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Generalization Of A Lie Algebra
In mathematics, a Lie algebra has been generalized in several ways. Graded Lie algebra and Lie superalgebra A graded Lie algebra is a Lie algebra with grading. When the grading is \mathbb/2, it is also known as a Lie superalgebra. Lie-isotopic algebra A Lie-isotopic algebra is a generalization of Lie algebras proposed by physicist R. M. Santilli in 1978. Definition Recall that a finite-dimensional Lie algebra L with generators X_1, X_2, ..., X_n and commutation rules : _i X_j= X_i X_j - X_j X_i = C_^k X_k, can be defined (particularly in physics) as the totally anti-symmetric algebra A(L)^- attached to the universal enveloping associative algebra A(L)=\ equipped with the associative product X_i \times X_j over a numeric field F with multiplicative unit 1. Consider now the axiom-preserving lifting of A(L) into the form A^*(L^*)=\, called universal enveloping isoassociative algebra, with isoproduct :X_i\times X_j = X_i T^* X_j, verifying the isoassociative law :X_i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, ,y= xy - yx . Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilinear Operator
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. A bilinear map can also be defined for modules. For that, see the article pairing. Definition Vector spaces Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function B : V \times W \to X such that for all w \in W, the map B_w v \mapsto B(v, w) is a linear map from V to X, and for all v \in V, the map B_v w \mapsto B(v, w) is a linear map from W to X. In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed. Such a map B satisfies the following properties. * For any \lambda \in F, B(\lambda v,w) = B(v, \lambda w) = \lambda B(v, w). * The map B is additive in both components: if v_1, v_2 \in V a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic (algebra)
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest positive number of copies of the ring's multiplicative identity () that will sum to the additive identity (). If no such number exists, the ring is said to have characteristic zero. That is, is the smallest positive number such that: : \underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer such that: : \underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). This definition applies in the more general class of rngs (see '); for (unital) rings the two definitions are equivalent due to their distributive law. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning. In mathematics, an ''axiom'' may be a " logical axiom" or a " non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example ''a'' + 0 = ''a'' in integer arithmetic. N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scientific Reports
''Scientific Reports'' is a peer-reviewed open-access scientific mega journal published by Nature Portfolio, covering all areas of the natural sciences. The journal was established in 2011. The journal states that their aim is to assess solely the scientific validity of a submitted paper, rather than its perceived importance, significance, or impact. In September 2016, the journal became the largest in the world by number of articles, overtaking '' PLOS ONE''. Abstracting and indexing The journal is abstracted and indexed in the Chemical Abstracts Service, the Science Citation Index Expanded, and selectively in Index Medicus/MEDLINE/PubMed. According to the ''Journal Citation Reports'', the journal has a 2023 impact factor 3.8. Reviewing policy The ''Guide to Referees'' states that to be published, "a paper must be scientifically valid and technically sound in methodology and analysis", and reviewers have to ensure manuscripts "are not assessed based on their perceived impor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phys
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." It is one of the most fundamental scientific disciplines. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Applicandae Mathematicae
''Acta Applicandae Mathematicae'' is a peer-reviewed mathematics journal published by Springer. Founded in 1983, the journal publishes articles on applied mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 1.215. According to SCImago Journal Rank (SJR), the journal h-index is 45, ranking it to Q2 in Applied Mathematics. References External links * Applied mathematics journals Academic journals established in 1983 English-language journals Springer Science+Business Media academic journals Triannual journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebras, Groups And Geometries
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 nuc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Generalized Lie Theories
A journal, from the Old French ''journal'' (meaning "daily"), may refer to: *Bullet journal, a method of personal organization *Diary, a record of personal secretive thoughts and as open book to personal therapy or used to feel connected to oneself. A record of what happened over the course of a day or other period *Daybook, also known as a general journal, a daily record of financial transactions *Logbook, a record of events important to the operation of a vehicle, facility, or otherwise *Transaction log, a chronological record of data processing *Travel journal, a record of the traveller's experience during the course of their journey In publishing, ''journal'' can refer to various periodicals or serials: *Academic journal, an academic or scholarly periodical **Scientific journal, an academic journal focusing on science **Medical journal, an academic journal focusing on medicine **Law review, a professional journal focusing on legal interpretation *Magazine, non-academic or scho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
