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https://en.wikipedia.org/wiki/Dependence%20relation | In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence.
Let be a set. A (binary) relation between an element of and a subset of is called a dependence relation, written , if it satisfies the following properties:
if , then ;
if , then there is a finite s... |
https://en.wikipedia.org/wiki/Kolmogorov%27s%20inequality | In probability theory, Kolmogorov's inequality is a so-called "maximal inequality" that gives a bound on the probability that the partial sums of a finite collection of independent random variables exceed some specified bound.
Statement of the inequality
Let X1, ..., Xn : Ω → R be independent random variables defined ... |
https://en.wikipedia.org/wiki/Nilpotent%20cone | In mathematics, the nilpotent cone of a finite-dimensional semisimple Lie algebra is the set of elements that act nilpotently in all representations of In other words,
The nilpotent cone is an irreducible subvariety of (considered as a vector space).
Example
The nilpotent cone of , the Lie algebra of 2×2 matr... |
https://en.wikipedia.org/wiki/Verma%20module | Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.
Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Specifically, although Verma modules themselves are infinite dimensional, qu... |
https://en.wikipedia.org/wiki/Lie%27s%20theorem | In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if is a finite-dimensional representation of a solvable Lie algebra, then there's a flag of invariant subspaces of with , meaning that for each and i.
Put in another way,... |
https://en.wikipedia.org/wiki/Przemys%C5%82aw%20Prusinkiewicz | Przemysław (Przemek) Prusinkiewicz is a Polish computer scientist who advanced the idea that Fibonacci numbers in nature can be in part understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars. Prusinkiewicz's main work is on the modeling of plant growt... |
https://en.wikipedia.org/wiki/Distribution%20%28differential%20geometry%29 | In differential geometry, a discipline within mathematics, a distribution on a manifold is an assignment of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle .
Distributions satisfying a further integrability conditio... |
https://en.wikipedia.org/wiki/Harnack%27s%20inequality | In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by . Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions. , and generalized Harnack's inequality to solutions of elliptic or p... |
https://en.wikipedia.org/wiki/Equivariant%20cohomology | In mathematics, equivariant cohomology (or Borel cohomology) is a cohomology theory from algebraic topology which applies to topological spaces with a group action. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a spac... |
https://en.wikipedia.org/wiki/Mary%20Ellen%20Rudin | Mary Ellen Rudin (December 7, 1924 – March 18, 2013) was an American mathematician known for her work in set-theoretic topology. In 2013, Elsevier established the Mary Ellen Rudin Young Researcher Award, which is awarded annually to a young researcher, mainly in fields adjacent to general topology.
Early life and educ... |
https://en.wikipedia.org/wiki/Hartogs%20number | In mathematics, specifically in axiomatic set theory, a Hartogs number is an ordinal number associated with a set. In particular, if X is any set, then the Hartogs number of X is the least ordinal α such that there is no injection from α into X. If X can be well-ordered then the cardinal number of α is a minimal cardi... |
https://en.wikipedia.org/wiki/Friedrich%20Hartogs | Friedrich Moritz "Fritz" Hartogs (20 May 1874 – 18 August 1943) was a German-Jewish mathematician, known for his work on set theory and foundational results on several complex variables.
Life
Hartogs was the son of the merchant Gustav Hartogs and his wife Elise Feist and grew up in Frankfurt am Main.
He studied at t... |
https://en.wikipedia.org/wiki/N-skeleton | In mathematics, particularly in algebraic topology, the of a topological space presented as a simplicial complex (resp. CW complex) refers to the subspace that is the union of the simplices of (resp. cells of ) of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping a... |
https://en.wikipedia.org/wiki/Charles%20%C3%89tienne%20Louis%20Camus | Charles Étienne Louis Camus (25 August 1699 – 2 February 1768), was a French mathematician and mechanician who was born at Crécy-en-Brie, near Meaux.
He studied mathematics, civil and military architecture, and astronomy after leaving Collège de Navarre in Paris. In 1730 he was appointed professor of architecture and... |
https://en.wikipedia.org/wiki/Normalization%20%28statistics%29 | In statistics and applications of statistics, normalization can have a range of meanings. In the simplest cases, normalization of ratings means adjusting values measured on different scales to a notionally common scale, often prior to averaging. In more complicated cases, normalization may refer to more sophisticated a... |
https://en.wikipedia.org/wiki/Hartogs%27s%20theorem%20on%20separate%20holomorphicity | In mathematics, Hartogs's theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. Roughly speaking, it states that a 'separately analytic' function is continuous. More precisely, if is a function which is analytic in each variable zi, 1 ≤ i ≤ n, while the other variables are h... |
https://en.wikipedia.org/wiki/Minkowski%E2%80%93Hlawka%20theorem | In mathematics, the Minkowski–Hlawka theorem is a result on the lattice packing of hyperspheres in dimension n > 1. It states that there is a lattice in Euclidean space of dimension n, such that the corresponding best packing of hyperspheres with centres at the lattice points has density Δ satisfying
with ζ the Rieman... |
https://en.wikipedia.org/wiki/Clifford%27s%20theorem%20on%20special%20divisors | In mathematics, Clifford's theorem on special divisors is a result of on algebraic curves, showing the constraints on special linear systems on a curve C.
Statement
A divisor on a Riemann surface C is a formal sum of points P on C with integer coefficients. One considers a divisor as a set of constraints on meromor... |
https://en.wikipedia.org/wiki/Finite%20topology | Finite topology is a mathematical concept which has several different meanings.
Finite topological space
A finite topological space is a topological space, the underlying set of which is finite.
In endomorphism rings and modules
If A and B are abelian groups then the finite topology on the group of homomorphisms Ho... |
https://en.wikipedia.org/wiki/Metaplectic%20group | In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It can be defined over either real or p-adic numbers. The construction covers more generally the case of an arbitrary local or finite field, and even the ring of adeles.
The metaplectic group has a particularly significant infin... |
https://en.wikipedia.org/wiki/Bred%20vector | In applied mathematics, bred vectors are perturbations related to Lyapunov vectors, that capture fast-growing dynamical instabilities of the solution of a numerical model. They are used, for example, as initial perturbations for ensemble forecasting in numerical weather prediction. They were introduced by Zoltan Toth a... |
https://en.wikipedia.org/wiki/Plimpton%20322 | Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University. This tablet, believed to have been written about 1800 BC, has a table of four columns and 15 rows of numbers in the cuneiform script of the pe... |
https://en.wikipedia.org/wiki/251%20%28number%29 | 251 (two hundred [and] fifty-one) is the natural number between 250 and 252. It is also a prime number.
In mathematics
251 is:
a Sophie Germain prime.
the sum of three consecutive primes (79 + 83 + 89) and seven consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47).
a Chen prime.
an Eisenstein prime with no imaginary ... |
https://en.wikipedia.org/wiki/Roger%20Heath-Brown | David Rodney "Roger" Heath-Brown (born 12 October 1952) is a British mathematician working in the field of analytic number theory.
Education
He was an undergraduate and graduate student of Trinity College, Cambridge; his research supervisor was Alan Baker.
Career and research
In 1979 he moved to the University of Ox... |
https://en.wikipedia.org/wiki/Hugh%20Lowell%20Montgomery | Hugh Lowell Montgomery (born August 26, 1944) is an American mathematician, working in the fields of analytic number theory and mathematical analysis. As a Marshall scholar, Montgomery earned his Ph.D. from the University of Cambridge. For many years, Montgomery has been teaching at the University of Michigan.
He is b... |
https://en.wikipedia.org/wiki/Weyl%27s%20theorem | In mathematics, Weyl's theorem or Weyl's lemma might refer to one of a number of results of Hermann Weyl. These include
the Peter–Weyl theorem
Weyl's theorem on complete reducibility, results originally derived from the unitarian trick on representation theory of semisimple groups and semisimple Lie algebras
Weyl's... |
https://en.wikipedia.org/wiki/Matrix%20population%20models | Matrix population models are a specific type of population model that uses matrix algebra. Population models are used in population ecology to model the dynamics of wildlife or human populations. Matrix algebra, in turn, is simply a form of algebraic shorthand for summarizing a larger number of often repetitious and te... |
https://en.wikipedia.org/wiki/Square-free%20element | In mathematics, a square-free element is an element r of a unique factorization domain R that is not divisible by a non-trivial square. This means that every s such that is a unit of R.
Alternate characterizations
Square-free elements may be also characterized using their prime decomposition. The unique factorization... |
https://en.wikipedia.org/wiki/Signed%20measure | In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values, i.e., to acquire sign.
Definition
There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. Signed meas... |
https://en.wikipedia.org/wiki/Hahn%20decomposition%20theorem | In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that for any measurable space and any signed measure defined on the -algebra , there exist two -measurable sets, and , of such that:
and .
For every such that , one has , i.e., is a positive set for .
Fo... |
https://en.wikipedia.org/wiki/Character%20sum | In mathematics, a character sum is a sum of values of a Dirichlet character χ modulo N, taken over a given range of values of n. Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in particular in the classical question of finding an upper bound for the least quadr... |
https://en.wikipedia.org/wiki/Kodaira%20embedding%20theorem | In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex manifolds are defined by homogeneous polynomials.
Kunihiko Kodaira's result is that for a compact Kähler manifold M, with... |
https://en.wikipedia.org/wiki/Product%20measure | In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two topological spaces, except that there can be many natural choices for t... |
https://en.wikipedia.org/wiki/Kodaira%20vanishing%20theorem | In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices q > 0 are automatically zero. The implications for the group with index q = 0 is usually that its dimension — the numb... |
https://en.wikipedia.org/wiki/Composite%20measure | Composite measure in statistics and research design refer to composite measures of variables, i.e. measurements based on multiple data items.
An example of a composite measure is an IQ test, which gives a single score based on a series of responses to various questions.
Three common composite measures include:
index... |
https://en.wikipedia.org/wiki/Church%20encoding | In mathematics, Church encoding is a means of representing data and operators in the lambda calculus. The Church numerals are a representation of the natural numbers using lambda notation. The method is named for Alonzo Church, who first encoded data in the lambda calculus this way.
Terms that are usually considered p... |
https://en.wikipedia.org/wiki/%CE%93-convergence | In the field of mathematical analysis for the calculus of variations, Γ-convergence (Gamma-convergence) is a notion of convergence for functionals. It was introduced by Ennio de Giorgi.
Definition
Let be a topological space and denote the set of all neighbourhoods of the point . Let further be a sequence of functio... |
https://en.wikipedia.org/wiki/263%20%28number%29 | 263 is the natural number between 262 and 264. It is also a prime number.
In mathematics
263 is
a balanced prime,
an irregular prime,
a Ramanujan prime, a Chen prime, and
a safe prime.
It is also a strictly non-palindromic number and a happy number.
References
Integers |
https://en.wikipedia.org/wiki/269%20%28number%29 | 269 (two hundred [and] sixty-nine) is the natural number between 268 and 270. It is also a prime number.
In mathematics
269 is a twin prime,
and a Ramanujan prime.
It is the largest prime factor of 9! + 1 = 362881,
and the smallest natural number that cannot be represented as the determinant of a 10 × 10 (0,1)-matrix.... |
https://en.wikipedia.org/wiki/AN/APQ-181 | The AN/APQ-181 is an all-weather, low probability of intercept (LPI) phased array radar system designed by Hughes Aircraft (now Raytheon) for the U.S. Air Force B-2A Spirit bomber aircraft. The system was developed in the mid-1980s and entered service in 1993. The APQ-181 provides a number of precision targeting modes,... |
https://en.wikipedia.org/wiki/Kloosterman%20sum | In mathematics, a Kloosterman sum is a particular kind of exponential sum. They are named for the Dutch mathematician Hendrik Kloosterman, who introduced them in 1926 when he adapted the Hardy–Littlewood circle method to tackle a problem involving positive definite diagonal quadratic forms in four as opposed to five or... |
https://en.wikipedia.org/wiki/Kuiper%27s%20theorem | In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H. It states that the space GL(H) of invertible bounded endomorphisms of H is such that all maps from any finite complex Y to GL(H) are homotopic to a constant, for the nor... |
https://en.wikipedia.org/wiki/Tensor%20product%20of%20modules | In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a ... |
https://en.wikipedia.org/wiki/A%E2%99%AF%20%28Axiom%29 | {{DISPLAYTITLE:A♯ (Axiom)}}
A♯ (pronounced: A sharp) is an object-oriented functional programming language distributed as a separable component of Version 2 of the Axiom computer algebra system. A# types and functions are first-class values and can be used freely together with an extensive library of data structures a... |
https://en.wikipedia.org/wiki/Strong%20prior | In Bayesian statistics, a strong prior is a preceding assumption, theory, concept or idea upon which, after taking account of new information, a current assumption, theory, concept or idea is founded. The term is used to contrast the case of a weak or uninformative prior probability. A strong prior would be a type of ... |
https://en.wikipedia.org/wiki/Throughput%20%28business%29 | Throughput is rate at which a product is moved through a production process and is consumed by the end-user, usually measured in the form of sales or use statistics. The goal of most organizations is to minimize the investment in inputs as well as operating expenses while increasing throughput of its production systems... |
https://en.wikipedia.org/wiki/Rho%20calculus | There are two different calculi that use the name rho-calculus:
The first is a formalism intended to combine the higher-order facilities of lambda calculus with the pattern matching of term rewriting.
The second is a reflective higher-order variant of the asynchronous polyadic pi calculus.
References
Lambda calcu... |
https://en.wikipedia.org/wiki/Zolotarev%27s%20lemma | In number theory, Zolotarev's lemma states that the Legendre symbol
for an integer a modulo an odd prime number p, where p does not divide a, can be computed as the sign of a permutation:
where ε denotes the signature of a permutation and πa is the permutation of the nonzero residue classes mod p induced by multiplic... |
https://en.wikipedia.org/wiki/Theta%20correspondence | In mathematics, the theta correspondence or Howe correspondence is a mathematical relation between representations of two groups of a reductive dual pair. The local theta correspondence relates irreducible admissible representations over a local field, while the global theta correspondence relates irreducible automorph... |
https://en.wikipedia.org/wiki/Weierstrass%E2%80%93Enneper%20parameterization | In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry.
Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863.
Let and be functions on either the entire complex plane or the unit disk, where is meromorphic and is analyt... |
https://en.wikipedia.org/wiki/Absolute%20Galois%20group | In mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group.
... |
https://en.wikipedia.org/wiki/United%20Nations%20Statistics%20Division | The United Nations Statistics Division (UNSD), formerly the United Nations Statistical Office, serves under the United Nations Department of Economic and Social Affairs (DESA) as the central mechanism within the Secretariat of the United Nations to supply the statistical needs and coordinating activities of the global ... |
https://en.wikipedia.org/wiki/Watchman%20route%20problem | The Watchman Problem is an optimization problem in computational geometry where the objective is to compute the shortest route a watchman should take to guard an entire area with obstacles given only a map of the area. The challenge is to make sure the watchman peeks behind every corner and to determine the best order ... |
https://en.wikipedia.org/wiki/Bauer%E2%80%93Fike%20theorem | In mathematics, the Bauer–Fike theorem is a standard result in the perturbation theory of the eigenvalue of a complex-valued diagonalizable matrix. In its substance, it states an absolute upper bound for the deviation of one perturbed matrix eigenvalue from a properly chosen eigenvalue of the exact matrix. Informally s... |
https://en.wikipedia.org/wiki/Asymptotic%20distribution | In mathematics and statistics, an asymptotic distribution is a probability distribution that is in a sense the "limiting" distribution of a sequence of distributions. One of the main uses of the idea of an asymptotic distribution is in providing approximations to the cumulative distribution functions of statistical est... |
https://en.wikipedia.org/wiki/Branched%20covering | In mathematics, a branched covering is a map that is almost a covering map, except on a small set.
In topology
In topology, a map is a branched covering if it is a covering map everywhere except for a nowhere dense set known as the branch set. Examples include the map from a wedge of circles to a single circle, where ... |
https://en.wikipedia.org/wiki/Torelli%20theorem | In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) C is determined by its Jacobian variety J(C), when the latter is given in the form of a principal... |
https://en.wikipedia.org/wiki/Cue%20validity | Cue validity is the conditional probability that an object falls in a particular category given a particular feature or cue. The term was popularized by , and especially by Eleanor Rosch in her investigations of the acquisition of so-called basic categories (;).
Definition of cue validity
Formally, the cue validity ... |
https://en.wikipedia.org/wiki/Noether%20normalization%20lemma | In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field k, and any finitely generated commutative k-algebra A, there exist algebraically independent elements y1, y2, ..., yd in A such that A is a finitely generated module over ... |
https://en.wikipedia.org/wiki/Kennesaw%20Mountain%20High%20School | Kennesaw Mountain High School is a public high school located in Kennesaw, Cobb County, Georgia, United States. It was founded in 2000 as a magnet school specializing in science and mathematics, and is one of sixteen high schools in the Cobb County School District.
History
Students
Kennesaw Mountain High School was f... |
https://en.wikipedia.org/wiki/Disjunctive%20sum | In the mathematics of combinatorial games, the sum or disjunctive sum of two games is a game in which the two games are played in parallel, with each player being allowed to move in just one of the games per turn. The sum game finishes when there are no moves left in either of the two parallel games, at which point (i... |
https://en.wikipedia.org/wiki/List%20of%20ARCA%20drivers | The following is a list of drivers who are currently competing in a series sanctioned by the Automobile Racing Club of America (ARCA).
ARCA Racing Series drivers
All statistics used in these tables are as of the end of the 2018 Lucas Oil 200 Driven by General Tire. (Race 1/20)
Full-time drivers
Part-time drivers
Ca... |
https://en.wikipedia.org/wiki/Mathematics%20Subject%20Classification | The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme that has collaboratively been produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH. The MSC is used by many mathematics journals, which ask au... |
https://en.wikipedia.org/wiki/Polish%20School%20of%20Mathematics | The Polish School of Mathematics was the mathematics community that flourished in Poland in the 20th century, particularly during the Interbellum between World Wars I and II.
Overview
The Polish School of Mathematics subsumed:
the Lwów School of Mathematics - mostly focused on functional analysis;
the Warsaw School ... |
https://en.wikipedia.org/wiki/Taubes%27s%20Gromov%20invariant | In mathematics, the Gromov invariant of Clifford Taubes counts embedded (possibly disconnected) pseudoholomorphic curves in a symplectic 4-manifold, where the curves are holomorphic with respect to an auxiliary compatible almost complex structure. (Multiple covers of 2-tori with self-intersection 0 are also counted.)
... |
https://en.wikipedia.org/wiki/Null%20%28mathematics%29 | In mathematics, the word null (from meaning "zero", which is from meaning "none") is often associated with the concept of zero or the concept of nothing. It is used in varying context from "having zero members in a set" (e.g., null set) to "having a value of zero" (e.g., null vector).
In a vector space, the null vec... |
https://en.wikipedia.org/wiki/Moduli%20of%20algebraic%20curves | In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, ... |
https://en.wikipedia.org/wiki/Biharmonic%20equation | In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. Specifically, it is used in the modeling of thin structures that react elastically to external forces.
Notation
It ... |
https://en.wikipedia.org/wiki/Euler%20method | In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is t... |
https://en.wikipedia.org/wiki/Gilbert%20Ames%20Bliss | Gilbert Ames Bliss, (9 May 1876 – 8 May 1951), was an American mathematician, known for his work on the calculus of variations.
Life
Bliss grew up in a Chicago family that eventually became affluent; in 1907, his father became president of the company supplying all of Chicago's electricity. The family was not affluent... |
https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch%20theorem%20for%20smooth%20manifolds | In mathematics, a Riemann–Roch theorem for smooth manifolds is a version of results such as the Hirzebruch–Riemann–Roch theorem or Grothendieck–Riemann–Roch theorem (GRR) without a hypothesis making the smooth manifolds involved carry a complex structure. Results of this kind were obtained by Michael Atiyah and Friedri... |
https://en.wikipedia.org/wiki/Upper%20and%20lower%20probabilities | Upper and lower probabilities are representations of imprecise probability. Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, this method uses two numbers: the upper probability of the event and the lower probability of the event.
Because frequentist statist... |
https://en.wikipedia.org/wiki/Helge%20Tverberg | Helge Arnulf Tverberg (March 6, 1935December 28, 2020) was a Norwegian mathematician. He was a professor in the Mathematics Department at the University of Bergen, his speciality being combinatorics; he retired at the mandatory age of seventy.
He was born in Bergen. He took the cand.real. degree at the University of B... |
https://en.wikipedia.org/wiki/Zeta%20function%20regularization | In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators. The technique is now commonly applied to proble... |
https://en.wikipedia.org/wiki/Carved%20turn | A carved turn is a skiing and snowboarding term for the technique of turning by shifting the ski or snowboard onto its edges. When edged, the sidecut geometry causes the ski (in the following, snowboard is implicit and not mentioned) to bend into an arc, and the ski naturally follows this arc shape to produce a turning... |
https://en.wikipedia.org/wiki/Grothendieck%20spectral%20sequence | In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced by Alexander Grothendieck in his Tôhoku paper, is a spectral sequence that computes the derived functors of the composition of two functors , from knowledge of the derived functors of and .
Many spectral sequences in a... |
https://en.wikipedia.org/wiki/Projective%20object | In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object.
Definition
An object in a category is projective if for any epimorphis... |
https://en.wikipedia.org/wiki/Leray%27s%20theorem | In algebraic topology and algebraic geometry, Leray's theorem (so named after Jean Leray) relates abstract sheaf cohomology with Čech cohomology.
Let be a sheaf on a topological space and an open cover of If is acyclic on every finite intersection of elements of , then
where is the -th Čech cohomology g... |
https://en.wikipedia.org/wiki/De%20Rham%E2%80%93Weil%20theorem | In algebraic topology, the De Rham–Weil theorem allows computation of sheaf cohomology using an acyclic resolution of the sheaf in question.
Let be a sheaf on a topological space and a resolution of by acyclic sheaves. Then
where denotes the -th sheaf cohomology group of with coefficients in
The De R... |
https://en.wikipedia.org/wiki/Gelfand%20pair | In mathematics, a Gelfand pair is a pair (G,K) consisting of a group G and a subgroup K (called an Euler subgroup of G) that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to... |
https://en.wikipedia.org/wiki/Formal%20moduli | In mathematics, formal moduli are an aspect of the theory of moduli spaces (of algebraic varieties or vector bundles, for example), closely linked to deformation theory and formal geometry. Roughly speaking, deformation theory can provide the Taylor polynomial level of information about deformations, while formal modul... |
https://en.wikipedia.org/wiki/Asian%20Pacific%20Mathematics%20Olympiad | The Asian Pacific Mathematics Olympiad (APMO) starting from 1989 is a regional mathematics competition which involves countries from the Asian Pacific region. The United States also takes part in the APMO. Every year, APMO is held in the afternoon of the second Monday of March for participating countries in the North ... |
https://en.wikipedia.org/wiki/Hilbert%20class%20field | In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime ideals in K.
In this c... |
https://en.wikipedia.org/wiki/Class%20number%20formula | In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function.
General statement of the class number formula
We start with the following data:
is a number field.
, where denotes the number of real embeddings of , and is the number... |
https://en.wikipedia.org/wiki/Current%20%28mathematics%29 | In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly supported differential k-forms, on a smooth manifold M. Currents formally behave like Schwartz distributions on a space of ... |
https://en.wikipedia.org/wiki/K-homology | In mathematics, K-homology is a homology theory on the category of locally compact Hausdorff spaces. It classifies the elliptic pseudo-differential operators acting on the vector bundles over a space. In terms of -algebras, it classifies the Fredholm modules over an algebra.
An operator homotopy between two Fredholm ... |
https://en.wikipedia.org/wiki/Length%20function | In the mathematical field of geometric group theory, a length function is a function that assigns a number to each element of a group.
Definition
A length function L : G → R+ on a group G is a function satisfying:
Compare with the axioms for a metric and a filtered algebra.
Word metric
An important example of a len... |
https://en.wikipedia.org/wiki/Van%20Aubel%27s%20theorem | In plane geometry, Van Aubel's theorem describes a relationship between squares constructed on the sides of a quadrilateral. Starting with a given convex quadrilateral, construct a square, external to the quadrilateral, on each side. Van Aubel's theorem states that the two line segments between the centers of opposite ... |
https://en.wikipedia.org/wiki/Primary%20ideal | In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n > 0. For example, in the ring of integers Z, (pn) is a primary ideal if p is a prime number.
The notion of primary ideals i... |
https://en.wikipedia.org/wiki/Jasmone | Jasmone is an organic compound, which is a volatile portion of the oil from jasmine flowers. It is a colorless to pale yellow liquid. Jasmone can exist in two isomeric forms with differing geometry around the pentenyl double bond, cis-jasmone and trans-jasmone. The natural extract contains only the cis form, while s... |
https://en.wikipedia.org/wiki/Direct%20image%20functor | In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf F defined on a topological space X and a continuous map f: X → Y, we can define a new sheaf f∗F ... |
https://en.wikipedia.org/wiki/%C3%89tale%20morphism | In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the... |
https://en.wikipedia.org/wiki/%C3%89tale%20fundamental%20group | The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.
Topological analogue/informal discussion
In algebraic topology, the fundamental group of a pointed topological space is defined as the group of homotopy classes of loops b... |
https://en.wikipedia.org/wiki/Discriminant%20of%20an%20algebraic%20number%20field | In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field. More specifically, it is proportional to the squared volume of the fundamental domain of the ring of integers, and it regulates which ... |
https://en.wikipedia.org/wiki/Negative%20relationship | In statistics, there is a negative relationship or inverse relationship between two variables if higher values of one variable tend to be associated with lower values of the other. A negative relationship between two variables usually implies that the correlation between them is negative, or — what is in some contexts ... |
https://en.wikipedia.org/wiki/Generalized%20Jacobian | In algebraic geometry a generalized Jacobian is a commutative algebraic group associated to a curve with a divisor, generalizing the Jacobian variety of a complete curve. They were introduced by Maxwell Rosenlicht in 1954, and can be used to study ramified coverings of a curve, with abelian Galois group. Generalized Ja... |
https://en.wikipedia.org/wiki/Octagonal%20antiprism | In geometry, the octagonal antiprism is the 6th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.
Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals.
In... |
https://en.wikipedia.org/wiki/Octagonal%20prism | In geometry, the octagonal prism is a prism comprising eight rectangular sides joining two regular octagon caps.
Symmetry
Images
The octagonal prism can also be seen as a tiling on a sphere:
Use
In optics, octagonal prisms are used to generate flicker-free images in movie projectors.
In uniform honeycombs and 4-... |
https://en.wikipedia.org/wiki/Boustrophedon%20transform | In mathematics, the boustrophedon transform is a procedure which maps one sequence to another. The transformed sequence is computed by an "addition" operation, implemented as if filling a triangular array in a boustrophedon (zigzag- or serpentine-like) manner—as opposed to a "Raster Scan" sawtooth-like manner.
Definit... |
https://en.wikipedia.org/wiki/786%20%28number%29 | 786 (seven hundred [and] eighty-six) is the natural number following 785 and preceding 787.
In mathematics
786 is:
a sphenic number.
a Harshad number in bases 4, 5, 7, 14 and 16.
the aliquot sum of 510.
part of the 321329-aliquot tree. The complete aliquot sequence starting at 498 is: 498, 510, 786, 798, 1122, 1470, 2... |
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