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ATIF JAVED: Hey, 3016. Welcome to my final video project for the class. My presentation will be on Modeling and Energy Analysis of Liquid Crystals using Mathematica. So liquid crystals pervade many areas of technology in our world today, including the screen you may be watching this on. They're used in LCD displays, thermometers, surfactants, polymers, detergents, and even Kevlar. So the versatility of this phase of matter is what definitely piqued my interest and made me especially enthusiastic about this project as an opportunity also to explore this subject as a scientist would, imposing my own questions and then finding ways to answer them. So let's move on to the subject matter. Liquid crystals are the state of matter which retains properties from the conventional crystalline solid and the isotropic liquid state. So here you see an example of the crystalline solid. LC molecules are anisotropic, meaning that they exhibit properties with different effects when ...

Hey, guys. Welcome back. Today, we're going to do another fun problem, which is a drill problem on joint PMFs. And the goal is that you will feel more comfortable by the end of this problem, manipulating joint PMFs. And we'll also review some ideas about independents in the process. So just to go over what I've drawn here, we are given an xy plane. And we're told what the PMF is. And it's plotted for you here. What these stars indicate is simply that there is a value there. But we don't know what it is. It could be anything between 0 and 1. And so we're given this list of questions. And we're just going to work through them linearly together. So we start off pretty simply. We want to compute, in part a, the probability that x takes on a value of 1. So for those of you who like formulas, I'm going to use the formula, which is usually referred to as marginalization. So the marginal over x is given by summing over the joint. So here we a...

PROFESSOR: Hey, we're back. Today we're going to do a singular value decomposition question. The problem is really simple to state: find the singular value decomposition of this matrix C equals [5, 5; -1, 7]. Hit pause, try it yourself, I'll be back in a minute and we can do it together. All right, we're back, now let's do it together. Now, I know Professor Strang has done a couple of these in lecture, but as he pointed out there, it's really easy to make a mistake, so you can never do enough examples of finding the SVD. So, what does the SVD look like? What do we want to end up with? Well, we want a decomposition C equals U sigma V transpose. U and V are going to be orthogonal matrices, that is, their columns are orthonormal sets. Sigma is going to be a diagonal matrix with non-negative entries. OK, good. So now, how do we find this decomposition? Well, we need two equations, OK? One is C transpose C is equal to V, sigma transpose, sigma, V transp...

PROFESSOR: Hey, we're back. Today we're going to do a singular value decomposition question. The problem is really simple to state: find the singular value decomposition of this matrix C equals [5, 5; -1, 7]. Hit pause, try it yourself, I'll be back in a minute and we can do it together. All right, we're back, now let's do it together. Now, I know Professor Strang has done a couple of these in lecture, but as he pointed out there, it's really easy to make a mistake, so you can never do enough examples of finding the SVD. So, what does the SVD look like? What do we want to end up with? Well, we want a decomposition C equals U sigma V transpose. U and V are going to be orthogonal matrices, that is, their columns are orthonormal sets. Sigma is going to be a diagonal matrix with non-negative entries. OK, good. So now, how do we find this decomposition? Well, we need two equations, OK? One is C transpose C is equal to V, sigma transpose, sigma, V transp...

PROFESSOR: Hey, everybody. Welcome to 6.849, Geometric Folding Algorithms. I am Erik Demaine. You can call me Erik. And we have as TA Jayson Lynch, who's right there. And this class is a bit unusual, at least for me, because I'm trying for the first time a new experiment, which is inverted lecturing. And I wrote this on the poster for the class, and everyone started asking me, what's inverted lectures? Well, it's not a new idea, but I've never tried it before. The concept is to make these in-class times, where we're all here together, more interactive by taking the lecture component of the class, which is covering all the material into videos that you watch online. So this class is basically going to alternate between real in-person things, as you are here-- of course, we are also being video recorded, so slightly a contradiction in terms-- but that's for the people in the interwebs to be able to watch this. So we'll alternate between the r...

Hey, everyone. Welcome back. Today, we're going to do another fun problem that has to do with a random number of coin flips. So the experiment we're going to run is as follows. We're given a fair six-sided die, and we roll it. And then we take a fair coin, and we flip it the number of times indicated by the die. That is to say, if I roll a four on my die, then I flip the coin four times. And then we're interested in some statistics regarding the number of heads that show up in our sequence. In particular, we want to compute the expectation and the variance of the number of heads that we see. So the first step of this problem is to translate the English to the math. So we have to define some notation. I went ahead and did that for us. I defined n to be the outcome of the die role. Now, since we flip the coin the number of times shown by the die roll, n is equivalently the number of flips that we perform. And n, of course, is a random variable, and I'v...