Let be a universal set and define a relation between subsets A, B…

Question Answered step-by-step Let be a universal set and define a relation between subsets A, B… Let Ω be a universal set and define a relation between subsets A, B ⊆ Ω by A ∼= B ⇔ ∃ a bijection f : A → B. Prove carefully that ∼= is an equivalence relation. [6 marks] What does it mean to say that a set is countable? [2 marks] State without proof the Schr¨oder-Bernstein theorem concerning the existence of a bijection between two sets. [2 marks] Show that the integers and the rational numbers are countable but that the real numbers are uncountable. [6 marks] An ML program consists of a finite sequence of characters drawn from a finite alphabet. Show that the set of ML programs is countable.converting A to a binary decision diagram (BDD) Briefly describe these alternative forms and state their respective advantages. [6 marks] (b) For each of the following formulas, present either a proof in a sequent or tableau calculus, or a falsifying interpretation. (i) [∀x ∃y Q(x, y)] ⇒ ∃y Q(y, y) [4 marks] (ii) [∀x (P(x) → ¬P(x))] ∧ [∃y P(y)] → ∃y Q(y) [4 marks][6 marks] 7 (TURN OVER) CST.2015.6.8 7 Mathematical Methods for Computer Science (a) An inner product space E containing piecewise continuous complex functions f(x) and g(x) on some interval is spanned by the orthonormal basis functions {ei} used in the Fourier series. Thus complex coefficients {αi} and {βi} exist such that f(x) = X i αiei(x) and g(x) = X i βiei(x). (i) Show that hf, gi = X i αiβi . [5 marks] (ii) Would the same result hold if the orthonormal basis functions {ei} that span E were not the Fourier basis? Justify your answer, and provide the name for coefficients {αi} and {βi} in such a case. In a specialist data communication network, digital data is sent between nodesusing air pressure in pipes. Each node has a small number of pipes leading to eachof its near neighbours. Each pipe may be either at atmospheric pressure or at someslightly higher pressure under control of, and detected by, transducers, fitted ateach end of every pipe. The time for a pressure change to propagate down a givenpipe depends on the length of the pipe and can vary from 1 to 10 milliseconds.Variations in the manufacturing of the transducers can sometimes add up to afurther millisecond to the time taken to change the pressure or detect that it haschanged.For peer-to-peer data transfer, nine pipes run between each connected neighbour.They are used as eight data pipes and one clock pipe, with data on the eight datapipes being latched into a broadside register at the receiving end on each positivepressure change on the clock pipe.(a) Design a timing diagram for reliable, continuous data transfer between a pairof nodes. [5 marks](b) What is the approximate, maximum, continuous data transfer rate achievablebetween adjacent nodes? (Assume all pipes between any pair of nodes areroughly the same length and allow a reasonable timing margin for safety.)[5 marks]A data packet structure is now imposed on the system, whereby packets of 8 bytesof data may be sent by a node to a neighbour from time to time. Pipes are usedbi-directionally, but only changing direction between packets. All pipes are to beat atmospheric pressure when there is no data to send.(c) Suggest how the start and end of a packet can be determined. [5 marks](d) What will happen if both ends of a link decide to transmit to each other atthe same time? Should an extra pipe be used to help solve this problem?[5 ma Answer all parts.(a) (i) Provide at least one argument in favour of computer cracking. [1 mark](ii) Name at least two of the three new offences created by the ComputerMisuse Act 1990. [2 marks](iii) State at least one of the two extensions of this basic law. [1 mark](b) What are the five stages of the Capability Maturity Model? [4 marks](c) In the context of loop design, what is a sentinel? Illustrate your answer bycoding a loop that finds the first non-negative number in an array. [4 marks](d) If a deterministic finite automaton accepts any string at all, why does it acceptone whose length is less than the number of states in the automaton?[4 marks](e) Give a logic symbol for a dual-port register file containing eight four-bitregisters. Be sure to name each input and output connection and give itswidth. What non-determinism can arise?(a) A 4-bit shift register constructed from edge-triggered D-type flip flops is shownbelow. If, on successive rising edges of the clock signal CLK, the input takeson the values 1, 0, 1, 0, 1, 1, 1, 0, what are the contents of the shift register aftereach edge of the clock? You may assume that the register contains all zeroesinitially.input D Q D Q D Q D Qx0 x1 x2 x3CLK[4 marks](b) Using a (possibly larger) shift register, show how one may detect a particularpattern in the input shown. As an example, use the 8-bit pattern 0xF0. Highorder bits precede low-order bits in the input stream. [4 marks](c) The input stream is framed by a one byte frame pattern (0xF0) every 256 bytes.However, the frame pattern may also appear at an arbitrary position in theinput stream.It is required to design a framing circuit which generates two outputs:framelock, which is asserted when the circuit “believes” it has determinedwhere the frame boundaries are, and frame pointer, which is asserted onthe clock edge immediately after the frame marker is detected. The circuit”believes” itself to be locked to the frame structure when two successive framepatterns have been found 256 bytes (i.e. 2048 bits) apart. The circuit shouldnot respond to unaligned frame patterns while it believes itself to be in lockor if, once in lock, it has missed fewer than two expected frame patterns.Draw a state diagram for the finite state control of the circuit. You may assumethe existence of an 11-bit resettable counter. You should consider the processof assuming lock, maintaining lock and the accommodation of a single missedframing pattern. State explicitly any additional assumptions you make.[10 marks](d) Outline the complexity in gates and flip flops for an implementation of theframing circuit. (a) One or more processors must be selected for inclusion in a portable device thathandles audio, image, video, telephone calls and simple word processing.Possible design combinations include:• a single processor;• a pair of basically similar processors but with different coprocessorand bus structures;• a standard processor and a custom VLIW processor.Explain whether one of these combinations is clearly the best. [2 marks](b) For each of the following processor technologies, justify whether it is applicablein the device:(i) Dynamically Scheduled Instruction Dispatch;(ii) Simultaneous Multi-Threading (SMT);(iii) SIMD extensions, similar to Intel’s MMX;(iv) Virtual Memory;(v) Dynamic Clock Frequency/Power Control;(vi) Snooping Cache. A majority gate with three inputs signals logic one on its output if two or more ofits inputs are one, and zero otherwise. A minority gate is its complement.(a) Give the boolean equation for a minority gate as a sum of products. [2 marks](b) Sketch the circuit diagram for a minority gate using NAND gates and inverters.[3 marks](c) Sketch the transistor-level circuit diagram for an alternative implementationas a single stage of CMOS logic. [3 marks](d) Calculate the number of transistors required for each implementation.[2 marks](e) Assuming that a conducting p-channel has a resistance twice that of a similarlysized n-channel, use logical effort to compare the performance of the twoimplementations.(a) The Transmission Control Protocol (TCP) employs a transmit windowand cumulative acknowledgement and timeout system, as well as sequencenumbering of packets, to achieve reliable delivery of data.(i) Outline the procedure for round-trip time estimation and the calculationof the retransmission timer. [8 marks](ii) Explain the function of buffering at the sender and receiver. [3 marks](iii) How do fast retransmit and fast recovery improve performance afterpacket loss? [3 marks](b) In a wireless network, delay to access the channel due to scheduling of themedia access control protocol, and random packet loss due to interference,may be non-negligible.(i) Explain how this can interfere with the round-trip time estimation processabove. [3 marks](ii) Explain how this can interfere with the congestion control scheme thatTCP employs.(a) Describe, with examples, the function of a naming service for a largescale distributed system. Include definitions for “name space” and “namingdomain”. [6 marks](b) Discuss consistency versus availability for naming data in large-scale systems.[4 marks](c) How can any distributed naming service be engineered so that invocations onbehalf of users can be resolved efficiently in the presence of failures and heavyload? [4 marks](d) Contrast the assumptions under which DNS was designed originally for theInternet, with the properties of dynamically formed groups of mobile hostsusing wireless communication (MANETS). How might DNS-like services beprovided for MANETS? [6 marks]5 Advanced Systems TopicsModern peer-to-peer (P2P) systems are typically described as structured orunstructured.(a) Compare and contrast these two approaches. Include a discussion of thegeneral topology, membership management and query mechanisms. Useexamples to support your answer. [6 marks](b) Which approach is more resilient to churn? Justify your answer. [2 marks](c) One early criticism of P2P systems was that they did not consider networklatencies. Describe how one can add proximity awareness to:(i) unstructured P2P systems; [1 mark](ii) structured P2P systems. [3 marks](d) Swarming P2P systems like BitTorrent are designed for efficient (and incentivecompatible) download of large files. However, it is typically not possible to usethe file until it has downloaded in its entirety. Sketch the design of a swarmingP2P system which supports streaming video – that is, allows playback of videoto overlap the ongoing download of the remainder of the stream. Commenton how efficient (in terms of network resources) your system would be incomparison with a system like BitTorrent. (a) Design a 2-bit multiplier for unsigned integers which takes input x1 x0representing the unsigned integer X, y1 y0 representing the unsigned integerY , and produces the output z3 z2 z1 z0 representing the unsigned integer Z.[4 marks](b) How can multipliers designed in part (a) be cascaded (with adders) to providea four-bit multiplier? [4 marks](c) Design a sequential 8-bit multiplier. You can assume that a 16-bit adder hasbeen provided. The finite state control can be described by a state diagram.[8 marks](d) Outline the design of a sequential divider which can divide 16-bit unsignedintegers by 8-bit unsigned integers.  Write  Java program to find the length of a given string S=”College”Java program to find the factorial of number. Input number should be given by user.) Write  Java program to read a file content line by line. (a) Explain the method of Active Contours. What are they used for, and how dothey work? What underlying trade-off governs the solutions they generate?How is that trade-off controlled? What mathematical methods are deployedin the computational implementation of Active Contours? [10 marks](b) When trying to detect and estimate visual motion in a scene, why is it usefulto relate spatial derivatives to temporal derivatives of the image data? Brieflydescribe how one motion model works by these principles. [5 marks](c) Provide a 3 × 3 discrete filter kernel array that approximates the Laplacianoperator. Explain what the Laplacian might be used for, and what is thesignificance of the sum of all of the taps in the filter. [3 marks](d) When visual sequences are encoded into an .mpeg video stream, typicallyabout what percentage of the compression achieved is intra-frame (compressionwithin individual still frames), and what percentage is inter-frame? Name akey feature that is extracted and estimated for purposes of prediction and,therefore, compression. [2 marks]7 SecurityThe Needham-Schroeder protocol is defined as1. A −→ S : A, B, NA2. S −→ A : {NA, B, KAB, {KAB, A}KBS }KAS3. A −→ B : {KAB, A}KBS4. B −→ A : {NB}KAB5. A −→ B : {NB − 1}KAB(a) Explain the symbolism, and the purpose of the messages. [5 marks](b) Explain the “bug” in the protocol. [5 marks](c) Is the bug actually a vulnerability if one can assume (as the Needham-Schroeder paper does) that all principals execute the protocol faithfully? Ifnot, why is it important? [5 marks](d) Describe how one modern protocol derived from Needham-Schroeder dealswith the issue.In order to test the integrity of a network of ducting, engineers have developed aninspection device which can be introduced at a node and which then finds its wayalong a length of ducting to an adjacent node.In a particular case, eight nodes are sited at the vertices (corners) of a cube and12 lengths of ducting are arranged along the edges of the cube.The inspection device is introduced at one node and equiprobably chooses one ofthe three lengths of ducting leading from that node for its first move. On arrivalat the adjacent node the device equiprobably chooses one of the three lengths ofducting leading from that node (including the length it has just inspected). Itcontinues in this fashion until the engineers stop its operation.kindly answer all the questions Computer Science Engineering & Technology C++ Programming CIVIL 123 Share QuestionEmailCopy link Comments (0)