To use it, the class is extended and a specialised void add(...) method should be provided for each acceptable type. public abstract class ConstrainedArray { protected ArrayList mArray = new ArrayList(); public Object get(int idx) {return mArray.get(idx);} public int size() { return mArray.size(); } } (i.i) Show how to a class IntListArray that extends this base class and accepts only Integer or LinkedList objects. Where appropriate, objects should be copied on insertion. [4 marks] (ii.i) Describe a sequence of events that would allow external modification of an object stored within an IntListArray, Give the transition rules from which transitions of the form P P 0 can be inferred, where is of the form a, a or . The rules should not use structural congruence (). [5 marks] Enumerate the ways in which a transition of the form P|Q R can be inferred from transitions of P and/or Q, and indicate the form of R in each case. [5 marks] Hence show that if P|Q R1, then there exists R2 such that Q|P R2 and R1 R2. [5 marks] Then, for a channel having the same bandwidth but whose signal-to-noise ratio P N0 is four times better, repeat your numerical estimate of capacity in bits per second. [5 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] (c) It is possible to sort an array of n elements using binary comparisons in (n log n) time. (d) It is possible to sort an array of n elements using binary comparisons using O(1) (i.e. constant independent of n) additional space. (e) Radix sorting can sort any set of integers in linear time. (f ) All straight lines from the inside of a polygon to the outside intersect the points on the edges forming its boundary an odd number of times. (g) It is always cheaper to find the shortest distance between two given nodes u, v of G than to find all N shortest distances from u to every other node. (h) It is possible to find the shortest paths between all N2 pairs of nodes of G in O(N3 ) time. (i) If G is connected then the minimal spanning subtree of G contains the N 1 edges whose weights are smallest. (j) Given n points (xi , yi), 1 6 i 6 n in a plane, then the four points (xa, ya), (xb, yb), (xc, yc), (xd, yd) such that xa is minimal of the xi , xb is maximal of the xi , yc is minimal of the yi , yd is maximal of the yi form a quadrilateral Q which can be used to speed up a convex hull algorithm by preprocessing to remove points which lie inside Q.