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These equations are called the Yule Walker equations. There are p + 1 equations in the p + 1 unknowns a1 , a2 , . . . , ap , 2 . Therefore, we can solve for the parameters of the process from the covariances. Fast algorithms such as the Levinson algorithm and the Durbin algorithm [433] have been devised to use the special structure of these equations to calculate the coef cients a1 , a2 , . . . , ap ef ciently from the covariances. (We set a0 = 1 for a consistent notation.) Not only do the Yule Walker equations provide a convenient set of linear equations for calculating the ak s and 2 from the R(k) s, they also indicate how the autocorrelations behave for lags greater than p. The autocorrelations for high lags are an extension of the values for lags less than p. These values are called the Yule Walker extension of the autocorrelations. The spectrum of the maximum entropy process is seen to be
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where the symbol {indicates a soft lower interval limit that includes values with lower indexes than (k Ka), which have relatively low contribution [32, 33]. For a variable
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Premuitiplication: Let A, B be (m x 71) and (71 xp) matrices respectively.
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proportion of the eld-measured K of an aquifer can be ascribed to intergranular ow, as opposed to ow through fractures (e.g. Price et al. 1982). Even where the absolute values of K determined in laboratory studies are dubious, they can at least reveal the relative permeabilities of a range of soils, i.e. identifying which are more permeable than others. However, in the vast majority of water resources investigations, laboratory measurements or estimates of K are unlikely to be of widespread practical use. Wherever possible, it will always be best to determine K and/or T from data obtained by eld testing of wells. The method of choice is test pumping, in which a borehole is pumped at a known rate and the resultant drawdown is measured. Essentially, test pumping is an exercise in quantifying the shape and rate of development of the cone of depression, from which it is possible to calculate transmissivity. There are two basic patterns of test pumping in common use: step-drawdown testing and constant-rate testing. Step-drawdown tests are essentially an engineering tool used for assessing the performance ef ciency of pumping wells. Section 7.3.1 brie y describes their implementation and interpretation. For purposes of determining accurate, representative values of transmissivity, the technique of choice is constant-rate test pumping. As the name of this method suggests, it involves pumping the well at a single constant rate, and monitoring the resultant drawdowns. It is possible to perform constant-rate tests by measuring drawdowns only in the pumped well, but far greater insights will be gained by monitoring drawdowns in one or more observation wells located within the radius of in uence of the pumping well. A wide range of analytical techniques has been developed to facilitate the interpretation of constant-rate pumping tests (see Kruseman and de Ridder 1991). The foundations for virtually all current methods for interpreting constant-rate pumping tests were laid by Theis (1935), who developed a method to calculate T and S by interpretation of time drawdown data for constant-rate pumping of a well fully penetrating a horizontal, con ned aquifer, which is assumed to have
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affected by fatigue, whereas the teams of other sizes were able to avoid this problem because they had enough players to rotate players in and out of play. The explanation of relative inferiority of performance by the teams of 9 was less clear, but Widmeyer et al. provided evidence that poor motivation as the explanation was unlikely. Instead, these authors argued that poor coordination was the most likely explanation, although coordination was not measured. Knowledge Requirements for Achieving Coordination In the studies just described, the process losses identified were often attributed to the need for team members in the team condition to coordinate their operations. In the studies by Kidd (1961) and Naylor and Briggs (1965), it was reported that the work required to achieve coordination impinged on the work associated with the task per se. By comparison, participants in the individual conditions were not required to coordinate their operations, and thus could focus exclusively on the task. In this regard, I /O psychologists have used the term taskwork to describe elements of a team member s task that are independent of fellow members operations, and thus do not introduce the need for coordination. Teamwork is the term used to describe elements that are interdependent with fellow members operations, and thus do introduce the need for coordination (McIntyre & Salas, 1995). Subsequently, the knowledge required to undertake taskwork has become known as taskwork knowledge, and that required to undertake teamwork as teamwork knowledge. Consider this distinction in relation to football. A quarterback must acquire taskwork knowledge to be able to aim a pass so that it lands accurately in a given target location and time a pass so that it arrives punctually at a given target time. However, the ability of the quarterback to execute the pass accurately and punctually also relies on the acquisition of teamwork knowledge pertaining to the operations of the quarterback s receivers, because these operations dictate the target location and timing of the pass. Similarly, a team member who transfers between teams is able to rely on previously acquired taskwork knowledge in the new team to undertake taskwork elements of the task because these elements are unrelated to fellow team members, and thus are relatively constant across teams. However, the team member s previously acquired teamwork knowledge is of limited use in the new team because the teamwork elements of the task have changed. Furthermore, although the individual performer needs only taskwork knowledge to undertake his or her task, he or she must gain teamwork
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mates indicate that 26% of girls and 20% of boys age 9 to 13 years do not participate in physical activity during their free time; only 38% participate in organized physical activity (Duke, Huhman, & Heitzler, 2003). Moreover, both leisure time physical activity (Kimm et al., 2002) and school-based physical activity (Grunbaum et al., 2004) decline during adolescence, especially among girls (Caspersen, Pereira, & Curran, 2000; Kimm et al., 2002). The pattern of activity for adults age 30 to 64 years is relatively stable until retirement, when there is some improvement until the final period of life (Caspersen et al., 2000). Older adults are less active than other age groups regardless of race/ethnicity (Crespo, Smit, Andersen, CarterPokras, & Ainsworth, 2000). Females are more likely than males to be physically inactive at most ages. Estimates from the Youth Risk Behavior Surveillance System are that 40% of U.S. girls, compared to 27% of boys, do not regularly engage in sufficient amounts of vigorous and moderate physical activity during the high school years (Grunbaum et al., 2004). Fifty
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FIGURE 1.7 A Davis cross staff sixteenth century. (Source: Peter I and, Taking the Stars, Celestial Navigation from Argonauts to Astronauts.)
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