From (30) it is obvious that the transversality condition ([frac {dmu}{dtau}]_{tau=hat{tau}}>0 ) for the occurrence of the Hopf bifurcation at (tau=hat{tau} ) is well satisfied provided (beta_{2}beta_{1}+alpha_{2}alpha_{1}>0 ).
We deduce from (34) that condition (ii) in Assumption 3 is satisfied, provided that no additional portfolio constraints are imposed (recall that condition (i) in Assumption 3 is always postulated to hold).
where the decision c ̂ is obtained by the decoding (5), and the condition, c ̂ ML ≠ c Tx, is satisfied provided that the vectors c ̂ ML and cTx differ in at least one component, i.e., c ̂ ML = c Tx if and only if all the components of these vectors are equal.
In particular, in the case of the problem (1.4), i.e., in the case that a 1 ( x ) = 1 | x | s 1 ( x ), a 2 ( x ) = 1 | x | s 2 ( x ), all conditions of Theorem 4.1 are satisfied provided (P), (A*), (A1), and (A2) hold.
If the fluid is air, it is adequately satisfied provided that the largest excursion in z is on the order of metres rather than kilometres and provided that the fluid velocity is everywhere less than about 100 metres per second.
One can check that all the properties given in (H) are satisfied provided that 0 < λ0 < 1. Case 1.
