Stable index

A more general definition is given by Feller [12]. We denote a Levy stable pdf dF(x) as La(x) and call a the Levy (stable) index. It can be shown that a stable law has a characteristic function of the form... 

The Langevin equation [Eq. (Ill)] was integrated numerically following the procedure developed in Ref. 90. Whence, we obtained the trajectories of the particle shown in Fig. 17. In the Brownian limit, we reproduce qualitatively the behavior found in Ref. 89. Accordingly, the fluctuations around the positions of the minima are localized in the sense that their width is clearly smaller than the distance between the minima and barrier. In contrast, for progressively smaller stable index a, characteristic spikes become visible, and the individual sojourn times in one of the potential wells decrease. In particular, we note that single spikes can be of the order of or larger than the distance between the two potential minima. 

Figure 17. Typical trajectories for different stable indexes a obtained from numerical integration of the Langevin equation [Eq. (111)]. The dashed lines represent the potential minima at 1. In the Brownian case a = 2, previously reported behavior is recovered [89]. In the Levy stable case, occasional long jumps of the order of or larger than the separation of the minima can be observed. Note the different scales.

Figure 20. Scaling exponent u as function of stable index a. The constant behavior t(a) 1 over the range 1 < a S 1.6 is followed by an increase above 1.6, and it eventually shows an apparent divergence close to a = 2, where Eq. (113) holds. Corresponding to the right ordinate, we also plot the decadic logarithm of the amplitude C(a).

The Citizen s Helper s Future. The Citizen s Helper has evolved from a stack that demonstrated the feasibility of various search and retrieval functions, to one in which many stable indexed lists of... 

Recall that, in general, the stable random variable Z involves four parameters the exponent (stable index) 0 < a < 2, the skewness —1 < < 1, the shift a e K,... 

The basic condition of the Standard application - the availability of stable coupled probabilistic or the multiple probabilistic relations between then controlled quality indexes and magnetic characteristics of steel. All the probabilistic estimates, used in the Standard, are applied at confidence level not less than 0,95. General requirements to the means of control and procedure of its performance are also stipulated. Engineers of standard development endeavoured take into consideration the existed practice of technical control performance and test at the enterprises that is why the preparation of object control for the performance of nondestructive test can be done during the process of ordinary acceptance test. It is suggested that every enterprise is operated in correspondence with direct and non-destructive tests, obtained exactly at it, for detailed process chart and definite product type, however the tests have long since been performed after development of the Standard displayed that process gives way to unification. 

It is important to know whether a polymer will be stable, that is, whether it will not decompose at a given temperature. There are several measures of thermal stability, the most important of which (from an economic standpoint) is the Underwriters Laboratories (UL) temperature index. 

Alkylated aromatics have excellent low temperature fluidity and low pour points. The viscosity indexes are lower than most mineral oils. These materials are less volatile than comparably viscous mineral oils, and more stable to high temperatures, hydrolysis, and nuclear radiation. Oxidation stabihty depends strongly on the stmcture of the alkyl groups (10). However it is difficult to incorporate inhibitors and the lubrication properties of specific stmctures maybe poor. The alkylated aromatics also are compatible with mineral oils and systems designed for mineral oils (see Benzene Toulene Xylenes and ethylbenzene). ... 

To achieve the maximum coating opacity the opacifter particle size should be between 0.2 and 0.3 ]lni. A good opacifter should not be soluble in the vitreous system, should have a refractive index substantially different from the refractive index of the system, should be inexpensive, easily milled to a submicrometer particle size, and thermally stable at the film s firing temperature. 

The pyrrolines or dihydropyrroles can exist in three isomeric forms 1-pyrroline (3,4-dihyro-2JT-pyrrole [5724-81-2]) (16) is an unstable material that resiniftes upon exposure to air 2-pyrroline (2,3-dihydro-lJT-pyrrole [638-31-3]) (17) is even more unstable only 3-pyrroline (2,5-dihydro-lJT-pyrrole [109-96-6]) (18) is reasonably stable. 3-Pyrroline bods at 91°C and has a density of 0.9097 g/cm and a refractive index of 1.4664. 

Cyclohexanecarboxaldehyde is stable at room temperature under nitrogen the submitters noted no appreciable variation in the refractive index after 30 days. 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

The capability index for a stable process defined as the quotient of tolerance width and process capability where process capability is the 6a range of a process s inherent variation. 

The capability index which accounts for process centering for a stable process using the minimum upper or lower capability index. 

Although the Langelier index is probably the most frequently quoted measure of a water s corrosivity, it is at best a not very reliable guide. All that the index can do, and all that its author claimed for it is to provide an indication of a water s thermodynamic tendency to precipitate calcium carbonate. It cannot indicate if sufficient material will be deposited to completely cover all exposed metal surfaces consequently a very soft water can have a strongly positive index but still be corrosive. Similarly the index cannot take into account if the precipitate will be in the appropriate physical form, i.e. a semi-amorphous egg-shell like deposit that spreads uniformly over all the exposed surfaces rather than forming isolated crystals at a limited number of nucleation sites. The egg-shell type of deposit has been shown to be associated with the presence of organic material which affects the growth mechanism of the calcium carbonate crystals . Where a substantial and stable deposit is produced on a metal surface, this is an effective anticorrosion barrier and forms the basis of a chemical treatment to protect water pipes . However, the conditions required for such a process are not likely to arise with any natural waters. 

The effect of pH on the corrosion of zinc has already been mentioned (p. 4.170). In the range of pH values from 5 -5 to 12, zinc is quite stable, and since most natural waters come within this range little difficulty is encountered in respect of pH. The pH does, however, affect the scale-forming properties of hard water (see Section 2.3 for a discussion of the Langelier index). If the pH is below the value at which the water is in equilibrium with calcium carbonate, the calcium carbonate will tend to dissolve rather than form a scale. The same effect is produced in the presence of considerable amounts of carbon dioxide, which also favours the dissolution of calcium carbonate. In addition, it is important to note that small amounts of metallic impurities (particularly copper) in the water can cause quite severe corrosion, and as little as 0-05 p.p.m. of copper in a domestic water system can be a source of considerable trouble with galvanised tanks and pipes. 

Phase composition changes as a function of temperature are similar to these that were observed for the lithium-containing system (see Fig. 21), except that individual stable ammonium-cobalt-oxyfluoroniobate occurs prior to the formation of CoNbOF5 (Fig.21, curve 4). It is assumed that the composition of this intermediate phase, formed at 330-350°C, is (NH CoNbOFy [129]. Complete removal of ammonium occurs at about 400°C (Fig. 21, curve 5) and leads to the formation of CoNbOFs. The compound is defined as practically isotropic rose-colored ciystals with a refractive index of N = 1.500. The... 

The Michaelis constant (fCM) is an index of the stability of an enzyme-substrate complex. Does a high Michaelis constant indicate a stable or an unstable enzyme-substrate complex Explain your reasoning. 

Sufficient stability of the hydrocarbon ions, as the salt or in the solution, is an obvious prerequisite for these procedures, and, in practice, selecting or designing the stable ions and choosing a proper solvent are tasks of primary importance. As an ordinary stability index for the ions, thermodynamic scales referred to the water molecule, i.e. p CR+ and pKa values, are chosen for the carbocation and carbanion, respectively. 

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