Prove that the grade-school multiplication algorithm (page 23), when applied to binary numbers,
always gives the right answer.
1.7. How long does the recursive multiplication algorithm (page 24) take to multiply an n-bit number
by an m-bit number? Justify your answer.
1.8. Justify the correctness of the recursive division algorithm given in page 25, and show that it
takes time O(n2) on n-bit inputs.
1.9. Starting from the definition of x ≡ y mod N (namely, that N divides x−y), justify the substitution
rule
x ≡ x′ mod N, y ≡ y′ mod N ⇒ x + y ≡ x′ + y′ mod N,
and also the corresponding rule for multiplication.
1.10. Show that if a ≡ b (mod N ) and if M divides N then a ≡ b (mod M ).
1.11. Is 41536 − 94824 divisible by 35?
1.12. What is 222006
(mod 3)?

1.13. Is the difference of 530,000 and 6123,456 a multiple of 31? 1.14. Suppose you want to compute the nth Fibonacci number Fn, modulo an integer p. Can you find an efficient way to do this? (Hint: Recall Exercise 0.4.)

The telephone network of today and the Resource Reservation Protocol (RSVP) and associated services proposed for the future Internet provide open loop control. Briefly explain the functions of signalling, admission control and policing in these types of networks. [3 marks each] (b) Traffic sources are described as elastic or inelastic. Outline how an inelastic traffic source can be characterised and how this relates to the type of network resource guarantees it may need. [5 marks] 2 CST.2006.7.3 3 Security (a) What is meant by Mandatory Access Control? Give an example. [5 marks] (b) How might you use mandatory access control to protect the safety-critical systems in a car (engine control unit, ABS, stability control, etc.) from userprogrammable systems (telephone, entertainment, navigation, etc.)? [5 marks] (c) What problems would you anticipate in keeping the implementation clean as these systems evolve? [5 marks] (d) What architecture might you therefore propose a car maker adopt for its nextgeneration networking? [5 marks] 4 Advanced Graphics (a) Describe, in outline, each of the implicit surface, NURBS surface, and constructive solid geometry methods for defining three-dimensional shapes. [4 marks each] (b) Compare and contrast the three methods. [8 marks] 3 (TURN OVER) CST.2006.7.4 5 Computer Systems Modelling (a) Describe the congruential methods for generating pseudo-random numbers from a Uniform (0, 1) distribution. [3 marks] (b) Let U be a Uniform (0, 1) random variable. Show that for any continuous distribution function, F(x), the random variable, X, defined by X = F ?1 (U) has the probability distribution function F(x). [3 marks] (c) Apply the method of part (b) to generate random variables with the following distributions. In each case, specify the distribution function F(x) that you use. (i) Uniform distribution on the interval (a, b), for a < b. [2 marks] (ii) Exponential distribution with parameter ?. [2 marks] (d) Define the Poisson process, N(t), (t ? 0) of rate ?. [2 marks] (e) Show that for each fixed t ? 0, N(t) is a Poisson random variable with parameter ?t. [3 marks] (f ) Show that the interarrival times of consecutive events in a Poisson process of rate ? are independent random variables each with the exponential distribution with parameter ?. Show how this leads to a method to simulate the events of a Poisson process. [5 marks] 4 CST.2006.7.5 6 Specification and Verification I (a) What is total about total correctness? [2 marks] (b) State the WHILE-Rule of Floyd-Hoare Logic. [2 marks] (c) Give a proof of {T} WHILE T DO SKIP {F}. [2 marks] (d) What does the truth of {T} WHILE T DO SKIP {F} show? [2 marks] (e) How are expressions like 1/0 handled in Floyd-Hoare Logic? [2 marks] (f ) What are verification conditions? [2 marks] (g) Must the verification conditions be true for correctness? Briefly justify your answer. [2 marks] (h) Name one method used to prove verification conditions. [2 marks] (i) What are the "hooked" variables in VDM used for? [2 marks] (j) What are weakest preconditions and weakest liberal preconditions? [2 marks] 7 Specification and Verification II (a) What are Binary Decision Diagrams and how are they used to represent statetransition functions symbolically? [4 marks] (b) What is temporal abstraction? How are models at different temporal abstraction levels related? [4 marks] (c) What is the difference between LTL and CTL? [4 marks] (d) How do the Verilog and VHDL simulation cycles differ? [4 marks] (e) What is the difference between formal verification using model checking and using theorem proving? [4 marks] 5 (TURN OVER) CST.2006.7.6 8 Information Theory and Coding (a) Give three different expressions for mutual information I(X; Y ) between two discrete random variables X and Y , in terms of their two conditional entropies H(X|Y ) and H(Y |X), their marginal entropies H(X) and H(Y ), and their joint entropy H(X, Y ). Explain in ordinary language the concept signified by each of the measures H(X|Y ), H(X), H(X, Y ), and I(X; Y ). Depict in a Venn diagram the relationships among all of the quantities mentioned here. [8 marks] (b) Suppose that women who live beyond the age of 70 outnumber men in the same age bracket by three to one. How much information, in bits, is gained by learning that a certain person who lives beyond 70 happens to be male? [2 marks] (c) What is the shortest possible code length, in bits per average symbol, that could be achieved for a six-letter alphabet whose symbols have the following probability distribution? 1 2 , 1 4 , 1 8 , 1 16 , 1 32 , 1 32 [3 marks] (d) If we wish to increase the transmission capacity of a noisy communication channel, is it more effective to increase its electronic bandwidth in Hertz, or to improve its signal-to-noise ratio? Briefly say why. [2 marks] (e) A continuous signal whose total bandwidth is 1 kHz and whose duration is 10 seconds may be perfectly represented (even at points in between the points at which it is sampled) by what minimal number of real numbers? [2 marks] (f ) Give the names of three functions (not necessarily their equations) which are self-Fourier. [3 marks]

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crime rate last year in counties with high school graduation percentage X= 80, (3) 510- (4) 0. 1.29. Refer to regression model (1.1). Assume that X = 0 is within the scope of the model. What is the implication for the regression function if fo = 0 so that the model is Y = BX, + &? How would the regression function plot on a graph? 1.30. Refer to regression model (1.1). What is the implication for the regression function if B = 0 so that the model is Y = Bo + 8? How would the regression function plot on a graph? 1.31. Refer to Plastic hardness Problem 1.22. Suppose one test item was molded from a single batch of plastic and the hardness of this one item was measured at 16 different points in time. Would the error term in the regression model for this case still reflect the same effects as for the experiment initially described? Would you expect the error terms for the different points in time to be uncorrelated? Discuss. 1.32. Derive the expression for b in (1.10a) from the normal equations in (1.9). 1.33. (Calculus needed.) Refer to the regression model Y = Bo + 8 in Exercise 1.30. Derive the least squares estimator of Bo for this model. 1.34. Prove that the least squares estimator of Bo obtained in Exercise 1.33 is unbiased. 1.35. Prove the result in (1.18)-that the sum of the Y observations is the same as the sum of the fitted values. 1.36. Prove the result in (1.20)-that the sum of the residuals weighted by the fitted values is zero. 1.37. Refer to Table 1.1 for the Toluca Company example. When asked to present a point estimate of the expected work hours for lot sizes of 30 pieces, a person gave the estimate 202 because this is the mean number of work hours in the three runs of size 30 in the study. A critic states that this person's approach "throws away" most of the data in the study because cases with lot sizes other than 30 are ignored. Comment. 1.38. In Airfreight breakage Problem 1.21, the least squares estimates are bo = 10.20 and b = 4.00, ande = 17.60. Evaluate the least squares criterion Q in (1.8) for the estimates (1) bo = 9, b = 3; (2) bo = 11, b = 5. Is the criterion Q larger for these estimates than for the least squares estimates? 1.39. Two observations on Y were obtained at each of three X levels, namely, at X = 5, X = 10, and X = 15. a. Show that the least squares regression line fitted to the three points (5, ), (10, ), and (15, Y3), where , 2, and Y3 denote the means of the Y observations at the three X levels, is identical to the least squares regression line fitted to the original six cases. crime rate last year in counties with high school graduation percentage X= 80, (3) 510- (4) 0. 1.29. Refer to regression model (1.1). Assume that X = 0 is within the scope of the model. What is the implication for the regression function if fo = 0 so that the model is Y = BX, + &? How would the regression function plot on a graph? 1.30. Refer to regression model (1.1). What is the implication for the regression function if B = 0 so that the model is Y = Bo + 8? How would the regression function plot on a graph? 1.31. Refer to Plastic hardness Problem 1.22. Suppose one test item was molded from a single batch of plastic and the hardness of this one item was measured at 16 different points in time. Would the error term in the regression model for this case still reflect the same effects as for the experiment initially described? Would you expect the error terms for the different points in time to be uncorrelated? Discuss. 1.32. Derive the expression for b in (1.10a) from the normal equations in (1.9). 1.33. (Calculus needed.) Refer to the regression model Y = Bo + 8 in Exercise 1.30. Derive the least squares estimator of Bo for this model. 1.34. Prove that the least squares estimator of Bo obtained in Exercise 1.33 is unbiased. 1.35. Prove the result in (1.18)-that the sum of the Y observations is the same as the sum of the fitted values. 1.36. Prove the result in (1.20)-that the sum of the residuals weighted by the fitted values is zero. 1.37. Refer to Table 1.1 for the Toluca Company example. When asked to present a point estimate of the expected work hours for lot sizes of 30 pieces, a person gave the estimate 202 because this is the mean number of work hours in the three runs of size 30 in the study. A critic states that this person's approach "throws away" most of the data in the study because cases with lot sizes other than 30 are ignored. Comment. 1.38. In Airfreight breakage Problem 1.21, the least squares estimates are bo = 10.20 and b = 4.00, ande = 17.60. Evaluate the least squares criterion Q in (1.8) for the estimates (1) bo = 9, b = 3; (2) bo = 11, b = 5. Is the criterion Q larger for these estimates than for the least squares estimates? 1.39. Two observations on Y were obtained at each of three X levels, namely, at X = 5, X = 10, and X = 15. a. Show that the least squares regression line fitted to the three points (5, ), (10, ), and (15, Y3), where , 2, and Y3 denote the means of the Y observations at the three X levels, is identical to the least squares regression line fitted to the original six cases.