Linear extension

The next step is, therefore, to extrapolate this provisional spectrum at both ends. At the short time side, the extension usually covers three decades of time, any farther extrapolation not resulting in a further improvement because the kernels in both Equations [5] and [6] approach zero. The extrapolation to longer times always consists of the two possible extremes a linear extension, and a sharpy deflecting one with ultimate slope of -1. Theoretical arguments for such a sharp increase in the slope of H(x) at long times were given in a molecular theory (20). 

The diagonal components. Yu. A i. and Aji are the coefficients of linear extension in the directions a a a , and a, respectively, while the nondiagonal components Ai - A i. An = An. and = S are called shear strains. For instance. 2A i2 is the change in the angle of the dihedron formed by the planes that before the deformation were respectively normal to the directions a and a . The shear strains are not essential for the complete representation of a deformation since they can be made to vanish by expressing S in the basis of its principal axes. 

Berman showed from a detailed analysis of the 13C-NMR spectra of several sialy-lated oligosaccharides that the conformation of an a-(2-3) bond gives a linear extension, whereas the attachment of the a-D-NeuiVAc to the 6-hydroxygroup yielded into a folded back conformation of the neuramic acid [188]. 

An overview on cycloproparenyl anions has also been reported.3 According to theoretical calculation, cyclopropabenzenyl anion is by ca 145 kJmol-1 more stable than the parent cyclopropenyl anion. It has been shown that the stability of the cyclopropabenzenyl anion could be considerably enhanced by substitution of the aromatic ring with fluorine and cyano groups, and also by a linear extension of the aromatic backbone. 

Table 2.4 Linear extension R as a function of filling rate (volume per time unit) of combustible liquids with a given flash point...

In practice, the crack tip yielding in polymers is often not of a circular zone type as described above, but is a co-linear extension of the crack. The deformed material within the zone often forms a porous structure with ligaments restraining the zone faces, as illustrated in Fig. 13. This porous material, usually termed the craze, can be regarded as providing cohesive forces over the zone length. The zone can then be... 

The method of epikernel principle seems to be incomplete due to its restriction to the 1st order perturbation theory and linear extension of the perturbation potential. Using more complete perturbation may produce the results comparable with the other method on account of higher elaborateness. The JT caused loss of planarity or of symmetry center in JT systems can be explained by pseudo-JT mechanisms only. Another problem is the applicability to the groups with complex characters (C , S , and C h for n > 2, T and Th). 

Carlsen, L., Lerche, D.B. and Sorensen, P.B. (2002) Improving the predicting power of partial order based QSARs through linear extensions. /. Chem. Inf. Comput. Sci., 42, 806-811. 

Lerche, D., Sorensen, P.B. and Briiggemann, R. (2003) Improved estimation of the ranking probabilities in partial orders using random linear extensions by approximation of the mutual ranking probability. J. Chem. Inf. Comput. Sci., 43, 1471-1480. 

Figure 2. Three snapshots of the system, for N = 100x100, showing the Wolff s clusters of correlated water molecules. For each molecule we show the states of the four arms and associate different colors to different arm s states. The state points are at pressure close to the critical value Pc (Pvfs = 0.72 Pcvfs) and T > Tc (toppanel, kBT/e= 0.053), T xsTc (middle panel, kfr/s = 0.0528), T < Tc (bottom pxinel, kfr/s = 0.052), showing the onset of the percolation at T Tc. At T fxTc (middle panel) there is one large cluster, in red on the right, with a linear size comparable to the system linear extension and spanning in the vertical direction.

Fig. 4 Fluorescence intensity of 13 with the addition of Co (dark circles). Inset shows linearity at low concentration. Solid line is linear extension of the initial slope, dashed line is expected if two Co are independently coordinated in 13 [35]...

In the chapter of Briiggemann and Carlsen some concepts introduced in the chapter of El-Basil are revitalized and explained in the context of the multivariate aspect. Basic concepts, like chain, anti-chain, hierarchies, levels, etc., as well as more sophisticated ones, like sensitivity studies, dimension theory, linear extensions and some basic elements of probability concepts are at the heart of this chapter. The difficult problem of equivalent objects, which lead to the items object sets vs. quotient sets are explained and exemplified. 

There are several possibilities to construct linear orders. Theoretically very important is the concept of linear extensions, which is explained later (vide infra). 

The linear extensions are the basis of the dimension theory of posets. Besides the dimension of posets other characterizations may be derived from linear extensions (Carlsen et al. 2002, Lerche et al. 2003, Lerche Sorensen 2003). 

Extensions are order-preserving maps from the ground set E into the ground set E see Davey Priestley 1990. Linear extensions (LEX(E ), <) are order-preserving maps from E to E, which assign to (E, <) a linear order. 

In Fig. 6 (b) an extension is shown (identify A with a, B with b, etc), but not a linear one. An additional preserving map leads to a linear order (Fig. 6 (c)). The diagram in Fig. 6 (c) is a linear extension of that in Fig. 6 (a). Given a poset (E, <) then several linear extensions (LEX(E),<) are possible. A systematic procedure is described by Atkinson (1989), especially for trees a closed formula can be derived Atkinson (1990). A useful formula to calculate the number of linear extensions is also given by Stanley (1986). 

In Fig. 14 the first column labels the linear extensions, which are represented as sequences in rows 1 - 14. A sequence a b c. .. is to be read as a > b > c. Furthermore there is a sequence (last row in the table) which is not a linear extension of (E, <). Vertical bold lines indicate jumps (see below). The last column indicates the number of jumps of each single linear extension. Consecutive elements in linear extensions (LEX(E ), <), which have no correspondence in (E, <) are called "jumps" (see Fig. 14, the vertical bold lines indicating jumps). The jump number, jump (LEX,(A , <)), obviously depends on the actual selected linear extension. The jump number of a poset (E, <), jump (E, <), is just min(jump(LEXj (E, <))), whereby the minimum is to be found by checking all linear extensions. Beside the jump - number there is also a bump - number. Once again the bump number is to be referenced to a specific linear extension. A bump is a consecutive pair of elements in a linear extension, which are comparable in the underlying poset. The bump number of a poset is the maximum about all bump numbers found for the linear extensions. If a linear extension of n elements is formed then n-1 consecutive relations are found in a linear extension. Therefore... 

If a specific element, say xe is selected then its spectrum is of interest (Atkinson 1990). It should be noted that other authors (for example Trotter 1991, Schroder 2003) also call the spectrum a projection. However, we favour "spectrum" as the more suitable name for the following construction. Thus, let LT be the number of linear extensions of a poset, then we can find the rank of an element x in the ith linear extension rank(i, x). Note that this construction should not be confused with the rank function, we discussed above. Conventionally, the bottom element of a linear extension is given the rank 1, thus the top element has the rank n (card E = n). However, if appropriate the top element may be assigned the first priority, such that bottom elements will get numbers > 1 (see for example chapter by Carlsen, p. 163). We call A,k(x) the frequency, how often x e E gets the rank k. The spectrum spec(x) is a tuple containing n components (Ii(x), I2(x),. .., A,n(x)). Thus for example the spectrum of element b in Fig. 14 as follows spec(b) = (0, 0, 3, 6, 5, 0). (i) There is no linear extension, where the rank of b is 1, 2 or 6. (ii) There are 3 linear extensions, where the rank of b is 3. (iii) There are 6 linear extensions, where the rank of b is 4. (iv) There are 5 linear extensions, where the rank of b is 5. Obviously ... 

The set of linear extensions is the basis for probability considerations Dividing A,k(x) by LT the quantity prob (rk(x) = k) = A,k(x)/LT can be interpreted as (ordinal) probability to get the rank k, sometimes also called absolute rank . Hence, an averaged rank, Rkav can be derived by... 

The dimension of a poset is based on the set of linear extensions. A linear extension can be considered as a set of ordered pairs. For example the linear extension no 1 in Fig. 14 (right side) ... 

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