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Symplectic Lie Algebra
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The term "symplectic" is a calque of "complex" introduced by Hermann Weyl in 1939. In mathematics it may refer to: * Symplectic Clifford algebra, see Weyl algebra * Symplectic geometry * Symplectic group * Symplectic integrator * Symplectic manifold * Symplectic matrix * Symplectic representation * Symplectic vector space It can also refer to: * Symplectic bone, a bone found in fish skulls * Symplectite, in reference to a mineral intergrowth texture See also * Metaplectic group * Symplectomorphism In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calque
In linguistics, a calque () or loan translation is a word or phrase borrowed from another language by literal word-for-word or root-for-root translation. When used as a verb, "to calque" means to borrow a word or phrase from another language while translating its components, so as to create a new lexeme in the target language. For instance, the English word "skyscraper" was calqued in dozens of other languages. Another notable example is the Latin weekday names, which came to be associated by ancient Germanic speakers with their own gods following a practice known as ''interpretatio germanica'': the Latin "Day of Mercury", ''Mercurii dies'' (later "mercredi" in modern French), was borrowed into Late Proto-Germanic as the "Day of Wōđanaz" (*''Wodanesdag''), which became ''Wōdnesdæg'' in Old English, then "Wednesday" in Modern English. The term ''calque'' itself is a loanword from the French noun ("tracing, imitation, close copy"), while the word ''loanword'' is a calq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by Carl Friedrich Gauss, David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines such as number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years. Weyl contributed to an exceptionally wide range of mathematical fields, including works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. Freeman Dyson wrote that Weyl alone b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Algebra
In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form : f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X). More precisely, let ''F'' be the underlying field, and let ''F'' 'X''be the ring of polynomials in one variable, ''X'', with coefficients in ''F''. Then each ''fi'' lies in ''F'' 'X'' ''∂X'' is the derivative with respect to ''X''. The algebra is generated by ''X'' and ''∂X''. The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension. The Weyl algebra is isomorphic to the quotient of the free algebra on two generators, ''X'' and ''Y'', by the ideal generated by the element :YX - XY = 1~. The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The ''n''-th Weyl algebra, ''An'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Geometry
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. The term "symplectic", introduced by Weyl, is a calque of "complex"; previously, the "symplectic group" had been called the "line complex group". "Complex" comes from the Latin ''com-plexus'', meaning "braided together" (co- + plexus), while symplectic comes from the corresponding Greek ''sym-plektikos'' (συμπλεκτικός); in both cases the stem comes from the Indo-European root *pleḱ- The name reflects the deep connections between complex and symplectic structures. By Darboux's Theorem, symplectic manifolds are isomorphic to the standard symplectic vector space locally, hence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic group and is also denoted by \mathrm(n). Many authors prefer slightly different notations, usually differing by factors of . The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group is denoted , and is the compact real form of . Note that when we refer to ''the'' (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension . The name "symplectic group" is due to Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex". The metaple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Integrator
In mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations. They are widely used in nonlinear dynamics, molecular dynamics, discrete element methods, accelerator physics, plasma physics, quantum physics, and celestial mechanics. Introduction Symplectic integrators are designed for the numerical solution of Hamilton's equations, which read :\dot p = -\frac \quad\mbox\quad \dot q = \frac, where q denotes the position coordinates, p the momentum coordinates, and H is the Hamiltonian. The set of position and momentum coordinates (q,p) are called canonical coordinates. (See Hamiltonian mechanics for more background.) The time evolution of Hamilton's equations is a symplectomorphism, meaning that it conserves the symplectic 2-form dp \wedge dq. A numerical scheme is a symplectic integrator if it also conserves this 2-fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Matrix
In mathematics, a symplectic matrix is a 2n\times 2n matrix M with real entries that satisfies the condition where M^\text denotes the transpose of M and \Omega is a fixed 2n\times 2n nonsingular, skew-symmetric matrix. This definition can be extended to 2n\times 2n matrices with entries in other field (mathematics), fields, such as the complex numbers, finite fields, p-adic number, ''p''-adic numbers, and Function field of an algebraic variety, function fields. Typically \Omega is chosen to be the block matrix \Omega = \begin 0 & I_n \\ -I_n & 0 \\ \end, where I_n is the n\times n identity matrix. The matrix \Omega has determinant +1 and its inverse is \Omega^ = \Omega^\text = -\Omega. Properties Generators for symplectic matrices Every symplectic matrix has determinant +1, and the 2n\times 2n symplectic matrices with real entries form a subgroup of the general linear group \mathrm(2n;\mathbb) under matrix multiplication since being symplectic is a property stable under m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Representation
In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (''V'', ''ω'') which preserves the symplectic form ''ω''. Here ''ω'' is a nondegenerate skew symmetric bilinear form :\omega\colon V\times V \to \mathbb F where F is the field of scalars. A representation of a group ''G'' preserves ''ω'' if :\omega(g\cdot v,g\cdot w)= \omega(v,w) for all ''g'' in ''G'' and ''v'', ''w'' in ''V'', whereas a representation of a Lie algebra g preserves ''ω'' if :\omega(\xi\cdot v,w)+\omega(v,\xi\cdot w)=0 for all ''ξ'' in g and ''v'', ''w'' in ''V''. Thus a representation of ''G'' or g is equivalently a group or Lie algebra homomorphism from ''G'' or g to the symplectic group Sp(''V'',''ω'') or its Lie algebra sp(''V'',''ω'') If ''G'' is a compact group (for example, a finite group), and F is the field of complex numbers, then by introducing a compatible unitary structure (which exists by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Vector Space
In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument separately; ; Alternating: holds for all ; and ; Non-degenerate: for all implies that . If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa. Working in a fixed basis, ''ω'' can be represented by a matrix. The conditions above are equivalent to this matrix being skew-symmetric, nonsingular, and hollow (all diagonal entries are zero). This should not be confused with a symplectic matrix, which represents a symplectic transformation of the space. If ''V'' is finite-dimensional, then its dimension must necessarily be even s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Bone
The skull is a bone protective cavity for the brain. The skull is composed of four types of bone i.e., cranial bones, facial bones, ear ossicles and hyoid bone. However two parts are more prominent: the cranium and the mandible. In humans, these two parts are the neurocranium and the viscerocranium (facial skeleton) that includes the mandible as its largest bone. The skull forms the anterior-most portion of the skeleton and is a product of cephalisation—housing the brain, and several sensory structures such as the eyes, ears, nose, and mouth. In humans these sensory structures are part of the facial skeleton. Functions of the skull include protection of the brain, fixing the distance between the eyes to allow stereoscopic vision, and fixing the position of the ears to enable sound localisation of the direction and distance of sounds. In some animals, such as horned ungulates (mammals with hooves), the skull also has a defensive function by providing the mount (on the fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectite
A symplectite (or symplektite) is a material texture: a micrometre-scale or submicrometre-scale intergrowth of two or more crystals. Symplectites form from the breakdown of unstable phases, and may be composed of minerals, ceramics, or metals. Fundamentally, their formation is the result of slow grain-boundary diffusion relative to interface propagation rate.Lee, H. J., G. Spanos, G. J. Shiflet, and H. I. Aaronson (1988), Mechanisms of the bainite (non-lamellar eutectoid) reaction and a fundamental distinction between the bainite and pearlite (lamellar eutectoid) reactions, Acta Metall., 36, 1129– 1140. If a material undergoes a change in temperature, pressure or other physical conditions (e.g., fluid composition or activity), one or more phases may be rendered unstable and recrystallize to more stable constituents. If the recrystallized minerals are fine grained and intergrown, this may be termed a symplectite. A cellular precipitation reaction, in which a reactant phase decompos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
