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An exact characteristic equation is derived.
An exact characteristic equation is derived giving the relationship between the non-dimensional frequency and the two non-dimensional parameters, i.e., the axially compressed force and the end mass.
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Furthermore, when an exact characteristics split flux formula is used, the spurious mode becomes nonexistent.
The explicit exact characteristic equation for eigenvalues is derived, which does not use the iteration approach.
The method has exact characteristic and can avoid information distortion and losing in the linguistic information processing.
In addition, the exact characteristic equations of the early paper can also be greatly simplified and the compact equations are presented herein.
The complete first six frequencies and mode shapes are determined for clamped, simply-supported, sliding, and ring supports using exact characteristic frequency equations.
The exact characteristic equations for the non-conservative elastic stability of general elastic restrained non-uniform columns governed by a fourth order ordinary differential equation with arbitrarily varying polynomial coefficients are derived.
For simplicity, instead of calculating the exact characteristic velocities, we follow the strategy of (Gammie et al. 2003) who propose a simplified dispersion relation for the fast MHD wave (omega^{2} = a^{2} k^{2}).
Although both types of information are from farmers' recollection and can hardly be "exact," characteristic information is more reliable than yields of specific years because it captures the feature of yield distribution (Gong et al. 2013).
It has to be noted that the exact characteristic of the noise in the bioluminescence profiles is not known.
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