Sentence examples for root of multiplicity from inspiring English sources

If Δ 3 = 0, Δ 2 = 0 and Δ 1 = 0 then (13) has one real root of multiplicity four.

(h) If Δ 3 = 0, Δ 2 = 0 and Δ 1 = 0 then (13) has one real root of multiplicity four.  .

If Δ 1 = 0 and Δ 2 = 0, then equation (7) has one pair of conjugate imaginary roots and one real root of multiplicity three.

(e) If Δ 1 = 0 and Δ 2 = 0, then equation (7) has one pair of conjugate imaginary roots and one real root of multiplicity three.  .

(d) If Δ 1 = 0 and Δ 2 ≠ 0, then equation (7) has one pair of conjugate imaginary roots and two real roots, one real root of multiplicity one and other one of multiplicity two.

If Δ 1 = 0 and Δ 2 ≠ 0, then equation (7) has one pair of conjugate imaginary roots and two real roots, one real root of multiplicity one and other one of multiplicity two.

Proof In view of Lemma 1, equation (7) has three positive roots of multiplicity one.

By Lemma 7, suppose that ({phi_{1},psi_{1}}) is a minimal period-two solution such that ({phi_{1},psi_{1}}) are roots of multiplicity two of (12) and ({phi_{2},psi_{2}}) are roots of multiplicity one of (12).

The CT study yields additional experimental insight into the role of capillary pressure and ultimately leads to the proposal of a capillary gate mechanism based on contact angle hysteresis as the root cause of multiplicity.

For an input vector, the algorithm computes a special bivariate polynomial such that each couple is a root of with multiplicity.

In view of Lemmas 6 and 7, from [27] we have that x ˜ ′ ( y ¯ 1 ) = 0, since y ¯ 1 is the root of (7) of multiplicity two.