Perpetual stability

The conclusion is that at least in the approximation of the secular system the orbit of Jupiter and Saturn is confined forever between two KAM invariant tori, thus assuring the perpetual stability of this system. Below we resume in brief the method and the result. For more details the reader is referred to Locatelli and Giorgilli (2000). 

The procedure above allows us to prove only that there is an invariant torus close to the initial conditions of the Sun-Jupiter-Saturn system, not that the orbit of the system actually lies on a torus. Since we can not exclude the possibility of Arnold s diffusion, this is not enough to prove the perpetual stability of the orbit of the secular system. Therefore, we make a more accurate analysis in order to prove that the orbit is actually confined in a gap between two invariant tori. The procedure is illustrated in Figure 3. 

Perpetual stability is a quite strong property that can be proved only in some special cases. The cases of an integrable system or of a system with two degrees of freedom that satisfies the hypotheses of KAM theorem have already been discussed. We can add the case of an elliptic equilibrium, which turns out to be stable in case the equilibrium corresponds to a minimum (or a maximum) of the Hamiltonian this result goes back to Dirichlet. 

Cans, at the turn of the 19lh century, synthesised aluminosilicate materials capable of water softening. He called them Permutits and for many years they were employed as domestic and industrial water softeners, as well as in the treatment of nuclear waste. This, together with the first identification of base exchange in soils (Way [2] and Thompson [3 J), led to the misunderstandings perpetuated in elementary texts that zeolites are responsible for the ion-exchange in soils, and that they are widely used as water softeners. In truth the clay minerals contribute mainly to the ability of soils to take up cations, such as ammonium, and Permutits were amorphous materials with low cation exchange capacities and limited chemical stability. 

The second reason that vitamin C is used as an electron donor is that the reaction product is fairly stable and unreactive. When vitamin C gives up an electron, it becomes a free radical called the ascorbyl radical. By free-radical standards, the ascorbyl radical is not very reactive. Its structure is stabilized by electron delocalization — the resonance effect first described by Linus Pauling in the late 1920s. This means that vitamin C can block free-radical chain reactions by donating an electron, while the reaction product, the ascorbyl radical, does not perpetuate the chain reaction itself. 

Posttreatment. Three planned phases are associated with consolidation, and somewhere in the future there will most likely be a fourth, unplanned, phase decomposition of the consolidant or further decomposition of the substrate. Because conservation is concerned with extending the stability of an artifact, theoretically into perpetuity, some pathway to future preservation should become part of every treatment. 

Since markets do not consider the laws of stoichiometry, the electrolytic chlor-alkali industry is in a state of perpetual imbalance. Nearly always, either chlorine or caustic soda is in long supply, while there is a shortage of the other. This situation is reflected in spot market prices, which can fluctuate wildly over short periods, usually in opposite directions for the two products. The combined price of an ecu is less volatile, and ecu customers who take nearly equivalent amounts of chlorine and caustic soda are a stabilizing influence in the industry. KOH, as a relatively minor product, is somewhat immune to these problems. Total KOH production is free to respond to market pressures the amount of chlorine produced along with KOH is a small part of the total and not a large factor in the market. The KOH producer who is also a chlorine merchant, however, is still subject to the fluctuating market prices of chlorine. 

Even though there are a number of metals that appear suitable for resistance thermometry, platinum has come to occupy a predominant position, partly because of excellent characteristics, such as chemical inertness and ease of fabrication, and partly because of custom that is, certain desirable features such as ready availability in high purity and the existence of a large body of knowledge about its behavior have come into being as its use grew and have tended to perpetuate that use. Its sensitivity down to 20 K and its stability are excellent. Its principal disadvantages are low resistivity, insensitivity below about 10 K, and a variation of the form of the resistance-temperature relation from specimen to specimen below about 30 K. 

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