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Splitting Cartan Subalgebra
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Splitting may refer to: * Splitting (psychology) * Lumpers and splitters, in classification or taxonomy * Wood splitting * Tongue splitting * Splitting, railway operation Mathematics * Heegaard splitting * Splitting field * Splitting principle * Splitting theorem * Splitting lemma * for the numerical method to solve differential equations, see Symplectic integrator See also * Split (other) * Splitter (other) Splitter or splitters may refer to: Technology * DSL filter or DSL splitter, in telecommunications * Fiber-optic splitter * Hybrid coil, a three windings transformer * Power dividers and directional couplers, in RF engineering * Siamese connect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Splitting
Splitting may refer to: * Splitting (psychology) * Lumpers and splitters, in classification or taxonomy * Wood splitting * Tongue splitting * Splitting, railway operation Mathematics * Heegaard splitting * Splitting field * Splitting principle * Splitting theorem * Splitting lemma * for the numerical method to solve differential equations, see Symplectic integrator See also * Split (other) Split(s) or The Split may refer to: Places * Split, Croatia, the largest coastal city in Croatia * Split Island, Canada, an island in the Hudson Bay * Split Island, Falkland Islands * Split Island, Fiji, better known as Hạfliua Arts, enterta ... * Splitter (other) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Splitting (psychology)
Splitting (also called black-and-white thinking or all-or-nothing thinking) is the failure in a person's thinking to bring together the dichotomy of both perceived positive and negative qualities of something into a cohesive, realistic whole. It is a common defense mechanism wherein the individual tends to think in extremes (e.g., an individual's actions and motivations are ''all'' good or ''all'' bad with no middle ground). This kind of dichotomous interpretation is contrasted by an acknowledgement of certain nuances known as "shades of gray". Splitting was first described by Ronald Fairbairn in his formulation of object relations theory; it begins as the inability of the infant to combine the fulfilling aspects of the parents (the good object) and their unresponsive aspects (the unsatisfying object) into the same individuals, instead seeing the good and bad as separate. In psychoanalytic theory this functions as a defense mechanism. Relationships Splitting creates instability i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lumpers And Splitters
Lumpers and splitters are opposing factions in any discipline that has to place individual examples into rigorously defined categories. The lumper–splitter problem occurs when there is the desire to create classifications and assign examples to them, for example schools of literature, biological taxa and so on. A "lumper" is a person who assigns examples broadly, assuming that differences are not as important as signature similarities. A "splitter" is one who makes precise definitions, and creates new categories to classify samples that differ in key ways. Origin of the terms The earliest known use of these terms was by Charles Darwin, in a letter to Joseph Dalton Hooker in 1857: ''It is good to have hair-splitters & lumpers''. They were introduced more widely by George G. Simpson in his 1945 work ''The Principles of Classification and a Classification of Mammals''. As he put it: A later use can be found in the title of a 1969 paper "On lumpers and splitters ..." by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wood Splitting
Wood splitting (''riving'',"Riving" def. 1.b. ''Oxford English Dictionary'' Second Edition on CD-ROM (v. 4.0) Oxford University Press 2009 cleaving) is an ancient technique used in carpentry to make lumber for making wooden objects, some basket weaving, and to make firewood. Unlike wood sawing, the wood is split along the grain using tools such as a hammer and wedges, splitting maul, cleaving axe, side knife, or froe. Woodworking In woodworking carpenters use a wooden siding which gets its name, clapboard, from originally being split from logs—the sound of the plank against the log being a clap. This is used in clapboard architecture and for wainscoting. Coopers use oak clapboards to make barrel staves. Split-rail fences are made with split wood. Basket making Some Native Americans traditionally make baskets from black ash by pounding the wood with a mallet and pulling long strips from the log. Firewood Log splitting is the act of splitting firewood from logs that have b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tongue Splitting
Tongue bifurcation, splitting or forking, is a type of body modification in which the tongue is cut centrally from its tip to as far back as the underside base, forking the end. Bifid tongue in humans may also be an unintended complication of tongue piercingsFleming, P., Flood, T. Bifid tongue — a complication of tongue piercing. Br Dent J 198, 265–266 (2005). https://doi.org/10.1038/sj.bdj.4812117 or a rare congenital malformation associated with maternal diabetes, orofaciodigital syndrome 1, Ellis–Van Creveld syndrome, Goldenhar syndrome, and Klippel–Feil syndrome. Practice Deliberate tongue splitting is a cosmetic body modification procedure that results in a ‘lizard-like’ bifid tongue. Tongue bifurcation has also been reported as an unintended complication of tongue piercing. According to Google Trends, search interest in tongue splitting peaked in 2004, and has the highest search interest in South America followed by the Post-Soviet States, then the Anglo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Splitting (raylway)
Splitting may refer to: * Splitting (psychology) * Lumpers and splitters, in classification or taxonomy * Wood splitting * Tongue splitting * Splitting, railway operation Mathematics * Heegaard splitting * Splitting field * Splitting principle * Splitting theorem * Splitting lemma * for the numerical method to solve differential equations, see Symplectic integrator See also * Split (other) Split(s) or The Split may refer to: Places * Split, Croatia, the largest coastal city in Croatia * Split Island, Canada, an island in the Hudson Bay * Split Island, Falkland Islands * Split Island, Fiji, better known as Hạfliua Arts, enterta ... * Splitter (other) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heegaard Splitting
In the mathematical field of geometric topology, a Heegaard splitting () is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies. Definitions Let ''V'' and ''W'' be handlebodies of genus ''g'', and let ƒ be an orientation reversing homeomorphism from the boundary of ''V'' to the boundary of ''W''. By gluing ''V'' to ''W'' along ƒ we obtain the compact oriented 3-manifold : M = V \cup_f W. Every closed, orientable three-manifold may be so obtained; this follows from deep results on the triangulability of three-manifolds due to Moise. This contrasts strongly with higher-dimensional manifolds which need not admit smooth or piecewise linear structures. Assuming smoothness the existence of a Heegaard splitting also follows from the work of Smale about handle decompositions from Morse theory. The decomposition of ''M'' into two handlebodies is called a Heegaard splitting, and their common boundary ''H'' is called the Heegaard surf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Splitting Field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a polynomial ''p''(''X'') over a field ''K'' is a field extension ''L'' of ''K'' over which ''p'' factors into linear factors :p(X) = c\prod_^ (X - a_i) where c\in K and for each i we have X - a_i \in L /math> with ''ai'' not necessarily distinct and such that the roots ''ai'' generate ''L'' over ''K''. The extension ''L'' is then an extension of minimal degree over ''K'' in which ''p'' splits. It can be shown that such splitting fields exist and are unique up to isomorphism. The amount of freedom in that isomorphism is known as the Galois group of ''p'' (if we assume it is separable). Properties An extension ''L'' which is a splitting field for a set of polynomials ''p''(''X'') over ''K'' is called a normal extension of ''K''. Given an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Splitting Principle
In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles. In the theory of vector bundles, one often wishes to simplify computations, say of Chern classes. Often computations are well understood for line bundles and for direct sums of line bundles. In this case the splitting principle can be quite useful. The theorem above holds for complex vector bundles and integer coefficients or for real vector bundles with \mathbb_2 coefficients. In the complex case, the line bundles L_i or their first characteristic classes are called Chern roots. The fact that p^*\colon H^*(X)\rightarrow H^*(Y) is injective means that any equation which holds in H^*(Y) (say between various Chern classes) also holds in H^*(X). The point is that these equations are easier to understand for direct sums of line bundles than for arbitrary vector bundles, so equations should be understood in Y and then pushed down to X. Since vector bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Splitting Theorem
In the mathematical field of differential geometry, there are various splitting theorems on when a pseudo-Riemannian manifold can be given as a metric product. The best-known is the Cheeger–Gromoll splitting theorem for Riemannian manifolds, although there has also been research into splitting of Lorentzian manifolds. Cheeger and Gromoll's Riemannian splitting theorem Any connected Riemannian manifold has an underlying metric space structure, and this allows the definition of a ''geodesic line'' as a map such that the distance from to equals for arbitrary and . This is to say that the restriction of to any bounded interval is a curve of minimal length which connects its endpoints. In 1971, Jeff Cheeger and Detlef Gromoll proved that, if a geodesically complete and connected Riemannian manifold of nonnegative Ricci curvature contains any geodesic line, then it must split isometrically as the product of a complete Riemannian manifold with . The proof was later simplified by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Splitting Lemma
In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent for a short exact sequence : 0 \longrightarrow A \mathrel B \mathrel C \longrightarrow 0. If any of these statements holds, the sequence is called a split exact sequence, and the sequence is said to ''split''. In the above short exact sequence, where the sequence splits, it allows one to refine the first isomorphism theorem, which states that: : (i.e., isomorphic to the coimage of or cokernel of ) to: : where the first isomorphism theorem is then just the projection onto . It is a categorical generalization of the rank–nullity theorem (in the form in linear algebra. Proof for the category of abelian groups and First, to show that 3. implies both 1. and 2., we assume 3. and take as the natural projection of the direct sum onto , and take as the natural injection of into the direct sum. To prove that 1 ... [...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-form. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
