Further, equation (7) has the unique positive root (omega_{n}), and the characteristic equation (4) has purely imaginary roots (pm iomega_{n}), where omega_{n}=sqrt{frac{-P_{n}+sqrt{P_{n}^{2}-4Q_{n}}{{2}}, quad n=0,1,2,ldots, N_{2}.
By a similar way, we can show that the equation (Delta_{2}(lambda)=0) has the unique simple root (lambda_{m}^) in the (delta=frac{1}{6} -neighborhood of the point (lambdelta=frac{1}{6} -neighborhood_{1}) and sofe (r_{1}inmatheb{N}) for which (lambda_{m}^=frac{m}{2}+O(frac{1}{m})), (forall mgeq r_{1}) holds.
Denote it − r 0. We get that g ′ ( t ) = 1 2 K t 2 + γ t − 1 has the unique positive root α = 2 γ + γ 2 + 2 K, and for t ≥ 0, g ″ ( t ) = K t + γ > 0. So, the necessary and sufficient condition that g ( t ) has two positive roots for t ≥ 0 is that the minimum of g ( t ) satisfies g ≤ 0, that is also η ≤ β.
Legumes have the unique capability to undergo root nodule and arbuscular mycorrhizal symbiosis.
Legumes have the unique capacity to undergo two important root endosymbioses: the root nodule symbiosis and the arbuscular mycorrhizal symbiosis.
Hairy roots have the unique property of being able to grow in vitro in the absence of exogenous plant growth regulators [ 8].
Then the equation (Delta_{1}(lambda)=0) may have at most a countable set of roots in the set (E_{alpha}), and all these roots may have the unique limit point at infinity.
At the end of iteration t, every root node in R (t) belongs to a distinct tree in the associated forest, and every tree in the forest has a unique root node which belongs to R (t).
Finally, assume that γ i, i = 1, …, p, satisfy (24), (25), and choose u ∈ B ¯. Then the function σ ( t ) = γ i ( u ( t ), u ′ ( t ), …, u ( n − 2 ) ( t ) ) − t, t ∈ [ a, b ], (29). is continuous and decreasing on [ a, b ] and it has a unique root in the interval ( a, b ), i.e. there exists a unique solution of the equation t = γ i ( u ( t ), …, u ( n − 2 ) ( t ) ). (30).
It is enough to show that Ψ ( R ) = 1 + R ln ( R ) 2 ( R − 1 ) < 2, and this holds true if and only if F ( R ) = R ln ( R ) − 2 R + 2 < 0. By simple computations one can show that the function F ( R ) has a unique root in ( 1, ∞ ).
