Marking points

Fig. 1 (right) shows upside an example of an input. The marked points are some of the scaled Dirac impulses. The belonging scaled impulse responses are shown downside. 

FIGURE 26.36 The side force coefficient of an OESBR black-fiUed tire tread compound on wet blunt Alumina 180 as function of log a v obtained at three speeds and five temperatures (black open squares) with a quadratic equation fitted to the data (black solid line). The red marked points were obtained at one speed for five temperatures with the dotted red line the best fitting quadratic equation, indicating the risk of extrapolation with a limited set of data. 

Mark point o on this line at 34 per cent acetone. 

Fig. 1. The potential energy curves with respect of the top angle in H3 isosceles triangle as illustration of 8 coordinate of the e E JT distortion. The marked points shows the CASSCF computed values, the continuous and dashed lines corresponding to the fit with Hamiltonian of equation (3).

FIGURE 3.7 A cryo-EM map of the Escherichia coli ribosome (complexed with fMet-tRNAf Met and mRNA) where fMet = formylmethionine obtained from 73,000 particles at a resolution of 11.5 A. (a-d) Four views of the map, with the ribosome 30S subunit painted in yellow, the ribosome 50S subunit in blue, helix 44 of 16S RNA in red, and fMet-tRNA at the P site in green. Inset on top juxtaposes the experimental tRNA mass (green, on left) with the appearance of the X-ray structure of tRNA at 11 A resolution (on right). Arrows mark points at which tRNA contacts the surrounding ribosome mass. Landmarks h = head and sp = spur of the 30S subunit. CP = central protuberance LI = LI stalk and St = L7/L12 stalk base of the 50S subunit. 

Consider Fig. 3.1 and assume that the marked points on the x-axis are equidistant with length h between them. For the moment, consider a simple first derivative around the region between x and x2 (to be made more precise in later sections). The function shown as a line is known only in the form of the fat points on it. Intuitively, one thinks of the approximation in that region as... 

Fig. 12.1 A radioactive decay curve of 36C1. P36 is the percentage of atmospheric 36C1 left in the sample. The time axis is expressed in half-lives and in 105-year units. The marked point of 56% P36, taken from the Lachlan case study, reveals water with an age of about 290,000 years. The mode of defining the P36 is explained in later sections.

If you choose to screen lines within a line graph (never the axes), assign the lines a 1.5 or 2 point weight, and do not use symbols to mark points on the lines. 

Figure 4.2 Neutral curve for the Blasius boundary layer identifying stable and unstable regions. The marked points are further investigated.

FIGURE 10.5 Several consecutive electron density sections, separated by a constant increment along the z axis, are stacked and displayed on a light box. The continuity of the protein, here the protease from streptomyces, is clear in many places. The dots are used in the early interpretation of the map to mark points, such as putative a carbon positions, along the chain. The first objective in interpreting an electron density map is generally to deduce the overall fold of the polypeptide. Only later are side chains identified and oriented. 

What the choice of a diagonal does is to imply a labelling, giving a correspondence between a sequence of points of the old polygon and a sequence of the refined one. In particular it implies a mark point which is an abscissa value which maps into itself under the map from old abscissa values to new ones. In the case of a primal binary scheme, the mark point is at a point of both new and old polygons. In the case of a dual scheme the mark point is at a mid-edge in both old and new. 

In the particular case shown as an example here, at the mark points constructed in this way other than at abscissae of original vertices, the fourth difference turns out to be zero, and so the discontinuities do not occur except at limit points corresponding to original control points. That ties up with our knowledge of B-splines, but it is a very special property. In general, subdivision schemes give limit curves with discontinuities of some derivative at all dyadic points. 

In general every power of the scheme considered introduces new mark points, and there is no guarantee in general that the 117th power will not show us places where the Holder continuity is lower than that found for lower powers. Thus this procedure can only ever give us upper bounds on the Holder continuity. 

By taking a high enough power of the scheme, any rational point can be determined as a mark point. The power needed is just the Euler function of the quotient when the denominator has all powers of 2 (in general of the arity) divided out. 

The first few powers and the denominators of the mark points that they can find are... 

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