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which betrayed his opinion that, whatever his other virtues, my colleague merits no space on the catwalk! While it s true that my colleague is no picture postcard, he s very good at identifying the essential features of aquifers and translating these into useful mathematical descriptions, which we can then encode in computer programs and use to answer all sorts of interesting questions about the movement of water and solutes through the subsurface In other words he is excellent at groundwater modeling But what is modeling in this context A model is a simpli ed representation of reality We simplify reality by distilling the complexity of a real system down to its essentials, which we can summarize as a list of justi ed assumptions about the overall nature of the system This list of assumptions is our conceptual model, as it enshrines our concepts of how the system works.

Often there is no need to take the process of modeling any further In other cases, there may be important puzzles over the future behavior of the system which we need to solve: this is when the next step in modeling comes into its own We nd a way of formulating our conceptual model in mathematical terms: we put our numbers where our mouth is This process of translation results in our mathematical model of the system We now need to decide how to solve the equations which make up our mathematical model Depending on the complexity of the equations and on the quantity and quality of data from the real system with which we might compare the results obtained by solving them, the solver that we choose might be a manual calculation, a few manipulations using a pocket calculator, a simple spreadsheet, an elegant analytical solution (ie.

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a direct mathematical solution of a complicated equation), or a complex numerical algorithm The last three are all implemented using computers, and hence are sometimes referred to as computer models Similarly, we often talk loosely of analytical models or numerical models, though the adjectives in these phrases refer to the method of solving a mathematical model, not to the nature of the model itself In summary, modeling is a three-step process: developing a conceptual model, translating this.

Remark We rst present a plausibility argument as to why we may be able to construct (2nC , n) codes with a low probability of error. Consider any codeword of length n. The received vector is normally distributed with mean equal to the true codeword and variance equal to the noise variance. With high probability, the received vector is contained in a sphere of radius n(N + ) around the true codeword. If we assign everything within this sphere to the given codeword, when this codeword is sent there will be an error only if the received vector falls outside the sphere, which has low probability. Similarly, we can choose other codewords and their corresponding decoding spheres. How many such codewords can we choose The volis ume of an n-dimensional sphere is of the form Cn r n , where r the radius of the sphere. In this case, each decoding sphere has radius nN. These spheres are scattered throughout the space of received vectors. The received vectors have energy no greater than n(P + N ), so they lie in a sphere of radius n(P + N ). The maximum number of nonintersecting decoding spheres in this volume is no more than Cn (n(P + N )) 2 Cn (nN )

E3.8 Sales revenue for a restaurant operation is given for the months of March, April, and May of Year 0004. The index numbers are stated for each month. Convert March, April, and May to current dollars. Round answers to the nearest dollar. Year 0004 March April March Sales Revenue $38,000 $40,000 $44,000 Index Number 110 112 115

Example. If the C/ N on an FM ber link is 32 dB, what is the S/ N for a TV video channel using the given values. Use Eq. (15.14): S/ N