Pointwise dimension

The notion of pointwise dimension allows us to quantify the local variations in scaling. Given a multifractal A, let be the subset of A consisting of all points with pointwise dimension a. If a is a typical scaling factor on A, then it will be represented often, so will be a relatively large set if a is unusual, then will be a small set. To be more quantitative, wenotethateach is itself a fractal, so it makes sense to measure its size by its fractal dimension. Thus, let /(a) denote the dimension of. Then /(a) is called the multifractal spectrum of A or the spectrum of scaling indices (Halsey et al. 1986). 

The parameter or variable V" will have the dimensions of permeability times pressure, as previously observed in terms of moles. However, in terms of the total gas volume permeated, and using the pointwise permeability and dealing with volume fractions, then V" would have the dimensions of... 

Clearly, all solutions have a few worse-than-l%-level species according to the pointwise embedding criterion. Thus, as a whole, D = 2 cannot be regarded as a good embedding dimension. One should note that the sum of the total... 

The measurement consists of a direct photometric evaluation. The TLC plate is scanned lane by lane by monochromatic light of the appropriate wavelength. The measuring slit is adjusted to give a light beam whose dimensions suit the size of the spot. Spots obtained by pointwise application are scaimed over their whole diameter. With bandwise application, which is preferable for quantitative evaluation, an aliquot of 50-75 % is taken from the homogeneous central part of the band. 

Proof. Since Y has a dualizing complex, Y has finite Krull dimension [17, Corollary V.7.2]. Since X is of finite type over Y, X has finite Krull dimension. By [17, Proposition V.8.2], it suffices to show that / (I) is pointwise dualizing. So we may assume that X = SpecB and Y = Specj4 are affine, both A and B are local, and / is induced by a local homomorphism from A to B. Then the assertion follows from [4, (5.1)]. ... 

Another dimension that has interesting properties is called the pointwise or mass dimension [e.g. 14 (p. 736), 32 (p. 334)]. To define it, one considers a curve in R and samples it uniformly at a large number No of points, i.e. one determines the spatial coordinates of No points uniformly distributed along the curve. Then, one centers a ball Br(x) of radius r at some location x on the curve, and one counts the number N(r, x) of sampled points within this ball. The probability of finding a point in this ball is given by the limit... 

In mixtures, the pressure dimension in the permeability coefficient becomes the partial pressure pp. That is, the pointwise permeability coefficient in a mixture will have the units or dimensions of... 

The practice here is not to attempt to adapt symbols to distinguish one kind of permeability coefficient from pother (e.g., a pointwise coefficient Pj from an overall coefficient, say P-). Rather, as noted earlier, all coefficients are simply denoted P, to be further distinguished by context of usage, with the required units or dimensions and magnitude dictated by that usage. 

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