Positively invariant set

It is immediate from the form of (4.3) that Q is a positively invariant set. We will refer to (4.3) as the reduced system. ... 

The invariant measure corresponding to Aj = 1 has already been shown in Fig. 6. Next, we discuss the information provided by the eigenmeasure U2 corresponding to A2. The box coverings in the two parts of Fig. 7 approximate two sets Bi and B2, where the discrete density of 1 2 is positive resp. negative. We observe, that for 7 > 4.5 in (15) the energy E = 4.5 of the system would not be sufficient to move from Bi to B2 or vice versa. That is, in this case Bi and B2 would be invariant sets. Thus, we are exactly in the situation illustrated in our Gedankenexperiment in Section 3.1. 

The third eigenmeasure 1 3 corresponding to A3 provides information about three additional almost invariant sets on the left hand side in Fig. 8 we have the set corresponding to the oscillation C D, whereas on the fight hand side the two almost invariant sets around the equilibria A and B are identified. Again the boxes shown in the two parts of Fig. 8 approximate two sets where the diserete density of 1/3 is positive resp. negative. In this case we can use Proposition 2 and the fact that A and B are symmetrically lelated to conclude that for all these almost invariant sets 5 > A3 = 0.9891. 

In most cases of 0 < d < 4, there are a set of bounded trajectories surrounding the stable fixed point and forming the main quasi-periodic islands. These regular trajectories are bounded by the largest invariant island. Outside the largest island there also exist smaller quasi-periodic islands, forming an invariant set of positive Lebesgue measure in the two-dimensional phase space. Besides, there exists a Cantor-like invariant set of unstable trajectories... 

The rest point Eq always exists, and 2 exists with xj = 1 — A2 and p the root of (3.5) if 0 < A2 < 1, which is contained in our basic assumption (3.6). The existence of 1 is a bit more delicate. In keeping with the definitions in (3.3), define Aq = fli/(mi/(l)-l). Then 0< Aq < 1 corresponds to the survivability of the first population in a chemostat under maximal levels of the inhibitor. Easy computations show that 1 = (1 — Aq, 0,1) will exist if Aq > 0 and will have positive coordinates and be asymptotically stable in the X -p plane if 0 < Aq < 1. If 1 — Aq is negative then [ is neither meaningful nor accessible from the given initial conditions, since thex2-p plane is an invariant set. The stability of either 1 or 2 will depend on comparisons between the subscripted As. The local stability of each rest point depends on the eigenvalues of the linearization around those points. The Jacobian matrix for the linearization of (3.2) at i = 1,2, takes the form... 

Conversely, since trajectories are bounded, the positive quadrant is invariant, and there are no limit cycles in the invariant set D T, so there must be a rest point of the form (i/i, U2) e if the origin is not an attractor. 

Lemma 4.3. (Martinyuk and Tsyganovsky, 1979a). Suppose that the above-mentioned conditions imposed on the right-hand side of (3.1) are satined. Then one can always find positive pj (0 Pi Po) such that for all p Pi the sequence (3.10) defines the sequence of invariant sets, each satisfying the inequality... 

In two-dimensional mappings, a global invariant set with a positive area is preserved by the escape dynamics. However, in four-dimensional mappings Arnold diffusion precludes the existence of complete barriers formed by invariant tori. Accordingly, no invariant set of positive Legesgue measure is expected to exist. Nevertheless, numerical integration shows that a quasi-invariant set persists for a very long time. This quasi-invariant set shows a property similar to the invariant... 

Definition 2 (Positive invariance). (Blanchini, 1999) A set P cz R" is said to be positively invariant... 

A. Reactor Checkout To assure safe operation of the reactor, a prescribed sequence of operations must be followed to start the reactor. This sequence, which is invariantly set by electrical interlocks in the control console, is briefly as follows (1) neutron source in, (2) the three safety rods completely withdrawn, (3) water-dump valve closed, (4) water pumped to operating level, and (5) the control rods moved to the desired positions. 

The information in Figure 14 was produced in the following way. The slope (or the Kurbatov coefficients shown in Table V) and position of the fractional absorption gdges in Figure 3 were used as the criteria of model fit. Kcd an< CdOH were used as the fitting parameters and all other parameters were held constant. Consequently, the intrinsic constants shown in Figure 14a represent best fit parameters and, given that all other surface and solution association constants are invariant, constitute a unique solution set for each adsorption density. 

Figure 3.14. Projections of unit cells are shown which correspond to cubic invariant complexes in their standard setting. The numhers indicate, in eighths of the unit cell edge a, the positions of the points along the third axis (perpendicular to the drawing). Therefore four represents a point at a height of 4/8 ( = Aa) and 26 represents two superimposed points at heights respectively of 2/8 and 6/8 of the edge. A few examples of representations with enlarged cells are shown (P2, P2, h)-Notice that with reference to these cells the shifting vector between P2 and P2 is A, A, A.

In conclusion, notice also that in terms of combinations of invariant lattice complexes, the positions of the atoms in the level X can be represented by 2A, A, A G, and those in the level % by A, A, M G (where G is the symbol of the graphitic net complex, here presented in non-standard settings by means of shifting vectors). 

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