... | | |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...
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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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...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 Textbook...
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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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...• 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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...Complete 12 questions below by choosing at least four from each section. · Ch. 11 of Discrete and Combinatorial Mathematics o Exercise 11.1, problems 3, 6, 8, 11, 15, & 16 · Ch. 11 of Discrete and Combinatorial Mathematics o Exercise 11.2, problems 1, 6, 12, & 13, o Exercise 11.3, problems 5, 20, 21, & 22 o Exercise 11.4, problems 14, 17, & 24 o Exercise 11.5, problems 4 & 7 o Exercise 11.6, problems 9 &10 · Ch. 12 of Discrete and Combinatorial Mathematics o Exercise 12.1, problems 2, 6, 7, & 11 o Exercise 12.2, problems 6 & 9 o Exercise 12.3, problems 2 & 3 * Exercise 12.5, problems 3 & 8 Section 11.1 3). For the graph in Fig. 11.7, how many paths are there from b to f ? 1). b a c d e g f 2). b a c d e f 3). b c d e g f 4). b c d e f 5). b e g f 6). b e f This would = 6 ways. 6). If a, b are distinct vertices in a connected undirected graph G, the distance from a to b is defined to be the length of a shortest path from a to b (when a = b the distance is defined to be 0). For the graph in Fig. 11.9, find the distances from d to (each of) the other vertices in G. d to e = 1 d to f = 1 d to c = 1 d to k = 2 d to g = 2 d to h = 3 d to j = 3 d to l = 3 d to m = 3 d to i = 4 Section 11.2 1). Let G be the undirected graph in Fig. 11.27(a). a) How many connected subgraphs of G have four vertices and include a cycle? 3 b) Describe the subgraph G1 (of...
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...Summary of Research Results One of the fundamental results of the project is a comprehensive 150-page survey article (Chen, Potts & Woeginger, 1998) published in the Handbook of Combinatorial Optimization. In this article, we described and explained the existing analytical as well as empirical approaches to solving sequencing and scheduling problems. This article has been cited by an increasing number of researchers in their work on algorithmic aspects of sequencing and scheduling. Chen (1999) surveys research on efficient on-line scheduling algorithms and their competitiveness with off-line algorithms, where on-line algorithms differ fundamentally from off-line ones: In on-line problems, partial solutions are required before all problem data are available and these data are made gradually available over time. Decisions based on partial solutions are irrevocable due to the passage of time. In another article (Chen, 2000a) published in the Encyclopaedia of Mathematics, both experience-based approach and sciencebased approach have been described and commented. One of the algorithms strikes an excellent balance between efficiency and effectiveness and has been used as a touchstone for more involved approaches. At a more basic level, several smaller articles (Chen, 2000b) to be published in The Informed Student Guide to the Management Sciences provide beginners in management sciences with basic knowledge of quantitative methods and analysis. All of the aforementioned articles have...
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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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...Chapter 1 Discrete Probability Distributions 1.1 Simulation of Discrete Probabilities Probability In this chapter, we shall first consider chance experiments with a finite number of possible outcomes ω1 , ω2 , . . . , ωn . For example, we roll a die and the possible outcomes are 1, 2, 3, 4, 5, 6 corresponding to the side that turns up. We toss a coin with possible outcomes H (heads) and T (tails). It is frequently useful to be able to refer to an outcome of an experiment. For example, we might want to write the mathematical expression which gives the sum of four rolls of a die. To do this, we could let Xi , i = 1, 2, 3, 4, represent the values of the outcomes of the four rolls, and then we could write the expression X 1 + X 2 + X 3 + X4 for the sum of the four rolls. The Xi ’s are called random variables. A random variable is simply an expression whose value is the outcome of a particular experiment. Just as in the case of other types of variables in mathematics, random variables can take on different values. Let X be the random variable which represents the roll of one die. We shall assign probabilities to the possible outcomes of this experiment. We do this by assigning to each outcome ωj a nonnegative number m(ωj ) in such a way that m(ω1 ) + m(ω2 ) + · · · + m(ω6 ) = 1 . The function m(ωj ) is called the distribution function of the random variable X. For the case of the roll of the die we would assign equal probabilities or probabilities 1/6 to each of the outcomes....
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...An Iterated Dynasearch Algorithm for the Single-Machine Total Weighted Tardiness Scheduling Problem Faculty of Mathematical Studies, University of Southampton, Southampton, SO17 1BJ, UK Faculty of Mathematical Studies, University of Southampton, Southampton, SO17 1BJ, UK Department of Decision and Information Sciences, Rotterdam School of Management, Erasmus University, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands Richard.Congram@paconsulting.com • C.N.Potts@maths.soton.ac.uk • S.Velde@fac.fbk.eur.nl Richard K. Congram • Chris N. Potts • Steef L. van de Velde T his paper introduces a new neighborhood search technique, called dynasearch, that uses dynamic programming to search an exponential size neighborhood in polynomial time. While traditional local search algorithms make a single move at each iteration, dynasearch allows a series of moves to be performed. The aim is for the lookahead capabilities of dynasearch to prevent the search from being attracted to poor local optima. We evaluate dynasearch by applying it to the problem of scheduling jobs on a single machine to minimize the total weighted tardiness of the jobs. Dynasearch is more effective than traditional first-improve or best-improve descent in our computational tests. Furthermore, this superiority is much greater for starting solutions close to previous local minima. Computational results also show that an iterated dynasearch algorithm in which descents are performed a few random moves away from previous...
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...Natural Computing Series Series Editors: G. Rozenberg Th. Bäck A.E. Eiben J.N. Kok H.P. Spaink Leiden Center for Natural Computing Advisory Board: S. Amari G. Brassard K.A. De Jong C.C.A.M. Gielen T. Head L. Kari L. Landweber T. Martinetz Z. Michalewicz M.C. Mozer E. Oja G. P˘ un J. Reif H. Rubin A. Salomaa M. Schoenauer H.-P. Schwefel C. Torras a D. Whitley E. Winfree J.M. Zurada For further volumes: www.springer.com/series/4190 Franz Rothlauf Design of Modern Heuristics Principles and Application Prof. Dr. Franz Rothlauf Chair of Information Systems and Business Administration Johannes Gutenberg Universität Mainz Gutenberg School of Management and Economics Jakob-Welder-Weg 9 55099 Mainz Germany rothlauf@uni-mainz.de Series Editors G. Rozenberg (Managing Editor) rozenber@liacs.nl Th. Bäck, J.N. Kok, H.P. Spaink Leiden Center for Natural Computing Leiden University Niels Bohrweg 1 2333 CA Leiden, The Netherlands A.E. Eiben Vrije Universiteit Amsterdam The Netherlands ISSN 1619-7127 Natural Computing Series ISBN 978-3-540-72961-7 e-ISBN 978-3-540-72962-4 DOI 10.1007/978-3-540-72962-4 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011934137 ACM Computing Classification (1998): I.2.8, G.1.6, H.4.2 © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations...
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...MATH 55 SOLUTION SET—SOLUTION SET #5 Note. Any typos or errors in this solution set should be reported to the GSI at isammis@math.berkeley.edu 4.1.8. How many different three-letter initials with none of the letters repeated can people have. Solution. One has 26 choices for the first initial, 25 for the second, and 24 for the third, for a total of (26)(25)(24) possible initials. 4.1.18. How many positive integers less than 1000 (a) are divisible by 7? (b) are divisible by 7 but not by 11? (c) are divisible by both 7 and 11? (d) are divisible by either 7 or 11? (e) are divisible by exactly one of 7 or 11? (f ) are divisible by neither 7 nor 11? (g) have distinct digits? (h) have distinct digits and are even? Solution. (a) Every 7th number is divisible by 7. Since 1000 = (7)(142) + 6, there are 142 multiples of seven less than 1000. (b) Every 77th number is divisible by 77. Since 1000 = (77)(12) + 76, there are 12 multiples of 77 less than 1000. We don’t want to count these, so there are 142 − 12 = 130 multiples of 7 but not 11 less than 1000. (c) We just figured this out to get (b)—there are 12. (d) Since 1000 = (11)(90) + 10, there are 90 multiples of 11 less than 1000. Now, if we add the 142 multiples of 7 to this, we get 232, but in doing this we’ve counted each multiple of 77 twice. We can correct for this by subtracting off the 12 items that we’ve counted twice. Thus, there are 232-12=220 positive integers less than 1000 divisible by 7 or 11. (e) If we want to exclude the multiples...
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...INFORMS Multiple Criteria Decision Making, Multiattribute Utility Theory: Recent Accomplishments and What Lies Ahead Author(s): Jyrki Wallenius, Peter C. Fishburn, Stanley Zionts, James S. Dyer, Ralph E. Steuer and Kalyanmoy Deb Source: Management Science, Vol. 54, No. 7 (Jul., 2008), pp. 1336-1349 Published by: INFORMS Stable URL: http://www.jstor.org/stable/20122479 Accessed: 15-10-2015 13:28 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. INFORMS is collaborating with JSTOR to digitize, preserve and extend access to Management Science. http://www.jstor.org This content downloaded from 130.243.57.230 on Thu, 15 Oct 2015 13:28:04 UTC All use subject to JSTOR Terms and Conditions SCIENCE MANAGEMENT WjEE. Vol. 54, No. 7, July 2008, 1336-1349 pp. DOI io.l287/nmsc.l070.0838 ISSN 0025-19091EISSN1526-55011081540711336@2008 INFORMS Criteria Decision Making, Multiattribute Multiple Utility Theory: Recent Accomplishments and What Lies Ahead School Helsinki Jyrki Wallenius of...
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...The problem of stacking a set of objects, such as bricks, books, or cards, on a tabletop to maximize the overhang is an attractive problem with a long history. J. G. Coffin [2] posed the problem in the “Problems and Solutions” section of this Monthly, but no solution was given there. The problem recurred from time to time over subsequent years, e.g., [18, 19], [12], [4]. Either deliberately or inadvertently, these authors all seem to have introduced the further restriction that there can be at most one object resting on top of another. Under this restriction, the harmonic stacks, described below, are easily seen to be optimal. ∗ A preliminary version of this paper [14] appeared in the Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’06), pages 231–240. This full version is to appear in the American Mathematical Monthly. † DIMAP and Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK. E-mail: msp@dcs.warwick.ac.uk ‡ School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: zwick@cs.tau.ac.il 1 Figure 2: Optimal stacks with 3 and 4 blocks compared with the corresponding harmonic stacks. The classical harmonic stack of size n is composed of n blocks stacked one on top of the other, 1 with the ith block from the top extending by 2i beyond the...
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... [4] Your SOP does not have to be too big. We at IITB tend to write long SOPs (1.5 to 2 pages) whereas IITM guys (who get better schools!)Usually write 0.75 page (max. 1 page). Basically, their SOPs are much more direct and to-the-point than ours. [5] REMEMBER THAT A SOP *MUST* BE ORIGINAL. Statement of Purpose I am applying to Stanford University for admission to the Ph.D. program in Computer Science. I am interested in Theoretical Computer Science, particularly in the Design and Analysis of Approximation Algorithms, Combinatory and Complexity Theory. My interest in Mathematics goes back to the time I was at school. This interest has only grown through my years in school and high school, as I have learnt more and more about the subject. Having represented India at the International Mathematical Olympiads on two occasions, I have been exposed to elements of Discrete Mathematics, particularly Combinatory and Graph Theory, outside the regular school curriculum at an early stage. The intensive training programs we were put through for the Olympiads have given me a lot of confidence in dealing with abstract mathematical problems. My exposure to Computer Science began after I entered the Indian Institute of Technology (IIT), Bombay. The excellent facilities,...
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