Hello, everybody. Welcome to this open networking and edge summit presentation about ONAP and O-RAN and how they're the perfect complement here. My name is Stephen Tirol. I'm a senior expert in automation and management at Ericsson and I was prior on the ONAP TSC and the ONAP architecture chair and had various roles in ONAP there. So together with my colleague, Mattias Sintton. I'm an expert in run traffic handling and performance at Ericsson and I've been working with O-RAN for the last two years. We're going to walk you through this presentation for the next 20 or 40 minutes or so. So in here what we believe that is and we see that O-RAN is defining an open and dis-advocated RAN. And at the same time, ONAP has been defining a network for automation platforms for multiple domains that includes the RAN. So we believe the question is not why these will meet, but how they should meet. Okay, so we start looking at O-RAN then. O-RAN is an industry initiative that works for additional desegregation of the RAN. It was announced at Mobile World Congress in 2018 and later in the year the first working group meetings was held. O-RAN has now 23 operator members and more than 100 non-operated companies contributing to the work. If we start to look at the O-RAN principles and what O-RAN would like to achieve, their aim is to create an industry initiative then, as said, for additional desegregation of the RAN. And it works with some principles where this openness then should bring service, agility and cloud-scale economics and intelligence for self-driving networks with AI optimized closed-loop automation. Through these principles, O-RAN works with three main objectives. The first one being desegregation and open interfaces. Within this objective, O-RAN is working on defining multivariate interfaces within the distributed unit defined by 3DPP. That means that there is another layer of desegregation introduced in O-RAN compared to 3DPP, which disaggregates the radio functionality and the baseline functionality. But here also, O-RAN works on making the 3DPP defined interfaces such as F1, truly multivander, and the initiative should try to make deployment of these multivander interfaces as easy as possible. The second objective of O-RAN then is automation and optimization. As part of this, O-RAN is defining two new control loops and two new radio intelligence controllers. The first one, the non-real-time rig placed in the SMO domain, the service and management and orchestration domain, is supposed to enable automation as part of the management domain. With the non-real-time rig comes an interface to work on a functionality called A1. The second of the new control loops is internal to the O-RAN and here O-RAN introduced the near real-time rig and the E2 interface for allowing AIML as a tool when trying to solve the radio resource management problem. The third objective of O-RAN is virtualization and open software. There is an aim to cloudify raw functionality as far as possible. And in doing that, O-RAN is creating a hardware and software reference architecture. And it also then looks on how to orchestrate cloud infrastructure for raw applications. If we look at the structure of O-RAN, there are nine different working groups defined. And these very much reflect the objectives of O-RAN. What we can see is that working group four and five very much relates to the objective number one with further desegregation. We're working group four works on the low-related script or on open front-all interface. And working group finds the standardization and work around the 3DP defined interfaces such as F1 and X2, for example. For objective number two, optimization or optimization, O-RAN has two working groups working on the new control loops and the controllers associated with that. And also, of course, the interfaces. If we look at the working groups six, seven and eight, those correspond to objective number three with virtualization and openness coming. Aside these working groups, working on specifications and details around the different objectives, we can see that there is an open source focus group in O-RAN. And this open source focus group is a collaboration between the O-RAN Alliance and Linux Foundation. We're the mission then to support creation of software for the radio access network as defined by the O-RAN architecture. Within the O-RAN software community, we can find currently 13 different projects running. And also here, those projects relate very much to the working groups defined by O-RAN. To the left here, you can see working groups related to work to the radio-intelligent controllers that O-RAN has defined. And to the right, then, you'll see projects related to the controller part of the management layer or the service management and orchestration domain. And those are the products then within the O-RAN open source community, which will be mostly related to the things that we discussed today and the things that O-RAN up to. If we look at the O-RAN architecture, this also then very much reflects the objectives of O-RAN. As we can see in this architecture, you can see how the different working groups map to this architecture. At the bottom here, we see the split in open-front all interface between the radio and base functionality. And in the middle, we can see the split between the 3DP defined functionality and O-RAN defined functionality in near real-time rig, which should allow for greater and simpler or more automated optimization of run functions. The near real-time rig should also be designed in such a way that it enables the usage of AI and ML as a tool to solve the radio resource management problems. And on top, we can see the service management orchestration functionality also then prepared to enable using AI and ML for use cases related to more management related tasks. And here we can also then see that there is an additional interface standardized by O-RAN called A1, which we will talk a lot more about later in this presentation. Also, O-RAN has just recently started to outline the internal structure of this SMO framework. And inside this, we can see that we can find the non real-time rig. And the non real-time rig is also, as part of the O-RAN work, split into two parts. One framework or platform part and then one part containing the intelligence that should solve automation use cases in the round domain. Within this split, O-RAN is then also talking about an R1 API, which will allow the split between platform functionality and the intelligence, which O-RAN called R-Apps. Thank you, Matthias. There's quite a lot going on in there. It must be very difficult to get track of it all. You've got the new interfaces with the A1 for the policy, the R1 for the apps and the traditional O-RAN for the configuration there. But with that already, I can see a little bit of how this relates into O-RAN App, where it's providing the automation platform. So it's a very good fit with the SMO concept as such there. I think that's really interesting. So I'm going to say a few words about O-RAN App then. As you know, O-RAN App started in 2017, it's been going for a while. So I assume that there's a lot of people that are very familiar with this already, but in case you're not, it could be good to say a few words about this as by way of introduction. So it actually stands for open in the sense of open source as well there, networking automation platform. And it's not really constrained to a particular domain. It goes across multi-domain. So O-RAN could be one of those domains if that's the way you choose to apply it. So there's I think a good touch point there. It creates a framework for model-driven service design and orchestration, policy-based control on analytics, and what that really is there to enable as well is sort of closed loop automation or control loop automation for virtualization and SDN. So this can be closed loop assurance, it could be open loop as well onto that there. And the way this is being built and organized, if you're not familiar with the O-RAN App there, it has this Linux Foundation Networking LFN board, that's a governing board. They deal with things like membership and they deal with things like money and marketing and so on goes into reporting into there. And they're very prominent, of course, in the ONES type of events and organizing those around the world there. Then when it comes down to the actual, the technical work, that's gathered and steered around what they call the TSC. So that's the Technical Steering Committee. And it has a number of subcommittees to support that, dealing with the requirements. So what's the requirements in the next release? So the TSC will be talking about what should be the content and what's the timing of the next releases and dealing with things like, yes, it's okay that this person, that we have the elections for the different subcommittees and projects and making sure the community is healthy there. Then the subcommittees have got requirements dealing with what should be the business requirements for the different projects, different for the releases, and that's looking at this from a cross-cutting perspective because the projects themselves actually produce the code, which is sitting here on the right, these are the code producing artifacts. So once this is for alignment of the contents, that will describe the impact into these. The architecture is looking for the architectural consistency. This is both someone has a new proposal, I think it should go like this, and this is very much related to how we can systemize these requirements. And of course, this should be a model driven platform. So modeling is very important. They're also dealing with the overall modeling. OWNAP has a security subcommittee and they're looking at both security requirements and security mechanisms and looking at how we can make this platform then secure. Control loop is looking at the cross-cutting the cross-project control loop and maturization there. As I said before, you've got the code producing projects. These are very similar to the architecture research. So you'll see things for service design and creation. You'll see one for policy, someone and so forth. I'm calling two out, one's dedicated to actually facilitating the documentation and there's also one that brings it together as this integration project. So that brings together a consistent release. We have a few coordinators and their purpose is that's a person and that can be a point of contact and they can bring things together. So one of them is the network management area and Magnus is working in that. He reaches out and deals, he's a single point of contact, but also reaches out to other communities. For example, with ORAN, but also with Etsy and making sure there's a good flow of information there. If we move on to the next page, so if we dig one level down, what is it? You can say that ONAP provides an automation platform for managing the services and resources throughout their entire lifecycle there. And so there's a, at a high level, a design time where you can onboard your resources and work with your service definition. When you're happy with that, you're pushing it over to the run time there. And that's taking feedback from the managed environment, which is the network, analyzing that, deciding what actions to be taken and pushing changes back down. It also has the ability here to, of course, instantiate or even modify a service definition. And then there is exposure, which is a lot about leveraging interact with other communities. So that you'll find TMF into there, you'll find little bits, well MEPH has come into there also via TMF. And we think about this from a number of particular dimensions, a number of particular aspects. So we think one aspect is that it provides a reference functional architecture. So we'll see, we'll introduce that on the next slide here, but you give expectations of the type of function, the functionality and architecture that could compromise a, that could make up an automation platform. When it has that architecture, there's a number of components that provides definitions for these reference components as well. And so far this is rather independent of the actual source code, but of course it is a reference source code as well. And you can take that piece by piece, or you can take that as the whole environment and use that there. It also comes with requirements on the VNFs and PNFs and CNFs in order to say, hey, here's what's necessary, here's what's recommended if you want to be managed by this network automation platform there. So it's looking for a bit of industry alignment around that way there. What you can see here is the most recent architecture from, from ONAP. I won't go into sort of, I won't go into all the details, I'll just pull out a couple of points. Here's the service design and creation part here. Actually, you have the design time, you have the overall runtime that was mentioned before, and here as we get down, we'll see actually, apart for actually watching and managing ONAP here. So this is making sure that the ONAP components are alive and can be instantiated and healthy there. What you see here is the service design and creation. There's a message bus for communication here. There's the data collection analytics and events for receiving and running for the receiving events, analyzing that, and actually an environment for different applications to be able to run on here. Adoption towards different cloud environments. We have these SDN control and application control, and now these are interfaces as down southbound towards the actual applications and understands the applications as well. And you could see that, okay, Matias was talking about the A1 interface, and that seems a good fit for what would go on in a controller here. Then we have the virtual function controller, and this deals with some of the software life cycle management as an NFEO and M capabilities. We have some infantry, service orchestration, policy, and actually managing the controller. In relation to the controllers, they have this common controller SDK, and that's a set of libraries and utilities for actually defining what you want on these controllers. So you don't have to be limited by these actual controller definitions. You've got other capabilities inside the CC SDK. This also applies for model utilities and the TOSCA password there as well. There's some common capabilities that are shared between this. Then, of course, you've got the external interfacing for both the human perspective and the external APIs. I said this also has component definitions, so here's just one example of a component definition. There's a definition for all of this, and this happens to be data collection analytics and events where you can see there's the south bound connections to the managed elements and also the services that are offered and construed. So you can see, for example, how it connects onto SO, how it deals, connects to CLAMP, which is closed loop automation and policy, so on and so forth. There is a description of this for each and every component. If we move along, one important aspect I think of ONAP is also the industry alignment, and it's a bit of a meeting point there, and here we see that both for ONAP and also for O-RAN. If O-RAN has the service management and orchestration that's focusing on the management of the RAN and ONAP is an automation platform that can cover different domains that includes the RAN, it's very natural that this can actually form an ecosystem, and ONAP could be one realisation, maybe there's others, but one realisation of the platform for the SMO. But together there's a broader community than that. Both ONAP, of course, can have a look at the 3GPP management interfaces, and O-RAN is, as Matias was mentioning, has obviously a great interaction with what is happening at 3GPP, both the network function side but also the management interfaces side. GM Forum has some capabilities for inventory management, service management, and that also relates to these scopes. Since they both ONAP and O-RAN deal with the virtual software as well, it's very natural that there's part of the ecosystem dealing with Etsy Mano, and ZSM, Zero Start Service Management, is also dealing with automation, and there's a good interaction between ONAP and ZSM, and I think there's even presentations about this in the recent layer 1, 2, 3, but there's interactions there, and that will of course take into account some of the O-RAN aspects as one domain. Of course, if you think about this from a connectivity perspective, both the ITF, MEF, and BBF have an interaction relationship here from the transport relation, and since this is not just specifications, since this is realisation, the CNCF is another important relationship there as well. And of course, this is interactions, and this is not always done through the formal means. A lot has been done, for example, as with the people that we have working together, both within the community and between the community. So for example, with the Etsy Mano work, the work we've been doing about the Etsy Alignment in ONAP, we've had presentations to the Etsy ISG about that, and there's the similar collaborations and presentations in between the ONAP and the O-RAN. Okay, so Matthias, you'll take us through some of the ONAP and O-RAN alignment examples here, I think. Yes, I will. Looking forward to that. You're welcome. As you pointed out, Steven, the scope of O-RAN is very, very big, but we thought, I mean, to exemplify this alignment, we'll touch upon two specific things where the scope of O-RAN and ONAP meets. Sorry, Matthias, I've lost your voice. Sorry, something happened there with the mute button. I'm terribly sorry. But I don't know where I was lost. The start of the page, I think, we can go back to. The start of the page, okay. Yes, so one specific example of where the scope of ONAP and O-RAN meets is then, as pointed out earlier, the A1 interface. And to understand the O-RAN interface, you need to understand the idea behind the non-real time rich. And the non-real time rich is being specified by O-RAN to add the capability to change the behavior of RAN based on analytics data and business policies on a per-UE basis and on a time scale that is down to several seconds. This should also be done then so that the non-real time rich is prepared for utilizing AI and ML as a tool to solve specific use cases. With this in mind then, the A1 interface is specified in O-RAN in addition to the ordinary configuration interface O-RAN. And this A1 interface then has the capability to issue policies for individual UEs down to the RAN so that you can affect the RAN behavior for individual users. And this is something that will be useful in cases where the default, when you understand from a management point of view that the default configuration is not working for a particular user. A good example of this is that with additional information of how fast the UE is moving, for example, you can trim mobility or set policies for mobility for the fast-moving UEs in an area to avoid ping pong effects or similar in RAN. But A1 also adds the capability to provide enrichment information to the RAN functionality from the management layer. So if you have analytics functionality in the management layer that gives an insight which would be valuable for the RAN function to work, this information can over the A1 interface be passed down to the RAN functionality as enrichment information. So that is kind of the overall objective of the non-return rate and the addition of the A1 interface. Okay, so for realizing this has been some work on the Converged A1 adapter, previously there was a little bit of work actually done in both the TSC inside of O-RAN and also there was an initiative started actually in ONAP as well. And now they've come together to put this together and created what's sort of a Converged A1 adapter. And this is done inside the common controller SDK to make the actual A1 adapter and then realized as part of the SDNC project and therefore part of the SDNC artifact there. This will then be upstreamed actually to the O-RAN OSC for integration. And this is described as actually I mentioned before there was a requirement subcommittee there is a requirement that's the ONAPs slash a 3GPP and O-RAN alignment A1 extensions. This is one example there are a few others but this is one example I think was worth highlighting and related to the A1 interface. Okay, the other specific defined by O-RAN is then the split between the the platform some platform functionality and R-Apps within the non real-time rick in O-RAN. And for this O-RAN is working on an R1 API that should allow for separate life cycle management of the different automation intelligence which O-RAN called R-Apps and a platform functionality. And this then decouples the automation into intelligence that addresses different kind of automation use cases from the platform functionality and is supposed to enable a simpler and more flexible handling of or a more flexible way to address different use cases with different kind of intelligence. Thanks Matias. So this R1 seems to be a very interesting interface. So I guess it needs access to data from the network and also be able to drive what you were mentioning describing before the A1 interface there. And to achieve that what we see is that there's a number of interfaces provided by own up components and I describe them the examples of those before. And what we see is there there can be an interaction and I think a sort of a subset of these could be quite valuable on the R1 interface. So we see and also thinking about from the R1 interface there could be needs that haven't been thought of today. So we see that there's a there'll be an influence between between O brand defining the R1 interface and a bit of a selection or adjustments to some of the the own app components interface. So this is a a cooperation that we an ecosystem in a way that we see that can be a very valuable land starting up now. Okay, now the if we move on to the concluding phase now the summary what we've been explaining is that O-RAN is creating an open and desaturated RAN and that includes the service management and orchestration layer there. We see that O-NAP provides a network automation platform and we see that there's a great opportunity for industry alignment here and we have and we see that's already started we provide a few examples of that. So with that I'd like to thank you for your attention or virtual attention today and I hope this was an enjoyable presentation from us and there was something in it for you. Yes, thank you very much for listening. All right, we have about four minutes for live Q&A with our speakers. Please feel free to submit your questions via the chat or Q&A panel and we will have our speakers answer them. Okay, can you can I be heard? This is Stephen. Yes, we can hear you. Great, thank you. I tried to answer some during the chat some of the questions that came up. Thank you for that. One was why is O-RAN linked so closely to O-NAP? What about OSM and other related solutions from big vendors like Nokia and Ericsson and so on there. And what I want to say there is O-RAN has a specification part and that's independent of any realisation and then it has an open source reference to the realisation work that they're starting up and working on and we think O-NAP is one possible reference platform for that. I'm not saying there can't be others and there's some activities that's ongoing like the A1 interface. As there's an example there and there could be others I think there was also a question from say a follow up question related to that as well as what about the front hall and the back hall and there's a few architectural options there to my understanding and of course O-NAP has transport orchestration capabilities I don't know that AT&T has been driving that thing for some parts as well and that could be a possibility I don't think it's at the top of the discussion tree at the moment but it certainly could be a possibility going forward as well. Yeah the first thing you guys I don't know maybe you have a better view. Yeah so well a slightly better view I would say because I guess we pointed out in the presentation on the scope of O-RUN it's very very big and I must admit that the lower layers here it's not my expertise perhaps but just to point out that there are open source projects in O-RUN related to development of the ODU and I guess it would be within those open source projects where the MAC schedule would be implemented. Okay I'll also note that not a question but apart from our colleague John Keaney who's actually will be demonstrating some of the ONF-A1 policy demos that has been done which is actually a representative of the work that we have in slides he has a link to that there so thanks for that now John. There was a question about the the slides and the quality of the format the quality of the text here the slides will be uploaded in the process sort of data. Was there anything else that people wanted to raise well I guess then we're at the end of our oh was there one coming in have you just have you just okay what about the relation to TMEF so the relation to MEF of course MEF has the legato and so on in general that relates into into ONF and it also brings in the the TM forum API is there and when it comes to the what the R1 interface could be that will be sort of influenced by the ONF platform there the part of that discussion I think could actually include what's the relationship towards the TMF APIs but it's very early very very mature to say exactly what R1 will do on on that one but that may come into into that play I don't know if there's anything else specific in R1 related to MEF other than other than that at the moment I've been answered the question Charles or addresses it maybe not answers it all right I think we have time for one more question it looks like it came through via the chat okay thank you okay so you're asking really about the A1 okay the the relationship about the A1 policy control and what happens with the 5QA it's an excellent question and something that can cause us a debate the way we see it and this is I can't speak for O-Round I can't speak for O-NAP I can only speak for the way we see this from every some point of view is that the R1 will be respecting the 5QIs that come from and the priority given by the 5QIs coming from the core network that's the business driven sort of requirement there but there are certain situations where you need to actually work at assuring the actual experience or doing slight optimizations while still respecting what comes from the core network and that's where they the cause nodging in the A1 could also come there are other cases that are independent of that for example policies for setting the frequencies for a fast-moving use and saying don't go to micro sale macro those are also but it's a very good conversation that one George all right well thank you to our sorry that's okay I just wanted to let everybody know that we did upload the PDF of the slides into the handout section so you should see a handout icon underneath your video panel where you can locate that PDF and I think that concludes our time today so thank you everybody for joining and please feel free to keep the conversation going using the Slack channel provided in the chat


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CHAPTER XXVIII. DREAMS AND DRESSMAKING Tato was now one of the family. They left Taormina the next day, and Frascati drove all the girls in his Victoria to the station. "'You must come again, Sr. Aine," he said, looking regretful at their departure. "'Next year the fountain of the ice-cream soda will be in operation, like those you have in Chicago, which is America. Our culture increases with our civilization. It is even hinted that Ilduca is to abandon our island forever. He has been interesting to us, but not popular, and you will not miss him when you come again to find he is not here. If this time he has caused you an inconvenience, I am sorry. It is regrettable, but—' "'But it is so,' said Patsy, laughing. Tato was again transformed. Patricia, who was the smallest of the three nieces, though not especially slim, had quickly altered one of her own pretty white gowns to fit the child, and as she was deft with her needle, and the others had enthusiastically assisted her. Tato now looked more like a fairy than ever. It was really wonderful what a suitable dress could do for the tiny Sicilian maid. She had lost her free and boyish manner, and become shy in retiring with strangers, although when in the society of the three nieces, she was as sweet and frank as ever. She wore her new gown gracefully, too, as if well accustomed to feminine attire all her life. The only thing now needed, as Patsy said, was time and which to grow her hair, which had always been cut short, in boyish fashion. They were a merry party when they boarded the train for Syracuse, and Uncle John arranged with the guard to secure two adjoining compartments all to themselves, that they might have plenty of room. "'Where did you put the money, Uncle John?' Beth whispered, when at last they were whirling along and skirting the base of Mount Etna toward the Catania side. "'I've hidden it in my trunk,' he replied, in the same confidential tone. There is no bank in this neighborhood to receive it, so I decided to carry it with us. "'But will it be safe in the trunk?' she inquired. "'Of course, my dear. Who would think of looking there for fifty thousand dollars? And no one knows we happen to have so much money with us.' "'What did the Count, I mean Mr. Weldon, do with his ransom?' "'Carries it in his satchel, so he can keep it with him, and have an eye on it. It's a great mistake, Beth, to do such a thing as that. It'll make him uneasy every minute, and he won't dare to let a faquino handle his grip. But in my case, on the other hand, I know it somewhere in the baggage-car, so I don't have to worry. The journey was a delightful one. The road skirted the coast through the oldest and most picturesque part of Sicily, and it amazed them to observe that however far they traveled Etna was always apparently next door, and within reaching distance. At Asicastello they were pointed out the seven aisles of the Cyclops, which the blind Polyphemus once hurled after the crafty Ulysses. Then they came to Catania, which is the second largest city in Sicily, but has little of historic interest. Here they were really at the nearest point to the mighty volcano, but did not realize it because it always seemed to be near them. Eighteen miles farther they passed the Antinoy, which in ancient days dared to rival Syracusa itself, and an hour later the train skirted the bay and Capo Santa Panagia and slowly came to a halt in that city which for centuries dominated all the known world and was more powerful and magnificent in its prime than Athens itself, Syracuse. The day had become cloudy and gray, and the wind whistled around them with a chill sweep as they left their coach at the station and waited for Kenneth to find carriages. Afterward they had a mile to drive to their hotel, for instead of stopping in the modern town Uncle John had telegraphed for rums at the Vila Paliti, which is located in the ancient Accordina at the edge of the Latomia di Capicini. By the time they arrived there they were blue with cold, and were glad to seek the warm rums prepared for them and pass the remainder of the afternoon, unpacking, and getting settled. "'I am afraid,' said Patsy dolefully, that we shall miss the bright sunshine and warmth of Taormina, Tato. "'Oh, it is not always warm there, nor is it always cold here,' replied the child. "'Indeed, Sr. Ina, I have heard that the climate of Syracusa is very delightful.' "'It doesn't look it,' returned Patsy, but it may improve. The interior of the hotel was comfortable, though, however bleak the weather might be outside. A good dinner put them all in better humor, and they passed the evening watching the strangers assembled in the parlors and wondering where they had come from and who they were. "'That money,' whispered Uncle John to Beth, as he kissed her good night, is still as safe as can be. I've lost the key to my trunk, and now I can't even get it at myself.' "'Lost it?' she exclaimed. "'Yes, but that won't matter. It's the big trunk that holds the things I don't often use, and if I can't unlock it, no one else can. That's certain. So I shall rest easy until I need something out of it, and then I'll get a locksmith to pick the lock.' "'But I wish you hadn't lost the key,' said the girl, thoughtfully. "'Strikes me it's good luck. Pleasant dreams, my dear. I can fancy Arthur Weldon lying awake all night with his dreadful thirty thousand tucked under his pillow. It's a great mistake to carry so much money with you, Beth, for you're sure to worry about it.' The next morning, when they came down to breakfast, they were all amazed at the gorgeous weather and the genial temperature that had followed the dreary afternoon of their arrival. Sarah Cuse was transformed, and from every window of the hotel the brilliant glow of countless flowers invited one to wander in the gardens, which are surpassed by few, if any, in the known world. The Velopoliti stood so near the edge of a monstrous quarry that it seemed as if it might topple into the abyss at any moment. Our friends were on historic ground, indeed, for these quarries, or Latomia, as they are called, supplied all the stone of which the five cities of ancient Syracuse were built. Cities which in our age have nearly, if not quite, passed out of existence. The walls of the quarry are a hundred feet in depth, and at the bottom are now acres upon acres of the most delightful gardens, whose luxuriance is attributable to the fact that they are shielded from the winds, while the sun reaches them nearly all the day. There are gardens on the level above, and beautiful ones, too. But these in the deep Latomia are the most fascinating. The girls could scarcely wait to finish breakfast before rushing out to descend the flights of iron steps that lead to the bottom of the vast excavation. And presently they were standing on the ground below and looking up at the vine-covered cliffs that shut out all the upper world. It was peaceful here, and soothing to tired nerves. Through blooming shrubbery and along quiet paths they might wander for hours, and at every step find something new to marvel at and to delight the senses. Here were ancient tombs cut from the solid rock, one of them that of an American midshipman, who died in Syracuse, and selected this impressive and lovely vault for his burial place. And there stood the famous statue of Archimedes, who used in life to wander in this very Latomia. Once, said Mr. Watson musingly, there were seven thousand Athenian prisoners confined in this very place, and allowed to perish through starvation and disease. The citizens of Syracuse, even the fine ladies and the little children, used to stand on the heights above and mock at the victims of their king's cruelty. Couldn't they climb out? asked Patsy, shuddering at the thought that some of the poor prisoners might have died on the very spot her feet now trod. No, dear, and it is said that the guards constantly patrolled the edge to slay any who might venture to make the attempt. Wasn't it dreadful, she exclaimed, but I'm glad they have made a flower garden of it now. Somehow it reminds me of a cemetery. But there were other interesting sites to be seen at Syracuse, and they laid out a systematic program of the places they would visit each morning while they remained there. The afternoons were supposed to be reserved for rest, but the girls were so eager to supply Tato with a fitting wardrobe that they at once began to devote the afternoons to shopping and dressmaking. The child had placed in Uncle John's keeping a liberally supplied purse, which the Duke wished to be applied to the purchase of whatever his daughter might need or desire. He wants me to dress as you do, said Tato simply, and because you will know what is fitting my station and will be required in my future life, he has burdened you with my society. It was selfish in my father, was it not? But I wanted so much to be with you, because you are good to me. And we're mighty glad to have you with us, answered Patsy. It's no end of fun getting a girl a whole new outfit from top to toe, and aside from that we already love you as if you were our little sister. Beth and Louise equally endorsed this statement, and indeed the child was so sweet and pretty, and so grateful for the least kindness bestowed upon her, that it was a pleasure to assist and counsel her. Tato looked even smaller in girls' clothing than in boys, and she improved so rapidly in her manners by constantly watching the nieces, that it was hard to imagine that she had until now been all on use to polite society. Already they began to dread the day when her father would come to claim her, and the girls and Uncle John had conceived a clever plan to induce the Duke to let his daughter travel with them on the continent, and then go for a brief visit to them in America. By that time, declared Louise, Tato's education will be accomplished, and she will be as refined and ladylike as any girl of her age we know. Blood will tell, they say, and the monk who taught her must have been an intelligent and careful man. She knows more of history and languages than all the rest of us put together, added Beth. And having adopted her, we mustn't do the thing by halves, concluded Patsy, so our darling little Brigandus must tease her papa to let her stay with us as long as possible. Tato smiled and blushed with pleasure. It was very delightful to know she had such enthusiastic friends. But she was afraid the Duke would not like to spare her for so long a time as a visit to America would require. You leave him to me, Sir Uncle John. I'll argue the case clearly and logically, and after that he will have to cave in gracefully. Meantime the dainty gowns and pretty costumes were one by one finished and sent to the hotel, and the girl's ransacked the rather inadequate shops of Syracuse for the smartest things in lingerie that could be procured. As they were determined to try everything on and see how their protégé looked in her finery, Tato was now obliged to dress for dinner and on every other possible occasion, and she not only astonished her friends by her loveliness, but drew the eye of every stranger as surely as the magna attracts the needle. Even in Sicily, where the Greek type of beauty today exists more perfectly than in Helen, there were few to compare with Tato, and it was only natural that the Americans should be very proud of her. Kenneth was sketching a bit of the quarry and the old monastery beyond it, with the blue sea glimmering in the gardens. Sometimes he would join the others in the morning trips to the catacombs, the cathedrals, or the museum. But the afternoons he devoted to his picture, and the others came to the gardens with him, and sat themselves down to sow or read beside his easel. Arthur Weldon was behaving very well indeed, and although a good deal of the credit belonged to Louise, who managed him with rare diplomatic ability, Uncle John grew to like the young man better each day, and had no fault whatever to find with him. He was still rather silent and reserved, but that seemed a part of his nature, inherited doubtless from his father, and when he chose to talk, his conversation was interesting and agreeable. Kenneth claimed that Arthur had a bad habit of making goo-goo eyes at Louise, but the young man's manner was always courteous and judicious when addressing her, and he managed to conceal his love with admirable discretion, at least when others were present. Uncle John's private opinion, confided in secret to his friend Mr. Watson, was that Louise really might do worse, that is, if they were both of the same mind when they grew up, and so the days passed pleasantly away, and the time for their departure from Syracuse drew near. On the last morning all of them, with the exception of Tato, who pleaded a headache, drove to the Latomia del Piridicio to see the celebrated ear of Dionysius, that vast cavern through which the tyrant is said to have overheard every whisper uttered by the prisoners who were confined in that quarry. There is a little room at the top of the cliff, also built from the rock, where it is claimed that Dionysius sat and played eavesdropper, and it is true that one in that place can hear the slightest sound uttered in the chamber below. Afterward the amphitheater and the ancient street of the tombs were paid a final visit, with a stop at San Giovanni, where St. Paul once preached. And at noon the tourist returned to the hotel hungry but enthusiastic, in time for the table du hôté luncheon. CHAPTER XXIX This is Fanny, Cried Petsy, appearing before Uncle John with a wide and startled face. I can't find Teto anywhere, and her drunk is gone from her room, as well as her gowns and everything she owns, continued best clear voice over her cousin's shoulder. Uncle John stared at them bewildered. Then an expression of anxiety cut over his kindly face. Are you sure? he asked. There can't be a mistake, Uncle. She's just gone. None of us has offended or annoyed the child, I suppose? Oh no, Uncle. She kissed us all very sweetly when we left her this morning. I can't understand it. Nor can we. Could her father have come for her, do you think? suggested Mr. Merrick after a moment's thought. I can't imagine her so ungrateful as to leave us without the word, said Petsy. I know Teto well, Uncle, and the dear child would not hurt our feelings for the world. She laughs us dearly.  And I don't trust her all together. Sometimes I've surprised a look in her eyes that wasn't as innocent and demure as she would have us imagine her. Oh, Louise! And there's another reason. What is it? She reformed too suddenly.  But next instant he drew a long-pressed and smiled again. It was lucky I lost a key to the trunk, he observed. Still a little ashamed of his temporary lack of confidence in Teto. It's been locked ever since we left our meaner, so the child couldn't be tempted by that. She wouldn't touch your money for the world, the Pets Teto is no thief. She comes off a race of thieves, though, best reminded her. I wonder if us, as money, is still safe, remarked Louise, following the line of swords suggested. As if with one accord the move down the hall to the door of the young man's room. Are you in, Arthur? asked Uncle John, knocking briskly. Yes, sir. He opened his door at once, and so we surprised a little group of anxious faces at sight. Is your money safe? asked Uncle John. Weldon gifted him a startled glance, and then ran to his dress and pulled open a drawer. After a moment's fumbling, he turned with a smile. All safe, sir. Uncle John and his nieces were visibly relieved. You see, continued Arthur, I've invented a clever hiding place, because the satchel could not be left alone, and they didn't wish to lug it with me every step I took. So I placed the packages of bills inside the leg of a pair of trousers, and put them in the drawer with some other closing on top and bottom. A dozen people might get damaged in the drawer without suspecting the fact that the money is hidden there. I've come to believe the place is as good as a bank, but you startled me for a minute with your question. What's wrong? Tattoo's gone. Gone? Departed back in baggage. But you're fifty-thousand, sir. Is it safe? It has to be, answered Uncle John. It is in a steel-bound double-locked trunk, to which I've lost the key. No bank can beat that, my boy. Then why did the child run away? They could not answer that. It's a mystery. Said Betsy, almost ready to weep. But I bet it's the cruel, wicked father of hers. Perhaps he came while we were out, and wouldn't wait a minute. What does the whole parter say? Ask Kenneth, who had joined the group in time to overhear the last speech and guess what had happened. Stupid, cried Uncle John. We never thought of the whole parter. Come back to our sitting room, and we'll have him up in a chibi. The parter answered his spell with electricity. The Americans were liberal guests. The young lady? Ah, she had driven away soon after they had themselves gone. A sin-faced, dark-eyed man had called for her and taken her away, placing her baggage on the top box of the carriage. Yes, she had paid her bail and tipped the servants liberally. Just as I suspected, cried Betsy, that horrid Duke has forced her to leave us. Perhaps he was jealous and feared we would want to keep her for always. Was she weeping a miserable parter? No, Signorina. She laughed and was very merry. And, but I had forgotten, there's a letter which she left for the young lady in your dole. Where? In the office. I'll bring it at once. He ran away and quickly returned, placing a rather bulky parcel in the girl's hands. You read it, Uncle John, she said. There can't be anything private in Tatu's letter, and perhaps she has explained everything. He put on his glasses and then took the missive and deliberately opened it. Tatu wrote the fine delicate hand, and although the English words were badly spelled, she expressed herself quite well in the foreign town. With the spelling and lack of punctuation corrected, her letter was as follows. Dear innocent foolish Betsy, how astonished you will be to find I've finished from your life forever, and what engrossed and indignant words you will hurl after Portetto. But they will not reach me, because you will not know in which direction to send them, and I will not care whether you are engrossed or not. You have been good to me, Betsy, and I really love you, fully as much as I have here of the true and pretty cousin of yours, whose cold eyes have made me tremble more than once. But the albass I forgive her, because she is the only clever one of the lot of you. Louis thinks she is clever, but her actions remind me of a juggler who explained his tricks before he did them, so that the audience would know how skillfully he was.  You tricked us that day in the mountain glen, and for the first terminal canter break and lost his prisoner, and his ransom money through being outwitted. But did you think that was the end? If so, you failed to appreciate us. Look, you, my dear, you could have done without the money, for our family has been robbing and accumulating for ages, and this little need to expend much from year to year. It is all in the bank of Italy too, and drawing the interest for my father is a wise man of business. That 400,000 lira was to have been our last ransom, and after we had fairly earned it, you tricked us and did not pay. So, my father and I determined to get even with you, as much through revenge as cupidity. We were obliged to desert the valley at once, because we were getting so rich that the government of fishers became uneasy and wanted us to go or be arrested. So, we consulted together and decided to ponder our little plot, which was so simple that it has worked perfectly.   But I didn't, for I can act. I played the child very nicely, I think, and you quite forgot I was a brilliant starter, with the wild-free blood of many brave outlaws cursing in my veins. Ah, I'm more proud of that than of my acting. Innocent as I seemed, I watched you all carefully, and knew from almost the first hour where the money had been put. I stole the key from Uncle John's trunk on the train, while we were going from Tarumina to Syracuse. But I did not take the money from it, because I had no better place to keep it, and the only danger was that he would force the lock some day. But Feralti's money?  At the first he would not let that little satchel out of his sight, and when he finally did, he removed the money to some other place. I searched his room many times, but could not find his hiding place until last night. While he was at dinner, I discovered the purse in one of the drawers of his dresser. But for this difficulty, I should have left your charming society before, as my father has been secretly waiting for me for three days. Having located Feralti's money, I waited until this morning, and when he had all left me, I signalled to my father from my window and prepared to disappear. It took but a few minutes to get the money from Uncle John's trunk and ask us to house a legs. Much obliged for it, I'm sure. Then I packed up all my pretty dresses in my new trunk, for part of our blood was to use your good taste in fitting me out properly. And now I'm writing this loving epistle before I leave. We shall go to Paris, or Vienna, or Cairo or London. Guess which? We shall have other names, very beautiful ones, and be rich and dignified and respected. When I grow old, I think I shall marry a prince and become a princess. But that will not interest you much, for you will not know that the great princess is your own little tattoo. Tell Uncle John I have left the key of his trunk on the mantle, behind the picture of the Madonna. I stuffed papers into us as trousers legs to deceive him. If he came back before, I had a chance to escape. But I hoped you would discover nothing until you read this letter, for I wanted to surprise you. Have I? Then I am content. You tricked me once, but I have tricked you at the last, and the final triumph is mine. In spite of all, Petsitea, I love you, for you are sweet and good, and although I would not be like you for the world, I can appreciate your excellent qualities. Remember this, when your anger is gone, I won't be able to visit you in America, but I shall always think of you in a more kindly way than I fear you will think of this Elysian tomboy tattoo. End of chapter 29, Recording by Ellie, December 2009 In his eyes blazed, Silas Watson stared blankly at his old friend, wondering if it was because he was growing old that he had been so easily hoodwinked by this saucy child. Beth was biting her lip to keep back the tears of humiliation that longed to trickle down her cheeks. Louis frowned because she remembered the hard things Tato had said of her. Patsy was softly crying at the loss of her friend. Then Kenneth laughed, and the sound sent a nervous shiver through the group. Tato's a brick, announced the boy audaciously.  That the thing is a good joke on us all, or are you too thin-skinned to laugh at your own expense? Oh, we can laugh, responded Uncle John gravely. But if Tato's a brick, it's because she's hard and insensible.  That made Patsy sob outright while Louis ejaculated with scorn, the little wretch. It serves us right for having confidence in a child reared to crime and murder from the cradle, said Arthur rather savagely. I don't know how much money I am worth, but I'd gladly spend another 30,000 to bring this wretched creature to justice. Money won't do it, decoyed the lawyer shaking his head regretfully. The rascals are too clever to be caught in Europe. It would be different at home. Well, the best thing to do is to grin and bear it, and forget the unpleasant incident as soon as possible, said Uncle John. I feel as if I'd had my pocket picked by my best friend, but it isn't nearly as disgraceful as being obliged to assist the thief by paying ransom money. The loss amounts to nothing to either of us, and such treachery, thank goodness, is rare in the world. We can't afford to let the thing make us unhappy, my friend, so cheer up all of you and don't dwell upon any more than you can help. They love Syracuse, a rather solemn group, in spite of this wise advice and journey back to Naples and then to Rome. There was much to see here, and they thought so energetically that when they boarded the train for Florence, they were all fagged at and could remember nothing clearly except the Colosseum and the Bass of Caricola. Florence was just now a bower of roses and very beautiful, but kind of flugged them to the galleries day after day until Uncle John declared he hated to look an old master in the face. After all, they're only dogs, he declared, or any 10-year-old boy in America can paint better pictures. Don't let anyone hear you say that, dear Caution Patsy. They'd think you don't know good art. But I do, he protested. If any of those pictures by old masters was used in a streetcar ad at home, it would be money wasted, for no one would look at them. The people wouldn't stand for it a minute. They're wonderful for the age in which they were painted, said Kenneth soberly. He must remember that we've had centuries in which to improve our art since then. Oh, I have proper respect for old age, I hope, replied Uncle John, but to fall down and worship a thing because it's gray-haired and out of date isn't just my style. All of those o's and a's over the old masters are rank humbug and I'm ashamed of the people that don't know better. And now Arthur Weldon was obliged to bid goodbye to Louise and her friends and take a train directly to Paris to catch the steamer for home. His attorney advised him that business demanded his immediate presence and he was obliged to return, however reluctantly. Kenneth and Mr. Watson also left the Part 8 Florence as the boy artist wished to remain there for a time to study the pictures that Uncle John so bitterly denounced. The others went on to Venice, which naturally proved to the nieces one of the most delightful places they had yet seen. Mr. Merrick loved it because he could ride in a gondola and rest his stubby legs, which had become wary with tramping through galleries and cathedrals. These last monuments, by the way, had grown to become the sort of nightmare to the little gentlemen. The girls were enthusiastic over cathedrals and allowed none to escape a visit for a time Uncle John had born up bravely, but the day of rebellion was soon coming. No cathedrals in Venice, I hope, he had settled their arrival. Oh yes, dear, the loveliest in the world, St. Mark's is here, you know. But no St. Paul's or St. Peter's? No, Uncle, there's this saluta and the never mind, we'll do that first one and then quit. What they build so many churches for, I can't imagine. Nobody goes to them but tourists, that I can see. He developed a streak of extravagance in Venice and purchased Venetian lace in Venetian glassware to such an extent that the nieces had to assure him that they were all supplied with enough to last them and their friends for all time to come. Major Doyle had asked for a Mirchand pipe and a Florentine leather pocket book, so Uncle John made a collection of 37 pipes of all shapes and sizes and bought so many pocket books that Patsy declared her father could use a different one every day in the month. But they're handy things to have, said her uncle, and we may not get to Europe again in a hurry. This was his excuse for purchasing many things, and it was only by reminding him of the duty you would have to pay in New York that the girls couldn't do some to desist. The customs tax worried the old gentlemen at times. For his trip he had always believed in a protective tariff, but now he referred to the United States customs as species of brigandage worse than that of Old Duce himself. They stopped at Milan to visit the great cathedral, and then raced through Spudsel and made a dash from Lucerne to Paris. Thank heaven, said Uncle John. There are no cathedrals in Gay Paris at any rate. Oh yes there are, they assured him. We must see Notre Dame, we must see Notre Dame anyway. And there are a dozen other famous cathedrals. Here's where Uncle John balked. See here, my dears, he announced. Not a cathedral will I visit from this time on. You can take a guide and go by yourselves if you feel you can't let any get away from you. Go and find another of Mike Angelo's last work. Every church has got one. For my part I've always been religiously inclined, but I've been to church enough lately to last me the rest of my natural life, and I'm fully determined not to darken the doors of another cathedral again. They're like circuses, anyhow. When you've seen one, you've seen them all. No argument would induce him to abandon this position, so the girls accepted his proposal and visited their beloved cathedrals in charge of a guide, whose well of information was practically inexhaustible if not remarkable for its clarity. The opera suited Uncle John better, and he freely reveled in the shops purchasing the most useless and preposterous things in spite of that growing bugbear of the customs duty. But finally this joyous holiday came to an end, as all good things will, and they sailed from Cherbourg for New York. Uncle John had six extra trunks, Patsey carried a French poodle that was as much trouble as an infant in arms, and Louise engineered several hat boxes that could not be packed at the last minute. But the girls embarked gay and rosy-cheeked and animated, and in spite of all the excitement and pleasure that had attended their trip, not one of the party was really sorry when the return voyage began. CHAPTER XXXI SAFE HOME To me, said Uncle John, as he stood on the deck and pointed proudly to the Statue of Liberty in New York Harbor, that is the prettiest sight I've seen since I left home. Prettier than the old masters, Uncle, asked Patsey mischievously. Yes, or the cathedrals, he retorted. When they reached the dock there was the major waiting to receive Patsey in a nude-checked suit with a big flower in his buttonhole, and a broad smile on his jolly face. And there was Mrs. Marrick too, with Arthur Weldon beside her, which proved to Louise that he had succeeded in making his peace with her mother. Also there were the stern featured, custom-house officials in their uniforms, and the sight of them sent the cold chills flying down Uncle John's spine. There was no one present to receive Patsey, but her uncle tucked her arm underneath his own with a proud gesture, and kept her clothes beside him. For the girl had quiet on his loving old heart on this trip, and she seemed to him more mature and far sweeter than when they had left home. But the greetings and the brigantage were soon over, and in a good time they were all assembled in the doiflet, where the joyous major had prepared an elaborate dinner to celebrate the return of the Wanderers. We have a million pipes and pocketbooks for you, Daddy, whispered Patsey, hugging him for the twentieth time. And I've got a thousand things to tell you about our adventures in strange lands. Say them till we're alone, said the Major. They are too good to waste on a crowd. Mr. Merrick was placed at the head of the table to make a speech. It was brief and to the point. I promised these young ladies to give them time of their lives, he said. Did I do it, girls? And in a lively chorus they answered, You did, Uncle John. End of CHAPTER XXXI. End of ON JANE JANE'S NECESS Broad by L. Frank Baum


Welcome back. So, in the last class we were discussing boundary value problems for general second order equations. So, this is y double dot equal to f of t. So, this is the equation and the boundary condition.  So, this is 2 a boundary conditions 1 b. The condition was this a 0 and a 1 do not vanish simultaneously and similarly b 0 and b 1 do not vanish simultaneously. A formal approach to a solution of this boundary value problem 1 a and 1 b totally called 1 was through an initial value problem discussed this last time u dot equal to u double dot equal to f t u u dot call it 2 a and now we impose conditions on u and u dot only at t equal to a. So, this the first one is sorry for the this thing. So, this is not u this is y by t y dot u a minus a 1 u dot a equal to alpha. So, there is no change in the first condition and now we want a since its second order equation want to impose one more initial condition and this we want it to be independent of the first one. So, this is minus c 1 u dot a equal to s. S is a parameter at our choice and again a real number and this is 2 b and by independence what I mean is this a 1 c 0 we have fixed c 0 and c 1 such that this is 1. So, let u be a solution of I v p in the interval a b. So, last time we stated a theorem where some sufficient conditions were imposed on the right hand side function f. So, that the solution of the initial value problem exists for all t in this interval and with this u we form this real valued function p of s remember the second condition on u depends on s. So, we form this b 0. So, u we emphasize the dependence on s. So, we say that u at t and f s is in the dependence on s. So, this is u b s plus b 1 u dot b s minus beta. So, just notice that this phi of s is nothing, but the second boundary condition. So, if there exists. So, this is the conclusion we do last time. So, if there exists s star such that p of s star is 0 then y given by y of t is equal to u of t s star is a solution of d v t and this is called shooting method. So, in this approach we attack this bound value problem through an insular problem and this is a big if. So, this in general for every s star satisfying this phi s star equal to 0 that means the roots of the function phi we get a solution of the boundary value problem. So, last time I stated a uniqueness result. So, let me just again recall. So, theorem 1.  This condition looks stronger, but that is sufficient for the solution of the initial value problem to exist in the entire interval a b. So, suppose f is uniformly lifted in u 1 u 2 variables in R.  Then d v p has as many solutions as the roots of. So, for every root of this function real valued function. So, remember this phi is from R to R. So, now we go to the second theorem some uniqueness result theorem 2 in addition to the hypothesis of theorem 1 assume.  So, this del f by del u 1 is positive and by del u 2 is bounded this is not norm just absolute value in R that is the reason in stated in theorem 1 and. So, a 0 b 0 a 0 a 1 have same sign and b 0 b 1 have same sign and a 0 and b 0 do not vanish simultaneously. So, I already assume that a 0 and a 1 do not vanish simultaneously and b 0 b 1 do not vanish simultaneously those are standing hypothesis on the boundary conditions. And in any addition to that now we are assuming that a 0 and a 1 have the both both have the same sign b 0 b 1 have the same sign and a 0 b 0 do not vanish simultaneously then the b v p has a unique solution. So, let me sketch a proof of this it is really simple and just use this one variable calculus, but little technical in detail. So, sufficient to show sufficient to show that phi has a unique root. So, if you just recall the conclusion of theorem 1 the b v p will have as many solutions as the roots of phi. So, if we show that phi has a unique root then b v p will have unique root. So, again let me recall. So, recall this phi of s is b 0 u b s plus b 1 u dot b s and u is the solution of the initial value problem. And since phi is a real valued function defined on real line. So, this is this is accomplished. So, we show phi has a unique root by showing that its derivative d phi by d s is bounded away from d phi by d s is bounded away that is d phi by d s bigger than or equal to c positive or d phi by d s less than or equal to minus is strictly less than 0 for all s. So, either it stays all the time all for all s it is for stays positive or it stays negative. So, just as an example if you take e s its derivative e is e to the s the derivative e to the s is bigger than or equal to 0 for all s in r, but it is not bounded away from 0. So, infimum e to the s s in r is 0. So, e to the s is not an example. So, once this is satisfied. So, you can show that. So, this is a simple calculus lemma. So, suppose phi from r to r be such that d phi by d s is bounded away from 0. So, then there exist unique s star such that phi of s star is 0. So, it is a very simple thing. So, it is just it is not hard to see that. Now, coming back to the so proof. So, you have to establish that d phi by d s is bounded away from 0. So, for this we introduce something. So, introduce xi t is d u by d s. So, I am taking partial differential equation derivative with respect to s remember s appears in the initial condition. So, now I am varying that s. So, I am considering u as a function of s initial condition it appears in the initial condition. And now look at the equation again for the u. So, look at recall that. So, u double dot equal to f of t u u dot. So, you know implicitly depends on this s. So, we will differentiate this equation with respect to s. So, differentiate with respect to s. So, this is this is what we learn in general theory the continuous dependence on the of the solution of the initial value problem with respect to initial data. So, if you do that thing then you get xi double dot equal to. So, let me just write it p t xi dot plus q t xi. So, remember this is xi the derivative of the solution of the initial value problem with respect to s. So, what is p? So, where p t just coming from this you are differentiating the right hand side with respect to s. And so you are using just the implicit differentiation. So, this is what you get.  So, now everything let me write once then it is understood and q t is just t does not depend on s. So, do not have to worry about that and same thing. So, this is called variation equation. So, this plays an important role now variation equation.  And we are trying to show the derivative of phi is bounded away from 0. So, what is the derivative of phi? Phi is nothing but this derivative of u with respect to s and that is precisely xi. So, that is why this role of xi is important. So, now you see that from hypothesis this q is positive. By hypothesis q is positive and p is bounded. So, we are going to make use of this. So, let me again rewrite here. So, xi double dot equal to p t xi dot. So, you keep on recalling this. So, this let me call it 3 a and similarly if you now differentiate the initial conditions to a what you get is xi of a is equal to you do that thing. So, let me just write what you get is by differentiating initial conditions to a with respect to s. So, everything with respect to s we obtain a 0 xi a minus a 1 xi dot a this is 0 because this is alpha there. So, alpha does not depend on s. So, that is we get that one. So, c 0 xi a minus c 1 xi dot a and there is an s here. So, when I differentiate with respect to s we will get 1 and by our choice of c 0 you immediately see that xi a if you solve these two equations you get a 1 and xi dot a is equal to a 0. So, let me call this as 3 b. Now, just concentrate on this. So, remember a 0 a 1 they have the same sign and they do not vanish simultaneously. So, let us fix. So, assume. So, the other case is similar assume a 0 is non negative and a 1 is non negative and look at the conditions 3 b. So, if a 1 is positive then xi a is positive and this implies xi t is positive in a small labour rule in this interval a 2 a epsilon. So, if a 1 is positive there is no problem there and if a 1 is 0 then a 0 is positive by our choice and look at the second condition xi dot a equal to a 0 a 0 is positive. So, xi is again increasing. So, again that implies xi t is in some a 2 a epsilon. This epsilon may be different. So, this may be different. What we are trying to say here is xi remains positive with our this assumption important. If we assume the both are negative then we will have negative thing here, but just for definiteness let us fix the signs of a 0 and a 1. So, that is what we get. So, therefore, xi t is positive for t in a a epsilon for some epsilon positive. So, claim. So, this is again very simple calculus argument xi t is positive. For all t in a epsilon. So, up to b the xi remains positive. So, only at a it may be 0 or it may be positive. So, that is the claim. So, once we have this thing it is the proof follows very easily and this is also very simple argument. So, suppose not a calculus argument suppose not. So, what does that mean? So, there exists some t star in a b such that xi t star is less than or equal to 0. So, may be better draw some diagram. So, there is a here, there is t star there, and there is b here. So, we already know that xi is positive. So, positive in some small neighborhood that is important is positive there. So, it may be 0 here and then it comes. So, looking at the picture. So, therefore, xi has a positive maximum it could be just relative maximum no problem in a t star. And we want to rule out this a t star is already ruled out because xi t star is less than or equal to 0. So, it cannot be a positive maximum. So, we want to rule out this positive maximum at a. So, how to do that thing? So, if a 0 is positive then xi dot a is a 0 positive. So, positive maximum cannot occur at a. And if a 0 is 0 then look at xi double dot a from the equation. So, this is go back to the equation. So, xi double dot a is a 0 positive. So, this is a 0 positive is equal to p t xi dot a plus q t xi a. And now we are assuming a 0 is 0. So, that means, this is 0 and this is q t xi a and xi a is a 1. So, this is strictly positive. So, and at a maximum the second derivative cannot be positive it can only be less than or equal to 0. So, therefore, again the positive maximum cannot occur at a. So, that means, what the positive maximum occurs at an interior point t 0. Let us go back. So, therefore, the positive maximum occurs at t call it t 0 or t 1 or t 1 in a t star. So, at an interior point. So, this is an interior it is not either a or t star. So, then we know that when a maximum occurs at an interior point. So, therefore, we have this xi of t 1 is strictly positive xi dot t 1 is 0 because it is a maximum. And one more condition this xi double dot t 1 is less than I put this is from calculus. But again you go back to the equation variation equation.  So, whole thing is positive, but at a maximum xi dot xi double dot t 1 is negative. So, this is a contradiction. So, therefore, our claim is true. So, therefore, what is our claim xi t is positive for all t up to b. So, that is important. So, this is the first step and the remaining steps are very straight forward. So, now, look at again go back to the look at the variation equation xi double dot equal to p t xi dot plus q t xi. So, everything at t. So, let me just write that. And just now, we have proved that this xi t is positive up to b and q is positive by assuming that this is a contradiction. So, this one is positive for all t in a. So, therefore, we have this inequality we can put greater than or just. And fortunately, we can integrate an inequality. We cannot differentiate an inequality, but we can certainly integrate an inequality. So, integrate this and preserving the sign of inequality integrate. You will not do directly. So, you have to multiply by exponential 0 to t. Let me write p s multiplied by this and then this is the integrating factor. So, multiply and integrate. So, what you obtain is. So, xi dot t is bigger than a 0. This is coming from the condition xi double dot a dot a equal to not 0 here a. Our point is this is exponential p s t s not p s. So, let me use some other variable may be eta. So, this is true for all t in a. And now, this integrate one more time. So, integrate one more time to obtain. So, finally, we have xi t is bigger than a 1 plus a 0. So, there is a double integral. So, let me write it first and then.  So, there is a double integral here. Now, you use the hypothesis. So, use the hypothesis mod p is less than equal to m. So, we are just using one sided.  And then if you plug in this hypothesis on p.  So, we have to do some integration there. Let me just write it e to the m t minus a divided by m. In the second inequality there are double integral. So, if you do that double integral, you get this. And this is valid for all t in a v. So, remember again. So, you have to just remember that you have to do some integration. So, you have to do some integration. So, you have to do some integration. So, you have to remember that thing.   And by our notation this is nothing, but xi of b s plus b 1 xi dot at b s. And our aim is to show that d phi by d s is bounded away from 0 that you remember that. So, now, we have got the expressions for xi and xi dot for all t. So, in particular for t equal to b and let us see whether they are bounded away from 0. So, that is our next task. So, if a 0. So, look again let me go back there. Just look at this, look at here. So, both xi dot b is greater than or equal to 0 and xi b since there is a 1 there and a 0 and a 1 are not simultaneously 0 xi b is always bounded away from 0. There is no problem about xi b. This is always bounded away from 0 and this in general not that is just I want to. So, note that xi b is let me write y b let me write t is bounded away from 0 for all t in a b. Of course, we are just interested in the final point b. The same is true for xi dot t if a 0 is strictly positive. So, the trouble occurs when a 0 is 0 and now again look at the conditions. So, if a 0 is 0 then b 0 is not 0. So, this is one of the hypothesis. In the second theorem we want this a 0 b 0 should not vanish simultaneously and also b 0 also b 0 b 1 have the same sign. If a 0 is 0 b 0 is not 0 and b 0 and b 1 have the same sign and again now look at phi s recall this b 0 u b s plus b 1. So, our only concern now is what happens if a 0 is 0. So, if a 0 is 0 then b 0 is not 0 and b 0 and b 1 they have same sign. So, if b 1 is 0 that is no problem if b 1 is not 0 b 1 and b 0 have same sign. And we are already seen that this derivative of u with respect to s is bounded away from 0 and this is just infimum could be 0, but in this case we have b 0 not 0. So, therefore, again d phi by d s d phi by d s is bounded away 0 even when a 0 is 0. So, let me again recall this. So, if a 0 is 0 by our hypothesis b 0 is not 0 and b 0 and b 1 have the same sign and that is important because the b 0 and b 1 they let me there is a beta here that does not matter. So, there is a beta there b 0 and when I take derivative with respect to s that beta anyhow goes away. So, important that b 0 and b 1 have the same sign and b 0 is not 0. So, when I take the derivative it again bound I already know that I already know that xi xi is bounded away from 0 I already know that I already know that the only problem with respect to xi dot xi dot and that is now taken care of by this additional hypothesis. So, remember this is additional hypothesis. So, when a 0 is 0 we required that b 0 different from 0 and we also required that b 0 b 1 have the same sign and that assures that d phi by d s is again bounded away from 0 even when a 0 0. So, therefore, d phi by d s bounded away from 0 in all the cases and this completes the problem. So, this very simple calculus one dimensional calculus tools very nice proof. Of course, one can always question this whether they are necessary or they can be weakened and that is we can always look into the literature and see if weaker hypothesis are possible. So, again let me recall this is the one. So, again last time I stated this example hope you have worked out. So, again let me recall that. So, this y double dot plus lambda E y equal to 0. So, y 0 equal to 0 y 1 equal to 0. So, this lambda is positive. So, I approach this boundary value problem via this initial value. So, this is b v p. So, this is I v p. So, u dot u double dot plus lambda E u equal to 0. So, you put one this initial condition same thing and now this is an independent initial condition I put a parameter s. So, last time I mentioned that this can be solved explicitly a solution of I v p can be found in explicit form. So, do it. So, it is similar to the one we did in case of conservative equations. So, this is also in conservative equation explicit form. So, using that explicit form obviously, it contains this parameter s. So, next question is is there an s star. So, you call this solution as u t s is there an s star such that u at 1 s star is 0 because that is the condition we want at t equal to 1. So, that is the question. So, if s if s then y of t is equal to u of t s star is a solution of b v. So, do this thing and also do this and also discuss uniqueness. So, why I am stating this example is that this e to the u in this e to the u is not globally lift. So, this condition is not really necessary. What we want is this solution to the initial value problem to exist for the interval in question. In this situation we have just 0 1. So, we are satisfied if a solution of the initial value problem exists for all t in this interval. We are not bothered whether it blows up outside that thing. So, there are I mean the possibilities of weaker hypothesis. So, with that thing I come to an end of this simple discussion on boundary value problems. So, we first discussed the linear boundary value problems in which we have the advantage of fundamental solutions from the linear system. But in the case of non-linear system our approach was through this IVP and you can now refer literature and even see more general problems involving this boundary values. With that brief discussion on the boundary value problems for second order equations we come to the end of this course on ordinary differential equations. We hope that you these listeners have enjoyed the course. The topics of the course were mainly based on the syllabus of ODE courses in most of our universities and other institutions. As you might have noticed we have omitted some important topics such as regular and singular storm value theory and corresponding Eigen value problem or spectral problems and also systems with periodic coefficients and the corresponding flow k theory. At the same time we have added as a trade off from discussion on qualitative theory of autonomous systems and shooting method for the boundary value problem of a general second order equations. The qualitative theory for autonomous systems is an important topic not only in physical and engineering sciences, but also in mathematical biology especially mathematical epidemiology where many problems have been modeled as first order autonomous ODE's. The subject field of ODE is quite vast and it is not at all possible to include all the topics in a single course to satisfy the various users of the subject. We however hope that those who have followed the course meticulously will be able to read and understand the cited differences further topics of their interest with our energy levels permitting and time permitting. We certainly hope to return with another course on ODE in future comprising of storm level theory, flow k theory and some advanced topics in qualitative theory and perhaps some topics in non-autonomous systems also. We would like to certainly hear from you regarding the contents and usefulness of this course. Your constructive criticisms are always welcome. Thank you.