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Bitangents Of A Quartic
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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Curve
In projective geometry, a dual curve of a given plane curve is a curve in the dual projective plane consisting of the set of lines tangent to . There is a map from a curve to its dual, sending each point to the point dual to its tangent line. If is algebraic then so is its dual and the degree of the dual is known as the ''class'' of the original curve. The equation of the dual of , given in line coordinates, is known as the ''tangential equation'' of . Duality is an involution: the dual of the dual of is the original curve . The construction of the dual curve is the geometrical underpinning for the Legendre transformation in the context of Hamiltonian mechanics. Equations Let be the equation of a curve in homogeneous coordinates on the projective plane. Let be the equation of a line, with being designated its line coordinatesin a dual projective plane. The condition that the line is tangent to the curve can be expressed in the form which is the tangential equation of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mathematical Intelligencer
''The Mathematical Intelligencer'' is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals. Volumes are released quarterly with a subset of open access articles. Springer also cross-publishes some of the articles in ''Scientific American''. Karen Parshall and Sergei Tabachnikov are currently the co-editors-in-chief. History The journal was started informally in 1971 by Walter Kaufman-Buehler, Alice Peters and Klaus Peters. "Intelligencer" was chosen by Kaufman-Buehler as a word that would appear slightly old-fashioned. An exploration of mathematically themed stamps, written by Robin Wilson, became one of its earliest columns. In 1978, the founders appointed Bruce Chandler and Harold "Ed" Edwards Jr. to serve jointly in the role of editor-in-chief. Prior to 1978, articles of the ''Intelligencer'' were not contained in regular volumes and were sent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canadian Mathematical Bulletin
The ''Canadian Mathematical Bulletin'' (french: Bulletin Canadien de Mathématiques) is a mathematics journal, established in 1958 and published quarterly by the Canadian Mathematical Society. The current editors-in-chief of the journal are Antonio Lei and Javad Mashreghi. The journal publishes short articles in all areas of mathematics that are of sufficient interest to the general mathematical public. Abstracting and indexing The journal is abstracted in: for the Canadian Mathematical Bulletin. * '''' * '' [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
McKay Correspondence
In mathematics, the McKay graph of a finite-dimensional representation of a finite group is a weighted quiver encoding the structure of the representation theory of . Each node represents an irreducible representation of . If are irreducible representations of , then there is an arrow from to if and only if is a constituent of the tensor product V\otimes\chi_i. Then the weight of the arrow is the number of times this constituent appears in V \otimes\chi_i. For finite subgroups of the McKay graph of is the McKay graph of the canonical representation of . If has irreducible characters, then the Cartan matrix of the representation of dimension is defined by c_V = (d\delta_ -n_)_ , where is the Kronecker delta. A result by Steinberg states that if is a representative of a conjugacy class of , then the vectors ((\chi_i(g))_i are the eigenvectors of to the eigenvalues d-\chi_V(g), where is the character of the representation . The McKay correspondence, named a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Arnold
Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, he made important contributions in several areas including dynamical systems theory, algebra, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics and singularity theory, including posing the ADE classification problem, since his first main result—the solution of Hilbert's thirteenth problem in 1957 at the age of 19. He co-founded two new branches of mathematics—KAM theory, and topological Galois theory (this, with his student Askold Khovanskii). Arnold was also known as a popularizer of mathematics. Through his lectures, seminars, and as the author of several textbooks (such as the fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ADE Classification
In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, rather than a posteriori verification of a parallelism, was posed in . The complete list of simply laced Dynkin diagrams comprises :A_n, \, D_n, \, E_6, \, E_7, \, E_8. Here "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the root system forming angles of \pi/2 = 90^\circ (no edge between the vertices) or 2\pi/3 = 120^\circ (single edge between the vertices). These are two of the four families of Dynkin diagrams (omitting B_n and C_n), and three of the five exceptional Dynkin diagrams (omitting F_4 and G_2). This list is non-redundant if one takes n \geq 4 for D_n. If one extends the families to include redundant terms, one obtains the exceptional isomorphisms :D_3 \cong A_3, E_4 \cong A_4 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sextic Equation
In algebra, a sextic (or hexic) polynomial is a polynomial of degree six. A sextic equation is a polynomial equation of degree six—that is, an equation whose left hand side is a sextic polynomial and whose right hand side is zero. More precisely, it has the form: :ax^6+bx^5+cx^4+dx^3+ex^2+fx+g=0,\, where and the ''coefficients'' may be integers, rational numbers, real numbers, complex numbers or, more generally, members of any field. A sextic function is a function defined by a sextic polynomial. Because they have an even degree, sextic functions appear similar to quartic functions when graphed, except they may possess an additional local maximum and local minimum each. The derivative of a sextic function is a quintic function. Since a sextic function is defined by a polynomial with even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If the leading coefficient is positive, then the function increases to positive infinity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theta Characteristic
In mathematics, a theta characteristic of a non-singular algebraic curve ''C'' is a divisor class Θ such that 2Θ is the canonical class. In terms of holomorphic line bundles ''L'' on a connected compact Riemann surface, it is therefore ''L'' such that ''L''2 is the canonical bundle, here also equivalently the holomorphic cotangent bundle. In terms of algebraic geometry, the equivalent definition is as an invertible sheaf, which squares to the sheaf of differentials of the first kind. Theta characteristics were introduced by History and genus 1 The importance of this concept was realised first in the analytic theory of theta functions, and geometrically in the theory of bitangents. In the analytic theory, there are four fundamental theta functions in the theory of Jacobian elliptic functions. Their labels are in effect the theta characteristics of an elliptic curve. For that case, the canonical class is trivial (zero in the divisor class group) and so the theta characteristic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Del Pezzo Surface
In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general type, whose canonical class is big. They are named for Pasquale del Pezzo who studied the surfaces with the more restrictive condition that they have a very ample anticanonical divisor class, or in his language the surfaces with a degree ''n'' embedding in ''n''-dimensional projective space , which are the del Pezzo surfaces of degree at least 3. Classification A del Pezzo surface is a complete non-singular surface with ample anticanonical bundle. There are some variations of this definition that are sometimes used. Sometimes del Pezzo surfaces are allowed to have singularities. They were originally assumed to be embedded in projective space by the anticanonical embedding, which restricts the degree to be at least 3. The degree ''d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Graph
In the mathematical field of graph theory, the Coxeter graph is a 3-regular graph with 28 vertices and 42 edges. It is one of the 13 known cubic distance-regular graphs. It is named after Harold Scott MacDonald Coxeter. Properties The Coxeter graph has chromatic number 3, chromatic index 3, radius 4, diameter 4 and girth 7. It is also a 3- vertex-connected graph and a 3- edge-connected graph. It has book thickness 3 and queue number 2. The Coxeter graph is hypohamiltonian: it does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from it is Hamiltonian. It has rectilinear crossing number 11, and is the smallest cubic graph with that crossing number . Construction The simplest construction of a Coxeter graph is from a Fano plane. Take the 7C3 = 35 possible 3-combinations on 7 objects. Discard the 7 triplets that correspond to the lines of the Fano plane, leaving 28 triplets. Link two triplets if they are disjoint. The result is the Cox ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heawood Graph
Heawood is a surname. Notable people with the surname include: * Jonathan Heawood, British journalist *Percy John Heawood (1861–1955), British mathematician **Heawood conjecture ** Heawood graph **Heawood number In mathematics, the Heawood number of a surface is an upper bound for the number of colors that suffice to color any graph embedded in the surface. In 1890 Heawood proved for all surfaces ''except'' the sphere that no more than : H(S)=\left\lfl ... See also * Heywood (surname) {{surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
