1 Coloring Graphs Each term, the MIT Schedules Office must assign a time slot for each final exam. This is not easy, because some students are taking several classes with finals, and a student can take only one test during a particular time slot. The Schedules Office wants to avoid all conflicts, but to make the exam period as short as possible

Number Theory I Number theory is the study of the integers. Number theory is right at the core of math ematics; even Ug the Caveman surely had some grasp of the integers- at least the posi tive ones. In fact, the integers are so elementary that one might ask, What's to study?

1 Unstacking Here is another wildly fun 6.042 game that's surely about to sweep the nation! You begin with a stack of n boxes. Then you make a sequence of moves. In each move, you divide one stack of boxes into two nonempty stacks. The game ends when you have n stacks, each containing a single box You earn points for each move; in particular, if you divide one stack of height a b into two stacks with heights a and b, then you score

It's really sort of amazing that people manage to communicate in the English language Here are some typical sentences: 1. You may have cake or you may have ice cream 2. If pigs can fly, then you can understand the Chernoff bound 3. If you can solve any problem we come up with then you get an a for the course. 4. Every American has a dream What precisely do these sentences mean? Can you have both cake and ice cream or must you choose just one desert? If the second sentence is true, then is the Chernoff bound incomprehensible? If you can solve some problems we come up with but not all, then do you get an a for the course? And can you still get an a even if you cant solve any of the problems? Does the last sentence imply that all Americans have the same dream or might

1 Strong Induction Recall the principle of strong induction: Principle of Strong Induction. Let(n) be a predicate. If ·P() is true,and for all n, P(O)A P(1)...A P(n) implies P(n+1), then P() is true for all n E N. As an example, let's derive the fundamental theorem of arithmetic

Random walks 1 Random walks a drunkard stumbles out of a bar staggers one step to the right, with a canal lies y steps to his right. Thi I equal p second, he either staggers one step to the left or probability. His home lies r steps to his left, and everal natural questions, including 1. What is the probability that the drunkard arrives safely at home instead of falling into the canal? 2. What is the expected duration of his journey however it ends? The drunkard's meandering path is called a random walk. Random walks are an im- portant subject, because they can model such a wide array of phenomena. For example

The expectation or expected value of a random variable is a single number that tells you a lot about the behavior of the variable. Roughly, the expectation is the average value, where each value is weighted according to the probability that it comes up. Formally, the expected value of a random variable r defined on a sample space s is: (B)=∑R()Pr(o) To appreciate its signficance, suppose S is the set of students in a class, and we select a student uniformly at random. Let r be the selected student's exam score. Then

1 Logic A proposition is a statement that is either true or false. Propositions can be joined by \and\, \or\, \not\, \implies\, or \if and only if\. For each of these connective, the defini- tion and notational shorthand are given in the table below. Here A and B denote arbitrary propositions

AutoCAD教案 1.1 AutoCAD简介 1.1.1什么是 AutoCAD cad(Computer Aided Design)的含义是指计算机辅助设计,是计算机技术的一个重 要的应用领域。 AutoCAD则是美国 Autodesk企业开发的一个交互式绘图软件,是用于二 维及三维设计、绘图的系统工具,用户可以使用它来创建、浏览、管理、打印、输出、共 享及准确复用富含信息的设计图形

CAD(computer aided design)是一种用计算机硬、软件系统辅助人们对产品或 工程进行设计的方法与技术,包括设计、绘图、工程分析与文档制作等设计活动, 它是一种新的设计方法,也是一门多学科综合应用的新技术。 传统CAD涉及以下一些基础技术: