I know I've been like, hello, good evening. I'm not Greg Roshke, as you can see. I'm Jill Sexton, Associate Director for the Digital Library at NC State. Greg was, his flight was canceled with the snow and Raleigh, and so I'm a stand-in, so please bear with me. I saw this presentation about 30 minutes ago. So I'll be talking through his part of it, but you're luckily in great hands. My colleague, Jason Evans Growth, is going to do the bulk of the talking about the work that he's done in the libraries and he's seen this presentation before. So I'm here today to talk to you about the library's a platform, technology, and the transformed libraries impact on teaching and learning. At NC State, we believe that the transformation of library spaces and the associated technologies provides significant opportunities to fundamentally engage and foster creative models of teaching and learning. A combination of high technology spaces, flexible learning environments, and library expertise that we have at NC State, has fostered an experiential learning environment that faculty and students have leveraged to enhance the educational experience. This session will explore several examples of experiential learning, creative uses of high technology spaces, and the important combination of ingredients that we use to maximize the experiential learning potential of new library spaces, which repositions our library as a fundamental difference maker in the educational environment provided by colleges and universities. One of the nice quotes that kind of encapsulates this is from one of our faculty members in the English department, Paul Fife, who in talking about the libraries has said that the libraries has been more than a service provider, even more than a collaborator and partner, they've really expanded his idea of the possible. And so what follows is a brief story of how we move from being considered a service provider to an expander of the possible on our campus. So as delivery of core curriculum and foundational learning increasingly moves online and in asynchronous environments, the libraries become a platform for deeper learning and a lead distinguishing element of the educational experience for the university. It's become a difference maker and why students choose to come to NC State to build their futures and how they actively learn while they're here. So the experiential library is a library where the full suite of a college or university's experiential and active learning offerings are available for the entire community and the next generation of experiential learning is pioneered for faculty and students. Today we know that things like maker spaces, high technology and immersive environments and virtual reality are part of active learning environments. Additional partnerships can add key experiential offerings like things like undergraduate research, the writing centers, career counseling and tutoring services. In the future, libraries should incorporate active learning classrooms embedded into library spaces and create more environments for the exploration of new technologies like virtual and augmented reality. Just see some examples of our work in this area coming up next. Things like student scholarships and fellowships funded by private support will help the library attract highly skilled students and promote robust peer learning networks and again, which we'll talk about further on in the presentation. Partnerships with and outreach to faculty by expert librarians will expand the volume and nature of how faculty and students use the library and library provided experience for learning spaces and services will be embedded into the curriculum at all phases in every college. This is obviously, this is Greg's aspiration, sorry. So we believe that the library needs to be the centerpiece for active and collaborative learning spaces on campus. Some of the opportunities that we can gain when we take this attack, we can reconnect faculty and students with library spaces through a chain of engagement and creativity. We incubate emerging technologies, have enhanced hands-on learning and engagement and with new kinds of creative pedagogy and we really provide a platform for programs and workshops. So I think, here we go. That's what I should be doing. And so what I'm gonna do next is turn it over to Jason who's gonna talk about some of the examples of the types of learning spaces and projects that we've got and he's gonna talk about some of the cool things that we're doing with these ideas. Thank you, Joe. Hi, everyone. I'm Jason Evans-Growth and I'm the Digital Media Librarian for the Learning Spaces and Services Department at NC State University Libraries. I started at NC State in 2013 as a Libraries Fellow, along with I think one of my Libraries Fellow classmates is in here also. And as a foundational standpoint, the Fellows Program I think is a good place to start with or at least a good example of some bedrock that lays the foundation for all of these aspirations to be the experiential library. In that, when Fellows come aboard, we're given time and resources and often projects that we wish could be done that haven't been done yet and then given some time and resources to do those projects. As if we're forging ahead, seeing if the stuff will work and often, if not always, pretty often, those projects and work that the Fellows do in that two-year appointment becomes a really big part of the library. So we're invested in this idea from the staff, which is where it's really important because that's who's doing the outreach in a lot of ways. And this is not just limited to Fellows, of course. It's just one of the things that comes out of this investment is that program. So I think that having staff on board with this idea of really becoming partners and building community and recognizing community and then showcasing it through all of the methods that we have at our disposal, it becomes a really a big driving force in making ours an experiential library. You see in this example page just a few things that we'll go over later, not the least of which is that picture of Chewbacca next to a bodybuilder from 1893. All right, so let's talk about immersive pedagogy. One of, that's one of the things that I think people see a lot of pictures of these giant video screens that we have at both of our libraries that we actually use to showcase student work and to engage with faculty and to create community programming. You'll see on the top is an example from a class with which I worked, English 585 graduate studies in film that was all about female directors. That is a 20 foot wide screen that's available, that's easy to see as you walk in to Hunt Library. 3,500 people are so pass it by every day and the group of people, most of the group of people watching this right now is the class as they're presenting their projects to each other about a single female director where they had to go into that person's filmography, their biography and create essentially a few things. A museum piece, a moving image look at all of the work that they had done and then some still images that talk about the director and behind the scene stuff too. So in this way, they've created something that they can put on their CV, they're engaging community at the library in film history and the idea of feminism in film, for example, they have to write something that is viewable by the public, they can put this on their resume, they have to take workshops to learn how to make movies and they have to work with research librarians and digital media librarians to put it together. That's a lot of engagement in one project that then lives in one place that anyone can come and experience. So we have the story behind it and then it tells its own story. A couple of more examples below of just some engagement and immersive learning. There was an animation class in our teaching and visualization lab. That's Dr. Matthew Booker on the right at the bottom talking about data and how it works with history and he looks dwarfed by the size of the projection there but he actually fits very well inside of it. Anyway, this is something that we've been working with in various forms forever. We learned it from museums, we learned it from grade schools, we learned it from PBS and now we can make it more interactive by making these technologies a little more democratized. It really started with this thing, our first big project which we're still talking about but I want to mention why we're talking about it now is Virtual Pulse Cross. Maybe it's been mentioned at CNI before. The cool thing about this was it was taking a very, a classic work and turning it into a 3D immersive world inside of a studio. I'm not gonna go into the details of it because you see it here but we recreated a cathedral in 3D and there was actors reading the words of John Dunn and all these beautiful reenactments and it took a team of so many people who were all experts and a long time to make it happen and it was super inspiring but it was also a little overwhelming for people, faculty members and students to come in and say if this is what is expected of me I'm not sure I can live up to it. Well they could and we wanted to be able to give them the resources to do it. It never has to be this but it can take the tenants of this and the tenants of this and make it their own which I think really helped. People were coming into the library on their own to see this happen. We proved it could work that way and then our job was to make it possible to do these kinds of projects with two people rather than like 20 and in Raleigh rather than internationally although both are welcome there. So we started there and then we started branching out. In 2014 we, Professor Marsha Gordon and myself worked with a graduate student class to take our video game lab which also has a 20 foot wide and five foot tall touchscreen or at least it did then and turn it into a video exposition room. The same way you might, the same thing you might find in a museum. So her students were asked to build a three screen simultaneously playing video project about American documentary filmmakers who experienced American conflicts. That's why they were shooting wars and to basically tell the story of a different American conflict. And they were asked to do this in a way that we had never used the room before. They were asked to do it in a way that they'd never experienced before by actually finding all these primary source materials, editing them together and building an interactive museum display inside of the library. And then they had to be docents for it and advertised for it too. But we didn't check the audio which was something that we probably should have. Classic digital media librarian move right there. But let's see what happens when a digital media librarian, I'm gonna play this because I think it's interesting to hear their voices. And when you can't hear their voices, I'm gonna do this real quick. Wait one second everyone. This is the other thing you learn is how to improvise. I'd love to sing you some lines from Hello Dolly for a moment as I reconnect this output to, may I have some sound in the back by the way? It looks like it is outputting by HDMI right now. So, do we have a volume control up here? Oh, it's set to output. If they don't talk, we'll just watch them for a moment. Maybe I can pretend, actually you know what, it'll be even more fun if I just guess at what they're saying. Yeah I do, oh yeah, perfect. So one of the things that we're also really concerned with is getting the voices of our community out there. And by that I mean like hearing them talk about their experience is a little more interesting than hearing me say how great of an experience they had. At least they think, let's find out. And you still can't hear them, so here's what they would say. That's Jason Buell, who's talking about how they used the library to both build and conceive the project. That's Professor Gordon, saying that she really wanted to do this kind of thing forever. This is an example of what it looked like, but imagine that 20 feet wide. So they had two moving images and text in the middle. Here Jason is saying, there we go. Let's try this again. All right, so this is important. Did you like what I did though? Thanks. So this project really, especially in the library, not just for research in the traditional sense, but for the places that research can go and gave me a lot of appreciation. Collaboration between students and librarians and faculty will define the times. People are gonna see it outside of the class. It was for more than a period of time. Things that might have once been tucked away and essays and journals can now get out much more easily. Just think about the potential, give meaning to a 21st century library. It was a nice experience. So they do say it better, but I think that's what we're trying to get at here. It's like using our resources to actually make something that gets out of the hands of the professor and into the world at large. People have things to say, we respect our students and faculty. We want them to use the library as a venue. To use the library as a platform. To use the library as a canvas and then we can create an experience and help collaborate on experiences to make that work too. I'm just gonna show this because that moved down into this where Professor Gordon then asked her undergraduate students to think about making movies using our technology lending program. This is them recreating the 1894 Sando picture that Thomas Edison Studios made. And in this short experience, the student makes comments about misogyny, femininity, the female body, the male body, but also who, like many of our students, have no idea that movies were being made in 1894 nor have they ever been asked to make a one shot film. And in this case, in-camera editing. And so people walking past this, not only see this, but they also see metadata about the class. Maybe they take that class, maybe they go to the Ask Us desk, which is what we call our front desk, and ask what camera they used or what kind of video editing software we can offer them and then we can check that stuff out to them. That is an experience that we can't often just predict. That's an experience that happens. So we have to use serendipity and then thoughtful curation of student work to make this go. We do this with physical spaces too. So we've afforded ourselves some physical spaces. These are some pictures from our experience in King project. Dr. Victoria Gallagher, Dr. Jason Miller and Keon Petaway put together after being inspired by finding a tape of Dr. King speaking in Rocky Mountain, North Carolina, an early version of speeches that we would then hear later and then coming to special collections while I worked there, asking us if we could digitize it, finding out he couldn't at our library, but then getting it digitized by an NC State alum and coming back with the idea for this project, meaning the library was there the whole time, which I think is awesome. He and Dr. Gallagher and Keon put together a whole day of events that involved walking tours, utilizing our big screens and then human speakers, analyzing this stuff for us. So that was physical, but then we also made some virtual experiences with this. You can see a motion capture suit on the actor that portrayed Dr. King in one of our virtual experiences. You can see Dr. Derek Ham using a personal VR rig to experience a rebuild of the church in which Dr. King spoke in North Carolina. So you can experience that with the original audio yourself. You can find out from that that we have a virtual reality studio at the library and the library can also now help you digitize your quarter inch cassette tapes if you need it. And then at the bottom, you can see the game lab again, used in a different way as a game simulation for people to feel what it was like to be part of the civil rights movement. So that faculty to recreate the experience that students could engage with, bringing back something that was lost to history through the library, through technology and through human engagement. We were the platform. There were lots of kinds of modes used here, AR, VR, preservation of audio. And we get to now spin this up. Unlike the St. Paul's Cross thing, we can actually recreate this in various modes in various places because of VR. But if we wanna bring the whole thing back up, we can. And we see about 800 people come through and experience this every semester now. So let's talk about other things. How do we get people to experience making? Well, maker spaces are helpful. You can see some examples of some of our early stuff in our Hunt library. We had a maker space, we have a maker space that's more of a service point. And on the top right there, you see our jar with a twist prototype. Some students decided to come up with a better way to get peanut butter out of a jar. And that's been displayed. That was prototyped in our maker space on 3D printing when 3D printing was just at the burgeoning point of being all the rage. And that has lived in our technology showcase forever because it shows that convergence of idea to an actual product. We have on the left here, we have the pulse dress with Jazelyn McNeil. And if Lauren's in here, I think she was helpful in making that happen too. But wearable electronics in fashion, we are a design school, we're a textile school. How do we get these things to work together? And then after the service point was popular, we built a really hands-on one to democratize that stuff further. On the bottom, you see some examples of people during our make-a-thon, which pairs with the Office of Sustainability to help students think about how they can use their hands-on skills to make things in the maker world that helps with sustainability. I think making is more than 3D printing, as we all do, I'm sure. And I'm the media librarian, we've been making media forever. And we still don't know how. We still learn how to do it better every year. One of the first coolest projects I think that ever showed the library as an experiential platform for sharing and community was our My Hunt Library Project, where we asked students to use Instagram to tell their story of our new library. And then we kept that stuff in our special collections, which they were excited about. They were thrilled we were archiving that, which was a surprise to us, although it shouldn't be. I don't know why we're such a humble group of people, but we are, and thankfully, I guess that's true. But anyway, we keep it, they like it, and then we show it back to them on that giant screen. To show you the size of that picture, the Contemporary American Female Directors Project, which I mentioned before, is the same width on that 20-foot screen as all the pictures that were taken in Instagram. So it's telling people that you can do stuff, even if you don't think you have the tools to do it, and we can help you figure that out. And we've moved that idea of collecting from our community and building it more to introduce them to each other through something called State of Sound, which I started to help pioneer about a year ago with a really great team of media folk. State of Sound is just that. We want to collect the state of soundmaking at NC State. We want to get podcasts and the music that gets made in our 11 audio and video recording studios. We want to partner with people like Moog on the right there and have a dial-tones event where I am Jill here, is improvising on a Moog workshop synthesizer, and then we write programs to visualize what she's doing behind her. And you can see below an image of our music studios too. We want to bridge disciplines by making people who maybe make music, but don't think that they can make a program or write in Python and have them find those two things in one place. Our make-a-thon's a good example of this. This group made a town out of paper that was controlled, that became a virtual environment by putting on a headset and connecting it to a GoPro. So while you're looking in 3D, you're actually seeing classic Walt Disney-invented paper 3D towns, crazy. Here you see Lucas who's animating 3D structures based on sine waves. And I think that what we can say is that we've learned a lot from all of this, but we're driven now even more to showcase the amazing work because the more the ball gets rolling, the faster it goes. So I'll have Jill close us out, thank you. So you've seen the kinds of things that we're working on in our library. Obviously, we didn't get there overnight. We faced a lot of challenges in getting to the point where we are. Some of those were an initial lack of awareness of the kinds of services and the skills and abilities that we can offer. Clearly, once we start offering these kinds of things, we have an increase in demand. So how do we meet that demand? We have to build up the staffing expertise and the capacity to support these kinds of new services that we want to offer. And then we need to work to build these kinds of activities that we're fostering in the libraries back into the classroom environment. So how do we do that? We do that by taking advantage of the opportunities that are unique and beautiful environments, technology, staff expertise, and their skill in providing programming workshops provide. So you can't just build a beautiful library and expect that people are gonna just show up. You have to draw users to your library and show them the possibilities to spark their imaginations. And I think you saw a lot of those examples with the kinds of projects that Jason talked about. We'll show some more examples next. So we do it with a rich portfolio of programs, exhibits, and events that drives greater usage, creates public engagement and the scholarship and impact of the university. And we've seen that time and time again. Much like museums, libraries now should have a constant flow of programming that connect various communities and bridges students, faculty, alumni, and the community. And if we do it really well, it adds to the diversity of the people and ideas who are engaging with the library. And finally, we wanna offer a portfolio of workshops and trainings that move faster than the curriculum of the university is able to to give students the latest skills and technical mastery to enhance their standing when they graduate and go off to seek their new jobs. I'm gonna give some examples around data visualization in all of these areas and how we engage with this subject area in the libraries to further it for the campus. So first, we offer a wide slate of workshops and data visualization, data science. These are some of the examples of the workshops that we've offered. I think this was last year's slate of workshops, which is kind of an example. I wanna quickly mention one of the ways that we're able to do this is by the creation of the program called the Peer Scholars Program, where we recruit graduate students who have specialized skills that library staff do not have. So particular statistics, expertise, LaTeX experience, and so on. And we give them opportunities to get paid to lead workshops for the library. And if it comes a really great opportunity for them, they have a resume builder. We create nice headshots for them. We record their workshop sessions. We can offer kind of peer critiques of their teaching ability. And then they have an online example of their teaching to show when they're on the job market and looking for work when they graduate. The Peer Scholars is one of the ways that we were able to, and just this is last year's workshop numbers in the library. We were able to deliver over 400 workshops, reaching over 6,000 students for 545 hours of workshop training in all these different areas. You can see we used 33 librarians, six staff members, and 11 students to accomplish that. In addition to workshops, we have an active slate of programs that engage the community and bring people in. This is a picture from one of our Coffee and Viz programs. It's one of our most successful public event series that we have ongoing in the library, where each month we bring in a faculty member to show off their work that they do in a wide variety of academic disciplines. Another example of a program, not necessarily in data visualization, but we not just have faculty research, we also make a point of showing off undergraduate research in the libraries. Every, I think every semester, so. At the end of the semester, we have our kind of undergraduate honors research slam where the top research papers written by undergraduates are those students have an opportunity to present their work in the libraries and share it with their peers. Another example of one of our most successful programs, Jason already covered this, so I won't share a ton, but the Experiencing King program, where we just brought in thousands of people from the public to tour the library. It was a real event. It was a really moving event that brought a lot of goodwill, I think, from the community as well, and again makes us a recognized force, not just on campus, but in the community of Raleigh. State of the Sciences is another really cool example of one of our really successful programs, public science outreach, where we invite people from the community come in and we have stations set up all over the library, different science related topics of research that's happening at NC State. This is a group of students from the dance department who are acting out the movement of ions around or electrons around a nucleus in an atom, so. Right, it was cool. Do we have time to play this? How long is this? I don't know. What time is it? D, if I can get this to play. You know, the first time I talk to Susan, Susan, I'll pause to provide a bit of a context. This is David Silver, a visiting faculty member from the University of San Francisco, University of San Francisco State, who came to the Hunt Library when it was new to kind of show off his work in the space. And so this is his example of, this is just a brief clip of him talking about why he wanted to come and the value he sees in a kind of library that does the kind of stuff we did. Is there any way I could come to the library to work with your video walls? I said that about two years ago. Last year I said something different. I said, is there any way I can come back to your library to work with your staff? Do you remember that? It's the staff that I'm after. This is a team collaboration. One of the things I've been really inspired about Hunt Library and about Hill Library too, if you all haven't been to Hill Library, that's an equally astonishing library. Is that something about the librarians here and administration have figured out that today's students do not work independently. They work in groups. And if you walk around this library, you'll see them working together. You'll see them moving the furniture. You'll see these once nice tiny desks turning to these circles. And it kind of proves to us that today's students understand that collaboration is key. Just a couple more slides. So I wanted to talk a little bit about how we grew capacity to be able to offer these kinds of experiences. This is kind of a cycle of like, I could spend probably 20 minutes, 30 minutes talking just about this. We don't have that much time. But it really kind of starts at the recruiting phase at an organizational level. And when we recruit, we recruit for people who are curious, who have an expectation that they're gonna continuously learn as part of their job. Then crucially, once they get here, we give them the time and the resources they need to be able to develop those new skills, time to experiment. And it's not just lip service to the idea of trying new things. We really support that through a variety of means, including funds to support new ideas through a program called the Good Ideas Grant. Then we build this organizational expertise and we grow it by teaching each other how to do things. We also provide formal learning and training opportunities. We do put a substantial amount of time and resources into professional development opportunities. We then provide, again, the opportunities for the people who are learning these new skills to apply them, to engage and gain a sense of purpose and mastery of those skills. And then it's just kind of a cycle that goes round and around. So that's kind of it in a nutshell. This is the final slide. We've got a couple of times for questions. This is, again, that quote from Paul Fife about the library being a platform to expand what researchers' ideas of what's possible in a library and that the university are. So thank you.


Well, thank you. I'm grateful for being here and I want to thank the organizers for this nice and interesting conference. I'm going to talk about conserved energies, and this is joint work, conserved energies for NLS, MKDV and KDV, and this is joint work with Daniel Tataro. And before I state exactly what we do, I want to give the equations. So there's the NLS equation. It's I, one-dimensional IDTU plus UXX is equal to plus or minus U squared U. So plus is, in the minus case it's focusing, then there are solitons, in the plus case it's defocusing, and there are no solitons. Then there is the M modified KDV equation, which is UT plus UXX, and I realize that the first thing I didn't want to put attention to, plus or minus 6, what do you want to write it?  Okay, now I want to add in a plus or minus, and then it's the focusing or the defocusing one. I want to write it with the absolute value here in order to allow complex solutions, then it's a complex MKDV, otherwise it's the real one. And then there's a KDV equation, which is UT plus UXX minus 6 U, UX is equal to 0. And all these equations are pretty connected, they are integrable, and the integrability shows in the existence of a lux pair, and I want to write it in the following version as a system, psi x, psi is a, let's say x derivative of a map to C2 is equal to, I know I only write it for the defocusing case, for the plus case, I C U U bar minus plus I C C, so this is the lux equation, C is a complex parameter, and the amazing fact is that if you complement it with an equation for the time derivative, is equal to I, I have to look it up because I can't remember these things, plus U squared value U squared to ICU plus UX minus 2 I C U bar minus U bar x to C squared plus U squared C. So, these are two first order equations, and the interesting, the very interesting structure is that if we want to solve them simultaneously for given data at one point, general, then this is only possible if a compatibility condition is satisfied, and the compatibility condition is exactly the defocusing NLS here on this side. So, and the connection to MK2V is the following, so for MK2V there's the same structure with the same operator here, the same equation here, and a different equation on this side, so there's a different time equation for the time derivative, and then there's the same structure, this is solvable for general, for given data at a point, let's say 0, 0, x equal to 0, t equal to 0 for general data, if and only if, then U satisfies the MK2V equation. And for the focusing case, there are some sign changes which I don't want to go into, so the only case, the only sign change here is a minor sign, and here it is more complex, but this equation doesn't play a big role for what I'm going to tell. Now, what is the effect of that? Well, suppose we take c, let's say we say imaginary part of c, larger equal to 0, then we might look for your solutions, I mean it's a 2 by 2 system, so there is a two-dimensional space of solutions, and I want to normalize them on the left. So if I want to have that c behaves like e to the minus i c x, 0 on the left, so the case to the left for x to minus infinity, then on the right, let's suppose that u is compactly supported, then on the right it decomposes in a linear combination of two, the similar fundamental system on the right, then on the right we get the decomposition that c behaves like t to the minus 1 of c e to the minus i c x, and well, here we have to be a bit more careful as x tends to plus infinity if c is a real number. So we get a linear combination of the two solutions, which you would have of a fundamental system, which would have a few would be equal to 0, and I use this to define t to the minus 1, which works well even if the imaginary part of c is positive, and there's a reflection coefficient, this is called the transmission coefficient, and the reflection coefficient, so t is the transmission coefficient. Now the fact that the equation, the analysis equation is the compatibility condition for solving these equations allows us to solve this equation at plus infinity, and check what happens with this plus infinity and minus infinity, and check what happens with the function c. So at plus or minus infinity we don't see the effect of u, so what we see here is a constant coefficient differential operator, it gives the time derivative of c in terms of c here, I know u, so we can check this, and if you check this on the left, then there are some time evolution, if you do it on the right, there's some time evolution, and what we see down is that t of c is independent of time, if you have a solution, if you have a solution to the analysis equation. So whatever, if you look for quantities which are conserved for NLS or MKTV or KTV, we simply have to look at the transmission coefficient, I have to see whether we can define it, and this is a conserved quantity at each point, we may integrate it, we may do whatever we want with it, and this gives a conserved quantity. So now this is some sort old, fairly old, it goes back to, well in this context to Aplowitz, the AK and S, this is Zeguer, Newell, and K-Kalp, sorry, so they did it for the non-inertial equation, and then there is an expansion of t of c, which is equal to minus one over two high i times a sum, this is formal, I should try to make this, this is a formal sum, and then there are the energies Ejc to the minus j minus one, well so this is at c tending to i infinity, or depending on what you assume on the initial data, anyhow it's the Laurent series at infinity, and then E1 is equal to the integral u squared dx, and E2 is equal to the integral ux, sorry u2 is equal to i times u u bar x momentum, and E3, that's some new stupid, I want to have this one, this 0, 1, and 2 equal to the integral ux squared plus 2 times u to the power 4, it's u to the power 4 dx, which one is it? So, and then you can continue, and it's classical to expand these things and to get recursive formulas, and what I want to do is I want to do it in the John Dewey-Pistanio, we sort of interpolated between these quantities in order to get a continuous family of conserved quantities, well that's not exactly what we did, so more or less more precisely we interpolated between the even ones, and not even that, we interpolated between linear combinations of the even ones, so here's what we get, well the things which we define, so let's first look at the focusing case for analysis in MKDV, then ES can be defined as an integral over, well at least for Schwarz functions, for Schwarz potentials as an integral over the real line, over the logarithm, it's always the logarithm which occurs of the transmission coefficient, evaluated at the real axis, so for that we need some decay and some Schwarz functions, and we have to study the decay which we don't do, but let's suppose that the logarithm of t is a Schwarz function, so say a bit more about when these things are defined later, so the one way to defining it is to integrate over the real axis, and to put a weight which is something like the Fourier weight for Sobolev spaces, and c to the 2s, so this is for s larger than minus a half, and then another way of expressing that is to do, to move the contour of integration to the line from i to i times infinity, and that gives the second definite, the second formula, which is equal to 4 sine pi s integral from 1 to infinity, so now we are writing this in a real fashion, torsqit minus 1 to the power s, time this logarithm evaluated at i tau over 2, I see I forgot factor 1 over 2, it should be evaluated at psi over 2 on the left hand side, so we evaluate it at i tau over 2, and then we have to do some corrections, because the weight c to the s goes at infinity polynomially, so we have to correct that in order to be able to do a contour integration, so we subtract the, I messed up, I wanted to call these guys h, so h1, h2, yeah, so we correct, we have to correct the behavior at infinity a little bit, thank you, oh sorry, yeah, so that's the only place where I wrote it, okay, so we have to correct the behavior at infinity in order to get, to be able to work with a contour integration, to move the contour, and if we do it in this fashion, then this doesn't depend on how many terms we correct if we have the regularity, and so we get this formula here, what you see here is that if s is an integer, then the sine pi s vanishes, and here you get the linear combination of the classical conserved quantities, so in that sense we interpolate between the classical conserved quantities, well then we can do the same thing for the focusing, nls if you do the same thing for the focusing nls, then we have to adjust for eigenvalues of the akns operator, so the akns operator has eigenvalues in the upper half plane, and we have to adjust for them, and there's a function, there's a way of doing that, so we have to add a sum over the eigenvalues with a multiplicity mj, over sum, with some function xi, and nothing changes on the right-hand side, now for kdv, that's the same structure, first if s is equal, is larger than minus one, then this is a generalization of the classical formula of a df, there's a different function xi, we have to adjust for the eigenvalues, so the eigenvalues in this case are minus, no, kj squared, and on the line from i to i to infinity, there's a similar expression as for the analysis equation, and again we have to adjust for the classical energies. Something special happens at s equal to minus one, so the left-hand side makes sense for minus one, and for the right-hand side there is this part of the integral, which makes, has the effect of converging to a delta function combined with a sine pi s, so the outcome is then eight times the transmission coefficient evaluated at a half minus this integral u, and here one has to do some normalization, which I don't want to take much about, talk much about, so these are conserved quantities for nls and mkdv, they go back to akns or to fadeev, this relation between the different contour integrals, and these are the objects we want to work with. Now defining these objects for Schwarz functions is trivial, because then the scattering transform becomes something nice, the logarithm of t is a Schwarz function on the real axis, and then it's an exercise in complex analysis, maybe not an entirely easy one to get these formulas, so as I said above, since t is independent of time whenever you have a solution whatever you do with t is going to give you a conserved quantity, so the question is how to relate them to things we work with in analysis, how to relate them to so-called spaces, and I want to give the results first and then explain a bit about the proof, thank you. You see how it works. Yeah, I tried it before, but it's always a bit different when I do it on the blackboard. Is it okay? I guess you still can have an impression of the formulas even if you don't see all of them. So the theorem, that's Daniel Tataru, so what we do is we, I mean we look at the right-hand side, we define the quantities by the right-hand side and then it turns out that it makes also sense with the left-hand side, so the first thing is if u is in hs then both sides are well defined, both sides are well defined, but basically you want to look at the more on the thing on the right-hand side and so the map from hs to es is continuous in u and also in s whenever u is sufficiently regular and it is analytic if i over 2 is not an eigenvalue, but you can't expect it to be an eigenvalue if this, so this is in the focusing case because then this function psi is not analytic at 1, so this has to be excluded. But otherwise it is analytic and it's jointly analytic in s and u in the appropriate sense, so whenever u is in hs, then it's analytic in s for s when es is analytic in u and sigma, u and s when s is less than sigma, so I don't want to write that, but in a appropriate sense it is as all the properties one would want to have it and the nice thing is we can play with both sides in order to get whatever is more convenient. Second, well how does it compare? I mean so this says that it's analytic, so whenever u is in hs then this is defined, but the question is does this control the hs now? Well so we can look at es of u minus u in hs where it's just basically the same as the left-hand side, but for the Fourier transforming value aided at I didn't correct all the xc over 2sc. So this is on the Fourier side similar to the integral on the left-hand side and this is less or equal to a constant times u in hs squared and now I want to use a strange norm here and I want to explain it later on if u in l2 d2 d u2 is less than delta for some delta depending on s. So this tells us that the quadratic part of I mean it's analytic so we can do an analytic expansion in u at zero and the quadratic part of that is exactly the hs norm. So for small if this quantity is small I'm going to say more about that in a minute, if this quantity is small then this quantity and the hs norm are pretty close together. And also what happens in that case is this function psi of the eigenvalue kappa j is larger equal to a constant times kappa j to the power 2 sorry the real part of kappa j, Japanese brackets to the power 2s times the imaginary part of kappa j. So all if this is small then all quantities on the left-hand side of the formula are non-negative. So there's a similar statement for kdv this is nls bigger than minus one half and there's a similar theorem for kdv where there is the same sort of statement for s larger than minus one that's the same statement and if s equal to minus one then it's the same statement but for h minus one we have to take out the potentials so that minus one over four is an eigenvalue of the Schrodinger operator which makes sense because then we want to evaluate the singular function this logarithm at one half and that we can the logarithm at one minus one at zero and that we can't. Okay now some corollaries the first corollary corollary if you use this on scaling then we get that if u zero is in hs I don't specify which problem but if you're in this range then the soup of u of t in hs is less than infinity and it could be made more precise for the solution for nls mkdv whatsoever. The second corollary is well and if you look at solitons solitons correspond to eigenvalues at prisms and anything else what you want to define in terms of the spectrum then they're all stable hs for all s in that range. So this is a question of scaling you might scale so if you look at the formulas now we look at the left hand side then in the left hand side if the smallest condition is satisfied then every term is positive and if you look at the second line then the soliton is the one where the logarithm of t vanishes on the real axis and there is a one contribution corresponding to the eigenvalue so this is the lowest value es can take whenever you have the soliton so this es is a lopinot functional and it's minimized at the soliton and similarly at the prism which is also characterized in terms of the spectrum. Okay so there's one thing on the blackboard which i didn't explain and i don't want to go into it i at least not too much but i want to give some information i have this du2 so what this u2 is it's something like h minus a half h plus a half if you don't get derivative so h a half is the space with half a derivative and it has doesn't have any good structure because there is no embedding into continuous functions l1 doesn't embed into h a half so this space doesn't have the homogeneous space this this space is not good for most things we do in analysis this u2 is a replacement of it and it has all the good properties one would wish to have or many of the good properties one would wish to have from the space with half a derivative and it's still on the same scale it's on the same scale also in the sense that if you look at the best of space then b one half one one embeds into u2 embeds into one half one infinity so it's it's a very close space to this h a half and du2 these are the formal derivatives there are like things with h minus a half same sort of embedding and then there come similar spaces for the same properties v2 and dv2 and there is a good duality between these things so what do you have here you have embeddings into bounded functions you have limits at infinity for these functions you have embeddings of l1 into du2 into dv2 and so on so all the things which don't work for which don't work for h a half or h minus a half many of those things work for u2 and v2 so in particular what is relevant here so if you look at this l2 du2 is defined by the norm so we take a function u and then we multiply it by a smooth cutoff function x minus j so this is something which lives with the bump of size one we take this thing into u2 and this is pieces and then we take an lp summation over the l2 summation over that and similarly with lp so we take small pieces we sum them in l2 and lp and then this l2 if you look at this l2 du2 norm then this is less or equal to a constant times u in hs for all s larger than minus a half so whenever you could possibly hope to get a good bound like that then this quantity is going to be bounded and it's smaller than it's controlled in terms of all double left spaces so I could replace it by any hs here h minus a half plus epsilon and then the state that's the same statement is true so can I ask a question though but if s if you take a u in s between like minus one and minus one half that could be infinity so you have a bunch of things but you don't have line two is that correct and i take u in hs purely in hs but s less than one half minus less than minus one half yeah so then you have the first part of the theorem you don't have the second part of the theorem then no so the assumption s larger than minus one half is for both parts oh i see so nothing of that makes sense i'm not quite sure about the sums for some sides of the equation but nothing makes sense if you and what's the analog for the kdv part of the u space ah so it should have given that so for the kdv part you have to replace you simply replace l2 du2 by h minus one half and then minus one okay um well what i present here is connected to a lot of literature and i can't go uh into all of that i mean the inverse scattering part for this akns system is due to uh up with carb nebel and ziko um fadeev and many others lax i mean i guess there's no point in repeating this the history inverse scattering and i'm sure that there are other people in the audience who can do that much better than i could do it um for the a priori estimate i should say there is a sort of a history which is different there is maybe i don't write it there is there are a priori estimates by christ coliander and tau for the uh non-linear schroedinger equation i did some a priori estimates not uniform in time but control over all for all times somehow with daniel to tau up to h minus fourth for the non-linear schroedinger equation um there is a lot of work on the kdv equation um i did some work on the kdv equation in h minus one with with tristan backmaster some uniform in time a priori estimate in h minus one which used a bit of inverse scattering but not much of the structure uh there is the work which was done simultaneously by kilibh vishan and jang kilibh vishan and jang and uh robin gave a talk on it last week uh on the kdv equation uh for s i think that the talk was for s between minus one less than zero but i guess he also some extension beyond zero probably one yeah what i what i think i said was one that's one uh and you worked with the so you worked with this term here and yeah that is important so um you worked with this term and this gives this allows to show an equivalence of this this single term here with the hs norm the integrated versions of that which lead probably to something similar than the formula on the right hand side uh with a single single correction at j equal to zero which is the l2 norm so uh it would probably give the same formula as this one and that's in the in that range okay so come up again now how do we prove it well the strategy is to bound the to control the integral part on the right hand side this is the most essential part once we have control of the integral part on the right hand side it also gives the control um by limits by certain limits um of the whole right hand side and then if you look at the default crossing part this allows to i mean first we look at the transmission coefficient not in the whole half plane but in the so we start with looking at the smallest condition so we control we're going we try to control the transmission coefficient t of c away from the real axis and the smallest condition ensures that we can rescale things so that whenever we get it somewhere we also get it uh let's say below one half and then at least morally we have a harmonic function the logarithm of t is a harmonic function which in the de-focusing case has a real part uh less than zero so then we are able to interpret this as in such a harmonic function on the half space and then the real part of that is going to be a measure and that allows to define the left hand side so if we use then that in the de-focusing part that t of c is lesser equal to one then the logarithm has the real part of the logarithm has the sign and then we can take the trace at the real part and that gives an interpretation of the formulas on the left hand side for the focus in case one also has to use then the background transform but i don't want to go into that what i want to do is i want to explain how we can prove how we can control the integrals well so we go back to the akns system so the akns system was c one prime is equal to minus i c c one plus u c two c two prime is equal to i c c two plus u bar c one and what we want is we want a solution so that we want c behaves like e to the minus i c x zero at x equal to minus infinity okay so then we can try to solve this equation recursively so we want to get a power expansion in terms of u so in the first step we integrate this equation and then we get c two one is equal to the integral from minus infinity to x times e to the i c x maybe you could take that up with a y y minus x u bar of x e to the minus i c x dx and then we take we look at c the so this stands for the iteration c one two and we get something similar and then so you have a prime i mean you have one and prime so this is that's a one and above it's a prime right so and then we get the first we get t is equal to one plus the limit as x tends to infinity c one one of x plus then we do this iteration and we call this guy t two of c e to the i multiplied by this and then we get t four and these terms have a pretty nice structure so if you do the the math then this t two j of c is equal to an integral and here we have x one less than y one so we iterate this thing up to y two j here we get e to the two i c the sum over the y chase by case minus the sum over the x case times u of y one u of y j u bar of x one u bar of x j dy dx so we do the iteration and then you get a pretty nice representation the imaginary part of c is positive so this is negative and ever this thing is larger than this one and we have the y's on the right hand side so that leads to exponential decay here whenever we are in the upper half plane okay so you do that then you try to control these things and I guess I'm running a bit out of time so I have to decide what I want to focus on so then the first estimate type of estimate is that this is that this t two j of c is bounded by I guess I don't want to go to go into that a constant times the u e to the minus i real part of c x u in l two and I put the imaginary part here c u d u two to the power j so this is where the properties of the spaces d u two come in nicely you have to take to put them also to a part into u and then they give very naturally this estimate here where this imaginary part of c corresponds to the localization in space two intervals of size imaginary part of c to the minus one this is exactly the scale on which this exponential decays so there is this this concentration I wanted to say a bit about that but I guess I'm going to skip that so this is the first estimate and it allows to get convergence provided provided this quantity is small and this is basically where the condition in the theorem comes in okay so far so good then we don't want to get estimates of in in terms of this thing we want to get estimates in terms of hs norms so how do we get go from here to hs norms uh well we can do embeddings we can use embeddings and then we get some variable and we can use embeddings here and look at the scaling parameter and get powers of the imaginary part and this is going to allow to get some of the integrals but it only goes up to a certain point and then we have to look at the logarithm you have to look at the logarithm of this thing and now I want to to uh go to the logarithm and well actually the way we did it I think I did make another sine error um anyhow so we get we want to take the logarithm of the of t to the minus one so this is t to the minus one is the sum of integrals which I write in this fashion so this is the trivial integral this here is the first integral which we had here this is the second integral so I wrote it here and read I hope that it's visible so this codifies the information that x1 is less than y1 less than x2 less than y2 and then we do the integral and then we have this with three things with four things and then if you take the logarithm we get a lot of combinatorics we can split the integration domains into the sets where we order the the variables and this is messy this is this is really messy so I did it for the first part I got the second part explain what this integral stands for in a minute I think I computed by hand these things here up to here and then I was convinced that the structure somehow persists but uh at MSRI I was stuck with that we were stuck with that and didn't know what to do but it was important for us to get this structure let me explain what this structure is so this calls so for every arc which is going up I put an x for every arc which is going down I put a y and then I get an here an ordering that x1 is less than x2 less than x2 x3 less than y1 this is going up that's x4 less than x4 less than y3 less than y4 the indices don't matter but this codifies this integral here the nice thing here is that we always have the last y on the right of every x so this here has a much better decay on the domain of integration as the original integral because the decays exponentially whenever two of the variables are far apart so whereas this this information here leads to decay of order basically at most imaginary part of c to the minus j because this is the size of the corresponding um simplex here we get a decay like imaginary part of c to the minus to the minus 2j plus 1 because the last y is on the right well so the last so we can't split these things into what I said it was not good so but no matter what we do is but this decays exponentially whenever we put things apart and here we could in in these integrals we could have the situation that y1 is close to x1 and y2 is close to x2 but x2 is far away from y1 and then it wouldn't get decay okay pay for that with the combinatorics I pay for that with the combinatorics right so at MSI there was the fortune situation that in this program there was people from probability and I showed this question to Martin Haier and Martin Haier told me that I should look at Hopf-Algebras and that it would be would be related to a famous formula in Hopf-Algebras so my colleagues in Bonn are still when I told them about Hopf-Algebras they were colleagues in Algebras they were excited well but but then I tried to get it into context with the millen amour theorem and that didn't work so the basic thing is it's not not a big effort to show that this is a Hopf-Algebras because it's very close to a standard Hopf-Algebras which I didn't know but that that can be that's not so difficult the Schaffler-Algebras and the millen amour theorem would give the conclusion if that would be a co-commodative Hopf-Algebras but it's a commodative one so it sounds almost the same but it is not but but then still if one tries to do things by foot basically then all integrals are connected then one gets the structures so where are we well we have this estimate here on the t's which transfers to the expansion of the logarithm of t so we get the estimates we need whenever j is large compared to s so if you look at the high order polynomial then things are easy but since we get that we have to look at the lower polynomials case by case and if you look at the conserved quantities then it's clear that something has to has to happen because if s is less than one less than zero then there is no conserved quantity which which hurts at s equal to zero you get the l2 norm so you have to handle the l2 norm this is easy this is easy because with the l2 norm if you check t2 well let's say if you check yeah if you check t2 then this is equal to u in hs squared so that requires simple calculation so the two term is okay now yeah now at the level of the at the next term at the t4 term that corresponds to the next at the at the level of the t t t four term you expect the l4 norm to the power four to show up at the level s equal to one so this has to be taken care of now with this connected structure you get up to this point and the idea to go beyond that is pretty simple this is do an integration by parts in the integration with respect to y1 if you do an integration by parts you get a derivative which falls on this um so if you do an integration by parts with respect to this this exponential so you gain a factor one over c and the derivative falls either on this then you get the same formula with the derivative or you get to the boundary terms and then you have an integral of one dimension less and then you continue to do that and if you end up with no with a one-dimensional integral in the end you get the conserved quantities and otherwise you do similar estimates as i explained before so i guess i went a bit over time i'm sorry for that thank you for the attention so this is the structure of freight home terminal i have no idea freight home determinant no i mean probably it has something to do with freight home determinant i have no idea i guess you know that better than i do i mean i mean in kdv so i knew that on the matrix case before right right for the scalar case right this this is exactly the operator i mean this is exactly the operation passing from the series of t i mean this yeah thing along the series is sort of restricting the technique diagrams just taking the the determinant yeah yeah but the real problem is make was okay so again the matrix sitting i haven't gone through detail but in the scalar setting it's actually a general phenomenon the transmission coefficient is the freight home determinant because the the integral kernel is semi-separable i mean it's a function of the lesser times a function of the greater right and then what is uh yes in that's if you ask that's something then might recognize it because that's exactly what makes processes on the line markovian well right i mean what why is the orange tunnel back process markovian it's because the kernel is semi-separable yeah so i'm not familiar with that with these things but what i think is true is that there's a lot of combinatorics behind which can be expressed in various ways and it should be it should be connected i mean one thing in the with the hope i mean the the reason why mattin hyra is interested in the hopfalje price i think different it's closer to what what i presented here and not not the freight home determinant but um the thing which is striking is there's this theorem it's not difficult to get and it's not difficult to usage in order to compute these things to any order but uh people i asked we are sort of stuck with explaining what the coefficients of this expansion should look like what the coefficients should look like so this is sort of big miracle it gives connected integrals which is what we need but we don't have any access to the size of the coefficients and maybe the freight home determinant to determines allowed to get that more explicitly there's a lot of combinatorics behind which i don't understand i have to go open right there's a stanley's enumerative combinatorics exactly has a theorem which says if you're if you have a generating function generated with coefficients of the number of objects of this type then the logarithm of your generating function is exactly the generating function of the number of connected objects of this type okay that sounds yeah so i would be interested in the reference i'd like to get that here i was happy to get to get that but it's clearly related to many things well just comment on on this estimate i think c is independent of j and if so it's articulating that uh that this is that you're really creating a the transmission coefficient is a holomorphic function of u and u r and these are the table coefficients essentially yes exactly so sort of the uh mckeen trubelitz picture of the world which is objects are holomorphic with regard to comments or questions so if not thank you again


took the terrain into work today. We're trying an experiment this week. The wife and I. We're gonna switch vehicles for the week and see how much gas we save. Cause I drive about 50 kilometers or 30 miles into work one way each day. So 100 kilometers or 60 miles every day. She drives three kilometers or like one and a half almost two miles. So she drives a lot less than I do. The pickup has a lot bigger of an engine, uses more fuel. This one's a lot better on fuel. We're gonna see how much money we actually saved if we switch vehicles. I'm in the process of debating, doing some math, doing some calculations, and doing some sleeping on it. In the process of making a decision of whether or not we should get a commuter vehicle for maintenance. If we were to get a Chevy Spark and get about seven liters per hundred kilometers, that's about half the fuel if not less than what the pickup uses. For my calculations right now, the savings alone would cover the insurance of the new car and the pickup. And also give us an extra $250 to $300 extra in our savings every month. So we'd actually be making money and have another asset in the family. I have a third vehicle. I mean, it's a depreciating asset, yeah, but if you get paid off right away, like these cars, they go for like $10,000 to $12,000. It's just stock, base model, as low as you can get. Just a commuter car. We were to be able to pick up one of those. I'm thinking about it. We would actually be ahead every month and have three vehicles. So that's why we're gonna switch vehicles this week, see how much this engine uses. It's a 2.2 liter, I believe, Eco-Tech Chevy four-cylinder or GM four-cylinder. See how much we save in this. And then we'll maybe go and take a little Chevy Spark for a test drive, do some research into it. My father-in-law works at GM, so he's got a lot of answers to our questions. And we'll see what happens. I mean, it seems like a win-win right now. We were talking about it this weekend. That's why I'm bringing it up now. Seems like a win-win extra asset plus extra money. So you actually make money buying a car, right? It's thinking about it. It's on my mind. So this week we're gonna switch vehicles, see what happens. And we'll probably talk more about this next year, 2022 in the new year and see what happens. No rash decisions, no quick movements, take our time.  I mean, we can get a Chevy Spark, then it'll be paid off relatively quickly, if not right away. And I'd be saving, what did I estimate again? Just on tires alone, I'd be saving only 1,500 bucks a year. Because the tires on my pickup are $3,600. They're not exactly the cheapest tires, but I like them. Tires on just a commuter car, they don't have to be fancy, it just has to be filled with air. 500 bucks for a set of tires. So I'm saving over $1,000 a year just on tires on a commuter car. I'd be saving about $2 to $300 a month on fuel. Maintenance, oil changes. I'd be saving for the first two years that have full warranty on everything on a new little car. And if you pay it out right, as quickly as possible, if not like right off the bat, just pay it off. You don't pay a lot of interest. That's where they'll get you. They'll get you with the interest, right? So I'm gonna try to avoid paying the interest and just pay it off either right off the bat when we take it home, or as quickly as possible like after that. We'll see. We'll think about that later. Right now we're gonna think about trucking. It's Monday morning, happy Monday. We're in the truck. We've no assignment yet, but one is coming down the pike pretty soon. Obviously getting another vehicle is a big decision. It's not something I just want to jump into right away. Though I am sharing it with you now because the idea of it looks promising, right? It looks like in the long term we could save money. And with it being a new car, like I said, I have two years of commuting to work worry-free. It'd be all under warranty. By that time, that car would be long paid off if we can't pay it off right away, but our first priority right now is our IVF treatment that we're going through that. Well, my wife is gonna have to go through. We're gonna be starting that hopefully in February. We're actually going in for our preliminary meetings and stuff about it in January next month, 2022. And we're hoping to get this procedure underway in February. We're able to get it going a few months earlier than we expected and we're very happy about that. We've gotten ourselves into a very good spot financially and I think we're ready. I think we'll be ready end of January. So I mean, getting a new vehicle isn't really high on the priority list right now. I just mentioned it now for something to talk about because I took the terrain in to work today. We're just gonna see if I can save some money driving this vehicle for a while. But you know, maybe towards spring or summer, we'll see, we'll start talking about it more seriously. Do a lot of our homework before then. And if all my calculations are right, I mean, why not, right? Why not? We get ourselves ahead. The maintenance on a pickup truck is very expensive. The oil changes I have to do every two and a half months. Tires are very expensive. I have to change those every two years with all the commuting I do. The gas is obviously getting more and more expensive. They keep raising the prices higher and higher. And you know, maintenance is expensive on a pickup. The suspension components, the shocks, the ball bearings, the U joints, everything on that wears out over time is way more expensive to fix than if it was on a tiny little commuter car. We'll see what happens. But first, our first priority financially is this IVF that we're gonna be going through early next year in 2022. So we'll focus on that first. This is, I just bring this up now just because it's on my mind. We'll see what happens. Time to get started. So we've got a 53-foot flat roll tide here. Hooked on. Gonna do the walk around on this yet. I forget this landing gear up. Fill it up with air. Make sure there's no leaks. Control the tires are good. Control the lights work. And we're gonna pull this thing up to Arbor. It's two hours north of here.  All these hoses are pretty stiff. This is cold out. When I was driving on the highway in the southern states, I noticed a lot of drivers have like plastic coil hoses here. They're airlines. And those snap in cold weather. They're no good up here. Get out of these rubber lines. Over lights on. Just gonna make sure everything's working. Let's make sure our brakes are working as well. Let's just hook our trailer brakes there on that. All right, so now our brake lights should be on. All of our signals should be flashing. Our marker lights should be on. Marker lights and signals. Tires, yeah, I should have made sure these tires were not flat before I rolled the landing gear up. So that would have been a waste of time. This is in here correctly. Brake lights are engaged. Signals, license plate light. Did any of you catch what I forgot to do before I did my walk around on here? If you answered, I forgot to fill the trailer with air so that I could listen for air leaks. You would be correct. So I'm gonna go back to the truck now, fill the trailer with air, and then listen for any air leaks. And make sure that the airbags are inflating. Don't need this anymore then. That in there. The air is flowing through that red line right now. There's no air leaks here. That's good. Let's just make our way to the back. Listening for air leaks along here. Seems like we're doing good. The airbags are not inflated yet, but they should be inflating soon as soon as the trailer air compression goes up. You can see under there, there's no air in there yet, but they should be filling up soon. There we go. You hear that? There. Watch that airbag. See how it's lifting up? Now I'm gonna move away from it, just in case if it is faulty and blows up, you don't wanna be right there.  Okay, the trailer's lifting up. See that? So it looks like we're all good. And I don't hear any air leaks. Fantastic. Good. We're all good to go. I've already checked the trailer, it's empty. But for your benefit, we'll go in there and check it again. Just to double triple check to make sure I'm not gonna pull a load down there or up there. Don't wanna show up there with a loaded trailer. There we go. See? Empty. It's kind of a waste of my time and waste of fuel if I drive all the way up there with a loaded trailer. Can't put a load on top of a load. So in a few hours, there will be freight in here and this trailer will be on a track to Tampa Bay, Florida. I hope it brings me back up palm tree. Lucky trailer gets to go and sit in the sun for a few days. I gotta stay up here in the cold. First off to turn right, though. Think she knows everything. A little bit chilly out there today. Well, I would say it's freezing. But my body's finally climatizing. It's finally getting used to this cold weather and now this is just a normal day. About minus 20 outside today. Two hours north. See how cold it is up there. They're usually just a couple of degrees colder and they got a little bit more snow than we do. Just a little bit. Cross here and get onto the highway. Don't see any vehicles coming. Fantastic. So while you're watching this vlog, the days have begun to get longer again. Well, technically they're exactly the same, but you know what I mean. Daylight hours have gotten longer. As of December 21st, we're finally on the way back to the summer solstice. This is a much better part of the year. Tied up here. Didn't use the whole trailer. I thought it was gonna be a full load. Better close this up and get in trouble if I'll leave that open on the road. There you go. Yeah, the weather's actually not bad today. As long as you dress for it, it's actually not bad at all. I got lined jeans on right now. And if the lined jeans, if the lined jeans aren't warm enough, I have the overalls that I put up that I've put over this yet that I can wear if it gets even colder, but really it's not that bad. My body's gotten used to the cold weather now. It's just like the first month of cold weather where you're just, everything feels so chilly and so freezing cold. Not for that, it's fine, not for that, it's fine. See if I can close this thing now, I might need to climb up there. This thing has given me problems before. I didn't have to undo the back at all, so don't have to worry about that. Just gotta get this thing closed. I had a little bit of a difficult time opening it, because the wheels, they're not stuck, they're greased. They're just frozen. See if I can do it from this side. Okay, there we go, there we go, that's, we got momentum. Come on, we gotta bring it all the way up there. Come on, come on. Oh, oh, I'm losing traction. Ow, and that's lost traction. Close. I'm gonna put you guys down, I need two hands for this. The other side, the other side is stuck now. Everything's cold, everything's frozen, but we're having fun. Can't stop me from having fun. Come on, come on, there we go. Geez, what's going on here? What's going on here? They're moving a little bit by little bit. Okay, here, one second, guys, stay there. I've been to work, but I snuck these babies in. Don't tell anybody. These things are amazing. Rick got this for me for Christmas like two years ago.  When you breathe in through this, you breathe in fresh, cool air. So it actually like keeps your body temperature at a good spot instead of like messing with your breath and stuff. Got icicles on my eyebrows. Can you see them? The sun is shining. The earth is spinning. Got my lunch right here. Had a good breakfast. What else do you really want? What else do you need? Got a nice warm home to go home to, thanks to our natural gas furnace. Thanks again to Alberta for the natural gas. I really appreciate it. Please keep sending it. I want some more, please. Felt so weird taking the train to work today. I wanna try this for the week. So far it looks like I'm going to save over 50 bucks a week in gas. It's like 200 bucks a month if we switch vehicles.  Or all month. So I don't know. I missed my pickup. I kinda just wanna jump back in it because I like driving it a lot better. And she likes this vehicle a lot better too. This is sort of like, they're both our vehicles, but this one's sort of hers. She keeps it nice and clean the way she likes it. And you know, it's the vehicle she picked out. It's her terrain. She really likes this vehicle and I like the pickup. That's the one I usually drive is my pickup. So we might just switch back. I don't know. Probably switch back tomorrow. I want my pick. I missed my pickup. But those savings are tempting. But what's the point of picking out the vehicle you wanna drive if you don't even get to drive it, right?  And she picked that one so she could drive that one. Oh no, we're not even gonna get to drive the vehicles we picked out for ourselves. But is it worth 50 bucks a week just to be in a vehicle that you wanted? I think it is. I don't know, we've been doing it this long. I gotta let the dogs out here. I had to park our vehicle on the end of our driveway because where we are in town, people like to use our driveway to turn around in. And that's very annoying, more annoying than what it usually would be for other people because no one likes people turning around in their driveway because I have to maintain that with my driveway. Wearing down my driveway, first of all. And second of all, our dogs go nuts every time someone turns into our driveway to turn around and it happens so often in the evenings. So I just started when I got home or when I get home, I just park our vehicle down at the end of the driveway to stop people from doing that. Even if people use our neighbor's driveway to turn around and our dogs go nuts. So if I could stop them from doing that too, I would do that. But at least I can control this one. So I park our vehicle right at the end of the driveway so no one can use it to turn around. It's our driveway. Stop using our driveway to turn around people. There's so many streets all around here. Just use the streets or the public streets. This is a private driveway. I know it's a little silly and petty but it's the way I'm doing it. Following through, stop using my driveway. So I'll have to talk about it with Britt when she gets home. She's supposed to get off work in over 45 minutes. Maybe she'll, she might get off a little early sometime she does. I miss my pickup. I was gonna, we were gonna at least try this for a week. I don't know if I'm gonna last the whole week. It's not that I don't like her terrain. Like the train's a great vehicle. It's actually, I call it a spaceship. There's so many buttons on the dash. I don't know what to do with myself. It took me forever to figure out how to hook up my Bluetooth phone to it so I could listen to my Joe Rogan podcast on the way in to work. Once I finally got that hooked up and then I guess I pressed the wrong button for the vents because the windshield and windows started fogging up. And then I realized that the air had somehow gotten on to recirculate. And I didn't want to recirculate. I wanted outside air coming in to unfog the air. I mean, unfog the windows and I wanted on defrost. I was blowing on the windshield on the windows. Oh boy. You gotta know like Morris code to try to like push all the right buttons in the right order to try to get what you want done, done. That car, the thing has a mind of its own. I like it though. It's a fancy vehicle. Britt picked it. It was her favorite one and she loves it. So it's her fancy terrain and I feel unworthy to be in it because I can't even figure out how to work the vents. And the Bluetooth, my pickup's so much easier. It's just, cause I ever had everything hooked up in my pickup already. I just get in the car and it's like, oh hey, it's you again. Here you want to listen to Joe Rogan here. Let me start that for you. The only thing it doesn't do is turn the volume down when I forget to turn it down the day before and then you know, it starts blasting music at me first thing in the morning and I gotta tell, whoa, whoa, whoa. I'm not the man I was last night. I'm not in a party mood. Oh, it's morning time. And that's the only thing. I wish it would automatically turn the volume down like some of the newer fancy vehicles. I think the terrain does that. And the train also has heated seats. That was nice. That's nice. Heated seats. My next vehicle will probably not have heated seats cause we're talking about getting the spark, right? And that's gonna be base of the base model. It's probably gonna have crank windows. Do they even make those still? I don't know, but I'm base of the base. It's gonna be a base, base model. Take the base model and then make it more base, base model. We'll see. We'll see if it's worth it in the end. I'm gonna do more calculations, more thinking. Thanks for watching today, everybody. And leave your opinions down below in the comment section. I've already decided it would be a Chevy Spark, okay? I'm a Chevy guy, so that's what it's gonna be. My father-in-law said that there's also the Chevy Sonic, but nah, I want the Spark. If I'm gonna get a commuter car, it's gotta be worth it.  We'll think about it. We'll give it a few months and see what happens. Let me know what you think down below. I'll see you tomorrow, though.


Welcome back. In the previous segment we discussed the organization of the main program and the classes which are related to the canvas and the graphics. Now we are going to talk about how the circuit is represented inside the computer. So this is done in a class called MathRep. So it keeps track of the data related to the circuit. So the data consists of the manner in which the components are connected, the conductance values and the current source values or what is sometimes referred to as current source strengths. Then the main member function in this is the function solve which solves the circuit built up until then. So let me talk about the representation a bit more precisely and this is discussed in section 23.4 of the book. So suppose we have an n node circuit then it will be represented using an n by n matrix and we are going to call this matrix A. In addition to that we will also have an n by 1 vector B. So these will keep evolving as our circuit gets built and then when we want the circuit to be solved we will perform mathematical operations on this matrix and vector and we will see all of that soon. So this matrix A is going to be used to represent the connections and the circuit values in the following manner. So whatever conductance there is between node i and node j we are going to make an entry of that value in the matrix, in the matrix element Aij. So just to make sure that this is understood. So let us say this is node i and let us say this is node j in our circuit and suppose connecting these two we have we put some conductance c. So then we want simultaneously in the representation to have Aij equal minus of c. So this will keep track of what the conductance is and of course it is symmetrical. So we will also have A of j i also minus c and it is 0 if no conductance is specified. So if there is no edge then this had better be 0 because no edge is equivalent to 0 conductance. So this is indicative of how much current can flow and if there is no edge no current can flow and therefore it is natural to say that the conductance is 0. Aii is going to be the sum of the conductance is connected to node i. So let me again give an example. So this is some ij, this is some k, this is some l and let us say this conductance is c, this is some d, this is some e then Aii we want to be equal to c plus d plus e. So note that given the circuit we can figure out what these values are and given these values we can figure out what the circuit is. So to that extent this is the representation of this circuit. Of course we are not completely done yet, we have not said how voltage sources are represented. So voltage sources are represented in the following manner. So we not only have this matrix but we also have that vector b. If you have a current source which say starts off at some vertex w and enters i and it has some strength j. So then b of i should be the sum of all such current values which are forced to enter. So we will see this in a minute and of course if there are some currents leaving then that is as good as negative entry. So over here we are subtracting the sum of the strength leaving i. So how does this circuit get built incrementally? Well what happens when a new node is added? So let me draw a picture over here, a new picture. So here is some circuit that we have created and now we create a new node. Well this new node does not have any connections at all. So what do you think will be present in its row and in its column? So if we add a new node we are going to have an extra column and an extra row added. And since the node is not connected to anything at all we have to make that initial row and column become 0. So again let me remind you how did we add a new node? Well suppose this circuit already had nodes 0 through n minus 1 then when we created a new node we created a new node n. So this is now going to correspond to increasing the number of rows and the number of columns. And so the nth row will corresponding to the new node and that has to be made 0. What about B of n? So if you remember or I guess you do not have to remember B of n is supposed to contain the sum of currents and currents leaving and entering through sources. But right now there are no sources connected. So as a result 0 gets added to the column vector B. So originally our column vector look like this so say B0 through B of n minus 1 and we are going to add a B of n to it and the value for this we are going to make it to B is 0 because there is no current source connected to this. So now what happens if a conductance C is added between nodes? So let us go back to this picture. So suppose a conductance C is added then what happens over here? Suppose let us say we are just adding this but if we add this aij should be minus c. As a result we are going to subtract c from aij remember aij was originally 0. So both of these were originally 0 so we are going to subtract c from them and we have to this c has to appear over here and so to aii there might have been other entries but we now add c as well. So that is how adding a conductance gets reflected in our data structure and if I add going from node i to node j so let us say this is node i and this is node j so I can even draw it over here so let me do it that way. So suppose I add a current source going from i to j over here so its strength is j well what should happen? So B of i should have the current sources entering i but this is current leaving i so as a result I should subtract j from B of i so which is exactly what I am doing and this j is entering j the node j and therefore I am adding capital J to B of j. So this is how exactly that circuit gets built up. So let us look at our code and look at the member functions add node add source add conductance. So these are operations in Mathrep. So add node first so we have to push back an entire zero row we have to append a new zero row so this is what happens this is what makes that happen notice that the vector class makes it very easy to append and append rows since this is a vector of vectors so a is a vector of vectors C over here and therefore this will just append an entire row and then we have to add a zero column so now what we do is we are just going to look at every row and we are going to append a zero to it that is it so this is a loop over every row of that vector and we are appending a zero and finally we want to append a zero to our B as well and so that is what this does. So what about add source so if I am adding a source you remember that if a source of value V is added then going from I to J then value has to be subtracted and to J the value has to be added so this is what happens what we need to do and if I want to add a conductance then to II the value has to be added see this and also to JJ because the conductance is connecting to and the treatment is symmetrical and from IJ the value has to be subtracted and also from GI so that is what add conductance does so quite simple. Alright so let us get back to our presentation so now we will talk about how to solve the circuit. So solving the circuit first of all requires us to find the voltages at the nodes so we do not know those voltages and let us say we represent the voltage at node I by X of I so X of I is so X is going to be a vector and I is going to be the ith index in it value at index I in it but right now this is unknown to us. So now I want to tell you a little bit about how circuits work so let me start on that so this is our vertex I and this is our vertex J and let us say that going from here to here there is conductance GIJ. Now how so what do we know about the current going from here to here okay so it turns out that this current from I to J is equal to the conductance of that times the voltage of I the voltage drop or the voltage at I minus the voltage at J so this is the current going from I to J. Now we do not know this we only know well yeah so everything here is an unknown so we do not know the current we do not know the voltages but what we have done is we have established a relationship between the voltages at these points and the current over here okay so this is just what is called Kerchoff's voltage law as you might have perhaps studied in physics okay all right so now I can ask what is the total current leaving I okay so there might be other edges okay so how do I calculate this well I just take the sum over let me let me do it in a different color just to keep track of it so I am going to take the sum over all the J all the all the conductance is connected to connected to I okay so if I just take the sum so that gives me the total current leaving okay now I can factor this out and that factoring is important so let me write it down a little bit carefully or I guess I can explain it on the screen okay so what is happening over here well there is this excite term this excite term is common to all the terms in this series so I am going to pull it out and what is the coefficient of the excite term well it is going to be the sum of the conductances gij so that is what I have written over here some of the conductances leaving I or these gij's and then I am left with this remaining term so gij minus xij and that I have just pulled that minus sign out over here okay all right so that is the total current leaving I through the conductances now this if you remember the sum of the conductances leaving I was just aii okay and therefore this is xi times aii and gij the negative of it is in fact aij so this is what the entire value is okay and in fact this is nothing but the ith component of the product of the matrix a and x okay so we do not know x yet but what we have done is we have established a relationship between the total current leaving I through the conductances and this matrix a and this unknown vector x so it is ith component of the product is indeed the current leaving I through the conductances okay and this this must equal I am sorry I should have said equal over here this must equal the total current entering I through the conductances right so so here is my here is my vertex I and I am talking about current leaving through conductances but there is current entering through voltage sources as well okay so what is the current entering through sorry through the current sources as well so what is the current entering through the current sources well the way we define b it is exactly bi okay so what does this tell us this tells us that this must be equal to this so the ith component of the product ax must equal the ith component of b but this is true for all ith components and therefore in fact this entire matrix product must exactly equal b okay so sorry we should be so we have to solve ax equal to b because we we know that ax must equal b and now x is the only unknown this vector of unknowns is there but we can get that if we solve this matrix equation so this is a simultaneous equation which we have to solve okay now there is a slight itch over here this system has n equations and n unknowns remember n is the number of nodes in our circuit okay unfortunately okay these equations are not independent okay but we will get an a solution if we set any unknown to 0 okay so after we set an unknown to 0 we will get a solution and the details of that I am not going to talk about right now but they are discussed in this section okay so that is what we are going to do we are going to set we are going to set xn minus 1 to 0 okay and this will effectively allow us to drop the last column as well okay so we will get a system which has n minus 1 equations and n minus 1 unknowns and these equations will all be independent that is they will contain distinct information and then we can solve it okay by the way if you are so many of you probably know circuit no electrical circuits a little bit and if we have voltages in our electrical circuit then voltages are all relative so I can fix any voltage to be 0 and that is exactly what we are doing over here okay but it relates to equations not being independent and that is discussed in this section so please read that okay so anyway so I have told you why this product ax must equal to b and I have told you that we are going to set one of the values xn minus 1 to 0 and then we just have to solve for the remaining unknowns so we have a smaller system well slightly smaller system which we have to solve okay so now how do we solve this system of equations so this is discussed in section 15.2.1 and I am going to discuss it very briefly okay so so we are going to solve ax equal to b and we are only going to look at the first n minus 1 unknowns and first n minus 1 equations so in fact let us forget that we even had that n minus xn minus 1 okay so basically the idea is that we are going to use equation i to eliminate xi from other equations so let me just write that down show that to you okay okay so our equations are a11 x1 plus a12 x2 plus a13 x3 and so on equal to b1 then a21 x1 plus a22 x2 plus a23 x3 and so on equals b2 and in general ak1 x1 plus ak2 x2 and somewhere here I will get akk xk and so on equals bk okay now what this is saying is that I am going to use this equation to eliminate x1 from everyone from every other equation over here okay so how do I do that okay so in general if I have say ai1 x1 plus ai2 x2 and ai i xi equals bi okay then this equation is going to be used to eliminate this ith unknown okay so if this is going to eliminate the ith unknown okay how do I do it well okay so let me let me write down what the equations these equations are so I am going to have a11 x1 plus a12 x2 and so on and say the typical term will be a1i xi and so on equals b1 okay then a21 x1 plus a22 x2 and so on and the term over here will be a2i xi all the way equals b2 okay and maybe the ith term is going to be the ith equation is going to be something like a j1 x1 plus a j2 x2 aji xi equals bj and let us say the ith equation is ai1 x1 plus ai2 x2 and here I am going to get aii xi equals bi okay so what our idea over here is saying is that we are going to use this equation to eliminate xi from all these other equations okay so how do we do that well if I want to get rid of this term over here in this jth equation what do I need to do I am going to multiply okay I am going I am going to multiply this by aji upon aii okay so this entire row is going to be multiplied by aji upon aii so what do I get over here so over here I will get aji xi if I now subtract this resulting row which has aji xi from this I will get a zero over here okay so that is how I have eliminated this so that is what I am going to do so I am going to do this for this row this row this row and also all the rows underneath it okay and now in this I am going to divide by aii okay so we in general we need to worry about whether aii is going to be non-zero but the way we have defined aii we will always be guaranteed that it will be non-zero and so we do not have to worry about it so another way of saying this is that we do not need to pivot in case you in case you have seen this heard this term pivot okay so what does this do so at the end of it once we do the elimination our matrix a is going to look like some non-zero entries over here along the diagonal and everything else is zero and b has changed suitably but b may have whatever entries over here zeros and non-zeros okay in general non-zeros okay so we are we have a diagonal matrix but now if we in fact make this one then this will exactly be the solution of ax sorry I should have x over here okay equal to so if this is made one then this will exactly be the solution okay so in other words if I divide this entry by this entry I will get the solution for xi okay so this is the ith row okay so that is in fact what we are going to do okay so xi is bi upon aii so that is what is involved in solving the circuit okay so at this point we will have found xi which are the voltages and so the program can print the voltage at all the nodes okay and once we know the voltage voltages we can also of course find the currents because we already said that they can be they are related as per this over here but we are not going to do that that is just a detail that you can take care of okay and you could also say that look I do not want the current printed on or the voltage printed on my on my shell window but I want it printed in the circuit in the canvas itself and that also could be done but again that is a detail that you can take care of okay all right so we will take a quick look at this member function solve okay so what does solve do so I have described it to you already so we are going to use the ith row okay to eliminate xi from every jth row okay so so long as of course i is not equal to ga so it has it is going to calculate this multiplier and then it is just going to subtract that row and that is that subtraction process that is all okay and then finally it is going to print out bi upon aii as the final voltage values okay and remember that the last so so we have we had thrown out one unknown and we had set that to 0 so that that was that had index a of size a dot size minus 1 and that value was 0 so we are going to print that out as well okay so that is it so that is what the solver looks like okay so we have this already and so what have we discussed so we have discussed how circuits are represented as a system of equations okay and we said that we are going to set 1 unknown to 0 and that gives us independent equations then we discussed how the system can be solved okay so we will take a quick break over here and then we will discuss extensions and we will conclude this lecture sequence thank you.