Span criteria

Uniform B-splines are much less sensitive to poor spacing of the control points, because the first derivative is given by a lower degree B-spline with control points the first differences of the control points. If the control points have first differences all more or less in the same direction, the curve cannot kink back on itself. 

The first derivatives can, of course, vary fairly wildly in magnitude if the first differences do, and there is a perfectly good theory of knot-insertion into non-uniform B-splines and this can indeed be expressed in terms of choosing the coefficients in a subdivision implementation. 

However, the step-independence criterion is still a relevant one, and, if we insist on it, the knot intervals have to be determined from local data at every 

There is a lot of subtlety here. Although it is still possible to define a basis function as being the effect of moving one point by an infinitesimal amount, this applies only to one specific original polygon. As soon as you start editing it, the basis functions change. 

This means that proofs really have to be hand-crafted from very fundamental principles. You cannot just wave your arms and say linearity because these schemes are no longer linear, and all the short-cuts which linearity brought can no longer be relied upon. 

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Another good property of D-optimal designs is the possibility of selecting a number of molecules higher than the minimum, depending upon the resources that one is prepared to invest in synthesis, still keeping the maximum span criterion. However, it was recognized recently [28] that it is always very appropriate to add a central point (or a few points near the center of the domain) in order to explore better the systematic variations within the variable space. 

The challenge in the synthesis of chemical libraries is the vast number of different, potentially drug-like small molecules which is estimated to be as high as 1060. As all of these molecules can never be synthesized and tested, it is essential to define criteria for the composition of libraries spanning the biologically relevant areas of the chemical space most efficiently. An important criterion of a compound library is its chemical diversity, a term describing the similarity or dissimilarity of all library components. Thus, chemical diversity expresses how well a library represents all theoretical possibilities within the chemical property space. A library with low... 

Sciubba (2003) proposed an approach called the extended exergy accounting (EEA), which calculates the real, resource-based value of a commodity product. The time span of EEA is the whole life of a plant. The EEA includes the exergetics flow sheets for non-energetic costs of labor and environmental remediation expenditures, and hence uses extended exergetic content. It also defines the criterion for an optimum process or operation (Sciubba, 2003). 

For initiated oxidation, the inhibitory criterion could be defined as the ratio v0/v or (v0/ v — v/v0), where v0 and v are the rates of initiated oxidation in the absence and presence of the fixed concentration of an inhibitor, respectively. Another criterion could be defined as the ratio of the inhibition coefficient of the combined action of a few antioxidants / to the sum of the inhibition coefficients of individual antioxidants when the conditions of oxidation are fixed (fx = IfiXi where f, and x, are the inhibition coefficient and molar fraction of z th antioxidant terminating the chain). It should, however, be noted that synergism during initiated oxidation seldom takes place and is typical of autoxidation, where the main source of radicals is formed hydroperoxide. It is virtually impossible to measure the initial rate in the presence of inhibitors in such experiments. Hence, inhibitory effects of individual inhibitors and their mixtures are usually evaluated from the duration of retardation (induction period), which equals the span of time elapsed from the onset of experiment to the moment of consumption of a certain amount of oxygen or attainment of a certain, well-measurable rate of oxidation. Then three aforementioned cases of autoxidation response to inhibitors can be described by the following inequalities (r is the induction period of a mixture of antioxidants). 

Criterion 4. Eliminated peaks More than 80% of the NOESY peaks should have been assigned by Candid, and less than 20% of the NOESY cross peaks with exclusively long-range assignments (spanning 5 or more residues) should have been eliminated by the peak filters of Candid. 

We will take a somewhat different but equivalent criterion in order to describe the crossover. As the crossover time r, we take the Rouse relaxation time of a polymer section, spanning the tube diameter ... 

Researchers at Unilever [128] discovered that the surfactant used to form the HIPEs must be of low HLB value (between 2 and 6), as would be expected for w/o emulsions. The optimum surfactant was sorbitan monooleate (Span 80), which has an HLB value of 4.3. However, the HLB number of the surfactant is not the only criterion for the preparation of stable HIPEs the chemical nature was also found to be of importance [105]. 

The properties of a material must dictate the applications in which it will best perform its intended use. All materials made to date with polymerized sulphur show time-dependent stress-strain behaviour. The reversion to the brittle behaviour of orthorhombic sulphur is inevitable as the sulphur transforms from the metastable polymeric forms to the thermodynamically stable crystalline structure. The time-span involved of at most 15 months (to date) would indicate that no such materials should be used in applications dependent on the strain softening behaviour. Design should not be based on the stress-strain relationships observed at an age of a few days. Since the strength of these materials is maintained, however, uses based on strength as the only mechanical criterion would be reasonable. 

These studies were followed by investigation of antitumorous activity of car-bofunctional derivatives of silatrane and homosilatrane79,80. The results obtained are listed in Table 19. The criterion of efficacy was life span of the test animals as compared to the untreated ones. The observations lasted 60 days. 

A criterion used to evaluate the subjectivity of choices characteristic of uncertainty. It focuses on spanning the alternative choices. 

For the nonionic surfactants, the maximum stabilization under either acidic (pH2) or basis (pH 10) conditions was by a factor of 3 albeit with different surfactants. At pH2, Brij 35 was the most effective, and was in the middle of the pack at pHlO. Thus, from the point of view of using only one surfactant at all values of the pH, Brij 35 is the best choice. As will be seen below, this remains true for the other (polish rate, surface roughness) criterion as well. The other nonionic surfactants employed were selected to span the entire HLB range in order to serve as a basis for comparison (Brij 35 has one of the highest HLB). Only two ionic surfactants were studied and both provided a maximum stabilization by a factor of 5 at appropriate pH anionic SDS at pH 2 and cationic CTAB at pHlO. Note that the isoelectric point of the alumina is about 9. These ionic surfactants were also examined in this study. 

In general the metal cation to substituent distances are found spanning a range of values. A woiking criterion for coordination to the metal cations is that the M—A distance not be greater than the sum of the van der Waals radii of M and A as listed by Pauling. This criterion is particularly convenient when the anion is a typical heteroatom, such as O or N, or a halide, X. In such cases it is usually possible to derive accurate estimates of these distances from compiled sources. However, the values of the M—A distance for cases where A is carbon and M is a Group la or Ha metal are not particularly well defined. Hence, Table 1 represents a recent search of the CSD for these values. ... 

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