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A route may be found that appears the shortest, but how do we know for sure that it is the shortest How long will it take to calculate the shortest route in such large-scale maps These may seem like hard questions to answer, so it may be decided to write programs to solve these problems The rst problem faced is how to represent the graph in Figure 31 to a computer Fortunately, there is an easy way to represent graphs to computers which involves the matrix method (Table 31) Since matrices are provided as standard in most programming languages, there is an easy way to represent the connectivity of the graph in Figure 31 Providing the information to a computer in a way that the computer understands is to provide a data structure to the computer.

This has the effect of placing the current value of the counter in the content object. For example, the following style de nition will display and the current value of the chapter counter at the beginning of each <h1> element:

Another advantage to the matrix method is that, if another node is added to the graph (another city is added to the map), it will be easy to add an extra row and column to the data structure and insert the distances between that node and all the nodes it connects to in the appropriate x,y entries, without needing to enter all the graph information again Once such a data structure is developed, an algorithm is required that will make use of this data structure to calculate shortest routes An algorithm is a sequence of steps that, if systematically and correctly executed, will produce the desired result For searching graphs there is a need to devise an algorithm that will explore paths rigorously, meaning that the solution is to be found as ef ciently as possible as well as guaranteeing to return the correct result.

Interpolation problem 1: Determine the smallest J such that the first J columns of V are linearly dependent.

Ra = gd4/k where is the coef cient of thermal expansion, the superadiabatic temperature gradient (the gradient in excess of that expected to be associated with the increasing pressure), the density of the uid, g the acceleration due to gravity, d the thickness of the convecting uid, k the thermal diffusivity (the ratio of the thermal conductivity to the product of density and speci c heat at constant volume), and the dynamic viscosity (Section 2103) For convection in the mantle, the Rayleigh number corresponding to the onset of convection is approximately 103 This corresponds to the minimum temperature gradient required for convection to occur For the actual temperature gradient the Rayleigh number is of the order of 106 or greater This implies very favorable conditions for convection and, as a consequence, thin boundary layers compared to the total layer thickness.

To ensure that there is an ef cient algorithm that doesn t explore a route that has been previously examined and found not to contain the desired solution, as well as exploring all possible routes that exist in the graph, a search algorithm is required that methodically searches routes one step at a time The conventional way to do this is to convert the type of search into a tree search Look again at S, the start node (Figure 31) There are three links from S to A, B and C Rather than decide arbitrarily to follow just one, all.

LiveConnect. Because not all Flash movies need the LiveConnect connection, you can prevent Java from being initialized by entering the swliveconnect attribute in the Parameters dialog box and setting its value to false. When the swliveconnect=false parameter is found by the browser, the Java is not initialized as part of the loading process and your movie loads more quickly.

The nature of the ow in a convecting uid can be judged by the magnitude of the Reynolds number (Re), which allows discrimination between laminar and turbulent ow Re is de ned: Re = vd/v where v is the velocity of ow and is the kinematic viscosity (the ratio of the dynamic viscosity, , to density) Taking v = 200 mm a 1 = 6 10 9 ms 1, d = 3000 km = 3 106 m and = 2 1017 m2 s 1, Re = 9 10 20 This very low value indicates that viscous forces dominate and hence the ow is laminar The effect of the Earth s rotation on convection can be judged by the magnitude of the Taylor number (T), which is de ned: T = (2wd2/ )2 where w is the angular velocity of rotation Putting w = 7.

age density percentage. This can be accomplished by installing racking systems that are dangerously high or by laying out aisles that are too narrow for efficient item movement.