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https://en.wikipedia.org/wiki/Proofs%20of%20quadratic%20reciprocity | In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred proofs of the law of quadratic reciprocity have been published.
Proof synopsis
Of the elementary combinatorial proofs, there are two which apply types of double cou... |
https://en.wikipedia.org/wiki/Logical%20equality | Logical equality is a logical operator that corresponds to equality in Boolean algebra and to the logical biconditional in propositional calculus. It gives the functional value true if both functional arguments have the same logical value, and false if they are different.
It is customary practice in various applicati... |
https://en.wikipedia.org/wiki/Intermediate%20Jacobian | In mathematics, the intermediate Jacobian of a compact Kähler manifold or Hodge structure is a complex torus that is a common generalization of the Jacobian variety of a curve and the Picard variety and the Albanese variety. It is obtained by putting a complex structure on the torus for n odd. There are several diffe... |
https://en.wikipedia.org/wiki/Degree%20of%20a%20continuous%20mapping | In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping. The degree is always an integer, but may be positive or negative depending on the orie... |
https://en.wikipedia.org/wiki/Castelnuovo%E2%80%93de%20Franchis%20theorem | In mathematics, the Castelnuovo–de Franchis theorem is a classical result on complex algebraic surfaces. Let X be such a surface, projective and non-singular, and let
ω1 and ω2
be two differentials of the first kind on X which are linearly independent but with wedge product 0. Then this data can be represented as a pul... |
https://en.wikipedia.org/wiki/De%20Franchis%20theorem | In mathematics, the de Franchis theorem is one of a number of closely related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, X and Y, in the case of genus g > 1. The simplest is that the automorphism group of X is finite (see though Hurwitz's automorphisms theorem). More generall... |
https://en.wikipedia.org/wiki/Enriques%20surface | In mathematics, Enriques surfaces are algebraic surfaces such that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial square. Enriques surfaces are all projective (and therefore Kähler over the complex numbers) and are elliptic surfaces of genus 0.
Over fields of characteristic not ... |
https://en.wikipedia.org/wiki/Tate%20module | In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative group scheme over a field K, Ks is the separable closure of K, and A = G(Ks) (the Ks-valued points of G). In this case,... |
https://en.wikipedia.org/wiki/Tate%20twist | In number theory and algebraic geometry, the Tate twist, named after John Tate, is an operation on Galois modules.
For example, if K is a field, GK is its absolute Galois group, and ρ : GK → AutQp(V) is a representation of GK on a finite-dimensional vector space V over the field Qp of p-adic numbers, then the Tate tw... |
https://en.wikipedia.org/wiki/Timothy%20Kanold | Dr. Timothy D. Kanold is a mathematics educator and author of textbooks. He was the president of the National Council of Supervisors of Mathematics (NCSM) from 2008 to 2009.
Dr. Kanold holds a bachelor's degree in Education and a master's degree in Mathematics from the University of Illinois, and a doctorate in Educat... |
https://en.wikipedia.org/wiki/Noether%27s%20theorem%20on%20rationality%20for%20surfaces | In mathematics, Noether's theorem on rationality for surfaces is a classical result of Max Noether on complex algebraic surfaces, giving a criterion for a rational surface. Let S be an algebraic surface that is non-singular and projective. Suppose there is a morphism φ from S to the projective line, with general fibre ... |
https://en.wikipedia.org/wiki/Nehari | Zeev Nehari, mathematician
Nehari manifold in mathematics
Nihari, South Asian stew |
https://en.wikipedia.org/wiki/Walter%20Plecker | Walter Ashby Plecker (April 2, 1861 – August 2, 1947) was an American physician and public health advocate who was the first registrar of Virginia's Bureau of Vital Statistics, serving from 1912 to 1946. He was a leader of the Anglo-Saxon Clubs of America, a white supremacist organization founded in Richmond, Virginia,... |
https://en.wikipedia.org/wiki/Axiality%20and%20rhombicity | In physics and mathematics, axiality and rhombicity are two characteristics of a symmetric second-rank tensor in three-dimensional Euclidean space, describing its directional asymmetry.
Let A denote a second-rank tensor in R3, which can be represented by a 3-by-3 matrix. We assume that A is symmetric. This implies tha... |
https://en.wikipedia.org/wiki/Fano%20variety | In algebraic geometry, a Fano variety, introduced by Gino Fano in , is a complete variety X whose anticanonical bundle KX* is ample. In this definition, one could assume that X is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as... |
https://en.wikipedia.org/wiki/The%20Number%20Devil | The Number Devil: A Mathematical Adventure () is a book for children and young adults that explores mathematics. It was originally written in 1997 in German by Hans Magnus Enzensberger and illustrated by Rotraut Susanne Berner. The book follows a young boy named Robert, who is taught mathematics by a sly "number devil"... |
https://en.wikipedia.org/wiki/Zariski%20surface | In algebraic geometry, a branch of mathematics, a Zariski surface is a surface over a field of characteristic p > 0 such that there is a dominant inseparable map of degree p from the projective plane to the surface. In particular, all Zariski surfaces are unirational. They were named by Piotr Blass in 1977 after Oscar ... |
https://en.wikipedia.org/wiki/Kerala%20school%20of%20astronomy%20and%20mathematics | The Kerala school of astronomy and mathematics or the Kerala school was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Tirur, Malappuram, Kerala, India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Ac... |
https://en.wikipedia.org/wiki/Gonality%20of%20an%20algebraic%20curve | In mathematics, the gonality of an algebraic curve C is defined as the lowest degree of a nonconstant rational map from C to the projective line. In more algebraic terms, if C is defined over the field K and K(C) denotes the function field of C, then the gonality is the minimum value taken by the degrees of field exten... |
https://en.wikipedia.org/wiki/K%C3%B6the%20conjecture | In mathematics, the Köthe conjecture is a problem in ring theory, open . It is formulated in various ways. Suppose that R is a ring. One way to state the conjecture is that if R has no nil ideal, other than {0}, then it has no nil one-sided ideal, other than {0}.
This question was posed in 1930 by Gottfried Köthe (190... |
https://en.wikipedia.org/wiki/Nil%20ideal | In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent.
The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil. Unfortuna... |
https://en.wikipedia.org/wiki/Locally%20nilpotent | In the mathematical field of commutative algebra, an ideal I in a commutative ring A is locally nilpotent at a prime ideal p if Ip, the localization of I at p, is a nilpotent ideal in Ap.
In non-commutative algebra and group theory, an algebra or group is locally nilpotent if and only if every finitely generated subal... |
https://en.wikipedia.org/wiki/Ruziewicz%20problem | In mathematics, the Ruziewicz problem (sometimes Banach–Ruziewicz problem) in measure theory asks whether the usual Lebesgue measure on the n-sphere is characterised, up to proportionality, by its properties of being finitely additive, invariant under rotations, and defined on all Lebesgue measurable sets.
This was a... |
https://en.wikipedia.org/wiki/Ian%20G.%20Macdonald | Ian Grant Macdonald (11 October 1928 – 8 August 2023) was a British mathematician known for his contributions to symmetric functions, special functions, Lie algebra theory and other aspects of algebra, algebraic combinatorics, and combinatorics.
Early life and education
Born in London, he was educated at Winchester C... |
https://en.wikipedia.org/wiki/Ore%20condition | In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The right Ore condition for a mult... |
https://en.wikipedia.org/wiki/Quasiregular%20representation | This article addresses the notion of quasiregularity in the context of representation theory and topological algebra. For other notions of quasiregularity in mathematics, see the disambiguation page quasiregular.
In mathematics, quasiregular representation is a concept of representation theory, for a locally compact g... |
https://en.wikipedia.org/wiki/Vector%20fields%20on%20spheres | In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras.
Specifically, the question is how many linearly independent smooth nowhere-zero vector fields can be constructed... |
https://en.wikipedia.org/wiki/Barlow%20surface | In mathematics, a Barlow surface is one of the complex surfaces introduced by . They are simply connected surfaces of general type with pg = 0. They are homeomorphic but not diffeomorphic to a projective plane blown up in 8 points. The Hodge diamond for the Barlow surfaces is:
See also
Hodge theory
References
Alg... |
https://en.wikipedia.org/wiki/Godeaux%20surface | In mathematics, a Godeaux surface is one of the surfaces of general type introduced by Lucien Godeaux in 1931.
Other surfaces constructed in a similar way with the same Hodge numbers are also sometimes called Godeaux surfaces. Surfaces with the same Hodge numbers (such as Barlow surfaces) are called numerical Godeaux... |
https://en.wikipedia.org/wiki/Stationary%20ergodic%20process | In probability theory, a stationary ergodic process is a stochastic process which exhibits both stationarity and ergodicity. In essence this implies that the random process will not change its statistical properties with time and that its statistical properties (such as the theoretical mean and variance of the process)... |
https://en.wikipedia.org/wiki/Horrocks%E2%80%93Mumford%20bundle | In algebraic geometry, the Horrocks–Mumford bundle is an indecomposable rank 2 vector bundle on 4-dimensional projective space P4 introduced by . It is the only such bundle known, although a generalized construction involving Paley graphs produces other rank 2 sheaves (Sasukara et al. 1993). The zero sets of sections o... |
https://en.wikipedia.org/wiki/Vector%20area | In 3-dimensional geometry and vector calculus, an area vector is a vector combining an area quantity with a direction, thus representing an oriented area in three dimensions.
Every bounded surface in three dimensions can be associated with a unique area vector called its vector area. It is equal to the surface integr... |
https://en.wikipedia.org/wiki/Contiguity | Contiguity or contiguous may refer to:
Contiguous data storage, in computer science
Contiguity (probability theory)
Contiguity (psychology)
Contiguous distribution of species, in biogeography
Geographic contiguity of territorial land
Contiguous zone in territorial waters
See also |
https://en.wikipedia.org/wiki/Petr%20Vop%C4%9Bnka | Petr Vopěnka (16 May 1935 – 20 March 2015) was a Czech mathematician. In the early seventies, he developed alternative set theory (i.e. alternative to the classical Cantor theory), which he subsequently developed in a series of articles and monographs. Vopěnka’s name is associated with many mathematical achievements, ... |
https://en.wikipedia.org/wiki/Klee%27s%20measure%20problem | In computational geometry, Klee's measure problem is the problem of determining how efficiently the measure of a union of (multidimensional) rectangular ranges can be computed. Here, a d-dimensional rectangular range is defined to be a Cartesian product of d intervals of real numbers, which is a subset of Rd.
The prob... |
https://en.wikipedia.org/wiki/Parallelizable%20manifold | In mathematics, a differentiable manifold of dimension n is called parallelizable if there exist smooth vector fields
on the manifold, such that at every point of the tangent vectors
provide a basis of the tangent space at . Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bu... |
https://en.wikipedia.org/wiki/Stunted%20projective%20space | In mathematics, a stunted projective space is a construction on a projective space of importance in homotopy theory, introduced by . Part of a conventional projective space is collapsed down to a point.
More concretely, in a real projective space, complex projective space or quaternionic projective space
KPn,
where ... |
https://en.wikipedia.org/wiki/Spanier%E2%80%93Whitehead%20duality | In mathematics, Spanier–Whitehead duality is a duality theory in homotopy theory, based on a geometrical idea that a topological space X may be considered as dual to its complement in the n-sphere, where n is large enough. Its origins lie in Alexander duality theory, in homology theory, concerning complements in manifo... |
https://en.wikipedia.org/wiki/Alexander%20duality | In mathematics, Alexander duality refers to a duality theory initiated by a result of J. W. Alexander in 1915, and subsequently further developed, particularly by Pavel Alexandrov and Lev Pontryagin. It applies to the homology theory properties of the complement of a subspace X in Euclidean space, a sphere, or other ma... |
https://en.wikipedia.org/wiki/Reduced%20homology | In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise statements to be made (as in Alexander duality) and eliminates many exceptio... |
https://en.wikipedia.org/wiki/Statgraphics | Statgraphics is a statistics package that performs and explains basic and advanced statistical functions.
History
The software was created in 1980 by Dr. Neil W. Polhemus while on the faculty at the Princeton University School of Engineering and Applied Science for use as a teaching tool for his statistics students. ... |
https://en.wikipedia.org/wiki/Nasrin%20Soltankhah | Nasrin Soltankhah () is an Iranian politician who was a Vice President under Mahmoud Ahmadinejad from 2009 to 2013.
Education
Soltankhan received a Bachelor of Science in Mathematics (1976), a Master of Science in Mathematics (1978), and a PhD in Mathematics (1994) from Sharif University of Technology.
Career
Cabine... |
https://en.wikipedia.org/wiki/OGF | OGF can refer to:
Open Gaming Foundation for role-playing games
Open Grid Forum for grid computing
Ordinary generating function in mathematics
Opioid growth factor, an alternative name for met-enkephalin |
https://en.wikipedia.org/wiki/Vector%20calculus%20identities | The following are important identities involving derivatives and integrals in vector calculus.
Operator notation
Gradient
For a function in three-dimensional Cartesian coordinate variables, the gradient is the vector field:
where i, j, k are the standard unit vectors for the x, y, z-axes. More generally, for a fun... |
https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Szekeres%20theorem | In mathematics, the Erdős–Szekeres theorem asserts that, given r, s, any sequence of distinct real numbers with length at least (r − 1)(s − 1) + 1 contains a monotonically increasing subsequence of length r or a monotonically decreasing subsequence of length s. The proof appeared in the same 1935 paper that mentions th... |
https://en.wikipedia.org/wiki/Alfred%20Horn | Alfred Horn (February 17, 1918 – April 16, 2001) was an American mathematician notable for his work in lattice theory and universal algebra. His 1951 paper "On sentences which are true of direct unions of algebras" described Horn clauses and Horn sentences, which later would form the foundation of logic programming.
B... |
https://en.wikipedia.org/wiki/Geometric%20genus | In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds.
Definition
The geometric genus can be defined for non-singular complex projective varieties and more generally for complex manifolds as the Hodge number (equal to by Serre duality), that is, th... |
https://en.wikipedia.org/wiki/Tverberg%27s%20theorem | In discrete geometry, Tverberg's theorem, first stated by Helge Tverberg in 1966, is the result that sufficiently many points in d-dimensional Euclidean space can be partitioned into subsets with intersecting convex hulls. Specifically, for any positive integers d, r and any set of
points there exists a point x (not n... |
https://en.wikipedia.org/wiki/Bockstein%20homomorphism | In homological algebra, the Bockstein homomorphism, introduced by , is a connecting homomorphism associated with a short exact sequence
of abelian groups, when they are introduced as coefficients into a chain complex C, and which appears in the homology groups as a homomorphism reducing degree by one,
To be more pre... |
https://en.wikipedia.org/wiki/Cartan%20formula | In mathematics, Cartan formula can mean:
one in differential geometry: , where , and are Lie derivative, exterior derivative, and interior product, respectively, acting on differential forms. See interior product for the detail. It is also called the Cartan homotopy formula or Cartan magic formula. This formula is n... |
https://en.wikipedia.org/wiki/Vlastimil%20Pt%C3%A1k | Vlastimil Pták (; November 8, 1925 in Prague – May 5 1999) was a Czech mathematician, who worked in functional analysis, theoretical numerical analysis, and linear algebra. Notable early work include generalizations of the open mapping theorem .
During 1945–49 Vlastimil Pták studied mathematics and physics at the Ch... |
https://en.wikipedia.org/wiki/Complete%20intersection | In mathematics, an algebraic variety V in projective space is a complete intersection if the ideal of V is generated by exactly codim V elements. That is, if V has dimension m and lies in projective space Pn, there should exist n − m homogeneous polynomials:
in the homogeneous coordinates Xj, which generate all other... |
https://en.wikipedia.org/wiki/Complete%20intersection%20ring | In commutative algebra, a complete intersection ring is a commutative ring similar to the coordinate rings of varieties that are complete intersections. Informally, they can be thought of roughly as the local rings that can be defined using the "minimum possible" number of relations.
For Noetherian local rings, there ... |
https://en.wikipedia.org/wiki/Altitude%20%28disambiguation%29 | Altitude is the height of an object over a datum.
It may also refer to:
Science and mathematics
Altitude (astronomy), one of the angular coordinates of the horizontal coordinate system
Altitude (triangle), in geometry, a line passing through one vertex of a triangle and perpendicular to the opposite side
Music
Altit... |
https://en.wikipedia.org/wiki/Open-access%20poll | An open-access poll is a type of opinion poll in which a nonprobability sample of participants self-select into participation. The term includes call-in, mail-in, and some online polls.
The most common examples of open-access polls ask people to phone a number, click a voting option on a website, or return a coupon cu... |
https://en.wikipedia.org/wiki/Grassmann%20number | In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1-dimensional algebra is known as a dual number. Grassmann numbers saw an early use in physics to express... |
https://en.wikipedia.org/wiki/Adjunction%20formula | In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedded in well-behaved spaces such as projective space or to prove theorems by... |
https://en.wikipedia.org/wiki/Rotation%20system | In combinatorial mathematics, rotation systems (also called combinatorial embeddings or combinatorial maps) encode embeddings of graphs onto orientable surfaces by describing the circular ordering of a graph's edges around each vertex.
A more formal definition of a rotation system involves pairs of permutations; such a... |
https://en.wikipedia.org/wiki/Peetre%20theorem | In mathematics, the (linear) Peetre theorem, named after Jaak Peetre, is a result of functional analysis that gives a characterisation of differential operators in terms of their effect on generalized function spaces, and without mentioning differentiation in explicit terms. The Peetre theorem is an example of a finit... |
https://en.wikipedia.org/wiki/Planar%20lamina | In mathematics, a planar lamina (or plane lamina) is a figure representing a thin, usually uniform, flat layer of the solid. It serves also as an idealized model of a planar cross section of a solid body in integration.
Planar laminas can be used to determine moments of inertia, or center of mass of flat figures, as w... |
https://en.wikipedia.org/wiki/Order%20dimension | In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order.
This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order.
first studied order dimension; for a more de... |
https://en.wikipedia.org/wiki/Jan%20Mauersberger | Jan Mauersberger (born 17 June 1985) is a retired German footballer who played as a defender.
Mauersberger retired at the end of the 2018/19 season.
Career statistics
References
External links
1985 births
Living people
German men's footballers
Germany men's youth international footballers
FC Bayern Munich II playe... |
https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Steenrod%20axioms | In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homology, developed by Samuel Eilenberg and Norman Steenrod.
One can define a h... |
https://en.wikipedia.org/wiki/%E2%89%A4 | ≤ may refer to:
Inequality (mathematics), relation between values; a ≤ b means "a is less than or equal to b"
Subgroup, a subset of a given group in group theory; H ≤ G is read as "H is a subgroup of G" |
https://en.wikipedia.org/wiki/Phil%20Moorby | Phil Moorby () was a British engineer and computer scientist. Moorby was born and brought up in Birmingham, England, and studied Mathematics at Southampton University, England. Moorby received his master's degree in computer science from Manchester University, England, in 1974. He moved to the United States in 1983.
W... |
https://en.wikipedia.org/wiki/Weierstrass%20point | In mathematics, a Weierstrass point on a nonsingular algebraic curve defined over the complex numbers is a point such that there are more functions on , with their poles restricted to only, than would be predicted by the Riemann–Roch theorem.
The concept is named after Karl Weierstrass.
Consider the vector spaces... |
https://en.wikipedia.org/wiki/Tate%20conjecture | In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The conjecture is a central problem in the theory of algebraic cycles. It can be co... |
https://en.wikipedia.org/wiki/Hedge%20%28linguistics%29 | In the linguistic sub-fields of applied linguistics and pragmatics, a hedge is a word or phrase used in a sentence to express ambiguity, probability, caution, or indecisiveness about the remainder of the sentence, rather than full accuracy, certainty, confidence, or decisiveness. Hedges can also allow speakers and writ... |
https://en.wikipedia.org/wiki/Azumaya%20algebra | In mathematics, an Azumaya algebra is a generalization of central simple algebras to -algebras where need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where is a commutative local ring. The notion was developed further in ring theory, and in algebraic geometry, where Alex... |
https://en.wikipedia.org/wiki/Left-right%20planarity%20test | In graph theory, a branch of mathematics, the left-right planarity test
or de Fraysseix–Rosenstiehl planarity criterion is a characterization of planar graphs based on the properties of the depth-first search trees, published by and used by them with Patrice Ossona de Mendez to develop a linear time planarity testing ... |
https://en.wikipedia.org/wiki/Scheinerman%27s%20conjecture | In mathematics, Scheinerman's conjecture, now a theorem, states that every planar graph is the intersection graph of a set of line segments in the plane. This conjecture was formulated by E. R. Scheinerman in his Ph.D. thesis (1984), following earlier results that every planar graph could be represented as the intersec... |
https://en.wikipedia.org/wiki/Whitney%27s%20planarity%20criterion | In mathematics, Whitney's planarity criterion is a matroid-theoretic characterization of planar graphs, named after Hassler Whitney. It states that a graph G is planar if and only if its graphic matroid is also cographic (that is, it is the dual matroid of another graphic matroid).
In purely graph-theoretic terms, thi... |
https://en.wikipedia.org/wiki/Home%20runs%20allowed | In baseball statistics, home runs allowed (HRA) signifies the total number of home runs a pitcher allowed.
The Major League Baseball record for the most home runs allowed by any pitcher belongs to Jamie Moyer (522 in his career). He gave up home runs while pitching for eight different teams across both leagues. Warren... |
https://en.wikipedia.org/wiki/Operad | In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad , one defines an algebra over to be a set together with concrete operations on this s... |
https://en.wikipedia.org/wiki/Beta%20prime%20distribution | In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind) is an absolutely continuous probability distribution. If has a beta distribution, then the odds has a beta prime distribution.
Definitions
Beta prime distribution is ... |
https://en.wikipedia.org/wiki/Join%20%28topology%29 | In topology, a field of mathematics, the join of two topological spaces and , often denoted by or , is a topological space formed by taking the disjoint union of the two spaces, and attaching line segments joining every point in to every point in . The join of a space with itself is denoted by . The join is defined... |
https://en.wikipedia.org/wiki/Pl%C3%BCcker%20formula | In mathematics, a Plücker formula, named after Julius Plücker, is one of a family of formulae, of a type first developed by Plücker in the 1830s, that relate certain numeric invariants of algebraic curves to corresponding invariants of their dual curves. The invariant called the genus, common to both the curve and its... |
https://en.wikipedia.org/wiki/Dual%20curve | In projective geometry, a dual curve of a given plane curve is a curve in the dual projective plane consisting of the set of lines tangent to . There is a map from a curve to its dual, sending each point to the point dual to its tangent line. If is algebraic then so is its dual and the degree of the dual is known as ... |
https://en.wikipedia.org/wiki/Vector%20bundles%20on%20algebraic%20curves | In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact Riemann surfaces, which is the classical approach, or as locally free sheaves on algebraic curves C in a more general, algebraic setting (which can for example admit singular points).
Some foundational results on... |
https://en.wikipedia.org/wiki/Opposite%20ring | In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring is the ring whose multiplication ∗ is defined by for all in R. The opposite ring ca... |
https://en.wikipedia.org/wiki/Quasi-Lie%20algebra | In mathematics, a quasi-Lie algebra in abstract algebra is just like a Lie algebra, but with the usual axiom
replaced by
(anti-symmetry).
In characteristic other than 2, these are equivalent (in the presence of bilinearity), so this distinction doesn't arise when considering real or complex Lie algebras. It can how... |
https://en.wikipedia.org/wiki/Whitehead%20product | In mathematics, the Whitehead product is a graded quasi-Lie algebra structure on the homotopy groups of a space. It was defined by J. H. C. Whitehead in .
The relevant MSC code is: 55Q15, Whitehead products and generalizations.
Definition
Given elements , the Whitehead bracket
is defined as follows:
The product ... |
https://en.wikipedia.org/wiki/B%C3%A9zout%20domain | In mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every finitely generated ideal is principal. Any principal ideal domain (PID) is a Béz... |
https://en.wikipedia.org/wiki/Holomorphic%20functional%20calculus | In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function f of a complex argument z and an operator T, the aim is to construct an operator, f(T), which naturally extends the function f from complex argument to operator argument. More ... |
https://en.wikipedia.org/wiki/Coframe | In mathematics, a coframe or coframe field on a smooth manifold is a system of one-forms or covectors which form a basis of the cotangent bundle at every point. In the exterior algebra of , one has a natural map from , given by . If is dimensional a coframe is given by a section of such that . The inverse image... |
https://en.wikipedia.org/wiki/CR%20manifold | In mathematics, a CR manifold, or Cauchy–Riemann manifold, is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge.
Formally, a CR manifold is a differentiable manifold M together with a preferred... |
https://en.wikipedia.org/wiki/Barry%20Simon | Barry Martin Simon (born 16 April 1946) is an American mathematical physicist and was the IBM professor of Mathematics and Theoretical Physics at Caltech, known for his prolific contributions in spectral theory, functional analysis, and nonrelativistic quantum mechanics (particularly Schrödinger operators), including... |
https://en.wikipedia.org/wiki/Nakai%20conjecture | In mathematics, the Nakai conjecture is an unproven characterization of smooth algebraic varieties, conjectured by Japanese mathematician Yoshikazu Nakai in 1961.
It states that if V is a complex algebraic variety, such that its ring of differential operators is generated by the derivations it contains, then V is a smo... |
https://en.wikipedia.org/wiki/Mordell%20curve | In algebra, a Mordell curve is an elliptic curve of the form y2 = x3 + n, where n is a fixed non-zero integer.
These curves were closely studied by Louis Mordell, from the point of view of determining their integer points. He showed that every Mordell curve contains only finitely many integer points (x, y). In other w... |
https://en.wikipedia.org/wiki/Zariski%20geometry | In mathematics, a Zariski geometry consists of an abstract structure introduced by Ehud Hrushovski and Boris Zilber, in order to give a characterisation of the Zariski topology on an algebraic curve, and all its powers. The Zariski topology on a product of algebraic varieties is very rarely the product topology, but ri... |
https://en.wikipedia.org/wiki/Theta%20divisor | In mathematics, the theta divisor Θ is the divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers (and principally polarized) by the zero locus of the associated Riemann theta-function. It is therefore an algebraic subvariety of A of dimension dim A − 1.
Classical theory
C... |
https://en.wikipedia.org/wiki/Grothendieck%E2%80%93Katz%20p-curvature%20conjecture | In mathematics, the Grothendieck–Katz p-curvature conjecture is a local-global principle for linear ordinary differential equations, related to differential Galois theory and in a loose sense analogous to the result in the Chebotarev density theorem considered as the polynomial case. It is a conjecture of Alexander Gro... |
https://en.wikipedia.org/wiki/Principal%20ideal%20theorem | In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the class group of an algebraic number field to the class group of its Hilbert class field, which sends all ideal classes to the class of a principal ideal. The phenomeno... |
https://en.wikipedia.org/wiki/Jet%20group | In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. A jet group is a group of jets that describes how a Taylor polynomial transforms under changes of coordinate systems (or, equivalently, diffeomorphisms).
Overview
The k-th ord... |
https://en.wikipedia.org/wiki/Kaufmann%20vortex | The Kaufmann vortex, also known as the Scully model, is a mathematical model for a vortex taking account of viscosity. It uses an algebraic velocity profile. This vortex is not a solution of the Navier–Stokes equations.
Kaufmann and Scully's model for the velocity in the Θ direction is:
The model was suggested by W. ... |
https://en.wikipedia.org/wiki/Symmetric%20product%20of%20an%20algebraic%20curve | In mathematics, the n-fold symmetric product of an algebraic curve C is the quotient space of the n-fold cartesian product
C × C × ... × C
or Cn by the group action of the symmetric group Sn on n letters permuting the factors. It exists as a smooth algebraic variety denoted by ΣnC. If C is a compact Riemann surface, ... |
https://en.wikipedia.org/wiki/Linearly%20disjoint | In mathematics, algebras A, B over a field k inside some field extension of k are said to be linearly disjoint over k if the following equivalent conditions are met:
(i) The map induced by is injective.
(ii) Any k-basis of A remains linearly independent over B.
(iii) If are k-bases for A, B, then the products are ... |
https://en.wikipedia.org/wiki/Prime%20decomposition%20of%203-manifolds | In mathematics, the prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) finite collection of prime 3-manifolds.
A manifold is prime if it cannot be presented as a connected sum of more than one manifold, none of which is th... |
https://en.wikipedia.org/wiki/Child%20mortality | Child mortality is the mortality of children under the age of five. The child mortality rate (also under-five mortality rate) refers to the probability of dying between birth and exactly five years of age expressed per 1,000 live births.
It encompasses neonatal mortality and infant mortality (the probability of death ... |
https://en.wikipedia.org/wiki/Normal%20surface | In mathematics, a normal surface is a surface inside a triangulated 3-manifold that intersects each tetrahedron so that each component of intersection is a triangle or a quad (see figure). A triangle cuts off a vertex of the tetrahedron while a quad separates pairs of vertices. A normal surface may have many componen... |
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