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one, B 0, is constant in time, the other one, ΔB z (t), may be time dependent. It is present in magnetic resonance imaging and helps with the spatial decoding of the NMR signal. M(t) × B(t) is the cross product of these two vectors. M 0 is the steady state nuclear magnetization (that is, for example, when t → ∞); it is in the z direction.
A static spacetime is one in which all metric components are independent of the time coordinate (so that =) and the geometry of the spacetime is unchanged under a time-reversal . A vacuum solution is one that satisfies the equation T a b = 0 {\displaystyle T_{ab}=0} .
It is used to analyse discrete dynamical systems by finding a new coordinate system in which the system (orbit) generated by h(x) looks simpler, a mere dilation.. More specifically, a system for which a discrete unit time step amounts to x → h(x), can have its smooth orbit (or flow) reconstructed from the solution of the above Schröder's equation, its conjugacy equation.
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One example is the planetary movement of three bodies: while there is no closed-form solution to the general problem, Poincaré showed for the first time that it exhibits deterministic chaos. Formally, a Hamiltonian system is a dynamical system characterised by the scalar function H ( q , p , t ) {\displaystyle H({\boldsymbol {q}},{\boldsymbol ...
A general guide has been provided as a design principle to achieve parameters (e.g., mesh size and physical parameters such as Poisson's ratio that appear in the nearly singular operator) independent convergence rate of the multigrid method applied to such nearly singular systems, [24] i.e., in each grid, a space decomposition based on which ...
The Madelung equations answer the question whether (,) obeys the continuity equations of hydrodynamics and, subsequently, what plays the role of the stress tensor. [ 9 ] The circulation of the flow velocity field along any closed path obeys the auxiliary quantization condition Γ ≐ ∮ m v ⋅ d l = 2 π n ℏ {\textstyle \Gamma \doteq \oint ...
The Rabi problem concerns the response of an atom to an applied harmonic electric field, with an applied frequency very close to the atom's natural frequency. It provides a simple and generally solvable example of light–atom interactions and is named after Isidor Isaac Rabi .