algebra
The Hilbert symbol can also be used to denote the central simple algebra over K with basis 1,i,j,k and multiplication rules , , . In this case the algebra represents an element of order 2 in the Brauer group of K, which is identified with -1 if it is a division algebra and +1 if it is isomorphic to the algebra of 2 by 2 matrices.
Hilbert symbols over the rationals
For a place v of the rational number field and rational numbers a, b we let (a, b)v denote the value of the Hilbert symbol in the corresponding completion Qv. As usual, if v is the valuation attached to a prime number p then the corresponding completion is the p-adic field and if v is the infinite place then the completion is the real number field.
Over the reals, (a, b)∞ is +1 if at least one of a or b is positive, and −1 if both are negative.
Over the p-adics with p odd, writing and , where u and v are integers coprime to p, we have
, where
and the expression involves two Legendre symbols.
Over the 2-adics, again writing and , where u and v are odd numbers, we have
, where
It is known that if v ranges over all places, (a, b)v is 1 for almost all places. Therefore, the following product formula
makes sense. It is equivalent to the law of quadratic reciprocity.
Kaplansky radical
The Hilbert symbol on a field F defines a map
where Br(F) is the Brauer group of F.
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https://en.wikipedia.org/wiki/Riesel
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Riesel may refer to:
People
Hans Riesel (1929–2014), Swedish mathematician who discovered a Mersenne prime
Victor Riesel (1913–1995), American labor union journalist
In Mathematics
Riesel number, an odd natural number k for which the integers of the form k·2n−1 are all composite
Riesel Sieve, a project to prove the smallest Riesel number
Places
Riesel, Texas
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https://en.wikipedia.org/wiki/Degree%20of%20a%20polynomial
|
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of degree but, nowadays, may refer to several other concepts (see Order of a polynomial (disambiguation)).
For example, the polynomial which can also be written as has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term.
To determine the degree of a polynomial that is not in standard form, such as , one can put it in standard form by expanding the products (by distributivity) and combining the like terms; for example, is of degree 1, even though each summand has degree 2. However, this is not needed when the polynomial is written as a product of polynomials in standard form, because the degree of a product is the sum of the degrees of the factors.
Names of polynomials by degree
The following names are assigned to polynomials according to their degree:
Special case – zero (see , below)
Degree 0 – non-zero constant
Degree 1 – linear
Degree 2 – quadratic
Degree 3 – cubic
Degree 4 – quartic (or, if all terms have even degree, biquadratic)
Degree 5 – quintic
Degree 6 – sextic (or, less commonly, hexic)
Degree 7 – septic (or, less commonly, heptic)
Degree 8 – octic
Degree 9 – nonic
Degree 10 – decic
Names for degree above three are based on Latin ordinal numbers, and end in -ic. This should be distinguished from the names used for the number of variables, the arity, which are based on Latin distributive numbers, and end in -ary. For example, a degree two polynomial in two variables, such as , is called a "binary quadratic": binary due to two variables, quadratic due to degree two. There are also names for the number of terms, which are also based on Latin distributive numbers, ending in -nomial; the common ones are monomial, binomial, and (less commonly) trinomial; thus is a "binary quadratic binomial".
Examples
The polynomial is a cubic polynomial: after multiplying out and collecting terms of the same degree, it becomes , with highest exponent 3.
The polynomial is a quintic polynomial: upon combining like terms, the two terms of degree 8 cancel, leaving , with highest exponent 5.
Behavior under polynomial operations
The degree of the sum, the product or the composition of two polynomials is strongly related to the degree of the input polynomials.
Addition
The degree of the sum (or difference) of two polynomials is less than or equal to the greater of their degrees; that is,
and .
For example, the degree of is 2, and 2
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https://en.wikipedia.org/wiki/Category%20of%20relations
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In mathematics, the category Rel has the class of sets as objects and binary relations as morphisms.
A morphism (or arrow) R : A → B in this category is a relation between the sets A and B, so .
The composition of two relations R: A → B and S: B → C is given by
(a, c) ∈ S o R ⇔ for some b ∈ B, (a, b) ∈ R and (b, c) ∈ S.
Rel has also been called the "category of correspondences of sets".
Properties
The category Rel has the category of sets Set as a (wide) subcategory, where the arrow in Set corresponds to the relation defined by .
A morphism in Rel is a relation, and the corresponding morphism in the opposite category to Rel has arrows reversed, so it is the converse relation. Thus Rel contains its opposite and is self-dual.
The involution represented by taking the converse relation provides the dagger to make Rel a dagger category.
The category has two functors into itself given by the hom functor: A binary relation R ⊆ A × B and its transpose RT ⊆ B × A may be composed either as R RT or as RT R. The first composition results in a homogeneous relation on A and the second is on B. Since the images of these hom functors are in Rel itself, in this case hom is an internal hom functor. With its internal hom functor, Rel is a closed category, and furthermore a dagger compact category.
The category Rel can be obtained from the category Set as the Kleisli category for the monad whose functor corresponds to power set, interpreted as a covariant functor.
Perhaps a bit surprising at first sight is the fact that product in Rel is given by the disjoint union (rather than the cartesian product as it is in Set), and so is the coproduct.
Rel is monoidal closed, if one defines both the monoidal product A ⊗ B and the internal hom A ⇒ B by the cartesian product of sets. It is also a monoidal category if one defines the monoidal product by the disjoint union of sets.
The category Rel was the prototype for the algebraic structure called an allegory by Peter J. Freyd and Andre Scedrov in 1990. Starting with a regular category and a functor F: A → B, they note properties of the induced functor Rel(A,B) → Rel(FA, FB). For instance, it preserves composition, conversion, and intersection. Such properties are then used to provide axioms for an allegory.
Relations as objects
David Rydeheard and Rod Burstall consider Rel to have objects that are homogeneous relations. For example, A is a set and R ⊆ A × A is a binary relation on A. The morphisms of this category are functions between sets that preserve a relation: Say S ⊆ B × B is a second relation and f: A → B is a function such that then f is a morphism.
The same idea is advanced by Adamek, Herrlich and Strecker, where they designate the objects (A, R) and (B, S), set and relation.
Notes
References
Binary relations
Monoidal categories
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https://en.wikipedia.org/wiki/AICC
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AICC may refer to:
AICc, a version of Akaike information criterion (AIC, which is used in statistics), that has a correction for small sample sizes
All India Congress Committee, the central presidium of the Congress Party
All India Christian Council, an alliance of Christian denominations, mission agencies, institutions, federations and Christian lay leaders in India.
Adiabatic Isochoric Complete Combustion
Arusha International Conference Centre, the leading conference venue in Tanzania
Aviation Industry Computer-Based Training Committee, an e-Learning group and a tracking specification
Aviation Industry Corporation of China, a Chinese state-owned aerospace and defense company
Accident Investigation Coordination Committee, subordinate to the Ministry of Civil Aviation and Communication Maldives
Arctic Icebreaker Coordinating Committee, a subcommittee of the University-National Oceanographic Laboratory System responsible for managing the US Research Icebreaker fleet
Association for Inherited Cardiac Conditions, a UK-based association of Geneticists and Cardiologists with expertise in inherited disease
|
https://en.wikipedia.org/wiki/James%20Harkins
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James Harkins (born 1905) was a Scottish professional footballer who played as an inside forward.
Career statistics
Source:
References
1905 births
People from Paisley, Renfrewshire
Scottish men's footballers
Men's association football inside forwards
Dalbeattie Star F.C. players
Petershill F.C. players
Third Lanark A.C. players
Solway Star F.C. players
Luton Town F.C. players
Port Vale F.C. players
Bo'ness F.C. players
Scottish Football League players
English Football League players
Year of death missing
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