commutative algebra

Given a linear differential operator the adjoint of this operator is defined as the operator T * such that where the notation is used for the scalar product or inner product. This definition therefore depends on the definition of the scalar product. In the functional space of square integrable functions, the scalar product is defined by . If one moreover adds the condition that f and g vanish for and , one ...

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Differential operator, Differential operator - Notations, Differential operator - Adjoint of an operator, Differential operator - Properties of differential operators, Differential operator - Several variables, Differential operator - Coordinate-independent description, Differential operator - Examples

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Again, let f(x) be a real-valued function of a real variable. Then f is odd if the following equation holds for all real x: f(−x) = −f(x) Geometrically, an odd function is symmetric with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin. The designation odd is also not due to the fact that the Taylor series of an odd function includes only odd powers. It is only by mathematical coincidence that this is true. Examples of odd functions are x, ...

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Even and odd functions, Even and odd functions - Even functions, Even and odd functions - Odd functions, Even and odd functions - Some facts, Even and odd functions - Basic properties, Even and odd functions - Series, Even and odd functions - Algebraic structure

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It is quite clear that the problem list, and its manner of discussion, were meant to be influential. Hilbert in no way fell short of the expectations of German academia on empire-building, programmatic verve, and the explicit setting of a direction and claiming of ground for a school. No one now talks of the 'Hilbert school' in quite those terms; nor did the Hilbert problems just have their moment as Felix Klein's Erlangen programme did. Klein was a colleague of Hilbert's, and in comparison the Hilbert list is far less prescriptive. M ...

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Hilbert's problems, Hilbert's problems - Nature and influence of the problems, Hilbert's problems - The problems as Hilbert's manifesto, Hilbert's problems - A round two dozen, Hilbert's problems - Summary, Hilbert's problems - Tabulated information, Hilbert's problems - Footnotes

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From 1959 onwards his interests turned towards number theory, in particular class field theory and the theory of complex multiplication. Amongst his most original contributions were: the concept of algebraic K-theory; the Galois representation theory for l-adic cohomology and the conceptions that these representations were 'large'; and the Serre conjecture on mod p representations that made Fermat's last theorem a c ...

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Jean-Pierre Serre, Jean-Pierre Serre - Life and career, Jean-Pierre Serre - Early work, Jean-Pierre Serre - Foundational work in algebraic geometry and the Weil conjectures, Jean-Pierre Serre - Other work, Jean-Pierre Serre - Awards, Jean-Pierre Serre - External link

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In the 1950s and 1960s, a fruitful collaboration between Serre and the two years younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were FAC (Faisceaux Algébriques Cohérents, on coherent cohomology) and GAGA. Serre had early on perceived a need to construct more general and refined cohomology theories to tackle these conjectures. In simple terms, the cohomology of a coherent sheaf over a finite field couldn't capture as much t ...

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Jean-Pierre Serre, Jean-Pierre Serre - Life and career, Jean-Pierre Serre - Early work, Jean-Pierre Serre - Foundational work in algebraic geometry and the Weil conjectures, Jean-Pierre Serre - Other work, Jean-Pierre Serre - Awards, Jean-Pierre Serre - External link

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Of the cleanly-formulated Hilbert problems, problems 3, 7, 10, 11, 13, 14, 17, 19 and 20 have a resolution that is accepted by consensus. On the other hand problems 1, 2, 5, 9, 15, 18+, 21, 22, have solutions that have partial acceptance, but where there exists some controversy as to whether it resolves the problem. The + on 18 denotes that the Kepler problem solution is a computer-assisted proof, a notion anachronistic for a Hilbert problem and also to some extent controversial because of its lack of verifia ...

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Hilbert's problems, Hilbert's problems - Nature and influence of the problems, Hilbert's problems - The problems as Hilbert's manifesto, Hilbert's problems - A round two dozen, Hilbert's problems - Summary, Hilbert's problems - Tabulated information, Hilbert's problems - Footnotes

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Public discussion of, and justification for, Bourbaki's thoughts has in general been through Jean Dieudonné, who initially was the 'scribe' of the group, writing under his own name. In a survey of le choix bourbachique written in 1977, he didn't shy away from a hierarchical development of the 'important' mathematics of the time. He also wrote extensively under his own name: nine volumes on analysis, perhaps in belated fulfilment of the original project or pretext; and also on other topics mostly connected with algebraic geomet ...

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Nicolas Bourbaki, Nicolas Bourbaki - Books by Bourbaki, Nicolas Bourbaki - Influence on mathematics in general, Nicolas Bourbaki - The group, Nicolas Bourbaki - The Bourbaki perspective and its limitations, Nicolas Bourbaki - Dieudonné as speaker for Bourbaki, Nicolas Bourbaki - The Bourbachique influence: education institutions trends

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The underlying drive, in Weil and Chevalley at least, was the perceived need for French mathematics to absorb the best ideas of the Göttingen school and the German algebraists. It is fairly clear that the Bourbaki point of view, while encyclopedic, was never intended as neutral. Quite the opposite, really: more a question of trying to make a consistent whole out of some enthusiasms, for example for Hilbert's legacy, with emphasis on formalism and axiomatics. But always through a transforming process ...

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Nicolas Bourbaki, Nicolas Bourbaki - Books by Bourbaki, Nicolas Bourbaki - Influence on mathematics in general, Nicolas Bourbaki - The group, Nicolas Bourbaki - The Bourbaki perspective and its limitations, Nicolas Bourbaki - Dieudonné as speaker for Bourbaki, Nicolas Bourbaki - The Bourbachique influence: education institutions trends

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Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on differentiable manifolds, there is a natural class of functions on an algebraic set, called regular functions. A regular function on an algebraic set V contained in is defined to be the restriction of a regular function on , in the sense we defined above. It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal ...

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Algebraic geometry, Algebraic geometry - Zeroes of simultaneous polynomials, Algebraic geometry - Affine varieties, Algebraic geometry - Regular functions, Algebraic geometry - The category of affine varieties, Algebraic geometry - Projective space, Algebraic geometry - The modern viewpoint, Algebraic geometry - Notes and history

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In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the two-dimensional sphere in three-dimensional Euclidean space could be defined as the set of all points (x,y,z) with x2See also:

Algebraic geometry, Algebraic geometry - Zeroes of simultaneous polynomials, Algebraic geometry - Affine varieties, Algebraic geometry - Regular functions, Algebraic geometry - The category of affine varieties, Algebraic geometry - Projective space, Algebraic geometry - The modern viewpoint, Algebraic geometry - Notes and history

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Alexander Grothendieck - Childhood and studies. Born to a Russian Jewish father and German Protestant mother in Berlin, he was a displaced person during much of his childhood due to the upheavals of World War II. Alexander lived with his father, Alexander Shapiro, and his mother, Hanka Grothendieck, both of whom were socialist revolutionaries. Until 1933 they lived together in Berlin. At the end of that year, Shapiro moved to Paris, and Hanka followed him the next year. They left Alexander with a family in Hambur ...

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Alexander Grothendieck, Alexander Grothendieck - Mathematical achievements, Alexander Grothendieck - Major mathematical topics from Récoltes et Semailles, Alexander Grothendieck - Life, Alexander Grothendieck - Childhood and studies, Alexander Grothendieck - Politics and retreat from scientific community, Alexander Grothendieck - Manuscripts written in the 1980s, Alexander Grothendieck - Disappearance

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Given any pair A, B of objects in an abelian category, there is a special zero morphism from A to B. This can be defined as the zero element of the hom-set Hom(A,B), since this is an abelian group. Alternatively, it can be defined as the unique composition A → 0 → B, where 0 is the zero object of the abelian category. In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism. This epimorphism is called the coimage of f, whil ...

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Abelian category, Abelian category - Definitions, Abelian category - Examples, Abelian category - Elementary properties, Abelian category - Related concepts, Abelian category - History

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Using regular functions from an affine variety to , we can define regular functions from one affine variety to another. First we will define a regular function from a variety into affine space: Let V be a variety contained in . Choose m regular functions on V, and call them f1,...,fm. We define a regular function f from V to by letting f(t1,...,tn)=(f1,...,fm). In other words, each fi determines one coordinate of the range of f. If V' is a variety contained in , we say that f is a ...

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Algebraic geometry, Algebraic geometry - Zeroes of simultaneous polynomials, Algebraic geometry - Affine varieties, Algebraic geometry - Regular functions, Algebraic geometry - The category of affine varieties, Algebraic geometry - Projective space, Algebraic geometry - The modern viewpoint, Algebraic geometry - Notes and history

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Consider the variety V(y=x2). If we draw it, we get a parabola. As x increases, the slope of the line from the origin to the point (x,x2) becomes larger and larger. As x decreases, the slope of the same line becomes smaller and smaller. Compare this to the variety V(y=x3). This is a cubic equation. As x increases, the slope of the line from the origin to the point (x,x3) becomes larger and larger just as before. But unlike before, as x decreases, the slope of the same line again becomes larg ...

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Algebraic geometry, Algebraic geometry - Zeroes of simultaneous polynomials, Algebraic geometry - Affine varieties, Algebraic geometry - Regular functions, Algebraic geometry - The category of affine varieties, Algebraic geometry - Projective space, Algebraic geometry - The modern viewpoint, Algebraic geometry - Notes and history

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Accounts of the early days vary, but original documents have now come to light. The founding members were all connected to the Ecole Normale Supérieure in Paris and included Henri Cartan, Claude Chevalley, Jean Coulomb, Jean Delsarte, Jean Dieudonné, Charles Ehresmann, René de Possel, Szolem Mandelbrojt and André Weil. There was a preliminary meeting, towards the end of 1934 (the minutes are in the Bourbaki archives — for a full description of the initial meeting consult Liliane Beaulieu in the Mathematical Intelligencer). Jean ...

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Nicolas Bourbaki, Nicolas Bourbaki - Books by Bourbaki, Nicolas Bourbaki - Influence on mathematics in general, Nicolas Bourbaki - The group, Nicolas Bourbaki - The Bourbaki perspective and its limitations, Nicolas Bourbaki - Dieudonné as speaker for Bourbaki, Nicolas Bourbaki - The Bourbachique influence: education institutions trends

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The emphasis on rigour may be seen as a reaction to the work of Jules-Henri Poincaré, who stressed the importance of free-flowing mathematical intuition, at a cost in completeness in presentation. The impact of Bourbaki's work initially was great on many active research mathematicians world-wide. It provoked some hostility, too, mostly on the side of classical analysts; they approved of rigour but not of high abstraction. Around 1950, also, some parts of geometry were still not fully axiomatic — in less prominent developments, one ...

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Nicolas Bourbaki, Nicolas Bourbaki - Books by Bourbaki, Nicolas Bourbaki - Influence on mathematics in general, Nicolas Bourbaki - The group, Nicolas Bourbaki - The Bourbaki perspective and its limitations, Nicolas Bourbaki - Dieudonné as speaker for Bourbaki, Nicolas Bourbaki - The Bourbachique influence: education institutions trends

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Algebraic geometry was developed largely by the Italian geometers in the early part of the 20th century. Enriques classified algebraic surfaces up to birational isomorphism. The style of the Italian school was very intuitive and does not meet the modern standards of rigor. By the 1930s and 1940s, Oscar Zariski, André Weil and others realized that algebraic geometry needed to be rebuilt on foundations of commutative algebra and valuation theory. Commutative algebra (earlier known as elimination theory and then ideal theory, and refoun ...

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Algebraic geometry, Algebraic geometry - Zeroes of simultaneous polynomials, Algebraic geometry - Affine varieties, Algebraic geometry - Regular functions, Algebraic geometry - The category of affine varieties, Algebraic geometry - Projective space, Algebraic geometry - The modern viewpoint, Algebraic geometry - Notes and history

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A ring R is a local ring if it has one (and therefore all) of the following equivalent properties: R has a unique maximal left ideal. R has a unique maximal right ideal. 1≠0 and the sum of any two non-units in a R is a non-unit. 1≠0 and if x is any element of R, then x or 1−x is a unit. If a finite sum is a unit, then so are some of its terms ...

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Local ring, Local ring - Definition and first consequences, Local ring - Examples, Local ring - Commutative, Local ring - Non-commutative, Local ring - Some facts and definitions, Local ring - Commutative, Local ring - General

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Local ring - Commutative. We also write (R, m) for a commutative local ring R with maximal ideal m. Every such ring becomes a topological ring in a natural way if one takes the powers of m as a neighborhood base of 0. This is the m-adic topology on R. If (R, m) and (S, n) are local rings, then a local ring homomorphism from R to S is a ring homomorphism f : R → S with the property < ...

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Local ring, Local ring - Definition and first consequences, Local ring - Examples, Local ring - Commutative, Local ring - Non-commutative, Local ring - Some facts and definitions, Local ring - Commutative, Local ring - General

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In abstract algebra, one must take care to distinguish between polynomials and polynomial functions. A polynomial f is defined to be a formal expression of the form where the coefficients a0, ..., an are elements of some ring R and X is considered to be a formal symbol. Two polynomials are considered to be equal if and only if the sequences of their coefficients are equal. Polynomials with coefficients in R can be added by simply ...

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Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

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In multivariate calculus, polynomials in several variables play an important role. These are the simplest multivariate functions and can be defined using addition and multiplication alone. An example of a polynomial in the variables x, y, and z is The total degree of such a multivariate polynomial can be found by adding the exponents of the variables in every term, and taking the maximum. The above polynomial f(x,&# ...

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Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

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Polynomial - Polynomials and calculus. One important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomials. The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Stone-Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial. Polynomials a ...

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Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

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