Sentence examples for for these formulas from inspiring English sources

Starting with infinite series expansions for sine, cosine, and π that had been introduced into China (without, however, knowledge of the calculus used to derive these series), Minggantu constructed proofs for these formulas and also derived series for some of the inverse trigonometric functions (arc sine and arc cosine).

We then design a quantifier elimination procedure for these formulas.

Deriving compact expressions for these formulas turns out to be extremely difficult; however, all the needed results are available in the literature.

For these formulas, recall that h ∗ denotes the value of h attained when cell 1 is about tojump down (i.e., cell 1 is active, cell 1 is not inhibited, and v 1 = θ I, see Figure 2A); similarly, m 2 ∗, m 3 ∗ denote the values of m 2, m 3 when cell 2 or cell 3 is about to jump down ( v 2 = θ I, v 3 = θ I ), respectively.

Costs and probabilities for these formulas were based primarily on published studies.

Costs and probabilities for these formulas were based on published studies and assumptions, as listed in Table 1 and described below.

For each of these formulas, several different orderings are observed depending on the relative charges and sizes of the A and B cations.

Conservative Reformers retained the shell of these formulas for worship, though they took great pains to bring these formulas into the tradition of evangelical teaching.

Some interesting consequences of these formulas, for the complete graph and other kinds of topologies, will be analyzed in Sects.

The aim of this work is to prove the validity or limitations of these formulas for various combinations of dusts and gases.

Unfortunately, the restricted use of these formulas for specific creatinine methods has not been an overriding concern in some studies.