Uniform subdivision

The appropriate denominators are straightforward to see in the uniform case, where we can say that the divided differences come half-way between the original values, and so we have well-defined abscissae associated with the difference values, and it is clear what is meant by saying that the second divided differences are divided by the difference in abscissa of the first divided differences. In the non-uniform context the convention often used is that the second divided differences are the differences of the first divided differences divided by half of the total width of the set of values taking part in the divided difference. In the uniform subdivision context we do not need that complication. 

A.Levin and D.Levin Analysis of quasi-uniform subdivision. Applied... 

A Levin The importance of polynomial reproduction in piecewise-uniform subdivision. pp272-307 in Mathematics of Surfaces XI, (eds Martin, Bez and Sabin) Springer LNCS3604 2005, ISBN 3-540-28225-... 

S.Schaefer and R.Goldman Non-uniform subdivision for B-splines of arbitrary degree. CAGD 26(1), pp75-81, 2009... 

Uniformity of final moisture content Decomposition of product Overdrying State of subdivision Product temperature h. Bulk density... 

Where scored tablets are provided, there is often discussion of the suitability of the product for halftablet dosing. This includes data on content uniformity and on the suitability of the finished product for subdivision—e.g., hardness specifications, data on ability to subdivide. 

A single evaluation set is the simplest and most rapid validation scheme. A fraction—usually between 50% and 90% of the total number—of the available samples constitutes the training set, while the remaining objects form the evaluation set. The subdivision may be arbitrary, random, or performed by way of a uniform design, such as the Kennard and Stone and the duplex algorithm (Kermard and Stone, 1969 Snee, 1977), which allows two subsets to be obtained that are imiformly distributed and representative of the total sample variability. 

In proposing these divisions, one must acknowledge that hybrid properties are certainly possible. Thus, an emulsifying property of a protein might be used as a sense enhancing one— to prepare a smooth food product—or a manipulative one—to provide even subdivision for uniform administration. 

It was difficult to manufacture since the nitroguanidine had to be introduced into the powder mass in a state of fine subdivision, otherwise the powder was not uniform. In some factories, therefore, methods were worked out to obtain nitroguanidine in the form of fine dust. 

Because the objective of this treatise is to form a uniform nomenclature that spans all of chemistry, many of the starting premises that are the foundation of the disparate subdivisions of chemistry have been re-examined. These premises were mainly of a historical, rather than a rational, origin, and, in many instances have been shown to be inconsistent. In their place, choices have been made with a global, rather than a local, and with an analytic, rather than a synthetic, approach in mind. Beginning with the selection in Chapter 1 of a "primary path", the nomenclature being developed eliminates ... 

Once a sample larger than 3x is collected, a question remains as to what to do with that sample prior to assay. The withdrawn draft guidance ANDAs Blend Uniformity Analysis recommends sample sizes l-3x to be collected, but that the sample weight tested (assayed) should be equivalent to the dosage used. Unfortunately any subdivision of the sample is likely to induce either potency losses or segregation,. so the logistics of how to subdivide such a sample remains a challenge. 

The body of this book describes five major questions which we are now able to answer about an arbitrary uniform, stationary, symmetric subdivision scheme. In each case there are additional issues which are conveniently addressed in the same chapter. 

The subdivision matrix of a binary uniform scheme has each column a copy of the mask, shifted down two rows for each step to the right. Every row (stencil) sums to 1, but the mask entries sum to 2. 

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. 

Handling non-uniformity is a known research topic at the time of writing. In the case of B-splines, the end conditions can elegantly be treated as a specific case of non-uniformity. Can this be extended neatly to other subdivision... 

Nira Dyn, John Gregory and David Levin Analysis of Uniform Binary subdivision schemes for curve design. Constructive Approximation 7 ppl27-147 1991... 

D. Levin Using Laurent polynomial representation for the analysis of non-uniform binary subdivision schemes. Adv. Comput. Math 11, pp41-54, 1999... 

T. J.Cashman, N.A.Dodgson and M.A.Sabin Non-uniform B-spline subdivision using rehne and smooth. ppl21-137 in Mathematics of Surfaces XII, (eds Martin, Sabin and Winkler), Springer LNCS4647, 2007 ISBN 3-540-73842-8... 

Zonal projections of the inferior olive to the nodulus (lobule X) and the flocculus have been substantiated in different species. Purkinje cells of these lobules are uniformly Zebrin-positive in rat, mouse, opossum, squirrel monkey (see Section 6.1.3.) and rabbit (Jaarsma unpublished observations). A compartmental subdivision of the white matter of the nodulus and the flocculus was demonstrated with AChE-histochemistry in the rabbit and the monkey (Section 6.1.5.) and correlated with the olivocerebellar projection to these lobules (Voogd et al., 1987a,b Tan et al., 1995a,b). 

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