CS 70 Discrete Mathematics for CS Spring 2008

CS 70 Discrete Mathematics for CS Spring 2007 Luca

Recall from your high school math that a polynomial in a single variable is of the form p(x) = adx + ad−1x + . . .+ a0. Here the variable x and the coefficients ai are usually real numbers. For example, p(x) = 5x3 +2x+1, is a polynomial of degree d = 3. Its coefficients are a3 = 5, a2 = 0, a1 = 2, and a0 = 1. Polynomials have some remarkably simple, elegant and powerful properties, which we wil...

CS 70 Discrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 11

Suppose that the message consists of n packets and suppose that k packets are lost during transmission. We will show how to encode the initial message consisting of n packets into a redundant encoding consisting of n+ k packets such that the recipient can reconstruct the message from any n received packets. Note that in this setting the packets are labelled and thus the recipient knows exactly ...

CS 70 Discrete Mathematics for CS Spring 2007 Luca Trevisan Lectures

Suppose we have a random variable X (it is helpful to think of the example of the random variable X that is +1 with probability 1/2 and −1 with probability 1/2) of finite expectation and finite variance, that describes the outcome of a certain process. Suppose we repeat the process several times independently, and so have random variables X1, . . . ,Xn, . . . that are mutually independent and h...

CS 70 Discrete Mathematics for CS Spring 2007

and so on, then P(n) must be true for all n. Intuitively, this seems quite reasonable. If the truth of P all the way up to n always implies the truth of P(n+1), then we immediately obtain the truth of P all the way up to n+1, which implies the truth of P(n+2), and so on ad infinitum. If we compare the Strong Induction axiom to the original Induction axiom from Lecture 2, we see that Strong Indu...

CS 468 ( Spring 2013 ) — Discrete Differential Geometry