We sat there, enthralled as he introduced his theory of plot ("the gradual perturbation of an unstable homeostatic system and its catastrophic restoration to a new and complexified equilibrium") or compared the structure of dramatic action to a love affair.
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These internal seachanges within NASA, along with shifts in its relationships with other federal space agencies and private companies, reflect a flux in the agency's identity and purpose that has become more pronounced in recent years as the global space landscape has complexified.
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There are other closed subgroups of the complexification of a compact connected Lie group G which have the same complexified Lie algebra. These are the other real forms of GC.
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Using complexification, rewrote the formula (1) into : where CS(S^3\backslash K) is called the Chern–Simons invariant. They showed that there is a clear relation between the complexified colored Jones polynomial and Chern–Simons theory from a mathematical point of view.
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Nigel Hitchin introduced the notion of a generalized almost complex structure on the manifold M, which was elaborated in the doctoral dissertations of his students Marco Gualtieri and Gil Cavalcanti. An ordinary almost complex structure is a choice of a half-dimensional subspace of each fiber of the complexified tangent bundle TM. A generalized almost complex structure is a choice of a half-dimensional isotropic subspace of each fiber of the direct sum of the complexified tangent and cotangent bundles. In both cases one demands that the direct sum of the subbundle and its complex conjugate yield the original bundle. An almost complex structure integrates to a complex structure if the half-dimensional subspace is closed under the Lie bracket.
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Geometries that are specializations of real projective geometry, such as Euclidean geometry, elliptic geometry or conformal geometry may be complexified, thus embedding the points of the geometry in a complex projective space, but retaining the identity of the original real space as special. Lines, planes etc. are expanded to the lines, etc. of the complex projective space.
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The presence/absence of homologs (or their effective count) can thus be used by programs to reconstruct the most likely evolutionary scenario along the species tree. Just as with reconciliation methods, this can be achieved through parsimonious or probabilistic estimation of the number of gain and loss events. Models can be complexified by adding processes, like the truncation of genes, but also by modelling the heterogeneity of rates of gain and loss across lineages and/or gene families.
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An abstract CR structure on a real manifold M of real dimension n consists of a complex subbundle L of the complexified tangent bundle which is formally integrable, in the sense that [L,L] ⊂ L, which has zero intersection with its complex conjugate. The CR codimension of the CR structure is k = n - 2 \dim L, where dim L is the complex dimension. In case k = 1, the CR structure is said to be of hypersurface type. Most examples of abstract CR structures are of hypersurface type.
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Put pij = xiyj \- xjyi. These are the homogeneous coordinates of the projective line joining x and y. There are six independent coordinates and they satisfy a single relation, the Plücker relation :p01 p23 \+ p02 p31 \+ p03 p12 = 0. It follows that there is a one-to-one correspondence between lines in RP3 and points on the Klein quadric, which is the quadric hypersurface of points [p01, p23, p02, p31, p03, p12] in RP5 satisfying the Plücker relation. The quadratic form defining the Plücker relation comes from a symmetric bilinear form of signature (3,3). In other words, the space of lines in RP3 is the quadric in P(R3,3). Although this is not the same as the Lie quadric, a "correspondence" can be defined between lines and spheres using the complex numbers: if x = (x0,x1,x2,x3,x4,x5) is a point on the (complexified) Lie quadric (i.e., the xi are taken to be complex numbers), then : p01 = x0 \+ x1, p23 = –x0 \+ x1 : p02 = x2 \+ ix3, p31 = x2 – ix1 : p03 = x4 , p12 = x5 defines a point on the complexified Klein quadric (where i2 = –1).
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Montonen–Olive duality applies to a very special type of gauge theory called N = 4 supersymmetric Yang–Mills theory, and it says that two such theories may be equivalent in a certain precise sense. If one of the theories has a gauge group G, then the dual theory has gauge group {^L}G where {^L}G denotes the Langlands dual group which is in general different from G.Frenkel 2009, p.5 An important quantity in quantum field theory is complexified coupling constant. This is a complex number defined by the formulaFrenkel 2009, p.
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The complexification of a real vector space results in a complex vector space (over the complex number field). To "complexify" a space means extending ordinary scalar multiplication of vectors by real numbers to scalar multiplication by complex numbers. For complexified inner product spaces, the complex inner product on vectors replaces the ordinary real-valued inner product, an example of the latter being the dot product. In mathematical physics, when we complexify a real coordinate space Rn we create a complex coordinate space Cn, referred to in differential geometry as a "complex manifold".
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The Clifford algebra over spacetime can be regarded as the set of real linear operators from to itself, , or more generally, when complexified to , as the set of linear operators from any 4-dimensional complex vector space to itself. More simply, given a basis for , is just the set of all complex matrices, but endowed with a Clifford algebra structure. Spacetime is assumed to be endowed with the Minkowski metric . A space of bispinors, , is also assumed at every point in spacetime, endowed with the bispinor representation of the Lorentz group.
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These are some of the many ways in which different quantizations of the same classical theory can result in inequivalent quantum theories, or even in the impossibility to carry quantization through. One can't distinguish between SO(3) and SU(2) or between SO(3,1) and SL(2,C) at this level: the respective Lie algebras are the same. In fact, all four groups have the same complexified Lie algebra, which makes matters even more confusing (these subtleties are usually ignored in elementary particle physics). The physical interpretation of the Lie algebra is that of infinitesimally small group transformations, and gauge bosons (such as the graviton) are Lie algebra representations, not Lie group representations.
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Mathematically LQG is local gauge theory of the self-dual subgroup of the complexified Lorentz group, which is related to the action of the Lorentz group on Weyl spinors commonly used in elementary particle physics. This is partly a matter of mathematical convenience, as it results in a compact group SO(3) or SU(2) as gauge group, as opposed to the non-compact groups SO(3,1) or SL(2.C). The compactness of the Lie group avoids some thus- far unsolved difficulties in the quantization of gauge theories of noncompact lie groups, and is responsible for the discreteness of the area and volume spectra. The theory involving the Immirzi parameter is necessary to resolve an ambiguity in the process of complexification.
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As with the inclusion of points at infinity and complexification of real polynomials, this allows some theorems to be stated more simply without exceptions and for a more regular algebraic analysis of the geometry. Viewed in terms of homogeneous coordinates, a real vector space of homogeneous coordinates of the original geometry is complexified. A point of the original geometric space is defined by an equivalence class of homogeneous vectors of the form , where is an nonzero complex value and is a real vector. A point of this form (and hence belongs to the original real space) is called a real point, whereas a point that has been added through the complexification and thus does not have this form is called an imaginary point.
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For example, the 6-dimensional sphere S6 has a natural almost complex structure arising from the fact that it is the orthogonal complement of i in the unit sphere of the octonions, but this is not a complex structure. (The question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf.) Using an almost complex structure we can make sense of holomorphic maps and ask about the existence of holomorphic coordinates on the manifold. The existence of holomorphic coordinates is equivalent to saying the manifold is complex (which is what the chart definition says). Tensoring the tangent bundle with the complex numbers we get the complexified tangent bundle, on which multiplication by complex numbers makes sense (even if we started with a real manifold).
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What this means for the Lorentz group is that, for sufficiently small velocity parameters, all four complexified Lie groups are indistinguishable in the absence of matter fields. To make matters more complicated, it can be shown that a positive cosmological constant can be realized in LQG by replacing the Lorentz group with the corresponding quantum group. At the level of the Lie algebra, this corresponds to what is called q-deforming the Lie algebra, and the parameter q is related to the value of the cosmological constant. The effect of replacing a Lie algebra by a q-deformed version is that the series of its representations is truncated (in the case of the rotation group, instead of having representations labelled by all half-integral spins, one is left with all representations with total spin j less than some constant).
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