The development was highly algebraic, and the more algebraic the technique, the further you get from having a geometrical intuition about what you're doing.
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One method arose in the mathematical field of algebraic geometry.
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On the whiteboard, basic algebraic equations were covered in blood.
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Nothing says 'I love you' like a customizable algebraic equation.
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"Sometimes you play completely on the algebraic side," he said.
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Understanding it more generally will force us to understand this algebraic structure.
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There's nothing more algebraic than spending your weekend building your favorite heroes.
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Now we've exposed the underlying algebraic structure inherent in this set of symmetries.
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Dr. Birkar's field is algebraic geometry, which investigates connections between numbers and shapes.
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But for many algebra students, the jumble of algebraic symbols is still confusing.
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They were no longer the plain, algebraic box score of a game already performed.
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Algebraic topologists have lived almost exclusively in multidimensional universes of their own calculation for decades.
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We've gotten a glimpse of the algebraic structure underlying the simple symmetries of the square.
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Everyone, it seemed, had a different view of the most important features of algebraic varieties.
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So for instance, in the same way that you would reduce an algebraic problem, e.g.
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His textbooks on algebraic geometry, translated into English, are regarded as classics in the field.
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Use Math to Make the Perfect Valentine "Nothing says 'I love you' like a customizable algebraic equation," writes The Times about Süss, the math widget above: Like many geometric figures, a heart can be captured in all its curvaceous glory by a single algebraic equation.
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Hilbert asked Noether to apply her work on algebraic invariants to the equations in Einstein's theory.
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They have a near algebraic ability to turn a positive into a negative, and vice versa.
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Instead of a square, they are an algebraic structure extracted from a special kind of elliptic curve.
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Worse, all their algebraic glitter rarely if ever adds any insight to the qualitative intuition behind them.
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Their stories play out in various permutations across 19 scenes in a scheme that seems almost algebraic.
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There are these two traditions, to take combinatorial objects and either make them geometric or make them algebraic.
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I think we found a really beautiful connection between the geometric and the algebraic structure of combinatorial objects.
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At Seoul National, he taught a yearlong lecture course in a broad area of mathematics called algebraic geometry.
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A vast and powerful language weaving algebraic patterns, calculating numbers like blooming flowers — fiery-hued hyacinth, glassy orchid.
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The process of identifying intriguing algebraic surfaces that possessed singularities sometimes took Dr. Hauser and his students days.
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In principle it's just a matter of algebraic manipulation to restate the equations with any given choice of variables.
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"Variables," together with givens and operators, could represent an algebraic equation — which aims to identify the unspecified, the unknown.
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I have dabbled in internet "brain games," solving algebraic problems flashing past and rerouting virtual trains to avoid crashes.
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Now he needs to reset computer screens, take work orders on a computer tablet and sometimes do algebraic calculations.
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Fractions are somewhat algebraic, although it's more of an equation, but one of my first things ... Equation of happiness.
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Setting this world problem up as algebraic equations 4a + 6b = 1.56 and 9a + 7b = 2.60 can help you solve it.
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"Algebraic geometry is a more rigid world, whereas symplectic geometry is more flexible," said Nick Sheridan, a research fellow at Cambridge.
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So in effect it's a mixture of the traditional algebraic approach from the 1700s and 1800s, together with modern numerical computation.
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The startup world is increasingly focused on applications for artificial intelligence, whose algebraic roots date back to ancient Syria and Iraq.
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We start by making a connection to real life and slowly build a foundation of knowledge for more abstract algebraic problems.
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When a textbook that he found on the topic began with algebraic formulas, he prodded his older brother to explain them.
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What's more, she had showed there was a deep connection between certain abstract algebraic structures — those that deal with symmetry — and physics.
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During our breaks, he'd demonstrate the many ways a Rubik's Cube could be solved and write out algebraic expressions: X = the corners.
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To keep track of the different symmetries, mathematicians might impose an algebraic structure on the collection of all ways to label the corners.
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He published his first book, "National Income Behavior: An Introduction to Algebraic Analysis," in 22011, the year he received his doctorate from Harvard.
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But in the process of creating it, he developed the idea of the Frobenioid, which is an algebraic structure extracted from a geometric object.
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But by the mid-26s there were also experiments in using them for algebraic computation, and particularly to automate the generation of series expansions.
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And one can imagine doing a whole hierarchy of algebraic transformations that in a sense give the numerical scheme as much help as possible.
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This article originally appeared on Noisey UK. Like algebraic geometry or thinking about where gusts of wind come from, love can be really confusing.
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Mathematicians Christopher Hacon from the University of Utah and James McKernan from the University of California, San Diego for contributions to birational algebraic geometry.
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But we know there's an algebraic structure underlying it—there's a mechanism for how the lower energy states can be related to higher energy states.
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Exams would test critical reflection (for example, awareness of where the results a student is "proving" might not hold true) as much as algebraic prowess.
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Thus the algebraic group of the symmetries of the labels actually contains three times as much information as the geometric object that gave rise to it.
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We know, however, that it is far more efficient to rearrange the given equation using algebraic rules and in just two computations we get an answer.
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Most of the familiar special functions of mathematical physics are differentially algebraic. All algebraic combinations of differentially algebraic functions are differentially algebraic. Furthermore, all compositions of differentially algebraic functions are differentially algebraic. Hölder’s Theorem simply states that the gamma function, \Gamma , is not differentially algebraic and is therefore transcendentally transcendental.
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The Riemann sphere is one of the simplest complex algebraic varieties. In algebraic geometry, a complex algebraic variety is an algebraic variety (in the scheme sense or otherwise) over the field of complex numbers.Parshin, Alexei N., and Igor Rostislavovich Shafarevich, eds. Algebraic Geometry III: Complex Algebraic Varieties.
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Algebraic numbers colored by leading coefficient (red signifies 1 for an algebraic integer) An algebraic integer is an algebraic number that is a root of a polynomial with integer coefficients with leading coefficient 1 (a monic polynomial). Examples of algebraic integers are , and . Therefore, the algebraic integers constitute a proper superset of the integers, as the latter are the roots of monic polynomials for all In this sense, algebraic integers are to algebraic numbers what integers are to rational numbers. The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a ring.
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For example, \C is an algebraic closure of \R, but not an algebraic closure of \Q, as it is not algebraic over \Q (for example is not algebraic over \Q).
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Many algebraic varieties are manifolds, but an algebraic variety may have singular points while a manifold cannot. Algebraic varieties can be characterized by their dimension. Algebraic varieties of dimension one are called algebraic curves and algebraic varieties of dimension two are called algebraic surfaces. In the context of modern scheme theory, an algebraic variety over a field is an integral (irreducible and reduced) scheme over that field whose structure morphism is separated and of finite type.
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Nonlinear algebra is closely related to algebraic geometry, where the main objects of study include algebraic equations, algebraic varieties, and schemes.
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They are used in algebraic number theory and algebraic topology.
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Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve.
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In mathematics, a differential algebraic group is a differential algebraic variety with a compatible group structure. Differential algebraic groups were introduced by .
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In algebraic models, the output variable is computed by solving algebraic equations.
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In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety. In terms of category theory, an algebraic group is a group object in the category of algebraic varieties.
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The set of the elements of L that are algebraic over K form a subextension, which is called the algebraic closure of K in L. This results from the preceding characterization: if s and t are algebraic, the extensions and are finite. Thus is also finite, as well as the sub extensions , and (if ). It follows that , st and 1/s are all algebraic. An algebraic extension is an extension such that every element of L is algebraic over K. Equivalently, an algebraic extension is an extension that is generated by algebraic elements.
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Differential algebraic geometry is an area of differential algebra that adapts concepts and methods from algebraic geometry and applies them to systems of differential equations/algebraic differential equations. Another way of generalizing ideas from algebraic geometry is diffiety theory.
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In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety V.
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His research interests have been in algebraic K-theory, commutative algebra and algebraic geometry, algebraic groups, geometric methods in group theory, and ζ functions on finite simple graphs.
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In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes. A morphism of algebraic stacks generalizes a morphism of schemes.
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Any finite extension is necessarily algebraic, as can be deduced from the above multiplicativity formula. The subfield generated by an element , as above, is an algebraic extension of if and only if is an algebraic element. That is to say, if is algebraic, all other elements of are necessarily algebraic as well. Moreover, the degree of the extension , i.e.
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In algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex algebraic geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions.
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In the mathematical field of knot theory, an algebraic link is a link formed by connecting the open ends of an algebraic tangle.. Algebraic links are also called arborescent links. Although algebraic links and algebraic tangles were originally defined by John H. Conway as having two pairs of open ends, they were subsequently generalized to more pairs..
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The class of algebraic extensions forms a distinguished class of field extensions, that is, the following three properties hold:Lang (2002) p.228 # If E is an algebraic extension of F and F is an algebraic extension of K then E is an algebraic extension of K. # If E and F are algebraic extensions of K in a common overfield C, then the compositum EF is an algebraic extension of K. # If E is an algebraic extension of F and E>K>F then E is an algebraic extension of K. These finitary results can be generalized using transfinite induction: This fact, together with Zorn's lemma (applied to an appropriately chosen poset), establishes the existence of algebraic closures.
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In algebraic geometry, Mnëv's universality theorem is a result which can be used to represent algebraic (or semi algebraic) varieties as realizations of oriented matroids, a notion of combinatorics.
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Introduction to commutative algebra. Addison-Wesley publishing Company. pp. 11–12.Kaplansky (1972) pp.74-76 or the weaker ultrafilter lemma,Mathoverflow discussion it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K. The algebraic closure of a field K can be thought of as the largest algebraic extension of K. To see this, note that if L is any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is contained within the algebraic closure of K. The algebraic closure of K is also the smallest algebraically closed field containing K, because if M is any algebraically closed field containing K, then the elements of M that are algebraic over K form an algebraic closure of K. The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite.
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Topics in this area include Riemann surfaces for algebraic functions and zeros for algebraic functions.
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In algebraic geometry, universal algebraic geometry is generalized from the geometry of rings to geometry of arbitrary varieties of algebras, so that every variety of algebra has its own algebraic geometry. The two terms algebraic variety and variety of algebra should not be confused.
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Algebraic geometry for groups, that was introduced by Baumslag, Myasnikov, Remeslennikov G. Baumslag, A. Miasnikov, V. N. Remeslennikov. Algebraic geometry over groups I. Algebraic sets and ideal theory. J. Algebra.
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In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal (for set inclusion) for this property. For example, the set of solutions of the equation is not irreducible, and its irreducible components are the two lines of equations and . It is a fundamental theorem of classical algebraic geometry that every algebraic set may be written in a unique way as a finite union of irreducible componenents.
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If a matroid is algebraic over a simple extension F(t) then it is algebraic over F. It follows that the class of algebraic matroids is closed under contraction,Oxley (1992) p.222 and that a matroid algebraic over F is algebraic over the prime field of F.Oxley (1992) p.224 The class of algebraic matroids is closed under truncation and matroid union. It is not known whether the dual of an algebraic matroid is always algebraicOxley (1992) p.223 and there is no excluded minor characterisation of the class.
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An algebraic number is any complex number that is a solution to some polynomial equation f(x)=0 with rational coefficients; for example, every solution x of x^5 + (11/2) x^3 - 7 x^2 + 9 = 0 (say) is an algebraic number. Fields of algebraic numbers are also called algebraic number fields, or shortly number fields. Algebraic number theory studies algebraic number fields. Thus, analytic and algebraic number theory can and do overlap: the former is defined by its methods, the latter by its objects of study.
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Algebraic integer: An algebraic number that is the root of a monic polynomial with integer coefficients.
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In particular, algebraic geometry may be viewed as the study of solution sets of algebraic equations.
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An algebraic manifold is an algebraic variety that is also an m-dimensional manifold, and hence every sufficiently small local patch is isomorphic to km. Equivalently, the variety is smooth (free from singular points). When is the real numbers, R, algebraic manifolds are called Nash manifolds. Algebraic manifolds can be defined as the zero set of a finite collection of analytic algebraic functions.
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In algebraic geometry, an algebraic generalization is given by the notion of a linear system of divisors.
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In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds.
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Many classifying stacks for algebraic groups are algebraic stacks. In fact, for an algebraic group space G over a scheme S which is flat of finite presentation, the stack BG is algebraictheorem 6.1.
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The Tschirnhausen cubic is an algebraic curve of degree three. In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial.
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In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties.
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There are numbers such as the cube root of 2 which are algebraic but not constructible. The real algebraic numbers form a subfield of the real numbers. This means that 0 and 1 are algebraic numbers and, moreover, if a and b are algebraic numbers, then so are a+b, a−b, ab and, if b is nonzero, a/b. The real algebraic numbers also have the property, which goes beyond being a subfield of the reals, that for each positive integer n and each real algebraic number a, all of the nth roots of a that are real numbers are also algebraic.
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The algebraic equations are the basis of a number of areas of modern mathematics: Algebraic number theory is the study of (univariate) algebraic equations over the rationals (that is, with rational coefficients). Galois theory was introduced by Évariste Galois to specify criteria for deciding if an algebraic equation may be solved in terms of radicals. In field theory, an algebraic extension is an extension such that every element is a root of an algebraic equation over the base field. Transcendental number theory is the study of the real numbers which are not solutions to an algebraic equation over the rationals.
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He was also a visiting scholar at MSRI (1986, 1990), Max Planck Institute for Mathematics in Bonn (1983, 1987), Tata Institute of Fundamental Research (1988), at University of Washington, Caltech, University Paris VI and Henri Poincaré Institute. Marc Levine (left) with Fabien Morel, Oberwolfach 2005 Levine works in algebraic geometry, in particular in the development of analogues of concepts from algebraic topology in algebraic geometry and the theory of motives (motivic cohomology, motivic homotopy, algebraic K theory). He developed, together with Fabien Morel, the theory of algebraic cobordism, an algebraic-geometry analog of the theory of cobordism in algebraic topology. In 2002, he was an invited speaker at ICM in Beijing (Algebraic Cobordism).
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In mathematics, especially algebraic geometry the decomposition theorem is a set of results concerning the cohomology of algebraic varieties.
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The behavior of algebraic cycles ranks among the most important open questions in modern mathematics. The Hodge conjecture, one of the Clay Mathematics Institute's Millenium Prize Problems, predicts that the topology of a complex algebraic variety forces the existence of certain algebraic cycles. The Tate conjecture makes a similar prediction for étale cohomology. Alexander Grothendieck's standard conjectures on algebraic cycles yield enough cycles to construct his category of motives and would imply that algebraic cycles play a vital role in any cohomology theory of algebraic varieties.
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To prove that is transcendental, we prove that it is not algebraic. If were algebraic, i would be algebraic as well, and then by the Lindemann–Weierstrass theorem (see Euler's identity) would be transcendental, a contradiction. Therefore is not algebraic, which means that it is transcendental. A slight variant on the same proof will show that if is a non- zero algebraic number then and their hyperbolic counterparts are also transcendental.
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It allows complex analytic methods to be used in algebraic geometry, and algebraic- geometric methods in complex analysis and field-theoretic methods to be used in both. This is characteristic of a much wider class of problems in algebraic geometry. See also algebraic geometry and analytic geometry for a more general theory.
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In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three.
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In algebraic geometry, a new scheme (e.g. an algebraic variety) can be obtained by gluing existing schemes through gluing maps.
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Equivalently, an algebraic space is the quotient of a scheme by an étale equivalence relation. A powerful result, the Artin representability theorem, gives simple conditions for a functor to be represented by an algebraic space.. A further generalization is the idea of a stack. Crudely speaking, algebraic stacks generalize algebraic spaces by having an algebraic group attached to each point, which is viewed as the automorphism group of that point. For example, any action of an algebraic group G on an algebraic variety X determines a quotient stack [X/G], which remembers the stabilizer subgroups for the action of G. More generally, moduli spaces in algebraic geometry are often best viewed as stacks, thereby keeping track of the automorphism groups of the objects being classified.
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The converse is not true however: there are infinite extensions which are algebraic. For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers. If a is algebraic over K, then K[a], the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. The converse is true as well, if K[a] is a field, then a is algebraic over K. In the special case where K = Q is the field of rational numbers, Q[a] is an example of an algebraic number field. A field with no proper algebraic extensions is called algebraically closed.
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For example, \Q(\sqrt 2, \sqrt 3) is an algebraic extension of \Q, because \sqrt 2 and \sqrt 3 are algebraic over \Q. A simple extension is algebraic if and only if it is finite. This implies that an extension is algebraic if and only if it is the union of its finite subextensions, and that every finite extension is algebraic. Every field K has an algebraic closure, which is up to an isomorphism the largest extension field of K which is algebraic over K, and also the smallest extension field such that every polynomial with coefficients in K has a root in it.
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Algebraic numbers are those that are a solution to a polynomial equation with integer coefficients. Real numbers that are not rational numbers are called irrational numbers. Complex numbers which are not algebraic are called transcendental numbers. The algebraic numbers that are solutions of a monic polynomial equation with integer coefficients are called algebraic integers.
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In calculus and mathematical analysis, algebraic operation is also used for the operations that may be defined by purely algebraic methods. For example, exponentiation with an integer or rational exponent is an algebraic operation, but not the general exponentiation with a real or complex exponent. Also, the derivative is an operation that is not algebraic.
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If, however, its coefficients are actually all integers, f is called an algebraic integer. Any (usual) integer z ∈ Z is an algebraic integer, as it is the zero of the linear monic polynomial: :p(t) = t − z. It can be shown that any algebraic integer that is also a rational number must actually be an integer, hence the name "algebraic integer". Again using abstract algebra, specifically the notion of a finitely generated module, it can be shown that the sum and the product of any two algebraic integers is still an algebraic integer.
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For fields of characteristic zero (such as the real numbers) linear and algebraic matroids coincide, but for other fields there may exist algebraic matroids that are not linear;. indeed the non-Pappus matroid is algebraic over any finite field, but not linear and not algebraic over any field of characteristic zero. However, if a matroid is algebraic over a field F of characteristic zero then it is linear over F(T) for some finite set of transcendentals T over FOxley (1992) p.221 and over the algebraic closure of F.
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Algebraic geometry is a branch of mathematics, classically studying solutions of polynomial equations. Modern algebraic geometry is based on more abstract techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves and quartic curves like lemniscates, and Cassini ovals.
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In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non- singular) and so of dimension four as a smooth manifold. The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of (real) dimension two). Many results were obtained, however, in the Italian school of algebraic geometry, and are up to 100 years old.
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In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined over that field.
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A torus, one of the most frequently studied objects in algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.
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A pivotal notion in the study of field extensions are algebraic elements. An element is algebraic over if it is a root of a polynomial with coefficients in , that is, if it satisfies a polynomial equation :, with in , and . For example, the imaginary unit in is algebraic over , and even over , since it satisfies the equation :. A field extension in which every element of is algebraic over is called an algebraic extension.
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In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
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Lurie's research interests started with logic and the theory of surreal numbers while he was still in high school. He is best known for his work, starting with his thesis, on infinity categories and derived algebraic geometry. Derived algebraic geometry is a way of infusing homotopical methods into algebraic geometry, with two purposes: deeper insight into algebraic geometry (e.g. into intersection theory) and the use of methods of algebraic geometry in stable homotopy theory.
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There are only countably many algebraic numbers, but there are uncountably many real numbers, so in the sense of cardinality most real numbers are not algebraic. This nonconstructive proof that not all real numbers are algebraic was first published by Georg Cantor in his 1874 paper "On a Property of the Collection of All Real Algebraic Numbers". Non-algebraic numbers are called transcendental numbers. Specific examples of transcendental numbers include π and Euler's number e.
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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, the corresponding moduli problem and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem.
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P is a transcendental number. :Proof. Suppose that P is algebraic. Then \\!\ P - \sqrt2 = \ln(1 + \sqrt2) must also be algebraic.
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In mathematics, an algebraic matroid is a matroid, a combinatorial structure, that expresses an abstraction of the relation of algebraic independence.
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Charles Waldo Rezk (born 26 January 1969) is an American mathematician, specializing in algebraic topology, category theory, and spectral algebraic geometry.
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Serguei Barannikov (; born April 16, 1972) is a mathematician, known for his works in algebraic topology, algebraic geometry and mathematical physics.
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In recent years, the term "algebraic statistics" has been used more restrictively, to label the use of algebraic geometry and commutative algebra to study problems related to discrete random variables with finite state spaces. Commutative algebra and algebraic geometry have applications in statistics because many commonly used classes of discrete random variables can be viewed as algebraic varieties.
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0, the project shifted its focus to building a backend-independent programming environment, code named "Samsara". The environment consists of an algebraic backend-independent optimizer and an algebraic Scala DSL unifying in-memory and distributed algebraic operators. Supported algebraic platforms are Apache Spark, H2O, and Apache Flink. Support for MapReduce algorithms started being gradually phased out in 2014.
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Some computer programs (and people) use a variant of algebraic chess notation termed long algebraic notation or fully expanded algebraic notation. In long algebraic notation, moves specify both the starting and ending squares. Sometimes it is separated by a hyphen (contrary to FIDE rules), for example: e2e4 or Nb1-c3 with hyphen. Captures are still indicated using "x": Rd3xd7.
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He was born and studied in Turin. Corrado Segre, his uncle, also served as his doctoral advisor. Among his main contributions to algebraic geometry are studies of birational invariants of algebraic varieties, singularities and algebraic surfaces. His work was in the style of the old Italian School, although he also appreciated the greater rigour of modern algebraic geometry.
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Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility.
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In algebraic geometry, a symmetric variety is an algebraic analogue of a symmetric space in differential geometry, given by a quotient G/H of a reductive algebraic group G by the subgroup H fixed by some involution of G.
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In 1968, he wrote Elements of the theory of algebraic curves, a book on algebraic geometry. He published several other important papers.
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In algebraic group theory, approximation theorems are an extension of the Chinese remainder theorem to algebraic groups G over global fields k.
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The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.
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Any field has an algebraic closure, which is moreover unique up to (non-unique) isomorphism. It is commonly referred to as the algebraic closure and denoted . For example, the algebraic closure of is called the field of algebraic numbers. The field is usually rather implicit since its construction requires the ultrafilter lemma, a set-theoretic axiom that is weaker than the axiom of choice.. Mathoverflow post In this regard, the algebraic closure of , is exceptionally simple.
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In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways. Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commutative algebra. Some are restricted to algebraic varieties while others apply also to any algebraic set. Some are intrinsic, as independent of any embedding of the variety into an affine or projective space, while other are related to such an embedding.
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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.
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For example, a circle and a line have always two intersection points (possibly not distinct) in the complex affine space. Therefore, most of algebraic geometry is built in complex affine spaces and affine spaces over algebraically closed fields. The shapes that are studied in algebraic geometry in these affine spaces are therefore called affine algebraic varieties. Affine spaces over the rational numbers and more generally over algebraic number fields provide a link between (algebraic) geometry and number theory.
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In mathematics, Belyi's theorem on algebraic curves states that any non- singular algebraic curve C, defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only. This is a result of G. V. Belyi from 1979. At the time it was considered surprising, and it spurred Grothendieck to develop his theory of dessins d'enfant, which describes nonsingular algebraic curves over the algebraic numbers using combinatorial data.
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One cannot define the concept of an algebraic surface in a space of dimension higher than three without a general definition of an algebraic variety and of the dimension of an algebraic variety. In fact, an algebraic surface is an algebraic variety of dimension two. More precisely, an algebraic surface in a space of dimension is the set of the common zeros of at least polynomials, but these polynomials must satisfy further conditions that may be not immediate to verify. Firstly, the polynomials must not define a variety or an algebraic set of higher dimension, which is typically the case if one of the polynomials is in the ideal generated by the others.
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In algebraic geometry a normal crossings singularity is a point in an algebraic variety that is locally isomorphic to a normal crossings divisor.
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Reference to a ground field may be common in the theory of Lie algebras (qua vector spaces) and algebraic groups (qua algebraic varieties).
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These include Deligne–Mumford stacks (similar to orbifolds in topology), for which the stabilizer groups are finite, and algebraic spaces, for which the stabilizer groups are trivial. The Keel–Mori theorem says that an algebraic stack with finite stabilizer groups has a coarse moduli space which is an algebraic space. Another type of generalization is to enrich the structure sheaf, bringing algebraic geometry closer to homotopy theory. In this setting, known as derived algebraic geometry or "spectral algebraic geometry", the structure sheaf is replaced by a homotopical analog of a sheaf of commutative rings (for example, a sheaf of E-infinity ring spectra).
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Algebraic spaces over the complex numbers are closely related to analytic spaces and Moishezon manifolds. Roughly speaking, the difference between complex algebraic spaces and analytic spaces is that complex algebraic spaces are formed by gluing affine pieces together using the étale topology, while analytic spaces are formed by gluing with the classical topology. In particular there is a functor from complex algebraic spaces of finite type to analytic spaces. Hopf manifolds give examples of analytic surfaces that do not come from a proper algebraic space (though one can construct non-proper and non-separated algebraic spaces whose analytic space is the Hopf surface).
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These are second-order structures. More complicated non-algebraic structures combine an algebraic component and a non- algebraic component. For example, the structure of a topological group consists of a topology and the structure of a group. Thus it belongs to the product of P(P(X)) and another ("algebraic") set in the scale; this product is again a set in the scale.
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Algebraic curves are the curves considered in algebraic geometry. A plane algebraic curve is the set of the points of coordinates such that , where is a polynomial in two variables defined over some field . One says that the curve is defined over . Algebraic geometry normally considers not only points with coordinates in but all the points with coordinates in an algebraically closed field .
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In algebraic geometry, the Keel–Mori theorem gives conditions for the existence of the quotient of an algebraic space by a group. The theorem was proved by . A consequence of the Keel–Mori theorem is the existence of a coarse moduli space of a separated algebraic stack, which is roughly a "best possible" approximation to the stack by a separated algebraic space.
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For any algebraic objects we can introduce the discrete topology, under which the algebraic operations are continuous functions. For any such structure that is not finite, we often have a natural topology compatible with the algebraic operations, in the sense that the algebraic operations are still continuous. This leads to concepts such as topological groups, topological vector spaces, topological rings and local fields.
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The focus of abstract algebraic logic shifted from the study of specific classes of algebras associated with specific logical systems (the focus of classical algebraic logic), to the study of: #Classes of algebras associated with classes of logical systems whose members all satisfy certain abstract logical properties; #The process by which a class of algebras becomes the "algebraic counterpart" of a given logical system; #The relation between metalogical properties satisfied by a class of logical systems, and the corresponding algebraic properties satisfied by their algebraic counterparts. The passage from classical algebraic logic to abstract algebraic logic may be compared to the passage from "modern" or abstract algebra (i.e., the study of groups, rings, modules, fields, etc.) to universal algebra (the study of classes of algebras of arbitrary similarity types (algebraic signatures) satisfying specific abstract properties). The two main motivations for the development of abstract algebraic logic are closely connected to (1) and (3) above.
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Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geometry and number theory are attempts to understand motivic cohomology.
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A far-reaching generalization of algebraic spaces is given by the algebraic stacks. In the category of stacks we can form even more quotients by group actions than in the category of algebraic spaces (the resulting quotient is called a quotient stack).
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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..
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In algebraic geometry, a homogeneous variety is an algebraic variety of the form G/P, G a linear algebraic group, P a parabolic subgroup. It is a smooth projective variety. If P is a Borel subgroup, it is usually called a flag variety.
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In algebraic geometry, the smooth completion (or smooth compactification) of a smooth affine algebraic curve X is a complete smooth algebraic curve which contains X as an open subset.Griffiths, 1972, p. 286. Smooth completions exist and are unique over a perfect field.
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In algebraic geometry, a 3-fold or threefold is a 3-dimensional algebraic variety. The Mori program showed that 3-folds have minimal models.
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In general, for an algebraic category C, an embedding between two C-algebraic structures X and Y is a C-morphism that is injective.
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The Italian school liked to reduce the geometry on an algebraic surface to that of linear systems cut out by surfaces in three-space; Zariski wrote his celebrated book Algebraic Surfaces to try to pull together the methods, involving linear systems with fixed base points. There was a controversy, one of the final issues in the conflict between 'old' and 'new' points of view in algebraic geometry, over Henri Poincaré's characteristic linear system of an algebraic family of curves on an algebraic surface.
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Algebraic Eraser (AE)Also referred to as the colored Burau key agreement protocol (CBKAP), Anshel-Anshel-Goldfeld-Lemieux key agreement protocol, Algebraic Eraser key agreement protocol (AEKAP), and Algebraic Eraser Diffie- Hellman (AEDH). is an anonymous key agreement protocol that allows two parties, each having an AE public–private key pair, to establish a shared secret over an insecure channel.Anshel I, Anshel M, Goldfeld D, Lemieux S. Key Agreement, The Algebraic Eraser and Lightweight Cryptography Algebraic methods in cryptography, Contemp. Math., vol.
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Another formal generalization is possible to universal algebraic geometry in which every variety of algebras has its own algebraic geometry. The term variety of algebras should not be confused with algebraic variety. The language of schemes, stacks and generalizations has proved to be a valuable way of dealing with geometric concepts and became cornerstones of modern algebraic geometry. Algebraic stacks can be further generalized and for many practical questions like deformation theory and intersection theory, this is often the most natural approach.
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Grothendieck approached algebraic geometry by clarifying the foundations of the field, and by developing mathematical tools intended to prove a number of notable conjectures. Algebraic geometry has traditionally meant the understanding of geometric objects, such as algebraic curves and surfaces, through the study of the algebraic equations for those objects. Properties of algebraic equations are in turn studied using the techniques of ring theory. In this approach, the properties of a geometric object are related to the properties of an associated ring.
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Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003.
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This is a glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary of ring theory and glossary of module theory. In this article, all rings are assumed to be commutative with identity 1.
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In 1882, Ferdinand von Lindemann published the first complete proof of the transcendence of . He first proved that is transcendental when is any non-zero algebraic number. Then, since is algebraic (see Euler's identity), must be transcendental. But since is algebraic, therefore must be transcendental.
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Bourbaki and Algebraic Topology by John McCleary (PDF) gives documentation (translated into English from French originals). Algebraic homology remains the primary method of classifying manifolds.
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In algebraic geometry, solution sets are called algebraic sets if there are no inequalities. Over the reals, and with inequalities, there are called semialgebraic sets.
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Recursive categorical syntax, also known as algebraic syntax, is an algebraic theory of syntax developed by Michael Brame as an alternative to transformational-generative grammar.
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Algebraic statistics is the use of algebra to advance statistics. Algebra has been useful for experimental design, parameter estimation, and hypothesis testing. Traditionally, algebraic statistics has been associated with the design of experiments and multivariate analysis (especially time series). In recent years, the term "algebraic statistics" has been sometimes restricted, sometimes being used to label the use of algebraic geometry and commutative algebra in statistics.
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In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also regular is called biregular, and they are isomorphisms in the category of algebraic varieties.
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The name algebraic integer comes from the fact that the only rational numbers that are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers. If is a number field, its ring of integers is the subring of algebraic integers in , and is frequently denoted as . These are the prototypical examples of Dedekind domains.
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Stacks are the underlying structure of algebraic stacks (also called Artin stacks) and Deligne–Mumford stacks, which generalize schemes and algebraic spaces and which are particularly useful in studying moduli spaces. There are inclusions: schemes ⊆ algebraic spaces ⊆ Deligne–Mumford stacks ⊆ algebraic stacks (Artin stacks) ⊆ stacks. and give a brief introductory accounts of stacks, , and give more detailed introductions, and describes the more advanced theory.
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The table below summarizes how algebraic expressions compare with several other types of mathematical expressions by the type of elements they may contain, according to common but not universal conventions. A rational algebraic expression (or rational expression) is an algebraic expression that can be written as a quotient of polynomials, such as . An irrational algebraic expression is one that is not rational, such as .
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Michael Artin defined an algebraic space as the quotient of a scheme by the equivalence relations that define étale morphisms. Algebraic spaces retain many of the useful properties of schemes while simultaneously being more flexible. For instance, the Keel–Mori theorem can be used to show that many moduli spaces are algebraic spaces. More general than an algebraic space is a Deligne–Mumford stack.
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The twisted cubic is a projective algebraic variety. Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.
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In mathematics, an algebraic cycle on an algebraic variety V is a formal linear combination of subvarieties of V. These are the part of the algebraic topology of V that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety. The most trivial case is codimension zero cycles, which are linear combinations of the irreducible components of the variety. The first non-trivial case is of codimension one subvarieties, called divisors.
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Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic have led to the notion of a valuation ring. The restriction of algebraic field extensions to subrings has led to the notions of integral extensions and integrally closed domains as well as the notion of ramification of an extension of valuation rings.
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Algebraic geometry is in many ways the mirror image of commutative algebra. This correspondence started with Hilbert's Nullstellensatz that establishes a one-to-one correspondence between the points of an algebraic variety, and the maximal ideals of its coordinate ring. This correspondence has been enlarged and systematized for translating (and proving) most geometrical properties of algebraic varieties into algebraic properties of associated commutative rings. Alexander Grothendieck completed this by introducing schemes, a generalization of algebraic varieties, which may be built from any commutative ring.
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Representation theory is a branch of mathematics that draws heavily on non-commutative rings. It studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication, which is non-commutative. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras.
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In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics for these deductive systems) and connected problems like representation and duality. Well known results like the representation theorem for Boolean algebras and Stone duality fall under the umbrella of classical algebraic logic . Works in the more recent abstract algebraic logic (AAL) focus on the process of algebraization itself, like classifying various forms of algebraizability using the Leibniz operator .
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In algebraic geometry, the dimension of an algebraic variety has received a number of apparently different definitions, which are all equivalent in the most common cases.
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Harada's research involves the symmetries of symplectic spaces and their connections to other areas of mathematics including algebraic geometry, representation theory, K-theory, and algebraic combinatorics.
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Ivan Aleksandrovich Panin (Иван Александрович Панин, born 2 July 1959 in Apatity, Russia) is a Russian mathematician, specializing in algebra, algebraic geometry, and algebraic K-theory.
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Margherita Piazzolla Beloch (12 July 1879 in Frascati - 28 September 1976 in Rome) was an Italian mathematician who worked in algebraic geometry, algebraic topology and photogrammetry.
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Alexander Schmidt (born 1965Regensburger Universitätszeitung 2004, pdf) is a German mathematician at the University of Heidelberg. His research interests include algebraic number theory and algebraic geometry.
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In computer science, algebraic semantics is a form of axiomatic semantics based on algebraic laws for describing and reasoning about program semantics in a formal manner.
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In algebraic geometry, a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain. Sums and products of periods remain periods, so the periods form a ring. gave a survey of periods and introduced some conjectures about them.
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In algebraic geometry, a toroidal embedding is an open embedding of algebraic varieties that locally looks like the embedding of the open torus into a toric variety. The notion was introduced by Mumford to prove the existence of semistable reductions of algebraic varieties over one-dimensional bases.
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216 A matroid that can be generated in this way is called algebraic or algebraically representable.Oxley (1992) p.218 No good characterization of algebraic matroids is known,Oxley (1992) p.215 but certain matroids are known to be non-algebraic; the smallest is the Vámos matroid..
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Every algebraic lattice is isomorphic to the congruence lattice of some algebra. The lattice Sub V of all subspaces of a vector space V is certainly an algebraic lattice. As the next result shows, these algebraic lattices are difficult to represent. Theorem (Freese, Lampe, and Taylor 1979).
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The Massey product is an algebraic generalization of the phenomenon of Borromean rings. In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product. The Massey product was created by William S. Massey, an American algebraic topologist.
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In mathematics, especially algebraic geometry, the Bass conjecture says that certain algebraic K-groups are supposed to be finitely generated. The conjecture was proposed by Hyman Bass.
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The original theory related to algebraic connectivity was produced by Miroslav Fiedler.M. Fiedler. "Algebraic connectivity of Graphs", Czechoslovak Mathematical Journal 23(98) (1973), 298–305.M. Fiedler.
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Burt James Totaro, FRS (b. 1967), is an American mathematician, currently a Professor at the University of California, Los Angeles, specializing in algebraic geometry and algebraic topology.
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Algebraic Geometry is an influentialMathSciNet lists more than 2500 citations of this book. algebraic geometry textbook written by Robin Hartshorne and published by Springer-Verlag in 1977.
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In the common case of a real algebraic curve, where is the field of real numbers, an algebraic curve is a finite union of topological curves. When complex zeros are considered, one has a complex algebraic curve, which, from the topologically point of view, is not a curve, but a surface, and is often called a Riemann surface. Although not being curves in the common sense, algebraic curves defined over other fields have been widely studied. In particular, algebraic curves over a finite field are widely used in modern cryptography.
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Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions leading to Artin stacks and, even finer, Deligne–Mumford stacks, both often called algebraic stacks. Sometimes other algebraic sites replace the category of affine schemes. For example, Nikolai Durov has introduced commutative algebraic monads as a generalization of local objects in a generalized algebraic geometry. Versions of a tropical geometry, of an absolute geometry over a field of one element and an algebraic analogue of Arakelov's geometry were realized in this setup.
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An algebraic hypersurface is an algebraic variety that may be defined by a single implicit equation of the form :p(x_1, \ldots, x_n)=0, where is a multivariate polynomial. Generally the polynomial is supposed to be irreducible. When this is not the case, the hypersurface is not an algebraic variety, but only an algebraic set. It may depend on the authors or the context whether a reducible polynomial defines a hypersurface.
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The general theory of algebraic structures has been formalized in universal algebra. The language of category theory is used to express and study relationships between different classes of algebraic and non-algebraic objects. This is because it is sometimes possible to find strong connections between some classes of objects, sometimes of different kinds. For example, Galois theory establishes a connection between certain fields and groups: two algebraic structures of different kinds.
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Heintz worked mainly in algebraic complexity theory, computational algebraic geometry, and semi algebraic geometry. For this purpose, he developed with his collaborators, different mathematical tools, e.g. the Bezout Inequality or the first effective Nullstellensatz in arbitrary characteristic. This allowed him and his collaborators to adapt Kronecker's elimination theory to the complexity requirements of modern computer algebra and to prove that all reasonable geometric (not algebraic) computation problems are solvable in PSPACE.
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Miroslav Fiedler (7 April 1926 – 20 November 2015) was a Czech mathematician known for his contributions to linear algebra, graph theory and algebraic graph theory. His article, "Algebraic Connectivity of Graphs", published in the Czechoslovak Math Journal in 1973, established the use of the eigenvalues of the Laplacian matrix of a graph to create tools for measuring algebraic connectivity in algebraic graph theory.Algebraic connectivity of graphs. Czechoslovak Math.
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In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra.
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In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2\. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in some sense most surfaces are in this class.
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Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces.Allen Hatcher, Algebraic topology. (2002) Cambridge University Press, xii+544 pp. . The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
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In algebraic geometry, the Reiss relation, introduced by , is a condition on the second-order elements of the points of a plane algebraic curve meeting a given line.
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The Virasoro algebra also has vertex algebraic and conformal algebraic counterparts, which basically come from arranging all the basis elements into generating series and working with single objects.
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In algebraic geometry and algebraic topology, branches of mathematics, homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to Fabien Morel and Vladimir Voevodsky. The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by replacing the unit interval , which is not an algebraic variety, with the affine line , which is. The theory requires a substantial amount of technique to set up, but has spectacular applications such as Voevodsky's construction of the derived category of mixed motives and the proof of the Milnor and Bloch-Kato conjectures.
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In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry, the same definition is used, but the defining function is required to be differentiable Algebraic geometry studies algebraic curves, which are defined as algebraic varieties of dimension one.
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If we only define sets using polynomial equations and not inequalities then we define algebraic sets rather than semialgebraic sets. For these sets the theorem fails, i.e. projections of algebraic sets need not be algebraic. As a simple example consider the hyperbola in R2 defined by the equation :xy-1=0.
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In algebraic number theory, the Ferrero–Washington theorem, proved first by and later by , states that Iwasawa's μ-invariant vanishes for cyclotomic Zp- extensions of abelian algebraic number fields.
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Jannsen in Oberwolfach 2009 Uwe Jannsen (born 11 March 1954, Meddewade)biography, pdf, University of Regensburg is a German mathematician, specializing in algebra, algebraic number theory, and algebraic geometry.
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Linear algebraic groups admit variants in several directions. Dropping the existence of the inverse map i\colon G \to G, one obtains the notion of a linear algebraic monoid..
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In mathematics, Weibel's conjecture gives a criterion for vanishing of negative algebraic K-theory groups. The conjecture was proposed by and proven by using methods from derived algebraic geometry.
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In functional programming, a generalized algebraic data type (GADT, also first-class phantom type, guarded recursive datatype, or equality-qualified type) is a generalization of parametric algebraic data types.
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In mathematics, the Demazure conjecture is a conjecture about representations of algebraic groups over the integers made by . The conjecture implies that many of the results of his paper can be extended from complex algebraic groups to algebraic groups over fields of other characteristics or over the integers. showed that Demazure's conjecture (for classical groups) follows from their work on standard monomial theory, and Peter Littelmann extended this to all reductive algebraic groups.
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Together with Gabriele Vezzosi and Jacob Lurie he has laid the foundations of the subject of derived algebraic geometry and higher category theory. His works establish several contributions to noncommutative algebraic geometry in the sense of Kontsevich and (shifted) symplectic geometry. He was an invited speaker at the International Congress of Mathematicians in 2014, speaking in the section on "Algebraic and Complex Geometry". with a talk "Derived Algebraic Geometry and Deformation Quantization".
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Hess has worked and written extensively on topics in algebraic topology including homotopy theory, model categories and algebraic K-theory. She has also used the methods of algebraic topology and category theory to investigate homotopical generalizations of descent theory and Hopf–Galois extensions. In particular, she has studied generalizations of these structures for ring spectra and differential graded algebras. She has more recently used algebraic topology to understand structures in neurology and materials science.
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Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, D. Much of the theory of differential Galois theory is parallel to algebraic Galois theory. One difference between the two constructions is that the Galois groups in differential Galois theory tend to be matrix Lie groups, as compared with the finite groups often encountered in algebraic Galois theory.
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The term "manifold" comes from German Mannigfaltigkeit, by Riemann. In English, "manifold" refers to spaces with a differentiable or topological structure, while "variety" refers to spaces with an algebraic structure, as in algebraic varieties. In Romance languages, manifold is translated as "variety" – such spaces with a differentiable structure are literally translated as "analytic varieties", while spaces with an algebraic structure are called "algebraic varieties". Thus for example, the French word "variété topologique" means topological manifold.
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In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology.
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The converse is clear: an integral domain has no nonzero nilpotent elements, and the zero ideal is the unique minimal prime ideal. This translates, in algebraic geometry, into the fact that the coordinate ring of an affine algebraic set is an integral domain if and only if the algebraic set is an algebraic variety. More generally, a commutative ring is an integral domain if and only if its spectrum is an integral affine scheme.
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In algebraic geometry a simple normal crossings singularity is a point in an algebraic variety, the latter having smooth irreducible components, that is locally isomorphic to a normal crossings divisor.
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Numerical algebraic geometry is a field of computational mathematics, particularly computational algebraic geometry, which uses methods from numerical analysis to study and manipulate the solutions of systems of polynomial equations.
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If one global condition, namely compactness, is added, the surface is necessarily algebraic. This feature of Riemann surfaces allows one to study them with either the means of analytic or algebraic geometry. The corresponding statement for higher-dimensional objects is false, i.e. there are compact complex 2-manifolds which are not algebraic.
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M. Cohn. (1981) Universal Algebra, Springer, p. 41. while, in other contexts, it is (somewhat ambiguously) called an algebraic structure, the term algebra being reserved for specific algebraic structures that are vector spaces over a field or modules over a commutative ring. The properties of specific algebraic structures are studied in abstract algebra.
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The most important of these invariants are homotopy groups, homology, and cohomology. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.
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The research of Wu includes the following fields: algebraic topology, algebraic geometry, game theory, history of mathematics, automated theorem proving. His most important contributions are to algebraic topology. The Wu class and the Wu formula are named after him. In the field of automated theorem proving, he is known for Wu's method.
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Dmitrii Andreevich Gudkov (1918–1992; alternative spelling Dmitry) was a Russian mathematician famous for his work on Hilbert's sixteenth problem and the related Gudkov's conjecture in algebraic geometry.V. Kharlamov – Topology of Real Algebraic Varieties and Related Topics, pp. 1–10 He was a student of Aleksandr Andronov.Vladimir I. Arnold – Real Algebraic Geometry, p.
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In algebraic geometry, a correspondence between algebraic varieties V and W is a subset R of V×W, that is closed in the Zariski topology. In set theory, a subset of a Cartesian product of two sets is called a binary relation or correspondence; thus, a correspondence here is a relation that is defined by algebraic equations. There are some important examples, even when V and W are algebraic curves: for example the Hecke operators of modular form theory may be considered as correspondences of modular curves. However, the definition of a correspondence in algebraic geometry is not completely standard.
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The informal definition of an algebraic function provides a number of clues about their properties. To gain an intuitive understanding, it may be helpful to regard algebraic functions as functions which can be formed by the usual algebraic operations: addition, multiplication, division, and taking an nth root. This is something of an oversimplification; because of the fundamental theorem of Galois theory, algebraic functions need not be expressible by radicals. First, note that any polynomial function y = p(x) is an algebraic function, since it is simply the solution y to the equation : y-p(x) = 0.
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Real algebraic geometry is the study of the real points of algebraic varieties. The fact that the field of the real numbers is an ordered field cannot be ignored in such a study. For example, the curve of equation x^2+y^2-a=0 is a circle if a>0, but does not have any real point if a<0. It follows that real algebraic geometry is not only the study of the real algebraic varieties, but has been generalized to the study of the semi-algebraic sets, which are the solutions of systems of polynomial equations and polynomial inequalities.
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Many examples of Noetherian topological spaces come from algebraic geometry, where for the Zariski topology an irreducible set has the intuitive property that any closed proper subset has smaller dimension. Since dimension can only 'jump down' a finite number of times, and algebraic sets are made up of finite unions of irreducible sets, descending chains of Zariski closed sets must eventually be constant. A more algebraic way to see this is that the associated ideals defining algebraic sets must satisfy the ascending chain condition. That follows because the rings of algebraic geometry, in the classical sense, are Noetherian rings.
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Because these three fields (algebraic geometry, algebraic number theory and commutative algebra) are so intimately connected it is usually difficult and meaningless to decide which field a particular result belongs to. For example, Hilbert's Nullstellensatz is a theorem which is fundamental for algebraic geometry, and is stated and proved in terms of commutative algebra. Similarly, Fermat's last theorem is stated in terms of elementary arithmetic, which is a part of commutative algebra, but its proof involves deep results of both algebraic number theory and algebraic geometry. Noncommutative rings are quite different in flavour, since more unusual behavior can arise.
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Anastasia Konstantinovna Stavrova () is a Russian mathematician specializing in algebraic groups, non-associative algebra, and algebraic K-theory. She is a researcher in the Chebyshev Laboratory at Saint Petersburg State University.
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Goodstein and Olenick reported that younger viewers tended to enjoy the "algebraic ballet" style "much more than older viewers, who are made uncomfortable by the algebraic manipulations they cannot quite follow".
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An Algebraic ApproachRafael Artzy (1992) Geometry. An Algebraic Approach, Bibliographisches Institute, , Review of Geometry by EJF Primrose Artzy had made 224 contributions to Mathematical Reviews by his last submission in 1995.
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Since any two modular functions are related by an algebraic relation, this implies that the functions κ, z, R, ρ are all algebraic functions of each other (of quite high degree) .
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These operations may be performed on numbers, in which case they are often called arithmetic operations. They may also be performed, in a similar way, on variables, algebraic expressions,William Smyth, Elementary algebra: for schools and academies, Publisher Bailey and Noyes, 1864, "Algebraic Operations" and more generally, on elements of algebraic structures, such as groups and fields.Horatio Nelson Robinson, New elementary algebra: containing the rudiments of science for schools and academies, Ivison, Phinney, Blakeman, & Co., 1866, page 7 An algebraic operation may also be defined simply as a function from a Cartesian power of a set to the same set. The term algebraic operation may also be used for operations that may be defined by compounding basic algebraic operations, such as the dot product.
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He generalized the classical approach to the classification of algebraic surfaces to the classification of algebraic three- folds. The classical approach used the concept of minimal models of algebraic surfaces. He found that the concept of minimal models can be applied to three- folds as well if we allow some singularities on them. The extension of Mori's results to dimensions higher than three is called the minimal model program and is an active area of research in algebraic geometry.
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In 1975 he earned his Ph.D. from the University of California, Berkeley as a student of Robion Kirby. In topology, he has worked on handlebody theory, low-dimensional manifolds, symplectic topology, G2 manifolds. In the topology of real-algebraic sets, he and Henry C. King proved that every compact piecewise-linear manifold is a real-algebraic set; they discovered new topological invariants of real- algebraic sets.S. Akbulut and H.C. King, Topology of real algebraic sets, MSRI Publications, 25.
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Schemes have become the basic objects of study for practitioners of modern algebraic geometry. Their use as a foundation allowed geometry to absorb technical advances from other fields. His generalization of the classical Riemann-Roch theorem related topological properties of complex algebraic curves to their algebraic structure. The tools he developed to prove this theorem started the study of algebraic and topological K-theory, which study the topological properties of objects by associating them with rings.
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More generally, for any two transcendental numbers a and b, at least one of a + b and ab must be transcendental. To see this, consider the polynomial (x − a)(x − b) = x2 − (a + b)x + ab. If (a + b) and ab were both algebraic, then this would be a polynomial with algebraic coefficients. Because algebraic numbers form an algebraically closed field, this would imply that the roots of the polynomial, a and b, must be algebraic.
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The term "algebraic combinatorics" was introduced in the late 1970s.Algebraic Combinatorics by Eiichi Bannai Through the early or mid-1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of symmetries (association schemes, strongly regular graphs, posets with a group action) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric functions, Young tableaux). This period is reflected in the area 05E, Algebraic combinatorics, of the AMS Mathematics Subject Classification, introduced in 1991.
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Most authors consider as an algebraic surface only algebraic varieties of dimension two, but some also consider as surfaces all algebraic sets whose irreducible components have the dimension two. In the case of surfaces in a space of dimension three, every surface is a complete intersection, and a surface is defined by a single polynomial, which is irreducible or not, depending on whether non-irreducible algebraic sets of dimension two are considered as surfaces or not.
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The research at the IMJ-PRG covers most of fundamental mathematics. It is subdivided in twelve team-projects: Algebraic Analysis; Complex Analysis and Geometry; Functional Analysis; Operator Algebra; Combinatorics and Optimization; Automorphic Forms; History of Mathematical Sciences; Geometry and Dynamics; Groups, Representations, and Geometry; Mathematical Logic; Number Theory; and Algebraic Topology and (Algebraic) Geometry.
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It gives a beautiful solution of an important problem. # His theorem that a compact analytic variety in a projective space is algebraic is justly famous. The theorem shows the close analogy between algebraic geometry and algebraic number theory. # Generalizing a result of Caratheodory on thermodynamics, he formulated a theorem on accessibility of differential spaces.
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In mathematics, a moduli scheme is a moduli space that exists in the category of schemes developed by Alexander Grothendieck. Some important moduli problems of algebraic geometry can be satisfactorily solved by means of scheme theory alone, while others require some extension of the 'geometric object' concept (algebraic spaces, algebraic stacks of Michael Artin).
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The truncated icosahedron or Buckminsterfullerene graph has a traditional connectivity of 3, and an algebraic connectivity of 0.243. The algebraic connectivity of a graph G can be positive or negative, even if G is a connected graph. Furthermore, the value of the algebraic connectivity is bounded above by the traditional (vertex) connectivity of the graph.J.
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In mathematical logic, an algebraic sentence is one that can be stated using only equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first-order logic involving only algebraic sentences. Saying that a sentence is algebraic is a stronger condition than saying it is elementary.
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An example is the field of complex numbers. Every field has an algebraic extension which is algebraically closed (called its algebraic closure), but proving this in general requires some form of the axiom of choice. An extension L/K is algebraic if and only if every sub K-algebra of L is a field.
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A special programming language, called 20-GATE,General Algebraic Translator Extended was developed for the G-20."20-GATE: Algebraic Compiler for the Bendix G-20", Carnegie Tech Computation Center, September 1962.
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In mathematics, differential refers to infinitesimal differences or to the derivatives of functions. The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology.
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In mathematics, the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number \tau(G) of a simply connected simple algebraic group defined over a number field is 1. In this case, simply connected means "not having a proper algebraic covering" in the algebraic group theory sense, which is not always the topologists' meaning.
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Algebraic structures are defined through different configurations of axioms. Universal algebra abstractly studies such objects. One major dichotomy is between structures that are axiomatized entirely by identities and structures that are not. If all axioms defining a class of algebras are identities, then this class is a variety (not to be confused with algebraic varieties of algebraic geometry).
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Given any algebraic function field K over k, we can consider the set of elements of K which are algebraic over k. These elements form a field, known as the field of constants of the algebraic function field. For instance, C(x) is a function field of one variable over R; its field of constants is C.
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Modern analytic geometry is essentially equivalent to real and complex algebraic geometry, as has been shown by Jean-Pierre Serre in his paper GAGA, the name of which is French for Algebraic geometry and analytic geometry. Nevertheless, the two fields remain distinct, as the methods of proof are quite different and algebraic geometry includes also geometry in finite characteristic.
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In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form.
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Nash's result was later (1973) completed by Alberto Tognoli who proved that any compact smooth manifold is diffeomorphic to some affine real algebraic manifold; actually, any Nash manifold is Nash diffeomorphic to an affine real algebraic manifold. These results exemplify the fact that the Nash category is somewhat intermediate between the smooth and the algebraic categories.
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These are now simply called varieties. The type of space that underlies most modern algebraic geometry is even more general than Weil's abstract algebraic varieties. It was introduced by Alexander Grothendieck and is called a scheme. One of the motivations for scheme theory is that polynomials are unusually structured among functions, and algebraic varieties are consequently rigid.
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Algebraic number theory interacts with many other mathematical disciplines. It uses tools from homological algebra. Via the analogy of function fields vs. number fields, it relies on techniques and ideas from algebraic geometry.
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While algebraic topologists largely continue to prefer CW complexes, there is a growing contingent of researchers interested in using simplicial sets for applications in algebraic geometry where CW complexes do not naturally exist.
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In algebraic geometry, a derived stack is, roughly, a stack together with a sheaf of commutative ring spectra. It generalizes a derived scheme. Derived stacks are the "spaces" studied in derived algebraic geometry.
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In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible.
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Solomon Lefschetz (; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.
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Algebraic varieties have continuous moduli spaces, hence their study is algebraic geometry. These are finite-dimensional moduli spaces. The space of Riemannian metrics on a given differentiable manifold is an infinite-dimensional space.
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In mathematics, a diffiety is a geometrical object introduced by Alexandre Mikhailovich Vinogradov (see ) playing the same role in the modern theory of partial differential equations as algebraic varieties play for algebraic equations.
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It combined both algebra and geometry under one specialty – algebraic geometry, now called analytic geometry, which involves reducing geometry to a form of arithmetic and algebra and translating geometric shapes into algebraic equations.
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Thus, Galois theory studies the symmetries inherent in algebraic equations.
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His professional interests include arithmetical algebraic geometry and mathematics education.
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On the other hand, Gauss sums have nicer algebraic properties.
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However, they are not algebraic surfaces (except in special cases).
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His research deals with differential algebraic equation and algorithmic algebra.
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He is known for the Puppe sequence in algebraic topology.
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Also \sin \theta is algebraic since it equals the algebraic number \cos(\theta-\pi /2). Finally, \tan \theta, where again \theta is a rational multiple of , is algebraic as being the ratio of two algebraic numbers. In a more elementary way, this can also be seen by equating the imaginary parts of the two sides of the expansion of the de Moivre equation to each other and dividing through by \cos^n \theta to obtain a polynomial equation in \tan \theta.
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A complex projective algebraic curve resides in n-dimensional complex projective space CPn. This has complex dimension n, but topological dimension, as a real manifold, 2n, and is compact, connected, and orientable. An algebraic curve over C likewise has topological dimension two; in other words, it is a surface. The topological genus of this surface, that is the number of handles or donut holes, is equal to the geometric genus of the algebraic curve that may be computed by algebraic means.
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The Éléments de géométrie algébrique ("Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné), or EGA for short, is a rigorous treatise, in French, on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the Institut des Hautes Études Scientifiques. In it, Grothendieck established systematic foundations of algebraic geometry, building upon the concept of schemes, which he defined. The work is now considered the foundation stone and basic reference of modern algebraic geometry.
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Techniques that have been found productive for helping such students include peer-assisted learning, explicit teaching with visual aids, instruction informed by formative assessment and encouraging students to think aloud. ;Algebraic reasoning :Elementary school children need to spend a long time learning to express algebraic properties without symbols before learning algebraic notation. When learning symbols, many students believe letters always represent unknowns and struggle with the concept of variable. They prefer arithmetic reasoning to algebraic equations for solving word problems.
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Their work generalizes algebraic geometry in a purely algebraic direction: instead of studying the prime ideals in a polynomial ring, one can study the prime ideals in any commutative ring. For example, Krull defined the dimension of any commutative ring in terms of prime ideals. At least when the ring is Noetherian, he proved many of the properties one would want from the geometric notion of dimension. Noether and Krull's commutative algebra can be viewed as an algebraic approach to affine algebraic varieties.
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The emphasis on algebraic surfaces--algebraic varieties of dimension two--followed on from an essentially complete geometric theory of algebraic curves (dimension 1). The position in around 1870 was that the curve theory had incorporated with Brill–Noether theory the Riemann–Roch theorem in all its refinements (via the detailed geometry of the theta-divisor). The classification of algebraic surfaces was a bold and successful attempt to repeat the division of algebraic curves by their genus g. The division of curves corresponds to the rough classification into the three types: g = 0 (projective line); g = 1 (elliptic curve); and g > 1 (Riemann surfaces with independent holomorphic differentials).
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In algebraic geometry, Chevalley's structure theorem states that a smooth connected algebraic group over a perfect field has a unique normal smooth connected affine algebraic subgroup such that the quotient is an abelian variety. It was proved by (though he had previously announced the result in 1953), , and . Chevalley's original proof, and the other early proofs by Barsotti and Rosenlicht, used the idea of mapping the algebraic group to its Albanese variety. The original proofs were based on Weil's book Foundations of algebraic geometry and are hard to follow for anyone unfamiliar with Weil's foundations, but later gave an exposition of Chevalley's proof in scheme- theoretic terminology.
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The transcendence of and are direct corollaries of this theorem. Suppose is a non-zero algebraic number; then is a linearly independent set over the rationals, and therefore by the first formulation of the theorem is an algebraically independent set; or in other words is transcendental. In particular, is transcendental. (A more elementary proof that is transcendental is outlined in the article on transcendental numbers.) Alternatively, by the second formulation of the theorem, if is a non-zero algebraic number, then is a set of distinct algebraic numbers, and so the set is linearly independent over the algebraic numbers and in particular cannot be algebraic and so it is transcendental.
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In classical algebraic geometry (that is, the part of algebraic geometry in which one does not use schemes, which were introduced by Grothendieck around 1960), the Zariski topology is defined on algebraic varieties. The Zariski topology, defined on the points of the variety, is the topology such that the closed sets are the algebraic subsets of the variety. As the most elementary algebraic varieties are affine and projective varieties, it is useful to make this definition more explicit in both cases. We assume that we are working over a fixed, algebraically closed field k (in classical geometry k is almost always the complex numbers).
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This implies that a trigonometric number is an algebraic number, and twice a trigonometric number is an algebraic integer. Ivan Niven gave proofs of theorems regarding these numbers.Niven, Ivan. Numbers: Rational and Irrational, 1961.
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Friedhelm Waldhausen (born 1938 in Millich, Hückelhoven, Rhine Province) is a German mathematician known for his work in algebraic topology. He made fundamental contributions in the fields of 3-manifolds and (algebraic) K-theory.
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Projective space plays a central role in algebraic geometry. The aim of this article is to define the notion in terms of abstract algebraic geometry and to describe some basic uses of projective space.
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Peter Schneider in Oberwolfach 2009 Peter Bernd Schneider (born 9 January 1953 in Karlsruhe) is a German mathematician, specializing in the p-adic aspects of algebraic number theory, arithmetic algebraic geometry, and representation theory.
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The vector space concept is at times referred to as the algebraic dimension.There are authors who use the term rank for algebraic dimension. Authors that do this frequently just use dimension when discussing geometric dimension.
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Because the algebraic numbers form a subfield of the real numbers, many irrational real numbers can be constructed by combining transcendental and algebraic numbers. For example, 3 + 2, + and e are irrational (and even transcendental).
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In algebraic geometry, the Bott–Samelson resolution of a Schubert variety is a resolution of singularities. It was introduced by in the context of compact Lie groups. The algebraic formulation is independently due to and .
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Accessed: May 5, 2013. The problem can be translated into algebraic terms,. but unfortunately there is no known algorithm to solve this algebraic problem. If a knot is invertible and amphichiral, it is fully amphichiral.
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In mathematics, specifically algebraic geometry, the valuative criteria are a collection of results that make it possible to decide whether a morphism of algebraic varieties, or more generally schemes, is universally closed, separated, or proper.
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Let us define algebraic systems used in the forthcoming symmetry definition.
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His main research was in algebraic geometry., MacTutor History of Mathematics.
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W. Frahm was a German mathematician who worked on algebraic geometry.
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This is a list of algebraic geometry topics, by Wikipedia page.
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A field containing is called an algebraic closure of if it is algebraic over (roughly speaking, not too big compared to ) and is algebraically closed (big enough to contain solutions of all polynomial equations). By the above, is an algebraic closure of . The situation that the algebraic closure is a finite extension of the field is quite special: by the Artin-Schreier theorem, the degree of this extension is necessarily 2, and is elementarily equivalent to . Such fields are also known as real closed fields.
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In algebraic number theory, the Albert–Brauer–Hasse–Noether theorem states that a central simple algebra over an algebraic number field K which splits over every completion Kv is a matrix algebra over K. The theorem is an example of a local-global principle in algebraic number theory and leads to a complete description of finite-dimensional division algebras over algebraic number fields in terms of their local invariants. It was proved independently by Richard Brauer, Helmut Hasse, and Emmy Noether and by Abraham Adrian Albert.
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In algebraic topology and algebraic geometry, a cyclic cover or cyclic covering is a covering space for which the set of covering transformations forms a cyclic group. As with cyclic groups, there may be both finite and infinite cyclic covers. Cyclic covers have proven useful in the descriptions of knot topology and the algebraic geometry of Calabi–Yau manifolds. In classical algebraic geometry, cyclic covers are a tool used to create new objects from existing ones through, for example, a field extension by a root element.
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Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension.
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To any irreducible algebraic variety is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the coordinate ring. The study of algebraic geometry makes heavy use of commutative algebra to study geometric concepts in terms of ring-theoretic properties. Birational geometry studies maps between the subrings of the function field.
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Informally in mathematical logic, an algebraic theory is one that uses axioms stated entirely in terms of equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first-order logic involving only algebraic sentences. The notion is very close to the notion of algebraic structure, which, arguably, may be just a synonym.
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In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its origins in the classical birational geometry of surfaces studied by the Italian school, and is currently an active research area within algebraic geometry.
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Her main scientific interests were in algebraic geometry, algebraic topology and photogrammetry. After her thesis, she worked on classification of algebraic surfaces studying the configurations of lines that could lie on surfaces. The next step was to study rational curves lying on surfaces and in this framework Beloch obtained the following important result:E. Strickland, Scienziate d'Italia: diciannove vite per la ricerca.
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The two-dimensional complex tori include the abelian surfaces. One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic. The algebraic ones are exactly the 2-dimensional abelian varieties. Most of their theory is a special case of the theory of higher- dimensional tori or abelian varieties.
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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 considered an arithmetic analog of the Hodge conjecture.
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This makes the transcendental numbers uncountable. No rational number is transcendental and all real transcendental numbers are irrational. The irrational numbers contain all the real transcendental numbers and a subset of the algebraic numbers, including the quadratic irrationals and other forms of algebraic irrationals. Any non-constant algebraic function of a single variable yields a transcendental value when applied to a transcendental argument.
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Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant. Thus the combinatorial topics may be enumerative in nature or involve matroids, polytopes, partially ordered sets, or finite geometries. On the algebraic side, besides group and representation theory, lattice theory and commutative algebra are common.
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It is also possible for different algebraic spaces to correspond to the same analytic space: for example, an elliptic curve and the quotient of C by the corresponding lattice are not isomorphic as algebraic spaces but the corresponding analytic spaces are isomorphic. Artin showed that proper algebraic spaces over the complex numbers are more or less the same as Moishezon spaces.
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He has been on the Yale University faculty since 1963, and became emeritus in 1996. He introduced the Tamagawa numbers, which are measures for algebraic groups over algebraic number fields. These measures play an essential role in conjectures on arithmetic algebraic geometry, such as those of Spencer Bloch and Kazuya Kato. Tamagawa's doctoral students included Doris Schattschneider and Audrey Terras.
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The theorem states that given any field F, an algebraic extension field E of F and an isomorphism \phi mapping F onto a field F' then \phi can be extended to an isomorphism \tau mapping E onto an algebraic extension E' of F' (a subfield of the algebraic closure of F'). The proof of the isomorphism extension theorem depends on Zorn's lemma.
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In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras.
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Spectral sequences can be constructed by various ways. In algebraic topology, an exact couple is perhaps the most common tool for the construction. In algebraic geometry, spectral sequences are usually constructed from filtrations of cochain complexes.
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Gerhard Hochschild Gerhard Paul Hochschild (April 29, 1915 in Berlin – July 8, 2010 in El Cerrito, California) was a German-born American mathematician who worked on Lie groups, algebraic groups, homological algebra and algebraic number theory.
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Julianna Sophia Tymoczko (born 1975) is an American mathematician whose research connects algebraic geometry and algebraic combinatorics, including representation theory, Schubert calculus, equivariant cohomology, and Hessenberg varieties. She is a professor of mathematics at Smith College.
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Anatoly LibgoberCURRICULUM VITAE and complete LIST of PUBLICATIONS at University of Illinois at Chicago web page (born 1949, in Moscow) is a Russian/American mathematician, known for work in algebraic geometry and topology of algebraic varieties.
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Abstract algebraic logic has become a well established subfield of algebraic logic, with many deep and interesting results. These results explain many properties of different classes of logical systems previously explained only in a case by case basis or shrouded in mystery. Perhaps the most important achievement of abstract algebraic logic has been the classification of propositional logics in a hierarchy, called the abstract algebraic hierarchy or Leibniz hierarchy, whose different levels roughly reflect the strength of the ties between a logic at a particular level and its associated class of algebras. The position of a logic in this hierarchy determines the extent to which that logic may be studied using known algebraic methods and techniques.
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In algebraic geometry, an affine algebraic set is defined as the set of the common zeros of an ideal of a polynomial ring R=k[x_1,\ldots, x_n]. An irredundant primary decomposition :I=Q_1\cap\cdots\cap Q_r of defines a decomposition of into a union of algebraic sets , which are irreducible, as not being the union of two smaller algebraic sets. If P_i is the associated prime of Q_i, then V(P_i)=V(Q_i), and Lasker–Noether theorem shows that has a unique irredundant decomposition into irreducible algebraic varieties :V(I)=\bigcup V(P_i), where the union is restricted to minimal associated primes. These minimal associated primes are the primary components of the radical of .
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In this case, one says that the polynomial vanishes at the corresponding point of Pn. This allows us to define a projective algebraic set in Pn as the set , where a finite set of homogeneous polynomials vanishes. Like for affine algebraic sets, there is a bijection between the projective algebraic sets and the reduced homogeneous ideals which define them. The projective varieties are the projective algebraic sets whose defining ideal is prime. In other words, a projective variety is a projective algebraic set, whose homogeneous coordinate ring is an integral domain, the projective coordinates ring being defined as the quotient of the graded ring or the polynomials in variables by the homogeneous (reduced) ideal defining the variety.
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Algebraic numbers on the complex plane colored by degree (red=1, green=2, blue=3, yellow=4) A real number r is called a real algebraic number if there is a polynomial p(x), with only integer coefficients, so that r is a root of p, that is, p(r)=0. Each real algebraic number can be defined individually using the order relation on the reals. For example, if a polynomial q(x) has 5 roots, the third one can be defined as the unique r such that q(r) = 0 and such that there are two distinct numbers less than r for which q is zero. All rational numbers are algebraic, and all constructible numbers are algebraic.
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The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial :x^2 - x - 1. Having degree 2, this polynomial actually has two roots, the other being the golden ratio conjugate.
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It has revolutionized the subject of homological algebra, a purely algebraic aspect of algebraic topology. It removed the need to distinguish the cases of modules over a ring and sheaves of abelian groups over a topological space.
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Abramo Giulio Umberto Federigo Enriques (5 January 1871 – 14 June 1946) was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry.
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In mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebraic curve. It was proved by .
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In algebraic geometry, one uses a normalization making CPn a Hodge manifold.
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These algebraic properties are easily obtained from Nevanlinna's definition and Jensen's formula.
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An algebraic curve (the Kampyle of Eudoxus) is also named after him.
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Many quantitative relationships in science and mathematics are expressed as algebraic equations.
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Algebraic specification is a software engineering technique for formally specifying system behavior.
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There is a general conjecture for algebraic surfaces, the Nagata–Biran conjecture.
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Yujiro Kawamata (born 1952) is a Japanese mathematician working in algebraic geometry.
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In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be normal. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable.
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André Weil's famous Weil conjectures proposed that certain properties of equations with integral coefficients should be understood as geometric properties of the algebraic variety that they define. His conjectures postulated that there should be a cohomology theory of algebraic varieties that gives number-theoretic information about their defining equations. This cohomology theory was known as the "Weil cohomology", but using the tools he had available, Weil was unable to construct it. In the early 1960s, Alexander Grothendieck introduced étale maps into algebraic geometry as algebraic analogues of local analytic isomorphisms in analytic geometry.
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An algebraic set being a finite union of algebraic varieties, its dimension is the maximum of the dimensions of its components. It is equal to the maximal length of the chains V_0\subsetneq V_1\subsetneq \cdots \subsetneq V_d of sub-varieties of the given algebraic set (the length of such a chain is the number of "\subsetneq"). Each variety can be considered as an algebraic stack, and its dimension as variety agrees with its dimension as stack. There are however many stacks which do not correspond to varieties, and some of these have negative dimension.
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In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group GLn (n x n invertible matrices), the subgroup of invertible upper triangular matrices is a Borel subgroup. For groups realized over algebraically closed fields, there is a single conjugacy class of Borel subgroups. Borel subgroups are one of the two key ingredients in understanding the structure of simple (more generally, reductive) algebraic groups, in Jacques Tits' theory of groups with a (B,N) pair.
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In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive.
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The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined by the set of its roots (a geometric object) in the complex plane. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is a defining feature of algebraic geometry.
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In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth.
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If G is a commutative algebraic group defined over an algebraic number field and A is a Lie subgroup of G with Lie algebra defined over the number field then A does not contain any non-zero algebraic point of G unless A contains a proper algebraic subgroup. One of the central new ingredients of the proof was the theory of multiplicity estimates of group varieties developed by David Masser and Gisbert Wüstholz in special cases and established by Wüstholz in the general case which was necessary for the proof of the analytic subgroup theorem.
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Commutative ring theory originated in algebraic number theory, algebraic geometry, and invariant theory. Central to the development of these subjects were the rings of integers in algebraic number fields and algebraic function fields, and the rings of polynomials in two or more variables. Noncommutative ring theory began with attempts to extend the complex numbers to various hypercomplex number systems. The genesis of the theories of commutative and noncommutative rings dates back to the early 19th century, while their maturity was achieved only in the third decade of the 20th century.
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Let denote the algebraic dual space of an -module . Let and be -modules. If is a linear map, then its algebraic adjoint or dual, is the map defined by . The resulting functional is called the pullback of by .
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In fact, as Davenport later admitted, his inherent prejudices against algebraic methods ("what can you do with algebra?") probably limited the amount he learned, in particular in the "new" algebraic geometry and Artin/Noether approach to abstract algebra.
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In algebraic geometry, a field of mathematics, the AF+BG theorem (also known as Max Noether's fundamental theorem) is a result of Max Noether that asserts that, if the equation of an algebraic curve in the complex projective plane belongs locally (at each intersection point) to the ideal generated by the equations of two other algebraic curves, then it belongs globally to this ideal.
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His first construction shows how to write the real algebraic numbersThe real algebraic numbers are the real roots of polynomial equations with integer coefficients. as a sequence a1, a2, a3, .... In other words, the real algebraic numbers are countable. Cantor starts his second construction with any sequence of real numbers. Using this sequence, he constructs nested intervals whose intersection contains a real number not in the sequence.
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One of seven Millennium Prize problems, the Hodge conjecture, is a question in algebraic geometry. Wiles' proof of Fermat's Last Theorem uses advanced methods of algebraic geometry for solving a long-standing problem of number theory. In general, algebraic geometry studies geometry through the use of concepts in commutative algebra such as multivariate polynomials. It has applications in many areas, including cryptography and string theory.
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From its very origins, homological algebra has played an enormous role in algebraic topology. Its influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes.
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In mathematics, homotopy theory is a systematic study of situations in which maps come with homotopies between them. It originated as a topic in algebraic topology but nowadays it is studied as an independent discipline. Besides algebraic topology, the theory has also been in used in other areas of mathematics such as algebraic geometry (e.g., A¹ homotopy theory) and category theory (specifically the study of higher categories).
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The four exponentials conjecture rules out a special case of non-trivial, homogeneous, quadratic relations between logarithms of algebraic numbers. But a conjectural extension of Baker's theorem implies that there should be no non-trivial algebraic relations between logarithms of algebraic numbers at all, homogeneous or not. One case of non-homogeneous quadratic relations is covered by the still open three exponentials conjecture.Waldschmidt, "Variations…" (2005), consequence 1.9.
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The unitary group of a quadratic module is a generalisation of the linear algebraic group U just defined, which incorporates as special cases many different classical algebraic groups. The definition goes back to Anthony Bak's thesis.Bak, Anthony (1969), "On modules with quadratic forms", Algebraic K-Theory and its Geometric Applications (editors--Moss R. M. F., Thomas C. B.) Lecture Notes in Mathematics, Vol. 108, pp.
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The projection method was introduced to enable an approximate solution of the algebraic transport equation of the Reynolds-stresses. In contrast to the approach of the tensor basis is not inserted in the algebraic equation, instead the algebraic equation is projected. Therefore, the chosen basis tensors does not need to form a complete integrity basis. However, the projection will fail if the basis tensor are linear dependent.
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Kai Behrend is a German mathematician. He is a professor at the University of British Columbia in Vancouver, British Columbia, Canada. His work is in algebraic geometry and he has made important contributions in the theory of algebraic stacks, Gromov–Witten invariants and Donaldson–Thomas theory (cf. Behrend function.) He is also known for Behrend's formula, the generalization of the Grothendieck–Lefschetz trace formula to algebraic stacks.
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In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are ultimately derived from the notion of divisibility in the integers and algebraic number fields. The background is that codimension-1 subvarieties are understood much better than higher- codimension subvarieties.
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He put forward the Waldhausen conjecture about Heegaard splitting. In the mid-seventies, he extended the connection between geometric topology and algebraic K-theory by introducing A-theory, a kind of algebraic K-theory for topological spaces. This led to new foundations for algebraic K-theory (using what are now called Waldhausen categories) and also gave new impetus to the study of highly structured ring spectra.
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The earliest successful attempt to define an algebraic variety abstractly, without an embedding, was made by André Weil. In his Foundations of Algebraic Geometry, Weil defined an abstract algebraic variety using valuations. Claude Chevalley made a definition of a scheme, which served a similar purpose, but was more general. However, Alexander Grothendieck's definition of a scheme is more general still and has received the most widespread acceptance.
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Another way of phrasing the Hodge conjecture involves the idea of an algebraic cycle. An algebraic cycle on X is a formal combination of subvarieties of X; that is, it is something of the form: : \sum_i c_iZ_i. The coefficients are usually taken to be integral or rational. We define the cohomology class of an algebraic cycle to be the sum of the cohomology classes of its components.
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In mathematics, graphs are useful in geometry and certain parts of topology such as knot theory. Algebraic graph theory has close links with group theory. Algebraic graph theory has been applied to many areas including dynamic systems and complexity.
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Patrick du Val (March 26, 1903 - January 22, 1987) was a British mathematician, known for his work on algebraic geometry, differential geometry, and general relativity. The concept of Du Val singularity of an algebraic surface is named after him.
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Claire Voisin (born 4 March 1962) is a French mathematician known for her work in algebraic geometry. She is a member of the French Academy of Sciences and holds the chair of Algebraic Geometry at the Collège de France.
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Each algebraic integer belongs to the ring of integers of some number field. A number is an algebraic integer if and only if the ring is finitely generated as an Abelian group, which is to say, as a -module.
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John Torrence Tate Jr. (March 13, 1925 – October 16, 2019) was an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry. He was awarded the Abel Prize in 2010.
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In algebraic geometry, the dimension of a scheme is a generalization of a dimension of an algebraic variety. Scheme theory emphasizes the relative point of view and, accordingly, the relative dimension of a morphism of schemes is also important.
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Biggs makes important connections with other branches of algebraic combinatorics and group theory.
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Subfields of topology include geometric topology, differential topology, algebraic topology and general topology.
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Hodge theory is a central part of algebraic geometry, proved using Kähler metrics.
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In algebraic graph theory, Babai's problem was proposed in 1979 by László Babai.
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Algebraic data types are particularly well-suited to the implementation of abstract syntax.
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In algebraic geometry, connectedness is generalized to the concept of a connected scheme.
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Most of the other examples above can also be evaluated using algebraic simplification.
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"Number theory and algebraic geometry." In Proc. Intern. Math. Congres., Cambridge, Mass., vol.
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Kleene introduced regular expressions and gave some of their algebraic laws. Here: sect.
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This can be generalized to curves of higher order as circular algebraic curves.
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In mathematics, a pseudo-canonical variety is an algebraic variety of "general type".
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In mathematical logic, an algebraic definition is one that can be given using only equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Saying that a definition is algebraic is a stronger condition than saying it is elementary.
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Rationalisation can be extended to all algebraic numbers and algebraic functions (as an application of norm forms). For example, to rationalise a cube root, two linear factors involving cube roots of unity should be used, or equivalently a quadratic factor.
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In 1992, he became a Fellow of the Royal Society of Edinburgh.Directory of Fellows , page 33. Since 1995, Ranicki has been the Chair of Algebraic Surgery at the University of Edinburgh.Chair of Algebraic Surgery, University of Edinburgh, Communications and Marketing.
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In algebraic geometry, in the foundational developments of André Weil the use of fields other than the complex numbers was essential to expand the definitions to include the idea of abstract algebraic variety over K, and generic point relative to K.
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For every algebraic number field and every finite field F there is a matroid M for which F is the minimal subfield of its algebraic closure over which M can be represented: M can be taken to be of rank 3.
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Zinovy Reichstein (born 1961) is a Russian-born American mathematician. He is a professor at the University of British Columbia in Vancouver. He studies mainly algebra, algebraic geometry and algebraic groups. He introduced (with J. Buhler) the concept of essential dimension.
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In mathematics, an ordinary singularity of an algebraic curve is a singular point of multiplicity r where the r tangents at the point are distinct . In higher dimensions the literature on algebraic geometry contains many inequivalent definitions of ordinary singular points.
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Quasi-separated morphisms are important for algebraic spaces and algebraic stacks, where many natural morphisms are quasi-separated but not separated. The condition that a morphism is quasi-separated often occurs together with the condition that it is quasi- compact.
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A linear algebraic group over a field k is defined as a smooth closed subgroup scheme of GL(n) over k, for some positive integer n. Equivalently, a linear algebraic group over k is a smooth affine group scheme over k.
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In algebraic geometry, a morphism of schemes f from X to Y is called quasi- separated if the diagonal map from X to X×YX is quasi-compact (meaning that the inverse image of any quasi-compact open set is quasi compact). A scheme X is called quasi-separated if the morphism to Spec Z is quasi-separated. Quasi- separated algebraic spaces and algebraic stacks and morphisms between them are defined in a similar way, though some authors include the condition that X is quasi-separated as part of the definition of an algebraic space or algebraic stack X. Quasi-separated morphisms were introduced by as a generalization of separated morphisms. All separated morphisms (and all morphisms of Noetherian schemes) are automatically quasi-separated.
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In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. More specifically, the conjecture states that certain de Rham cohomology classes are algebraic; that is, they are sums of Poincaré duals of the homology classes of subvarieties. It was formulated by the Scottish mathematician William Vallance Douglas Hodge as a result of a work in between 1930 and 1940 to enrich the description of de Rham cohomology to include extra structure that is present in the case of complex algebraic varieties. It received little attention before Hodge presented it in an address during the 1950 International Congress of Mathematicians, held in Cambridge, Massachusetts.
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In mathematics, specifically category theory, a functor is a map between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.
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Maxwell Alexander Rosenlicht (April 15, 1924 – January 22, 1999) was an American mathematician known for works in algebraic geometry, algebraic groups, and differential algebra. Rosenlicht went to school in Brooklyn (Erasmus High School) and studied at Columbia University (B.A. 1947) and at Harvard University, where he studied under Zariski and was awarded in his doctorate on an Algebraic Curve Equivalence Concepts in 1950. In 1952, he went to Northwestern University.
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Universal algebra has also been studied using the techniques of category theory. In this approach, instead of writing a list of operations and equations obeyed by those operations, one can describe an algebraic structure using categories of a special sort, known as Lawvere theories or more generally algebraic theories. Alternatively, one can describe algebraic structures using monads. The two approaches are closely related, with each having their own advantages.
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Christopher David Godsil is a professor and the former Chair at the Department of Combinatorics and Optimization in the faculty of mathematics at the University of Waterloo. He wrote the popular textbook on algebraic graph theory, entitled Algebraic Graph Theory, with Gordon Royle,Robin J. Wilson's review of for Mathematical Reviews (). His earlier textbook on algebraic combinatorics discussed distance-regular graphs and association schemes.Andrew Woldar's review of for Mathematical Reviews ().
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In mathematics, an abelian surface is 2-dimensional abelian variety. One- dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic. The algebraic ones are called abelian surfaces and are exactly the 2-dimensional abelian varieties. Most of their theory is a special case of the theory of higher-dimensional tori or abelian varieties.
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Unlike the traditional connectivity, the algebraic connectivity is dependent on the number of vertices, as well as the way in which vertices are connected. In random graphs, the algebraic connectivity decreases with the number of vertices, and increases with the average degree.Synchronization and Connectivity of Discrete Complex Systems, Michael Holroyd, International Conference on Complex Systems, 2006. The exact definition of the algebraic connectivity depends on the type of Laplacian used.
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They studied mathematics, physics, and propaedeutics at Berlin and Göttingen. Löbenstein's field of expertise was algebraic geometry. Together with Kahn she made a contribution to Hilbert's sixteenth problem. Hilbert's sixteenth problem concerned the topology of algebraic curves in the complex projective plane; as a difficult special case in his formulation of the problem Hilbert proposed that there are no algebraic curves of degree 6 consisting of 11 separate ovals.
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The roots of a polynomial expression of degree n, or equivalently the solutions of a polynomial equation, can always be written as algebraic expressions if n < 5 (see quadratic formula, cubic function, and quartic equation). Such a solution of an equation is called an algebraic solution. But the Abel–Ruffini theorem states that algebraic solutions do not exist for all such equations (just for some of them) if n \ge 5\.
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For any finite extension of fields, the restriction of scalars takes quasiprojective varieties to quasiprojective varieties. The dimension of the resulting variety is multiplied by the degree of the extension. Under appropriate hypotheses (e.g., flat, proper, finitely presented), any morphism T \to S of algebraic spaces yields a restriction of scalars functor that takes algebraic stacks to algebraic stacks, preserving properties such as Artin, Deligne-Mumford, and representability.
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Examples of block codes are Reed–Solomon codes, Hamming codes, Hadamard codes, Expander codes, Golay codes, and Reed–Muller codes. These examples also belong to the class of linear codes, and hence they are called linear block codes. More particularly, these codes are known as algebraic block codes, or cyclic block codes, because they can be generated using boolean polynomials. Algebraic block codes are typically hard-decoded using algebraic decoders.
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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 , and can be used to study ramified coverings of a curve, with abelian Galois group. Generalized Jacobians of a curve are extensions of the Jacobian of the curve by a commutative affine algebraic group, giving nontrivial examples of Chevalley's structure theorem.
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A complex K3 surface has real dimension 4, and it plays an important role in the study of smooth 4-manifolds. K3 surfaces have been applied to Kac–Moody algebras, mirror symmetry and string theory. It can be useful to think of complex algebraic K3 surfaces as part of the broader family of complex analytic K3 surfaces. Many other types of algebraic varieties do not have such non-algebraic deformations.
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Vladimir P. Gerdt (born January 21, 1947) is a Russian mathematician and a full professor at the Joint Institute for Nuclear Research (JINR) where he is the head of the Group of Algebraic and Quantum Computations. His research interests are concentrated in computer algebra, symbolic and algebraic computations, algebraic and numerical analysis of nonlinear differential equations, polynomial equations, applications to mathematics and physics, and quantum computation with over 210 published articles.
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Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance, algebraic topology uses algebraic objects to study topologies. The Poincaré conjecture, proved in 2003, asserts that the fundamental group of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not. Algebraic number theory studies various number rings that generalize the set of integers.
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In computer science, double pushout graph rewriting (or DPO graph rewriting) refers to a mathematical framework for graph rewriting. It was introduced as one of the first algebraic approaches to graph rewriting in the article "Graph-grammars: An algebraic approach" (1973)."Graph-grammars: An algebraic approach", Ehrig, Hartmut and Pfender, Michael and Schneider, Hans-Jürgen, Switching and Automata Theory, 1973. SWAT'08. IEEE Conference Record of 14th Annual Symposium on, pp.
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One advantage of defining varieties over arbitrary fields through the theory of schemes is that such definitions are intrinsic and free of embeddings into ambient affine n-space. A k-algebraic set is a separated and reduced scheme of finite type over Spec(k). A k-variety is an irreducible k-algebraic set. A k-morphism is a morphism between k-algebraic sets regarded as schemes over Spec(k).
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However, no irrational algebraic number has been proven to be normal in any base.
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In algebraic topology, several types of products are defined on homological and cohomological theories.
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Springer Science & Business Media. Miranda, R. (1995). Algebraic curves and Riemann surfaces (Vol. 5).
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Cf. Shafarevich (1995). This construction plays a role in algebraic geometry and conformal geometry.
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This article is a survey for some important types of kernels in algebraic structures.
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The symbol is often used to indicate isomorphic algebraic structures or congruent geometric figures.
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There was a school in India which delighted in geometric demonstrations of algebraic results.
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Computational complexity can be easily eliminated, when basing S5 signal analysis on algebraic invariants.
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János Kollár (born June 7, 1956) is a Hungarian mathematician, specializing in algebraic geometry.
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An affine algebraic variety whose coordinate ring is a Cohen–Macaulay ring is equidimensional.
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In mathematics, the Zahlbericht (number report) was a report on algebraic number theory by .
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In mathematics, the additive polynomials are an important topic in classical algebraic number theory.
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Frans Oort (born 17 July 1935) is a Dutch mathematician, specializing in algebraic geometry.
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Motivic integration is a notion in algebraic geometry that was introduced by Maxim Kontsevich in 1995 and was developed by Jan Denef and François Loeser. Since its introduction it has proved to be quite useful in various branches of algebraic geometry, most notably birational geometry and singularity theory. Roughly speaking, motivic integration assigns to subsets of the arc space of an algebraic variety, a volume living in the Grothendieck ring of algebraic varieties. The naming 'motivic' mirrors the fact that unlike ordinary integration, for which the values are real numbers, in motivic integration the values are geometric in nature.
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In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism : is a closed map, i.e. maps closed sets onto closed sets.Here the product variety X × Y does not carry the product topology, in general; the Zariski topology on it will have more closed sets (except in very simple cases). This can be seen as an analogue of compactness in algebraic geometry: a topological space X is compact if and only if the above projection map is closed with respect to topological products.
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The square root of 2 is an algebraic number equal to the length of the hypotenuse of a right triangle with legs of length 1. An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently—by clearing denominators—with integer coefficients). All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as and , are called transcendental numbers.
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They were instrumental in the proof of Fermat's last theorem and are also used in elliptic-curve cryptography. In parallel with the abstract trend of the algebraic geometry, which is concerned with general statements about varieties, methods for effective computation with concretely-given varieties have also been developed, which lead to the new area of computational algebraic geometry. One of the founding methods of this area is the theory of Gröbner bases, introduced by Bruno Buchberger in 1965. Another founding method, more specially devoted to real algebraic geometry, is the cylindrical algebraic decomposition, introduced by George E. Collins in 1973.
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In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e. which contain transcendental elements, are called transcendental. For example, the field extension R/Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental, while the field extensions C/R and Q()/Q are algebraic, where C is the field of complex numbers.
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Grothendieck's construction of new cohomology theories, which use algebraic techniques to study topological objects, has influenced the development of algebraic number theory, algebraic topology, and representation theory. As part of this project, his creation of topos theory, a category-theoretic generalization of point-set topology, has influenced the fields of set theory and mathematical logic. The Weil conjectures were formulated in the later 1940s as a set of mathematical problems in arithmetic geometry. They describe properties of analytic invariants, called local zeta functions, of the number of points on an algebraic curve or variety of higher dimension.
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The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on algebraic curves are formal linear combinations of points on the curve. Classical work on algebraic curves related these to intrinsic data, such as the regular differentials on a compact Riemann surface, and to extrinsic properties, such as embeddings of the curve into projective space. While divisors on higher- dimensional varieties continue to play an important role in determining the structure of the variety, on varieties of dimension two or more there are also higher codimension cycles to consider.
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In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field.
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He is a member of the U.S. National Academy of SciencesBloch, Spencer J. U.S. National Academy of Sciences. Accessed January 12, 2010. Election Citation: "Bloch has done pioneering work in the application of higher algebraic K-theory to algebraic geometry, particularly in problems related to algebraic cycles, and is regarded as the world's leader in this field. His work has firmly established higher K-theory as a fundamental tool in algebraic geometry." and a Fellow of the American Academy of Arts and Sciences American Academy of Arts & Sciences, NEWLY ELECTED MEMBERS, APRIL 2009, American Academy of Arts and Sciences.
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This is the doctrine of multi-sorted algebraic theories. If one only wanted 1-sorted algebraic theories, one would restrict the objects to only those categories that are generated under products by a single object. Doctrines were discovered by Jonathan Mock Beck.
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The real numbers are a superset containing the algebraic and the transcendental numbers. For some numbers, it is not known whether they are algebraic or transcendental. The following list includes real numbers that have not been proved to be irrational, nor transcendental.
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Hélène Esnault Hélène Esnault (born 1953 in Paris) is a French and German mathematician, specializing in algebraic geometry. She received her PhD in 1976 under Professor Lê Dũng Tráng, writing her dissertation on Singularites rationnelles et groupes algebriques (Rational singularities and algebraic groups).
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In particular, quasi-projective varieties are Noetherian schemes. This class includes algebraic curves, elliptic curves, abelian varieties, calabi-yau schemes, shimura varieties, K3 surfaces, and cubic surfaces. Basically all of the objects from classical algebraic geometry fit into this class of examples.
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Informally, a free object over a set A can be thought of as being a "generic" algebraic structure over A: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure.
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Claude Chevalley (; 11 February 1909 – 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory, and the theory of algebraic groups. He was a founding member of the Bourbaki group.
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In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21M. F. Atiyah and I. G. Macdonald (1969).
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Free objects are all examples of a left adjoint to a forgetful functor which assigns to an algebraic object its underlying set. These algebraic free functors have generally the same description as in the detailed description of the free group situation above.
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An algebraic Hecke character is a Hecke character taking algebraic values: they were introduced by Weil in 1947 under the name type A0. Such characters occur in class field theory and the theory of complex multiplication.Husemoller (1987) pp. 299–300; (2002) p.
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His research focuses on topics in algebraic graph theory, particularly the symmetry of graphs and the action of finite groups on combinatorial objects. He is regarded as the founder of the Slovenian school of research in algebraic graph theory and permutation groups.
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Oleg Viro in 2008 at the Mathematical Research Institute of Oberwolfach Oleg Yanovich Viro () (b. 13 May 1948, Leningrad, USSR) is a Russian mathematician in the fields of topology and algebraic geometry, most notably real algebraic geometry, tropical geometry and knot theory.
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Over the complex numbers, some authors consider only the algebraic K3 surfaces. (An algebraic K3 surface is automatically projective.Huybrechts (2016), Remark 1.1.2) Or one may allow K3 surfaces to have du Val singularities (the canonical singularities of dimension 2), rather than being smooth.
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In mathematics, an (equivalence class of an) inner form of an algebraic group G over a field K is another algebraic group associated to an element of H1(Gal(/K), Inn(G)) where Inn(G) is the group of inner automorphisms of G.
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The founders of the theory of algebraic groups include Maurer, Chevalley, and . In the 1950s, Armand Borel constructed much of the theory of algebraic groups as it exists today. One of the first uses for the theory was to define the Chevalley groups.
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Algebraic decision trees are a generalization of linear decision trees that allow the test functions to be polynomials of degree d. Geometrically, the space is divided into semi-algebraic sets (a generalization of hyperplane). The evaluation of the complexity is typically more difficult.
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In mathematics, a fake projective plane (or Mumford surface) is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective plane, but are not isomorphic to it. Such objects are always algebraic surfaces of general type.
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Goguen, J. A., "Tossing Algebraic Flowers Down the Great Divide", University of California, San Diego.
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Kishinev Shtiintsa 1971. 3\. n-ary quasigroup. Kishinev, Ştiinţa 1972. 4\. Changes in algebraic networks.
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Douglas Conner Ravenel (born 1947) is an American mathematician known for work in algebraic topology.
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Mei-Chu Chang is a mathematician who works in algebraic geometry and combinatorial number theory.
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Saying that a theory is algebraic is a stronger condition than saying it is elementary.
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Samit Dasgupta is a professor of mathematics at Duke University working in algebraic number theory.
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All groups that have been considered in this section are Lie groups and algebraic groups.
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William Edgar Fulton (born August 29, 1939) is an American mathematician, specializing in algebraic geometry.
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Large categories on the other hand can be used to create "structures" of algebraic structures.
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Baker turned his interest towards algebraic geometry, and he entered Trinity College, Cambridge in 1927.
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Skip Garibaldi is an American mathematician doing research on algebraic groups and especially exceptional groups.
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The algebraic formulation of a model does not contain any hints how to process it.
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13–14Fulton, William. Introduction to intersection theory in algebraic geometry. No. 54. American Mathematical Soc.
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In mathematics, rigidity of K-theory encompasses results relating algebraic K-theory of different rings.
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Compatible systems of ℓ-adic representations are a fundamental concept in contemporary algebraic number theory.
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Quantifier elimination over the reals is another example, which is fundamental in computational algebraic geometry.
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There are a lot of conoids with singular points, which are investigated in algebraic geometry.
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Toroidal embeddings have recently led to advances in algebraic geometry, in particular resolution of singularities.
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However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.
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In 1974, Biggs published Algebraic Graph Theory which articulates properties of graphs in algebraic terms, then works out theorems regarding them. In the first section, he tackles the applications of linear algebra and matrix theory; algebraic constructions such as adjacency matrix and the incidence matrix and their applications are discussed in depth. Next, there is and wide-ranging description of the theory of chromatic polynomials. The last section discusses symmetry and regularity properties.
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Universal algebra defines a notion of kernel for homomorphisms between two algebraic structures of the same kind. This concept of kernel measures how far the given homomorphism is from being injective. There is some overlap between this algebraic notion and the categorical notion of kernel since both generalize the situation of groups and modules mentioned above. In general, however, the universal-algebraic notion of kernel is more like the category-theoretic concept of kernel pair.
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Young diagram of a partition (5,4,1). Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. Algebraic combinatorics is continuously expanding its scope, in both topics and techniques, and can be seen as the area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant.
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More special versions of both are continuous and algebraic cpos. Adding even further completeness properties one obtains continuous lattices and algebraic lattices, which are just complete lattices with the respective properties. For the algebraic case, one finds broader classes of posets that are still worth studying: historically, the Scott domains were the first structures to be studied in domain theory. Still wider classes of domains are constituted by SFP-domains, L-domains, and bifinite domains.
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An algebraic closure Kalg of K contains a unique separable extension Ksep of K containing all (algebraic) separable extensions of K within Kalg. This subextension is called a separable closure of K. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of Ksep, of degree > 1. Saying this another way, K is contained in a separably- closed algebraic extension field. It is unique (up to isomorphism).
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For example, a branch of the hyperbola of equation x y-1 = 0 is not an algebraic variety, but is a semi-algebraic set defined by x y-1=0 and x>0 or by x y-1=0 and x+y>0. One of the challenging problems of real algebraic geometry is the unsolved Hilbert's sixteenth problem: Decide which respective positions are possible for the ovals of a nonsingular plane curve of degree 8.
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In algebra, the theory of equations is the study of algebraic equations (also called “polynomial equations”), which are equations defined by a polynomial. The main problem of the theory of equations was to know when an algebraic equation has an algebraic solution. This problem was completely solved in 1830 by Évariste Galois, by introducing what is now called Galois theory. Before Galois, there was no clear distinction between the “theory of equations” and “algebra”.
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The typical use of discriminants in algebraic geometry is for studying algebraic curve and, more generally algebraic hypersurfaces. Let be such a curve or hypersurface; is defined as the zero set of a multivariate polynomial. This polynomial may be considered as a univariate polynomial in one of the indeterminates, with polynomials in the other indeterminates as coefficients. The discriminant with respect to the selected indeterminate defines a hypersurface in the space of the other indeterminates.
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Some of these properties are connectedness (is the space in one or several pieces), the number of holes the space has, the knottedness of the space, and so on. Spaces are then studied by assigning algebraic constructions to them. This is similar to what is done in high school analytic geometry whereby to certain curves in the plane (geometric objects) are assigned equations (algebraic constructions). The most common algebraic constructions are groups.
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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 G module is a module that can be expressed as a sum (not necessarily direct) of rational representations.
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Durov introduced commutative algebraic monads as a generalization of local objects in a generalized algebraic geometry. Versions of a tropical geometry, of an absolute geometry over a field with one element and an algebraic analogue of Arakelov geometry were realized in this setup. He holds the position of a senior research fellow at the laboratory of algebra at the St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences.
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Based on the concept of the homotopy, computation methods for algebraic and differential equations have been developed. The methods for algebraic equations include the homotopy continuation method and the continuation method (see numerical continuation). The methods for differential equations include the homotopy analysis method.
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Ulrike Luise Tillmann FRS is a mathematician specializing in algebraic topology, who has made important contributions to the study of the moduli space of algebraic curves. She is titular Professor of Mathematics at the University of Oxford and a Fellow of Merton College, Oxford..
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Au-Yang H and Perk J. H. H. (2011), "Spontaneous magnetization of the integrable chiral Potts model", Journal of Physics A 44, 445005 (20pp), arXiv:1003.4805. an algebraic (Ising-like) way of obtaining order parameter has been given, giving more insight into the algebraic structure.
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Let denote the algebraic dual space of a vector space . Let and be vector spaces over the same field . If is a linear map, then its algebraic adjoint or dual, is the map defined by . The resulting functional is called the pullback of by .
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If F(x,y,z) is polynomial in x, y and z, the surface is called algebraic. Example 5 is non-algebraic. Despite difficulty of visualization, implicit surfaces provide relatively simple techniques to generate theoretically (e.g. Steiner surface) and practically (see below) interesting surfaces.
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In algebraic geometry, determinantal varieties are spaces of matrices with a given upper bound on their ranks. Their significance comes from the fact that many examples in algebraic geometry are of this form, such as the Segre embedding of a product of two projective spaces.
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In mathematics, algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall, with L being used as the letter after K. Algebraic L-theory, also known as 'hermitian K-theory', is important in surgery theory.
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In algebraic geometry, a group-stack is an algebraic stack whose categories of points have group structures or even groupoid structures in a compatible way. It generalizes a group scheme, which is a scheme whose sets of points have group structures in a compatible way.
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T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005, , p. 11 He also wrote on algebraic geometry, number theory, and integral equations. At Chicago, Moore supervised 31 doctoral dissertations, including those of George Birkhoff, Leonard Dickson, Robert Lee Moore (no relation), and Oswald Veblen.
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One of the earliest examples of a non-quasiprojective algebraic variety were given by Nagata. Nagata's example was not complete (the analog of compactness), but soon afterwards he found an algebraic surface that was complete and non- projective. Since then other examples have been found.
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Tsen's theorem is about the function field of an algebraic curve over an algebraically closed field.
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For elliptic curves, potential good reduction is equivalent to the j-invariant being an algebraic integer.
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She is best known for her works in group theory, algebraic graph theory and combinatorial designs.
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Since it is a projective space, algebraic techniques can also be effective tools in its study.
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Complex topics such as; proofs, algebraic functions and sets will be introduced during studies of CIS.
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Tara Suzanne Holm is a mathematician at Cornell University specializing in algebraic geometry and symplectic geometry.
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He was a pioneer of study of merged continual algebraic logic (parametrical) and topological (structural) models.
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Miles Anthony Reid FRS (born 30 January 1948) is a mathematician who works in algebraic geometry.
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This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves.
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In mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomology theory for topological spaces.
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Moreover, Grothendieck introduced K-groups, as discussed above, which paved the way for algebraic K-theory.
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Ravi D. Vakil (born February 22, 1970) is a Canadian-American mathematician working in algebraic geometry.
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Projective algebraic manifolds are an equivalent definition for projective varieties. The Riemann sphere is one example.
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These functions, their negativity and minima have a direct interpretation in algebraic topology (Baudot & Bennequin, 2015).
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Generally, in abstract algebra, a field extension F / E is algebraic if every element f of the bigger field F is the zero of a polynomial with coefficients e0, ..., em in E: :p(f) = emfm \+ em−1fm−1 \+ ... + e1f + e0 = 0. Every field extension of finite degree is algebraic. (Proof: for x in F, simply consider 1, x, x2, x3, ... – we get a linear dependence, i.e. a polynomial that x is a root of.) In particular this applies to algebraic number fields, so any element f of an algebraic number field F can be written as a zero of a polynomial with rational coefficients.
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Example: In the theory of algebraically closed fields of characteristic 0, there is a 1-type represented by elements that are transcendental over the prime field. This is a non-isolated point of the Stone space (in fact, the only non-isolated point). The field of algebraic numbers is a model omitting this type, and the algebraic closure of any transcendental extension of the rationals is a model realizing this type. All the other types are "algebraic numbers" (more precisely, they are the sets of first-order statements satisfied by some given algebraic number), and all such types are realized in all algebraically closed fields of characteristic 0.
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Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.
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All algebraic spaces are assumed of finite type over a locally Noetherian base. Suppose that j:R→X×X is a flat groupoid whose stabilizer j−1Δ is finite over X (where Δ is the diagonal of X×X). The Keel–Mori theorem states that there is an algebraic space that is a geometric and uniform categorical quotient of X by j, which is separated if j is finite. A corollary is that for any flat group scheme G acting properly on an algebraic space X with finite stabilizers there is a uniform geometric and uniform categorical quotient X/G which is a separated algebraic space.
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In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general.
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If V′ is a variety contained in Am, we say that f is a regular map from V to V′ if the range of f is contained in V′. The definition of the regular maps apply also to algebraic sets. The regular maps are also called morphisms, as they make the collection of all affine algebraic sets into a category, where the objects are the affine algebraic sets and the morphisms are the regular maps. The affine varieties is a subcategory of the category of the algebraic sets. Given a regular map g from V to V′ and a regular function f of k[V′], then .
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The modern approaches to algebraic geometry redefine and effectively extend the range of basic objects in various levels of generality to schemes, formal schemes, ind-schemes, algebraic spaces, algebraic stacks and so on. The need for this arises already from the useful ideas within theory of varieties, e.g. the formal functions of Zariski can be accommodated by introducing nilpotent elements in structure rings; considering spaces of loops and arcs, constructing quotients by group actions and developing formal grounds for natural intersection theory and deformation theory lead to some of the further extensions. Most remarkably, in late 1950s, algebraic varieties were subsumed into Alexander Grothendieck's concept of a scheme.
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Shafarevich made fundamental contributions to several parts of mathematics including algebraic number theory, algebraic geometry and arithmetic algebraic geometry. In algebraic number theory, the Shafarevich–Weil theorem extends the commutative reciprocity map to the case of Galois groups, which are extensions of abelian groups by finite groups. Shafarevich was the first to give a completely self-contained formula for the pairing, which coincides with the wild Hilbert symbol on local fields, thus initiating an important branch of the study of explicit formulas in number theory. Another famous result is Shafarevich's theorem on solvable Galois groups, giving the realization of every finite solvable group as a Galois group over the rationals.
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In algebraic geometry, a localized Chern class is a variant of a Chern class, that is defined for a chain complex of vector bundles as opposed to a single vector bundle. It was originally introduced in Fulton's intersection theory, as an algebraic counterpart of the similar construction in algebraic topology. The notion is used in particular in the Riemann–Roch-type theorem. S. Bloch later generalized the notion in the context of arithmetic schemes (schemes over a Dedekind domain) for the purpose of giving #Bloch's conductor formula that computes the non-constancy of Euler characteristic of a degenerating family of algebraic varieties (in the mixed characteristic case).
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Waring proved the fundamental theorem of symmetric polynomials, and specially considered the relation between the roots of a quartic equation and its resolvent cubic. Mémoire sur la résolution des équations (Memoire on the Solving of Equations) of Alexandre Vandermonde (1771) developed the theory of symmetric functions from a slightly different angle, but like Lagrange, with the goal of understanding solvability of algebraic equations. Paolo Ruffini was the first person to develop the theory of permutation groups, and like his predecessors, also in the context of solving algebraic equations. His goal was to establish the impossibility of an algebraic solution to a general algebraic equation of degree greater than four.
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It is usually not the case that the general solution of an algebraic differential equation is an algebraic function: solving equations typically produces novel transcendental functions. The case of algebraic solutions is however of considerable interest; the classical Schwarz list deals with the case of the hypergeometric equation. In differential Galois theory the case of algebraic solutions is that in which the differential Galois group G is finite (equivalently, of dimension 0, or of a finite monodromy group for the case of Riemann surfaces and linear equations). This case stands in relation with the whole theory roughly as invariant theory does to group representation theory.
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Algebraic number: Any number that is the root of a non- zero polynomial with rational coefficients. Transcendental number: Any real or complex number that is not algebraic. Examples include and . Trigonometric number: Any number that is the sine or cosine of a rational multiple of pi.
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The fundamental group of a root system is defined, in analogy to the computation for Lie groups. This allows to define and use the fundamental group of a semisimple linear algebraic group G, which is a useful basic tool in the classification of linear algebraic groups.
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Georges-Henri Halphen (; 30 October 1844, Rouen – 23 May 1889, Versailles) was a French mathematician. He was known for his work in geometry, particularly in enumerative geometry and the singularity theory of algebraic curves, in algebraic geometry. He also worked on invariant theory and projective differential geometry.
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In its logarithmic form it is the following conjecture. Let λ1, λ2, and λ3 be any three logarithms of algebraic numbers and γ be a non-zero algebraic number, and suppose that λ1λ2 = γλ3. Then λ1λ2 = γλ3 = 0\. The exponential form of this conjecture is the following.
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The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques not available to Fermat.
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This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e.
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William Edward Hodgson Berwick (11 March 1888 in Dudley Hill, Bradford - 13 May 1944 in Bangor, Gwynedd) was a British mathematician, specializing in algebra, who worked on the problem of computing an integral basis for the algebraic integers in a simple algebraic extension of the rationals.
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"Laplacian of graphs and algebraic connectivity", Combinatorics and Graph Theory (Warsaw, 1987), Banach Center Publications 25(1) (1989), 57–70. In his honor the eigenvector associated with the algebraic connectivity has been named the Fiedler vector. The Fiedler vector can be used to partition a graph.
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Using the Lie correspondence between Lie groups and Lie algebras, the notion of a real form can be defined for Lie groups. In the case of linear algebraic groups, the notions of complexification and real form have a natural description in the language of algebraic geometry.
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Georgescu George. Department of Mathematics and Informatics of the Bucharest His student also published extensive, original work on algebraic logic, MV-algebra, algebra, algebraic topology, categories of MV-algebras, category theory and Łukasiewicz–Moisil algebra.Algebraic Mathematics and Logics. 2009. GNUL contributed book of 500+ contributing authors.
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On the other hand, limits in general, and integrals in particular, are typically excluded. If an analytic expression involves only the algebraic operations (addition, subtraction, multiplication, division, and exponentiation to a rational exponent) and rational constants then it is more specifically referred to as an algebraic expression.
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Igor Rostislavovich Shafarevich (; 3 June 1923 – 19 February 2017) was a Russian mathematician who contributed to algebraic number theory and algebraic geometry. He wrote books and articles that criticised socialism, and he was an important dissident during the Soviet regime. Shafarevich died at the age of 93.
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A conjectural example in the theory of motives is the so-called motivic t-structure. Its (conjectural) existence is closely related to certain standard conjectures on algebraic cycles and vanishing conjectures, such as the Beilinson-Soulé conjecture.Hanamura, Masaki. Mixed motives and algebraic cycles. III. Math. Res. Lett.
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Max Noether (24 September 1844 – 13 December 1921) was a German mathematician who worked on algebraic geometry and the theory of algebraic functions. He has been called "one of the finest mathematicians of the nineteenth century".Lederman, p. 69. He was the father of Emmy Noether.
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Algebraic groups which are not affine behave very differently. In particular, a smooth connected group scheme which is a projective variety over a field is called an abelian variety. In contrast to linear algebraic groups, every abelian variety is commutative. Nonetheless, abelian varieties have a rich theory.
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Every sufficiently small local patch of an algebraic manifold is isomorphic to km where k is the ground field. Equivalently the variety is smooth (free from singular points). The Riemann sphere is one example of a complex algebraic manifold, since it is the complex projective line.
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Once a logic is assigned to a level of this hierarchy, one may draw on the powerful arsenal of results, accumulated over the past 30-odd years, governing the algebras situated at the same level of the hierarchy. The above terminology can be misleading. 'Abstract Algebraic Logic' is often used to indicate the approach of the Hungarian School including Hajnal Andréka, István Németi and others. What is termed 'Abstract Algebraic Logic' in the above paragraphs should be 'Algebraic Logic'.
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Given a polynomial, it may be that some of the roots are connected by various algebraic equations. For example, it may be that for two of the roots, say and , that . The central idea of Galois' theory is to consider permutations (or rearrangements) of the roots such that any algebraic equation satisfied by the roots is still satisfied after the roots have been permuted. Originally, the theory had been developed for algebraic equations whose coefficients are rational numbers.
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Karen Ellen Smith (born 1965, Red Bank, New Jersey) is an American mathematician, specializing in commutative algebra and algebraic geometry. She completed her bachelor's degree in mathematics at Princeton University before earning her PhD in mathematics at the University of Michigan in 1993. Currently she is the Keeler Professor of Mathematics at the University of Michigan. In addition to being a researcher in algebraic geometry and commutative algebra, Smith with others wrote the textbook An Invitation to Algebraic Geometry.
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Algebraic holography, also sometimes called Rehren duality, is an attempt to understand the holographic principle of quantum gravity within the framework of algebraic quantum field theory, due to Karl-Henning Rehren. It is sometimes described as an alternative formulation of the AdS/CFT correspondence of string theory, but some string theorists reject this statement . The theories discussed in algebraic holography do not satisfy the usual holographic principle because their entropy follows a higher-dimensional power law.
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Commutative algebraic geometry begins by constructing the spectrum of a ring. The points of the algebraic variety (or more generally, scheme) are the prime ideals of the ring, and the functions on the algebraic variety are the elements of the ring. A noncommutative ring, however, may not have any proper non-zero two-sided prime ideals. For instance, this is true of the Weyl algebra of polynomial differential operators on affine space: The Weyl algebra is a simple ring.
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The value-level approach to programming invites the study of the space of values under the value-forming operations, and of the algebraic properties of those operations. This is what is called the study of data types, and it has advanced from focusing on the values themselves and their structure, to a primary concern with the value-forming operations and their structure, as given by certain axioms and algebraic laws, that is, to the algebraic study of data types.
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104 The modular function j(τ) is algebraic on imaginary quadratic numbers τ:Serre (1967) p. 293 these are the only algebraic numbers in the upper half-plane for which j is algebraic. If Λ is a lattice with period ratio τ then we write j(Λ) for j(τ). If further Λ is an ideal a in the ring of integers OK of a quadratic imaginary field K then we write j(a) for the corresponding singular modulus.
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This work also gave rise to the ideas of an algebraic space and algebraic stack, and has proved very influential in moduli theory. Additionally, he has made important contributions to the deformation theory of algebraic varieties. With Peter Swinnerton-Dyer, he provided a resolution of the Shafarevich-Tate conjecture for elliptic K3 surfaces and the pencil of elliptic curves over finite fields. Artin contributed to the theory of surface singularities which are both fundamental and seminal.
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Algebra tiles allow both an algebraic and geometric approach to algebraic concepts. They give students another way to solve algebraic problems other than just abstract manipulation. The National Council of Teachers of Mathematics (NCTM) recommends a decreased emphasis on the memorization of the rules of algebra and the symbol manipulation of algebra in their Curriculum and Evaluation Standards for Mathematics. According to the NCTM 1989 standards "[r]elating models to one another builds a better understanding of each".
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The exact inner product used does not matter, because a different inner product will give an equivalent norm on , and so give an equivalent metric. If the ground field is arbitrary and is considered as an algebraic group, then this construction shows that the Grassmannian is a non- singular algebraic variety. It follows from the existence of the Plücker embedding that the Grassmannian is complete as an algebraic variety. In particular, is a parabolic subgroup of .
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In mathematics, directed algebraic topology is a refinement of algebraic topology for directed spaces, topological spaces and their combinatorial counterparts equipped with some notion of direction. Some common examples of directed spaces are spacetimes and simplicial sets. The basic goal is to find algebraic invariants that classify directed spaces up to directed analogues of homotopy equivalence. For example, homotopy groups and fundamental n-groupoids of spaces generalize to homotopy monoids and fundamental n-categories of directed spaces.
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Results and definitions stated below, for affine varieties, can be translated to projective varieties, by replacing An(kalg) with projective space of dimension n − 1 over kalg, and by insisting that all polynomials be homogeneous. A k-algebraic set is the zero-locus in An(kalg) of a subset of the polynomial ring k[x1, …, xn]. A k-variety is a k-algebraic set that is irreducible, i.e. is not the union of two strictly smaller k-algebraic sets.
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Such techniques of applying geometrical constructions to algebraic problems were also adopted by a number of Renaissance mathematicians such as Gerolamo Cardano and Niccolò Fontana "Tartaglia" on their studies of the cubic equation. The geometrical approach to construction problems, rather than the algebraic one, was favored by most 16th and 17th century mathematicians, notably Blaise Pascal who argued against the use of algebraic and analytical methods in geometry. The French mathematicians Franciscus Vieta and later René Descartes and Pierre de Fermat revolutionized the conventional way of thinking about construction problems through the introduction of coordinate geometry. They were interested primarily in the properties of algebraic curves, such as those defined by Diophantine equations (in the case of Fermat), and the algebraic reformulation of the classical Greek works on conics and cubics (in the case of Descartes).
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The study of algebraic curves can be reduced to the study of irreducible algebraic curves: those curves that cannot be written as the union of two smaller curves. Up to birational equivalence, the irreducible curves over a field F are categorically equivalent to algebraic function fields in one variable over F. Such an algebraic function field is a field extension K of F that contains an element x which is transcendental over F, and such that K is a finite algebraic extension of F(x), which is the field of rational functions in the indeterminate x over F. For example, consider the field C of complex numbers, over which we may define the field C(x) of rational functions in C. If y2 = x3 − x − 1, then the field C(x, y) is an elliptic function field. The element x is not uniquely determined; the field can also be regarded, for instance, as an extension of C(y). The algebraic curve corresponding to the function field is simply the set of points (x, y) in C2 satisfying y2 = x3 − x − 1\.
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In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field.... For example, x^2+y^2-1 is absolutely irreducible, but while x^2+y^2 is irreducible over the integers and the reals, it is reducible over the complex numbers as x^2+y^2 = (x+iy)(x-iy), and thus not absolutely irreducible. More generally, a polynomial defined over a field K is absolutely irreducible if it is irreducible over every algebraic extension of K,. and an affine algebraic set defined by equations with coefficients in a field K is absolutely irreducible if it is not the union of two algebraic sets defined by equations in an algebraically closed extension of K. In other words, an absolutely irreducible algebraic set is a synonym of an algebraic variety,. which emphasizes that the coefficients of the defining equations may not belong to an algebraically closed field. Absolutely irreducible is also applied, with the same meaning to linear representations of algebraic groups.
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Sometimes the term "inductive data type" is used for algebraic data types which are not necessarily recursive.
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The usual algebraic constructions for ordinary vector spaces have their counterpart in the super vector space setting.
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The topics of his research include geometric invariant theory and moduli of vector bundles over algebraic curves.
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1\. Fundamentals of the theory of quasigroups and loops. M .: Nauka, 1967. 2\. Algebraic networks and quasigroups.
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S-equivalence is an equivalence relation on the families of semistable vector bundles on an algebraic curve.
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Alexandru Dimca is a Romanian mathematician, who works in algebraic geometry at University of Nice Sophia Antipolis.
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J. Rhodes, Keynote talk at the International Conference on Semigroups & Algebraic Engineering (Aizu, Japan), 26 March 1997.
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The FX-502P series use the algebraic logic as was state of the art at the time.
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Michael Jerome Hopkins (born April 18, 1958) is an American mathematician known for work in algebraic topology.
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The Algebraic Reconstruction Technique (ART) was the first iterative reconstruction technique used for computed tomography by Hounsfield.
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Christophe Breuil (; born 1968) is a French mathematician, who works in arithmetic geometry and algebraic number theory.
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If matter is supplied at several places we have to take the algebraic sum of these contributions.
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For example, any complex algebraic variety X, with its classical (Euclidean) topology, is compactifiable in this sense.
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The Hardy-Weinberg law describes the genetic equilibrium in a population by means of an algebraic equation.
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As in algebraic notation, each cell is identified by a letter+number combination. are horizontal and identified by numbers 1–8. are straight and 30° oblique to the vertical, identified by letters a–l. Moves can be recorded in long algebraic notation (LAN) to avoid confusion, for example: 1.
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Ivan Cherednik (Иван Владимирович Чередник) is a Russian mathematician. He introduced double affine Hecke algebras, and used them to prove Macdonald's constant term conjecture in . He has also dealt with algebraic geometry, number theory and Soliton equations. His research interests include representation theory, mathematical physics, and algebraic combinatorics.
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Grothendieck & Raynaud, SGA 1, Exposé XII. (The first group here is defined using the Zariski topology, and the second using the classical (Euclidean) topology.) For example, the equivalence between algebraic and analytic coherent sheaves on projective space implies Chow's theorem that every closed analytic subspace of CPn is algebraic.
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Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study.
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In mathematics, the Hodge bundle, named after W. V. D. Hodge, appears in the study of families of curves, where it provides an invariant in the moduli theory of algebraic curves. Furthermore, it has applications to the theory of modular forms on reductive algebraic groups and string theory.
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Cyclic cohomology of the commutative algebra A of regular functions on an affine algebraic variety over a field k of characteristic zero can be computed in terms of Grothendieck's algebraic de Rham complex.Boris L. Fegin and Boris L. Tsygan. Additive K-theory and crystalline cohomology. Funktsional. Anal. i Prilozhen.
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Available on-line at: Mocavo.com hence founding the field of algebraic topology. In 1916 Oswald Veblen applied the algebraic topology of Poincaré to Kirchhoff's analysis.Oswald Veblen, The Cambridge Colloquium 1916, (New York : American Mathematical Society, 1918-1922), vol 5, pt. 2 : Analysis Situs, "Matrices of orientation", pp. 25-27.
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Pointwise operations inherit such properties as associativity, commutativity and distributivity from corresponding operations on the codomain. If A is some algebraic structure, the set of all functions X to the carrier set of A can be turned into an algebraic structure of the same type in an analogous way.
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There are no further algebraic constraints (by definition). In particular, the x_i cannot (must not) be taken to be matrices or other algebraic objects; they are only symbols, devoid of further properties. Strings of symbols, such as x_1^2 x_2 x_1^6 x_3^2, cannot be further reduced.
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In mathematics, in the theory of algebraic curves, a delta invariant measures the number of double points concentrated at a point.John Milnor, Singular Points of Hypersurfaces, p. 85 It is a non-negative integer. Delta invariants are discussed in the "Classification of singularities" section of the algebraic curve article.
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In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a and its absolute Galois group. It is an interdisciplinary subject as it uses tools from algebraic number theory, arithmetic geometry, algebraic geometry, model theory, the theory of finite groups and of profinite groups.
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In either algebraic or descriptive chess notation, en passant captures are sometimes denoted by "e.p." or similar, but such notation is not required. In algebraic notation, the capturing move is written as if the captured pawn advanced only one square; for example, ...bxa3 or ...bxa3e.p. (see beginning illustration).
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David Eisenbud (left), Frank-Olaf Schreyer (middle), Joseph Daniel Harris (right), Oberwolfach 2006 Frank-Olaf Schreyer is a German mathematician, specializing in algebraic geometry and algorithmic algebraic geometry. Schreyer received in 1983 his PhD from Brandeis University with thesis Syzgies of Curves with Special Pencils under the supervision of David Eisenbud. Schreyer was a professor at University of Bayreuth and is since 2002 a professor at Saarland University. He is involved in the development of (algorithmic) algebraic geometry advanced by David Eisenbud.
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In the context of algebraic geometry, the notion of branch points can be generalized to mappings between arbitrary algebraic curves. Let ƒ:X → Y be a morphism of algebraic curves. By pulling back rational functions on Y to rational functions on X, K(X) is a field extension of K(Y). The degree of ƒ is defined to be the degree of this field extension [K(X):K(Y)], and ƒ is said to be finite if the degree is finite.
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The Lisp Algebraic Manipulator (also known as LAM) was created by Ray d'Inverno, who had written Atlas LISP Algebraic Manipulation (ALAM was designed in 1970).Entry at hopl.murdoch.edu.au Computer Algebra: from the Visible to the Invisible, R. A. d'Inverno, General Relativity and Gravitation, Volume 38, Number 6, June 2006 Algebraic computing in general relativity, Raymon A. d'Inverno, General Relativity and Gravitation, Volume 6, Number 6, December, 1975Entry at people.ku.edu LAM later became the basis for the interactive computer package SHEEP.
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The field F is algebraically closed if and only if it has no proper algebraic extension. If F has no proper algebraic extension, let p(x) be some irreducible polynomial in F[x]. Then the quotient of F[x] modulo the ideal generated by p(x) is an algebraic extension of F whose degree is equal to the degree of p(x). Since it is not a proper extension, its degree is 1 and therefore the degree of p(x) is 1\.
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A topological field of sets is called algebraic if and only if there is a base for its topology consisting of complexes. If a topological field of sets is both compact and algebraic then its topology is compact and its compact open sets are precisely the open complexes. Moreover, the open complexes form a base for the topology. Topological fields of sets that are separative, compact and algebraic are called Stone fields and provide a generalization of the Stone representation of Boolean algebras.
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In algebraic geometry, a mixed Hodge structure is an algebraic structure containing information about the cohomology of general algebraic varieties. It is a generalization of a Hodge structure, which is used to study smooth projective varieties. In mixed Hodge theory, where the decomposition of a cohomology group H^k(X) may have subspaces of different weights, i.e. as a direct sum of Hodge structures :H^k(X) = \bigoplus_i (H_i, F_i^\bullet) where each of the Hodge structures have weight k_i.
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Gabriele Vezzosi (born 1966) is an Italian mathematician, born in Florence (Italy). His main interest is algebraic geometry. Vezzosi earned an MS degree in Physics at the University of Florence, under the supervision of Alexandre M. Vinogradov, and a PhD in Mathematics at the Scuola Normale Superiore in Pisa, under the supervision of Angelo Vistoli. His first papers dealt with differential calculus over commutative rings, intersection theory, (equivariant) algebraic K-theory, motivic homotopy theory, and existence of vector bundles on singular algebraic surfaces.
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In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to two Austrian mathematicians, Walther Mayer and Leopold Vietoris. The method consists of splitting a space into subspaces, for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces.
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In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.
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The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. Certain special classes of manifolds also have additional algebraic structure; they may behave like groups, for instance. In that case, they are called Lie Groups. Alternatively, they may be described by polynomial equations, in which case they are called algebraic varieties, and if they additionally carry a group structure, they are called algebraic groups.
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One way that leads to generalizations is to allow reducible algebraic sets (and fields that aren't algebraically closed), so the rings R may not be integral domains. A more significant modification is to allow nilpotents in the sheaf of rings, that is, rings which are not reduced. This is one of several generalizations of classical algebraic geometry that are built into Grothendieck's theory of schemes. Allowing nilpotent elements in rings is related to keeping track of "multiplicities" in algebraic geometry.
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Bernd Siebert (born 5 March 1964 in Berlin-Wilmersdorf) is a German mathematician who researches in algebraic geometry.
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Porteous wrote his thesis Algebraic Geometry under W.V.D. Hodge and Michael Atiyah at University of Cambridge in 1961.
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Much of algebraic geometry and complex analytic geometry is formulated in terms of coherent sheaves and their cohomology.
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Marino Pannelli (16 November 1855, Macerata – 16 April 1934, Macerata) was an Italian mathematician, specializing in algebraic geometry.
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An algebraic lattice is complete. (def) 10\. A complete lattice is bounded. 11\. A heyting algebra is bounded.
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Oberwolfach in 2004 Ofer Gabber (עופר גאבר; born May 16, 1958) is a mathematician working in algebraic geometry.
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Algebraization of Gentzen systems by Ramon Jansana, J. Font and others is a significant improvement over 'algebraic logic'.
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Prakash Belkale (born May 1973, Bangalore) is an Indian-American mathematician, specializing in algebraic geometry and representation theory.
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Lawrence Ein Lawrence Man Hou Ein (born 18 November 1955) is a mathematician who works in algebraic geometry.
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In mathematics, a real plane curve is usually a real algebraic curve defined in the real projective plane.
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Functions of finite support are used in defining algebraic structures such as group rings and free abelian groups.
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Examples are homomorphisms, which preserve algebraic structures; homeomorphisms, which preserve topological structures; and diffeomorphisms, which preserve differential structures.
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Calegari works in algebraic number theory, including Langlands reciprocity and torsion classes in the cohomology of arithmetic groups.
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J. London Math. Soc. (2) 8 (1974) 539–544. # (with R. A. Main) ‘Non- algebraic matroids exist’. Bull.
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Springer Science & Business Media.Albrecht, R. (2012). Computer algebra: symbolic and algebraic computation (Vol. 4). Springer Science & Business Media.
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13, pp. 147–158., algebraic-geometric codes and even general linear block codesS. Bulygin and R. Pellikaan (2009).
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In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought.
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Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients). For example, noncommutative algebraic geometry is supposed to extend a notion of an algebraic scheme by suitable gluing of spectra of noncommutative rings; depending on how literally and how generally this aim (and a notion of spectrum) is understood in noncommutative setting, this has been achieved in various level of success. The noncommutative ring generalizes here a commutative ring of regular functions on a commutative scheme.
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In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety X embedded in projective space and a hyperplane section Y, the homology, cohomology, and homotopy groups of X determine those of Y. A result of this kind was first stated by Solomon Lefschetz for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories. A far-reaching generalization of the hard Lefschetz theorem is given by the decomposition theorem.
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What is true for the generic point is true for "most" points of the variety. In Weil's Foundations of Algebraic Geometry (1946), generic points are constructed by taking points in a very large algebraically closed field, called a universal domain. Although this worked as a foundation, it was awkward: there were many different generic points for the same variety. (In the later theory of schemes, each algebraic variety has a single generic point.) In the 1950s, Claude Chevalley, Masayoshi Nagata and Jean-Pierre Serre, motivated in part by the Weil conjectures relating number theory and algebraic geometry, further extended the objects of algebraic geometry, for example by generalizing the base rings allowed.
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Indeed, there is an equivalence of categories between the category of smooth projective algebraic curves over R (with regular maps as morphisms) and the category of compact connected Klein surfaces. This is akin to the correspondence between smooth projective algebraic curves over the complex numbers and compact connected Riemann surfaces. (Note that the algebraic curves considered here are abstract curves: integral, separated one-dimensional schemes of finite type over R. Such a curve need not have any R-rational points (like the curve X2+Y2+1=0 over R), in which case its Klein surface will have empty boundary.) There is also a one-to-one correspondence between compact connected Klein surfaces (up to equivalence) and algebraic function fields in one variable over R (up to R-isomorphism). This correspondence is akin to the one between compact connected Riemann surfaces and algebraic function fields over the complex numbers.
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The main directions of I. A. Panin's work are the theory of oriented cohomology on algebraic varieties, algebraic K-theory of homogeneous varieties, Gersten's conjecture, the Grothendieck-Serre conjecture on principal G-bundles, and purity in algebraic geometry.. I. A. Panin proved (together with A. L. Smirnov) theorems of the Riemann-Roch type for oriented cohomology theories and Riemann-Roch type theorems for the Adams operation. Panin found a proof of Gersten's conjecture in the case of equal characteristic and an affirmative solution (jointly with Manuel Ojanguren) of the "purity" problem for quadratic forms. Panin computed the algebraic K-groups of all twisted forms of flag varieties and all principal homogeneous spaces over the inner forms of semisimple algebraic groups. He, jointly with A. S. Merkurjev and A. R. Wadsworth, generalized, to arbitrary Borel varieties, results proved by David Tao concerning index reduction formulas for the function fields of involution varieties.
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More precisely, let V be an algebraic variety over K (assumptions here are: V is an irreducible set, a quasi-projective variety, and K has characteristic zero). A type I thin set is a subset of V(K) that is not Zariski-dense. That means it lies in an algebraic set that is a finite union of algebraic varieties of dimension lower than d, the dimension of V. A type II thin set is an image of an algebraic morphism (essentially a polynomial mapping) φ, applied to the K-points of some other d-dimensional algebraic variety V′, that maps essentially onto V as a ramified covering with degree e > 1. Saying this more technically, a thin set of type II is any subset of :φ(V′(K)) where V′ satisfies the same assumptions as V and φ is generically surjective from the geometer's point of view.
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The main contribution of Moshe Jarden in algebra, and in mathematics in general, is his research on families of large algebraic extensions of Hilbertian fields (in particular global fields), parametrized by the automorphisms of the absolute Galois group of the base field. Notable results in this domain are the zero theorem, the transfer theorem, the free generators theorem, the Frey-Jarden theorem about the rank of algebraic varieties over large algebraic fields, Geyer-Jarden theorem about torsion points on elliptic curves over large algebraic fields, and the strong approximation theorem over such fields. The remarkable development of Galois theory over a class of large fields, called ample fields,These fields, previously called large fields by Florian Pop, were so named by M. Jarden because the term large fields was already used in another context. is described in the second book of Jarden: Algebraic Patching.
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Following on from the example, in algebraic geometry one studies algebraic sets, i.e. subsets of Kn (where K is an algebraically closed field) that are defined as the common zeros of a set of polynomials in n variables. If A is such an algebraic set, one considers the commutative ring R of all polynomial functions A → K. The maximal ideals of R correspond to the points of A (because K is algebraically closed), and the prime ideals of R correspond to the subvarieties of A (an algebraic set is called irreducible or a variety if it cannot be written as the union of two proper algebraic subsets). The spectrum of R therefore consists of the points of A together with elements for all subvarieties of A. The points of A are closed in the spectrum, while the elements corresponding to subvarieties have a closure consisting of all their points and subvarieties.
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Algebraic Combinatorics is a peer-reviewed open access mathematical journal specializing in the field of algebraic combinatorics. It is published by the Centre Mersenne. The editors-in-chief are Akihiro Munemasa (Tohoku University), Satoshi Murai (Waseda University), Hugh Thomas (Université du Québec à Montréal), and Hendrik Van Maldeghem (Ghent University).
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A lattice is called algebraic if it is complete and compactly generated. In 1963, Grätzer and Schmidt proved that every algebraic lattice is isomorphic to the congruence lattice of some algebra.G. Grätzer and E. T. Schmidt, Characterizations of congruence lattices of abstract algebras, Acta Sci. Math. (Szeged) 24 (1963), 34–59.
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In algebraic geometry, the Néron–Severi group of a variety is the group of divisors modulo algebraic equivalence; in other words it is the group of components of the Picard scheme of a variety. Its rank is called the Picard number. It is named after Francesco Severi and André Néron.
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In mathematics, a Severi variety is an algebraic variety in a Hilbert scheme that parametrizes curves in projective space with given degree and geometric genus and at most node singularities. Its dimension is 3d + g − 1\. It is a theorem that Severi varieties are algebraic varieties, i.e. it is irreducible.
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Kirwan's research interests include moduli spaces in algebraic geometry, geometric invariant theory (GIT), and in the link between GIT and moment maps in symplectic geometry.Prof Kirwan profile , europeanwomeninmaths.org; accessed 9 May 2014. Her work endeavours to understand the structure of geometric objects by investigation of their algebraic and topological properties.
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However, the word fraction can also be used to describe mathematical expressions that are not rational numbers. Examples of these usages include algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as /2 (see square root of 2) and π/4 (see proof that π is irrational).
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Pantelides algorithm gives a systematic method for reducing high-index systems of differential-algebraic equations to lower index, by selectively adding differentiated forms of the equations already present in the system.C Pantelides, The Consistent Initialization of Differential-Algebraic Systems, SIAM J. Sci. and Stat. Comput. Volume 9, Issue 2, pp.
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175 The category having these branes as its objects is called the Fukaya category.Aspinwal et al. 2009, p. 575 The derived category of coherent sheaves is constructed using tools from complex geometry, a branch of mathematics that describes geometric curves in algebraic terms and solves geometric problems using algebraic equations.
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An algebraic number is a number that is a solution of a non-zero polynomial equation in one variable with rational coefficients (or equivalently — by clearing denominators — with integer coefficients). Numbers such as that are not algebraic are said to be transcendental. Almost all real and complex numbers are transcendental.
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He received the Mathematical Association of America's first Chauvenet Prize, in 1925, for his article "Algebraic functions and their divisors," which culminated in his 1933 book Algebraic functions. Bliss once headed a government commission that devised rules for apportioning seats in the U.S. House of Representatives among the several states.
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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\.
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In algebraic geometry, the cohomology of a stack is a generalization of étale cohomology. In a sense, it is a theory that is coarser than the Chow group of a stack. The cohomology of a quotient stack (e.g., classifying stack) can be thought of as an algebraic counterpart of equivariant cohomology.
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The algebraic integers in a finite extension field k of the rationals Q form a subring of k, called the ring of integers of k, a central object of study in algebraic number theory. In this article, the term ring will be understood to mean commutative ring with a multiplicative identity.
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The Fano matroid, derived from the Fano plane. Matroids are one of many areas studied in algebraic combinatorics. Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
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Sergio Doplicher, Klaus Fredenhagen, John E. Roberts: The quantum structure of spacetime at the Planck scale and quantum fields, Commun.Math.Phys. , Vol. 172, 1995, pp. 187-220 Dorothea Bahns, Sergio Doplicher, Gerardo Morsella, Gherardo Piacitelli: Quantum Spacetime and Algebraic Quantum Field Theory] , in: Advances in Algebraic Quantum Field Theory, Springer 2015, pp.
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Because the singleton is the zero object of Mag, and because Mag is algebraic, Mag is pointed and complete.
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Griffiths, P., & Harris, J. (2014). Principles of algebraic geometry. John Wiley & Sons.Wells, R. O. N., & García-Prada, O. (1980).
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Mark Edward Mahowald (December 1, 1931 – July 20, 2013) was an American mathematician known for work in algebraic topology.
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In addition, kinematics applies algebraic geometry to the study of the mechanical advantage of a mechanical system or mechanism.
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Benkart's work on noncommutative algebras related to algebraic combinatorics became a basic tool in the construction of tensor categories.
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In algebraic geometry, Reider's theorem gives conditions for a line bundle on a projective surface to be very ample.
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Charles Minshall Jessop (1861 – March 9, 1939) was a mathematician at the University of Durham working in algebraic geometry.
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By the late 1960s, computer systems could perform symbolic algebraic manipulations well enough to pass college-level calculus courses.
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Ralph Louis Cohen Ralph Louis Cohen (born 1952) is an American mathematician, specializing in algebraic topology and differential topology.
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In mathematics, noncommutative projective geometry is a noncommutative analog of projective geometry in the setting of noncommutative algebraic geometry.
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In mathematics, the Hochschild–Mostow group, introduced by , is the universal pro-affine algebraic group generated by a group.
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Eff is a functional programming language similar in syntax to OCaml which integrates the functionality of algebraic effect handlers.
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This is a list of polynomial topics, by Wikipedia page. See also trigonometric polynomial, list of algebraic geometry topics.
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Carolina Bhering de Araujo is a Brazilian mathematician specializing in algebraic geometry, including birational geometry, Fano varieties, and foliations.
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There are a number of basic definitions of K-theory: two coming from topology and two from algebraic geometry.
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Several approaches to Springer correspondence have been developed. T. A. Springer's original construction (1976) proceeded by defining an action of W on the top-dimensional l-adic cohomology groups of the algebraic variety Bu of the Borel subgroups of G containing a given unipotent element u of a semisimple algebraic group G over a finite field. This construction was generalized by Lusztig (1981), who also eliminated some technical assumptions. Springer later gave a different construction (1978), using the ordinary cohomology with rational coefficients and complex algebraic groups.
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In algebraic geometry, Chow's moving lemma, proved by , states: given algebraic cycles Y, Z on a nonsingular quasi-projective variety X, there is another algebraic cycle Z' on X such that Z' is rationally equivalent to Z and Y and Z' intersect properly. The lemma is one of key ingredients in developing the intersection theory, as it is used to show the uniqueness of the theory. Even if Z is an effective cycle, it is not, in general, possible to choose the cycle Z' to be effective.
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A generic point of the topological space X is a point P whose closure is all of X, that is, a point that is dense in X.David Mumford, The Red Book of Varieties and Schemes, Springer 1999 The terminology arises from the case of the Zariski topology on the set of subvarieties of an algebraic set: the algebraic set is irreducible (that is, it is not the union of two proper algebraic subsets) if and only if the topological space of the subvarieties has a generic point.
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In mathematics, homologyin part from Greek ὁμός homos "identical" is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes.
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In mathematics, a J-structure is an algebraic structure over a field related to a Jordan algebra. The concept was introduced by to develop a theory of Jordan algebras using linear algebraic groups and axioms taking the Jordan inversion as basic operation and Hua's identity as a basic relation. There is a classification of simple structures deriving from the classification of semisimple algebraic groups. Over fields of characteristic not equal to 2, the theory of J-structures is essentially the same as that of Jordan algebras.
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Given a polynomial, it may be that some of the roots are connected by various algebraic equations. For example, it may be that for two of the roots, say A and B, that . The central idea of Galois theory is to consider those permutations (or rearrangements) of the roots having the property that any algebraic equation satisfied by the roots is still satisfied after the roots have been permuted. An important proviso is that we restrict ourselves to algebraic equations whose coefficients are rational numbers.
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An algebraic dual of a connected graph is a graph such that and have the same set of edges, any cycle of is a cut of , and any cut of is a cycle of . Every planar graph has an algebraic dual, which is in general not unique (any dual defined by a plane embedding will do). The converse is actually true, as settled by Hassler Whitney in Whitney's planarity criterion:. : A connected graph is planar if and only if it has an algebraic dual.
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The conventional term Hodge cycle therefore is slightly inaccurate, in that x is considered as a class (modulo boundaries); but this is normal usage. The importance of Hodge cycles lies primarily in the Hodge conjecture, to the effect that Hodge cycles should always be algebraic cycles, for V a complete algebraic variety. This is an unsolved problem, ; it is known that being a Hodge cycle is a necessary condition to be an algebraic cycle that is rational, and numerous particular cases of the conjecture are known.
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For ensuring more regularity, the function that defines a curve is often supposed to be differentiable, and the curve is then said to be a differentiable curve. A plane algebraic curve is the zero set of a polynomial in two indeterminates. More generally, an algebraic curve is the zero set of a finite set of polynomials, which satisfies the further condition of being an algebraic variety of dimension one. If the coefficients of the polynomials belong to a field , the curve is said to be defined over .
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He began his academic career at the Tata Institute of Fundamental Research, Mumbai as a Visiting Fellow in 1983. Srinivas has worked mainly in algebraic geometry specialising in the study of algebraic cycles on singular algebraic varieties. He has also worked on the interface with commutative algebra: on projective modules, divisor class groups, unique factorization domains, and Hilbert functions and multiplicity. He was awarded the Shanti Swarup Bhatnagar Prize for Science and Technology in 2003, the highest science award in India, in the mathematical sciences category.
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An A-structure on a PL manifold is a structure which gives an inductive way of resolving the PL manifold to a smooth manifold. Compact PL manifolds admit A-structures. Compact PL manifolds are homeomorphic to real-algebraic sets. Put another way, A-category sits over the PL-category as a richer category with no obstruction to lifting, that is BA → BPL is a product fibration with BA = BPL × PL/A, and PL manifolds are real algebraic sets because A-manifolds are real algebraic sets.
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Linear algebraic groups (or more generally, affine group schemes) are analogues in algebraic geometry of Lie groups, but over more general fields than just R or C. In particular, over finite fields, they give rise to finite groups of Lie type. Although linear algebraic groups have a classification that is very similar to that of Lie groups, their representation theory is rather different (and much less well understood) and requires different techniques, since the Zariski topology is relatively weak, and techniques from analysis are no longer available., .
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The rational singularity and fundamental cycle are such examples of his sheer originality and thinking. He began to turn his interest from algebraic geometry to noncommutative algebra (noncommutative ring theory), especially geometric aspects, after a talk by Shimshon Amitsur and an encounter in Chicago with Claudio Procesi and Lance W. Small, "which prompted [his] first foray into ring theory".From the MacTutor biography: "His main research area changed from algebraic geometry to noncommutative ring theory". Today, he is a recognized world leader in noncommutative algebraic geometry.
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The most common type of equation is an algebraic equation, in which the two sides are algebraic expressions. Each side of an algebraic equation will contain one or more terms. For example, the equation : Ax^2 +Bx + C = y has left-hand side Ax^2 +Bx + C , which has three terms, and right-hand side y , consisting of just one term. The unknowns are x and y, and the parameters are A, B, and C. An equation is analogous to a scale into which weights are placed.
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In the same period began the algebraization of the algebraic geometry through commutative algebra. The prominent results in this direction are Hilbert's basis theorem and Hilbert's Nullstellensatz, which are the basis of the connexion between algebraic geometry and commutative algebra, and Macaulay's multivariate resultant, which is the basis of elimination theory. Probably because of the size of the computation which is implied by multivariate resultants, elimination theory was forgotten during the middle of the 20th century until it was renewed by singularity theory and computational algebraic geometry.
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It appeared in print in a paper written by Armand Borel with Serre. This result was his first work in algebraic geometry. He went on to plan and execute a programme for rebuilding the foundations of algebraic geometry, which were then in a state of flux and under discussion in Claude Chevalley's seminar; he outlined his programme in his talk at the 1958 International Congress of Mathematicians. His foundational work on algebraic geometry is at a higher level of abstraction than all prior versions.
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In algebraic geometry, a formal holomorphic function along a subvariety V of an algebraic variety W is an algebraic analog of a holomorphic function defined in a neighborhood of V. They are sometimes just called holomorphic functions when no confusion can arise. They were introduced by . The theory of formal holomorphic functions has largely been replaced by the theory of formal schemes which generalizes it: a formal holomorphic function on a variety is essentially just a section of the structure sheaf of a related formal scheme.
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In mathematics, in particular, in algebraic geometry, an isogeny is a morphism of algebraic groups (a.k.a group varieties) that is surjective and has a finite kernel. If the groups are abelian varieties, then any morphism f : A → B of the underlying algebraic varieties which is surjective with finite fibres is automatically an isogeny, provided that f(1A) = 1B. Such an isogeny f then provides a group homomorphism between the groups of k-valued points of A and B, for any field k over which f is defined.
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Galina Nikolaevna Tyurina (July 19, 1938 – July 21, 1970) was a Soviet mathematician specializing in algebraic geometry. Despite dying young, she was known for "a series of brilliant papers" on the classification of complex or algebraic structures on topological spaces, on K3 surfaces, on singular points of algebraic varieties, and on the rigidity of complex structures. She was the only woman among a group of "exceptionally brilliant" Soviet mathematicians who became active in the 1960s and "quickly became the leaders and the driving forces of Soviet mathematics".
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With Israel Gelfand and Andrei Zelevinsky, Kapranov investigated generalized Euler integrals, A-hypergeometric functions, A-discriminants, and hyperdeterminants, and authored Discriminants, Resultants, and Multidimensional Determinants in 1994. According to Gelfand, Kapranov, and Zelevinsky: In 1995 Kapranov provided a framework for a Langlands program for higher-dimensional schemes, and with, Victor Ginzburg and Eric Vasserot, extended the "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. In 1998 Kapranov was an Invited Speaker with talk Operads and Algebraic Geometry at the International Congress of Mathematicians in Berlin.
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In mathematics, the Burnside ring of a finite group is an algebraic construction that encodes the different ways the group can act on finite sets. The ideas were introduced by William Burnside at the end of the nineteenth century. The algebraic ring structure is a more recent development, due to Solomon (1967).
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Vector logicMizraji, E. (1992). Vector logics: the matrix-vector representation of logical calculus. Fuzzy Sets and Systems, 50, 179–185Mizraji, E. (2008) Vector logic: a natural algebraic representation of the fundamental logical gates. Journal of Logic and Computation, 18, 97–121 is an algebraic model of elementary logic based on matrix algebra.
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Thus the dreams of the early topologists have long been regarded as a mirage. Cubical higher homotopy groupoids are constructed for filtered spaces in the book Nonabelian algebraic topology cited below, which develops basic algebraic topology, including higher analogues to the Seifert–van Kampen theorem, without using singular homology or simplicial approximation.
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In mathematics, in the field of algebraic number theory, a modulus (plural moduli) (or cycle, or extended ideal) is a formal product of places of a global field (i.e. an algebraic number field or a global function field). It is used to encode ramification data for abelian extensions of a global field.
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A Dieudonné crystal is a crystal D together with homomorphisms F:Dp→D and V :D→Dp satisfying the relations VF=p (on Dp), FV=p (on D). Dieudonné crystals were introduced by . They play the same role for classifying algebraic groups over schemes that Dieudonné modules play for classifying algebraic groups over fields.
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Group schemes that are not algebraic groups play a significant role in arithmetic geometry and algebraic topology, since they come up in contexts of Galois representations and moduli problems. The initial development of the theory of group schemes was due to Alexander Grothendieck, Michel Raynaud and Michel Demazure in the early 1960s.
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The system must also include encoding methods to: address each cell; assign quantized data to cells; and perform algebraic operations on the cells and the data assigned to them. Main concepts of the DGGS Core Conceptual Data Model: # reference frame elements, and, # functional algorithm elements; comprising: ## quantization operations, ## algebraic operations, and ## interoperability operations.
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The subsequent notion of "automorphic representation" has proved of great technical value for dealing with the case that G is an algebraic group, treated as an adelic algebraic group. As a result, an entire philosophy, the Langlands program has developed around the relation between representation and number theoretic properties of automorphic forms..
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Hodge also wrote, with Daniel Pedoe, a three-volume work Methods of Algebraic Geometry, on classical algebraic geometry, with much concrete content – illustrating though what Élie Cartan called 'the debauch of indices' in its component notation. According to Atiyah, this was intended to update and replace H. F. Baker's Principles of Geometry.
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The Stacks Project is an open source collaborative mathematics textbook writing project with the aim to cover "algebraic stacks and the algebraic geometry needed to define them". , the book consists of 113 chapters spreading over 6900 pages. The maintainer of the project, who reviews and accepts the changes, is Aise Johan de Jong.
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Another theorem of his concerns the constructible sets in algebraic geometry, i.e. those in the Boolean algebra generated by the Zariski-open and Zariski-closed sets. It states that the image of such a set by a morphism of algebraic varieties is of the same type. Logicians call this an elimination of quantifiers.
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Algebraic structures occur as both discrete examples and continuous examples. Discrete algebras include: boolean algebra used in logic gates and programming; relational algebra used in databases; discrete and finite versions of groups, rings and fields are important in algebraic coding theory; discrete semigroups and monoids appear in the theory of formal languages.
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In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives. It was originally introduced by Yevsey Nisnevich, who was motivated by the theory of adeles.
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In the 1980s he turned to the application of characteristic classes and other topological and algebraic concepts in mathematical physics, first in the algebraic structure of anomalies in quantum field theory, where he worked with among others, Tom Kephart and Paolo Cotta- Ramusino. He referred to the research field as cohomological physics.
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A highly symmetrical graph, the Petersen graph, which is vertex-transitive, symmetric, distance-transitive, and distance-regular. It has diameter 2. Its automorphism group has 120 elements, and is in fact the symmetric group S_5. Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs.
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An algebraic subgroup of an algebraic group is a Zariski-closed subgroup. Generally these are taken to be connected (or irreducible as a variety) as well. Another way of expressing the condition is as a subgroup that is also a subvariety. This may also be generalized by allowing schemes in place of varieties.
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William Schumacher Massey (August 23, 1920 \- June 17, 2017) was an American mathematician, known for his work in algebraic topology. The Massey product is named for him. He worked also on the formulation of spectral sequences by means of exact couples, and wrote several textbooks, including A Basic Course in Algebraic Topology ().
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Examples of topologies include the Zariski topology in algebraic geometry that reflects the algebraic nature of varieties, and the topology on a differential manifold in differential topology where each point within the space is contained in an open set that is homeomorphic to an open ball in a finite-dimensional Euclidean space.
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The Poincaré conjecture, proved in 2002/2003 by Grigori Perelman, is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of Eilenberg–MacLane spaces which are spaces with prescribed homotopy groups. Similarly algebraic K-theory relies in a way on classifying spaces of groups.
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Bridgeland's research interest is in algebraic geometry, focusing on properties of derived categories of coherent sheaves on algebraic varieties. His most-cited papers are on stability conditions, on triangulated categories and K3 surfaces; in the first he defines the idea of a stability condition on a triangulated category, and demonstrates that the set of all stability conditions on a fixed category form a manifold, whilst in the second he describes one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic K3 surface. Bridgeland's work helped to establish the coherent derived category as a key invariant of algebraic varieties and stimulated world-wide enthusiasm for what had previously been a technical backwater. His results on Fourier- Mukai transforms solve many problems within algebraic geometry, and have been influential in homological and commutative algebra, the theory of moduli spaces, representation theory and combinatorics.
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The mathematical discipline of topological combinatorics is the application of topological and algebraic topological methods to solving problems in combinatorics.
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However, the algebraic structure is given by an asymmetric cone gluing construction, suggesting that it is not the last word.
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Jürgen Neukirch (24 July 1937 – 5 February 1997) was a German mathematician known for his work on algebraic number theory.
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Paul Olum (August 16, 1918 – January 19, 2001) was an American mathematician (algebraic topology), professor of mathematics, and university administrator.
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The laws of inheritance are then encoded as algebraic properties of the algebra. For surveys of genetic algebras see , and .
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In all cases, being absolutely irreducible is the same as being irreducible over the algebraic closure of the ground field.
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The hypersimplices were first studied and named in the computation of characteristic classes (an important topic in algebraic topology), by ...
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Pohlig, S. Algebraic and combinatoric aspects of cryptography. Tech. Rep. No. 6602-1, Stanford Electron. Labs., Stanford, Calif., Oct. 1977.
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Crystalline cohomology and many other cohomology theories in algebraic geometry are also defined as sheaf cohomology on an appropriate site.
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For higher Chow groups (precursor of motivic homologies) of algebraic stacks, see Roy Joshua's Intersection Theory on Stacks:I and II.
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The algorithm is implemented for algebraic number fields in the PARI/GP computer algebra system, available through the function elllocalred.
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This is a list of surfaces, by Wikipedia page. See also List of algebraic surfaces, List of curves, Riemann surface.
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Jun-Muk Hwang (; born 27 October 1963) is a South Korean mathematician, specializing in algebraic geometry and complex differential geometry.
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Steiner's other work was primarily on the properties of algebraic curves and surfaces and on the solution of isoperimetric problems.
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Classical algebraic logic, which comprises all work in algebraic logic until about 1960, studied the properties of specific classes of algebras used to "algebraize" specific logical systems of particular interest to specific logical investigations. Generally, the algebra associated with a logical system was found to be a type of lattice, possibly enriched with one or more unary operations other than lattice complementation. Abstract algebraic logic is a modern subarea of algebraic logic that emerged in Poland during the 1950s and 60s with the work of Helena Rasiowa, Roman Sikorski, Jerzy Łoś, and Roman Suszko (to name but a few). It reached maturity in the 1980s with the seminal publications of the Polish logician Janusz Czelakowski, the Dutch logician Wim Blok and the American logician Don Pigozzi.
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At the university, she prepared and defended her PhD thesis, Algebraic Treatment of the Functional Calculi of Lewis and Heyting, in 1950 under the guidance of Prof. Andrzej Mostowski. This thesis on algebraic logic initiated her career contributing to the Lwów–Warsaw school of logic: In 1956, she took her second academic degree, doktor nauk (equivalent to habilitation today) in the Institute of Mathematics of the Polish Academy of Sciences, where between 1954 and 1957, she held a post of Associate Professor, becoming a Professor in 1957 and subsequently Full Professor in 1967. For the degree, she submitted two papers, Algebraic Models of Axiomatic Theories and Constructive Theories, which together formed a thesis named Algebraic Models of Elementary Theories and their Applications.
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An algebraic plane curve is a curve in an affine or projective plane given by one polynomial equation (or , where F is a homogeneous polynomial, in the projective case.) Algebraic curves have been studied extensively since the 18th century. Every algebraic plane curve has a degree, the degree of the defining equation, which is equal, in case of an algebraically closed field, to the number of intersections of the curve with a line in general position. For example, the circle given by the equation has degree 2. The non-singular plane algebraic curves of degree 2 are called conic sections, and their projective completion are all isomorphic to the projective completion of the circle (that is the projective curve of equation ).
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Room's PhD work concerned generalizations of the Schläfli double six, a configuration formed by the 27 lines on a cubic algebraic surface. In 1938 he published the book The geometry of determinantal loci through the Cambridge University Press. Nearly 500 pages long, the book combines methods of synthetic geometry and algebraic geometry to study higher- dimensional generalizations of quartic surfaces and cubic surfaces. It describes many infinite families of algebraic varieties, and individual varieties in these families, following a unifying principle that nearly all loci arising in algebraic geometry can be expressed as the solution to an equation involving the determinant of an appropriate matrix.Review of The geometry of determinantal loci by Virgil Snyder (1939), Bulletin of the AMS 45: 499–501, .
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The ideas were developed in the period 1955–1965 (which was roughly the time at which the requirements of algebraic topology were met but those of algebraic geometry were not). From the point of view of abstract category theory the work of comonads of Beck was a summation of those ideas; see Beck's monadicity theorem. The difficulties of algebraic geometry with passage to the quotient are acute. The urgency (to put it that way) of the problem for the geometers accounts for the title of the 1959 Grothendieck seminar TDTE on theorems of descent and techniques of existence (see FGA) connecting the descent question with the representable functor question in algebraic geometry in general, and the moduli problem in particular.
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When F is the algebraic closure of a finite field, the result follows from Hilbert's Nullstellensatz. The Ax–Grothendieck theorem for complex numbers can therefore be proven by showing that a counterexample over C would translate into a counterexample in some algebraic extension of a finite field. This method of proof is noteworthy in that it is an example of the idea that finitistic algebraic relations in fields of characteristic 0 translate into algebraic relations over finite fields with large characteristic. Thus, one can use the arithmetic of finite fields to prove a statement about C even though there is no homomorphism from any finite field to C. The proof thus uses model- theoretic principles to prove an elementary statement about polynomials.
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The discovery of algebraic invariants with Gaussian processes is based on David Hilbert's „Über die vollen Invariantensysteme“ and the studies of Grace and Young from 1903. Algebraic invariants with Gaussian processes were discovered by Clemens Par in 2010 in conjunction with the depicted vertical plane. S5 'Signal Analysis' is not specified. It may be either based on statistical methods, which require extensive computational power, or on the discovery of algebraic invariants with Gaussian processes by Clemens Par in 2010 (after having been averted to this classical problem by Rudolf E. Kálmán), based on German mathematician David Hilbert's published proof of the invariant field in 1893 and the apolarity behavior of algebraic cones as extensively studied by Grace and Young in 1903.
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The way forward was otherwise. The topos definition first appeared somewhat obliquely, in or about 1960. General problems of so-called 'descent' in algebraic geometry were considered, at the same period when the fundamental group was generalised to the algebraic geometry setting (as a pro-finite group). In the light of later work (c.
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Francesco Severi (photo by Konrad Jacobs) Francesco Severi (13 April 1879 – 8 December 1961) was an Italian mathematician. Severi was born in Arezzo, Italy. He is famous for his contributions to algebraic geometry and the theory of functions of several complex variables. He became the effective leader of the Italian school of algebraic geometry.
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At first, it is not clear how to show that there are any algebraic varieties which are not rational. In order to prove this, some birational invariants of algebraic varieties are needed. A birational invariant is any kind of number, ring, etc which is the same, or isomorphic, for all varieties that are birationally equivalent.
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As seen in the examples above tori can be represented as linear groups. An alternative definition for tori is: :A linear algebraic group is a torus if and only if it is diagonalisable over an algebraic closure. The torus is split over a field if and only if it is diagonalisable over this field.
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An important method for calculating the various groups is the concept of stable algebraic topology, which finds properties that are independent of the dimensions. Typically these only hold for larger dimensions. The first such result was Hans Freudenthal's suspension theorem, published in 1937. Stable algebraic topology flourished between 1945 and 1966 with many important results .
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In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V (and more generally). It does not use differential calculus, being based directly on abstract algebra, and in the most concrete cases just the theory of a system of linear equations.
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Contemporary algebraic geometry treats blowing up as an intrinsic operation on an algebraic variety. From this perspective, a blowup is the universal (in the sense of category theory) way to turn a subvariety into a Cartier divisor. A blowup can also be called monoidal transformation, locally quadratic transformation, dilatation, σ-process, or Hopf map.
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Eduard Jacob Neven Looijenga (born 30 September 1948, Zaandam)Prof. dr. E.J.N. Looijenga, 1948 - at Album Academicum, Universiteit van Amsterdam is a Dutch mathematician who works in algebraic geometry and the theory of algebraic groups.Member profile, KNAW, retrieved 2015-01-26. He was a professor of mathematics at Utrecht University until his retirement in 2013.
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Acta Numerica, 26, pp.591-721. by Jinchao Xu and Ludmil Zikatanov, the "algebraic multigrid" methods are understood from an abstract point of view. They developed a unified framework and existing algebraic multigrid methods can be derived coherently. Abstract theory about how to construct optimal coarse space as well as quasi-optimal spaces was derived.
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The correct proof was published in May 1995. The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques not available to Fermat.
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In a different direction, it is finer than the qfh topology, so h locally, algebraic correspondences are finite sums of morphisms.Suslin, Voevodsky, Singular homology of abstract algebraic varieties Finally, every proper surjective morphism is an h covering, so in any situation where de Jong's theorem on alterations is valid, h locally all schemes are regular.
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Other contexts where real-valued functions and their special properties are used include monotonic functions (on ordered sets), convex functions (on vector and affine spaces), harmonic and subharmonic functions (on Riemannian manifolds), analytic functions (usually of one or more real variables), algebraic functions (on real algebraic varieties), and polynomials (of one or more real variables).
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His research is concerned with commutative algebra, Galois theory of rings, algebraic geometry, algebraic groups, representations of groups and differential Galois theory. He has also published on mathematics education. He is the author or coauthor of over 85 research papers and 5 books. In 2012 Magid was elected a Fellow of the American Mathematical Society.
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Ruan and Tian's results are in a somewhat more general setting. With Jun Li, Tian gave a purely algebraic adaptation of these results to the setting of algebraic varieties. This was done at the same time as Kai Behrend and Barbara Fantechi, using a different approach.Behrend, K.; Fantechi, B. The intrinsic normal cone. Invent. Math.
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Generally, polynomials define an algebraic set of dimension two or higher. If the dimension is two, the algebraic set may have several irreducible components. If there is only one component the polynomials define a surface, which is a complete intersection. If there are several components, then one needs further polynomials for selecting a specific component.
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Algebraic differential equations are widely used in computer algebra and number theory. A simple concept is that of a polynomial vector field, in other words a vector field expressed with respect to a standard co-ordinate basis as the first partial derivatives with polynomial coefficients. This is a type of first-order algebraic differential operator.
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In his An Investigation into the laws of thought Boole now defined a function in terms of a symbol x as follows: :"8. Definition. – Any algebraic expression involving symbol x is termed a function of x, and may be represented by the abbreviated form f(x)" Boole then used algebraic expressions to define both algebraic and logical notions, e.g., 1 − x is logical NOT(x), xy is the logical AND(x,y), x + y is the logical OR(x, y), x(x + y) is xx + xy, and "the special law" xx = x2 = x.cf .
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Schmidt was elected in 1954 a member of the Heidelberger Akademie der WissenschaftenGabriele Dörflinger: Mathematik in der Heidelberger Akademie der Wissenschaften. 2014, pp. 68–70 and was made in 1968 an honorary doctor of the Free University of Berlin. Schmidt is known for his contributions to the theory of algebraic function fields and in particular for his definition of a zeta function for algebraic function fields and his proof of the generalized Riemann–Roch theorem for algebraic function fields (where the base field can be an arbitrary perfect field).
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Over an algebraically closed field of characteristic 0, any unipotent abelian connected algebraic group is isomorphic to a product of copies of the additive group G_a. The analogue of this for fields of characteristic p is false: the truncated Witt schemes are counterexamples. (We make them into algebraic groups by forgetting the multiplication and just using the additive structure.) However, these are essentially the only counterexamples: over an algebraically closed field of characteristic p, any unipotent abelian connected algebraic group is isogenous to a product of truncated Witt group schemes.
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The methods of noncommutative algebraic geometry are analogs of the methods of commutative algebraic geometry, but frequently the foundations are different. Local behavior in commutative algebraic geometry is captured by commutative algebra and especially the study of local rings. These do not have a ring-theoretic analogue in the noncommutative setting; though in a categorical setup one can talk about stacks of local categories of quasicoherent sheaves over noncommutative spectra. Global properties such as those arising from homological algebra and K-theory more frequently carry over to the noncommutative setting.
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Berwick was an algebraist, and worked on the problem of computing an integral basis for the algebraic integers in a simple algebraic extension of the rationals, and studied rings in algebraic integers. In 1927 he published Integral Bases, an ambitious account that used heavy numerical computations in place of practical proofs. He published sixteen papers, ten of them -- including a 1915 paper giving sufficient conditions for a quintic expression to be solved by radicals -- in Proceedings of the London Mathematical Society. Much of his work gained recognition only in the 1960s, when it was republished.
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The ideas surrounding algebraic functions go back at least as far as René Descartes. The first discussion of algebraic functions appears to have been in Edward Waring's 1794 An Essay on the Principles of Human Knowledge in which he writes: :let a quantity denoting the ordinate, be an algebraic function of the abscissa x, by the common methods of division and extraction of roots, reduce it into an infinite series ascending or descending according to the dimensions of x, and then find the integral of each of the resulting terms.
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Foundations of Algebraic Geometry is a book by that develops algebraic geometry over fields of any characteristic. In particular it gives a careful treatment of intersection theory by defining the local intersection multiplicity of two subvarieties. Weil was motivated by the need for a rigorous theory of correspondences on algebraic curves in positive characteristic, which he used in his proof of the Riemann hypothesis for curves over a finite field. Weil introduced abstract rather than projective varieties partly so that he could construct the Jacobian of a curve.
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For example, consider the self-map of the affine plane defined in terms of a pair of affine coordinates by :(z_1,z_2)\mapsto (z_1,z_2+f(z_1)) where f is a polynomial in a single variable. This is an automorphism of the algebraic variety, but not an automorphism of the affine structure. The Jacobian determinant of such an algebraic automorphism is necessarily a non-zero constant. It is believed that if the Jacobian of a self-map of a complex affine space is non-zero constant, then the map is an (algebraic) automorphism.
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In algebraic geometry, a generic point of an algebraic variety is a point whose coordinates do not satisfy any other algebraic relation than those satisfied by every point of the variety. For example, a generic point of an affine space over a field is a point whose coordinates are algebraically independent over . In scheme theory, where the points are the sub varieties, a generic point of a variety is a point whose closure for the Zariski topology is the whole variety. A generic property is a property of the generic point.
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Figurine algebraic notation (or FAN) is a widely used variation of algebraic notation which substitutes a piece symbol for the letter representing a piece, for example: ♞c6 in place of Nc6. (Pawns are unlabeled, just like in regular algebraic notation.) This enables moves to be read independent of language. The Unicode Miscellaneous Symbols set includes all the symbols necessary for FAN. In order to display or print these symbols, one has to have one or more fonts with good Unicode support installed on the computer, that the Web page, or word processor document, etc.
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In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
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A groupoid scheme is also called an algebraic groupoid, for example in , to convey the idea it is a generalization of algebraic groups and their actions. When the term "groupoid" can naturally refer to a groupoid object in some particular category in mind, the term groupoid set is used to refer to a groupoid object in the category of sets. Example: Suppose an algebraic group G acts from the right on a scheme U. Then take R = U \times G, s the projection, t the given action. This determines a groupoid scheme.
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Descartes also devised an algebraic method for finding the normal at any point of a curve whose equation is known. The construction of the tangents to the curve then easily follows and Descartes applied this algebraic procedure for finding tangents to several curves. The third book, On the Construction of Solid and Supersolid Problems, is more properly algebraic than geometric and concerns the nature of equations and how they may be solved. He recommends that all terms of an equation be placed on one side and set equal to 0 to facilitate solution.
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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 Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology.
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Du Val's early work before becoming a research student was on relativity, including a paper on the De Sitter model of the universe and Grassmann's tensor calculus. His doctorate was on algebraic geometry and in his thesis he generalised a result of Schoute. He worked on algebraic surfaces and later in his career became interested in elliptic functions. He received his Ph.D. with a thesis entitled 'On Certain Configurations of Algebraic Geometry Having Groups of Self-Transformations Representable by Symmetry Groups of Certain Polygons' under Baker's supervision in 1930.
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If X and Y are algebraic structures of some fixed type (such as groups, rings, or vector spaces), and if the function f from X to Y is a homomorphism, then ker f is a congruence relation (that is an equivalence relation that is compatible with the algebraic structure), and the coimage of f is a quotient of X. The bijection between the coimage and the image of f is an isomorphism in the algebraic sense; this is the most general form of the first isomorphism theorem. See also Kernel (algebra).
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Closure with a twist is a property of subsets of an algebraic structure. A subset Y of an algebraic structure X is said to exhibit closure with a twist if for every two elements : y_1, y_2 \in Y there exists an automorphism \phi of X and an element y_3 \in Y such that : y_1 \cdot y_2 = \phi(y_3) where "\cdot" is notation for an operation on X preserved by \phi. Two examples of algebraic structures which exhibit closure with a twist are the cwatset and the generalized cwatset, or GC-set.
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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 classification were known in the 1950s. The result of , that holomorphic vector bundles on the Riemann sphere are sums of line bundles, is now often called the Birkhoff–Grothendieck theorem, since it is implicit in much earlier work of on the Riemann–Hilbert problem.
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This theory is still foundational, and also had an influence on the (technically very different) scheme theory of Grothendieck. Spencer then continued this work, applying the techniques to structures other than complex ones, such as G-structures. In a third major part of his work, Kodaira worked again from around 1960 through the classification of algebraic surfaces from the point of view of birational geometry of complex manifolds. This resulted in a typology of seven kinds of two-dimensional compact complex manifolds, recovering the five algebraic types known classically; the other two being non-algebraic.
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Modeling algebraic expressions with algebra tiles is very similar to modeling addition and subtraction of integers using algebra tiles. In an expression such as 5x-3 one would group five positive x tiles together and then three negative unit tiles together to represent this algebraic expression. Along with modeling these expressions, algebra tiles can also be used to simplify algebraic expressions. For instance, if one has 4x+5-2x-3 they can combine the positive and negative x tiles and unit tiles to form zero pairs to leave the student with the expression 2x+2.
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Brill and Max Noether developed alternative proofs using algebraic methods for much of Riemann's work on Riemann surfaces. Brill–Noether theory went further by estimating the dimension of the space of maps of given degree d from an algebraic curve to projective space Pn. In birational geometry, Noether introduced the fundamental technique of blowing up in order to prove resolution of singularities for plane curves. Noether made major contributions to the theory of algebraic surfaces. Noether's formula is the first case of the Riemann-Roch theorem for surfaces.
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In algebraic number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in (the set of integers). The set of all algebraic integers, , is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers. The ring is the integral closure of regular integers in complex numbers. The ring of integers of a number field , denoted by , is the intersection of and : it can also be characterised as the maximal order of the field .
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He worked on the deformation theory for mappings to groups, which led to the solution of the Novikov problem on multiplicative subgroups in operator doubles, and to construction of the quantum group of complex cobordisms. He went on to treat problems related both with algebraic geometry and integrable systems. He is also well known for his work on sigma-functions on universal spaces of Jacobian varieties of algebraic curves that give effective solutions of important integrable systems. Buchstaber created an algebro-functional theory of symmetric products of spaces and described algebraic varieties of polysymmetric polynomials.
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Mumford's work in geometry combined traditional geometric insights with the latest algebraic techniques. He published on moduli spaces, with a theory summed up in his book Geometric Invariant Theory, on the equations defining an abelian variety, and on algebraic surfaces. His books Abelian Varieties (with C. P. Ramanujam) and Curves on an Algebraic Surface combined the old and new theories. His lecture notes on scheme theory circulated for years in unpublished form, at a time when they were, beside the treatise Éléments de géométrie algébrique, the only accessible introduction.
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In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory. Geometric invariant theory studies an action of a group G on an algebraic variety (or scheme) X and provides techniques for forming the 'quotient' of X by G as a scheme with reasonable properties. One motivation was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects.
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A morphism of abelian varieties is a morphism of the underlying algebraic varieties that preserves the identity element for the group structure. An isogeny is a finite-to-one morphism. When a complex torus carries the structure of an algebraic variety, this structure is necessarily unique. In the case g = 1, the notion of abelian variety is the same as that of elliptic curve, and every complex torus gives rise to such a curve; for g > 1 it has been known since Riemann that the algebraic variety condition imposes extra constraints on a complex torus.
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The sets being studied may also be subsets of algebraic structures other than the integers, for example, groups, rings and fields.
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In algebraic geometry, a Coble curve is an irreducible degree-6 planar curve with 10 double points. They were studied by .
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In algebraic geometry, the Segre cubic is a cubic threefold embedded in 4 (or sometimes 5) dimensional projective space, studied by .
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Norman Linstead Biggs (born 2 January 1941) is a leading British mathematician focusing on discrete mathematics and in particular algebraic combinatorics..
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In algebraic geometry, a Steinerian of a hypersurface, introduced by , is the locus of the singular points of its polar quadrics.
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Peter Bürgisser (born 1962) is a German mathematician and theoretical computer scientist who deals with algorithmic algebra and algebraic complexity theory.
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Bernardo Uribe Jongbloed (born 1975) is a Colombian mathematician. Uribe's research deals with algebraic geometry and topology with string theory applications.
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Dedekind's contribution would become the basis of ring theory and abstract algebra, while Kronecker's would become major tools in algebraic geometry.
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In mathematics, especially in real algebraic geometry, a semialgebraic space is a space which is locally isomorphic to a semialgebraic set.
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The Schröder–Bernstein theorem from set theory has analogs in the context operator algebras. This article discusses such operator-algebraic results.
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Emma Previato (born 1952) is a professor of mathematics at Boston University. Her research concerns algebraic geometry and partial differential equations.
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Reduced rings play an elementary role in algebraic geometry, where this concept is generalized to the concept of a reduced scheme.
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Via the Penrose-Ward transform these solutions give the holomorphic vector bundles often seen in the context of algebraic integrable systems.
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That is, if the angle is an algebraic, but non-rational, number of degrees, the trigonometric functions all have transcendental values.
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This is a list of named algebraic surfaces, compact complex surfaces, and families thereof, sorted according to the Enriques–Kodaira classification.
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Mark William Gross (30 November 1965)Anon (2017) is an American mathematician, specializing in differential geometry, algebraic geometry, and mirror symmetry.
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Krohn–Rhodes theory, sometimes also called algebraic automata theory, gives powerful decomposition results for finite transformation semigroups by cascading simpler components.
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In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures.
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The poset can be a partially ordered algebraic structure.Fujishige, Satoru Submodular functions and optimization. Second edition. Annals of Discrete Mathematics, 58.
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Andreas Thom is a German mathematician, working on geometric group theory, algebraic topology, ergodic theory of group actions, and operator algebras.
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Arnold worked on dynamical systems theory, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics and singularity theory.
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Adrian Ioviță (born 28 June 1954) is a Romanian-Canadian mathematician, specializing in arithmetic algebraic geometry and p-adic cohomology theories.
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The main methods for this renewing of elimination theory are Gröbner bases and cylindrical algebraic decomposition, which were introduced around 1970.
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The Riemann–Hurwitz formula concerning (ramified) maps between Riemann surfaces or algebraic curves is a consequence of the Riemann–Roch theorem.
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The homotopy colimit of a sequence of spaces :X_1 \to X_2 \to \cdots, is the mapping telescope.Hatcher's Algebraic Topology, 4.G.
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An algebraic curve is an algebraic variety of dimension one. This implies that an affine curve in an affine space of dimension n is defined by, at least, n−1 polynomials in n variables. To define a curve, these polynomials must generate a prime ideal of Krull dimension 1. This condition is not easy to test in practice.
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Work in the spirit of Riemann was carried out by the Italian school of algebraic geometry in the early 1900s. Contemporary treatment of complex geometry began with the work of Jean-Pierre Serre, who introduced the concept of sheaves to the subject, and illuminated the relations between complex geometry and algebraic geometry.Serre, J. P. (1955). Faisceaux algébriques cohérents.
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In the theory of algebraic groups, a special group is a linear algebraic group G with the property that every principal G-bundle is locally trivial in the Zariski topology. Special groups include the general linear group, the special linear group, and the symplectic group. Special groups are necessarily connected. Products of special groups are special.
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In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the height of the defining ideal. For this reason, the height of an ideal is often called its codimension. The dual concept is relative dimension.
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This 'illustrative' approach uses examples or narrative to explain management accounting procedures. This format, though useful when communicating with humans, can be difficult to translate into an algebraic form, suitable for computer model building. Mepham Mepham, M. (1980). Accounting Models, London: Pitmans extended the algebraic, or deductive, approach to cost accounting to cover many more techniques.
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In mathematics, a Klein surface is a dianalytic manifold of complex dimension 1. Klein surfaces may have a boundary and need not be orientable. Klein surfaces generalize Riemann surfaces. While the latter are used to study algebraic curves over the complex numbers analytically, the former are used to study algebraic curves over the real numbers analytically.
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Every simplicial complex has an associated face ring, also called its Stanley–Reisner ring. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in algebraic combinatorics. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of simplicial polytopes.
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Alexandre Mikhailovich Vinogradov (; 18 February 1938 – 20 September 2019) was a Russian and Italian mathematician. He made important contributions to the areas of differential calculus over commutative algebras, the algebraic theory of differential operators, homological algebra, differential geometry and algebraic topology, mechanics and mathematical physics, the geometrical theory of nonlinear partial differential equations and secondary calculus.
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Gillet's research deals with differential geometry, algebraic und arithmetic geometry, in particular Arakelov theory and algebraic K-theory. He collaborated with Christophe Soulé and Jean-Michel Bismut. Gillet and Soulé proved in 1992 an arithmetic Riemann-Roch theorem. Gillet was in 2008 a Senior Fellow at the Clay Mathematics Institute and from 1986 to 1989 a Sloan Fellow.
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But to extend split-complex arithmetic to four dimensions leads to hyperbolic quaternions, and opens the door to abstract algebra's hypercomplex numbers. Reviewing the expressions of Lorentz and Einstein, one observes that the Lorentz factor is an algebraic function of velocity. For readers uncomfortable with transcendental functions cosh and sinh, algebraic functions may be more to their liking.
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His research interests include homotopy theory, algebraic K-theory, and arithmetic algebraic geometry. Hesselholt was born in Vejrumbro, a village in the Viborg Municipality of Denmark. He studied at Aarhus University, earning a bachelor's degree in 1988, a master's degree in 1992, and a Ph.D. in 1994; his dissertation, supervised by Ib Madsen, concerned K-theory.
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Richard Gordon Swan (; born 1933) is an American mathematician who is known for the Serre–Swan theorem relating the geometric notion of vector bundles to the algebraic concept of projective modules,. and for the Swan representation, an l-adic projective representation of a Galois group.. His work has mainly been in the area of algebraic K-theory.
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Clebsch's diagonal cubic surface. In mathematics, the Manin conjecture describes the conjectural distribution of rational points on an algebraic variety relative to a suitable height function. It was proposed by Yuri I. Manin and his collaborators in 1989 when they initiated a program with the aim of describing the distribution of rational points on suitable algebraic varieties.
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In mathematics, Artin's criteria are a collection of related necessary and sufficient conditions on deformation functors which proving their representability of these functors as either Algebraic spaces or as Algebraic stacks. In particular, these conditions are used in the construction of the moduli stack of elliptic curves and the construction of the moduli stack of pointed curves.
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A slice of the Consani–Scholten quintic In the mathematical fields of algebraic geometry and arithmetic geometry, the Consani–Scholten quintic is an algebraic hypersurface (the set of solutions to a single polynomial equation in multiple variables) studied in 2001 by Caterina Consani and Jasper Scholten. It has been used as a test case for the Langlands program.
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In algebraic geometry, a singularity of an algebraic variety is a point of the variety where the tangent space may not be regularly defined. The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like cusps. For example, the equation defines a curve that has a cusp at the origin .
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Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element. The chromatic polynomial of a graph counts the number of proper colourings of that graph.
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Antonio Luigi Gaudenzio Giuseppe Cremona (7 December 1830 – 10 June 1903) was an Italian mathematician. His life was devoted to the study of geometry and reforming advanced mathematical teaching in Italy. His reputation mainly rests on his Introduzione ad una teoria geometrica delle curve piane. He notably enriched our knowledge of algebraic curves and algebraic surfaces.
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Quite generally, as this example illustrates, for a compact Riemann surface or algebraic curve, the Hodge number is the genus g. For the case of algebraic surfaces, this is the quantity known classically as the irregularity q. It is also, in general, the dimension of the Albanese variety, which takes the place of the Jacobian variety.
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The semicubical parabola was discovered in 1657 by William Neile who computed its arc length. Although the lengths of some other non-algebraic curves including the logarithmic spiral and cycloid had already been computed (that is, those curves had been rectified), the semicubical parabola was the first algebraic curve (excluding the line and circle) to be rectified.
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Journal of Algebraic Combinatorics is a peer-reviewed scientific journal covering algebraic combinatorics. It was established in 1992 and is published by Springer Science+Business Media. The editor-in-chief is Ilias S. Kotsireas (Wilfrid Laurier University). In 2017, the journal's four editors-in-chief and editorial board resigned to protest the publisher's high prices and limited accessibility.
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In algebraic geometry, especially in scheme theory, a property is said to hold geometrically over a field if it also holds over the algebraic closure of the field. In other words, a property holds geometrically if it holds after a base change to a geometric point. For example, a smooth variety is a variety that is geometrically regular.
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Spencer Janney Bloch (b. May 22, 1944; New York CitySpencer Bloch CV, Department of Mathematics, University of Chicago. Accessed January 12, 2010) is an American mathematician known for his contributions to algebraic geometry and algebraic K-theory. Bloch is a R. M. Hutchins Distinguished Service Professor Emeritus in the Department of Mathematics of the University of Chicago.
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Among the reconstruction methods one can find algebraic reconstruction techniques (e.g., DART W. van Aarle, K J. Batenburg, and J. Sijbers, Automatic parameter estimation for the Discrete Algebraic Reconstruction Technique (DART), IEEE Transactions on Image Processing, 2012 or K. J. Batenburg, and J. Sijbers, "Generic iterative subset algorithms for discrete tomography", Discrete Applied Mathematics, vol. 157, no. 3, pp.
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In algebraic geometry, given a reductive algebraic group G and a Borel subgroup B, a spherical variety is a G-variety with an open dense B-orbit. It is sometimes also assumed to be normal. Examples are flag varieties, symmetric spaces and (affine or projective) toric varieties. There is also a notion of real spherical varieties.
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In Grothendieck's language, an abstract algebraic variety is usually defined to be an integral, separated scheme of finite type over an algebraically closed field, although some authors drop the irreducibility or the reducedness or the separateness condition or allow the underlying field to be not algebraically closed.Liu, Qing. Algebraic Geometry and Arithmetic Curves, p. 55 Definition 2.3.
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A locally Noetherian scheme is a locally Noetherian formal scheme in the canonical way: the formal completion along itself. In other words, the category of locally Noetherian formal schemes contains all locally Noetherian schemes. Formal schemes were motivated by and generalize Zariski's theory of formal holomorphic functions. Algebraic geometry based on formal schemes is called formal algebraic geometry.
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Calculations could be performed in two modes CAL and RUN. In the former, the PC-1403 would behave like a normal scientific calculator at the time with formulas entered in algebraic logic. In RUN mode, BASIC statements could be entered for immediate execution and print out. The latter was similar to the Direct algebraic logic employed by modern calculators.
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Kanakanahalli Ramachandra (18 August 1933 – 17 January 2011) was an Indian mathematician working in both analytic number theory and algebraic number theory.
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The resulting construct is a set of pairs of conditions and values, called a "value set". See narrowing of algebraic value sets.
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Greenberg's conjecture is either of two conjectures in algebraic number theory proposed by Ralph Greenberg. Both are still unsolved as of 2020.
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In algebraic geometry, Horrocks bundles are certain indecomposable rank 3 vector bundles (locally free sheaves) on 5-dimensional projective space, found by .
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Helena Rasiowa (20 June 1917 – 9 August 1994) was a Polish mathematician. She worked in the foundations of mathematics and algebraic logic.
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Note the previous construction can be proven in Algebraic geometry using the affine cone of a projective variety X using Local cohomology.
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Ron Donagi, Berkeley 1990 Ron Yehuda Donagi (born March 9, 1956) is an American mathematician, working in algebraic geometry and string theory.
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In 2018, the Association for Women in Mathematics gave Bergner the Ruth I. Michler Memorial Prize for her research on algebraic -theory.
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In algebraic topology, a branch of mathematics, a connective spectrum is a spectrum whose homotopy sets \pi_k of negative degrees are zero..
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Besides the FIDE standard (or short) algebraic notation (SAN) already described, several similar systems are in use for their own particular advantages.
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In algebraic topology, cobordism theories are fundamental extraordinary cohomology theories, and categories of cobordisms are the domains of topological quantum field theories.
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A typical algebra problem. The following sections lay out examples of some of the types of algebraic equations that may be encountered.
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Branched coverings are easily constructed as Kummer extensions, i.e. as algebraic extension of the function field. The hyperelliptic curves are prototypic examples.
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Another (noncommutative) version of derived algebraic geometry, using A-infinity categories has been developed from early 1990s by Maxim Kontsevich and followers.
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An algebraic cycle is a higher-codimension generalization of a divisor; by definition, a Weil divisor is a cycle of codimension 1.
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A great deal has been worked out about their theory: algebraic geometry, local zeta-functions via Jacobi sums, Hardy-Littlewood circle method.
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However it is the Gauss sum for which the algebraic properties hold. Such cubic exponential sums are also now called Kummer sums.
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Mikhail Mikhailovich Postnikov (; 27 October 1927 – 27 May 2004) was a Soviet mathematician, known for his work in algebraic and differential topology.
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The International Symposium on Algebraic Geometry held in Rome in 1965. Enrico Bompiani talking to Giovanni Battista Rizza and Vittorio Dalla Volta.
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The group G is in general difficult to compute, the understanding of algebraic solutions is an indication of upper bounds for G.
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Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology.
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47, and p. 88 Example 3.2.3 Classical algebraic varieties are the quasiprojective integral separated finite type schemes over an algebraically closed field.
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In the Zariski topology on the affine plane, this graph of a polynomial is closed. In algebraic geometry and commutative algebra, the Zariski topology is a topology on algebraic varieties, introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space, called the spectrum of the ring. The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces.
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Commutative algebra (in the form of polynomial rings and their quotients, used in the definition of algebraic varieties) has always been a part of algebraic geometry. However, in the late 1950s, algebraic varieties were subsumed into Alexander Grothendieck's concept of a scheme. Their local objects are affine schemes or prime spectra, which are locally ringed spaces, which form a category that is antiequivalent (dual) to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field k, and the category of finitely generated reduced k-algebras. The gluing is along the Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes.
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Algebraic geometry has also recently found applications to statistical learning theory, including a generalization of the Akaike information criterion to singular statistical models.
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Carlo Rosati (Livorno, 24 April 1876 – Pisa, 19 August 1929) was an Italian mathematician working on algebraic geometry who introduced the Rosati involution.
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Chenyang Xu (; born 1981) is a Chinese mathematician in the area of algebraic geometry and a professor at the Massachusetts Institute of Technology.
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In 2009, Harizanov received a grant from the National Science Foundation to research how algebraic, topological, and algorithmic properties of mathematical structures relate.
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The Hasse principle for algebraic groups was used in the proofs of the Weil conjecture for Tamagawa numbers and the strong approximation theorem.
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In his original paper, Shannon presented a result which stated that functions computable by the GPAC are those functions which are differentially algebraic.
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He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to noncommutative algebra and noncommutative algebraic geometry".
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In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.
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Proceedings of the Royal Society. which showed that for an arbitrary algebraic plane curve a linkage can be constructed that draws the curve.
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In mathematics, potential good reduction is a property of the reduction modulo a prime or, more generally, prime ideal, of an algebraic variety.
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In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra.
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Thomas Schick, Oberwolfach 2012 Thomas Schick (born 22 May 1969 in Alzey) is a German mathematician, specializing in algebraic topology and differential geometry.
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Michael Liam McQuillan is a Scottish mathematician studying algebraic geometry. As of 2019 he is Professor at the University of Rome Tor Vergata.
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The pairing is used in number theory and algebraic geometry, and has also been applied in elliptic curve cryptography and identity based encryption.
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Henri Gillet (2006) Henri Antoine Gillet (born 8 July 1953, Tangier) is a European-American mathematician, specializing in arithmetic geometry and algebraic geometry.
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John Greenlees Semple (10 June 1904 in Belfast, Ireland – 23 October 1985 in London, England) was a British mathematician working in algebraic geometry.
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Rosa C. Orellana is an American mathematician specializing in algebraic combinatorics and representation theory. She is a professor of mathematics at Dartmouth College.
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This is one of the steps used in the proof of the Weil conjectures. Behrend's trace formula generalizes the formula to algebraic stacks.
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Kharlampovich, and A. Myasnikov. "Algebraic geometry over free groups: lifting solutions into generic points." Groups, languages, algorithms, pp. 213–318, Contemporary Mathematics, vol.
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In many such cases, the poset has additional structure: For example, the poset can be a lattice or a partially ordered algebraic structure.
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To every algebraic extension L of k, the L-algebraic set associated to a given k-algebraic set V is the fiber product of schemes V ×Spec(k) Spec(L). A k-variety is absolutely irreducible if the associated kalg-algebraic set is an irreducible scheme; in this case, the k-variety is called a variety. An absolutely irreducible k-variety is defined over k if the associated kalg-algebraic set is a reduced scheme. A field of definition of a variety V is a subfield L of kalg such that there exists a k∩L-variety W such that W ×Spec(k∩L) Spec(k) is isomorphic to V and the final object in the category of reduced schemes over W ×Spec(k∩L) Spec(L) is an L-variety defined over L. Analogously to the definitions for affine and projective varieties, a k-variety is a variety defined over k if the stalk of the structure sheaf at the generic point is a regular extension of k; furthermore, every variety has a minimal field of definition.
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Oberwolfach in 2004 Charles Alexander Weibel (born October 28, 1950 in Terre Haute, Indiana) is an American mathematician working on algebraic K-theory, algebraic geometry and homological algebra. Weibel studied physics and mathematics at the University of Michigan, earning bachelor's degrees in both subjects in 1972. He was awarded a master's degree by the University of Chicago in 1973 and achieved his doctorate in 1977 under the supervision of Richard Swan (Homotopy in Algebraic K-Theory). From 1970 to 1976 he was an "Operations Research Analyst" at Standard Oil of Indiana, and from 1977 to 1978 was at the Institute for Advanced Study.
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To define the Chow coordinates, take the intersection of an algebraic variety Z, inside a projective space, of degree d and dimension m by linear subspaces U of codimension m. When U is in general position, the intersection will be a finite set of d distinct points. Then the coordinates of the d points of intersection are algebraic functions of the Plücker coordinates of U, and by taking a symmetric function of the algebraic functions, a homogeneous polynomial known as the Chow form (or Cayley form) of Z is obtained. The Chow coordinates are then the coefficients of the Chow form.
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If G is a semisimple algebraic group (or Lie group) and V is a (finite dimensional) highest weight representation of G, then the highest weight space is a point in the projective space P(V) and its orbit under the action of G is a projective algebraic variety. This variety is a (generalized) flag variety, and furthermore, every (generalized) flag variety for G arises in this way. Armand Borel showed that this characterizes the flag varieties of a general semisimple algebraic group G: they are precisely the complete homogeneous spaces of G, or equivalently (in this context), the projective homogeneous G-varieties.
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From an algebraic perspective, complex numbers enter quite naturally into the study of algebraic functions. First of all, by the fundamental theorem of algebra, the complex numbers are an algebraically closed field. Hence any polynomial relation p(y, x) = 0 is guaranteed to have at least one solution (and in general a number of solutions not exceeding the degree of p in y) for y at each point x, provided we allow y to assume complex as well as real values. Thus, problems to do with the domain of an algebraic function can safely be minimized.
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In algebraic geometry, the Mumford–Tate group (or Hodge group) MT(F) constructed from a Hodge structure F is a certain algebraic group G. When F is given by a rational representation of an algebraic torus, the definition of G is as the Zariski closure of the image in the representation of the circle group, over the rational numbers. introduced Mumford–Tate groups over the complex numbers under the name of Hodge groups. introduced the p-adic analogue of Mumford's construction for Hodge–Tate modules, using the work of on p-divisible groups, and named them Mumford–Tate groups.
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In this identity one can assume that F and G are vector-valued (have several components). An algebraic addition theorem is one in which G can be taken to be a vector of polynomials, in some set of variables. The conclusion of the mathematicians of the time was that the theory of abelian functions essentially exhausted the interesting possibilities: considered as a functional equation to be solved with polynomials, or indeed rational functions or algebraic functions, there were no further types of solution. In more contemporary language this appears as part of the theory of algebraic groups, dealing with commutative groups.
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The first clear sign of a possible generalization from Chow groups to a more general motivic cohomology theory for algebraic varieties was Quillen's definition and development of algebraic K-theory (1973), generalizing the Grothendieck group K0 of vector bundles. In the early 1980s, Beilinson and Soulé observed that Adams operations gave a splitting of algebraic K-theory tensored with the rationals; the summands are now called motivic cohomology (with rational coefficients). Beilinson and Lichtenbaum made influential conjectures predicting the existence and properties of motivic cohomology. Most but not all of their conjectures have now been proved.
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In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties are not algebraic varieties but are families of algebraic varieties. Shimura curves are the one-dimensional Shimura varieties. Hilbert modular surfaces and Siegel modular varieties are among the best known classes of Shimura varieties. Special instances of Shimura varieties were originally introduced by Goro Shimura in the course of his generalization of the complex multiplication theory.
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The Hodge index theorem was a result on the intersection number theory for curves on an algebraic surface: it determines the signature of the corresponding quadratic form. This result was sought by the Italian school of algebraic geometry, but was proved by the topological methods of Lefschetz. The Theory and Applications of Harmonic Integrals summed up Hodge's development during the 1930s of his general theory. This starts with the existence for any Kähler metric of a theory of Laplacians – it applies to an algebraic variety V (assumed complex, projective and non-singular) because projective space itself carries such a metric.
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Algebraic numbers are all numbers that can be defined explicitly or implicitly in terms of polynomials, starting from the rational numbers. One may generalize this to "closed-form numbers", which may be defined in various ways. Most broadly, all numbers that can be defined explicitly or implicitly in terms of polynomials, exponentials, and logarithms are called "elementary numbers", and these include the algebraic numbers, plus some transcendental numbers. Most narrowly, one may consider numbers explicitly defined in terms of polynomials, exponentials, and logarithms – this does not include all algebraic numbers, but does include some simple transcendental numbers such as or ln 2\.
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For applications to number theory, van der Waerden and Weil formulated algebraic geometry over any field, not necessarily algebraically closed. Weil was the first to define an abstract variety (not embedded in projective space), by gluing affine varieties along open subsets, on the model of manifolds in topology. He needed this generality for his construction of the Jacobian variety of a curve over any field. (Later, Jacobians were shown to be projective varieties by Weil, Chow and Matsusaka.) The algebraic geometers of the Italian school had often used the somewhat foggy concept of the generic point of an algebraic variety.
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In algebraic geometry, a Seshadri constant is an invariant of an ample line bundle L at a point P on an algebraic variety. It was introduced by Demailly to measure a certain rate of growth, of the tensor powers of L, in terms of the jets of the sections of the Lk. The object was the study of the Fujita conjecture. The name is in honour of the Indian mathematician C. S. Seshadri. It is known that Nagata's conjecture on algebraic curves is equivalent to the assertion that for more than nine general points, the Seshadri constants of the projective plane are maximal.
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Algebraic varieties are locally defined as the common zero sets of polynomials and since polynomials over the complex numbers are holomorphic functions, algebraic varieties over C can be interpreted as analytic spaces. Similarly, regular morphisms between varieties are interpreted as holomorphic mappings between analytic spaces. Somewhat surprisingly, it is often possible to go the other way, to interpret analytic objects in an algebraic way. For example, it is easy to prove that the analytic functions from the Riemann sphere to itself are either the rational functions or the identically infinity function (an extension of Liouville's theorem).
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In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras and (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called finitary algebraic categories.
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An action of a linear algebraic group G on a variety (or scheme) X over a field k is a morphism :G \times_k X \to X that satisfies the axioms of a group action. As in other types of group theory, it is important to study group actions, since groups arise naturally as symmetries of geometric objects. Part of the theory of group actions is geometric invariant theory, which aims to construct a quotient variety X/G, describing the set of orbits of a linear algebraic group G on X as an algebraic variety. Various complications arise.
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The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings. Initially proved as Riemann's inequality by , the theorem reached its definitive form for Riemann surfaces after work of Riemann's short-lived student . It was later generalized to algebraic curves, to higher-dimensional varieties and beyond.
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Many number systems, such as the integers and the rationals enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields. Further abstract algebraic concepts such as modules, vector spaces and algebras also form groups.
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Quadratic surd: An algebraic number that is the root of a quadratic equation. Such a number can be expressed as the sum of a rational number and the square root of a rational. Constructible number: A number representing a length that can be constructed using a compass and straightedge. These are a subset of the algebraic numbers, and include the quadratic surds.
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If such a structure has also a greatest element (the meet of the empty set), it is also a complete lattice. Thus a complete semilattice turns out to be "a complete lattice possibly lacking a top". This definition is of interest specifically in domain theory, where bounded complete algebraic cpos are studied as Scott domains. Hence Scott domains have been called algebraic semilattices.
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Visual representations, manipulatives, gestures, and to some degree grids, can support qualitative reasoning about mathematics. Instead of only emphasizing computational skills, multiple representations can help students make the conceptual shift to the meaning and use of, and to develop algebraic thinking. By focusing more on the conceptual representations of algebraic problems, students have a better chance of improving their problem solving skills.
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Axel Harnack Carl Gustav Axel Harnack (, Dorpat (now – 3 April 1888, Dresden) was a Baltic German mathematician who contributed to potential theory. Harnack's inequality applied to harmonic functions. He also worked on the real algebraic geometry of plane curves, proving Harnack's curve theorem for real plane algebraic curves. He was the son of the theologian Theodosius HarnackHarnack, Axel; George L. Cathcart.
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This fundamental idea first surfaced in algebraic topology. Difficult topological questions can be translated into algebraic questions which are often easier to solve. Basic constructions, such as the fundamental group or the fundamental groupoid of a topological space, can be expressed as functors to the category of groupoids in this way, and the concept is pervasive in algebra and its applications.
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In mathematics, a stacky curve is an object in algebraic geometry that is roughly an algebraic curve with potentially "fractional points" called stacky points. A stacky curve is a type of stack used in studying Gromov–Witten theory, enumerative geometry, and rings of modular forms. Stacky curves are deeply related to 1-dimensional orbifolds and therefore sometimes called orbifold curves or orbicurves.
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In a UFD, the prime elements occurring in a factorization are only expected to be unique up to units and their ordering. However, even with this weaker definition, many rings of integers in algebraic number fields do not admit unique factorization. There is an algebraic obstruction called the ideal class group. When the ideal class group is trivial, the ring is a UFD.
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In mathematics, the Serre group S is the pro-algebraic group whose representations correspond to CM-motives over the algebraic closure of the rationals, or to polarizable rational Hodge structures with abelian Mumford–Tate groups. It is a projective limit of finite dimensional tori, so in particular is abelian. It was introduced by . It is a subgroup of the Taniyama group.
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The new algebraic geometry that would succeed the Italian school was distinguished also by the intensive use of algebraic topology. The founder of that tendency was Henri Poincaré; during the 1930s it was developed by Lefschetz, Hodge and Todd. The modern synthesis brought together their work, that of the Cartan school, and of W.L. Chow and Kunihiko Kodaira, with the traditional material.
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86 (Infobase Publishing 2006). Later, Persian and Arabic mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi's contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero, thus he had to distinguish several types of equations.
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In number theory, an aurifeuillean factorization, or aurifeuillian factorization, named after Léon-François-Antoine Aurifeuille, is a special type of algebraic factorization that comes from non-trivial factorizations of cyclotomic polynomials over the integers. Although cyclotomic polynomials themselves are irreducible over the integers, when restricted to particular integer values they may have an algebraic factorization, as in the examples below.
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The algebraic stress model arises in computational fluid dynamics. Two main approaches can be undertaken. In the first, the transport of the turbulent stresses is assumed proportional to the turbulent kinetic energy; while in the second, convective and diffusive effects are assumed to be negligible. Algebraic stress models can only be used where convective and diffusive fluxes are negligible, i.e.
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The answer is provided by Belyi's theorem, which states that the Riemann surfaces that can be described by dessins are precisely those that can be defined as algebraic curves over the field of algebraic numbers. The absolute Galois group transforms these particular curves into each other, and thereby also transforms the underlying dessins. For a more detailed treatment of this subject, see or .
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The periods are intended to bridge the gap between the algebraic numbers and the transcendental numbers. The class of algebraic numbers is too narrow to include many common mathematical constants, while the set of transcendental numbers is not countable, and its members are not generally computable. The set of all periods is countable, and all periods are computable , and in particular definable.
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Glossa has been cited as a predecessor for Algebraic Combinatorics, a mathematics journal established under similar circumstances when the editorial board of Springer's Journal of Algebraic Combinatorics resigned en masse in 2017. The editorial board criticized Springer for "double-dipping", that is, charging large subscription fees to libraries in addition to high fees for authors who wished to make their publications open access.
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Algebraic curves can also be space curves, or curves in a space of higher dimension, say . They are defined as algebraic varieties of dimension one. They may be obtained as the common solutions of at least polynomial equations in variables. If polynomials are sufficient to define a curve in a space of dimension , the curve is said to be a complete intersection.
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Kathryn Hess (born 1967) is a professor of mathematics at École Polytechnique Fédérale de Lausanne (EPFL) and is known for her work on homotopy theory, category theory, and algebraic topology, both pure and applied. In particular, she applies the methods of algebraic topology to better understanding neurology, cancer biology, and materials science. She is a fellow of the American Mathematical Society.
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AMG is regarded as advantageous mainly where geometric multigrid is too difficult to apply, but is often used simply because it avoids the coding necessary for a true multigrid implementation. While classical AMG was developed first, a related algebraic method is known as smoothed aggregation (SA). In a recent overview paper Xu, J. and Zikatanov, L., 2017. Algebraic multigrid methods.
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In mathematical logic, predicate functor logic (PFL) is one of several ways to express first-order logic (also known as predicate logic) by purely algebraic means, i.e., without quantified variables. PFL employs a small number of algebraic devices called predicate functors (or predicate modifiers)Johannes Stern, Toward Predicate Approaches to Modality, Springer, 2015, p. 11. that operate on terms to yield terms.
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There are general existence theorems that apply; the most basic of them guarantees that :Whenever C is a variety, then for every set X there is a free object F(X) in C. Here, a variety is a synonym for a finitary algebraic category, thus implying that the set of relations are finitary, and algebraic because it is monadic over Set.
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Such an extension is called an algebraically closed extension. Among all such extensions there is one and only one (up to isomorphism, but not unique isomorphism) which is an algebraic extension of F;See Lang's Algebra, §VII.2 or van der Waerden's Algebra I, §10.1. it is called the algebraic closure of F. The theory of algebraically closed fields has quantifier elimination.
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Andrei Suslin (, sometimes transliterated Souslin) was a Russian mathematician who contributed to algebraic K-theory and its connections with algebraic geometry. He was a Trustee Chair and Professor of mathematics at Northwestern University.Andrei Suslin, faculty profile , Department of Mathematics, Northwestern University He was born on 27 December 1950 in St. Petersburg, Russia. As a young person, he was an "all Leningrad" gymnast.
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In mathematics and computer science, a rational series is a generalisation of the concept of formal power series over a ring to the case when the basic algebraic structure is no longer a ring but a semiring, and the indeterminates adjoined are not assumed to commute. They can be regarded as algebraic expressions of a formal language over a finite alphabet.
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For example, in studying algebraic surfaces, it can be useful to consider families of algebraic surfaces over any scheme Y. In many cases, the family of all varieties of a given type can itself be viewed as a variety or scheme, known as a moduli space. For some of the detailed definitions in the theory of schemes, see the glossary of scheme theory.
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Consider a formal algebraic specification for the boolean data type. One possible algebraic specification may provide two constructor functions for the data-element: a true constructor and a false constructor. Thus, a boolean data element could be declared, constructed, and initialized to a value. In this scenario, all other connective elements, such as XOR and AND, would be additional functions.
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The lemniscate of Gerono In algebraic geometry, the lemniscate of Gerono, or lemniscate of Huygens, or figure-eight curve, is a plane algebraic curve of degree four and genus zero and is a lemniscate curve shaped like an \infty symbol, or figure eight. It has equation :x^4-x^2+y^2 = 0. It was studied by Camille-Christophe Gerono.
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Chess notations are various systems that have developed to record either the moves made in a game of chess or the position of pieces on a chessboard. The earliest systems of notation used lengthy narratives to describe each move; these gradually evolved into terser notation systems. Currently algebraic chess notation is the accepted standard and is widely used. Algebraic notation has several variations.
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In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology.
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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 phenomenon has also been called principalization, or sometimes capitulation.
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A special feature of Kosterlitz–Thouless transitions is the exponential divergence of correlation times and length (instead of algebraic ones). This serves a transcendental equation which can be solved numerically. The figure shows a comparison of the Kibble–Zurek scaling with algebraic and exponential divergences. The data illustrate, that the Kibble–Zurek mechanism also works for transitions of the Kosterlitz–Thoules universality class.
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65, No. 9 (Nov., 1958), pp. 685–690; 'Gerhard Frey and Hans-Georg Rück, The strong Lefschetz principle in algebraic geometry, Manuscripta Mathematica, Volume 55, Numbers 3–4, September, 1986, pp. 385–401. This principle permits the carrying over of some results obtained using analytic or topological methods for algebraic varieties over C to other algebraically closed ground fields of characteristic 0.
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In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their interactions with Cartier divisors, they play a central role in the study of algebraic varieties.
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"EL" stands both for "Exponential-Logarithmic" and as an abbreviation for "elementary". Whether a number is a closed-form number is related to whether a number is transcendental. Formally, Liouville numbers and elementary numbers contain the algebraic numbers, and they include some but not all transcendental numbers. In contrast, EL numbers do not contain all algebraic numbers, but do include some transcendental numbers.
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Gross works on complex geometry, algebraic geometry, and mirror symmetry. Gross and Bernd Siebert jointly developed a program (known as the Gross–Siebert Program) for studying mirror symmetry within algebraic geometry. His previous doctoral students have included Daniel Budreau, Andrei Caldararu, Ricardo Castano-Bernard, Man Wai Cheung, Karl Fredrickson, Michael Kasa, Diego Matessi, Brandon Meredith, Peter Overholser, Simone Pavanelli and Michael Slawinski.
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A connected linear algebraic group G over an algebraically closed field is called semisimple if every smooth connected solvable normal subgroup of G is trivial. More generally, a connected linear algebraic group G over an algebraically closed field is called reductive if the largest smooth connected unipotent normal subgroup of G is trivial.SGA 3 (2011), v. 3, Définition XIX.1.6.1.
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As with any compact Riemann surface, the sphere may also be viewed as a projective algebraic curve, making it a fundamental example in algebraic geometry. It also finds utility in other disciplines that depend on analysis and geometry, such as the Bloch sphere of quantum mechanics and in other branches of physics. The extended complex plane is also called closed complex plane.
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Given some algebraic structure, an identity is an equation in which all variables are tacitly universally quantified, and in which all operations are among the primitive operations proper to the structure. Algebraic structures axiomatized solely by identities are called varieties. Many standard results in universal algebra hold only for varieties. Quasigroups are varieties if left and right division are taken as primitive.
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Vakil is an algebraic geometer and his research work spans over enumerative geometry, topology, Gromov-Witten theory, and classical algebraic geometry. He has solved several old problems in Schubert calculus. Among other results, he proved that all Schubert problems are enumerative over the real numbers, a result that resolves an issue mathematicians have worked on for at least two decades.
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In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation \oplus, a unary operation eg, and the constant 0, satisfying certain axioms. MV-algebras are the algebraic semantics of Łukasiewicz logic; the letters MV refer to the many-valued logic of Łukasiewicz. MV-algebras coincide with the class of bounded commutative BCK algebras.
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A Maths24 card with the inclusion of variables, being used in an algebraic expression, in fractional form and with powers Finally, cards were printed with x and y variables that could appear in many forms, including fractions, powers and algebraic expressions. Participants would be required to find positive integers less than ten that the variables could represent, as well as the solution.
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For historical development of the word vector, see and The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from to . Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity.
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At Moscow State University he initially worked in the fields of algebraic geometry and topology and derived what is now known as Gamkrelidze's formula .Computation of Chern cycles of Algebraic Manifolds. (Russian). Dokl. Nauk. SSSR 90(1953), No.4 719-722 In 1954 he began his work on optimal control. He wrote Mathematical Theory of Optimal Processes with Pontryagin, Boltyanskii and Mishchenko.
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Fig. 10: Relations between mathematical spaces: schemes, stacks etc Algebraic geometry studies the geometric properties of polynomial equations. Polynomials are a type of function defined from the basic arithmetic operations of addition and multiplication. Because of this, they are closely tied to algebra. Algebraic geometry offers a way to apply geometric techniques to questions of pure algebra, and vice versa.
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In this generality, character varieties are only algebraic sets and are not actual varieties. To avoid technical issues, one often considers the associated reduced space by dividing by the radical of 0 (eliminating nilpotents). However, this does not necessarily yield an irreducible space either. Moreover, if we replace the complex group by a real group we may not even get an algebraic set.
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In algebraic geometry, Sumihiro's theorem, introduced by , states that a normal algebraic variety with an action of a torus can be covered by torus- invariant affine open subsets. The "normality" in the hypothesis cannot be relaxed.Toric Varieties The hypothesis that the group acting on the variety is a torus can also not be relaxed.Bialynicki-Birula decomposition of a non- singular quasi-projective scheme.
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Let k be an algebraically closed field and G a linear algebraic group (that is, affine algebraic group) over k. By definition, Lie(G) is the Lie algebra of all derivations of k[G] that commute with the left action of G. As in the Lie group case, it can be identified with the tangent space to G at the identity element.
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Geiser retired in 1913 as professor emeritus due to health problems, and his successor in the professorial chair of synthetic geometry was Hermann Weyl. From 1881 to 1887 and from 1891 to 1895 Geiser was the director of the Zürich Polytechnikum. Geiser taught algebraic geometry, differential geometry, and the theory of invariants. He published research on algebraic geometry and minimal surfaces.
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Throughout this article, k denotes a field. The algebraic closure of a field is denoted by adding a superscript of "alg", e.g. the algebraic closure of k is kalg. The symbols Q, R, C, and Fp represent, respectively, the field of rational numbers, the field of real numbers, the field of complex numbers, and the finite field containing p elements.
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The term "geometric mechanics" occasionally refers to 17th-century mechanics.Sébastien Maronne, Marco Panza. "Euler, Reader of Newton: Mechanics and Algebraic Analysis". In: Raffaelle Pisano.
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In mathematics, the residue formula says that the sum of the residues of a meromorphic differential form on a smooth proper algebraic curve vanishes.
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Any field extension has a transcendence basis. Thus, field extensions can be split into ones of the form (purely transcendental extensions) and algebraic extensions.
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In algebraic geometry, a Plücker surface, studied by , is a quartic surface in 3-dimensional projective space with a double line and 8 nodes.
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A groupoid is an algebraic structure (G,\ast) consisting of a non-empty set G and a binary partial function '\ast' defined on G.
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String topology, a branch of mathematics, is the study of algebraic structures on the homology of free loop spaces. The field was started by .
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The Birch–Tate conjecture is a conjecture in mathematics (more specifically in algebraic K-theory) proposed by both Bryan John Birch and John Tate.
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In algebraic topology, the mapping spectrum F(X, Y) of spectra X, Y is characterized by :[X \wedge Y, Z] = [X, F(Y, Z)].
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There are, a bit weaker, reconstruction theorems from the derived categories of (quasi)coherent sheaves motivating the derived noncommutative algebraic geometry (see just below).
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Heinrich Wilhelm Ewald Jung (4 May 1876, Essen – 12 March 1953, Halle (Saale)) was a German mathematician, who specialized in geometry and algebraic geometry.
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In algebraic combinatorics, a Bender–Knuth involution is an involution on the set of semistandard tableaux, introduced by in their study of plane partitions.
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Mireille Martin-Deschamps is a French mathematician who studies the algebraic geometry of space curves. She was president of the Société mathématique de France.
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Oberwolfach in 2006 Wolfgang Lück (born 19 February 1957 in Herford) is a German mathematician who is an internationally recognized expert in Algebraic topology.
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In mathematics, the Hasse derivative is a generalisation of the derivative which allows the formulation of Taylor's theorem in coordinate rings of algebraic varieties.
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In mathematics, an arithmetic variety is the quotient space of a Hermitian symmetric space by an arithmetic subgroup of the associated algebraic Lie group.
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In algebraic topology, Johnson–Wilson theory E(n) is a generalized cohomology theory introduced by . Real Johnson–Wilson theory ER(n) was introduced by .
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This isomorphism is true for topological line bundles, the obstruction to injectivity of the Chern class for algebraic vector bundles is the Jacobian variety.
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Another generalization of division to algebraic structures is the quotient group, in which the result of 'division' is a group rather than a number.
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Michel Raynaud (; 16 June 1938 – 10 March 2018) was a French mathematician working in algebraic geometry and a professor at Paris-Sud 11 University.
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In algebra, a Calabi–Yau algebra was introduced by Victor Ginzburg to transport the geometry of a Calabi–Yau manifold to noncommutative algebraic geometry.
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In a unipotent affine algebraic group, all elements are unipotent (see below for the definition of an element being unipotent in such a group).
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Frontiers of Economic Research. Princeton University Press, Princeton, NJ, 1998. xii+272 pp. Zimmermann, U. Linear and combinatorial optimization in ordered algebraic structures. Ann.
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Kenkichi Iwasawa ( Iwasawa Kenkichi, September 11, 1917 – October 26, 1998) was a Japanese mathematician who is known for his influence on algebraic number theory.
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If distinguishes points of and if denotes the range of the injection then is a vector subspace of the algebraic dual space of and the pairing becomes canonically identified with the canonical pairing (where is the natural evaluation map). In particular, in this situation we can assume without loss of generality that is a vector subspace of 's algebraic dual and is the evaluation map. :Convention: Often, whenever is injective (especially when forms a dual pair) then we will use the common practice of assuming without loss of generality that is a vector subspace of the algebraic dual space of , that is the natural evaluation map, and we may also denote by . In a completely analogous manner, if distinguishes points of then it is possible for to be identified as a vector subspace of 's algebraic dual space.
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If the field F is not algebraically closed, the point of view of function fields is a little more general than that of considering the locus of points, since we include, for instance, "curves" with no points on them. For example, if the base field F is the field R of real numbers, then x2 + y2 = −1 defines an algebraic extension field of R(x), but the corresponding curve considered as a subset of R2 has no points. The equation x2 + y2 = −1 does define an irreducible algebraic curve over R in the scheme sense (an integral, separated one-dimensional schemes of finite type over R). In this sense, the one-to-one correspondence between irreducible algebraic curves over F (up to birational equivalence) and algebraic function fields in one variable over F holds in general. Two curves can be birationally equivalent (i.e.
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In algebraic geometry, a generic point P of an algebraic variety X is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic geometry, a generic point of an affine or projective algebraic variety of dimension d is a point such that the field generated by its coordinates has transcendence degree d over the field generated by the coefficients of the equations of the variety. In scheme theory, the spectrum of an integral domain has a unique generic point, which is the minimal prime ideal. As the closure of this point for the Zariski topology is the whole spectrum, the definition has been extended to general topology, where a generic point of a topological space X is a point whose closure is X.
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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 smooth variety. The converse statement, that smooth algebraic varieties have rings of differential operators that are generated by their derivations, is a result of Alexander Grothendieck.. Schreiner cites this converse to EGA 16.11.2. The Nakai conjecture is known to be true for algebraic curves. and Stanley–Reisner rings.. A proof of the conjecture would also establish the Zariski–Lipman conjecture, for a complex variety V with coordinate ring R. This conjecture states that if the derivations of R are a free module over R, then V is smooth..
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For any local ring A there is a universal Henselian ring B generated by A, called the Henselization of A, introduced by , such that any local homomorphism from A to a Henselian ring can be extended uniquely to B. The Henselization of A is unique up to unique isomorphism. The Henselization of A is an algebraic substitute for the completion of A. The Henselization of A has the same completion and residue field as A and is a flat module over A. If A is Noetherian, reduced, normal, regular, or excellent then so is its Henselization. For example, the Henselization of the ring of polynomials k[x,y,...] localized at the point (0,0,...) is the ring of algebraic formal power series (the formal power series satisfying an algebraic equation). This can be thought of as the "algebraic" part of the completion.
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Chow Wei-Liang (; October 1, 1911, Shanghai – August 10, 1995, Baltimore) was a Chinese mathematician born in Shanghai, known for his work in algebraic geometry.
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The cover topics in elementary number theory, algebraic number theory and analytic number theory, including modular arithmetic, quadratic congruences, quadratic reciprocity and binary quadratic forms.
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In category theory the usage of "left" is "right" has some algebraic resemblance, but refers to left and right sides of morphisms. See adjoint functors.
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We will first give the definition for algebraic structures like groups and modules, and then the general definition, which can be used in any category.
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This is in general stronger than being irreducible over the field k, and implies the module is irreducible even over the algebraic closure of k.
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Franklin Paul Peterson (1930–2000) was an American mathematician specializing in algebraic topology. He was a professor of mathematics at the Massachusetts Institute of Technology...
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Yuri Tschinkel (Юрий Чинкель, born 31 May 1964 in Moscow) is a Russian-German- American mathematician, specializing in algebraic geometry, automorphic forms and number theory.
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Cognition and Instruction, 18(2), 209-237. Nathan, M. J., and Koedinger, K. R. (2000). Teachers’ and researchers’ beliefs about the development of algebraic reasoning.
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The first prize in 1991 was awarded to Linda Petzold for DASSL, a differential algebraic equation solver. This code is available in the public domain.
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He is a specialist of algebraic geometry and is best known for his work on motivic integration, part of it in collaboration with Jan Denef.
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In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.
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Paul Dirac used algebraic constructions to produce a relativistic model for the electron, predicting its magnetic moment and the existence of its antiparticle, the positron.
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In good situations, the algebraic character is multiplicative, i.e., the character of the tensor product of two weight modules is the product of their characters.
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Geometric group theory studies the connections between algebraic properties of finitely generated groups and topological and geometric properties of spaces on which these groups act.
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Mónica Clapp is a mathematician at the Universidad Nacional Autónoma de México (UNAM) known for her work in nonlinear partial differential equations and algebraic topology.
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In mathematics, the Borel fixed-point theorem is a fixed-point theorem in algebraic geometry generalizing the Lie–Kolchin theorem. The result was proved by .
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By the spectral theorem a symmetric matrix over the reals is always diagonalizable, and has therefore exactly n real eigenvalues (counted with algebraic multiplicity). Thus .
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Edgar Henry Brown, Jr. (born 27 December 1926) is an American mathematician specializing in algebraic topology, and for many years a professor at Brandeis University.
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Binary operations are the keystone of most algebraic structures, that are studied in algebra, in particular in semigroups, monoids, groups, rings, fields, and vector spaces.
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It was then a short step to define the general notion of a singular point of an algebraic variety; that is, to allow higher dimensions.
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On a theorem of Kontsevich. Algebraic and Geometric Topology, vol. 3 (2003), pp. 1167–1224James Conant, and Karen Vogtmann, Infinitesimal operations on complexes of graphs.
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In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities. They were studied by .
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Landsburg's articles in academic journals have dealt with many fields, including algebraic K-theory, module patching, philosophy of science and, moral philosophy. Selected publications follow.
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In mathematics, a generalized cwatset (GC-set) is an algebraic structure generalizing the notion of closure with a twist, the defining characteristic of the cwatset.
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A triangle center P is called a transcendental triangle center if P has no trilinear representation using only algebraic functions of a, b and c.
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Léon César Autonne (28 July 1859, Odessa – 12 January 1916) was a French engineer and mathematician, specializing in algebraic geometry, differential equations, and linear algebra.
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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.
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Since 1994 he is there Duncan Professor for mathematics. Sommese deals with numerical algebraic geometry (solution of polynomial equation systems) with applications, e.g. in robotics.
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In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field determines a variety over a bigger field, or the pullback of a family of varieties, or a fiber of a family of varieties. Base change is a closely related notion.
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In functional analysis, a branch of mathematics, the algebraic interior or radial kernel of a subset of a vector space is a refinement of the concept of the interior. It is the subset of points contained in a given set with respect to which it is absorbing, i.e. the radial points of the set. The elements of the algebraic interior are often referred to as internal points.
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Once again, the field extension discussed above is a key example: if is not algebraic (i.e., is not a root of a polynomial with coefficients in ), then is isomorphic to . This isomorphism is obtained by substituting to in rational fractions. A subset of a field is a transcendence basis if it is algebraically independent (don't satisfy any polynomial relations) over and if is an algebraic extension of .
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In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case. It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves.
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Queries are parsed, optimised, and executed in the rasdaman server. The parser receives the query string and generates the operation tree. Further, it applies algebraic optimisation rules to the query tree where applicable; of the 150 algebraic rewriting rules, 110 are actually optimising while the other 40 serve to transform the query into canonical form. Parsing and optimization together take less than a millisecond on a laptop.
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The purpose of defining a group homomorphism is to create functions that preserve the algebraic structure. An equivalent definition of group homomorphism is: The function h : G → H is a group homomorphism if whenever a ∗ b = c we have h(a) ⋅ h(b) = h(c). In other words, the group H in some sense has a similar algebraic structure as G and the homomorphism h preserves that.
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The Lighthill-Freeman model is based upon a single ordinary differential equation and one algebraic equation. The five species model is based upon 5 ordinary differential equations and 17 algebraic equations. Because the 5 ordinary differential equations are tightly coupled, the system is numerically "stiff" and difficult to solve. The five species model is only usable for entry from low Earth orbit where entry velocity is approximately .
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In algebraic topology, simplicial complexes are often useful for concrete calculations. For the definition of homology groups of a simplicial complex, one can read the corresponding chain complex directly, provided that consistent orientations are made of all simplices. The requirements of homotopy theory lead to the use of more general spaces, the CW complexes. Infinite complexes are a technical tool basic in algebraic topology.
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They wrote many articles together and had a major impact on the development of algebraic combinatorics. They succeeded in giving a combinatorial understanding of various algebraic and geometric questions in representation theory. Thus they introduced many new objects related to both fields like Schubert polynomials and Grothendieck polynomials. They were also the first to define the crystal graph structure on Young tableaux (though not under this name).
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Nicolas Tadeusz Courtois (born 14 November 1971) is a cryptographer and senior lecturer in computer science at University College London. Courtois was one of the co-authors of both the XSL attack against block ciphers, such as the Advanced Encryption Standard,.. and the XL system for solving systems of algebraic equations. used in the attack. Other cryptographic results of Courtois include algebraic attacks on stream ciphers,.
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One of the classical results in algebraic number theory is that the ideal class group of an algebraic number field K is finite. This is a consequence of Minkowski's theorem since there are only finitely many Integral ideals with norm less than a fixed positive integer page 78. The order of the class group is called the class number, and is often denoted by the letter h.
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He dealt with the problems of cardinality in set theory. During his visiting professorship in Halle, East Germany he contributed to the discovery of the mathematical achievements of Georg Cantor, too. He was the important scholar of the Debrecen algebraic school founded by Tibor Szele. At the Martin Luther University of Halle- Wittenberg he had a great role in the establishment of the Modern algebraic school.
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This is formalized by the notion of Kodaira dimension of a variety, and by this measure projective spaces are the most special varieties, though there are other equally special ones, meaning having negative Kodaira dimension. For algebraic curves, the resulting classification is: projective line, torus, higher genus surfaces (g \geq 2), and similar classifications occur in higher dimensions, notably the Enriques–Kodaira classification of algebraic surfaces.
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An important application of the extended GCD algorithm is that it allows one to compute division in algebraic field extensions. Let an algebraic extension of a field , generated by an element whose minimal polynomial has degree . The elements of are usually represented by univariate polynomials over of degree less than . The addition in is simply the addition of polynomials: :a+_Lb=a+_{K[X]}b.
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Acta Mathematica profile at the Springer Verlag website (retrieved 19 October 2009) Björner is a recognized expert in algebraic and topological combinatorics.Abstract of the CBMS Regional Conference in the Mathematical Sciences - Algebraic and Topological Combinatorics of Ordered Sets - 18 - 22 July 2005 He is a 1983 recipient of the Pólya Prize, and is a member of the Royal Swedish Academy of Sciences since 1999.
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Wallace received degrees in mathematics from Columbia University (BA), University of California at Berkeley (MA) and a Ph.D in 1975 at Tulane University with his dissertation Permutation Groupoids and Circuit Bases: An Algebraic Resolution of Some Graph Structures. Permutation Groupoids and Circuit Bases: An Algebraic Resolution of Some Graph Structures, 1975. Advised by Karl Hofmann. He was a professor at Emory University, DePauw and Boston University.
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In mathematics, an algebraic function field (often abbreviated as function field) of n variables over the field k is a finitely generated field extension K/k which has transcendence degree n over k. Equivalently, an algebraic function field of n variables over k may be defined as a finite field extension of the field K = k(x1,...,xn) of rational functions in n variables over k.
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See also for various other examples in degree 5. Évariste Galois introduced a criterion allowing one to decide which equations are solvable in radicals. See Radical extension for the precise formulation of his result. Algebraic solutions form a subset of closed-form expressions, because the latter permit transcendental functions (non-algebraic functions) such as the exponential function, the logarithmic function, and the trigonometric functions and their inverses.
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In computer vision, Vidal has made many contributions to rigid motion segmentation, activity recognition and dynamic textures. In medical image computing, Vidal developed algorithms for recognition of surgical gestures. In robotics, Vidal developed algorithms for distributed control of unmanned vehicles. In control theory, Vidal studied algebraic conditions for observability of hybrid systems as well as algebraic geometric approaches for the identification of hybrid systems.
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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.
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Armand Borel (21 May 1923 - 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in algebraic topology, in the theory of Lie groups, and was one of the creators of the contemporary theory of linear algebraic groups.
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Every algebraic extension of a field of characteristic zero is separable, and every algebraic extension of a finite field is separable.Isaacs, Theorem 18.11, p. 281 It follows that most extensions that are considered in mathematics are separable. Nevertheless, the concept of separability is important, as the existence of inseparable extensions is the main obstacle for extending many theorems proved in characteristic zero to non-zero characteristic.
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300 The separable closure of in an algebraic closure of is simply called the separable closure of . Like the algebraic closure, it is unique up to an isomorphism, and in general, this isomorphism is not unique. A field extension E\supseteq F is separable, if is the separable closure of in . This is the case if and only if is generated over by separable elements.
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Gunnar E. Carlsson (born August 22, 1952 in Stockholm, Sweden) is an American mathematician, working in algebraic topology. He is known for his work on the Segal conjecture, and for his work on applied algebraic topology, especially topological data analysis. He is a Professor Emeritus in the Department of Mathematics at Stanford University. He is the founder and president of the predictive technology company Ayasdi.
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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 also a projective line. Then the theorem states that S is rational.
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Every connected closed 2-dimensional manifold (surface) admits a constant curvature metric, by the uniformization theorem. There are 3 such curvatures (positive, zero, and negative). This is a classical result, and as stated, easy (the full uniformization theorem is subtler). The study of surfaces is deeply connected with complex analysis and algebraic geometry, as every orientable surface can be considered a Riemann surface or complex algebraic curve.
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In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation :, such that is a point on the curve. Here denotes the -invariant. The curve is sometimes called , though often that is used for the abstract algebraic curve for which there exist various models. A related object is the classical modular polynomial, a polynomial in one variable defined as .
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Hironaka's variety is a smooth 3-dimensional complete variety but is not projective as it has a non-trivial curve algebraically equivalent to 0. Any 2-dimensional smooth complete variety is projective, so 3 is the smallest possible dimension for such an example. There are plenty of 2-dimensional complex manifolds that are not algebraic, such as Hopf surfaces (non Kähler) and non-algebraic tori (Kähler).
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In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants of spaces, such as their fundamental weak ∞-groupoid.
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Over characteristic 0 there is a nice decomposition theorem of an algebraic group G relating its structure to the structure of a linear algebraic group and an Abelian variety. There is a short exact sequence of groupspage 8 > 0 \to M\times U \to G \to A \to 0 where A is an abelian variety, M is of multiplicative type, meaning, and U is a unipotent group.
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Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called variety of groups.
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Because classical and tropical geometry are closely related, results and methods can be converted between them. Algebraic varieties can be mapped to a tropical counterpart and, since this process still retains some geometric information about the original variety, it can be used to help prove and generalize classical results from algebraic geometry, such as the Brill–Noether theorem, using the tools of tropical geometry.
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Libgober's early work studies the diffeomorphism type of complete intersections in complex projective space. This later led to the discovery of relations between Hodge and Chern numbers.A.Libgober, J.Wood, Differentiable structures on complete intersections I, Topology, 21 (1982),469-482 He introduced the technique of Alexander polynomialA.Libgober,Development of the theory of Alexander invariants in algebraic geometry, Topology of algebraic varieties and singularities, 3–17, Contemp. Math.
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Originally, algebraic geometry was the study of common zeros of sets of multivariate polynomials. These common zeros, called algebraic varieties belong to an affine space. It appeared soon, that in the case of real coefficients, one must consider all the complex zeros for having accurate results. For example, the fundamental theorem of algebra asserts that a univariate square-free polynomial of degree has exactly complex roots.
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Ringed spaces appear in analysis as well as complex algebraic geometry and scheme theory of algebraic geometry. Note: In the definition of a ringed space, most expositions tend to restrict the rings to be commutative rings, including Hartshorne and Wikipedia. "Éléments de géométrie algébrique", on the other hand, does not impose the commutativity assumption, although the book mostly considers the commutative case.EGA, Ch 0, 4.1.1.
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The following example is neither a hypersurface, nor a linear space, nor a single point. Let A3 be the three-dimensional affine space over C. The set of points (x, x2, x3) for x in C is an algebraic variety, and more precisely an algebraic curve that is not contained in any plane.Harris, p.9; that it is irreducible is stated as an exercise in Hartshorne p.
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In the mid-1970s, Enright introduced new methods that led him to an algebraic way of looking at discrete series (which were fundamental representations constructed by Harish-Chandra in the early 1960s), and to an algebraic proof of the Blattner multiplicity formula. He was known for Enright–Varadarajan modules, Enright resolutions, and the Enright completion functor, which has had a lasting influence in algebra.
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Among the many books that Grave wrote were Theory of Finite Groups (1910) and A Course in Algebraic Analysis (1932). He also studied the history of algebraic analysis. Among the honours that were given to him was election to the Academy of Sciences of Ukraine in 1919, election to the Shevchenko Scientific Society in 1923 and election to the Academy of Sciences of the USSR in 1929.
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Later mathematicians introduced algebraic methods, which transform a geometric problem into algebraic equations. These methods were simplified by exploiting symmetries inherent in the problem of Apollonius: for instance solution circles generically occur in pairs, with one solution enclosing the given circles that the other excludes (Figure 2). Joseph Diaz Gergonne used this symmetry to provide an elegant straightedge and compass solution, while other mathematicians used geometrical transformations such as reflection in a circle to simplify the configuration of the given circles. These developments provide a geometrical setting for algebraic methods (using Lie sphere geometry) and a classification of solutions according to 33 essentially different configurations of the given circles.
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A linear system of divisors algebraicizes the classic geometric notion of a family of curves, as in the Apollonian circles. In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family. These arose first in the form of a linear system of algebraic curves in the projective plane. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of divisors D on a general scheme or even a ringed space (X, OX).
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Using the intrinsic concept of tangent space, points P on an algebraic curve C are classified as smooth (synonymous: non- singular), or else singular. Given n−1 homogeneous polynomials in n+1 variables, we may find the Jacobian matrix as the (n−1)×(n+1) matrix of the partial derivatives. If the rank of this matrix is n−1, then the polynomials define an algebraic curve (otherwise they define an algebraic variety of higher dimension). If the rank remains n−1 when the Jacobian matrix is evaluated at a point P on the curve, then the point is a smooth or regular point; otherwise it is a singular point.
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Hypersurfaces have some specific properties that are not shared with other algebraic varieties. One of the main such properties is Hilbert's Nullstellensatz, which asserts that a hypersurface contains an algebraic set if and only if the defining polynomial of the hypersurface has a power that belongs to the ideal generated by the defining polynomials of the algebraic set. A corollary of this theorem is that, if two irreducible polynomials (or more generally two square-free polynomials) define the same hypersurface, then one is the product of the other by a nonzero constant. Hypersurfaces are exactly the subvarieties of dimension of an affine space of dimension of .
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When a new problem can be shown to follow the laws of one of these algebraic structures, all the work that has been done on that category in the past can be applied to the new problem. In full generality, algebraic structures may involve an arbitrary collection of operations, including operations that combine more than two elements (higher arity operations) and operations that take only one argument (unary operations). The examples used here are by no means a complete list, but they are meant to be a representative list and include the most common structures. Longer lists of algebraic structures may be found in the external links and within :Category:Algebraic structures.
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The terminology of algebraic geometry changed drastically during the twentieth century, with the introduction of the general methods, initiated by David Hilbert and the Italian school of algebraic geometry in the beginning of the century, and later formalized by André Weil, Jean-Pierre Serre and Alexander Grothendieck. Much of the classical terminology, mainly based on case study, was simply abandoned, with the result that books and papers written before this time can be hard to read. This article lists some of this classical terminology, and describes some of the changes in conventions. translates many of the classical terms in algebraic geometry into scheme-theoretic terminology.
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The feature that the numeric lacks is the ability to solve algebraic equations such as indefinite integrals and derivatives. To fill in the gap of needing an algebraic calculator, Texas Instruments introduced the second model with the name TI-Nspire CAS. The CAS is designed for college and university students, giving them the feature of calculating many algebraic equations like the Voyage 200 and TI-89 (which the TI-Nspire was intended to replace). However, the TI-Nspire does lack part of the ability of programming and installing additional apps that the previous models had, although a limited version of TI-BASIC is supported, along with Lua in later versions.
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The conjecture of Kontsevich and Zagier would imply that equality of periods is also decidable: inequality of computable reals is known recursively enumerable; and conversely if two integrals agree, then an algorithm could confirm so by trying all possible ways to transform one of them into the other one. It is not expected that Euler's number e and Euler-Mascheroni constant γ are periods. The periods can be extended to exponential periods by permitting the product of an algebraic function and the exponential function of an algebraic function as an integrand. This extension includes all algebraic powers of e, the gamma function of rational arguments, and values of Bessel functions.
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In mathematics, the Neukirch–Uchida theorem shows that all problems about algebraic number fields can be reduced to problems about their absolute Galois groups. showed that two algebraic number fields with the same absolute Galois group are isomorphic, and strengthened this by proving Neukirch's conjecture that automorphisms of the algebraic number field correspond to outer automorphisms of its absolute Galois group. extended the result to infinite fields that are finitely generated over prime fields. The Neukirch–Uchida theorem is one of the foundational results of anabelian geometry, whose main theme is to reduce properties of geometric objects to properties of their fundamental groups, provided these fundamental groups are sufficiently non- abelian.
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In 1876 Alfred B. Kempe published his article On a General Method of describing Plane Curves of the nth degree by Linkwork, which showed that for an arbitrary algebraic plane curve a linkage can be constructed that draws the curve. This direct connection between linkages and algebraic curves has been named Kempe's universality theoremA. Saxena (2011) Kempe’s Linkages and the Universality Theorem , RESONANCE that any bounded subset of an algebraic curve may be traced out by the motion of one of the joints in a suitably chosen linkage. Kempe's proof was flawed and the first complete proof was provided in 2002 based on his ideas.
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A set can be regarded as an algebra having no nontrivial operations or defining equations. From this perspective, the idea of the power set of as the set of subsets of generalizes naturally to the subalgebras of an algebraic structure or algebra. The power set of a set, when ordered by inclusion, is always a complete atomic Boolean algebra, and every complete atomic Boolean algebra arises as the lattice of all subsets of some set. The generalization to arbitrary algebras is that the set of subalgebras of an algebra, again ordered by inclusion, is always an algebraic lattice, and every algebraic lattice arises as the lattice of subalgebras of some algebra.
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36 T: H ⊗ H → H ⊗ H is defined by T(x ⊗ y) = y ⊗ x). Other interesting Hopf algebras are certain "deformations" or "quantizations" of those from example 3 which are neither commutative nor co- commutative. These Hopf algebras are often called quantum groups, a term that is so far only loosely defined. They are important in noncommutative geometry, the idea being the following: a standard algebraic group is well described by its standard Hopf algebra of regular functions; we can then think of the deformed version of this Hopf algebra as describing a certain "non-standard" or "quantized" algebraic group (which is not an algebraic group at all).
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Like polynomials, rational expressions can also be generalized to n indeterminates X1,..., Xn, by taking the field of fractions of F[X1,..., Xn], which is denoted by F(X1,..., Xn). An extended version of the abstract idea of rational function is used in algebraic geometry. There the function field of an algebraic variety V is formed as the field of fractions of the coordinate ring of V (more accurately said, of a Zariski-dense affine open set in V). Its elements f are considered as regular functions in the sense of algebraic geometry on non-empty open sets U, and also may be seen as morphisms to the projective line.
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Another fundamental result is the Golod–Shafarevich theorem on towers of unramified extensions of number fields. Shafarevich and his school greatly contributed to the study of algebraic geometry of surfaces. He initiated a Moscow seminar on classification of algebraic surfaces that updated the treatment of birational geometry around 1960 and was largely responsible for the early introduction of the scheme theory approach to algebraic geometry in the Soviet school. His investigation in arithmetic of elliptic curves led him, independently of John Tate, to the introduction of the most mysterious group related to elliptic curves over number fields, the Tate–Shafarevich group (usually called 'Sha', written 'Ш', his Cyrillic initial).
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At each point of the projective variety, all the polynomials in the set were required to equal zero. The complement of the zero set of a linear polynomial is an affine space, and an affine variety was the intersection of a projective variety with an affine space. André Weil saw that geometric reasoning could sometimes be applied in number-theoretic situations where the spaces in question might be discrete or even finite. In pursuit of this idea, Weil rewrote the foundations of algebraic geometry, both freeing algebraic geometry from its reliance on complex numbers and introducing abstract algebraic varieties which were not embedded in projective space.
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"Max Deuring's lectures gave him a taste for algebraic number theory. He studied not only algebraic geometry and analytic number theory of which he displayed a deep knowledge but he became an expert in several other allied subjects as well". On the suggestion of his doctoral advisor, K. G. Ramanathan, he began working on a problem relating to the work of the German number theorist Carl Ludwig Siegel. In the course of proving the main result to the effect that every cubic form in 54 variables over any algebraic number field K had a non-trivial zero over that field, he had also simplified the earlier method of Siegel.
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Towards the end of the century, the Italian school of algebraic geometry (Enriques, Segre, Severi) broke out of the traditional subject matter into an area demanding deeper techniques. During the later part of the 19th century, the detailed study of projective geometry became less fashionable, although the literature is voluminous. Some important work was done in enumerative geometry in particular, by Schubert, that is now considered as anticipating the theory of Chern classes, taken as representing the algebraic topology of Grassmannians. Paul Dirac studied projective geometry and used it as a basis for developing his concepts of quantum mechanics, although his published results were always in algebraic form.
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In mathematics, a Raynaud surface is a particular kind of algebraic surface that was introduced in and named for . To be precise, a Raynaud surface is a quasi-elliptic surface over an algebraic curve of genus g greater than 1, such that all fibers are irreducible and the fibration has a section. The Kodaira vanishing theorem fails for such surfaces; in other words the Kodaira theorem, valid in algebraic geometry over the complex numbers, has such surfaces as counterexamples, and these can only exist in characteristic p. Generalized Raynaud surfaces were introduced in , and give examples of surfaces of general type with global vector fields.
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In the 20th century designs were applied to the design of experiments, notably Latin squares, finite geometry, and association schemes, yielding the field of algebraic statistics.
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In algebraic data types, a constructor is one of many tags that wrap data. If a constructor does not take any data arguments, it is nullary.
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Ergodic theory has fruitful connections with harmonic analysis, Lie theory (representation theory, lattices in algebraic groups), and number theory (the theory of diophantine approximations, L-functions).
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In both the and elements, coordinate-dependent formulas can be used instead of constants. These formulas can use various standard algebraic and mathematical operators and expressions.
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In mathematics, the Schottky problem, named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties.
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The Council of Scientific and Industrial Research awarded him the Shanti Swarup Bhatnagar Prize for Science and Technology in 1991 for his work in algebraic geometry.
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Formulas for local and global differents and discriminants, ramification theory, and the formula for the genus of an algebraic extension of a function field are developed.
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In algebraic geometry, the Cayleyan is a variety associated to a hypersurface by , who named it the pippian in and also called it the Steiner–Hessian.
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He co-authored the book "Algebraic Set Theory" with Ieke Moerdijk and recently started a web-based expositional project Joyal's CatLab Joyal's CatLab on categorical mathematics.
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