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Hall Plane Of Order 9
In mathematics, a Hall plane is a non-Desarguesian projective plane constructed by Marshall Hall Jr. (1943). There are examples of order ''p''2''n'' for every prime ''p'' and every positive integer ''n'' provided ''p''2''n'' > 4. Algebraic construction via Hall systems The original construction of Hall planes was based on the Hall quasifield (also called a ''Hall system''), H of order p^ for ''p'' a prime. The creation of the plane from the quasifield follows the standard construction (see quasifield for details). To build a Hall quasifield, start with a Galois field, F = \operatorname(p^n) for ''p'' a prime and a quadratic irreducible polynomial f(x) = x^2 - rx - s over ''F''. Extend H=F\times F, a two-dimensional vector space over ''F'', to a quasifield by defining a multiplication on the vectors by (a,b)\circ (c,d) = (ac -bd^f(c), ad - bc + br) when d \neq 0 and (a,b) \circ (c,0) = (ac, bc) otherwise. Writing the elements of ''H'' in terms of a basis , that is, identifying ( ... [...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] |
Duality (projective Geometry)
In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of duality, one through language () and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a ''duality''. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry. Principle of duality A projective plane may be defined axiomatically as an incidence structure, in terms of a set of ''points'', a set of ''lines'', and an incidence relation that determines which points lie on which lines. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 for the combined June/July issue. The first volume appeared in 1953. Each issue of the magazine since January 1995 is available in its entirety on the journal web site. Articles are peer-reviewed by an editorial board of mathematical experts. Since 2019, the editor-in-chief is Erica Flapan. The cover regularly features mathematical visualization Mathematical phenomena can be understood and explored via visualization. Classically this consisted of two-dimensional drawings or building three-dimensional models (particularly plaster models in the 19th and early 20th century), while today it ...s. The ''Notices'' is self-described to be the world's most widely read mathematical journal. As the membership journal of the American Mathematical Society, the ''Notices'' is sent to the approximately 30,000 AMS members worldwide, one-third of whom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two '' directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative. Quaternions are generally represented in the form :a + b\ \mathbf i + c\ \mathbf j +d\ \mathbf k where , and are real numbers; and , and are the ''basic quaternions''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unital (geometry)
In geometry, a unital is a set of ''n''3 + 1 points arranged into subsets of size ''n'' + 1 so that every pair of distinct points of the set are contained in exactly one subset. This is equivalent to saying that a unital is a 2-(''n''3 + 1, ''n'' + 1, 1) block design. Some unitals may be embedded in a projective plane of order ''n''2 (the subsets of the design become sets of collinear points in the projective plane). In this case of ''embedded unitals'', every line of the plane intersects the unital in either 1 or ''n'' + 1 points. In the Desarguesian planes, PG(2,''q''2), the classical examples of unitals are given by nondegenerate Hermitian curves. There are also many non-classical examples. The first and the only known unital with non prime power parameters, ''n''=''6'', was constructed by Bhaskar Bagchi and Sunanda Bagchi. It is still unknown if this unital can be embedded in a projective plane of order ''36'', if such a plane exists. Unitals Classical We review some termi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Permutation Group
In mathematics, a permutation group ''G'' acting on a non-empty finite set ''X'' is called primitive if ''G'' acts transitively on ''X'' and the only partitions the ''G''-action preserves are the trivial partitions into either a single set or into , ''X'', singleton sets. Otherwise, if ''G'' is transitive and ''G'' does preserve a nontrivial partition, ''G'' is called imprimitive. While primitive permutation groups are transitive, not all transitive permutation groups are primitive. The simplest example is the Klein four-group acting on the vertices of a square, which preserves the partition into diagonals. On the other hand, if a permutation group preserves only trivial partitions, it is transitive, except in the case of the trivial group acting on a 2-element set. This is because for a non-transitive action, either the orbits of ''G'' form a nontrivial partition preserved by ''G'', or the group action is trivial, in which case ''all'' nontrivial partitions of ''X'' (which exis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism Group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the group of invertible linear transformations from ''X'' to itself (the general linear group of ''X''). If instead ''X'' is a group, then its automorphism group \operatorname(X) is the group consisting of all group automorphisms of ''X''. Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups are studied in a general way in the field of category theory. Examples If ''X'' is a set with no additional structure, then any bijection from ''X'' to itself is an automorphism, and hence the automorphism group of ''X'' in this case is precisely the symmetric group of ''X''. If the set ''X'' has additional struct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-Desarguesian 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] |
Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. See also * ''Bulletin of the American Mathematical Society'' * '' Journal of the American Mathematical Society'' * ''Memoirs of the American Mathematical Society'' * ''Notices of the American Mathematical Society'' * ''Proceedings of the American Mathematical Society'' External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR (; short for ''Journal Storage'') is a digital library founded in 1995 in New York City. Originally containing digitized back issues of academic journals, it now encompasses books and other primary sources as well as current issues of j ... American Mathematical Society academic journals Mathematics journals Publications established in 1900 {{math-journal-st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Wedderburn
Joseph Henry Maclagan Wedderburn FRSE FRS (2 February 1882 – 9 October 1948) was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a field, and part of the Artin–Wedderburn theorem on simple algebras. He also worked on group theory and matrix algebra. His younger brother was the lawyer Ernest Wedderburn. Life Joseph Wedderburn was the tenth of fourteen children of Alexander Wedderburn of Pearsie, a physician, and Anne Ogilvie. He was educated at Forfar Academy then in 1895 his parents sent Joseph and his younger brother Ernest to live in Edinburgh with their paternal uncle, J R Maclagan Wedderburn, allowing them to attend George Watson's College. This house was at 3 Glencairn Crescent in the West End of the city. In 1898 Joseph entered the University of Edinburgh. In 1903, he published his first three papers, worked as an assistant in the Physical Laboratory of the U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oswald Veblen
Oswald Veblen (June 24, 1880 – August 10, 1960) was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in .... He proved the Jordan curve theorem in 1905; while this was long considered the first rigorous proof of the theorem, many now also consider Camille Jordan's original proof rigorous. Early life Veblen was born in Decorah, Iowa. His parents were Andrew Anderson Veblen (1848–1932), Professor of Physics at the University of Iowa, and Kirsti (Hougen) Veblen (1851–1908). Veblen's uncle was Thorstein Veblen, noted economist and sociologist. Oswald went to school in Iowa City. He did his undergraduate studies at the University of Iowa, where he received a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
