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PROFESSOR: Two particles is interesting in many cases. But in order to see what really is happening and how much structure you have to go to more than two particles. Two particles is a little too special. So we need to go beyond two particles and see what happens with this operator. So I'll do that. So we'll add more particles. So let's do that. So if you have n particles, n particles, capital n, we can consider also permutations. And how many permutations are you going to have? You're going to have n factorial permutations. And notation of Hilbert space, n factorial permutation operators. So why you say n factorial? Because a permutation is a reordering of your n objects. And therefore, in order to define the reordering, you must decide which object is going to be the new first. You have your n objects there. So you have n choices to pick a first and n minus 1 choices to be a second, n minus 2. So n factorial is the number of permutations. The permutation...

An interesting random variable associated with the Bernoulli process is the time of the kth success or the time of the kth arrival, depending on what kind of context we have in mind. So the picture is as follows. The process starts and we wait until the first arrival occurs, and the time that it occurs, we call that time Y1. Then we keep observing the process, and there's a time at which a second arrival comes. We call that time Y2. The process continues, and there is a certain time that the third arrival comes. We call that time Y3. Now, the time that the first arrival comes, this is also what we called T1. T1 is this length. It's the time until the first arrival. Let us give a name to the time it takes from the first to the second arrival, and we call that time T2, which is the second inter-arrival time. And similarly, we will call T3 the time between the second and the third arrival. So we define in general Tk to be the difference between two consecutive arriva...

An interesting question for computer architects is what capabilities must be included in the ISA? When we studied Boolean gates in Part 1 of the course, we were able to prove that NAND gates were universal, i.e., that we could implement any Boolean function using only circuits constructed from NAND gates. We can ask the corresponding question of our ISA: is it universal, i.e., can it be used to perform any computation? What problems can we solve with a von Neumann computer? Can the Beta solve any problem FSMs can solve? Are there problems FSMs can't solve? If so, can the Beta solve those problems? Do the answers to these questions depend on the particular ISA? To provide some answers, we need a mathematical model of computation. Reasoning about the model, we should be able to prove what can be computed and what can't. And hopefully we can ensure that the Beta ISA has the functionality needed to perform any computation. The roots of computer science stem from the e...