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12 Pages

### l5_induction3

Course: CS 237, Spring 2011
School: BU
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Word Count: 3401

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Mathematics 6.042/18.062J for Computer Science Srini Devadas and Eric Lehman February 15, 2005 Lecture Notes Induction III 1 Two Puzzles Here are two challenging puzzles. 1.1 The 9Number Puzzle The numbers 1, 2, . . . , 9 are arranged in a 3 3 grid as shown below: 1 2 3 4 5 6 7 8 9 You can rearrange the numbers by rotating rows and columns. For example, rotating the rst row to the right gives: 1 2 3 4 5 6 7...

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BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanFebruary 17, 2005 Lecture NotesNumber Theory INumber theory is the study of the integers. Number theory is right at the core of math ematics; even Ug the Caveman surely had so
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanFebruary 24, 2005 Lecture NotesNumber Theory IIImage of Alan Turing removed for copyright reasons.The man pictured above is Alan Turing, the most important gure in the histor
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanMarch 1, 2005 Lecture NotesGraph Theory 1 IntroductionInformally, a graph is a bunch of dots connected by lines. Here is an example of a graph:B A F D G I C E HSadly, this de
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanMarch 3, 2005 Lecture NotesGraph Theory II 1 Coloring GraphsEach term, the MIT Schedules Office must assign a time slot for each final exam. This is not easy, because some stu
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanMarch 10, 2005 Lecture NotesSums and ApproximationsWhen you analyze the running time of an algorithm, the probability some procedure succeeds, or the behavior of a loadbalanci
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanMarch 15, 2005 Lecture NotesSums, Approximations, and Asymptotics II 1 Block StackingHow far can a stack of identical blocks overhang the end of a table without toppling over?
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanMarch 17, 2005 Lecture NotesRecurrencesRecursion- breaking an object down into smaller objects of the same type- is a ma jor theme in mathematics and computer science. For exa
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanMarch 25, 2005 Lecture NotesCounting I20480135385502964448038 489445991866915676240992 1082662032430379651370981 1178480894769706178994993 1253127351683239693851327 1301505129
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanMarch 31, 2005 Lecture NotesCounting IIWe realize everyone has been working pretty hard this term1 , and were considering awarding some prizes for truly exceptional coursework
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanApril 5, 2005 Lecture NotesCounting IIIToday well briey review some facts you dervied in recitation on Friday and then turn to some applications of counting.1 The Bookkeeper R
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanApril 7, 2005 Lecture NotesGenerating FunctionsGenerating functions are one of the most surprising, useful, and clever inventions in discrete math. Roughly speaking, generatin
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanApril 14, 2005 Lecture NotesIntroduction to ProbabilityProbability is the last topic in this course and perhaps the most important. Many algorithms rely on randomization. Inve
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanApril 21, 2005 Lecture NotesConditional ProbabilitySuppose that we pick a random person in the world. Everyone has an equal chance of being selected. Let A be the event that t
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanApril 26, 2005 Lecture NotesIndependence 1 Independent EventsSuppose that we flip two fair coins simultaneously on opposite sides of a room. Intu itively, the way one coin lan
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanApril 28, 2005 Lecture NotesRandom VariablesWe've used probablity to model a variety of experiments, games, and tests. Through out, we have tried to compute probabilities of e
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanMay 3, 2005 Lecture NotesExpected Value IThe expectation or expected value of a random variable is a single number that tells you a lot about the behavior of the variable. Rou
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanMay 5, 2005 Lecture NotesExpected Value II 1 The NumberPicking GameHere is a game that you and I could play that reveals a strange property of expectation. First, you think of
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanMay 10, 2005 Lecture NotesRandom Walks 1 Random WalksA drunkard stumbles out of a bar. Each second, he either staggers one step to the left or staggers one step to the right,
BU - CS - 237
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric LehmanMay 12, 2005 Lecture NotesSpecial Topics 1 StreaksWas the table of H s and T s below generated by ipping a fair coin 100 times, or by someone tapping the H and T keys in a what
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 17A Bird's Eye View of System AbstractionsIn this chapter, we will present a bird-eyes view of computer organization in order to understand how issues of concurrency and synchronization a
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 13Processes as Resource ConsumersSources of Concurrency in a Computing SystemAs we alluded to before, the management of concurrency (coordination, resource management, synchronization, e
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 14Performance Metrics of Computing SystemsIntroductionPerformance metrics of computing systems are needed in order to evaluate the goodness of a particular solution (architecture, protoc
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 7Performance Evaluation of Computing SystemsIntroductionThe various performance metrics we have examined so far enable us to speak of many aspects of a systems performance (e.g., utiliza
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 15A Basic Introduction to ProbabilityThe study of probability is all about the study of chancethe quantification of the chance or the likelihood that something (e.g. an event, an outcome)
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 13Probability Distributions as Modeling ToolsRecall that a probability distribution provides a characterization of a random variableby enabling us to calculate the probability of a random
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 9Elementary Queuing AnalysisNotation In discussing various server queues, it will be necessary to talk about various random variables associated with these queues. The Figure 1 below prov
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 4Variations of the M/M/1 Queuing SystemIn the analysis of the M/M/1 system, we have been concerned with exponential service times. Recall that an exponential distribution is memoryless (i
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 10Discrete Event SimulationAs it should be obvious by now, models that are amenable to analysis, such as queuing models, ignore (or abstract out) many of the details often present in real
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 11Elementary Queuing AnalysisMultiple Server Queuing Systems Multi-server QueuesAssume that there are N servers in a system. We consider two scenarios. In the first scenario, (illustrate
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 4General Processor SharingFrom non-preemptive FCFS to preemptive round-robin schedulingThe queuing systems (and networks) we have studied so far helped us model how processes consume res
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 12Resource Management BasicsWhen we discussed the simple M/M/1 system earlier in this class, we assumed that upon completion of service, the server (e.g., CPU or Disk) would simply pick t
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 10Resource Management Basics (continued)So far we have only examined one non-preemptive scheduler-namely FCFS-and we have noted that it was not &quot;friendly&quot; to short jobs. One approach we h
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 11Stateful Resource SchedulingSo far, we have only examined scheduling approaches for state-less resources. Recall that we defined such resources as those whose service time (for a given
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 13Real-Time SchedulingIn most of the systems we have been concerned with so far in this course, it was assumed that the &quot;correctness&quot; of the system's response to a request was independent
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 13Concurrency and Synchronization: Two-Process Mutual ExclusionSo far our consideration of resource management has made a fundamental assumption-that the processes competing for a shared
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 5Concurrency and Synchronization: N-Process Mutual ExclusionRecall that we solved the 2-process mutual exclusion problem using Dekkers and Petersons algorithms. We now turn our attention
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 9Concurrency and Synchronization: Basic Primitives and ConstructsSo far, we have established that hardware approaches relying on disabling interrupts do not solve the mutual exclusion pro
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 12Concurrency and Synchronization: Deadlocks and StarvationAs we have observed when discussing the dining philosophers problem, deadlock and thus the possibility of starvation should be a
BU - CS - 350
CS-350: Fundamentals of Computing Systems Lecture NotesPage 1 of 9The Concept of TransactionsThe notion of a &quot;transaction&quot; is a recurring one when building computing systems in general, and database systems in particular. Consider, for example, the tas
BU - CS - 350
Computer ScienceComputer SpeedupComputer ScienceCS-350Distributed ComputingAzer BestavrosComputer Science Department Boston UniversityMoores Law: The density of transistors on a chip doubles every 18 months, for the same cost (1965) Copyright Azer
BU - CS - 350
Computer ScienceDistributed SystemsComputer ScienceCS-350Distributed SynchronizationAzer BestavrosComputer Science Department Boston University Examples:Client-Server SystemsDistributed File systems (e.g. NFS, AFS, etc.) Web Applications (e.g. HT
BU - CS - 350
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BU - ASTRO - 101
Known Asteroids in our Solar System by yearThe Sun seen by an X-ray cameraYear KnownJohannes KeplerHans LippersheyAlbert EinsteinIsaac NewtonGalileo GalileeOur Solar SystemSaturnian Moon EnceladusNewest: 2003 UB313 ?
BU - ASTRO - 101
AS 101 Lecture 2 ASClassLogistics: Ifyoudidnotgetonelasttime,pickup:Syllabus ProblemSet1 Everyoneshouldattendadaylabstartingthisweek Note:Willcallonstudents.LastClassTodaysClass IntroductiontoAS101 Syllabus NatureofScience&amp; Astronomy TimeandDista
BU - ASTRO - 101
ClassLogistics: ProblemSet1AS 101 Lecture 3 AS Labs:BuildingaStar CatalogDueTue PickupTimeouttothinkcardoffandrunning!LastClass NatureofScience&amp;Astronomy TimeandDistanceinAstronomy TodaysClass TimeandDistanceintheUniverse TheCelestialSphere Obser
BU - ASTRO - 101
ClassLogistics:ClassLogistics: AS 101 Lecture 4 AS ProblemSet1: ProblemSet2: Duetodayinclass SolutionSetavailable Webbasedseeemail Ifyoucannotaccessweb emailprofOppenheimorTF.TodaysClass ObservingtheNightSky coordinatesandperspective Anglesands
BU - ASTRO - 101
AS 101 Lecture 5 ASLogistics ProblemSet2:LastClass:ObservingtheNightSkyII ChangingSkywithSeasons Earthsorbit ObservingtheSun TodaysClass:ObservingtheNightSkyIII CoordinatesandTimes PrecessionoftheEquinox ObservingtheMoon EclipsemaybeWebbased:gotomas
BU - ASTRO - 101
AS 101 Lecture 6Logistics Problem Set 3:Web based: go to masteringastronomy.com Course ID: AS101SP10OPPENHEIMLast Class: Observing the Night Sky III Coordinates and Times Precession of the Equinox Day and Night Todays Class: Observing the MoonEclip
BU - ASTRO - 101
AS101 Lecture 7Logistics: Pr. Set 2: Extended At MasteringAstronomy Do the NIGHT LAB ASAPSpecial Announcement: The AS101 class roster has 75 students, the Mastering Astronomy site has 63 12 students are missing in action!Last Lecture: Observing The Mo
BU - ASTRO - 101
AS101 Lecture 8Logistics: Pr. Set 3: Due Oct 5 Make sure that you are on MasteringAstronomy!Last Lecture: Solar Eclipses Planetary motions Historic development of AstronomyTodays Lecture: Greek astronomy Heliocentric universe Ptolemy, Copernicus, Br
BU - ASTRO - 101
AS101 Lecture 9Logistics: Do not forget Night Lab Homework: Read Chapters 3 and 4Last Lecture: Greek astronomy Heliocentric universe Ptolemy, Copernicus, BraheTodays Lecture: Kepler Galileo NewtonNews Items: The NY TimesNew Planet May Be Able to N
BU - ASTRO - 101
AS101 Lecture 10Logistics: Do not forget Night Lab Homework 3 due today. Homework 4 availableLast Lecture: Kepler GalileoTodays Lecture: NewtonIsaac Newton (1642-1727) Greatest Scientist in HistoryHistory Born middle class Educated in Oxford Fled
BU - ASTRO - 101
AS101 Lecture 11Logistics: Homework 4 available Today is the last day to drop this class with a W gradeLast Lecture: Newtons Laws of Motion Newtons Law of Universal GravitationTodays Lecture: Applications of Newtons Laws Weight, Orbits, Tides Conser
BU - ASTRO - 101
AS101 Lecture 12Logistics: Homework 5 is available Start reading Chapter 5MIDTERM 1 on Thursday Oct 21In class Covers material up to todays Last Lecture: class Applications of Newtons Laws Close Book, Notes etc. Bring calculator No cell Weight, Orbits
BU - ASTRO - 101
AS101 Lecture 13Logistics: Homework 5 due today Homework 6 available Review end-of-chapter Questions Review homework problemsMIDTERM 1 on Thursday Oct 21In class Covers material up to last class Close Book, Notes etc. Bring calculator No cell phones o
BU - ASTRO - 101
AS101 Lecture 14Logistics: Homework 6 due today Homework 7 available Do night lab as soon as you canLast Lecture: Atom LightTodays Lecture: Light and matter Doppler Effect Telescopes Spectrometer. interferometerLight and Matter When light interact
BU - ASTRO - 101
AS101 Lecture 15Logistics: Do the night lab as soon as you canLast Lecture: Light and matter Doppler EffectTodays Lecture: Telescopes Spectrometer, interferometerCurved mirrorIf the mirror is a sphere:Sphere Since all rays do not converge to the
BU - ASTRO - 101
AS101 Lecture 16Logistics: Do the night lab as soon as you can Last Day to WITHDRWA is Nov 5.Last Lecture: Telescopes Spectrometer, interferometerTodays Lecture: Overview of the solar system Solar System formationMajor characteristics of the Sun
BU - ASTRO - 101
AS101 Lecture 17Logistics: Do the night lab as soon as you can Last Day to WITHDRW is Nov 5.Last Lecture: Overview of the solar system Solar System formationTodays Lecture: Solar System formation Dating the solar system Geology of Terrestrial Planet
BU - ASTRO - 101
AS101 Lecture 18Logistics: Do the night lab as soon as you canLast Lecture: Solar System formation Dating the solar systemTodays Lecture: Geology of Terrestrial PlanetsComparative Planetary GeologyGeology: The word geology comes from the Greek geo
BU - ASTRO - 101
AS101 Lecture 19Logistics: Do the night lab as soon as you can Midterm 2: Nov 18 same formatCovers Chapters 5 - 9Last Lecture: Geology of Terrestrial PlanetsTodays Lecture: Geology of Terrestrial Planets (Contd).Searching for geologic cause and ef