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We need to show that there exists a trajectory connecting the two singular points ((a_{2k},0)) and ((b_{2k},0)).
According to the formulation, the most probable dynamical trajectory connecting the initial and final atomic configurations, is determined through minimizing an appropriate object function that is derived from the classical action by imposing dynamical conditions as required as the conservations of total energy.
Indeed, Cory's shearwater tracks did not follow the shortest (great circle) trajectory connecting breeding and wintering areas, but flyways were 1.6 3.1 times longer; neither followed geographic or magnetic loxodromes, nor the direct sun compass directions.
From the productive trajectories, i.e., the stretches of trajectory connecting the most populated states (N,I,D), several snapshots were selected and Pcommit was computed for each.
The gradient of this field at each point forms a unique trajectory connecting the 2 boundaries, and the thickness at each point is calculated by integrating along these trajectories.
In ADMD, the most probable transition trajectory connecting two given boundary states is generated by minimizing the action of the system (see Section 2 for more details).
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The existence of fixed points, closed orbits and the unions of fixed points and the trajectories connecting them is shown.
First of all, the trajectory is partitioned into so-called event segments, which represent the partial trajectories connecting two successive HO or LAU positions.
By combining in the same 3D graphical environment several geometric entities associated to the PPRM, such as workspace volume, singularity surface, and the different solutions of the FKP, this approach allows designers, in a user-friendly way, to generate the singularity-free trajectories connecting different assembly modes.
Here we report the development of a novel, predictive Bayesian approach that is able to infer likely evolutionary trajectories connecting phenotypes from sparsely sampled, highly stochastic data.
A volume of linear size L includes roughly N 3 = 2 (L / a ) 3 fields of spacing a in either an FCC or HCP arrangement (and 6 2 (L / a ) 3 trajectories connecting neighboring fields).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com