Inequality triangle

The most important consequence of bound smoothing is the transfer of infonnation from those atoms for which NMR data are available to those that cannot be observed directly in NMR experiments. Within the original experimental bounds, the minimal distance intervals are identified for which all triangle inequalities can be satisfied. A distance chosen outside these intervals would violate at least one triangle inequality. Eor example, an NOE between protons pi and pj and the covalent bond between pj and carbon Cj imposes upper and lower bounds on the distance between pi and Cy, although this distance is not observable experimentally nor is it part of E hem-... 

Note that although the bounds on the distances satisfy the triangle inequalities, particular choices of distances between these bounds will in general violate them. Therefore, if all distances are chosen within their bounds independently of each other (the method that is used in most applications of distance geometry for NMR strucmre determination), the final distance matrix will contain many violations of the triangle inequalities. The main consequence is a very limited sampling of the conformational space of the embedded structures for very sparse data sets [48,50,51] despite the intrinsic randomness of the tech-... 

Figure 3 Flow of a distance geometry calculation. On the left is shown the development of the data on the right, the operations, d , is the distance between atoms / and j Z. , and Ujj are lower and upper bounds on the distance Z. and ZZj, are the smoothed bounds after application of the triangle inequality is the distance between atom / and the geometric center N is the number of atoms (Mj,) is the metric matrix is the positional vector of atom / 2, is the first eigenvector of (M ,) with eigenvalue Xf,. V , r- , and ate the y-, and -coordinates of atom /. (1-5 correspond to the numbered list on pg. 258.)...

If the distances satisfy the triangle inequalities, they are embeddable in some dimension. One possible solution is therefore to try to start refinement in four dimensions and use the allowed deviation into the fourth dimension as an additional annealing parameter [43,54]. The advantages of refinement in higher dimensions are similar to those of soft atoms discussed below. 

Indeed, with the aid of the triangle inequality and the relation one can write down... 

Making use of (97) and the triangle inequality, we derive the estimate for a solution of problem (84b)... 

The possible values of k are all the integers included in the triangle inequality... 

The triangle inequality follows from the Schwarz inequality. ... 

With the aid of the well-known triangle inequality v + w < v + w we establish... 

Remark The accurate account of error zh = yh — uh can be done in a number of different ways. In concluding Section 11 the usual way of proper evaluation of the error zh was recommended for an additive scheme. Another way of proceeding is connected with the triangle inequality... 

Frechet [63] made an abstract formulation of the notion of distance in 1906. Hausdorff [64] proposed the term metric space, where he introduced the function d that assigns a nonnegative real number d p, q) (the distance between p and q) to every pair ip. q) of elements (points) of a nonempty set S. A metric space is a pair (S,d) if the function d satisfies several conditions, such as triangle inequality. In 1942, Menger [65] proposed that if we replace d(p, q) by a real function Fpq whose value is Fpq(x) for any real number x, this can be interpreted as the probability that the distance between p and q is less than x. Since probabilities can be neither negative nor greater than 1, we have... 

The distance between two points Ri and Rj in the representation space can be any nonnegative, real, commutative function that satisfies the triangle inequality (ref. 8). Usually, when comparing spectra Euclidean or Manhattan distances are employed. The generalized form of both, the Minkowski distance, can be written as follows ... 

A corresponding inverse triangle inequality can be applied to each triplet to raise values in the lower bound matrix L. Now, a distance matrix D, usually referred to as the trial distance matrix, can be constructed by simply choosing elements dt/ randomly between w/ and lif and used to construct a metric matrix G. A matrix so constructed might be some approximation to the distances in the real molecule, but probably not a very good one. Clearly, every time an element d is selected, it puts limits on subsequent selected distances. This problem of correlated distances is discussed further in the section Systematic Errors and Bias. 

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