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Harnack's Curve Theorem
In real algebraic geometry, Harnack's curve theorem, named after Axel Harnack, gives the possible numbers of connected components that an algebraic curve can have, in terms of the degree of the curve. For any algebraic curve of degree in the real projective plane, the number of components is bounded by :\frac \le c \le \frac+1.\ The maximum number is one more than the maximum genus of a curve of degree , attained when the curve is nonsingular. Moreover, any number of components in this range of possible values can be attained. A curve which attains the maximum number of real components is called an M-curve (from "maximum") – for example, an elliptic curve with two components, such as y^2=x^3-x, or the Trott curve, a quartic with four components, are examples of M-curves. This theorem formed the background to Hilbert's sixteenth problem. In a recent development a Harnack curve is shown to be a curve whose amoeba has area equal to the Newton polygon of the polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quartic Plane Curve
In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one of not equal to zero. This equation has 15 constants. However, it can be multiplied by any non-zero constant without changing the curve; thus by the choice of an appropriate constant of multiplication, any one of the coefficients can be set to 1, leaving only 14 constants. Therefore, the space of quartic curves can be identified with the real projective space It also follows, from Cramer's theorem on algebraic curves, that there is exactly one quartic curve that passes through a set of 14 distinct points in general position, since a quartic has 14 degrees of freedom. A quartic curve can have a maximum of: * Four connected components * Twenty-eight bi-tangents * Three ordinary double points. One may also consider quartic curves over othe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology (journal)
''Topology'' was a peer-reviewed mathematical journal covering topology and geometry. It was established in 1962 and was published by Elsevier. The last issue of ''Topology'' appeared in 2009. Pricing dispute On 10 August 2006, after months of unsuccessful negotiations with Elsevier about the price policy of library subscriptions, the entire editorial board of the journal handed in their resignation, effective 31 December 2006. Subsequently, two more issues appeared in 2007 with papers that had been accepted before the resignation of the editors. In early January the former editors instructed Elsevier to remove their names from the website of the journal, but Elsevier refused to comply, justifying their decision by saying that the editorial board should remain on the journal until all of the papers accepted during its tenure had been published. In 2007 the former editors of ''Topology'' announced the launch of the ''Journal of Topology'', published by Oxford University Press ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dmitry Gudkov (mathematician)
Dmitrii Andreevich Gudkov (1918–1992; alternative spelling Dmitry) was a Soviet mathematician famous for his work on Hilbert's sixteenth problem and the related Gudkov's conjecture in algebraic geometry. He was a student of Aleksandr Andronov.Jeremy Gray – ''The Hilbert Challenge'', p. 147 Selected papers *D. A. Gudkov, "The topology of real projective algebraic varieties", ''Russian Mathematical Surveys'', 1974, 29 (4), pp. 1–79 (translated from the Russian original). *D. A. Gudkov "Periodicity of the Euler characteristic of real algebraic (M—1)-manifolds", ''Functional Analysis and Its Applications'', April–June, 1973, Volume 7, Issue 2, pp. 98–102 (translated from the Russian original). *D.A Gudkov. "Ovals of sixth order curves". in the book ''Nine Papers on Hilbert's 16th Problem'' ''American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimer Model
In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by dominoes, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a perfect matching in the grid graph formed by placing a vertex at the center of each square of the region and connecting two vertices when they correspond to adjacent squares. Height functions For some classes of tilings on a regular grid in two dimensions, it is possible to define a height function associating an integer to the vertices of the grid. For instance, draw a chessboard, fix a node A_0 with height 0, then for any node there is a path from A_0 to it. On this path define the height of each node A_ (i.e. corners of the squares) to be the height of the previous node A_n plus one if the square on the right of the path from A_n to A_ is black, and minus one otherwise. More details can be found in . Thurston's height condition describes a test for determining whether a simply- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' join ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton Polygon
In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields. In the original case, the local field of interest was ''essentially'' the field of formal Laurent series in the indeterminate ''X'', i.e. the field of fractions of the formal power series ring K X, over K, where K was the real number or complex number field. This is still of considerable utility with respect to Puiseux expansions. The Newton polygon is an effective device for understanding the leading terms aX^r of the power series expansion solutions to equations P(F(X)) = 0 where P is a polynomial with coefficients in K /math>, the polynomial ring; that is, implicitly defined algebraic functions. The exponents r here are certain rational numbers, depending on the branch chosen; and the solutions themselves are power series in K Y with Y = X^ for a denominator d corresponding to the branch. The Newton polygon gives an effective, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amoeba (mathematics)
In complex analysis, a branch of mathematics, an amoeba is a set associated with a polynomial in one or more complex variables. Amoebas have applications in algebraic geometry, especially tropical geometry. Definition Consider the function : \operatorname: \big( \setminus \\big)^n \to \mathbb R^n defined on the set of all ''n''-tuples z = (z_1, z_2, \dots, z_n) of non-zero complex numbers with values in the Euclidean space \mathbb R^n, given by the formula : \operatorname(z_1, z_2, \dots, z_n) = \big(\log, z_1, , \log, z_2, , \dots, \log, z_n, \big). Here, log denotes the natural logarithm. If ''p''(''z'') is a polynomial in n complex variables, its amoeba \mathcal A_p is defined as the image of the set of zeros of ''p'' under Log, so : \mathcal A_p = \left\. Amoebas were introduced in 1994 in a book by Gelfand, Kapranov, and Zelevinsky. Properties * Any amoeba is a closed set. * Any connected component of the complement \mathbb R^n \setminus \mathcal A_p is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harnack Curve
Harnack is the surname of a German family of intellectuals, artists, mathematicians, scientists, theologians and those in other fields. Several family members were executed by the Nazis during the last years of the Third Reich. * Theodosius Harnack (1817–1889), German theologian :*Anna Harnack (1849–?) :* Adolf von Harnack (1851–1930), German liberal theologian and historian of religion ::*Agnes von Zahn-Harnack (1884–1950), German writer and women's rights activist ::* Ernst von Harnack (1888–1945), German anti-Nazi resistance fighter :::* Gustav-Adolf von Harnack (1917-2010), German pediatrician ::* Elisabet von Harnack (1892–1976), German social worker ::* Axel von Harnack (1895–1974), German historian and philologist :* Carl Gustav Axel Harnack (1851–1888), German mathematician :* Erich Harnack, professor of pharmacology :* Otto Harnack, literature historian ::* Clara Harnack, painter and wife of Otto :::* Arvid Harnack (1901–1942), German anti-Nazi resista ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Sixteenth Problem
Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, as part of his list of 23 problems in mathematics. The original problem was posed as the ''Problem of the topology of algebraic curves and surfaces'' (''Problem der Topologie algebraischer Kurven und Flächen''). Actually the problem consists of two similar problems in different branches of mathematics: * An investigation of the relative positions of the branches of real algebraic curves of degree ''n'' (and similarly for algebraic surfaces). * The determination of the upper bound for the number of limit cycles in two-dimensional polynomial vector fields of degree ''n'' and an investigation of their relative positions. The first problem is yet unsolved for ''n'' = 8. Therefore, this problem is what usually is meant when talking about Hilbert's sixteenth problem in real algebraic geometry. The second problem also remains unsolved: no upp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trott Curve
In the theory of algebraic plane curves, a general quartic plane curve has 28 bitangent lines, lines that are tangent to the curve in two places. These lines exist in the complex projective plane, but it is possible to define quartic curves for which all 28 of these lines have real numbers as their coordinates and therefore belong to the Euclidean plane. An explicit quartic with twenty-eight real bitangents was first given by As Plücker showed, the number of real bitangents of any quartic must be 28, 16, or a number less than 9. Another quartic with 28 real bitangents can be formed by the locus of centers of ellipses with fixed axis lengths, tangent to two non-parallel lines. gave a different construction of a quartic with twenty-eight bitangents, formed by projecting a cubic surface; twenty-seven of the bitangents to Shioda's curve are real while the twenty-eighth is the line at infinity in the projective plane. Example The Trott curve, another curve with 28 real bitangents, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
