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18.4.3.1 WA Problem Given a set P of directed paths on Cn , P can be partitioned into two subsets, one is the subset of clockwise paths and the other is the subset of counter-clockwise paths. A clockwise path only uses links of Cn in the clockwise direction and does not share a link with any counter-clockwise path. So the WA problem on Cn can be solved for each subset independently. The WA problem on Cn for each subset can be studied as the WA problem on the undirected ring Cn . Given a set P of paths on Cn , the con ict graph GP is a circular arc graph [42]. It is known that for circular arc graph GP the clique number (GP ) can be computed in polynomial time [24] but nding the chromatic number (GP ) is NP-hard [22]. Algorithms that use at most 2L 1 and 3 (GP )/2 colors are known [31,42]. To describe those algorithms, we rst introduce some new notation. A path p on Cn is identi ed by the segment [p ,p ], where p and p are end nodes of p. Given a node u, a b (resp. a b) if a [u, b] (resp. a [b, u]). Theorem 4 (Tucker [42]) The WA problem for a set P of paths on Cn with link load L can be solved by at most 2L 1 colors.

Corporate R&D efforts, if successful, lead to an increase in value of the corporation and the price of its stock. However, not all research efforts are successful or of equal impact on a firm s future earnings. Empirical data supports the use of three figures of merit as valid predictors of future profitability of a firm as compared to others in the same industry. Patent count, which is the relative size of a firm s patent portfolio as compared to other firms in the industry, has been found to correlate well with that firm s future profitability. Citation impact, which is a measure of the frequency with which the patents of one firm are cited in later issued patents of others, is an indication of how basic is the firm s technology. The more often its patents are so cited, the more basic and valuable its technology is likely to be. Science linkage is a measure of the frequency with which scientific articles are cited as prior art in a firm s own patents. The higher a firm s science linkage, the closer its technology is to basic research, as opposed to product development. Patents pertaining to basic research are, generally, more valuable than patents on product developments.

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Note: + + = Repeatedly documented positive associations with physical activity; + = Weak or mixed evidence of positive association with physical activity; 0 0 = Repeatedly documented lack of association with physical activity; 0 = Weak or mixed evidence of no association with physical activity; = Repeatedly documented negative associations with physical activity; = Weak or mixed evidence of negative association with physical activity. Blank spaces indicate no data available. Adapted from The Determinants of Physical Activity and Exercise, by R. K. Dishman, J. F. Sallis, and D. R. Orenstein, 1985, Public Health Reports, 100, pp. 158 171; Exercise psychology, by J. Buckworth and R. K. Dishman, 2002, Champaign, IL: Human Kinetics. Adapted with permission.

where ZS R jo L ; YSH G jo C ; l g2 ; U is the unit matrix, and [0] is the null vector. Equations (3.6.3) and (3.6.4) represent the generalized matrix eigenvalue and eigenvector problem. If [Mv ] denotes the complex eigenvector matrix associated with the characteristic matrix [ZS ][YSH ], then, following the same procedure as in the last section, it can be shown that the voltage eigenvector e and the current eigenvector j are solutions of the following set of decoupled equations: ! d g 3:6:5 e diag k j yk dz d 3:6:6 j diag gk yk e dz where gk is the propagation constant of the kth mode equal to the square root of the kth eigenvalue of [ZS ][YSH ], yk is the characteristic admittance of the kth mode equal to the corresponding element of the diagonal matrix [Yk ] given by Yk Mv 1 YSH Mv and V I ! Mv 0 0 Mv T 1 ! e j ! 3:6:8 3:6:7

B.2.2

Symptoms Double vision and ptosis Difficulty in chewing, swallowing and talking Difficulty in lifting head up from the lying position Difficulty in lifting arms above shoulder level, and in standing from low chairs and out of the bath Breathing problems and difficulty in sitting from the lying position Weak hand-grips, ankles and feet

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