Differentiable Manifolds

Choquet–Bruhat, Yvonne

doi:10.1016/b978-044450473-9/50005-1

Cited by 147 publications

(294 citation statements)

References 2 publications

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“…Unfortunately, the mathematical tools to handle especially distributed systems are quite advanced. An almost complete introduction to these methods can be found in [2], the needed differential geometric methods are also presented in [4,9]. As mentioned above, also the study of partial differential equations requires advanced methods, see, e.g.…”

Section: Discussionmentioning

confidence: 98%

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Mathematical modeling for nonlinear control: a Hamiltonian approach

Schlacher

2008

Mathematics and Computers in Simulation

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Section: Discussionmentioning

confidence: 98%

“…Let us consider the curve x(t + τ) = e Aτ x(t) given by (2), then its tangent vector at t is given bẏ…”

Section: Linear Systemsmentioning

confidence: 99%

Mathematical modeling for nonlinear control: a Hamiltonian approach

Schlacher

2008

Mathematics and Computers in Simulation

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“…Hence we may apply the implicit function theorem [4] to obtain the following Theorem 1 For all ω ∈ (−1, 1) there exist ǫ > 0 and a C 1 map [0, ǫ) → ℓ 2 , denoted α → θ α , such that θ α is a frequency ω, coupling α solution of (2.5) and θ 0 = θ b . The map α → θ α is unique provided ǫ is chosen sufficiently small.…”

Section: Analytic Results On Discrete Breathersmentioning

confidence: 99%

Discrete breathers in anisotropic ferromagnetic spin chains

Speight¹,

Sutcliffe²

2001

J. Phys. A: Math. Gen.

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We prove the existence of discrete breathers (time-periodic, spatially localized solutions) in weakly coupled ferromagnetic spin chains with easy-axis anisotropy. Using numerical methods we then investigate the continuation of discrete breather solutions as the intersite coupling is increased. We find a band of frequencies for which the 1-site breather continues all the way to the soliton solution in the continuum. There is a second band, which abuts the first, in which the 1-site breather does not continue to the soliton solution, but a certain multi-site breather does. This banded structure continues, so that in each band there is a particular multi-site breather which continues to the soliton solution. A detailed analysis is presented, including an exposition of how the bifurcation pattern changes as a band is crossed. The linear stability of breathers is analyzed. It is proved that 1-site breathers are stable at small coupling, provided a non-resonance condition holds, and an extensive numerical stability analysis of 1-site and multisite breathers is performed. The results show alternating bands of stability and instability as the coupling increases. * Email : J.M. Speight@leeds.ac.uk

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“…Recall that continuous quantum (Lie) symmetries appear in the analysis of manifolds using the commutator of the covariant derivatives ∇ k φ l (≡ ∂ x k φ l − l kj φ j ) with ∇ l acting on the contravariant components of a vector, φ l , [15]: …”

Section: Analysis Of Quantum Dynamical Manifolds Into Lie Group Symmementioning

confidence: 99%

Quantum dynamical manifolds 5. Hydrogen mass-spacetime

Scofield

2000

Int. J. Quantum Chem.

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The newly developed methods of quantum dynamical manifold theory (QDMT) are used to determine a theory of the differential geometry and topology of the quantum dynamical manifold of the hydrogen nuclear‐electron system. This provides an ab initio theory of the structure of nuclear‐atomic system including the quarks, gluons, and (W+,Z0,W−) in the proton and the atomic orbital electron in a way that treats the strong, electromagnetic, weak, and gravitational forces from a unified QDMT viewpoint. The calculations proposed resemble self‐consistent Green's function ones in Fourier space. The quasi‐particles provide the basis for computing the globally defined connection needed for determining the self‐consistent gauge potentials. This approach allows the simultaneous determination of coupling constants and masses along with the quasi‐particle amplitudes. All integrals needed can be analytically computed using a Gaussian‐type orbital basis. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 369–393, 2000

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Differentiable Manifolds

Cited by 147 publications

References 2 publications

Mathematical modeling for nonlinear control: a Hamiltonian approach

Mathematical modeling for nonlinear control: a Hamiltonian approach

Discrete breathers in anisotropic ferromagnetic spin chains

Quantum dynamical manifolds 5. Hydrogen mass-spacetime

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