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PROFESSOR: So as a more technical aside, let's analyze more carefully the problem of release of viral load from a drop by process of diffusion. So here, again, I sketch a droplet, which would typically be an aerosol droplet in the size range of, let's say, microns. And the virion of interest has a size that is much smaller than that on the order of, let's say, 100 nanometers. And this white path is showing how such a virion would go from its initial position R, let's say, as a radio position, to the boundary. Now, the general problem of finding the expected first passage time from the point inside a domain to a boundary is a classical problem in the theory of stochastic processes and random walks. And it has the following representation. So the mean are expected first passage time from a point to an absorbing surface solves the following problem. Is the Laplacian of that time with a minus sign is 1 over D where D is the diffusivity. And so this is the DV t...

PROFESSOR: So one more thing let's do with this operator. So we're getting accustomed to these operators, and these permutation operators can also act on operators themselves. So that's important. So consider the action on operators. So for example, an operator B that acts belonging to the linear operator some V, when acting on V tensor V, we define two operators, B1 and B2. And you define them in an obvious way, like B1 acting on Ui 1 tensor Uj 2. OK, B as an operator knows how to act on every vector on the vector space capital V. So when you say 1, you're meaning that this operator acts on the first Hilbert space. So this is equal to B times Ui 1 tensor Uj 2. So it just acts on the first state. How does it act? Via B, that is an operator, in the vector space V. Similarly, if you have B2 of Ui tensor Uj 2 you have Ui 1 tensor BUj 2 to OK. So these are operators that act either on the first state or in the second state. So the permutation operators can do ...

As a preparation for more complex and more difficult models, we will start by looking at the simplest model that there is, that involves a linear relation and normal random variables. The specifics of the model are as follows. There's an unknown parameter modeled as a random variable, Theta, that we wish to estimate. What we have in our hands is Theta plus some additive noise, W. And this sum is our observation, X. The assumptions that we make are that Theta and W are normal random variables. And to keep the calculations simple, we assume that they're standard normal random variables. Furthermore, we assume that Theta and W are independent of each other. According to the Bayesian program, inference about Theta is essentially the calculation of the posterior distribution of Theta if I tell you that the observation, capital X takes on a specific value little x. To calculate this posterior distribution, we invoke the appropriate form of the Bayes rule. We have the pr...

GILBERT STRANG: OK. More about eigenvalues and eigenvectors. Well, actually, it's going to be the same thing about eigenvalues and eigenvectors but I'm going to use matrix notation. So, you remember I have a matrix A, 2 by 2 for example. It's got two eigenvectors. Each eigenvector has its eigenvalue. So I could write the eigenvalue world that way. I want to write it in matrix form. I want to create an eigenvector matrix by taking the two eigenvectors and putting them in the columns of my matrix. If I have n of them, that allows me to give one name. The eigenvector matrix, maybe I'll call it V for vectors. So that's A times V. And now, just bear with me while I do that multiplication of A times the eigenvector matrix. So what do I get? I get a matrix. That's 2 by 2. That's 2 by 2. You get a 2 by 2 matrix. What's the first column? The first column of the output is A times the first column of the input. And what is A times x1? Well, A times x1...

PROFESSOR: So connectivity is more than just an all or nothing affair. We can talk about how connected a graph is. So let's begin with two vertices. Two vertices are said to be k-edge connected if they remain connected if you remove fewer than k edges from the graph. Let's look at an example. So here's a graph, and let's focus on those two vertices that I've highlighted in magenta. They are 1-edge connected because they're connected, and if you remove one edge they become disconnected. So they're 1-edge connected, but they're not 2-edge connected. In particular, if I delete that edge, then they no longer is a path between the two magenta vertices. Here's an example of two vertices, these two green vertices, that are 2-edge connected. That means that I can remove any number of edges less than two, which is to say one edge, and they'll stay connected. But if I do remove two vertices, they become disconnected. So with those two edges j...