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(a) In the quantum teleportation protocol, Alice and Bob are every in possession of one qubit of a couple in the joint country00i + statei. Explain how the protocol works. In precise, display that it involves the transmission of precisely two classical bits of information from Alice to Bob and reveal how, at the quit of the protocol, Bob is in possession of a qubit in country

Perlin Noise, including how it differs from white noise. [3 marks] (ii) Explain Barycentric Co-ordinates, including how they are calculated. Diagrams are encouraged. [3 marks] (c) Given a ray R(t) = O + Dt and a unit sphere S, initially centred on the origin and subsequently transformed by affine matrix M, where M represents the transformation of the sphere from local to world coordinates: (i) state the centre of the sphere in local co-ordinates and in world co-ordinates; (ii) give an expression in terms of t for the local co-ordinates of the intersections between R and S; (iii) give an expression for the world co-ordinates of the same intersections; and (iv) give an expression for the world co-ordinates of the normal at those intersections.

(a) Describe the differences in complexity and usage between parsimony and distance phylogenetic methods. Give an example of the usage of both methods. [6 marks] (b) Describe the differences in complexity and usage between hierarchical clustering and the Markov clustering (MCL) algorithm. [5 marks] (c) Explain how to identify different gene features using Hidden Markov Model methods such as Genescan. [5 marks] (d) Explain how you could identify a regulatory network involving a set of genes. [4 mark

(a) State the Fermat-Euler theorem, carefully defining any terms that you use. Deduce that 2p 2 (mod p) for any prime p. [5 marks] (b) Explain how this result can be used to show that a number is composite without actually finding a factor. Give an example. [3 marks] (c) Let Mm = 2m 1 be the mth Mersenne number. Suppose that m is composite. Prove that Mm is composite. [3 marks] (d) A composite number m that satisfies 2m 2 (mod m) is known as a pseudo-prime. (i) Suppose that m is prime. Prove that Mm is either prime or a pseudoprime. [3 marks] (ii) Suppose that m is a pseudo-prime. Prove that Mm is a pseudo-prime. [3 marks] (iii) Deduce that there are infinitely many pseudo-primes.

(a) Give the two definitions of the complexity class NP, one using the term Turing machine and one using the term verifier. [4 marks] (b) For each of the following statements, state whether it is true, false or unknown. In each case, give justification for your answer. In particular, if the truth statement is unknown, state any implications that might follow from it being true or false. [2 marks each] (i) 3SAT P CLIQUE (ii) TSP P (iii) NL P (iv) PSPACE 6= NPSPACE (c) Let = {0, 1}. Prove that and {0, 1} are the only languages in P which are not complete for P with respect to polynomial-time reductions.

(a) State precisely what it means for a language (i) to be co-NP-complete, (ii) to be in NL and (iii) to be in PSPACE. [6 marks] (b) Consider the following two decision problems. Problem 1: Given an undirected graph G = (V, E) with |V | even, does G contain a clique with at least |V |/2 vertices? Problem 2: Given an undirected graph G = (V, E), does G contain a clique with at least |V | 3 vertices? (i) Which of the two problems is in P and which one is NPcomplete? [2 marks] (ii) For the problem in P, describe a polynomial-time algorithm. [4 marks] (iii) For the other problem, prove that it is NP-complete. [8 marks] 3 (TURN OVER) CST.2014.6.410 Designing Interactive Applications Some of today's photocopiers are connected by networks to repair centres so that technicians can monitor their performance and detect problems without visiting customer premises. Although this offers cost savings, it can have a negative impact on customer relations. Suggest an explanation for this, drawing on your knowledge of the service technician's job. [5 marks] You have been asked to design a modification to a networked photocopier, to enable users to send messages to the repair centre when they encounter problems. Again drawing on your understanding of the nature of photocopier repair work, produce a rough design for the message-system user interface. Include a one-sentence problem statement, a mental-model description and an outline of the design of the user interface itself. [15 marks] 11 Introduction to Functional Programming Describe how recursive definitions are modelled in the -calculus using Y. [6 marks] Consider the following attempt to define Y in ML: val Y = fn f => (fn x => f (x x)) (fn x => f (x x)) Explain why this doesn't work. [6 marks] Give a satisfactory definition of Y in ML and illustrate its use by defining the factorial function. [4 + 4 marks] 12 Computer Vision Discuss the problem of face recognition and face detection based on wavelet encodings of facial structure and facial features. How can one distinguish between those facial undulations that are generic (universal, or normally present in all faces), and those which are particular to a given face and which therefore distinguish it from others? How can statistical decision theory formalize these two pattern recognition problems - face detection and face recognition? What are the main advantages and disadvantages of using wavelets for the encoding of faces? [20 marks] CST.96.12.7 13 Complexity Theory (a) Show that the problem 3-SAT is at least as hard to solve as n-SAT. [5 marks] (b) Show that the task of finding a minimum cost closed circuit in a weighted directed graph (a Travelling Salesman Problem of the minimization variety) is at least as hard as the Hamiltonian Circuit Problem. [5 marks] (c) Show that the class NP-complete is contained in the class P-space. [5 marks] (d) Show that the class P-space is contained in the class EXP-time. [5 marks] In each case ensure that your answer makes it clear what the problems and classes involved are. Standard results do not need to be proved provided they are clearly stated. 14 Numerical Analysis II In Peano's theorem, if a quadrature rule integrates polynomials of degree N exactly over an interval [a,b], then the error in integrating f CN+1[a,b] is conventionally expressed as where Explain the notation ( and Ex. [3 marks] It follows directly from Taylor's theorem that . Explain clearly, in simple stages, how to complete the proof of Peano's theorem. [8 marks] For the mid-point rule, what is N? If K(t) does not change sign in [a,b] then [1 mark] What is a branch delay slot and why does it arise? [7 marks] How can branch delays be avoided? [7 marks] If a processor exhibited one branch delay slot how would you reorder (and possibly modify) the instructions in the following loop to gain a performance advantage? loop ldr r2,r3,#4 % r2=load(r3), r3=r3+4 add r4,r4,r2 % r4=r4+r2 add r1,r1,#1 % r1=r1+1 cmp r1,#10 % compare r1 with constant 10 bne loop % branch if not equal to loop [6 marks] 2 Computer Architecture Write short notes on each of the following parameters of cache design: (a) size [4 marks] (b) mapping function [4 marks] (c) replacement algorithm [4 marks] (d) write policy [4 marks] (e) block size [4 marks] CST.96.12.2 3 Digital Communication I Hosts X and Y are communicating through the data network provided by the switches A, B, C and D and the links interconnecting them as shown above. Initially all packets are travelling through switches A, C and D. (a) A packet is corrupted on the link between C and D. Describe the events that take place to recover from the error when (i) an end to end flow and error control protocol is in operation [5 marks] (ii) flow and error control are performed on a hop by hop basis (b) Switch C fails. Describe the events that follow to recover when [5 marks] (i) the network is a datagram network [5 marks] (ii) the network is connection oriented [5 marks] 4 Graphics Consider the control of detail in a curve represented as a sequence of straight line segments. Describe Douglas and Peucker's algorithm for removing superfluous points. [10 marks] Describe how Overhauser interpolation can be used to introduce additional points. [10 marks]

C; conversely a variable of type D can be assigned a value of type C using a cast. By considering storage layouts, explain why the former assignment is always valid and the latter sometimes invalid. [4 marks] (c) A new programming language has the notion of "statically scoped exceptions" in which the program exception foo; void f() { try { void g() { raise foo; } try { g(); } except (foo) { C2 } } except (foo) { C1 } } would execute C1 rather than C2 as the former was in scope at the raise point. By analogy with statically scoped variables, or otherwise, explain how such exceptions might be implemented on a stack. [10 marks]

In the following, N is a feedforward neural network architecture taking a vector xT = ( x1 x2 xn ) of n inputs. The complete collection of weights for the network is denoted w and the output produced by the network when applied to input x using weights w is denoted N(w,x). The number of outputs is arbitrary. We have a sequence s of m labelled training examples s = ((x1,l1),(x2,l2),...,(xm,lm)) where the li denote vectors of desired outputs. Let E(w;(xi,li)) denote some measure of the error that N makes when applied to the ith labelled training example. Assuming that each node in the network computes a weighted summation of its inputs, followed by an activation function, such that the node j in the network computes a function g w(j) 0 + k X i=1 w(j) i input(i)! of its k inputs, where g is some activation function, derive in full the backpropagation algorithm for calculating the gradient E w = E w1 E w2 E wW T for the ith labelled example, where w1,...,wW denotes the complete collection of W weights in the network.[20 marks]

30940919 (a) Dene the operators of the core relational algebra. [5 marks] (b) Let R be a relation with schema (A1,...,An,B1,...,Bm) and S be a relation with schema (B1,...,Bm). The quotient of R and S, written RS, is the set of tuples t over attributes (A1,...,An) such that for every tuple s in S, the tuple ts (i.e. the concatenation of tuples t and s) is a member of R. Dene the quotient operator using the operators of the core relational algebra. [8 marks] (c) The core relational algebra can be extended with a duplicate elimination operator, and a grouping operator. (i) Dene carefully these two operators. [3 marks] (ii) Assuming the grouping operator, show how the duplicate elimination operator is, in fact, unnecessary. [2 marks] (iii) Can the grouping operator be used to dene the projection operator? Justify your answer. [2 marks] 1. Define analog transmission. 2. Define carrier signal and its role in analog transmission. 3. Describe digital-to-analog conversion. 4. Which characteristics of an analog signal are changed to represent the digital signal in each of the following digital-to-analog conversion? a. ASK b. FSK c. PSK d. QAM 5. Which of the four digital-to-analog conversion techniques (ASK, FSK, PSK or QAM) is the most susceptible to noise? Juxtapose your answer. 6. Define constellation diagram and its role in analog transmission. 7. What are the two components of a signal when the signal is represented on a constellation diagram? Which component is shown on the horizontal axis? Which is shown on the vertical axis?

In the late 90s it was observed that the relative price of equipment (capital) has declined at an average annual rate of more than 3 percent. There has also been a negative correlation (-0.46) between the relative price of new equipment and new equipment investment. This can be interpreted as evidence that there has been significant technological change in the production of new equipment. Technological advances have made equipment less expensive, triggering increases in the accumulation of equipment both in the short and long run. Concrete examples in support of this interpretation abound: new and more powerful computers, faster and more efficient means of telecommunication and transportation, robotization of assembly lines, and so on. In this problem we are going to extend the Solow Growth Model to allow for such investment specific technological progress. Start with the standard Solow model with population growth and assume for simplicity that the production function is Cobb-Douglas: Yt = Kt L1t , where the population growth rate is deltaLt/Lt= n. Similarly, just as in the basic model, assume that investment and consumption are constant fractions of output It = sYt and Ct = (1 s)Yt. However, assume that the relationship between investment and capital accumulation is modified to:

Kt+1 Kt = qtIt Kt

where the variable qt represents the level of technology in the production of capital equipment and grows at an exogenously given rate , i.e. deltaqt / qt= . Intuitively, when qt is high, the same investment expenditure translates into a greater increase in the capital stock. (Note: another way to interpret qt is as the inverse of the relative price between machinery and output: when qt is high, machinery is relatively cheaper). (a) Transform the model (the production function, the equations for consumption and investment, and the capital accumulation equation) in per-worker form.

c) Suppose that capital per worker kt grows at a constant rate (we do not know that yet, but we will make a guess). Divide the capital accumulation equation by kt and use this assumption to prove that qtk1t has to be constant over time.

Suppose that there are three states of the world, a, b, and c. The probabilities of the three states are 1 = 0.25, 2 = 0.5, and 3 = 0.25. Let A, B, and C denote the Arrow-Debreu securities that pay $1 in states a, b, and c, respectively. That is, A = (1,0,0), B = (0,1,0) and C = (0,0,1). Let pA = 0.4, pB = 0.5 and pC = 0.2 denote the prices of A, B, and C.

Consider a security X which is worth $2 in state a, $3 in state b, and $1 in state c. If there are liquid markets for A, B, C and X, what is the price of X?

a) Suppose that the US is currently in a balance of payments equilibrium and receives shocks that reduce both Y1 and Y2 by 3%. In the intertemporal model, the US balance of payments moves into ___ and world prices P* ___.

deficit, rise 3 Computation Theory (a) Explain how to code register machine programs P as numbers pPq N so that each e N can be decoded to a unique register machine program prog(e). [10 marks] (b) Find a number e1 N for which prog(e1) is a register machine program for computing the function one N N with one(x) = 1 for all x N. [2 marks] (c) Why is it important for the theory of computation that the functions involved in the coding and decoding given in part (a) are themselves register machine computable? (You are not required to prove that they are computable.)