Sentence examples for and the computations of from inspiring English sources

Thanks to Lemma 4.2 and the computations of Proposition 4.11, we can show the assertion.

There has been extensive study of both the theories and the computations of generalized matrix eigenvalue problems, and one can refer to for more details.

Furthermore, the existence of nontrivial solutions of discrete Kirchhoff-type problem (1.1) with resonance at both zero and infinity is also studied by employing the critical point theory, especially the local linking, Morse theory and the computations of critical groups.

In the 4-point RFFT, due to redundancy, the bottom butterfly at the second stage is removed and the computations of real and imaginary parts of X[1] are separated as shown in Fig. 6.

In this paper, we study the existence of nontrivial solutions for discrete Kirchhoff-type problems with resonance at both zero and infinity by using variational methods and the computations of critical groups.

Here, the geometries, electronic structures, as well as the dipole moments and polarizabilities of organic dye sensitizers C343 and 20 kinds of NKX derivatives were calculated using density functional theory (DFT), and the computations of the time dependent DFT with different functionals were performed to explore the electronic absorption properties.

In either event, the purchase price is known, and the computation of basis is easy.

Also, the computation of nonsaturated throughput and the computation of saturated throughput are carried out separately.

In what follows, we shall discuss the growth and the computation of π chronologically.

In each step, we save one evaluation of the derivative and the computation of one inverse.

Our system synchronizes the image capturing and the computation of gaze direction.