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PG(2,3)
In geometry, the Hesse configuration is a configuration of 9 points and 12 lines with three points per line and four lines through each point. It can be denoted as (94 123) or configuration matrix \left begin9 & 4 \\ 3 & 12 \\ \end\right /math>. It is symmetric (point and line transitive) with 432 automorphisms. It can be realized in the complex projective plane as the set of inflection points of an elliptic curve, but it has no realization in the Euclidean plane. It was introduced by Colin Maclaurin and studied by , and is also known as Young's geometry, named after the later work of John Wesley Young on finite geometry. Description The Hesse configuration has the same incidence relations as the lines and points of the affine plane over the field of 3 elements. That is, the points of the Hesse configuration may be identified with ordered pairs of numbers modulo 3, and the lines of the configuration may correspondingly be identified with the triples of points satisfying a line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus ''any'' two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in Perspective (graphical)#Renaissance, perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by , RP2, or P2(R), among other notations. There are many other projective planes, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Broken Diagonal
In recreational mathematics and the theory of magic squares, a broken diagonal is a set of ''n'' cells forming two parallel diagonal lines in the square. Alternatively, these two lines can be thought of as wrapping around the boundaries of the square to form a single sequence. In pandiagonal magic squares A magic square in which the broken diagonals have the same sum as the rows, columns, and diagonals is called a pandiagonal magic square. Examples of broken diagonals from the number square in the image are as follows: 3,12,14,5; 10,1,7,16; 10,13,7,4; 15,8,2,9; 15,12,2,5; and 6,13,11,4. The fact that this square is a pandiagonal magic square can be verified by checking that all of its broken diagonals add up to the same constant: : 3+12+14+5 = 34 : 10+1+7+16 = 34 : 10+13+7+4 = 34 One way to visualize a broken diagonal is to imagine a "ghost image" of the panmagic square adjacent to the original: The set of numbers of a broken diagonal, wrapped around the original square, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hesse Configuration Hierarchy Chart
Hesse or Hessen ( ), officially the State of Hesse (), is a States of Germany, state in Germany. Its capital city is Wiesbaden, and the largest urban area is Frankfurt, which is also the country's principal financial centre. Two other major historic cities are Darmstadt and Kassel. With an area of 21,114.73 square kilometers and a population of over six million, it ranks seventh and fifth, respectively, among the sixteen German states. Frankfurt Rhine-Main, Germany's second-largest metropolitan area (after Rhine-Ruhr), is mainly located in Hesse. As a cultural region, Hesse also includes the area known as Rhenish Hesse (Rheinhessen) in the neighboring state of Rhineland-Palatinate. Etymology The German name , like the names of other German regions ( "Swabia", "Franconia", "Bavaria", "Saxony"), derives from the dative plural form of the name of the inhabitants or German tribes, eponymous tribe, the Hessians (, singular ). The geographical name represents a short equivalent o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hesse Pencil
In mathematics, the syzygetic pencil or Hesse pencil, named for Otto Hesse, is a pencil (one-dimensional family) of cubic plane elliptic curves in the complex projective plane, defined by the equation :\lambda(x^3+y^3+z^3) + \mu xyz =0. Each curve in the family is determined by a pair of parameter values (\lambda,\mu) (not both zero) and consists of the points in the plane whose homogeneous coordinates (x,y,z) satisfy the equation for those parameters. Multiplying both \lambda and \mu by the same scalar does not change the curve, so there is only one degree of freedom in selecting a curve from the pencil, but the two-parameter form given above allows either \lambda or \mu (but not both) to be set to zero. Each curve in the pencil passes through the nine points of the complex projective plane whose homogeneous coordinates are some permutation of 0, –1, and a cube root of unity. There are three roots of unity, and six permutations per root, giving 18 choices for the homogeneous coo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hessian Matrix
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued Function (mathematics), function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Otto Hesse, Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants". The Hessian is sometimes denoted by H or \nabla\nabla or \nabla^2 or \nabla\otimes\nabla or D^2. Definitions and properties Suppose f : \R^n \to \R is a function taking as input a vector \mathbf \in \R^n and outputting a scalar f(\mathbf) \in \R. If all second-order partial derivatives of f exist, then the Hessian matrix \mathbf of f is a square n \times n matrix, usually defined and arranged as \mathbf H_f= \begin \dfrac & \dfrac & \cdots & \dfrac \\[2.2ex] \dfrac & \dfrac & \cdots & \dfrac \\[2.2ex] \vdots & \vdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pencil (mathematics)
In geometry, a pencil is a family of geometric objects with a common property, for example the set of Line (geometry), lines that pass through a given point in a plane (mathematics), plane, or the set of circles that pass through two given points in a plane. Although the definition of a pencil is rather vague, the common characteristic is that the pencil is completely determined by any two of its members. Analogously, a set of geometric objects that are determined by any three of its members is called a bundle. Thus, the set of all lines through a point in three-space is a bundle of lines, any two of which determine a pencil of lines. To emphasize the two-dimensional nature of such a pencil, it is sometimes referred to as a ''flat pencil''. Any geometric object can be used in a pencil. The common ones are lines, planes, circles, conics, spheres, and general curves. Even points can be used. A pencil of points is the set of all points on a given line. A more common term for this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The American Mathematical Monthly
''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. The editor-in-chief is Vadim Ponomarenko (San Diego State University). The journal gives the Lester R. Ford Award annually to "authors of articles of expository excellence" published in the journal. Editors-in-chief The following persons are or have been editor-in-chief: See also *''Mathematics Magazine'' *''Notices of the American Mathematical Society ''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
13 4 Configuration
Thirteen or 13 may refer to: * 13 (number) * Any of the years 13 BC, AD 13, 1913, or 2013 Music Albums * ''13'' (Black Sabbath album), 2013 * ''13'' (Blur album), 1999 * ''13'' (Borgeous album), 2016 * ''13'' (Brian Setzer album), 2006 * ''13'' (Die Ärzte album), 1998 * ''13'' (The Doors album), 1970 * ''13'' (Havoc album), 2013 * ''13'' (HLAH album), 1993 * ''13'' (Indochine album), 2017 * ''13'' (Marta Savić album), 2011 * ''13'' (Norman Westberg album), 2015 * ''13'' (Ozark Mountain Daredevils album), 1997 * ''13'' (Six Feet Under album), 2005 * ''13'' (Suicidal Tendencies album), 2013 * ''13'' (Solace album), 2003 * ''13'' (Second Coming album), 2003 * 13 (Timati album), 2013 * ''13'' (Ces Cru EP), 2012 * ''13'' (Denzel Curry EP), 2017 * ''Thirteen'' (CJ & The Satellites album), 2007 * ''Thirteen'' (Emmylou Harris album), 1986 * ''Thirteen'' (Harem Scarem album), 2014 * ''Thirteen'' (James Reyne album), 2012 * ''Thirteen'' (Megadeth album), 2011 * ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utility Graph
In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings. * In a normative context, utility refers to a goal or objective that we wish to maximize, i.e., an objective function. This kind of utility bears a closer resemblance to the original utilitarian concept, developed by moral philosophers such as Jeremy Bentham and John Stuart Mill. * In a descriptive context, the term refers to an ''apparent'' objective function; such a function is revealed by a person's behavior, and specifically by their preferences over lotteries, which can be any quantified choice. The relationship between these two kinds of utility functions has been a source of controversy among both economists and ethicists, with most maintaining that the two are distinct but generally related. Utility function Consider a set of alternatives among which a person has a preference ordering. A utility functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Bipartite Graph
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher. Llull himself had made similar drawings of complete graphs three centuries earlier.. Definition A complete bipartite graph is a graph whose vertices can be partitioned into two subsets and such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a bipartite graph such that for every two vertices and, is an edge in . A complete bipartite graph w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Complex Polygon
In geometry, a regular complex polygon is a generalization of a regular polygon in real coordinate space, real space to an analogous structure in a Complex number, complex Hilbert space, where each real dimension is accompanied by an imaginary number, imaginary one. A regular polygon exists in 2 real dimensions, \mathbb^2, while a complex polygon exists in two complex dimensions, \mathbb^2, which can be given real representations in 4 dimensions, \mathbb^4, which then must be projected down to 2 or 3 real dimensions to be visualized. A ''complex polygon'' is generalized as a complex polytope in \mathbb^n. A complex polygon may be understood as a collection of complex points, lines, planes, and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on. The ''regular complex polygons'' have been completely characterized, and can be described using a symbolic notation developed by Harold Scott MacDonald Coxeter, Coxeter. A ''regular comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
