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Here you see a circuit powered by a battery and connected through a light bulb. This section of the circuit is made from copper foil. We can put a cut into the foil to make this circuit incomplete. Taking advantage of the thermal expansion properties of copper, we can place some candles under the foil to provide enough heat to expand the foil and complete the circuit again! Many materials, like metals, expand when you heat them. But other materials, like polymers, shrink when heated. The macroscopic properties of both of these materials are highly dependent on temperature. The difference in the macroscopic behavior of these two materials is determined by very different microscopic structure. Statistical Mechanics is the method used to describe and predict behavior at the macro-scale based on statistical models of microscopic behavior. The first step to understanding the power of Statistical Mechanics, is to use this method to define and understand temperature in terms of ...

GILBERT STRONG: Well, you see spring has finally come to Boston. My sweater is gone and it's April the 16th, I think. It's getting late spring. So today, this video is the nice case, constant coefficient, linear equations, and the right hand side is an exponential. Those are the best. And we've seen that before. In fact, let me extend, we saw it for first order equations, here it is for second order equations, and it could be an nth order equation. We could have the nth derivative and all lower derivatives. The first derivative, the function itself, 0-th derivative with coefficients, constant coefficients, equalling e to the st. That's what makes it easy. And what do we do when the right hand side is e to the st? We look for a solution, a multiple of e to the st. Capital Y times e to the st, that's going to work. We just plug into the equation to find that transfer function capital Y. Can I just do that? I'll do it for the nth degree. Why not? So w...

In this video, we'll see how to color our points by region and how to add a linear regression line to our plot. Here we have our plot from the last video, where we've colored the points dark red. Now, let's color the points by region instead. This time, we want to add a color option to our aesthetic, since we're assigning a variable in our data set to the colors. To do this, we can type ggplot, and then first give the name of our data, like before, WHO, and then in our aesthetic, we again specify that x = GNI and y = FertilityRate. But then we want to add the option color = Region, which will color the points by the Region variable. And then we just want to add the geom_point function and hit Enter. Now, in our plot, we should see that each point is colored corresponding to the region that country belongs in. So the countries in Africa are colored red, the countries in the Americas are colored gold, the countries in the Eastern Mediterranean are colored gr...

In this video, we'll see how to build a CART model in R. Let's start by reading in the data file "stevens.csv". We'll call our data frame stevens and use the read.csv function to read in the data file "stevens.csv". Remember to navigate to the directory on your computer containing the file "stevens.csv" first. Now, let's take a look at our data using the str function. We have 566 observations, or Supreme Court cases, and nine different variables. Docket is just a unique identifier for each case, and Term is the year of the case. Then we have our six independent variables: the circuit court of origin, the issue area of the case, the type of petitioner, the type of respondent, the lower court direction, and whether or not the petitioner argued that a law or practice was unconstitutional. The last variable is our dependent variable, whether or not Justice Stevens voted to reverse the case: 1 for reverse, and 0 for affirm. Now bef...