Affine invariance

A further affine invariant is established by the projective vector... 

The additional condition needed to ensure (3.15) is the affine invariance property (or invariance with respect to the specific affine transformation z Rz). 

Let zo.Zi,... be a sequence of points obtained by application of the numerical method % to (3.16). If the same method, when applied to (3.17) produces the set of points Azo, Azi,..then we say that the numerical method is invariant with respect to the transformation z = Az. An affine invariant method is one which is invariant under any such transformation. 

If a differential equation is time-reversible with respect to the involution R, and we apply a method which is affine invariant and symmetric, then the method will be time-reversible (3.15). 

All Runge-Kutta methods and Partitioned Runge-Kutta methods are affine invariant, thus, if they are also symmetric, then they preserve time-reversal symmetry. 

Show that any Runge-Kutta method is affine invariant, thus the Trapezoidal Rule is time-reversible. 

DH79] Deuflhard P. and Heindl G. (1979) Affine invariant convergence theorems for Newton s method and extensions to related methods. SIAM J. Numer. Anal. 16 501-516. 

Mikolajczyk, K. and Schmid, C. (2004) Scale and affine invariant interest point detectors./ntemor/ona/yottma/... 

Lazebiuk, S., Schmid, C. and Ponce, J. (2003) Sparse texture representation using affine-invariant neighbourhoods. Proceedings of Computer Vision and Pattern Recognition (CVPR), 2 319-324. 

The first and second affine invariant moments are defined in Equation 7 and 8, respectively [12]. 

Chapter 5 describes a study of the effect of micelles on the Diels-Alder reaction of 1 with 2. Literature studies on micellar catalysis of Diels-Alder reactions invariably failed to reveal significant accelerations. These results are unexpected, since most Diels-Alder reactants have a high affinity for... 

Difficulties of incompatibility can arise with mixtures of basic dyes on acrylic fibres because of competition for the limited number of dyeing sites available and the differences between dyes in terms of affinity and rate of diffusion. The rate of uptake of each dye when applied in admixture with another is invariably slower than when the dye is applied alone at the same concentration. Competition effects of this kind can lead to serious practical problems unless the dyes are carefully designed and selected to have similar dyeing characteristics [97,98,104,105]. Dyes with exceptionally low affinity and rapid rates of diffusion have been developed, offering improved migration on acrylic fibres [106]. These dyes have migration properties not unlike those of monosulphonated acid dyes on nylon. 

Parallel synthesis of 62 different fucosylated tripeptides resulted in two ligands with submicromolar affinity for the P-selectin however, the desired activity for the E-selectin was not observed.98 For the E-selectin selectivity, it was necessary to incorporate a hydroxyl group that mimics the 4-hydroxyl of the central Gal in SLex in addition to a Fuc-residue and a carboxylate to obtain ligands with > 10-fold increased activity over that of the SLex tetrasaccharide.81 One of the best ligands was obtained from Thr(a-Fuc)-OEt, which was first /V-acylated with a hydroxyl amino acid and then elongated with a di-acid to furnish the acid mimic of the sialic acid carboxylate (Fig. 14.4). This approach was further developed as a solid-phase method where the molecule was linked to a solid support through the invariable fucosyl moiety.99... 

The word fractal was coined by Mandelbrot in his fundamental book.1 It is from the Latin adjective fractus which means broken and it is used to describe objects that are too irregular to fit into a traditional geometrical setting. The most representative property of fractal is its invariant shape under self-similar or self-affine scaling. In other words, fractal is a shape made of parts similar to the whole in some way.61 If the objects are invariant under isotropic scale transformations, they are self-similar fractals. In contrast, the real objects in nature are generally invariant under anisotropic transformations. In this case, they are self-affine fractals. Self-affine fractals have a broader sense than self-similar fractals. The distinction between the self-similarity and the selfaffinity is important to characterize the real surface in terms of the surface fractal dimension. 

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