Annulus principle

It follows from the annulus principle (Theorems 4.2 and 4.5) that there exists a smooth stable invariant curve C of the form... 

It follows from the proof of the annulus principle that the curve C is a limit of a sequence of the curves obtained as iterations of an arbitrary initial curve R = const., for example, R = 1. Observe from (11.6.9) that for any small (5 > 0, all iterations of this curve lie inside the annulus i2 — 1 < provided that /X is small enough. Indeed, if /x is sufficiently small, then for i = 1 -h we have... 

It follows from the annulus principle (see the proof of Theorem 11.4) that the function depends smoothly on lj for p > 0. Moreover, as we have done with the derivatives of Tp with respect to (p we can verify that the derivatives of with respect to u tend to zero as 0. This implies that is a smooth function of a for all p>0. 

Afraimovich, V. S. and Shilnikov, L. P. [1977] The annulus principle and problems on interaction of two self-oscillation systems, Prikladnaja Matematika i Mehanika 41, 618-627. 

The fundamental principle of this method is equating the pit volume change with the corresponding change in annulus pressure. We write... 

A typical spouted bed has a substantial depth, which in the case of a cylindrical vessel is usually at least of the order of two column diameters, measured from the inlet orifice to the surface of the annulus. If the bed is much shallower, the system becomes hydrodynamically different from true spouting. The situation in this respect is similar to that for gas-fluidized beds, where the generally formulated principles of fluidization are not applicable to very shallow beds. 

Throughout the life of the reactor both the main and standby PCTL pumps have seized frequently due to sodium rising too high in the pump vessel. The pump vessel sodium levels are controlled by venting gas from the space above the sodium or injecting gas into it. In principle the sodium level was raised or lowered in 10 mm steps by operation of the gas valves in an automatic sequence. In practice, however, it was often necessary to intervene manually because the gas valves often passed or were blocked. In these circumstances operator error could easily lead to the sodium level into the annulus round the pump shafts where it would solidify causing the pump to seize. 

Rayleigh (i6) succeeded in the technically difficult task of mounting a mica plate for diffusion experiments at temperatures up to 415° C. The principle of his method was to keep an outer annulus of the plate cold and thermally insulated as far... 

When describing these various O-alkylation products the terms cone, partial cone, 1,2- and 1,3-altemate should be used only for the tetraethers, where the mutual arrangement of the O-alkyl groups unambiguously defines one of these basic conformations. Since the OH group can pass through the annulus a partially O-alkylated calixarene can still assume different conformations (at least in principle). This has been demonstrated by the formation of different conformational isomers in subsequent O-alkylation steps. For instance for 12 cone and partial... 

The second governing principle relates to the simultaneous reciprocation of atrioventricular volumes during the cardiac cycle such that the four-chambered heart (i.e., the contents of the pericardial sack) operates as a near-perfect constant-volume pump [4]. That means that while the two ventricles eject and the closed atrioventricular valves move toward the apex, the two atria fill. This is accomplished by keeping the epicardial apical portion of the heart fixed in space within the pericardial sack, while the backs of the two atria are similarly anchored in the mediastinum by the pulmonary veins and cavae. The nearperfect constant-volume physiology by which the epicardial surface has minimal radial displacement in the normal heart provides the mechanism that relates diastolic transmitral flow to simultaneous, mandatory ascent of the mitral annulus. 

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