This novel study further treats the NCHX itself as a global porous medium characterized with global permeability (Kg) and form coefficient (Cg), which is being influenced by the local tube-to-tube inter-connector porous medium, characterized with a local permeability (Ki).
The underidentification tests in all four 2SLS regressions reject the null hypothesis indicating that the reduced form coefficient matrix is full rank and the instruments are able to identify the model, satisfying the relevance assumption.
Within each transformed block, the coefficients in position ( are selected to form coefficient groups and for every group an histogram is created. Histogram modification is then applied to insert the watermark only to AC groups.
Following the algorithms given in Kuznetsov [8], the critical normal form coefficient c ( 0 ), that determines the nondegeneracy of the period-doubling bifurcation and the stability of period-doubling cycle, is given by the following formula: c ( 0 ) = 1 6 〈 v, C ( w, w, w ) 〉 − 1 2 〈 v, B ( w, ( H − I ) − 1 B ( w, w ) ) 〉. (4.3).
Following the algorithms given in [24], the sign of the critical normal form coefficient (c(delta_{F})), which determines the direction of the flip bifurcation, is given by the following formula: c(delta_{F}) = frac{1}{6} bigllangle p, C q, q, q) bigrrangle - frac{1}{2} bigllangle p, Bbigl q, (A-I)^{-1} B q, q bigr) bigrrangle.
Following the algorithms given in Kuznetsov [10], the critical normal form coefficient (c(0)), that determines the nondegeneracy of period-doubling bifurcation and the stability of period-doubling cycle, is given by the following formula: c(0)=frac{1}{6}bigllangle v,C w,w,w bigrrangle - frac{1}{2}bigllangle v,Bbigl w, H-I)^{-1}B(w, H-Igr)bigrrangle.
An alternate approach provided is to tabulate the digital filter coefficients for different settings of the controls (a 25 x 25 x 25 matrix) and use the Gray-Markel algorithm (tf2latc in Matlab) to convert the 3rd order direct form coefficients to a lattice filter implementation.
The obtained floating-point numerical values for the normal form coefficients and the stability criterion confirm the results by Markeev (1969) and Coppola and Rand (1989), while the obtained analytical and exact numeric expressions confirm the results by Meyer and Schmidt (1986) and Schmidt (1989).
(c) Study of the dynamics of the normal form for the bifurcation under generic assumptions on the normal form coefficients.
Therefore, if there are multiple solutions which make the reduced form coefficients compatible with the structural coefficients, the model is underidentified.
(c′): Study of the dynamics of the normal form for the symmetric bifurcation under generic assumptions on the normal form coefficients (the symmetries mean that some coefficients may be zero, some are constrained to be equal while others may be forced to satisfy nontrivial and sometimes obscure algebraic relationships).
