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In particular, Hilbert space theory can be used to develop a rich theory of weak solutions of partial differential equations and a corresponding theory of finite element approximations to weak solutions. An orthogonal set of unit vectors (i.e., vectors of norm one) is called an orthonormal set...
Because the path is an integration throughout the dataset, the accuracy of the cell interpolation functions and the accuracy of the original vector data play important roles in realizing accurate solutions. No definitive study that relates cell size or interpolation function characteristics to visualizati...
2.2 Non-degeneracy The non-degeneracy of solutions near the free boundary points allows us to obtain several results on the measure-theoretic structure of the free boundary via the blow-up analysis. We start by proving the following weak non-degeneracy condition.Proposition 2.7...
This paper tries to reveal the influence of the external source terms f and g on the number of the solutions. It is shown that the level set of the corresponding functional has a quite rich topology and the system admits k spikes synchronized vector solutions for any k∈Z+ when f and g...
where βk+1 has two solutions βa and βb according to the WORD method. Then, one of the solutions βa or βb which one can minimize the variation function F(β)=‖P[w(k)]⊥wk+1/‖wk+1‖2‖22 will be selected. It is noticed that the appropriate β is determined by a selecti...
With NeurVec, we just need to estimate the solution on every k steps instead of step by step as in Eq. (3). NeurVec is trained from the more accurate solutions with fine step size \Delta t to characterize the error caused by the use of the coarse step size k\Delta t. The paramete...