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Akin: And today I will present to you one of the most interesting topics in my field of study, which is graph theory. Um, I know the topic might sound a little bit intimidating, but what you need to understand this presentation is quite simple. So, if you can do the basic math, like simple additions or simple multiplication, you are pretty good to go. And what you'll need is the observation skills and that's probably all you need to have for understanding this presentation. Before I jump in, can I ask like, if any of you guys have been familiar with this topic before? Student: A little bit, a little bit. Akin: Okay, um, yeah, I hope you get something out of today's presentation. Hope it will be useful. Okay. Okay about today's agenda, we'll go over some definitions to see what the graph is. And next we are going to explore one of the most interesting questions in graph theory, which is the shortest path problem. And then we are going to see some exampl...

The phone and content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ETHAN MEYERS: What I'm talking about today is neural population decoding, which is very similar to what Rebecca was talking about, except for I'm talking about now more on the single neuron level and also talk a bit about some MEG at the end. But kind of to tie it to what was previously discussed, Rebecca talked a lot about, at the end, the big catastrophe. Well, you don't know if something is not there in the fMRI signal because things could be masked when you're averaging over a large region as you do when you're recording from those bold signals. And when you're doing decoding on single neurons, that is not really an issue because you're actually going do...

An N type and I take a P type and I put them together, some really important things happen. So important that it led to this, which is the very first transistor. That's a transistor. It's a dope semiconductor, one type, next to a dope semiconductor, another type. Now, those are the three people-- Bardeen, Shockley, and Brattain-- who won the Nobel Prize for this work. And that's the first transistor. Does anybody know how many transistors we make today? Per second. That's the only time metric we can use. Yeah, it's something around 10 trillion. We make 10 trillion transistors per second today. This is a big deal. This started it all, and it's all about chemistry and the physics of this device. This led to diodes and photodiodes. This led to the whole revolution. And what I want to point out here is that this also led-- it called on chemistry. So I love this chart. Unfortunately, they didn't update it, but this is an Intel chart that they used t...

DAVID JORDAN: Hello, and welcome back to recitation. Today the problem I'd like to work with you is about computing partial derivatives and the total differential. So we have a function z which is x squared plus y squared. So it depends on the two variables x and y. Now the variables x and y themselves depend on two auxiliary variables, u and v. So that's the setup that we have. So in part a, we just want to compute the total differential dz in terms of dx and dy. So u and v aren't going to enter into the picture. And then in part b, we're going to compute the partial derivative partial z partial u in two different ways. First, we're going to compute it using the chain rule. And then we're going to compute it using total differentials. And so we'll substitute in some of the work that we had in a to solve that part. So why don't you pause the video now and work on the problem. We'll check back and we'll do it together. Hi, and welcom...

DAVID JORDAN: Hello, and welcome back to recitation. So today, the problem I'd like to work with you is about taking partial derivatives in the presence of constraints. So this is a pretty subtle business. So take your time when you work these problems. So what we have is we have this function w, and it's a function of four variables: x, y, z, and t. OK? But it's not really a function of these four variables because we have a constraint. So we want to study how w changes as we vary the parameters, except that we have imposed this constraint here. So that really we kind of only have three variables, because we have four variables and one constraint. So that's what partial derivatives with constraints help us do. So let's explain first the notation. OK? So it says partial w partial z, and then we have the subscripts x and y. So what's important about this notation is not what you see as much as what you don't see. What you don't see is the va...

SUMMER MORRILL: Being unsure and not knowing is actually a great thing to show your students, right? Because it models that curiosity, it models that question-asking. And you can come back to them and say, here's what I learned and how I learned it. SARAH HANSEN: Today on Chalk Radio, how thinking like scientists can make us better teachers and how teachers can be powerful mentors for future scientists. I'm Sarah Hansen. For this episode, I sat down with Dr. Summer Morrill. SUMMER MORRILL: I started off as a graduate student at MIT in biology, but I'm currently the Instructor of TA Training in the Biology Department SARAH HANSEN: I actually spoke with Summer right before she left MIT. Now she's a science instructor at a secondary school in New Hampshire. One of the reasons we were excited to bring Summer on the show was her unique and innovative approach to thinking about teaching. SUMMER MORRILL: I guess what I'm most interested in is thinking about h...