...CARLETON UNIVERSITY SCHOOL OF COMPUTER SCIENCE WINTER 2013 COMP. 3803 - DISCRETE STRUCTURES: II ASSIGNMENT I DUE: FRIDAY JAN 25, 2013 __________________________________________________________________________________ Assignment Policy: Late assignments will not be accepted. You are expected to work on the assignments on your own. Past experience has shown conclusively that those who do not put adequate effort into the assignments do not learn the material and have a probability near 1 of doing poorly on the exams. Important note: When writing your solutions, you must follow the guidelines below. • • • • The answers should be concise, clear and neat. Make sure that your TA can read your solution. Please submit the solutions in the order of the problems, the solution to Problem 1, then to Problem 2 and so on. When presenting proofs, every step should be justified. Assignments should be stapled or placed in an unsealed envelope with your name and student number. Substantial departures from the above guidelines will not be graded. 1. 2. 3. 4. 5. Prove that the sum of n real rational numbers is rational if all of them are rational. Does the reverse hold true? What can you say about the rationality of the product of n rational numbers? Prove that if n is a positive integer, then n is odd if and only if 5n+6 is odd. Show by induction that n5 – n is divisible by 5 for all n ≥ 0. Show by induction that n2 – 1 is divisible by 8 whenever n is an odd positive integer. If possible...
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...Course Design Guide MTH/221 Version 1 1 Course Design Guide College of Information Systems & Technology MTH/221 Version 1 Discrete Math for Information Technology Copyright © 2010 by University of Phoenix. All rights reserved. Course Description Discrete (as opposed to continuous) mathematics is of direct importance to the fields of Computer Science and Information Technology. This branch of mathematics includes studying areas such as set theory, logic, relations, graph theory, and analysis of algorithms. This course is intended to provide students with an understanding of these areas and their use in the field of Information Technology. Policies Faculty and students/learners will be held responsible for understanding and adhering to all policies contained within the following two documents: University policies: You must be logged into the student website to view this document. Instructor policies: This document is posted in the Course Materials forum. University policies are subject to change. Be sure to read the policies at the beginning of each class. Policies may be slightly different depending on the modality in which you attend class. If you have recently changed modalities, read the policies governing your current class modality. Course Materials Grimaldi, R. P. (2004). Discrete and combinatorial mathematics: An applied introduction. (5th ed.). Boston, MA: Pearson Addison Wesley. Article References Albert, I. Thakar, J., Li, S., Zhang, R., & Albert, R...
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...Discrete geometry Apolinario G. Sanger III Submitted to : Professor Rody G. Balete MT-31 Chapter I The Problem and It’s Background Introduction: Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object. Discrete geometry has large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory,toric geometry, and combinatorial topology. Although polyhedra and tessellations have been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics studied were: the density of circle packings by Thue,projective configurations by Reye and Steinitz, the geometry of numbers by Minkowski, and map colourings by Tait, Heawood, and Hadwiger. László Fejes Tóth, H.S.M. Coxeter and Paul Erdős, laid the foundations of discrete geometry. "This is an introduction to the field of discrete geometry understood as the investigation...
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...Econometric Reviews, 27(1–3):10–45, 2008 Copyright © Taylor & Francis Group, LLC ISSN: 0747-4938 print/1532-4168 online DOI: 10.1080/07474930701853509 REALIZED VOLATILITY: A REVIEW Michael McAleer1 and Marcelo C. Medeiros2 2 School of Economics and Commerce, University of Western Australia Department of Economics, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Brasil 1 Downloaded At: 15:53 5 September 2008 This article reviews the exciting and rapidly expanding literature on realized volatility. After presenting a general univariate framework for estimating realized volatilities, a simple discrete time model is presented in order to motivate the main results. A continuous time specification provides the theoretical foundation for the main results in this literature. Cases with and without microstructure noise are considered, and it is shown how microstructure noise can cause severe problems in terms of consistent estimation of the daily realized volatility. Independent and dependent noise processes are examined. The most important methods for providing consistent estimators are presented, and a critical exposition of different techniques is given. The finite sample properties are discussed in comparison with their asymptotic properties. A multivariate model is presented to discuss estimation of the realized covariances. Various issues relating to modelling and forecasting realized volatilities are considered. The main empirical findings using...
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...Signal may be either continuous-time or discrete-time, with either analog or digital values [1]. The signals which are represented by a continuous function are called continuous signals and those which are described by number sequences are called discrete signals [2]. We have seen about a signal in brief. The second component in signal processing is a system which is a process whose input and output are signals. Signal processing is a vast area comprising the concepts of electrical engineering, systems engineering and applied mathematics that deals with both the analog and discrete time signals, represented by variation in time or spatial physical quantities. Precise statistical depiction is required for the development of improved signal processing algorithms of natural signals [3]. The major operations of Signal processing includes 1) signal acquisition and reconstruction, 2) Quality improvement including filtering, smoothing and digitization, 3) feature extraction 4) signal compression and 5) prediction [4] [5]. Analog signal processing, Discrete-time signal processing, Non-linear signal processing and Digital signal processing are the four major categories of signal processing. The signal processing performed over analog signals for the purpose of any of the major operations of signal processing is known to be analog signal processing and the same concept is applied for discrete-time signal processing, where the only difference is discrete signal is employed. An analog signal...
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...CHAPTER 1 1.1 to 1.41 - part of text 1.42 (a) Periodic: Fundamental period = 0.5s (b) Nonperiodic (c) Periodic Fundamental period = 3s (d) Periodic Fundamental period = 2 samples (e) Nonperiodic (f) Periodic: Fundamental period = 10 samples (g) Nonperiodic (h) Nonperiodic (i) Periodic: Fundamental period = 1 sample l.43 π 2 y ( t ) = 3 cos 200t + -- 6 2 π = 9 cos 200t + -- 6 9 π = -- cos 400t + -- 1 2 3 9 (a) DC component = -2 9 π (b) Sinusoidal component = -- cos 400t + -- 2 3 9 Amplitude = -2 1 200 Fundamental frequency = -------- Hz π 1.44 The RMS value of sinusoidal x(t) is A ⁄ 2 . Hence, the average power of x(t) in a 1-ohm 2 resistor is ( A ⁄ 2 ) = A2/2. 1.45 Let N denote the fundamental period of x[N]. which is defined by 2π N = ----Ω The average power of x[n] is therefore N -1 1 2 P = --- ∑ x [ n ] N 1 = --N n=0 N -1 ∑A n=0 2 N -1 A = ----N 1.46 2 2 2πn cos --------- + φ N ∑ cos n=0 2 2πn + φ -------- N The energy of the raised cosine pulse is E = π⁄ω 1 ∫–π ⁄ ω -- ( cos ( ωt ) + 1 ) 4 2 dt 1 π⁄ω 2 = -- ∫ ( cos ( ωt ) + 2 cos ( ωt ) + 1 ) dt 2 0 1 π ⁄ ω 1 1 -- cos ( 2ωt ) + -- + 2 cos ( ωt ) + 1 dt = -- ∫ 2 0 2 2 1 3 π - - = -- -- --- = 3π ⁄ 4ω 2 2 ω 1.47 The signal x(t) is even; its total energy is therefore 5 2 E = 2 ∫ x ( t ) dt 0 2 4 5 = 2 ∫ ( 1 ) dt + 2 ∫ ( 5 – t ) dt 2 0 2 4 5 1 4 3 = 2 [ t ] t=0 + 2 – -- ( 5 – t ) 3 t=4 2 26 = 8 + -- = ----3 3 1.48 (a)...
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...PRE LAB #2 % This script creates a signal, and then quantizes it to a specified number % of bits. It then calculates the quantization error. clc clear all close all Change the #of bits means change in resolution Change the #of bits means change in resolution fprintf(' Sampling and Quantization\n'); b=5; % Number of bits. N=100; % Number of samples in final signal. n=0:(N-1); %Index Change the value of N means change in sampling rate of sig Change the value of N means change in sampling rate of sig % Choose the input type. choice = questdlg('Choose input','Input',... 'Sine','Sawtooth','Random','Random'); fprintf('Bits = %g, levels = %g, signal = %s.\n', b, 2^b, choice); % Create the input data sequence. switch choice case 'Sine' x=sin(2*pi*n/N); case 'Sawtooth' x=sawtooth(2*pi*n/N); case 'Random' x=randn(1,N); % Random data x=x/max(abs(x)); % Scale to +/- 1 end % Signal is restricted to between -1 and +1. x(x>=1)=(1-eps); % Make signal from -1 to just less than 1. x(x<-1)=-1; % Quantize a signal to "b" bits. xq=floor((x+1)*2^(b-1)); % Signal is one of 2^n int values (0 to 2^n-1) xq=xq/(2^(b-1)); % Signal is from 0 to 2 (quantized) xq=xq-(2^(b)-1)/2^(b); % Shift signal down (rounding) xe=x-xq; % Quantization...
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...decrypt the message.This system doesn’t require secure key transmission.So, it resolves the one of the problem faced by symmetric key cryptosystem. If someone is able to compute respective private key from a given public key, then this system is no more secure. So, Public key cryptosystem requires that calculation of respective private key is computationally impossible from given public key. In most of the Public key cryptosystem, private key is related to public key via Discrete Logarithm. Examples are Diffie-Hellman Key Exchange, Digital Signature Algorithm (DSA), Elgamal which are based on DLP in finite multiplicative group. 1 2. Discrete logarithm problem The Discrete Logarithm Problem (DLP)is the problem of finding an exponent x such that g x ≡ h (mod p) where, g is a primitive root for Fp and h is a non-zero element of Fp . Let, n be the order of g. Then solution x is unique up to multiples of n and x is called discrete logarithm of h to the base g (i.e.) x = logg h. In cryptosystem based on Discrete Logarithm , x is used...
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...• Ch. 1 of Discrete and Combinatorial Mathematics o Supplementary Exercises 1, 2, 7, & 8 1. In the manufacture of a certain type of automobile, four kinds of major defects and seven kinds of minor defects can occur. For those situations in which defects do occur, in how many ways can there be twice as many minor defects as there are major ones? 2. A machine has nine different dials, each with five settings labeled 0, 1, 2, 3, and 4. a) In how many ways can all the dials on the machine be set? b) If the nine dials are arranged in a line at the top of the machine, how many of the machine settings have no two adjacent dials with the same setting? 7. There are 12 men at a dance. (a) In how many ways can eight of them be selected to form a cleanup crew? (b) How many ways are there to pair off eight women at the dance with eight of these 12 men? 8. In how many ways can the letters in WONDERING be arranged with exactly two consecutive vowels? • Ch. 2 of Discrete and Combinatorial Mathematics o Exercise 2.1, problems 2 o Exercise 2.2, problems 3 o Exercise 2.4, problems 1 o Exercise 2.5, problems 1 2. Identify the primitive statements in Exercise 1 below: Exercise 1. Determine whether each of the following sentences is a statement. a) In 2003 GeorgeW. Bush was the president of the United States. b) x + 3 is a positive integer. c) Fifteen is an even number. d) If Jennifer is late for the party, then her cousin Zachary will be quite angry...
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...Week 17 : Signal Processing Test - Attempt #1 Top of Form |Time Remaining: [pic] | | |Page: 1 2 | Page 1 [pic] |Question 1. 1. (TCO 3) What is the expression for the transform admittance for an unfluxed inductance of 4H? (Points : 6) | | | | [pic] 0.25/s | | [pic] 4/s | | [pic] 0.25s | | [pic] 4s | | | | ...
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...[pic] DATA COMMUNICATION THEORY LAB REPORT Report on Digital Transmission of Analogue Signals Pulse Amplitude Modulation INTRODUCTION: The experiment uses the L.J.Electronics Modicom-1 board to investigate the sampling of signals, and the filter effects on the reconstruction of the original signal from the sampled input. The sample/Hold operation is also investigated in this experiment. The Modicum -1 board allows an analogue signal to be sampled at a number of different rate(2kHz,4kHZ,8kHZ,16kHz,32kHz).The pulse width is varied in the steps of 0% from 0 to 90 of the sampling interval. Second order and fourth order low pass filter are available with cut-off frequencies set at 3.4 kHz. BACKGROUND: L.J.Electronics MODICOM-1 board is used to investigate the sampling of signals, and the effects of filter on the reconstruction of the original signal from the sampled input. In addition to this Sample/Hold operation is also investigated. [pic] MODICOM-1 board The MODICOM-1 board is considered as five different blocks namely 1.) Power Input 2.) Sampling Control Logic 3.) Sampling Circuit 4.) Second Order Low Pass Filter and 5.) Fourth Order Low Pass Filter Sampling Control logic: It is used to generate the timing and control signals that sample the input waveform, and also creates a sinusoidal...
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...* MTH/221 Week Four Individual problems: * * Ch. 11 of Discrete and Combinatorial Mathematics * Exercise 11.1, problems 8, 11 , text-pg:519 Exercise 11.2, problems 1, 6, text-pg:528 Exercise 11.3, problems 5, 20 , text-pg:537 Exercise 11.4, problems 14 , text-pg:553 Exercise 11.5, problems 7 , text-pg:563 * Ch. 12 of Discrete and Combinatorial Mathematics * Exercise 12.1, problems 11 , text-pg:585 Exercise 12.2, problems 6 , text-pg:604 Exercise 12.3, problems 2 , text-pg:609 Exercise 12.5, problems 3 , text-pg:621 Chapter 11 Exercise 11.1 Problem 8: Figure 11.10 shows an undirected graph representing a section of a department store. The vertices indicate where cashiers are located; the edges denote unblocked aisles between cashiers. The department store wants to set up a security system where (plainclothes) guards are placed at certain cashier locations so that each cashier either has a guard at his or her location or is only one aisle away from a cashier who has a guard. What is the smallest number of guards needed? Figure 11.10 Problem 11: Let G be a graph that satisfies the condition in Exercise 10. (a) Must G be loop-free? (b) Could G be a multigraph? (c) If G has n vertices, can we determine how many edges it has? Exercise 11.2 Problem 1: Let G be the undirected graph in Fig. 11.27(a). a) How many connected subgraphs ofGhave four vertices and include a cycle? b) Describe the...
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...Selected Textbook Exercises (UOP Course) For more course tutorials visit www.tutorialrank.com Tutorial Purchased: 3 Times, Rating: A+ Mathematics - Discrete Mathematics Complete 12 questions below by choosing at least four from each section. • Ch. 1 of Discrete and Combinatorial Mathematics o Supplementary Exercises 1, 2, 7, 8, 9, 10, 15(a), 18, 24, & 25(a & b) • Ch. 2 of Discrete and Combinatorial Mathematics o Exercise 2.1, problems 2, 3, 10, & 13, o Exercise 2.2, problems 3, 4, & 17 o Exercise 2.3, problems 1 & 4 o Exercise 2.4, problems 1, 2, & 6 o Exercise 2.5, problems 1, 2, & 4 • Ch. 3 of Discrete and Combinatorial Mathematics o Exercise 3.1, problems 1, 2, 18, & 21 o Exercise 3.2, problems 3 & 8 ----------------------------------------------- MTH 221 Week 1 Individual Assignment Selected Textbook Exercises (UOP Course) For more course tutorials visit www.tutorialrank.com Tutorial Purchased:2 Times, Rating: No rating Complete the six questions listed below: • Ch. 1 of Discrete and Combinatorial Mathematics o Supplementary Exercise 2 • Ch. 2 of Discrete and Combinatorial Mathematics o Exercise 2.1, problem 10; p 54 o Exercise 2.2, problem 4; p 66 o Exercise 2.3, problem 4; p 84 • Ch. 3 of Discrete and Combinatorial Mathematics o Exercise 3.1, problem 18; p 135 ----------------------------------------------- MTH 221 Week 2 Individual and Team Assignment Selected...
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...ISBN 0-558-83970-3 Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, by Ralph P. Grimaldi. Published by Addison Wesley. Copyright © 2004 by Pearson Education, Inc. ISBN 0-558-83970-3 Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, by Ralph P. Grimaldi. Published by Addison Wesley. Copyright © 2004 by Pearson Education, Inc. ISBN 0-558-83970-3 Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, by Ralph P. Grimaldi. Published by Addison Wesley. Copyright © 2004 by Pearson Education, Inc. ISBN 0-558-83970-3 Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, by Ralph P. Grimaldi. Published by Addison Wesley. Copyright © 2004 by Pearson Education, Inc. ISBN 0-558-83970-3 Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, by Ralph P. Grimaldi. Published by Addison Wesley. Copyright © 2004 by Pearson Education, Inc. ISBN 0-558-83970-3 Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, by Ralph P. Grimaldi. Published by Addison Wesley. Copyright © 2004 by Pearson Education, Inc. ISBN 0-558-83970-3 Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, by Ralph P. Grimaldi. Published by Addison Wesley. Copyright © 2004 by Pearson Education, Inc. ISBN 0-558-83970-3 Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition...
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...price? This is an important question. If not then there is risk-free money to be made. If C < 50p buy it and hedge to make pro…t. If C > 50p sell it and hedge, make a guaranteed pro…t. Supply and demand should make this price converge to 50p. How do I know to sell 1/2 the stock for hedging (and not another value)? means the amount of stock sold for hedging purposes. The right choice for hedging means that the value of the portfolio does not depend on the direction of the stock. Earlier we had 1 101 = 99 1 0 = = 101 99 1 2 Note it is purely a coincident in this example that delta has the same value as the option. Note = V+ S+ V S = Range of option payo¤s Range of stock prices This model is discrete time, discrete stock. When we go to continuous time continuous stock delta will become @V : @S How does this change if interest rates are non-zero? Everything is as before but we now have a discount factor. Consider the earlier example but with r = 10% over one day, i.e. 1 1 = 1 + rt 1 + 0:1=252 0:9996 Now discount tomorrow’ value to get to todays s V 0:5 100 = 0:5 99 V = 0:51963 0:9996 So the portfolio value today must be the Present Value of the portfolio value tomorrow. Consider a portfolio asset price Si : ; long an option and short assets. Vi denotes the option value corresponding to S+ V+ S0 V0 S 0 V = V0 S0 At time T there are two possible outcomes: + Choosing so ...
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