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the following content is provided under a Creative Commons license your support will help MIT open courseware continue to offer highquality educational resources for free to make a donation or view additional materials from hundreds of MIT courses visit mitop courseware at ocw.mit.edu so what is an 18821 p um well it's no more and no less than uh a presentation of the project that you've been working on as you've defined it and an account of the results that you've obtained in uh in studying that Pro problem and so these findings can come in many different forms um uh theoretical mathematics uh um you know the the gold standard is a proof a rigorous proof uh that's great if you can do it um that's great if the problem admits that kind of thing um but there are many other kinds of uh findings that you may want to report on in this report as well you may very well come up with things that you are damn sure are true but you can't figure out aof of...

One of the central tasks of this course has been to unpack the ways in which work in today's economy has changed over time. Often, when we talk about this topic, one of the first things that comes to mind is the role of technology in transforming work. Over the past few years, we have seen an explosion in what's known as the on-demand economy. It's visible everywhere. We as consumers are able to have our food delivered, grab a ride home, or even have someone run their errands, all purchased and arranged in a matter of minutes from our smartphones. Our colleague John McCarthy talks in more detail about these new ways of organizing work in a different video lecture in this course. What we're going to talk about now is how these employment arrangements are connected to two issues-- one, how we think about workers' rights in the on-demand economy, and two, how technology can promote new forms of worker advocacy in settings where gigs, not full-time jobs fo...

The following 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 to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. MICHAEL: We're going to talk about the Wigner distribution function and integrate imaging. AUDIENCE: [INAUDIBLE]. MICHAEL: Yeah. So I'm going to start with a description of what a conventional camera does. A conventional camera produces one view of something. It does a pretty good job of imaging within a small range of distance from the camera depending on how you focus it. And what we can do is create optical systems which allow image sensor to capture multiple views. So basically, let's say, you have an array of tiny cameras that share one image sensor in the back, And one way of doing this is to use a pinhole or microlens array. So think of the pinhole cameras of yore...

PROFESSOR: So we go back to the integral. We think of k. We'll write it as k naught plus k tilde. And then we have psi of x0 equal 1 over square root of 2pi e to the ik naught x-- that part goes out-- integral dk tilde phi of k naught plus k tilde e to the ik tilde x dk. OK. So we're doing this integral. And now we're focusing on the integration near k naught, where the contribution is large. So we write k as k naught plus a little fluctuation. dk will be dk tilde. Wherever you see a k, you must put k naught plus k tilde. And that's it. And why do we have to worry? Well, we basically have now this peak over here, k naught. And we're going to be integrating k tilde, which is the fluctuation, all over the width of this profile. So the relevant region of integration for k tilde is the range from delta k over 2 to minus delta k over 2. So maybe I'll make this picture a little bigger. Here is k naught. And here we're going to be going and integrate ...

Hi. In this problem, we'll get more practice using conditioning to help us calculate expectations of variances. We'll see that in this problem, which deals with widgets and crates, it's actually similar in flavor to an earlier problem that we did, involving breaking a stick twice. And you'll see that in this problem, we'll again use the law of iterated expectations and the law of total variance to help us calculate expectations of variances. And again, we'll be taking the approach of attacking the problem by splitting into the stages and building up from the bottom up. So in this problem, what we have is a crate, which contains some number of boxes. And we don't know how many boxes are. It's random. And it's given by some discrete random variable, n. And in each box, there are some number of widgets. And again, this is also random. And in each box, say for Box I, there are xi number of widgets in each one. What we're really intere...

[MUSIC PLAYING] DENNIS MCLAUGHLIN: Students come in with opinions already, and they can say things in a seminar that are based upon just their common knowledge and not based on the readings. CURT NEWTON: Today on Chalk Radio, we're exploring global food production, water, and climate change, and the challenge of teaching when polarized opinions come to class. DENNIS MCLAUGHLIN: I'd say generally, the students learn that most of these issues are more subtle and complex than they thought when they came in the class. So one of the things that happens at the end of the class is, people's opinions are usually not as polarized as they were in the beginning. CURT NEWTON: I'm Curt Newton, your guest host for this episode of Chalk Radio. At MIT, officially, I'm director of MIT OpenCourseWare. But for so many years at MIT and in my personal life, I'm also doing a lot of work on climate change. So when Sarah Hansen, your usual host, lined up this episode deal...