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Hughes Plane
In mathematics, a Hughes plane is one of the non-Desarguesian projective planes found by . There are examples of order ''p''2''n'' for every odd prime ''p'' and every positive integer ''n''. Construction The construction of a Hughes plane is based on a nearfield N of order ''p''''2n'' for ''p'' an odd prime whose kernel K has order ''p''''n'' and coincides with the center of N. Properties A Hughes plane H: # is a non-Desarguesian projective plane of odd square prime power order of Lenz-Barlotti type I.1, # has a Desarguesian Baer subplane H0, # is a self-dual plane in which every orthogonal polarity of H0 can be extended to a polarity of H, # every central collineation of H0 extends to a central collineation of H, and # the full collineation group of H has two point orbits (one of which is H0), two line orbits, and four flag orbits. The smallest Hughes Plane (order 9) The Hughes plane of order 9 was actually found earlier by Veblen and Wedderburn in 1907. A construction of this p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-Desarguesian Projective Plane
In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective spaces of dimension not 2; in other words, the only projective spaces of dimension not equal to 2 are the classical projective geometries over a field (or division ring). However, David Hilbert found that some projective planes do not satisfy it. The current state of knowledge of these examples is not complete. Examples There are many examples of both finite and infinite non-Desarguesian planes. Some of the known examples of infinite non-Desarguesian planes include: *The Moulton plane. *Moufang planes over alternative division rings that are not associative, such as the projective plane over the octonions. Since all finite alternative division rings are fields (Artin–Zorn theorem), the only non-Desarguesian Moufang planes are infinite. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Near-field (mathematics)
In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws. Alternatively, a near-field is a near-ring in which there is a multiplicative identity and every non-zero element has a multiplicative inverse. Definition A near-field is a set Q together with two binary operations, + (addition) and \cdot (multiplication), satisfying the following axioms: :A1: (Q, +) is an abelian group. :A2: (a \cdot b) \cdot c = a \cdot (b \cdot c) for all elements a, b, c of Q (The associative law for multiplication). :A3: (a + b) \cdot c = a \cdot c + b \cdot c for all elements a, b, c of Q (The right distributive law). :A4: Q contains an element 1 such that 1 \cdot a = a \cdot 1 = a for every element a of Q (Multiplicative identity). :A5: For every non-zero element a of Q there exists an element a^ such that a \cdot a^ = 1 = a^ \cdot a (Multiplicative inverse). Notes on the definition # The above is, strictly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice-versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
