Admissible pair

We ultimately need to determine whether in the vicnity of the optimal steady state pair (y, u), the objective functional I could reduce further for some admissible pair of state and corresponding control, (y, u), where... 

We shall return to the above equation after coming up with an admissible pair (y, u), or equivalently, an admissible variational pair (Jy, Ju). 

Thus, there are infinitely many admissible pairs (y, u). Each y has continuous first and second partial derivatives with respect to... 

Theorem 13.12 For any system sufficiently close to a system with the homoclinic-S with a negative saddle value the periodic orbits or homoclinic loops in U must have admissible codes. If there is a pair of periodic orbits or a pair of homoclinic loops or a limit cycle and a homoclinic loop their codes must form an admissible pair. If the separatrices Fi and F2 are P -stable i.e, N is a quasi-minimal set) then their codes form an admissible pair of infinite sequences. 

Here, /xi = 0 corresponds to the curve Ci the curve C21 also leaves the point (0,0) towards negative values of H2 the curves C21 and L intersect at infinitely many points. Then, one may continue inductively according to this rule let (p, s) be an admissible pair of words ended by 1 then Cp and Cs intersect at infinitely many points, and if P and Q are neighboring points of intersection of Cp and Cs such that hp > hs on the interval between P and Q, then P and Q are further connected by arcs of Cps and Csp which intersect Cs and, respectively, Cp infinitely many times, etc. In the region bounded by these arcs of Cps and a stable limit cycle exists with the code ps. On the contrary, if hp < hs on the interval between P and Q, then for these values... 

For each admissible pair (p, s) of infinite sequences, there exists a curve C(p, s) corresponding to the existence of a quasiminimal attractor with the rotation number / (p) = /3(s). The curve C(p, s) is found from... 

In general for a set of nonlinear equations the necessary condition for determinancy is that there exists at least one admissible set of output variables for the equations (C7, S4). We can think of an output variable as that variable for which a given equation is solved either by an iteration process or by an elimination process. The set of all such assigned pairs of variables and equations is called the output set. Clearly an admissible output set must satisfy the conditions (i) each equation has exactly one output variable, (ii) each variable appears as the output variable of exactly one equation, (iii) each output variable must occur in the assigned equations in such a manner that it can be solved for uniquely. Such an output set (circled entities) is illustrated in terms of an occurrence matrix in Fig. 4. Algorithms for finding output sets have been published by Steward (S4) and Gupta et al. (G8). 

The diversity in attitude between the different societies towards the admission of women is striking. It can be interpreted in terms of how much the society was a pure professional society and how much a men s club. In particular, as the Royal Institute of Chemistry was the accrediting body, the role of the Chemical Society was much more towards the socialising end of the spectrum and therefore more hostile to women s admission. What comes through strongly is the importance of key individuals, particularly the pair of Ida Smedley and Martha Whiteley,100 who had each established themselves as respected chemists. However, as women were excluded from the governing body, it was supportive male chemists, such as William Tilden and William Ramsay, who had to argue the women s case in the Council of the Society. 

The Maternal Lifestyle Study has reported that the prevalence of adverse perinatal complications associated with the use of cocaine or opiates during pregnancy was lower than has been previously reported (249). In 11 811 mother-infant pairs followed prospectively, 11% of the exposed and non-exposed groups were hospitalized at least once. However, violence was a factor (20%) in admissions among the cocaine-exposed women. 

In the simplest admissible physical description of H20 neglecting hybridization, we concentrate attention on the four-electron valence problem involving 2pz = z, 2py = y AOs on the oxygen atom, lsi = h, si = hi AOs on the two hydrogen atoms, while 2s = s and 2px = x are doubly occupied AOs on oxygen making the two lone pairs. It... 

The MCSCF wavefunctions discussed in this chapter consists of linear expansions in terms of JV-electron basis functions. These N-electron functions depend on 4N variables and are spanned by the space of the Cartesian product of the spatial and spin coordinates for each electron. The total wavefunction must be antisymmetric with respect to interchange of the coordinates of any pair of electrons. This is because electrons are half-integer spin particles called fermions and all fermion wavefunctions must satisfy this property. This is treated as a constraint on the class of admissible basis functions in the MCSCF... 

Reactivity of the PP mechano-radicals wdth oxygen was studied in more detail. As stated in Section III-4.1, the two kinds of radicals were produced in a pair by a scission of PP main chain. Spectrum a in Fig. 19 is the superposition of the two spectra from the paired radicals. The milled PP sample, which was brought into contact with oxygen, changed ESR spectrum from the pattern a to b in Fig. 19 very quickly after admission of air into the sample tube. The asymmetric peak is undoubtly the qiectrum of peroxy radical. But half of the observed spectrum at the high field side appears as a half of the octet, which comes from one of the mechano-radical,... 

In RNA, the original set consisting of the Watson-Crick base pairs is complemented by the G-U wobble pair, which is admissible in RNA double helices. Other admissible bps include U-U in internal loops as well as A-A, G-A or G-G (purine-purine closing pairs) at the ends of double helical regions or in multiloops. The secondary structures, which can be drawn in two dimensions without knots or pseudoknots, are indispensable for the... 

Lemma 3.33. Let S be the category of ringed sites and admissible ringed continuous functors. Then (L( ) ° , . R( )Mod) monoidal adjoint pair of... 

If (y, u) is admissible, then the related variational pair Sy, Ju) is called admissible as well. This pair must satisfy some constraints, which are determined as follows. For (y, u) the constraints in terms of sufficiently small variations 5y and Ju near (y, u) become... 

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