The hematopoietic system of mammals is a convenient model to use to study development and differentiation. The bone marrow transplantation experiment discussed in lecture indicates that stem cells are more than a figment of some biologist's fertile imagination

1 The pulverizer We saw in lecture that the greatest common divisor(GCD)of two numbers can be written as a linear combination of them. That is, no matter which pair of integers a and b we are given, there is always a pair of integer coefficients s and t such that

Guessing a particular solution. Recall that a general linear recurrence has the form: f(n)=a1f(n-1)+a2f(n-2)+…+aaf(n-d)+g(n) As explained in lecture, one step in solving this recurrence is finding a particular solu- tion; i.e., a function f(n)that satisfies the recurrence, but may not be consistent with the boundary conditions. Here's a recipe to help you guess a particular solution:

Problem 1. Sammy the Shark is a financial service provider who offers loans on the fol lowing terms. Sammy loans a client m dollars in the morning This puts the client m dollars in debt to Sammy. Each evening, Sammy first charges\service fee\, which increases the client's debt by f dollars, and then Sammy charges interest, which multiplies the debt by a factor

An explorer is trying to reach the Holy Grail, which she believes is located in a desert shrine d days walk from the nearest oasis. In the desert heat, the explorer must drink continuously. She can carry at most 1 gallon of water, which is enough for 1 day. However, she is free to create water caches out in the desert. For example, if the shrine were 2/3 of day's walk into the desert, then she could recover

Srini devadas and Eric Lehman Lecture notes Number theory ll Image of Alan Turing removed for copyright reasons s The man pictured above is Alan Turing, the most important figure in the history of mputer science. For decades, his

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?

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

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

1 Induction Recall the principle of induction: Principle of Induction. Let P(n) be a predicate. If ·P(0) is true,an for all nE N, P(n) implies P(n+1), then P(n) is true for all nE N As an example let's try to find a simple expression equal to the following sum and then use induction to prove our guess correct 1·2+2·3+3:4+…+n·(mn+1) To help find an equivalent expression, we could try evaluating the sum for some small n and(with the help of a computer) some larger n sum