Formalise and prove, using your second formula, that on the n th cycle the value of r is the sum of the values of a on all the cycles up to the n th. You may assume that values are natural numbers and ignore the possibility of overflow. [8 marks] 11 Advanced Graphics Give a parametric definition of a torus centred at the origin and aligned with the coordinate axes. [4 marks] Give a casual verification that each practical exhibit should be addressed by a adjusted tree. You might expect that the cluster contains no holes, for example the dened addendums are 1; : : : ; n for some n > 0. [7 marks] Continuous Mathematics

The complex form of the Fourier series is:

where ck is a complex number and ck = ck.

(a) Prove that the complex coeffiffifficient, ck, encodes the amplitude and phase

coeffiffifficients, Ak and k, in the alternative form:

An interprocess communication environment is based on synchronous message

passing. A server is to be designed to support a moderate number of simultaneous

client requests.

Clients send a request message to the server, continue in parallel with server

operation, then wait for the server's reply message.

Discuss the design of the server's interaction with the clients. Include any problems

you foresee and discuss alternative solutions to them. [20 marks]

2CST.2001.4.3

3 Further Java

(a) Describe how mutual-exclusion locks provided by the synchronized keyword

can be used to control access to shared data structures. In particular you

should be clear about the behaviour of concurrent invocations of difffferent

synchronized methods on the same object, or of the same synchronized method

on difffferent objects. [6 marks]

(b) Consider the following class defifinition:

class Example implements Runnable {

public static Object o = new Object();

int count = 0;

public void run() {

while (true) {

synchronized (o) { Let the profundity of a tree be dened by depth(Lf) = 0 and depth(Br(v; t1; t2)) = 1 + maxfdepth(t1); depth(t2)g: State and legitimize exact lower and upper limits for the quantity of Br hubs contained in a fair tree of profundity d, for erratic d > 0. [6 marks] The solution needs to include a domain controller, a storage server, a failover cluster with at least two nodes, an application server running the service, a scenario of live migration of the application server from one cluster node to another, and a scenario of failover when the cluster node hosting the application server was down. In each of the following cases, establish whether the set defined is recursively enumerable: (a) the set of all recursive subsets of N [5 marks] (b) the set of all recursive sequences of natural numbers [2 marks] In the process of using functional dependencies to normalise a schema, what is meant by a lossless join decomposition and how is such a decomposition guaranteed? [4 marks] 6 CST.2000.4.7 9 Numerical Analysis I Define the absolute error x and relative error x in representing a number x. How are these errors related? Which type of error is associated with the term loss of significance? Define machine epsilon m. (d) (c) the set of all finite sequences of natural numbers [5 marks] Review that Reach is the issue of choosing, given a diagram G a source vertex s and an objective vertex t, whether G contains a way from s to t; and Sat is the issue of concluding whether a given Boolean recipe is satisable. (a) A two state Markov process emits the letters {A, B, C, D, E} with the probabilities shown for each state. Changes of state can occur when some of the symbols are generated, as indicated by the arrows.