Euclidean algorithm

Modular inverses can be computed efficiently with the extended Euclidean algorithm. However, this is somewhat slower than modular multiplication and asymptotically 0(P). 

As mentioned, Jacobi symbols can be evaluated using the law of quadratic reciprocity, which yields an algorithm similar to the Euclidean algorithm and of asymptotic complexity OQr ). 

The encryption exponent e must be chosen relatively prime to 660, say e = 91. The decryption exponent is d = mod (p(n) = 313. It can be found using the extended euclidean algorithm. 

The same idea can be developed in the case of a non-Euclidean metric such as the city-block metric or L,-norm (Section 31.6.1). Here we find that the trajectories, traced out by the variable coefficient kj are curvilinear, rather than linear. Markers between equidistant values on the original scales of the columns of X are usually not equidistant on the corresponding curvilinear trajectories of the nonlinear biplot (Fig. 31.17b). Although the curvilinear trajectories intersect at the origin of space, the latter does not necessarily coincide with the centroid of the row-points of X. We briefly describe here the basic steps of the algorithm and we refer to the original work of Gower [53,54] for a formal proof. 

In non-metric MDS the analysis takes into account the measurement level of the raw data (nominal, ordinal, interval or ratio scale see Section 2.1.2). This is most relevant for sensory testing where often the scale of scores is not well-defined and the differences derived may not represent Euclidean distances. For this reason one may rank-order the distances and analyze the rank numbers with, for example, the popular method and algorithm for non-metric MDS that is due to Kruskal [7]. Here one defines a non-linear loss function, called STRESS, which is to be minimized ... 

In the above equation, the norm is usually the Euclidean norm. We have a linear convergence rate when 0 is equal to 1. Superlinear convergence rate refers to the case where 0=1 and the limit is equal to zero. When 0=2 the convergence rate is called quadratic. In general, the value of 0 depends on the algorithm while the value of the limit depends upon the function that is being minimized. 

The simplest formulation of the packing problem is to give some collection of distance constraints and to calculate these coordinates in ordinary three-dimensional Euclidean space for the atoms of a molecule. This embedding problem - the Fundamental Problem of Distance Geometry - has been proven to be NP-hard [116]. However, this does not mean that practical algorithms for its solution do not exist [117-119]. 

There may be some great future algorithm or approach developed using some of these concepts, but for now how about the Euclidean Distance approach (equation 74-8) where ... 

Instead of using the Euclidean distance, also other distance measures can be considered. Moreover, another power than 2 could be used for the membership coefficients, which will change the characteristics of the procedure (degree of fuzzification). Similar to fc-means, the number of clusters k has to be provided as an input, and the algorithm also uses cluster centroids Cj which are now computed by... 

In order to characterize the interaction between different clusters, it is necessary to consider the mechanism of cluster identification during the process of the DA algorithm. As the temperature (Tk) is reduced after every iteration, the system undergoes a series of phase transitions (see (18) for details). In this annealing process, at high temperatures that are above a pre-computable critical value, all the lead compounds are located at the centroid of the entire descriptor space, thereby there is only one distinct location for the lead compounds. As the temperature is decreased, a critical temperature value is reached where a phase transition occurs, which results in a greater number of distinct locations for lead compounds and consequently finer clusters are formed. This provides us with a tool to control the number of clusters we want in our final selection. It is shown (18) for a square Euclidean distance d(xi,rj) = x, — rj that a cluster Rj splits at a critical temperature Tc when twice the maximum eigenvalue of the posterior covariance matrix, defined by Cx rj =... 

The simplest form of pattern comparison (euclidean distance in 16-diraensional space) was used to design and build an operational portable gas monitoring unit (15). With the limited computer power of a portable instrument, any one of about a dozen gases could be identified In less than one minute of computational time. This algorithm was evaluated using a data set for repeated runs of 16 different chemicals in 2 different sensor arrays (14). The results... 

Prior to analysis, the Raman shift axes of the spectra were calibrated using the Raman spectrum of 4-acetamidophenol. Pretreatment of the raw spectra, such as vector normalization and calculation of derivatives were done using Matlab (The Mathworks, Inc.) or OPUS (Bruker) software. OPUS NT software (Bruker, Ettlingen, Germany) was used to perform the HCA. The first derivatives of the spectra were used over the range from 380 cm-1 to 1700 cm-1. To calculate the distance matrix, Euclidean distances were used and for clustering, Ward s algorithm was applied [59]. 

Fig. 4.3. Dendrogram resulting from cluster analysis containing 91 spectra from 15 tree species (see also Table 4.2). Cluster analysis was done on first derivatives over the spectral range 380 cm-1 to 1700 cm-1). The distance matrix was calculated using Euclidean distance and Ward s algorithm was applied for clustering. Spectra were measured after decomposition of carotenoid molecules with 633 nm irradiation. For example, spectra of each species are shown in Fig. 4.1. Reprinted with permission from [52]...

Finally, applying MaxMin algorithm II, the sets of molecules for which the Euclidean distance is maxima have been determined. 

Search algorithms have advanced over the years to the point that most of the spectral data are used in the search. The methods are referred to as full-spectra searches because the entire spectral pattern is used in the matching procedure. Again, a number of similarity metrics are used, but most produce similar results. Typically, the spectral range for the search is selectable, and the library and target spectra are all normalized so that the total spectral area is 1.0. Either the Euclidean distance or the dot product between the target and library entries is calculated. The Euclidean distance is defined as... 

The best-known relocation method is the k-means method, for which there exist many variants and different algorithms for its implementation. The k-means algorithm minimizes the sum of the squared Euclidean distances between each item in a cluster and the cluster centroid. The basic method used most frequently in chemical applications proceeds as follows ... 

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