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Bioctonion
In mathematics, a bioctonion, or complex octonion, is a pair (''p,q'') where ''p'' and ''q'' are biquaternions. The product of two bioctonions is defined using biquaternion multiplication and the biconjugate p → p*: :(p,q)(r,s) = (pr - s^* q,\ sp + q r^*). The bioctonion ''z'' = (''p,q'') has conjugate ''z''* = (''p''*, – ''q''). Then norm ''N''(''z'') of bioctonion ''z'' is ''z z''* = ''p p''* + ''q q''*, which is a complex quadratic form with eight terms. The bioctonion algebra is sometimes introduced as simply the complexification of real octonions, but in abstract algebra it is the result of the Cayley–Dickson construction that begins with the field of complex numbers, the trivial involution, and quadratic form z2. The algebra of bioctonions is an example of an octonion algebra. For any pair of bioctonions ''y'' and ''z'', : N(y z) = N(y) N(z), showing that ''N'' is a quadratic form admitting composition, and hence the bioctonions form a composition algebra. Comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley–Dickson Construction
In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as Cayley–Dickson algebras, for example complex numbers, quaternions, and octonions. These examples are useful composition algebras frequently applied in mathematical physics. The Cayley–Dickson construction defines a new algebra as a Cartesian product of an algebra with itself, with multiplication defined in a specific way (different from the componentwise multiplication) and an involution known as conjugation. The product of an element and its conjugate (or sometimes the square root of this product) is called the norm. The symmetries of the real field disappear as the Cayley–Dickson construction is repeatedly applied: first losing order, then commutativity of multiplication, associativity of multipli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quark
A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly observable matter is composed of up quarks, down quarks and electrons. Owing to a phenomenon known as ''color confinement'', quarks are never found in isolation; they can be found only within hadrons, which include baryons (such as protons and neutrons) and mesons, or in quark–gluon plasmas. There is also the theoretical possibility of more exotic phases of quark matter. For this reason, much of what is known about quarks has been drawn from observations of hadrons. Quarks have various intrinsic properties, including electric charge, mass, color charge, and spin. They are the only elementary particles in the Standard Model of particle physics to experience all four fundamental interactions, also known as ''fundamental forces'' (electro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metrika
The Lithuanian Metrica or the Metrica of the Grand Duchy of Lithuania ( la, Acta Magni Ducatus Lithuaniae, lt, Lietuvos Metrika, pl, Metryka Litewska, or ''Metryka Wielkiego Księstwa Litewskiego''; be, Літоўская Метрыка, uk, Литовська метрика) is a collection of the 14–18th century legal documents of the Chancellery of the Grand Duchy of Lithuania (GDL). Maintained systematically since the 2nd half the 15th century, metrica consisted, initially and primarily, of the copies of the documents issued by the Grand Duke, Lithuanian Council of Lords, and Seimas. The Metrica also included some important externally originated documents (like translations of the issues (''yarlyks'') of the Crimea Khans, copies of the Muscovy diplomatic documents etc.), the office-keeping documental materials (like registers of acts, inventories of the Metrica itself etc.) The selection of the classes of the documents included in the Metrica had increased since the 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Physical Society
The American Physical Society (APS) is a not-for-profit membership organization of professionals in physics and related disciplines, comprising nearly fifty divisions, sections, and other units. Its mission is the advancement and diffusion of knowledge of physics. The society publishes more than a dozen scientific journals, including the prestigious '' Physical Review'' and ''Physical Review Letters'', and organizes more than twenty science meetings each year. APS is a member society of the American Institute of Physics. Since January 2021 the organization has been led by chief executive officer Jonathan Bagger. History The American Physical Society was founded on May 20, 1899, when thirty-six physicists gathered at Columbia University for that purpose. They proclaimed the mission of the new Society to be "to advance and diffuse the knowledge of physics", and in one way or another the APS has been at that task ever since. In the early years, virtually the sole activity of the AP ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PhilPapers
PhilPapers is an interactive academic database of journal articles in philosophy. It is maintained by the Centre for Digital Philosophy at the University of Western Ontario, and as of 2022, it has "394,867 registered users, including the majority of professional philosophers and graduate students." The general editors are its founders, David Bourget and David Chalmers. PhilPapers receives financial support from other organizations, including a substantial grant in early 2009 from the Joint Information Systems Committee Jisc is a United Kingdom not-for-profit company that provides network and IT services and digital resources in support of further and higher education institutions and research as well as not-for-profits and the public sector. History T ... in the United Kingdom. The archive is praised for its comprehensiveness and organization, and for its regular updates. In addition to archiving papers, the editors run and publish the most extensive ongoing survey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foundations Of Physics
''Foundations of Physics'' is a monthly journal "devoted to the conceptual bases and fundamental theories of modern physics and cosmology, emphasizing the logical, methodological, and philosophical premises of modern physical theories and procedures". The journal publishes results and observations based on fundamental questions from all fields of physics, including: quantum mechanics, quantum field theory, special relativity, general relativity, string theory, M-theory, cosmology, thermodynamics, statistical physics, and quantum gravity ''Foundations of Physics'' has been published since 1970. Its founding editors were Henry Margenau and Wolfgang Yourgrau. The 1999 Nobel laureate Gerard 't Hooft was editor-in-chief from January 2007. At that stage, it absorbed the associated journal for shorter submissions ''Foundations of Physics Letters'', which had been edited by Alwyn Van der Merwe since its foundation in 1988. Past editorial board members (which include several Nobel laureate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lepton
In particle physics, a lepton is an elementary particle of half-integer spin ( spin ) that does not undergo strong interactions. Two main classes of leptons exist: charged leptons (also known as the electron-like leptons or muons), and neutral leptons (better known as neutrinos). Charged leptons can combine with other particles to form various composite particles such as atoms and positronium, while neutrinos rarely interact with anything, and are consequently rarely observed. The best known of all leptons is the electron. There are six types of leptons, known as '' flavours'', grouped in three '' generations''. The first-generation leptons, also called ''electronic leptons'', comprise the electron () and the electron neutrino (); the second are the ''muonic leptons'', comprising the muon () and the muon neutrino (); and the third are the ''tauonic leptons'', comprising the tau () and the tau neutrino (). Electrons have the least mass of all the charged leptons. The heavi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octonion Algebra
In mathematics, an octonion algebra or Cayley algebra over a field ''F'' is a composition algebra over ''F'' that has dimension 8 over ''F''. In other words, it is a unital non-associative algebra ''A'' over ''F'' with a non-degenerate quadratic form ''N'' (called the ''norm form'') such that :N(xy) = N(x)N(y) for all ''x'' and ''y'' in ''A''. The most well-known example of an octonion algebra is the classical octonions, which are an octonion algebra over R, the field of real numbers. The split-octonions also form an octonion algebra over R. Up to R-algebra isomorphism, these are the only octonion algebras over the reals. The algebra of bioctonions is the octonion algebra over the complex numbers C. The octonion algebra for ''N'' is a division algebra if and only if the form ''N'' is anisotropic. A split octonion algebra is one for which the quadratic form ''N'' is isotropic (i.e., there exists a non-zero vector ''x'' with ''N''(''x'') = 0). Up to ''F''-algebra isomorphism, there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composition Algebra
In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involution called a conjugation: x \mapsto x^*. The quadratic form N(x) = x x^* is called the norm of the algebra. A composition algebra (''A'', ∗, ''N'') is either a division algebra or a split algebra, depending on the existence of a non-zero ''v'' in ''A'' such that ''N''(''v'') = 0, called a null vector. When ''x'' is ''not'' a null vector, the multiplicative inverse of ''x'' is When there is a non-zero null vector, ''N'' is an isotropic quadratic form, and "the algebra splits". Structure theorem Every unital composition algebra over a field can be obtained by repeated application of the Cayley–Dickson construction starting from (if the characteristic of is different from ) or a 2-dimensional composition subalgebra (if ). The poss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof: * Biquaternions when the coefficients are complex numbers. * Split-biquaternions when the coefficients are split-complex numbers. * Dual quaternions when the coefficients are dual numbers. This article is about the ''ordinary biquaternions'' named by William Rowan Hamilton in 1844 (see ''Proceedings of the Royal Irish Academy'' 1844 & 1850 page 388). Some of the more prominent proponents of these biquaternions include Alexander Macfarlane, Arthur W. Conway, Ludwik Silberstein, and Cornelius Lanczos. As developed below, the unit quasi-sphere of the biquaternions provides a representation of the Lorentz group, which is the foundation of special relativity. The algebra of biquatern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
