So, I'm Gavin Clinch, carina's here beside me, Oruendority, from Ledicenni, unfortunately, can't make it. Today is in Scotland on an RPL tour. So the project then addresses recognition of prior learning and the coalition, or the Convent Altar Alliance partners have built an E-portfolio and we've developed this over the last year and a half Llywodraeth yn ddweud â'r sydd yng Nghymru a'i fath i ddim yn bwysig i fynd eu cyframe. Jog fel y dynion o'r Seisyddion Llywodraeth, a bod yna'r grwpiau sydd yn fermento. Rhyw gael ei wneud yr Alp Llywodraeth wedi cael ei bodi eu cynnig o'r Grwpiau. There's a big demand from part-time learners to move quickly through the education process and to achieve the qualifications that they often need. So we need to address this and what we've done is an attempt to do that. So the RPL roadshow is going to build upon this work and we've built a website. This is the front page of the website. So this is a protocol, information about the RPL process. What is RPL? It kind of gets away from or gets around the fact that a lot of institutions don't have RPL coordinators. They don't have anybody there that can actually talk about RPL. So there's an information site here and it goes through the process of what RPL is and how to apply for it. So what we have is an applicant applies for a programme on the course. Maybe they get told by a head of department or by a programme coordinator that they don't meet the standard entry requirements. They need to make an RPL application. So they look through the website, they move on to the application process, they log in here. This is now in Moodle. We selected Moodle because it allows for the assessment of learning. Once they log in, they skip through this process, which is just a number of tells them the steps that they have to go through. They have to tick a box to say that they have actually applied for the programme because we don't want people just applying for RPL. We don't even know that they're in the system. Then they come to this page here. This really is predicated on some of the practice that we had and that they had on that Kenny. There are five tabs here along the top. The first one is that they give a profile of themselves and they can say something about themselves in a couple of hundred words here and say who they are. I don't have all of these, these are just screen grabs. I'm not going to spend too much time on this, but they can talk about their work experience, they can talk about their educational experience. They can upload all of the information that they have so they can scan in copies of the certificates or they can upload any work projects that they've done, any formats that we're kind of accepting. So it's all here in one location. This kind of collection then that the heads of departments can look into and that we can manage in a much better way than any kind of iniosyncratic practice that goes on within an institute that different departments have taken different approaches to RPL. So this is the assessor view. The assessor goes in once an application has been made and they've been notified that somebody has applied for their programme. The assessor goes in, clicks on the applicant, can make notes in here and then effectively they're either approving or not approving the application. If they approve the application then the action can be, we're missing the end of that slide, but they can sign off on that and then they can archive it once it's been ratified by the institute itself. So it's a simple enough process to go through. So the tell week itself is designed to develop the RPL assessor and mentoring skills to engage with the e-portfolio. So how does an assessor actually work with the e-portfolio, work with this technology, work the way through it and how does an institute manage it as well. This is, as I say, it's in Moodle, it's branded Connacht Ulster Alliance, that's our grouping. But we can hand this over to any institute or cluster that they can take it, they can rebrand it, they can use the technology in behind it and they can host it themselves. So the tell week is going to cover that, the skills development roadshow and we're going to reach four, the plan is to reach four higher education institutes across the country in the tell week. In dates really in April and May, not one single week. So the institutes that sign up for the roadshow and that engage with this will be given this copy of this Moodle plug-in, which they can host themselves. And it's about 600 euro per annually I think to host, so it's not expensive. OK. Good morning, my name is Karina Ginti, I'm from Galway Mayo Institute of Technology. So just to continue on from Gavin's overview there of our tool, what are we actually going to do? So what we have proposed is that we would work with four higher education institutes in Ireland and one in particular that we have been discussing this project with in advance of coming here today. Because it's been moving a little bit further is DCU and Mark Glenn is here with us today. So we have tools that we want to share with each other. And there is also other institutes that are interested in this but we're not going to talk about that today because we want to see how this project goes and if it gets funding and all of that. So what we plan to do is in order to roll out this tool across four higher education institutes we need to provide a set of open educational learning resources for the staff that are involved in engaging and developing this tool within their own institute. So we have proposed developing an assessment type open educational resource and one that's linked to mentoring as well because the role of assessor and role of mentor is quite different. And when you're managing their journey through the RPL process we need to understand the roles and responsibilities of each of those people. So that's one element. There obviously will be a site visit built in to that program. A site visit with that higher education institute and the staff involved. We also want to involve an international case. We've worked quite closely with people in Scotland who are very well advanced in the whole area of RPL and also we have contacts in France and Australia and in the US. So collectively we'd love to have four lovely webinar pieces or open educational resources that would be prerecorded and we could share them with the partners involved. We'd like to say that we could share this with everybody in the country but I suppose we have to be reasonable at what can we do in a short space of time and can we build a case with four institutes and that might be a case and a stepping stone for the future. So also we think it's really important here to build this online community of RPL practitioners who are engaged with using this type of technology or engage with assessing RPL applicants within their own institute. And you're probably familiar with some of the figures. Lifelong learning participation rates is at 7.3% in Ireland compared to 10.5% which is the EU average. And if we look at the employed sector it's only at 6.2% compared to the EU average of 11.2%. So straight away we see that we need to provide technology tools like this to enable access into higher education and try and build up those numbers. So by doing that we can't just do that in the CUA. We want to be able to share these tools, these technology tools to other providers so that we can try and bring up these rates in the future. So and then creating this online community for RPL practitioners it's quite timely. I know that the QQI in the last couple of months they've been trying to work out a network of practitioners in this space and trying to get them to work together and share resources. So perhaps that can tie in with that as well. Then we'd like to see that participants gain a digital badge for their participation in this. And the project all aboard which has already exists which is running very well. And NUI Galway have a close relationship with us as well because they're part of the cluster. So we'd like to be able to tap in with them and see how can we create a digital badge for people that engage on our RPL roadshow and engage with this tool. Where do we need money or how will we do it? We obviously need technology support. There's only the three of us. There's myself, there's Gavin and there's Oren in letter Kenny. And we've been driving this project for it's actually over two years now from the start, the idea and then building, testing the site. We've been piloting it over a year. We've had about 64 applicants that have engaged in this tool. We've got feedback from it. We've changed the tool. We've developed it further. It's now a lovely tool that can plug into other institutes and working very well. We've adapted it even to include a challenge exam or assessment piece where if there is a gap in somebody's portfolio that they can actually complete a challenge assignment or an exam that is set by that assessor to ensure that are we checking everything. You know, we're not just dismissing that candidate, we're actually testing or evaluating them. So to roll this out we would need some technology support and that's where the funding would go. And we would also need some support for learning technologist. We've engaged a Moodle technologist programmer for some of the project and some of the features that we've needed to bring it to this stage. If we were to share it across four institutes we would need some support to ensure that it went seamlessly into those institutes. That's really it. We need funding to pay for obviously the OERs and making sure that they're lovely professionally branded that can be used across any institute. How will we promote it? Well, we already have the site established, myexperience.ie. We have, through social media, we would set up the Twitter hashtag. There's also LinkedIn groups, institution websites through the forum and through the associates, the new associates group that has been renamed. And there was another group, Gavin, as well that we were talking about. Helen, Helen as well. And how will we evaluate the impact? Again, we would like to see a survey built into an evaluation tool and a survey tool built into myexperience.ie website. We would do some participant interviews, focus group session and looking at coverage around it. Because it is a big national topic as well and looking at participation rates in higher education, this is a good news story. This is a tool that will help build engagement and interest in taking on higher education programmes. If you have institutes across the country from the west, the east, south and in the centre of Ireland, engaging in trialling and testing a tool that's going to give access to higher education, we would see that it would perhaps give some pure coverage around that as well. So, any questions? Thank you.


QuickBooks Online 2024, increase screen size, duplicate tabs and open multiple browsers. Get ready and pack some trail mix because we're hiking on QuickBooks Online, our audit trail to success. Here we are at the Intuit owner of QuickBooks website that's at intuit.com that's intuit.com. We're going to be looking for the QuickBooks Online test drive. The primary tool we'll be using for the first part of the course. There's two primary ways we can get access to this primary tool. The first way, the easy way, the way we're going to be using after this first presentation is to just go to your favorite browser, type in QuickBooks Online test drive and search through the results. We'll see this shortly but before we do that let's go to the second way which would be going to the QuickBooks web page or I would go first to the intuit.com web page intuit being the owner of QuickBooks. This way takes a little bit longer but some people might feel that they're a little bit more secure going this way considering you're not just doing a search but going directly to the source, the owner of the software and finding it from their direct web page. So this way we would go into intuit.com. I'm going to search for QuickBooks and then I'm going to go all the way to the bottom of the QuickBooks page because this page changes often times but if I go to the bottom we've had this products down here for a very long time and we're going to go on to the QuickBooks Online. Note that we're not going to be using the 30 day free trial for the first part of the course. That would give us the blank file, the file with nothing in it which would be great if we're starting from scratch but for the first part of the course that's not what we want to do. We want to start with a file that has stuff in it so we can deconstruct before we then create something from scratch. So what we want to do is scroll down and you have it down here. Let's keep going down and there it is. Here's the test drive. So you go into the test drive here and then it takes you into this screen. You've got to verify that you're not a robot. That's the first way to do it. First a word from our sponsor.  Like our crunching numbers is my cardio product line. Now I'm not saying that subscribing to this channel, crunching numbers with us will make you thin, fit and healthy or anything. However it does seem like it worked for her. Just saying. So subscribe, hit the bell thing and buy some merchandise so you can make the world a better place by sharing your accounting instruction exercise routine. If you would like a commercial free experience consider subscribing to our website at accountinginstruction.com or accountinginstruction.thinkific.com and so I'm going to close this out and show you the second way to do it. We just search in our search engine for QuickBooks Online test drive such as this and then it might not be the first option. It's the first option for me because I've done this a few times but if you look at the URL you can still be fairly secure that you're going to the right spot because it should be a QuickBooks website or an Intuit.com website. So this one says test drive QuickBooks Online so I feel fairly confident and you can see that's a little bit faster to get in there. So then down here we have the available QuickBooks test drive so you got the United States one that's the one we're going to use and then you have the advanced. We're just going to use the normal test drive which I believe is going to be equivalent to the standard QuickBooks Pro and we're going to go into that. We're going to verify that we're not a robot. I think it's QuickBooks Plus but we're going to verify that we're not a robot and that should take us into the file. Now sometimes you might have some issues getting into this file. One of the reasons that might be is because either you already have a QuickBooks Online account and you have the QuickBooks open somewhere else right. So if you have QuickBooks open somewhere else then I don't believe you can get into here from the same browser because QuickBooks is trying to log you in and the universal browser and you're logged in under your username and whatnot. So if you want to use that browser you might have to log out of QuickBooks. Now even if you're not logged into a separate account within into it or QuickBooks at the time your browser might have cookies in it that are still basically have you logged in so that might be a reason so you'd have to log out of your file for QuickBooks so that you can log into the test drive which basically logs you in as a universal as though you're logging in as a universal user as opposed to your particular account. Now that's a little bit annoying actually because sometimes you might want to have this sample company open at the same time that you have your actual file open. Is there a way to do that? There is. So the way we can do that is we have to use a different browser. If you use the same browser then QuickBooks is going to think that you're using your username and they're not going to let you into this file because they're going to try to get you to go to your company files and whatnot. So you can use a different browser entirely like I'm using Chrome here so you could use like Microsoft Edge or you can use the incognito. So if you're in Chrome you have the three dots up top. So we want to hit the three dots and you can see this new incognito window. So that opens up a new window but now it's supposed to be one that doesn't have any cookies in it and therefore it's not going to have you logged in to the browser even though I'm still using a Google browser in essence right. So now if I logged in here QuickBooks online test drive I should be able to log in again in a separate browser. In doing this you can have your actual company file open and you can have your separate browser open which is very nice. Now the other way you can do this is you can open a completely different browser. So I have Microsoft Edge or search engine of Edge and Chrome open now so in Microsoft Edge then I shouldn't have the same cookies I shouldn't be logged in here so I can type in QuickBooks online test drive and again I should have these two these two windows should be able to be open at the same time. I find the incognito thing to be the best because most people probably don't use multiple browsers and so this incognito window opening up the test drive in there can be a great resource if you want both of them open at the same time. Now just a couple other things on the navigation that we want to look at here. So later in a following presentation we're going to get into more details on how to navigate around here look get into the details of what everything is on the left hand side in the search area. Right now we just want to look at a few things with regards to the navigation and if you're used to the desktop version at all or if you're used to desktop software in general like Microsoft Office type of software for example then you have a little bit different feel of those kind of softwares than you do on an online type of software. So I'm just going to do some comparisons to QuickBooks desktop even if you have never used QuickBooks desktop it's just a good comparison between you know what's on a desktop version between what's on an online version. So when we look at a desktop such as this the navigation here what you're the tools that you have to navigate is usually you're going to have some kind of open windows list on the left hand side and then they had the classic dropdowns which you might remember in say Microsoft Office products before the ribbon happened so if I look at Word here for example or Excel instead of the dropdowns now they put this ribbon so this is another tool that can be standard kind of tool for a desktop type of software and then of course the thing that you're working on will be displayed in the middle so if I open up a report for example and I go to my company and financial and I open up my balance sheet for example it's going to be displayed in the middle and now I've got my two open items on the left I can toggle between the home page and the balance sheet on the left hand side so we and in order to increase the size of something like this we actually have to increase the display size of the whole thing meaning I can go into my settings and increase the display which will increase a lot of the buttons on the home page and make them larger or I have some settings that I can go up top and go into my reports and adjust the fonts of the report we'll get into that concepts later but if I go into the web-based kind of system notice it's not practical typically to have the dropdowns you could have all the dropdowns in a web-based type of system but oftentimes they're going to put their information in this case on the left hand side similar to the dropdowns but they're on the left hand side and they're going to be expanding to the right and then if you go into any of these items then up top you have your tabs so a lot of times on a web-based application you're going to have these tabs as opposed to the dropdowns and they go back and forth to be using these different kind of dropdown methods which I think are going to be more or less useful depending on the kind of thing you're you're operating with is it a tablet is it a phone is it a computer so we've got basically the tab dropdown thing on the left hand side it opens the screen that you're working on in the middle so if I did the same thing and I opened up my reports for example if I went into my reports and I opened up a balance sheet then that's going to be displaying in the middle however note that I can't easily toggle from this balance sheet to the home page it didn't open up a new page it just opened up over the top in the middle so what if I want to have the balance sheet open and I want to have the profit and loss open well in the desktop version you can talk you can open both of them and toggle between them with this open items how can I do that similar kind of thing with the online can I only open one thing at a time well here you can use the tabs now tabs I mean up top the actual open window that we have up top so with a lot of web-based applications we can have multiple windows open up top and we just have to make sure that we refresh the windows so that we're using the you know the latest and greatest version of our reports so what we will typically do every time that we open up a file is we're going to open up our normal reports profit loss and the income statement and and have them in a separate tab so usually the first thing we'll do is I'll go up top right click on the tab and then we go to duplicate it so now I've got two tabs in the same browser and this is going to be where our profit losses and then I'll do that again right click and duplicate so now we've got the income statement here and this one or what did I make this the let's make this the balance sheet reports and I know I'm going through the reports we haven't really looked at that over here yet but we'll just have the balance sheet and then the income statement and then the first tab is usually what I think of as my data input tab that's my default method whenever I'm working on something typically I open something up I open up the major reports that we're going to be using that being the financial statements balance sheet income statement and then I have my tab on the left where I can do my data input things like invoices and whatnot that will then have an impact on the end result balance sheet and income statement and I can then jump over here refresh the screen run the report and then and then I can go from here so that's going to be the method we can use to have that same flexibility of having multiple windows open and not being having to toggle back and forth now the other great thing about an online version is that you can actually disconnect these I can take this tag of this tab and put it over here I can put them side by side right I picked up the wrong one here and there it is so I can put them side by side here and I can if I'm doing my data input I can have my report on the right and I can do the data input on the left so that's great and then I can grab this tab if I need to I could just grab it left click on it and then pull it and I'm just going to click it right in there to put it back where it was and then make it full screen so so that take if you if you're used to desktop software this tab and duplicating tabs and making sure you refresh the tabs can be confusing at first but once you get to do it it's pretty it's not too bad to to work with and the websites have gotten a lot better at being able to refresh everything so that you have everything basically up to date so these things are a lot are all up to date although if you're if you're working for a long time sometimes it might log you out and you have to log back in and so on and refresh the screen so some of those types of things happen more often online than on a desktop version but it works pretty well so the other thing that's a little bit different with the desktop than the online version is that we can we can zoom into the screen on a web-based kind of system so for example if I'm in a report and I want to zoom in to this report I'm going to close this is called a hamburger over here so we can also often close this hamburger and if I want to zoom in I can hold down control and scroll up so now I'm at 150% zoom in so I can really so that's really nice to be able to zoom into particular things that works great with reports but you have to be a little bit careful with it as well because it does distort the screen so for example if I'm working on a web-based application if I do that over here and I scroll in it might actually adjust you know the things up top the items so now I've got this drop down that has all the items in it that were up top so if you're working in a screen that has a different zoom in than I'm working in or possibly even just a different size of a screen such as you're using a phone or a tablet and I'm using a normal computer screen then you might have different layouts due to the fact that the web page is trying to optimize how it's going to fit on the page based on the size of the screen that you're using and how zoomed in you are into the screen so that could create a little bit more confusion when we're trying to follow along with a presentation if you have a different you know size of the screen than I have for example also note that QuickBooks has gotten way better at this but if you zoom into something like an invoice if I add an invoice now I have a data input form it didn't open the invoice there it goes and if you zoom in if you zoom into an invoice then it might mess things up right it could it could if I'm zoomed all the way in notice they can't really fit everything on the screen and if you do the data input this way then sometimes it can get a little bit glitchy like it won't it won't actually record or something like that now it's gotten way better than it used to be at that so it records pretty good even if you're zoomed in but if that happens you just close out the invoice and then scroll back down so that you're at 100 percent and do the data input at the 100 percent so that's so that's going to be the the zooming on it now if you're in the reports over here the other option you have on the reports is to on the on the desktop version was to change the font size you don't have as much options on the online version to change the font size in color as the desktop version so that's one of the things the desktop actually does a little bit better but again you kind of compensate with that by being able to scroll in and do this do this zoom in and zoom out feature so those are just a few tips that we want to first look at and next time we'll go in and we'll start to really analyze all of these options on the left hand side and and dig into how you navigate around around the screen so this time we just want to say how can we get into the software this is the sample company Greg's design and landscaping services remember every time you do data input into this company file it will basically refresh to the default once you log out and go back into it that's basically how we said how it's set up we looked at how to increase the size of the screen duplicating the tabs make sure you have that down we'll do that often so we'll have a lot of practice with that but that's a really useful tool when I first started using QuickBooks online a long time ago I didn't even realize you could do that so I was kind of frustrated that I couldn't go to multiple windows because I didn't know how to duplicate the screen and the duplicate of the screen works pretty well and then and so you can have the multiple tabs and remember that if you're using this sample file you might want to open it up in the incognito window if you have another company file open or you have a into it account open so we'll get into the navigation next time


Last time I gave you a brief introduction to water waves and today for the first five minutes I'll remind you just a few things from last time. So remember that we're looking at the equations of motion for a fluid in some domain omega of t and we're not looking at general fluid in this in this fluid domain but we're looking at the fluid that's either irrotational or has constant vorticity and because of that the equations of motion for the fluid are reduced to equations of motions on this free interface and on this free interface you're looking at two objects first you're looking at the motion of the interface itself and second you're looking at the velocity potential on this interface. So you have this surface gamma of t and your unknowns will be first the interface gamma of t and secondly the velocity potential the velocity potential we called it phi restricted to gamma of t. So you expect this water wave equations to be a system a system describing the motion of these two variables. Now in studying this evolution there's a couple of parameters that I introduced last time which I'll remind you so one parameter that appears is g which is the gravity a second parameter that I will use is sigma this is the surface tension. The third parameter that we'll use in one case at least will be the c which I'll use for the vorticity. So one of the models I wanted to tell you about is a model of fluid in this domain which has constant vorticity rather than b irrotational and the last parameter that will come up will be h which will be the depth of the fluid. So I'll mention two interesting situations one when the depth is infinite and the other one the depth is finite. Now for my purposes here the bottom of the fluid is going to be flat and somebody asked me about this last time what happens when you trade a flat interface for a non-flat interface is all in terms of the long-time behavior of the flow. For the short-time behavior it makes no difference but for the long-time behavior it is important that the bottom is flat. All right and now in terms of this four parameters in here let me remind you what are the four problems that we were looking at. So the first problem was when gravity is greater than zero surface tension is equal to zero the depth well vorticity is zero if you want and the depth is infinite so these are the pure gravity waves in an infinite in infinite depth. The second problem was when gravity is equal to zero sigma greater than zero again c is equal to zero and h is infinite these are capillary waves of infinite depth so waves which are driven by by the surface tension. The third problem was when we have gravity and we have no surface tension and now we were going to allow for non-zero vorticity and again in infinite depth. Here the vorticity does not have a sign and the reason for this is because you can always change the sign of the vorticity by reversing the time direction. Introducing vorticity in here somehow breaks the symmetry between waves that move to the left and waves that move to the left to the right and so this means that the equations are no longer symmetric with respect to time reversal otherwise the water wave equations are fully reversible and when you change the time direction you get the very same thing but not in this case. Okay and the fourth problem that I will mention is the case when you have gravity again no surface tension so sigma is equal to zero no vorticity but now the depth of the fluid is less than infinite so still gravity waves but gravity waves on a finite bottom and maybe I should also remind you that all of this projects are joint work with Mihail Ifrim and then in the first project one part of it is also joined with John Hunter and on the last project this is joint work also with Benjamin Harrop Griffiths. Now one thing that I will not do today is write down four sets of equations for you that would take way too much time but what I will try to do is to focus more on the first model just because it's simpler and maybe this is 2d everything that I say is 2d so I'll focus more on the first model and and maybe tell you what what changes to when you go to the other models and the first idea that I wanted to introduce to you in terms of studying this models which I mentioned last time is the use of holomorphic coordinates and of course this is not not our idea this is as I said last time almost a hundred years old but it's an idea that turns out to be very useful in working with the this equations and studying their long time dynamics and Claude's question answering also Claude's question this this is also the reason we stay in two dimensions because these coordinates are well suited to the two dimensional problem so the idea here is the following you have your fluid domain either with finite or infinite bottom and part of the problem when you're looking at this domain is that you have to deal with the Dirichlet to Neumann map associated to the Laplace equation in this domain that's so so the differential operator is not a multiplier has variable coefficients and further the coefficients depend on the unknown functions in in your problem so that's difficult to deal not impossible to deal in this problem but difficult to deal with and so it would be more it would be nicer if one could diagonalize this Dirichlet to Neumann operator and so the idea is instead of working with this domain you pick up a flat domain and I'm drawing this flat domain as a strip but really you have two scenarios one when you have finite bottom and here you have your water domain and here you have a strip and the other scenario is where you have infinite bottom so you push the bottom out and then here you have your infinite bottom here you have a half plane so on the right here you're either working on a half plane or on a strip and so by the Riemann mapping theorem you can find a conformal transformational conformal map between these two domains just to set the notations I'll use here maybe the letter zeta which is alpha plus i beta and here I'll use the letter z and the coordinates are going to be denoted by x plus i y so z is a function of zeta and instead of writing our equation in this domain I'm going to reparameterize the domain and in particular I'm going to reparameterize the surface using this conformal map and the main point of this is again to diagonalize the Dirichlet to Neumann map and so what are going to be the new variables under this holomorphic coordinates so one variable here is going to be the conformal map itself that will describe the shape of the domain but we don't need the full conformal map to describe this domain we just need the restriction of this conformal map to the top in here so this is let's call this line beta is equal to 0 this would be beta is equal to minus h and here this is y is equal to minus h and by the way so in this conformal map we're looking at not all possible conformal maps so looking at maps that take the bottom to the bottom and the top to the top and the infinities to the appropriate infinities and when you do this you're really left with the one single degree of freedom in the finite bottom case and this is the horizontal translations of this domain so you have one ambiguity in here whereas when you look at the infinite bottom case you have two degrees of freedom you have horizontal translations and scaling if you want in the choice of this conformal map and you want to fix this two degrees of freedom so that your equation is uniquely determined okay so coming back to our variables the first variable will be the function z of alpha which is z the conformal map itself restricted to the top restricted to the region where beta is equal to 0 and then our second variable so this describes the surface and our second variable should describe the as I said before the velocity potential on the top but now it's very convenient to think of this velocity potential also as coming from a holomorphic function in this domain and then the simple observation maybe let me write it here is that since the velocity potential phi satisfies the Laplace equation this means that it has a harmonic conjugate and the harmonic conjugate turns out to be exactly the stream function that I also introduced last time associated to the fluid and so this means that if I'm looking at phi plus i theta this will be a holomorphic function in the fluid domain in here okay but now I already have my change of coordinates from here to here so composing with this change of coordinates I can think of this as a single holomorphic function of the variable zeta in in in this domain so this requires fixing the the conformal transformation from here to here and lastly I don't again also in terms of q I don't need to look at all values of q in this strip or in this half plane but I only need to look at q on the top that will determine what happens on the bottom and so my second variable will be q of alpha which is phi plus i theta restricted to the top to beta is equal to zero so z describes the surface and q describes the velocity potential together with its complex conjugate now in terms of the size of these objects naively in a reasonable setting you think of q as being something bounded but z will not be something bounded alright and so because of this because infinity is mapped into infinity and because of this it is convenient to replace this variable z by a different variable where you take out the leading part so let me find that w of alpha to be z of alpha minus alpha and I want to assume that this guy is bounded and you will immediately realize that the moment I'm assuming that this guy is bounding bounded I have removed scaling from the degrees of freedom in here right because alpha if I rescale z and not rescale alpha then this will no longer be a bounded function okay so I'm still left to the horizontal translations and that has to do with constants and constants are much easier to handle in this theory now what is the state space what are the class of functions for z and q that we're looking at here and so the important observation is you don't want to look at arbitrary functions z and q z and q instead will will be functions which are defined on the top but which have holomorphic extensions in the fluid in the model domain okay so z and q have holomorphic maybe let me use w in here and you'll see in a moment why in in model domain and furthermore they have some some decay conditions which vary depending on they have in some sense some boundary conditions on the bottom and to describe this boundary conditions I'll have to discriminate between finite depth and infinite depth so if the depth is infinite then I expect w and q to go to zero more like their holomorphic extensions at as if you want beta goes to minus infinity so I have holomorphic functions in a half plane and they go to zero at infinity and if instead the depth is finite then this functions w and q of course they'll not be equal to zero in here that will be way too much to ask because if you have a holomorphic function that's equal to zero here has to be zero everywhere right but instead the piece of information that you get for q because of the boundary condition on the bottom and for w by construction is that w and q are real on the bottom okay so from here on our functions will be functions which live in this class they have holomorphic extensions in the model domain and they have either decay at on the bottom or they are real on the bottom and because of these conditions what in particular what happens on the top uniquely determines what happens in the entire domain and so I'm going to make this a definition we call such functions this is a slight abuse of terminology holomorphic okay normally you call holomorphic functions functions in open domain but here we're going to just to keep terminology short to call homomorphic functions functions which are the traces on the top of holomorphic functions which satisfy certain boundary condition on the bottom and carefully observing here that if you look at the space of holomorphic functions that's obviously an algebra okay but in the infinite depth case this is a complex algebra whereas in the finite bottom case this is no longer a complex algebra it's only a real algebra so multiplication I is not the holomorphic function in the finite bottom case just for the sake of okay and of course you can also describe this functions in in this class of holomorphic functions in a in a Fourier fashion and so in again I will discuss the two scenarios if the depth is equal is is infinite then these functions have the property that the Fourier transform view is supported in the region where psi is smaller than 0 so it has only negative frequencies and if you have a function that has only negative frequencies then the real and the imaginary part of this two functions are related and precisely the the relation between these two is that the imaginary part of you is the Hilbert transform of the real part of you so we're looking at functions specifically which satisfy this relation and in this context that you can also talk about anti-holomorphic functions and anti- holomorphic functions are functions which are satisfied the same condition but have anti-holomorphic extensions of course if functions holomorphic then its complex conjugate will be anti-holomorphic so you have the same relation but with the minus sign and then the interesting object to work within here is the projector the projector to negative frequencies okay so P is the projector to negative frequencies and you can relate this projector to the Hilbert transform so P will be 1 minus identity i times the Hilbert transform divided by 2 and here I should caution you that various parts of the literature is different conventions for the sign in the Hilbert transform so if you're if you might be slightly confused in here this may be because I'm just using a different convention from one you have seen in the past so this is what happens in the infinite depth case now in the finite depth case so when the depth is finite the situation changes a little bit because now instead of solving the Laplace equation in the infinite domain you solve the Laplace equation in a strip and so this relation will be replaced by the following relation so now you no longer have the Fourier transform of you supported in sine negative instead you're going to have the following relation you had of I'll try hard to get the signs right minus C is equal to e to the power I think h c times u hat of C bar so this is the relation you're looking at so now these functions have both positive and negative frequencies and instead of having this relation which involves the Hilbert transform you're going to have a different relation that's the imaginary part of you is equal to something that also depends on height till h of a real part of you and this is what's called the tilbert transform apparently I learned this name from my collaborators but when I tried to look it up in the literature I could not really find it except in some software package so but it's a good name it fits Hilbert Hilbert apparently tilbert is also a name I found by during my search okay and and this operator is defined by its symbol it's a multiplier is minus I hyperbolic tangent of h psi so don't confuse the two h's right let me connect the two and so then the projector you can also talk about the anti holomorphic functions with the opposite sign obviously and so you have a similar projector to make to to to this space of holomorphic functions I'll write it down for you so p this depends on h of u is equal to one half this is never going to be used but just for your curiosity well of u plus one half I one plus I to the transform inverse this is real part of you and this is the imaginary part of you maybe when you talk when you think about this this projector I should say in this case this the reason we call this a project this is project this is an orthogonal projector in L2 so here you're thinking of the L2 setting but here it doesn't make so much sense to think of the L2 setting because in particular this guy will be unbounded in L2 because you're inverting this symbol and so here you're you have to work with the different Hilbert space I'll denote it by this with a norm norm of u in this space will be equal to norm of imaginary well h times real part of you in L2 plus norm of imaginary part of you in L2 okay so for imaginary part use L2 but for the real part use this multiplier and what this multiplier means this is still an L2 norm at high frequencies but when you look at this guy at low frequencies at low frequency this is like a differentiation operator so you're at low frequencies here you're you're only controlling the derivative of you and in particular this means that you you don't have any access to the constants in the function you okay but this this space will be very nice and consistent with what I want to tell you later all right and this this projector here and this projector here will play absolutely identical roles in writing down the equation so when I write down the equations in particular for one case 1 and case 4 the equations will be exactly the same with just different meanings for everything that I'm writing okay so one thing that let's say one thing that I will not have time to to to do for you is to show you how one actually derives the equations in holomorphic coordinates so instead of that I'll just write down the equations and then I'll spend some time explaining their their structure so we have these two variables W and Q these are holomorphic variables and so in this space of holomorphic functions we're going to look at the following evolution so wt plus f times 1 plus w alpha is equal to 0 so this is the equation for the evolution of w and the equation for the evolution of Q f Q alpha minus ijw plus the projection of Q alpha square over j I'll explain in a moment all the notations plus i sigma projection of w alpha alpha divided by j to the power one half one plus w alpha minus its complex conjugate just for brevity is equal to 0 so this is an equation that this is a system of equations that describes at least the cases one three and four in here and if you wanted to also look at the case when you have a nonzero vorticity instead of 0 here you're going to put the vorticity times some expression and here you're going to have also a linear term in the vorticity times some longer expression all right so this is for the case three my green color coding if you want and important to remember to to to to observe in here is that this corrections when you have vorticity also includes linear terms and because it includes linear terms it will change the dispersion relation for the problem and I'll comment more on that later so what are the objects in this long expression so j is a natural object this is the Jacobian of the change of coordinates from one setting to the other so it's one plus w alpha square and then f is equal to the projection of q alpha minus q alpha bar divided by j now if you if you just look at these this equation let's say when I looked at this equations for the first time look at them you pretty much see nothing okay you see that they're nonlinear equations you see that they're complicated you can simplify them by setting some of the parameters to zero so for instance if you set to the vorticity to zero and the surface tension to zero you left only with this part of the equations which might seem more manageable and it is more manageable I'm getting to that yes so what you don't see in this equation is the Dirichlet to Neumann map and the reason you don't see the Dirichlet to Neumann map is that in these equations we're talking about the Dirichlet to Neumann map on the model domain and on the model domain the Dirichlet to Neumann map is just the absolute value of d okay and so in some sense the role of the Dirichlet to Neumann map in this equation is played by all of these projectors that you see projectors time the derivative that's something very close to Dirichlet to Neumann map all right one thing that might be misleading actually in this formulation is the following you look at the Euler equations and what is the the leading term in the Euler equation is a transport right and then there's some coupling because of the incompressibility condition you might say oh this looks like a transport equation have to be careful because this f in here is a holomorphic function it's a projection in particular this it cannot be real right if it were real it had to be constant had to be zero actually so this is not a transport equation not yet but still eventually we want to think of this as some sort of transport equation and when you have a fully nonlinear equation how do you understand its character well the obvious thing to do is to linearize it right and you look at the linearization try to understand what what type of equation is the linearized equation and this is the next thing I'll try to show you and I could actually use this board and of course if I wanted to write the all the full linearized equations my choice of this board would be very poor but first first thing you want to do with the linearized equations is you want to understand what is the leading part of the linearized equations so let me call by w and q the linearized variables okay and so now I would like to write down an equation for dt of w and q and in first approximation one thing that you're going to observe in here is that the expressions that you're going to get on the right will contain the following variables they'll contain of course w alpha q alpha maybe you'll see a little bit of w and q maybe you'll see also some w alpha bar and q alpha bar right there's complex conjugates arising in here complex conjugates are anti-holomorphic we're not very much worried about the anti-holomorphic components because those get killed by the projections so really the leading part when you compute the linearized equation is the part that has the derivative and so at the level of the linearized equation this will look like some matrix times d alpha w q plus some lower order terms and so when you see a system like this you'll think oh this is this may well be something hyperbolic would like this to be something hyperbolic given that we started with the Euler equation and so important object is to understand what this matrix is now this matrix actually is not very hard to compute what you have to do is you have to differentiate this equations with respect to a big w and big q but you only care about the differentiation with respect to the holomorphic component not with respect to anti-holomorphic component so for instance when I'm looking at j I only care about the one plus w alpha component of j when I'm looking at this expression I'm going to care only about q alpha and the one plus w alpha plus from from j so it's an easy exercise to do the computation and so I'll write down this matrix for you and I'll write the write it down in a form that's easy to to read okay hopefully I can fit it and so so what you get in here you inherit the f obviously but this f gets coupled with another guy q alpha bar over j in some fraction here this is a little bit long so I can divide it by j and here 1 plus over 1 plus w alpha bar minus q alpha square divided by j times 1 plus w alpha it seems tedious but believe me this is worth it over j and then plus q alpha over j all right so we want to understand what what kind of matrix is this right in particular want to understand what are its eigenvalues and one obvious remark in here is that in this matrix you have a multiple of the identity right the multiple of the identity is the object that contains this guys right and now what happens with the remaining matrix suppose I take this guys out well the remaining matrix first of all has trace 0 okay and then it's very easy to see that it also has determinant 0 so the remaining matrix has a double 0 eigenvalue so but of course it's not diagonalizable right it's going to be if you diagonalize it you're going to get the obvious Jordan block and so this matrix has a double eigenvalue and the double eigenvalue is given by this object f plus q alpha bar over j we're not yet out of the woods we'd like this to be a hyperbolic equation so we'd like this object to be real right so first thing we do always is we baptize it so I'm going to use the letter B for this and maybe I can put that on this board actually I'll write it down here because I might want to save it for later okay so B is going to turn out to be again a brief computation two times the real part of the projection of q alpha over j okay so this is indeed a real object so this means that our problem has at heart is a hyperbolic problem but the hyperbolic one with the double speed this is no surprise right because if you think of the Euler equations everything gets transported with the same speed now what is the meaning of this speed B that we see in here one thing I was telling you about last time was the fact that you have this gauge freedom having to do with the changes of coordinates and in particular we choose some non-standard coordinates in here so this B really is the velocity that comes from the change of coordinates if you want B is the difference between our coordinates and the Lagrangian coordinates in the same way as the standard velocity is the difference between the Eulerian coordinates and the Lagrangian coordinates okay so so so far so good and I'll tell you that we were really happy when we got this guy to be real when we were computing the equations because that meant we are not making any any big mistakes but now coming back to our matrix in here one thing you want to you always want to do when you see a matrix like this with the double eigenvalues you want to diagonalize of course will not be diagonalizable but you want to put it in some good Jordan form right and there's there's one very simple change of variable that does that and so that's from w and q going to the pair of variables w and r where small r is equal to q minus q alpha 1 plus w alpha times w and this idea of diagonalizing things of course is not new and you maybe eventually will see how to implement this not only at the level of the linearized equation but also at the level of the full equation so anticipating a little bit for those of you who are familiar with these things I'll say that this is very closely related to ideas that first arose in non-linear wave equations in the work of Alina who sort of pioneered this idea of choosing the good variable when you solve the equation and then work of David Land closer to water waves work of Alazard Burkin Zuley again for for water waves equation so you want to work with the right variables you want to diagonalize your system properly okay so what happens when you diagonalize your system properly so let me write down a little bit more of the linearized system so you have dt so suppose I now write the system for w and r so I'll have the system dt w plus b d alpha w plus and the next term I'll get in here is r alpha divided by 1 plus w bar alpha and equals and here I'm not going to elaborate this this will be some lower order terms and and at least in the sense of local theory but more interesting it will be to write the equation for r dt of r plus b the alpha of r so we start with the transport which is the leading part and here comes the coupling term with the first equation and this coupling term has to do with the with the gravity let me make sure I get the correct sign and this will be i times g plus a g yeah g plus a 1 plus w alpha multiplied by w and this is again some lower order terms okay so so in order here you see the the main features of of this equation the first feature the leading feature is the transport the second feature is the coupling between these two between these two equations and if you want to think of something to call a coupling coefficient that should not be this coefficient or this coefficient but really should be the product of these two coefficients and the product of these two coefficients in here if I multiply them and I can write that here I look at g plus a divided by j all right and this is the object that I have that plays a key role in this analysis and that I have mentioned yesterday and it's it's a relatively easy computation to show that this is the normal derivative of the pressure restricted to the free boundary so the sign of this quantity the fact that this g plus a quantity is positive is place a key role in the well-postedness of this system and that's what's called the Taylor stability condition so you want this guy to be positive all right else this equation is no longer locally well-post yes yes I'm getting to that so a is equal to two times the imaginary part of a projection and this projection is of r times r alpha bar sorry and and I was waiting and r is q alpha over one plus w alpha okay so you first my hand a little bit here I had to give something away before it was ready but in any case so let me finish this this discussion in the following way so if you look at this equation one obvious symmetry of this equation is the translation the invariance with respect to spatial translations right and since this equation is invariant with respect to spatial translations then you have one special solution to the linearized equation and that's the derivative of the solution so if I'm looking at w alpha and q alpha this object will solve the linearized equation all right but if I want to study this object properly using the linearized equation then I should diagonalize it right so what do I get when I diagonalize it well w alpha stays unchanged okay but again a very simple computation will show you that when you diagonalize this you're going to get the pair of variables w alpha and q alpha divided by one plus w alpha so this means that w alpha and q alpha is really not the right variable to study this equation in instead you want to study this pair of variables and that justifies giving a name to this guy in the first place all right and once you give it a name you have to understand what it is right and this turns out to have a very simple interpretation this is really nothing but the two components of r are the velocity vector field uh the two components of the velocity vector field so r represents the velocity vector field written in a complex form okay um exactly if you want to write it down r would be u minus i v where u is the horizontal velocity v is the vertical velocity so one thing we take away from this this computation is that um even though this equation is written as an equation in w and q it's natural in the first place to differentiate it because it's fully non-linear and when you differentiate it and diagonalize it then you get an equation a system of equations in these two variables w alpha and r and this would be the good variable salla alina let's say okay um and so to relate this perhaps with work that was done by people in the audience let's say using uh using the uh your Eulerian setting let's say in the Eulerian setting this diagonalization happens at the level of the undifferentiated variables and it involves uh to do it properly have to use paradeferential calculus uh which uh if we like paradeferential calculus is nice and good but here it's it's it's nice that you get this diagonalization very simple and algebraically all right so um the last thing going along this line that I want to show you is the system of equations for this differentiated variables for for w alpha and r all right let's see how this works out I know that no no but it's trying to figure out how hard to push on this yeah okay do you see everything there all right one thing what which you may have noticed while computing the linearized equations is that I have neglected the effect of the surface tension and that's because that one does not play any role in this discussion about diagonalization diagonalization has nothing to do with the surface tension has nothing to do with also with vorticity so that those variables will remain the same in all cases so now I promise you the system for w alpha and r and here I'm I'm I run into a bit of problem with my notations because uh I don't want to really well let me not make things more complicated than I should be um so so here's the system dt of w alpha plus b d alpha w alpha plus one plus w one plus w bar um r alpha is equal to one plus w m just be a little bit patient alpha thank you um and then uh dt of r plus b d alpha of r um minus i times uh w minus a g w minus a and i is in front divided by one plus w alpha is equal to zero so I see this this is a pretty manageable system I wrote this system in this first case and it no longer applies in the case of finite bottom you have to make some changes there and one obvious reason why you see it it doesn't apply it has this i in here right i is bad i is not holomorphic multiplication by i makes sense in the infinite bottom case but has no meaning in the finite bottom case all right but in any case this is our system and now uh I'll I'll try to to give you some idea about what happens when you try to look at the local well postedness theory for this equation so now I'm really getting into the problem um but before I get to local well postedness I need to tell you a little bit about the function spaces and the first thing that I have to show you then is what is the energy for the system so this system has an energy uh and the energy is uh integral of w square actually this is the energy for the system above okay because it's at the level of w and q times one plus real part of w alpha plus the imaginary part of q or q alpha bar I might have some wrong signs in here so don't panic if some things are not positive okay they should be positive so two comments about this energy so there's a corresponding energy obviously in Eulerian coordinates this energy as was discovered by Zakharov plays a role of the Hamiltonian for the system one thing I'm not going to do for you is I'm not going to write down the symplectic form in holomorphic coordinates that's if you want one downside of using holomorphic coordinates the symplectic form is ugly uh symplectic form is simpler in uh Eulerian coordinates but but here it doesn't look so good so this energy really corresponds to Nether's theorem because of the time translation invariance of the system uh because uh you also have a spatial invariance with respect to spatial translations you're also going to have some horizontal momentum which I'm not going to write down so you you can play a little bit with the Hamiltonian structure if you wanted all of these problems that we're looking at are Hamiltonian and they all have the same symplectic form uh do they no uh this one has a different symplectic form uh the one with the vorticity but the other ones have the same symplectic form one thing that we read and I should say one last thing that this energy uh consists well consists of two parts that's obvious uh this part uh is like the norm of q in h one half homogeneous square okay it's exactly this well up to a constant perhaps a factor of two uh this one you you want to relate this to uh norm of w in l2 square right um you have this guy in here uh so is this positive or not um and the answer is to this is very easy or this energy will posit it will be positive at least as long as uh the simplest thing to say is that the fluid surface does not have self-intersections um in effect this energy will still be positive sometimes when you have self-intersections but you can allow some kinds of self-intersections and not others I'm not going to elaborate that but this is essentially a positive object all right the energy is positive definite all right and so one thing that we read of this energy is the kind of spaces that we need to work in in so from here we see that uh the the spaces to study our revolution in are spaces of the form l2 cross h one h one half homogeneous and this applies to the first problem this will change when you move to each of the other problems they'll have different slightly different setups and in particular this business that I was discussing before with the projector in the case of finite bottom means that in the case of finite bottom you will no longer control the low frequencies of w properly so the energy will will instead of having the l2 norm of w and here the h one half norm of q will contain norms which are with respect to this space and with respect to the associated inner product so you lose some low frequency control in the finite bottom case all right so so we you want to use this space so you might say when you look at local well positeness uh you want to work with say w q in uh l let me call this space script h which is l2 cross h one half and then when you look at the differentiated variables w alpha and r from here on for further derivatives this is the variables you want to work with so this should belong to maybe some hs and let me discuss a little bit this exponent s all right and again this discussion will be specific to problem one and in the high frequency limit also specific to three and four but not to two because two has a very different scaling at high frequency all problems one three and four have the same scaling at high frequency whereas problem two the surface tension term uh takes over at high frequencies all right so so what what can we say about this exponent s and uh one uh thing to look at uh is whether the problem has scaling and this problem one in particular does have scaling uh so the scaling is uh if you want written at the level of w alpha w alpha of alpha and t goes into of t and alpha goes into w alpha of lambda t and lambda square alpha and uh q and r of t and alpha goes into lambda r of lambda t lambda square alpha and once you have some scaling you can compute what is the critical exponent in here that's very easy to see so the critical exponent uh corresponds to s so um maybe let's write this here so s critical is equal to one half all right um and uh you might say oh perhaps we should be looking at solving this problem at this regularity level but then you remember that this problem is a quasi-linear problem and to my knowledge hardly any quasi-linear problem was solved at the scaling level so instead we're we're going to give up a little bit and the exponent that I'll be uh dealing with primarily today is s is equal to one so it will be one uh rather than one half half derivative about the critical level and for those of you who are familiar with what happens with the non-linear wave equation is exactly the same balance of forces the state of the art so to speak is more or less about half uh derivative above uh the scaling you can one can expand this discussion a lot I don't want to do that um so let's see all right so um maybe before I tell you about uh local rule posedness I'll tell you about the energy estimates and so one sort of twist we introduce in in this business of uh energy estimates is actually not a new twist is a twist that one has seen all over the place in connection with non-linear wave equations namely that when you control the evolution of energy you don't want to control the evolution of energy in terms of sobolev norms of the solution rather you want to control the evolution of energy in terms of point-wise norms of the solutions on grounds that this point-wise norms of the solutions are weaker objects and they really tell you that the energy will budge only when you have point-wise concentration and not just a large l2 norm or large sobolev norm and so in order to do that one thing that we do in all of these models we introduce some sort of control parameters and I will warn you that I'll oversimplify things just a tiny bit in here by less than a log to put it this way just to keep things short so we're going to use two control parameters our first control parameter will be uh what I'll call a um and this will be a scale invariant norm so this is norm of w alpha in l infinity plus norm of d one half r in l infinity okay um so so this corresponds exactly to the critical exponent one half and so this means that this constant a can appear in any combination you want in in your estimates more or less and the second control parameter that we're going to use is half a derivative higher but half a derivative higher so that would correspond to s is equal to one I'll remind you that we work in one space dimension and in one space dimension the embedding h one half into l infinity does not work but one thing we were very happy to be able to do in here is to relax this l infinity requirement to a bm o requirement because h one half homogeneous does embed into bm o so in here output be a more norms d one half w alpha in bm o plus r alpha in bm o and maybe so this is as I said half derivative higher and to get another way of understanding the the level of regularity for this I I'll remind you that when I defined that coefficient the the transport coefficient b which was somewhere did I erase it okay all right so so you see that I can rewrite this as two times the real part of r projection projection of r divided by one plus w alpha bar so the leading part of this is the real part of r okay so the fact that you control r alpha in bm o means that you're controlling the derivative of your velocity in bm o so you don't control the velocity in l infinity you only ask to control the derivative of the velocity in bm o and so if you do this then by no means you're going to have energy estimates okay for for even a transport equation and so in order to make this work we have to make use somehow of the fine cancellation structure that's buried into into the water wave equation um and so what what is the good energy estimate then for for our equations I'll write two types of energy estimates first of all energy estimates for the linearized equation and then energy estimates for for for the differentiated equation now when you look at the linearized equation what is a good energy so so you look in here and you know that the l two cross the h one half norm of w and r should be the correct object so let's define this energy a two of w and r to be equal to integral of w square plus the imaginary part of r r alpha bar um d alpha so this would be the the good energy for the linearized problem around zero um maybe I should put a g in here all right but here we're looking at the non-linear problem and the coupling between these two equations has the g plus a coefficient in it and so this g plus a should really appear in your energy and this is where it appears and so this is the natural energy for the non-linear equation now you see also another way where where this coefficient becomes important you make sure that your energy is positive definite okay and so what is the our energy estimate for this functional d dt of e two of w and r just to give you an idea of the flavor of the estimates that we're proving is smaller than and here we're going to have an implicit constant and your implicit constant depends on a and smaller than b e two of w and r okay and now in terms of n adjustments we primarily tend to think of the n adjustments for the linearized equation rather than for the original equation because there's more information if you can prove n adjustments for the linearized equation and so if you look at the original equation you can have a similar n adjustment which says that d dt of the energy same energy if you want d k w alpha and d k r smaller than a b e two of d k w alpha and d k r with respect to the diagonal variable so you see the same dependence on the parameter b and the implicit dependence on the parameter a and at least in the case one but we can also implement this in in the second case for the second problem these estimates are invariant with respect to scaling okay so there isn't much to improve here in terms of of the energy estimates this this is as good as they get and so if you just use this n adjustments both for the differentiated equation and for the linearized equation what you're going to end up with is the following theorem maybe i can use that board so the theorem would be that the equations one two and four so these are equations which share this scaling in the high frequency limit are locally well posed for w alpha and r belonging to this space h one so this corresponds to choosing s is equal to one in the air and some some comments are in order here so of course this is not the first time a local well-posedness theorem was proved the first local well-posedness results for water waves are due to na limov for small data in the case one and two of sianikov in some sort of gevrey setting for the problem four of sianikov was also the guy to introduce this holomorphic coordinates in the dynamic problem the same theorem and then this guy also na limov work with fairly high regularity the next important result was the result of cj u cj proved well-posedness again in high sobole of regularity but for large data and her contribution one of her contributions was to figure out the positivity of that normal derivative of the pressure then I should mention the work of al-azard burq and zuyili their initial work if I remember correctly was just a little bit above this epsilon above okay and so that's one one we were able to get the first result so we managed to beat them by an epsilon but then they came back and they said oh but this problem is also a dispersive problem it has three cards estimates and if you work hard then use those three cards estimates you can improve this result a little bit more and and and then after them it came and I'll apologize if I misspelled the name guian and the boy fair and they improved a little bit more the threshold and I don't know what the current threshold is maybe it's I'll guess it's one eight better than am I right something like that one eight better than that by using three cards estimates however our n adjustment still remains the the better one so we have the right and I just not just no no no no we did not try to use any any three cards estimates and and there's a reason why why we stuck to the n adjustments and that's because the n adjustments are the object that's crucial when you look at the problem I really wanted to look and that's the long time behavior of this equation now one thing I should say maybe a few more comments I have in here when you look at when you state a local wall post and as a result the correct setting for the this is very vague setting vague terminology if you want you have to qualify what means local wall post illness this is a quasi-linear problem so local wall postness includes existence let me write this here existence uniqueness means continuous dependence on the initial data but what you don't get because the problem is quasi-linear you don't get better you don't get any sort of uniformly continuous dependence on the initial data so this is in some sense optimal another thing we were keen to do and and we were able to do is in here we have these parameters we have let's say when you move from the first case to the first case you have the depth and we wanted to have this results uniform with respect to at least some of the parameters in here and so what I can add in here is that this result is uniform as the depth goes to infinity and it's also uniform as the vorticity goes to zero so you can take those limits if you want and have uniform estimates for them let's see another okay maybe this concludes the remark about the local wall post illness theorem and that brings me to the topic of next lecture where I'll try to tell you about the long time estimates for the equation my plan was to get to that today I'll confess but it did not happen and I did not want to rush the first part because I know that many of you have not seen much of this stuff before so if you lose it early on then the last lecture will be useless but nevertheless next time I'll try to tell you as much as I can about the long time estimates for for this problem and that's it for today this right so once you write down these sets of equations you no longer care whether your initial data came from a graph or not and maybe one thing that I was planning to show to you but again I did not have time to is the proof of the positivity of the normal derivative of the pressure our proof not cj's proof because that proof applies regardless of whether your curve has self-intersections or anything like that so one can formally continue these solutions past self-intersections there's no absolutely no issue with that except you lose the physical interpretation right what your model is no longer a wave it's just your favorite pd let's say but or most hated pd is depending but has nothing to do with the self-intersections so you say you have local world personas for one two and four but all of those cases have c equals zero so I'm a little confused about that sorry sorry sorry one three and four thank you very much one three and four right and so since you asked the question or pointed this I'll say that there is the corresponding result for problem two is also true except that you have to fiddle a little bit with the numbers because that problem has a different scaling okay can if I can make another remark I meant to make this remark early on and I forgot about it so as you see we have these four parameters that we're looking at but we're looking at just the four four of these problems and actually there is a much larger variety of problems than one can study in here that's this is one nice thing about water waves is a great playground you can find lots and lots of equations very different dispersion relations very different dynamical properties lots of interesting problems to to study


to the cloud. Done. Hello everyone. We are on the 28th of August 2023. And it's GSOC Jenkins Docker base quick start weekly meeting. It's the last official one as we are reaching the end of the GSOC 2023. We haven't asked for any extensions. So this week is the final week. And we will have as mentors, so Bea Viento, Jean-Marc and myself have to begin the evaluation of Ashutosh before September 4. And it's starting tonight my time at 8pm. So be afraid Ashutosh, be very afraid. That should go fine. We have another deadline, which is important. It's the 14th of September when you will have to give a presentation, a 10-minute presentation and five minutes for questions about the whole work you did for the GSOC project. And this morning I had another talk with Damian DuPoto from the infra team because as you know it we have a chicken and egg problem. We can't have PRs merged into Jenkins IO as long as we don't have our images in the Jenkins CI or Jenkins infra organization. So he asked us to open a ticket on the infra help desk to start the beginning of the discussion about where should the repo be integrated, should it be Jenkins CI or Jenkins infra with this, because that in a few minutes from now. So last meeting Ashutosh, we were okay with you beginning the presentation at the beginning of September. We were not in a hurry. That's still valid for me. What about you? It's fine with you. So from next week I would say beginning of next week. Yes. That's cool. And you already have some material from the midterm presentation. So that should go fine. Regarding the extension. So we didn't ask for any extension because we are more or less done with what we had in mind. And what we'd like is to bring this project to Hacktoberfest and to be part of Hacktoberfest, we would like, we will have to have it integrated into Jenkins CI or Jenkins infra. We are not an open source community yet. It's just the four of us. So that does not work for Hacktoberfest. That's why I'd like us to focus on having your work integrated into Jenkins IO and Jenkins CI or infra before Hacktoberfest so that we can tag along the month of September, good first issue, Hacktoberfest and so on if people would like to participate. So the discussion I had with Damian this morning was about should it be into Jenkins CI organization or Jenkins infra. As you already know, Jenkins IO is in Jenkins infra. And that's because it's run on Jenkins infrastructure. So that's why it's there. As for Jenkins CI, most of the projects which lie into the Jenkins CI GitHub organization are things that will run in the end on the end user machines. And that's a rule that we have. And this project is in the middle of the bridge, I would say, because some parts of this project will finish into Jenkins IO. So it's part of infra. But some parts of that will run on the end users machines because they will run Docker compose with our images. So I think we have to discuss that. And the first step is just to declare a ticket on Jenkins infra support. That's not an easy one, but that's okay. So the first thing to do will be in September. I don't know when, but in September, have your repo or fork of your repo, maybe a stripped down version of the repo without documentation. We'll see something maybe much more concise, maybe cleaner. I mean, with not that many rhythmifies and so on. We'll see something more concise into Jenkins CI. Maybe we'll see. As for the pull request we had last week, I saw six plugin update PRs. So your work works beautifully. It's a good thing to see and plugins are updated regularly. That's a lot of work that we don't have to do manually. And that's a good thing. We also have seen nine depend about updates. So this has proved that it works, which is pretty cool. I've seen some version updates today on the agents. You know, the agent image Docker agent images. So hopefully within the week, we should have some new dependable updates, because I think it's round weekly. Now, as for the open PRs, there is a big one, the end to end multi branch. And could you give us a status on this PR, please? Yes. So the tutorial was more or less complete, but I was getting another error while checking for the tutorials from the GitHub actions. The main GitHub action files that test the running container and checks if the job is running that was giving error. I looked into it and the error was coming because the conflict dot XML file and that creates the job, but not being accepted into the Jenkins for some reason, it was being created, but it was not showing as the job. So for that, I I searched a little bit and found out that we can curl and create a new job with curl command too. So for that, I have the XML file is still there, but we are not transferring it directly into the container with we are transferring it after the containers are created with the curl command now. And it's working now. That's cool. Well done. So it could be reviewed and merged this week, hopefully. I think, but we want to review it a little bit. He asked a question I replied. Okay. And then weekend came. So that's cool. I think he replied on weekend. I haven't seen that. It was about because right now, we are using images with the branch name right now, because it's in separate branch that we need to remove when we merge it and remain too big. It's too big. Well, okay, we need to have the merge in which is right. Now should I remove it? I guess so, yeah, but we're almost good to go. That's cool. Now there is a conversation going on. It's about the layout of the documentation. So you asked the question as we asked you to do last week. So yeah, should we keep the same set of Docker campus files? Should we do another one for the second persona or not? And I gave my insight on that and Bavian to also. So I don't know what you think about that, but we may go with a smaller file for the second persona, which could be done automatically or by hand for the time being. I don't know. I tried it by doing it by hand and it worked fine. It was not that complicated. Yeah, please. And we just have to think of keeping them up to date. Yeah, okay. We need to use them as files. Okay, I didn't think about that. I was thinking something like this, like we will give the git clone command, which will lead to our repository, which will contain all the tutorials and users can just do this and skip this part. But second part with the files, this is for the first persona. But if someone wants to do it manually, like second persona, they can, this is just for the main part because I've removed all the other services and it works with the same command, doc composer, Maven. And I have edited the new images, I'm sure. This one is too big. I edited it, but I was thinking like this. What do you guys think? I don't know. I think it's maybe too much information for the first persona. It's very nice to have it there, but it's maybe too much. What do you think, Jean-Marc? First site, it also looks very busy. Looks very busy. The question is, what could be simplified? Yes, all should we create another documentation elsewhere? The channel is, if you want to know more additional information. Could you please, yeah, go ahead. The additional information, like should we do it, like just the link here for the first persona and if second persona wants to use it, you can get to somewhere else, like GitHub read me or something like that. I think it would be better. Yeah, we can add instructions for building images and all that. Yeah, I think so. Okay, so like here only we'll give the link for the GitHub list and then we'll continue from here. I would prefer that if you don't mind. So for like, for making the PR for an event tutorial, all we need to do is remove some stuff. So like, where should we add the documentation like right now? GitHub read me or because right now read me file is not that verbose. Yeah, I got it. Maybe you should create another file next to this one, but it should also be part of Jenkins.io in the end. So I don't know how you could articulate that, but maybe you have between some somehow of this for the second persona, but I don't think it's placed in the readme.md. I think it should be part of the official documentation also in a way or another. So we shouldn't lose it. We have to have it somewhere, but I'm not so sure readme.md is the right place for it to be. Mm-hmm. Great. Jenkins.io used a lot of the direction too. Mm-hmm. Maybe you can, there are a lot of different files. I was getting confused with them when I was trying to do this too. I think some more can be there too. But how will we, we'll be keeping them updated so they... That's a question. With ASCII.doc, you can have some include files. Yes. You know, if you want to reference a code file somewhere, instead of copy paste into the document, you can do it via an include instruction. So that's maybe what we should do. Oh, sorry. Somebody is ringing at the door. I'll be right back. John-Marc, please go ahead. Well, I a little bit lost the... I prefer that we pause the recording. We'll pick up again when Bruno comes back. Is that okay for you? It's okay. So we are back. Thank you, John-Marc. So we were discussing should we create a twin file on some sort in order to have the official documentation for the first persona and a more detailed one for the second persona. And we were also discussing that we can include some source code files within the ASCII.doc. So maybe that's the solution we should be aiming for. You know, keep the Docker files up to date on their own and then include them in the ASCII.doc files. But we have to be very careful about keeping the files up to date nonetheless because the files that we will include are not the main files we are working on. The main files are the Docker compose that are kept up up to date things to depend about. But we have to make an excerpt of them and then create those small files for the second persona. And that's where we have to be very careful. I would love to have an automatic way of doing that instead of relying on our brains. I don't know how to do that, frankly, for the time being. We could maybe use something like YQ because it's YML. That could work, but I'm not so sure about that. Never tried that. So YQ is a YAML parser and that we already use a little bit in the GitHub actions, I think, and for Github in order to find the right endpoint, I would say. Does it work on volumes too? Volumes? I think that should be the issue. Yeah, it should work. Hopefully, yes. I will try it today. My son is not working anymore. There are days like that. Yeah, should stay in bed these days. Do you manage? Screen is off. Yeah, his problem is he is the meeting organizer. If he drops out, the meeting dies. Yes. And I don't know how to reclaim that. I had the opportunity once, reclaim host, there I am. I got the message that you have now. Going to post the recording. Yeah, thanks a lot. Zoom crashed. Can you see my screen, by the way? Yes. Okay, thank you. So regarding the layout of the documentation, do you know what to do next, Ashutosh? Yes, I think I got that. I will be experimenting with YQ today. So I'll let you guys know how it turns out. Okay, thank you. YQ and source code includes within ASCII. Nice. And then we'll see if we can open the first PR on Jenkins.io. Even if we know it won't be merged until a few weeks from now. Thank you. Now, regarding the tutorial about end-to-end multi-branch pipeline project creation, so we're almost ready to merge. You still have a few things to do, but I think that by the end of the week, we should be able to merge it. You managed to get rid of the issues you were having with the jobs in the test. So that's fine. We still have one which is not mandatory, which was used dependabot and update CLI to keep the samples up-to-date. As for dependabot, it's working. I think it's working. We have seen quite a few PRs, but we will have to keep it up-to-date because you added another directory for the very last example. So I think you should add a section about that into the existing dependabot file. And as for update CLI, frankly, I'm not so sure we will have the time to experiment with it. And to be honest, dependabot already makes almost all the work we need it to do. I was pushing a few weeks from now for update CLI because we did not manage to get it to work. And so update CLI was a nice replacement for dependabot. But now that dependabot works, well, let's keep it working this way. And maybe later on we could have update CLI for more touchy parts that dependabot can't handle. We'll see. So no hurry on this one, not mandatory. And I added another one minutes before the meeting, thinking of usual mark. I wrote move test from shell to goss. I haven't described it properly. I just get three links and that's all. But goss is a project written in Go that Jane Keane's infra already uses. So the whole thing you're doing as you know with curl and so on in a big bash file can be simplified with this tool which is called goss. So I don't know what it stands for. The first part Go is because it's written in Go, I guess. But it has a few advantages. It works on Windows, Linux, macOS. It can work with Docker compose. It's well written, I would say. Documentation is pretty well done. And yeah, I'd like that we try this one in order to simplify the existing GitHub actions. Your GitHub actions work beautifully. But frankly, six months from now, if I have to make modification in those files, I will have a hard time. Really, not because you wrote it badly. It's just because it's a bunch of shell commands. Yeah, it does a job. But frankly, maybe we tweaked things a little bit too far. And yes, it works. But we could have a framework to help us do that. So it could be goss. It could be bats. But frankly, I told with people using bats and goss, and they do prefer goss. So we don't have to do that. But maybe we should experiment just to see if we can get it to work easily. And if that could help us have something that is more maintainable six months from now. So I thought of you, Jean-Marc, because I know you're curious. Just looking at it. Yeah, just looking at the page here. Yeah. And for the Jenkins slow and Jenkins fast, there are already working examples. Fast and slow is because on some architectures, you will have Jenkins starting very fast. So you don't have that many steps, you know, wait for that and wait for that and wait for that. And on machines that are pretty slow, you have to be to have a few more steps, I guess, in order to wait for the several parts of Jenkins starting. Any help? That's just my idea. Yeah, I'm not seeing the right thing here. So I don't know if it's a standalone tool. It's something to include how to start. No, it's a standalone tool. You can install it within your Docker images if you want to test directly via Docker, or you can install it on your machine and test whatever you want on the machine you are running the things on. It could be a real service. It could be a service running through Docker. Do you have the reference to it? Because my Google foo doesn't give a good result. Oh, you mean the repo? Yeah, the repo or where the tool is located so it can have... Sorry, it's on GitHub. I'll send you the link via the chat if you don't mind. Yep. Let me find the right window. Can you see it? Oh, G-O-S-S. Goss, my bad. How would you prudence that? I have no idea. I made it French like Goss, which means child, by the way, in French. Well, I wrote it with a single S. So, Goss Orc, yeah, that looks much more... Oops, I need to check the time. There it is, yeah. Well, have a look to it. Thank you. Yeah, so I don't want to force push. I would just say that maybe we should experiment with that and see if it helps us having something more maintainable and more easy to write. It would be good to write now for the latest issue. I was editing that file and it is hard to read, even though I created it. So, yeah. Only thing that bothers me is that it relies on YAML once more. John Mark, you are about to say... No, looking at it. And so, this is a recommended replacement for bats? At least from the Jenkins infrared team, that's where I heard that, because they were thinking of getting rid of bats wherever they can and replace that with Goss. Yeah, the problem is with bats is that you need to be good at bash programming. Yes. And for Goss, you have to be good at YAML indentation. But most of the time, our IDE help us with YAML indentation. So, we're almost safe. And there are already processes running for the infrared team thanks to Goss. So, we can hope that it's a sturdy enough tool for us to use it. Yeah. Good. I learned something. Yeah, I never heard before this morning of this tool. So, Damian Dupoldo, if you'll read us. Thank you. And we're at the end of the agenda I had. John Mark, while you were away, I think I talked about the presentation that will have to be done by September the 14th. Is there anything you would like to insist on before we wrap it up? First recommendation is to be on time. So, not going over the budget. So, 10 minutes for questions and also for presentation and questions. So, the questions will be the adjustment variable. I am also interested. Maybe one or two minutes. And a couple of bullet points is what did you learn and what do you wish to tell the people that want to be part of GSOC for next year? So, in trying to make one single advice to have a successful candidature, but also a successful GSOC. Now, focus is something I will say. It had been a long time since the last time you say focus, focus, focus. Yeah. Does that help, Ashutosh? Thank you, John Mark. Ashutosh, anything you'd like to share? Or would you have a question, doubts or? No, that's it. I think to John Mark, maybe John Mark was not there. I will not be available for like three days, two or three days, because there is a festival here. So, yeah, and I will be traveling. Yeah. Okay. I heard that you were explaining when I joined the meeting, you were explaining what the festival was about. At least I understood that it was a festival, so it's no need to re-explain. I will do my homework there and knew that you were going to be offline or very difficult to reach for three days. Yeah. Go ahead. Normally for Ashutosh now, in four hours, he's done. Yeah. You didn't hear that, Ashutosh. You see, I have three weeks to go. No, no, just kidding. But yes, we are supposed, John Mark, to evaluate your work by, no, not by tonight, from tonight, 8 p.m. our time. And we have until the fourth of September, I think, to evaluate. John Mark, would you like to set up, that we set up a meeting to discuss that with Belvianto or you, or do you want that we do that asynchronously? Oh, I fear I will have a busy time. So, asynchronously. Okay. No problem with me. Here, trying to do it sync, the problem is I don't want to influence Belvianto. Yeah. Yeah. And here, started asynchronously. And if there is a tie or there are arguments that we need to discuss, let's make it together. Okay. Got it. Thank you, John Mark. Let's do it this way. We can discuss it tomorrow morning. Why not? Let's do that, of course. Thank you. You will be available standard time, like every week, because you're back home tomorrow morning? Tomorrow morning, normally, it should be available at standard time. Now, I found the house in because of the long absence, and I need to get a handyman here quickly. Oh, yeah, got it. So, we'll discuss that tomorrow. And no problem if you have to change the time slot. That's fine with me. You know my agenda. That's okay. Ashutosh, best of luck for the rest of the work this week. Enjoy the festival and see you whenever you feel like it, because no mandatory meeting anymore. But we could keep this time slot. If you ever want to, we can keep it for a few weeks or whatever, or if we all the week, we'll discuss that maybe on matrix element, whatever, as you feel. Okay. Ashutosh, you wanted to say something? I was saying that I will be on the Docs Office hours and Docs Office hours. Despite the festival? No, the festival. When is the festival? The festival is on 30th. So I should be able to make it on Thursday, 31, because it's a night for me. Okay. Unless you have eaten too much. I'll let you guys know. Okay, great. Enjoy the festival. Yeah, and thanks a lot for your help. Zoom related, Jean-Marc. I was lost today. Thanks a bunch, folks, and see you when I see you. Bye-bye. Okay, need to drop too. Yeah, bye-bye.


There's kids who are unknown, who are getting way more streams than vets in this game. And it's just all because of the climate and the temperature of where we're at.  They'll look, yeah. And don't work in reverse. So you just got to let the guys do what they do at some point.