Omega Point

In the late twentieth century such thoughts about computers ensuring our survival were incorporated by some scientific philosophers into visions of the very distant future. This idea of healthcare ultimately placed in the context of computers that will sequester the universe and ourselves with it, will be resurrected in Chapter 10, along with mention of such thinkers as the Jesuit teacher Paul Teilhard de Chardin s thesis on the Omega Point, and other thinkers specializing in eschatology, the destiny of the universe, and see a role for humankind in that. 

Teilhard s viewpoint allows him to depict an imaginary evolution of the noosphere. The psychic, interior side of matter or so-called radial energy" directs matter to higher levels of organisation which culminate in the end of the evolutionary process. This end is external to the evolution itself The Earth s noosphere will be replaced by a supermind and will coalesce into a so-called Omega-Point. As Teilhard put it (1961, pp. 273, 287-288) ... 

Teilhard saw the noosphere as a transitional stage of evolution from the biosphere to the Omega-Point. He describes the noosphere as a layer over the biosphere, because to him it is the begiiming of a separation process. The radial energy enters a stage of visible dominance and partial separation on the way to total independence. 

The noosphere is said to be a cogitative layer of the Earth. It is a transitional state between non-reflective life and the Omega point. The last stage of the noosphere evolution is tied up with the death of Earth. 

Noogenesis is aimed at a psychic centre (the Omega-Point), which transcends space and time. 

This value of co can be called the saturation value or ( ). since it applies only with flashing liquids (i.e., in the flashing region for pressures less than the bubble point, as seen in Fig. 23-32). A generalization defines omega to apply also with noncondensable gases by using Oo= x%v%/v0... 

For a system of constant diameter (giving a single potential choke point at the end of the pipe), and if the gas is ideal, then the Design Charts for adiabatic flow of gases, given in Perry111 or the Omega method with = 1 (see Annex 8) can be used to determine the flow capacity. is a parameter within these charts. 

Omega is a correlating parameter in an "equation of state" (EOS) which links the specific volume of a two-phase mixture flowing in a relief system with the pressure at any point. Such an EOS is required to evaluate the HEM without performing repeated flash calculations. The EOS used by the Omega method is ... 

Before going on to list these families, one further point needs to be made. This arises because there are several instances where a unique zeolite framework has been first characterized from a laboratory preparation and later identified as a natural species. The first of these was mazzite, prepared first as synthetic zeolite ZSM-4 (or omega) but known in naMe prior to the allocation of the lUPAC code hence its inclusion in Table 7. The other species have been discovered in naMe after an lUPAC code had been allocated to their synthetic analogs. The details for these materials are given in Table 8. 

If the omega limit set is particularly simple - a rest point or a periodic orbit - this gives information about the asymptotic behavior of the trajectory. An invariant set which is the omega limit set of a neighborhood of itself is called a (local) attractor. If (3.1) is two-dimensional then the following theorem is very useful, since it severely restricts the structure of possible attractors. 

If an omega limit set contains an asymptotically stable rest point P, then that point is the entire omega limit set. If all of the eigenvalues of the variational matrix have positive real part then the rest point is said to be a repeller such a rest point cannot be in the omega limit set of any trajectory other than itself. If k eigenvalues have positive real part and n-k eigenvalues have negative real part then there exist two sets M P), called the stable manifold and defined by... 

Theorem (Butler-McGehee). Suppose that P is a hyperbolic rest point of (3.1) which is in w(x), the omega limit set of but is not the en-... 

Since 1 is a local attractor, to prove the theorem it remains only to show that it is a global attractor. This is taken care of by the Poincare-Bendixson theorem. As noted previously, stability conditions preclude a trajectory with positive initial conditions from having 0 or 2 in its omega limit set. The system is dissipative and the omega limit set is not empty. Thus, by the Poincare-Bendixson theorem, the omega limit set of any such trajectory must be an interior periodic orbit or a rest point. However, if there were a periodic orbit then it would have to have a rest point in its interior, and there are no such rest points. Hence every orbit with positive initial conditions must tend to j. (Actually, two-dimensional competitive systems cannot have periodic orbits.) Figure 5.1 shows the X1-X2 plane. 

Proof. Note that M (Eq), the stable manifold of Eq, is either the p axis if El exists or the x -p plane if Ei does not exist. The manifold M E2) is the X2 p plane less the p axis if E exists, M (Ei) is the Xi p plane less the p axis. Since (Xi(0), X2(0), p(0)) does not belong to any of these stable manifolds, its omega limit set (denoted by w) cannot be any of the three rest points. Moreover, w cannot contain any of these rest points by the Butler-McGehee theorem (see Chapter 1). (By arguments that we have used several times before, if w did then it would have to contain Eq or an unbounded orbit.) If w contains a point of the boundary of then, by the invariance of w, it must contain one of the rest points Eq,Ei,E2 or an unbounded trajectory. Since none of these alternatives are possible, CO must lie in the interior of the positive cone. This completes the proof. 

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