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Malcev-admissible Algebra
In algebra, a Malcev-admissible algebra, introduced by , is a (possibly non-associative) algebra that becomes a Malcev algebra under the bracket 'a'', ''b''= ''ab'' − ''ba''. Examples include alternative algebras, Malcev algebras and Lie-admissible algebras. See also * Jordan-admissible algebra References * * * *{{citation , last=Myung , first=Hyo Chul , year=1986 , title=Malcev-admissible algebras , url=https://books.google.com/books?id=PBvvAAAAMAAJ , series= Progress in Mathematics , volume=64 , publisher=Birkhäuser Boston Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields: * Springer continues to publish science (parti ... , place=Boston, MA , isbn= 0-8176-3345-6 , mr=0885089 , doi=10.1007/978-1-4899-6661-2 Non-associative algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''algebra'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-associative Algebra
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' if it is a vector space over ''K'' and is equipped with a ''K''- bilinear binary multiplication operation ''A'' × ''A'' → ''A'' which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (''ab'')(''cd''), (''a''(''bc''))''d'' and ''a''(''b''(''cd'')) may all yield different answers. While this use of ''non-associative'' means that associativity is not assumed, it does not mean that associativity is disallowed. In other words, "non-associative" means "not ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Over A Field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear". The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and non-associative algebras. Given an integer ''n'', the ring of real square matrices of order ''n'' is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Malcev Algebra
In mathematics, a Malcev algebra (or Maltsev algebra or Moufang– Lie algebra) over a field is a nonassociative algebra that is antisymmetric, so that :xy = -yx and satisfies the Malcev identity :(xy)(xz) = ((xy)z)x + ((yz)x)x + ((zx)x)y. They were first defined by Anatoly Maltsev (1955). Malcev algebras play a role in the theory of Moufang loops that generalizes the role of Lie algebras in the theory of groups. Namely, just as the tangent space of the identity element of a Lie group forms a Lie algebra, the tangent space of the identity of a smooth Moufang loop forms a Malcev algebra. Moreover, just as a Lie group can be recovered from its Lie algebra under certain supplementary conditions, a smooth Moufang loop can be recovered from its Malcev algebra if certain supplementary conditions hold. For example, this is true for a connected, simply connected real-analytic Moufang loop. Examples *Any Lie algebra is a Malcev algebra. *Any alternative algebra may be made into a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternative Algebra
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the algebra. Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions. The associator Alternative algebras are so named because they are the algebras for which the associator is alternating. The associator is a trilinear map given by : ,y,z= (xy)z - x(yz). By definition, a multilinear map is alternating if it vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent toSchafer (1995) p. 27 : ,x,y= 0 : ,x,x= 0. Both of these identities together imply that : ,y,x= , x, x+ , y, x- , x+y, x+y= , x+y, -y= , x, -y- , y, y= 0 for all x and y. This is equivalent to the ''flexible identity''Schafer (1995) p. 28 :(xy)x = x(yx). The associator of an al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie-admissible Algebra
In algebra, a Lie-admissible algebra, introduced by , is a (possibly non-associative) algebra that becomes a Lie algebra under the bracket 'a'', ''b''= ''ab'' − ''ba''. Examples include associative algebras, Lie algebras, and Okubo algebra In abstract algebra, algebra, an Okubo algebra or pseudo-octonion algebra is an 8-dimensional algebra over a field, non-associative algebra similar to the one studied by Susumu Okubo. Okubo algebras are composition algebras, flexible algebras (''A'' ...s. See also * Malcev-admissible algebra * Jordan-admissible algebra References * * * * {{Authority control Non-associative algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan-admissible Algebra
In algebra, a noncommutative Jordan algebra is an algebra, usually over a field of characteristic not 2, such that the four operations of left and right multiplication by ''x'' and ''x''2 all commute with each other. Examples include associative algebras and Jordan algebras. Over fields of characteristic not 2, noncommutative Jordan algebras are the same as flexible Jordan-admissible algebras, where a Jordan-admissible algebra – introduced by and named after Pascual Jordan – is a (possibly non-associative) algebra that becomes a Jordan algebra under the product ''a'' ∘ ''b'' = ''ab'' + ''ba''. See also *Malcev-admissible algebra *Lie-admissible algebra In algebra, a Lie-admissible algebra, introduced by , is a (possibly non-associative) algebra that becomes a Lie algebra under the bracket 'a'', ''b''= ''ab'' − ''ba''. Examples include associative algebras, Lie algebras, and Okubo algebr ... References * * * {{refend Non-a ... [...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] |
Hadronic Journal
Ruggero Maria Santilli (born September 8, 1935) is an Italo-American nuclear physicist. Mainstream scientists dismiss his theories as fringe science. Biography Ruggero Maria Santilli was born September 8, 1935) in Capracotta. He studied physics at the University of Naples and earned his PhD in physics from the University of Turin, graduating in 1965. He held various academic positions in Italy until 1967, when he took a position at University of Miami; a year later he moved to Boston University, and subsequently held visiting scientist positions at Massachusetts Institute of Technology and Harvard University. In September 1981, Santilli established a one-man organization, the Institute for Basic Research in Boston; he told a reporter from '' St. Petersburg Times'' in 2007 that he left Harvard because scientists there viewed his work as "heresy". In 1982 Austrian-British philosopher Karl Popper wrote that Santilli's calls for tests on the validity of quantum mechanics within n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Progress In Mathematics
Progress is the movement towards a refined, improved, or otherwise desired state. In the context of progressivism, it refers to the proposition that advancements in technology, science, and social organization have resulted, and by extension will continue to result, in an improved human condition; the latter may happen as a result of direct human action, as in social enterprise or through activism, or as a natural part of sociocultural evolution. The concept of progress was introduced in the early-19th-century social theories, especially social evolution as described by Auguste Comte and Herbert Spencer. It was present in the Enlightenment's philosophies of history. As a goal, social progress has been advocated by varying realms of political ideologies with different theories on how it is to be achieved. Measuring progress Specific indicators for measuring progress can range from economic data, technical innovations, change in the political or legal system, and questions beari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhäuser Boston
Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields: * Springer continues to publish science (particularly: history of science, geosciences, computer science) and mathematics books and journals under the Birkhäuser imprint (with a leaf logo) sometimes called Birkhäuser Science. * Birkhäuser Verlag – an architecture and design publishing company was (re)created in 2010 when Springer sold its design and architecture segment to ACTAR. The resulting Spanish-Swiss company was then called ActarBirkhäuser. After a bankruptcy, in 2012 Birkhäuser Verlag was sold again, this time to De Gruyter. Additionally, the Reinach-based printer Birkhäuser+GBC operates independently of the above, being now owned by ''Basler Zeitung''. History The original Swiss publishers program focused on regional literature. In the 1920s the sons of Emil Birk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
