Row-pattern

Perhaps the most fascinating detail is the surface reconstruction that occurs with CO adsorption (see Refs. 311 and 312 for more general discussions of chemisorption-induced reconstructions of metal surfaces). As shown in Fig. XVI-8, for example, the Pt(lOO) bare surface reconstructs itself to a hexagonal pattern, but on CO adsorption this reconstruction is lifted [306] CO adsorption on Pd( 110) reconstructs the surface to a missing-row pattern [309]. These reconstructions are reversible and as a result, oscillatory behavior can be observed. Returning to the Pt(lOO) case, as CO is adsorbed patches of the simple 1 x 1 structure (the structure of an undistorted (100) face) form. Oxygen adsorbs on any bare 1 x 1 spots, reacts with adjacent CO to remove it as CO2, and at a certain point, the surface reverts to toe hexagonal stmcture. The presumed sequence of events is shown in Fig. XVIII-28. 

Figure 11. Row patterns in a baffled tank with centrally mounted impeller.

If air is supplied horizontally through a wall, the supply air has to have exactly the same temperature as the room air. A temperature difference of 1 C or more will create Row patterns like those shown in Fig. 8.28. 

In the illustration of Fig. 29.4 we regard the matrix X as either built up from n horizontal rows of dimension p, or as built up from p vertical columns x,.of dimension n. This exemplifies the duality of the interpretation of a matrix [9]. From a geometrical point of view, and according to the concept of duality, we can interpret a matrix with n rows and p columns either as a pattern of n points in a p-dimensional space, or as a pattern of p points in an n-dimensional space. The former defines a row-pattern P" in column-space 5, while the latter defines a column-pattern P in row-space S". The two patterns and spaces are called dual (or conjugate). The term dual space also possesses a specific meaning in another... 

Fig. 29.5. Geometrical interpretation of an nxp matrix X as either a row-pattern of n points P" in p-dimensional column-space S (left panel) or as a column-pattern of p points / in n-dimensional row-space S" (right panel). The p vectors Uy form a basis of 5 and the n vectors v, form a basis of 5".

In Chapter 29 we introduced the concept of the two dual data spaces. Each of the n rows of the data table X can be represented as a point in the p-dimensional column-space S . In Fig. 31.2a we have represented the n rows of X by means of the row-pattern F. The curved contour represents an equiprobability envelope, e.g. a curve that encloses 99% of the points. In the case of multinormally distributed data this envelope takes the form of an ellipsoid. For convenience we have only represented two of the p dimensions of SP which is in reality a multidimensional space rather than a two-dimensional one. One must also imagine the equiprobability envelope as an ellipsoidal (hyper)surface rather than the elliptical curve in the figure. The assumption that the data are distributed in a multinormal way is seldom fulfilled in practice, and the patterns of points often possess more complex structure than is shown in our illustrations. In Fig. 31.2a the centroid or center of mass of the pattern of points appears at the origin of the space, but in the general case this needs not to be so. 

In Fig. 31.2a we have represented the ith row x, of the data table X as a point of the row-pattern F in column-space S . The additional axes v, and V2 correspond with the columns of V which are the column-latent vectors of X. They define the orientation of the latent vectors in column-space S. In the case of a symmetrical pattern such as in Fig. 31.2, one can interpret the latent vectors as the axes of symmetry or principal axes of the elliptic equiprobability envelopes. In the special case of multinormally distributed data, Vj and V2 appear as the major and minor... 

Fig. 31.2. Geometrical example of the duality of data space and the concept of a common factor space, (a) Representation of n rows (circles) of a data table X in a space Sf spanned by p columns. The pattern P" is shown in the form of an equiprobabi lity ellipse. The latent vectors V define the orientations of the principal axes of inertia of the row-pattern, (b) Representation of p columns (squares) of a data table X in a space y spanned by n rows. The pattern / is shown in the form of an equiprobability ellipse. The latent vectors U define the orientations of the principal axes of inertia of the column-pattern, (c) Result of rotation of the original column-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the score matrix S and the geometric representation is called a score plot, (d) Result of rotation of the original row-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the loading table L and the geometric representation is referred to as a loading plot, (e) Superposition of the score and loading plot into a biplot.

Fig. 31.3. (a,b) Reproduction of distances D and angular distances 0 in a score plot (a = 1) or loading plot (p = 1) in the common factor-space (c,d) Unipolar axis through the representation of a row or column and through the origin 0 of space. Reproduction of the data X is obtained by perpendicular projection of the column- or row-pattern upon the unipolar axis (a + P = 1). (e,0 Bipolar axis through the representation of two rows or two columns. Reproduction of differences (contrasts) in the data X is obtained by perpendicular projection of the column- or row-pattern upon the bipolar axis (a + P = 1). 

The vector of column-means nip defines the coordinates of the centroid (or center of mass) of the row-pattern P" that represents the rows in column-space Sf . Similarly, the vector of row-means m defines the coordinates of the center of mass of the column-pattern that represents the columns in row-space S". If the column-means are zero, then the centroid will coincide with the origin of SP and the data are said to be column-centered. If both row- and column-means are zero then the centroids are coincident with the origin of both 5" and S . In this case, the data are double-centered (i.e. centered with respect to both rows and columns). In this chapter we assume that all points possess unit mass (or weight), although one can extend the definitions to variable masses as is explained in Chapter 32. 

FIGURE 7-25 Row patterns for staggered and in-line tube banks (photos by R. D. WilUs). 

Figure 13.31 Grout curtain in stratified deposits using two- and three-row patterns.

A three-row pattern is generally adequate. Spacing of holes in each row is determined by economic factors. Wide spacing reduces drilling costs and increases chemical costs. Generally, spacings of about 3 feet turn out to be most economical. Assume this is so. 

A cement-clay grout followed by a cement-bentonite grout was used to fill the coarser voids. Over 1 million ft of cement-based grout was placed in a three-row pattern over the entire curtain length, with two additional rows of holes near the dam. The effectiveness of the cutoff was monitored by piezometers placed upstream and downstream of the curtain. The difference in elevation between two specific holes, compared with the difference in river levels (actually, headwater and tailwater elevations) opposite those holes, is an index of effectiveness, and this index is an excellent way to assess quickly the effects of grouting. This index for the Rocky Reach Dam, plotted against time, is shown in Fig. 17.7. The chart makes it clear that cutoff effectiveness leveled off at about 80% for cement-based grouts. 

The combination of membrane configurations and input-output port arrangements results in many possible feed and permeant Row patterns. For simplicity, several idealized Row patterns shown in Fig. 20.6-2 have been used to model these different situaUous. In Fig. 20.6-2o-c. both the feed and permeant streams are... 

L.E. Murr, and A.C. Nunes, Jr., Row Patterns during Friction Stir Welding, Mater. Charact, Vol 49, 2003, p 95-101... 

Tableaux containing this common fixed part , followed by an arbitrary 2-row pattern for the open-shell electrons, will be called acceptable in terms of the branching diagram (Fig. 4.3), for the example g = 2, n = 5, they correspond to coupling schemes such as...

---