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My Links .ATasteOfTWiki view a short introductory presentation on TWiki for beginners .WelcomeGuest starting points on TWiki .TWikiUsersGuide ...

My Links .ATasteOfTWiki view a short introductory presentation on TWiki for beginners .WelcomeGuest starting points on TWiki .TWikiUsersGuide ...

My Links .ATasteOfTWiki view a short introductory presentation on TWiki for beginners .WelcomeGuest starting points on TWiki .TWikiUsersGuide ...

My Links .ATasteOfTWiki view a short introductory presentation on TWiki for beginners .WelcomeGuest starting points on TWiki .TWikiUsersGuide ...

My Links ATasteOfTWiki view a short introductory presentation on TWiki for beginners WelcomeGuest starting points on TWiki TWikiUsersGuide complete ...

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My Links ATasteOfTWiki view a short introductory presentation on TWiki for beginners WelcomeGuest starting points on TWiki TWikiUsersGuide complete ...

My Links .ATasteOfTWiki view a short introductory presentation on TWiki for beginners .WelcomeGuest starting points on TWiki .TWikiUsersGuide ...

Welcome to the 15 251 Wiki If you want to contribute to the wiki, please read Information for Editors Lecture Notes Course Syllabus Contains grading ...

14. Modular Arithmetic and the RSA Cryptosystem Conventional modern day cryptography typically requires a key #151;a string of numbers #151;shared by both sender and ...

LaTeX is a system of typesetting mathematical notation. You enter LaTeX markup language in a plain text file, and then typeset it using the program pdflatex . You ...

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Induction: One Step at a Time The most useful proof technique in computer science is mathematical induction. It is the primary way in which we: (A) Prove theorems ...

15. Algebraic Structures: Groups, Rings, and Fields In this chapter, we are going to study the abstract properties of binary operations. Algebraic structures known ...

The Dating Game Imagine that you are a matchmaker, with 100 female clients and 100 male clients. Each of the women gives you a complete list of the 100 men, ordered ...

The Golden Ratio, Fibonacci Numbers, and Other Recurrences Fibonacci numbers have links to many different kinds of mathematics. They can be defined recursively as ...

Alternatives to LaTeX Here are some tips for those of you that don't like LaTeX Microsoft Word 2007 Overview The Microsoft Equation Editor has improved by leaps ...

FTP/SFTP FTP is a transfer protocol used to move files from one machine to another. In this class, you will probably use FTP to transfer your homework to the handin ...

Lecture Notes Many of these lecture notes were based on a draft version of an upcoming textbook developed from 15 251. Please do not distribute them outside of CMU ...

Homework Submission The rules for submitting written homework are as follows: You must hand in a single file in pdf format . See the Converting Between File ...

One Minute to Learn Programming: Finite Automata Today, we're gonna talk about an incredibly simple programming system that has a ton of real world applications: Finite ...

My Links .ATasteOfTWiki view a short introductory presentation on TWiki for beginners .WelcomeGuest starting points on TWiki .TWikiUsersGuide ...

Thales and Gödel's Legacy: What is Proof? Thales #8217; Legacy: What Is A Proof? Thales Of Miletus (600 BC) Insisted on Proofs! #8220;first mathematician #8221; ...

Cantor's Legacy: Infinity and Diagonalization Georg Cantor (1845 1918) The Ideal Computer The ideal computer has infinite memory. There are two ways to look at this ...

On Time Versus Input Size Introduction Computer scientists seek not only to find correct solutions to a particular problem, but also to efficiently solve a problem ...

Turing's Legacy: The Limits Of Computation Alan Turing (1912 1954) Grading Scripts and the HELLO Assignment Suppose we give you the following assignment: write a ...

On Raising a Number to a Power The Egyptians, as well as other people in both ancient and modern times, spent a lot of time trying to figure out how to do calculations ...

Primes, Euclid #8217;s GCD, and Continued Fractions Determining the greatest common divisor is an example of finding an integer relation between two numbers. Dating ...

Ancient Wisdom: Unary and Binary One of the key insights in problem solving is that we don't have to stick with the representation in which we encounter a problem ...

Random Walks on Graphs At any node, go to one of the neighbors of the node with equal probability. Let #8217;s start simple #8230;We #8217;ll just walk in a straight ...

Counting in Terms of Proportions Distributions For any random event, like flipping a coin, each outcome has a value and a probability. A probability distribution ...

Solving Problems, Writing Proofs, and Enjoying the Pain: How to Succeed in This Class While humans are capable of rudimentary abstraction, our brains did not evolve ...

Drawing Automata LaTeX Within LaTeX, the and http://www.lsv.ens cachan.fr/~gastin/gastex/gastex.html GasTeX packages may be useful to draw automata. The former ...

Pancakes with a Problem The cook at the Sunrise Cafe is sloppy: when she prepares pancakes, they come out all different sizes. The waiter, however, is tidy. Before ...

Combinatorial Games Combinatorial games are two person games with perfect information and no chance moves, and with a win or lose outcome. Such a game is determined ...

Combinatorial Games Combinatorial games are two person games with perfect information and no chance moves, and with a win or lose outcome. Such a game is determined ...

Random Variables and Great Expectations Some problems involving probability can seem really messy. For example, if we randomly put 100 letters into 100 addressed envelopes ...

Basic Information on Circuits You should already be familiar with logical functions like OR and AND. A gate is just a visual representation of these functions. For ...

Your Guide to 15 251 Homeworks This section is about how to succeed on the homework. The point of the homework exercises is to give you experience using the techniques ...

Using SFTP to Transfer Files on Windows SFTP is set up so you can connect to a remote server simply by entering the name of the server, or its Internet address in ...

Using SSH to Connect Remotely on Windows Computing Services supports the use of the SSH.com Secure Shell client on Windows. Its graphic interface follows most of the ...

Converting Between File Types If you are using LaTeX to typeset your solutions, you can get PDF files by using pdflatex (if you want to get postscript, you can run ...

Technical Support Essentials Converting files to PDF Submitting Homework Accessing the 15 251 bulletin board Andrew Basic UNIX commands ...

Java: Reading from Standard In Here is a bit of starter code, just as a quick aid to help java programmers that have never dealt with file input/output before. The ...

Telnet (SSH) Due to security restrictions, you must use SSH to connect to andrew machines. Windows The officially supported SSH client is SSH Secure Shell: it supports ...

Basic Unix Commands Some common UNIX gotchas.. If you want to run a program that's in your current directory, you need to preface it with ./ , e.g. ./tetris ...