Stability with respect

The standard NF T 65-011 distinguishes the bitumen emulsions by their ionic nature (anionic or cationic), their stability with respect to agglomerates and weight content of base binder. There are 20 grades of emulsions. 

A novel optimization approach based on the Newton-Kantorovich iterative scheme applied to the Riccati equation describing the reflection from the inhomogeneous half-space was proposed recently [7]. The method works well with complicated highly contrasted dielectric profiles and retains stability with respect to the noise in the input data. However, this algorithm like others needs the measurement data to be given in a broad frequency band. In this work, the method is improved to be valid for the input data obtained in an essentially restricted frequency band, i.e. when both low and high frequency data are not available. This... 

The described approach is suitable for the reconstruction of complicated dielectric profiles of high contrast and demonstrates good stability with respect to the noise in the input data. However, the convergence and the stability of the solution deteriorate if the low-frequency information is lacking. Thus, the method needs to be modified before using in praetiee with real microwave and millimeter wave sourees and antennas, whieh are usually essentially band-limited elements. 

We describe a simple computational example to demonstrate two key features of the new protocol Stability with respect to a large time step and filtering of high frequency modes. In the present manuscript we do not discuss examples of rate calculations. These calculations will be described in future publications. 

Sodium iodide [7681-82-5] Nal, occurs as colorless crystals or as a white crystalline solid. It has a salty and slightly bitter taste. In moist air, it gradually absorbs as much as 5% water, which causes caking or even Hquefaction (dehquescence). The soHd slowly becomes brown when exposed to air because some iodide is oxidized to iodine. Water solutions are neutral or slightly alkaline and gradually become brown for the same reason. Aqueous solutions are stabilized with respect to oxidation by raisiag the pH to 8—9.5 (see Iodine and iodine compounds). 

The deprotonation of 132 is favored at Ni and the coordination of 135 occurs preferentially at 82- A second entity of 135 coordinates at N3. A computational study of thiouracil derivatives of the tungsten(O) hexacarbonyl shows that the sulfur-bound thiouracil is serving as a ir-donor during the CO dissociation (Scheme 91) [99IC4715]. DFT calculations show that 137 is significantly stabilized with respect to the alternative reaction product 138. 

Trioxane 210 has been used as a model system by Gu and coworkers to study the antimalarial drug artemisinin 211 (Scheme 137) [97CPL234, 99JST103]. It is the boat/twist form rather than the chair conformer of 210 that describes the subunit in 211. Moreover, geometric parameters and vibrational frequencies can only reliably be computed at the DFT level and by post-Hartree-Fock methods. B3-LYP/6-31G calculations on the conformers of 3,3,6,6-tetramethyl-1,2,4,5-tetroxane show that the chair conformer is stabilized with respect to the twisted conformer by about -2.8 kcal/mol [00JST85]. No corresponding boat conformer was found. 

Secondly, the Llo structure must be stable with respect to both small and large strains. The former is assured by the positive definiteness of elastic moduli. Clearly, an exhaustive test of the stability with respect to large strains cannot be made and the following two types of tests were performed. First, the energy of the structure was calculated as a function of strains corresponding to 20% changes in the lattice parameters a and... 

K State Cycle Length Number of State Cycle Attractors Stability with respect to minimal perturbations... 

Table 2.3 Stability with respect to pH of some metal-EDTA complexes...

There is a substantial literature on the thermal and photochemical degradation of PS and it is well established that polymer properties are sensitive to the manner in which a particular sample of PS is prepared. For example, it has been reported that PS prepared by anionic polymerization shows enhanced stability with respect to that prepared by a radical mechanism.2 10 This has often been attributed to the presence of "weak links" in the latter polymers. However, the precise nature of the "weak links" remains the subject of some controversy. T he situation is further confused by all PS prepared by radical mechanisms often being considered as a class without reference to the particular polymerization conditions employed in their preparation. In many cases the polymers are "commercial samples" with details of the method of preparation incomplete or unstated. 

The preceding examples provide enough reason to conclude that the concept of stability with respect to the input data is identical with the concept of continuous dependence of the solution of a difference problem upon... 

Here we give the general formulation of the concept of stability with respect to coefficients for the difference scheme in Section 1 by having recourse to an operator equation of the first kind... 

It is natural to require a solution of problem (11) to depend continuously not only on perturbations of the right-hand side, but also on perturbations of the operator A (for instance, on the coefficients of a difference operator). As in the case of difference schemes arising in Section 1, this property of operator equations is to be understood as stability with respect to coefficients or co-stability of an operator equation. 

Stability with respect to the initial data. Let us investigate the stability of scheme (II) with homogeneous boundary conditions by the method of separation of variables. For this, we proceed as usual. This amounts to applying to scheme (II) with homogeneous boundary conditions the identities... 

Stability with respect to the right-hand side. 

The energy inequality method. An investigation of difference schemes for the string vibration equation may be carried out by means of the energy inequality method (see Section 1). Here we restrict ourselves to stability with respect to the initial data with regard to the problem... 

In this chapter we study the stability with respect to the initial data and the right-hand side of two-layer and three-layer difference schemes that are treated as operator-difference schemes with operators in Hilbert space. Necessary and sufficient stability conditions are discovered and then the corresponding a priori estimates are obtained through such an analysis by means of the energy inequality method. A regularization method for the further development of various difference schemes of a desired quality (in accuracy and economy) in the class of stability schemes is well-established. Numerous concrete schemes for equations of parabolic and hyperbolic types are available as possible applications, bring out the indisputable merit of these methods and unveil their potential. 

In conformity with the superposition principle ( is a linear operator), the stability of the Cauchy problem with respect to the right-hand side follows from the uniform stability with respect to the initial data... 

Inequality (12) expresses the property of continuous dependence which is uniform in h and t of the Cauchy problem (4) upon the input data. Here and below the meaning of this property is stability. A difference scheme is said to be absolutely stable if it is stable for any r and h (not only for all sufficiently small ones). It is fairly common to distinguish the notion of stability with respect to the initial data and that with respect to the right-hand side. Scheme (4) is said to be stable with respect to the initial data if a solution to the homogeneous equation... 

Quite often, the reader can encounter the statement that stability with respect to the initial data implies stability with respect to the right-hand side . How is one to understand the nature of this assertion ... 

It is worth noting here that condition (21) is sufficient for uniform stability with respect to the initial data. 

On the other hand, (28) yields 2/ +i < i / (i)- Comparison of these inequalities gives (27), from which the stability with respect to the initial data immediately follows. 

Stability with respect to the initial data in Ha- Stability of scheme (1) with respect to the initial data is investigated with further estimation of a solution of problem (la). 

Stability with respect to the initial data in Hb- Let us write down the second energy identity for scheme (la) assuming B also to be a self-adjoint operator 5 = 5 > 0. At the first stage we take the inner product of (la) and 2ry ... 

Stability with respect to the right-hand side. We now consider problem (lb) under conditions (9), (10) and (27), which are put together for later use. Since A and R are constant operators, identity (18) for (lb) is of the form... 

Stability with respect to the initial data. Since the conditions a > A/4 and S > 0 of Theorem 1 are fulfilled, we have for a solution of problem (65) with (p = 0 the estimate... 

Substituting (72) into (63) and taking into account (78), we conclude that (72) is just the solution of problem (63). As we stated in (71), on the strength of stability with respect to the initial data we have for ... 

Observe that the stability with respect to the right-hand side and boundary conditions implies that the moments t and t can be arbitrarily taken from the interval... 

Methods for the convergence rates of additive schemes. So far we have established many times that approximation and stability of a difference scheme provide its convergence. For additive schemes we shall need stability with respect to the right-hand side so that it follows from the condition of summarized approximation... 

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