So welcome everyone to this Impact Engineer concurrent session, Meet the Rising Stars. My name is Aaron Weinerman, manager of Global Public Affairs at ASME, the American Society of Mechanical Engineers. I'm based in Washington DC, and I will be your host for this session. I'm delighted to introduce you to some incredible special guests. But before we get started, I'd like to recognize a senior volunteer at ASME and a very close colleague of mine, Kailin Geile. Kailin serves as the senior vice president of public affairs and outreach at ASME, which includes our engineering for global development enterprise. So in Kailin's day job, he serves as the senior manager of global system safety at Boeing in Washington state. So Kailin, thank you so much for joining us today and thank you especially for your support in advancing engineering for global development at ASME. The floor is yours for some opening remarks. Absolutely. Well, thank you Aaron and thank you everybody for being here. It's my pleasure to take a minute here to welcome you and thank you for joining this session as well as attending Impact Engineer overall. I'm just as anxious as all of you are, I'm sure, to hear from our rising star nominees. So I'll try to keep my remarks brief here. As Aaron mentioned, I lead the ASME public affairs and outreach sector, or PANO as we call it, where ASME's core constituencies, industry, government, and academia come together to expand global awareness, knowledge, and application of engineering and technology through education, outreach, and advocacy. ASME's engineering for global development, as Aaron mentioned, is one of the externally facing mission focused programs that falls within PANO. ASME, as many of you know, is also a founder and a major supporter of engineering for change. And it's been really inspiring to see the work that EGD and E4C have done to advance ASME's mission to advance engineering for the benefit of humanity. As you are all well aware, and as many of the speakers this morning have reinforced, we're facing a lot of really complex challenges at the moment. COVID-19 in particular has exposed humankind's collective vulnerability and has highlighted the increasing criticality of science and technology-based solutions. The pandemic's negative effect on our progress towards the SGGs, as was discussed this morning. And it demonstrates the disproportionate impact that this pandemic is having on those around the world who lack reliable access to food, water, internet, education, and other basic needs. ASME's been around for 140 years, and in that history we witnessed a number of really pressing global challenges. And that history has taught us that meeting those challenges locally and systemically demands innovative and collaborative solutions. Hence the theme of this year's impact engineered for partnerships advancing the decade of action. This is a continuation and extension of ASME's engineering for global developments and engineering for changes effort to cultivate a social entrepreneurship ecosystem and to advance innovation broadly for the benefit of humankind. Impact engineered along with the engineering for change fellows program, the innovation showcase, or ISO, and innovate for impact, our new sustainable design challenge in partnership with Siemens, are really manifestations of this commitment to catalyze collaborations that incubate creative solutions in service of a better, safer, and more egalitarian world. I continue to be inspired by this event's impact and potential by all the amazing work that all of you are doing and by the commitment and dedication and passion that I see among all of you and I that I know we'll see in the sessions presenters today. And I don't want to delay that any longer so thank you all for being here back to you Aaron. Caitlin, thanks so much. Before we meet our rising stars, I should note what the rising star award is all about. This award is a recognition of great honor that features one emerging leader or organization, leveraging technology to achieve social impact. Each nominee has been nominated by a prestigious sponsor. I would also remind our attendance, our audience that they can use the q&a chat function to engage with our audience and I will ask your questions, and we can engage with our speakers. So let's meet our rising stars. Number one, we have Steven long, who is representing household farming system. Steven, I'll ask you where you're located right now, what attracted you to engineering, and just give us a really quick overview of your technology. And Steven, if you can hear me, you're on mute. There we go. Thank you very much. As was mentioned, my name is Steven long. I'm from Diana. That's in South America. We asked the part of the Caribbean, right in Diana. The city of Georgetown is located on the course on the course third plane. We're at and also we're six feet below sea level. So there are a lot of things that affect agriculture there. Right. So, as an engineering engineer in mind, as our engineer do we look for solutions. We look for ways to make life better for others. We build, we take, we construct. So, in light of the challenge that that farmers face and the challenges that affects food supply, I thought about the idea of building this device that will make agriculture not just wasteful because what happens also in the in Georgetown that is located close to the sea, and we are six feet below sea level is that there's a lot of flooding and it's highly populated. So there's not sufficient sufficient space for persons to plant and we have to travel on distances or food has gone from the rural areas. And that takes up a lot of time. So, with this device, the household farming system, finding simple ways to make to bring agriculture to the urban areas. We build we create and manufacture this system. We are fine. We not just only put plants in the pots, but we find ways in which to ensure that it is sustainable. We know that it needs a certain amount of sunlight water. So we ensure that it's also space efficient.  So it's vertical use of the vertical right now this system is curry some very attractive features. One of them is that it has a history best repellent lights. Or, you know these lights that repel pass right forgive me right I'm not a public speaker. No that's that's perfect. Thank you Stephen I'm going to I'm going to move on to our other panelists and we'll have more of an opportunity to go into each of your technologies as well. So with that I'm going to move to Zahin Razeen, who's representing Hydroquo plus Zahin the floor is yours to tell us where you're based right now, what attracted you to engineering and a very quick overview of your technology. So Zahin did we lose your audio. We did so I'm going to go now to a sagoon six Anna, who is representing cocoa networks. So again you know the questions and the floor is yours. Sure, thanks Aaron. Yeah so I'm normally based in Nairobi in Kenya, although I'm traveling at the moment. And really what attracted me to engineering was the ability to really understand how things work, and to try to find ways to find good solutions better solutions to the problems that exist and then seeing those solutions get out into the world. Right and have people use them that's that's really what exciting was a power to impact people through through through good engineering. Basically, you know despite decades of development effort. Most households in urban Africa still use dirty cooking fuels. Right, and this is this is the problem that for almost a decade now has has been a focus for me and my colleagues and so what we've done is basically designed an affordable, convenient safe new cooking solution that's based around the idea of clean fuel ATMs right so we create networks of these fuel ATMs, and these are relatively quick and inexpensive to deploy at scale. Okay, and the solution that we developed is one that has met consumer needs in a really unprecedented way we've seen rapid growth over the last year, since we launched. And it does this in a way that also addresses two really important issues. One is indoor air pollution. And the second is climate change. And so that's really the power of the solution. And that's really the progress that we've been making over the last few years with this technology. Excellent thank you very much. I see that Zahin is back Zahin do you have audio and video. Can you hear me. Yes, the floor is yours to tell us where you're based, what attracted you to engineering originally and a very quick overview of your technology. So hey everyone. So I was very much trained as a mathematician in pure mathematics but then I realized the only way I'd be able to offer value to the people around me is by leveraging technology as an engine for social good so I guess engineering was the fastest way to do it. So at the Hydro Co we basically what we do is leverage AI to hydrometric information systems that act as a prescriptive tool for water utility sectors in relation to water quality demand, leakage identification, the forecasting of demands and in prediction events so this is what we do as a whole. So yeah, that's us. Excellent. Jasmine I'll now turn to you Jasmine Sheen is representing reconnect. Can you tell us a little bit about your technology, what attracted you to engineering and where you're located. My name is Jasmine Sheen I am currently located in Cambridge, Massachusetts, and originally I grew up in China and I came to the US for college and I've always been at the intersection of science engineering design and business management and urban planning and I've seen and worked all over the world and have learned that communities really have already embodied the innovation and strength and resourcefulness to solve a lot of the challenges that we are facing and we are as we can be facilitators and enablers of these communities to help them bring changes to to their own communities and so in 2017. I think I really changed the face of Puerto Rico and that's where we started and realized that there is a major operation information and coordination gap between local communities and humanitarian aid agencies. So right now we're building reconnect that's leveraging human networks data and machine intelligence to close the last mile disaster relief gaps and build long term resilience for underserved communities. My name is Jasmine. I'll now turn last but not least Isaac sassy who's representing sassy technologies so Isaac, let us know where you are, what attracted you to engineering and just a very quick overview of your technology. My name is Aaron. My name is Isaac and I am from, I'm speaking from Kumasi Ghana, West Africa, which is really exciting. So what attracted me to engineering was the fact that I realized that being able to develop solutions that people use that solve problems that they had was a very powerful thing. It could be a tool for good and a tool to solve lots of societal problems and that was the reason why I went into engineering because it was at the intersection of my passion and my skills and my interest and it could be a force that I could use to change the world. And that is what I am doing now with my startup sassy technologies, where we are looking to empower farmers with affordable technologies. We realize that when you look at post office losses over 30% of the food we produce. And we realize that most of the food produce is produced by smallholder farmers who do not have the technologies needed to reduce their losses and improve their yields. So what we have done is we have developed a suite of technologies from a local grain moist and bundled it together with other technologies that solve post service losses and presented it to farmers in a way that they can afford, which is really exciting for the African context. Thank you. Thanks Isaac and thanks to all of our panelists for providing such interesting context about your technologies and where you can go with them. So we're going to head into Q&A portion here so for our audience watching please engage with our chat function at the bottom of your screen. And as we kick off here I think I'll ask everyone to comment on social entrepreneurship specifically. So we talked about you know what brought you to engineering but in the realm of social entrepreneurship what are some opportunities or challenges you're seeing in this field. We're actually getting a lot more attention these days than compared with what Kailin was talking about. You know we're seeing some very stark and existential challenges as a society, a global society. So in the context of what brought you to social entrepreneurship. I would hope that we can have a nice conversation that might let in our audience a little bit about what brought you to that space. So let's start with Segun if that's okay. For your story I know you're based in Singapore right that's also a very highly tech, almost Silicon Valley atmosphere so the floor is yours. So for me, you know what's interesting when you look at tech when you look at Silicon Valley tech is the idea that, you know, innovations can scale rapidly, right so why can't we do that in the social impact space or the environmental space. That's that's kind of what got me thinking about, if I want to do something impactful let me try to find a business model that can really scale. So the idea is by scaling, we can reach people we can bring more capital in. And if the model that we're creating for whatever revenge you're working on is is actually profitable that will just accelerate the flow of capital into the venture which will accelerate the scale and the impact right so so the idea of achieving impact with the idea of running a business like a business made a lot of sense and so really for for us and Cocoa Networks. What we do is with a commercial context. We are a technology company seeking to deliver an attractive return on investment to our shareholders, while significantly improving the lives of the people who are serving in terms of giving them a much faster cheaper cleaner way to cook, and in a way of doing that that doesn't just a damaged the environment right those two things go hand in hand. You know we make more money for every liter of fuel that sold right and we help the planet for every liter of fuel we're sold and so there's absolutely no reason to compromise on those two things. And that's what social entrepreneurship means to me. So Zahin, I guess in the same context you know focusing on your technology, water security for future generations reading your slogan here. Clean water is of course a main tenant of engineering a safer and cleaner world for people so in that sense, what do you want to see for social entrepreneurship going forward and and for hydro quo plus specifically and to build on what Segrin said is basically where I want to be able to offer scalable solutions in water management and the water related. I would say the fields, and I guess the best way to do it is kind of engaging with sectors in a way where you'd be able to offer let's say the services that essentially the governments are like, in general, due to the bureaucracy are unable to offer due to the lack that exists in emerging economies, so that we'd be able to do it. And I guess like again social entrepreneurship is the best way to do it and I guess water being one of the bedrock of civilizations. You know, there's there was a saying I guess the next world war will be fought over water and that's why there's so many countries that are trying to build aquifers so that when the water stress kind of goes out of control they're able to kind of extract water from there so I guess like if you're able to leverage again emerging technologies are all the skill technologies that are available. So that's the way to provide security for water and also kind of eradicate scarcity in the process I think it would be a great thing and I think everyone will come in a unified way. And I think everyone will be impacted because again at the end of the day I guess if I'm on mistake and once every month I guess four billion people are affected from water scarcity. So yeah so like that's the idea. Absolutely. Thank you so much for that. Jasmine I'll now turn to you. You talked about, you know, in the context of natural disasters and certainly we've all been attuned to the coven 19 pandemic which is not in the same category but in that time natural disasters have been popping up with more rapidity as well so talk to us a little bit about in that context of where reconnect started and where you'd like to see it go and as a social entrepreneurship firm. Right, absolutely. So I went to Puerto Rico before and after Hurricane Maria and at least 4000 people die because of the storm and it's aftermath 90% of the homes were had suffered damages and many people endure months without water and electricity and over 100,000 people left the island of 3 million. And during my two years at MIT and I was really able to spend time on the island to I was volunteering rebuilding homes I was with community leaders cooking hot meals for the elderly and I was with humanitarian aid agencies, distributing the different donation goods and services to affect a community. And what really struck me was that, again communities really already have their, they're so resourceful they're so optimistic and they're dedicated to, to their communities when around the world that because of the growing severity and frequency of disasters that there is no entity or government will be able to meet all the needs. So there's this major push that we need to build real is resilience at the individual and community level. And also the humanitarian space it's a very tricky and difficult space to be in for innovation because there is a deep and broad set of principles and directives that people have to follow because it is people's lives that are on the line. We can't just break things and an experiment. And also there is limited amount of resources. But we can either spend money on innovation or building new things or we can also have the money going to buying supplying food and water and the basic essentials that people already need. So we have to be very, very careful how we do that and the way that we're doing it is that we're doing it very, very closely with our community partners with our users to elevate their voices and their expertise in, in the area because a lot of the times we put a lot more focus on infrastructure, rather than the more disadvantaged social inequalities that we're facing. So that's where we are seeing this is not just engineering and data problem but it's very much a people problem and we say that there is no natural disaster ever. It's, it's a choice, it's the choices that we make as a society. It's an interesting way of framing it. I mean, certainly, there are a lot of decisions to be made and reactions that we can take as a society in the face of disasters, and but also your point about infrastructure is an important one. Turning now to Isaac, Isaac, you know, in the context of your technology, what are some of the challenges that you faced in social entrepreneurship but also why in the first place are you involved in this, because you could have gone and probably many other different directions so talk to us a little bit about where you'd like to see SESI technologies go. Aaron, let me start with your second question first, the why? So I personally, I grew up in, in the midst of farmers, my, my dad is a farmer, my mom is a farmer. And so agriculture was something that I grew up with. And it was something that hits home close to home because now I grew up in agriculture, which is supposed to provide abundance of food, yet I went to bed. There are so many times I went to bed without eating dinner. So then growing up through all of these hardships and being fortunate to be able to go through junior high school, senior high school, university on scholarships and on the good role of other people and building skills and engineering. I thought that I believe that if there was one place I could give back, then that was in, that was in agriculture where I came from. And being able to, you know, do my parts to ensure that the kid coming after me doesn't have to go to bed on an empty stomach was something that resonated really, really well with me. And so this is my passion. It's a combination of my passion, my skill, and in the context of social entrepreneurship, it's something that can bring profit. Because when you're solving problems for people and you're creating value for people, value that people appreciate and are willing to pay for, then that is also profit. So then there's a combination of skill, passion, and then profit in social entrepreneurship. And in Africa, one of the biggest challenges is that, you know, social entrepreneurship is seen with the mindset of nonprofits, right.   Because when you look at agriculture in Africa, especially given the fact that it's dominated by small order farmers, the notion or the mentality that the farmers have is that give me this for free. And so then you have the extra responsibility of creating that mindset shift where people are now able to realize that, hey, this is a product or this is a solution that is providing me value. Once I'm getting value out of this, then I should be willing to pay for it. And so what is one of the biggest challenges we've had to overcome the mindset shift.  And so you really have to think around how you can design around all of those limitations and in terms of their ability to pay. So these, these all bring about challenges, but I am excited that we're crossing that barrier and we hope that very soon, every farmer in Africa will have something that is made from technology or powered by technology. I like the ambition thanks Isaac and thank you for letting us in on your story as well. Stephen, I'll ask you the same question. Your video has been having a little bit of trouble it might make more sense to do the audio if it cuts out but we'll see how it goes, but what attracted you to social entrepreneurship specifically and and what are some of the impacts you're looking to make with household farming system. Floor is yours. First of all, I believe in ensuring that people's lives are better. Now, in this generation, not, I'm not waiting to do it for the next generation. I want to make an impact now. If people are wrong me, I must be able to do something to ensure that equality of lives, the standard of lives are better now in my lifetime. Right. I want them to have the best health, the best food, best lifetime to just put everything in a cup. Right. We know in economics, we know the different eras. They had the time when they was farming. It was solely farming then it went to industrial manufacturing. Then it went into services and then what start to happen is that from services start it we entered into the information age. And then we know entering to the health care, the health care age, which health care is a big, big topic. People want to know how to live long. Right. But while going through all these different phases, something happened with our food supply. Right. It has deteriorated the quality. And, and, and even though let's say the food of some in some cases where food supply had increased the quality. So you may take in some some stuff, eat some food, some produce. But then in the next couple of years, you have to buy medicine to offset the effects. Right. And that's why health care is having such a bang now. Baby boomers have retired. Right. Though all those forces come coming from the baby boom age, they retired and they put in a strain on health care system. And we forget to talk about healthy lifestyle, what we put in, right with our food, right, will determine how much train we put on a health care system eventually right. So I want to make a difference in my generation now to ensure that portions eat healthy. Right. It is a challenge because it's global warming, there's flooding, there are many changes in the, in the environment. So, to give Jack his jacket food is not growing at a fast rate. There are some challenges. There's a lot of pests. Right. And in our and then because of the motive of profit, persons who want to cut corners and put a lot of chemicals so that it could make money to live. Right. Or to support their families. Right. So, with this engineering, this divisive type engineer, right and working towards is to ensure that we go back to the basics. Right. To help persons farm, and it's not worrisome. They don't have to think about coming home from a hard day work eight to four or two shifts, or two jobs, and then they have to run to the supermarket. The thing is right there in their house, right there on either windows in their balconies in the verandahs. Right. As you can see here, they could get a cabbage. Right. And you're looking at disposable income. Right. Once this is sustained, it adds to the disposable income of the household. Right. So it has held factors. Right. It saves on time. Right. It adds to the health of the family, the household. We have things like celery here. We know important celery is with things like coals and so. So, in forming parts of the scientific qualities of this, the science. Sorry there. In forming portions of the scientific. Yeah, sorry there. I'll end that. Okay, thank you. Thank you, Steven. I think that's a good segue into, I mean, we were talking a lot about the societal impact here. Each of you just touched on. No problem. I'm just going to. Did you hear that? There we go. So, going back to the impact here, I know that Kailin Geile has a question on social impact and Kailin, the floor is yours to ask for our panelists. Yeah, thanks, Aaron. So I was actually just going to ask exactly what Aaron just said, you know, obviously a big component to all of your businesses is not just financial performance but the actual impact you're having on quality of life around the world. How do you go about trying to measure your social impact? So let's start with Segun on that one. Yeah, sure. So so measuring social impact is is a huge is a huge thing. It's important for us. It's important for our investors and other stakeholders. The way about the way we go about doing that is relatively simple in our case because we're really looking to achieve a transition from the use of a dirty fuel to use of a clean fuel. And because of the way our technology works, we essentially have a metering system for each individual customer. So I can, I can look up in our databases and see exactly how much fuel each individual consumer has purchased when they purchased it, etc. So with that, plus some sampling regarding the baseline, we're able to really accurately identify what kind of a transition we've achieved. And of course, once we know that transition, we can assess the carbon, the carbon emissions reductions, we can, we can estimate the deforestation that was avoided all those other measures. And so that's how the model sort of fills itself in, but the ability to measure at an individual level is what gives us a huge advantage in being able to report very, very accurate numbers. Excellent. Thank you. And as a hen, I'll ask the same question. Obviously, your technology has to do with with measurement in a different way but can you also touch on how partnerships are important here, because obviously, none of these technologies can happen in a vacuum and so each of you has partnerships that inform your your progress and how you measure it so the floor is yours. Right. Thank you again, Aaron. So I basically, I knew the best way to commercialize the technology and I think made this viable product for Bangladesh, a country that hasn't even been able to I guess like optimize on their e-commerce industries by kind of working with the government of hands-on. So when I went to the government and I said, hey, you have a lot of water quality issues and, you know, the one of the leading causes of water quality issues is due to the decaying pipelines and aging infrastructure. So irrespective of you building new pipelines, the water is always going to get contaminated. So why don't I just, you know, deploy a hydrometric device that will act as a decision support system for you to act on, I think, make a different decision based on region specific areas. And so they kind of agreed to it and I got lucky and then, you know, I went on about deploying a device as an area that was able to offer, I guess, the data on based on 100,000 house connections. So at a time we were able to kind of map out where the sources of turbidity spikes were coming from or the hotspot areas where, let's say, the E. coli content was coming from the total peak of color from the essential water quality index parameters according to the World Health Organization. So by that we were able to, by definition, working with the on field engineers and with our technology, we were able to get ahead of things and also reduce the optics and also kind of increase non revenue water profits of the water utility sector. And then to give you also like a shorter example is we're using radio metric imaging using computational hydraulics. So the satellite data, we kind of ingest that with the ground data as well as the satellite data and then we're able to kind of map out where again the sources of anomaly they're coming from. So let's say, for instance, someone might say that this is where on my while my house connection is possibly going from a water leakage and I'm not getting water supply. So essentially the area would have been according to assumptions would be in a different location. And we're able to like specifically point out where the sources of leakage is coming from. So by that, we are kind of also reducing the efficiency. I think the impact is again built on the utility sectors and as the value that we provide through that the distribution system. So yeah, excellent. Thank you very informative. Jasmine I'll turn to you I know you work with a lot of partners and have they been instrumental or what's been the process in helping you get that measurement of your impact. Is that something you can do in house or is that something you lean more on for with with your partnerships. Yeah, absolutely. Measuring impact in the disaster space is very complicated. And so we essentially work to bring positive change to enhance disaster resilience meaning the capacity to withstand and recover from shocks. So both at individual community and disaster management levels. And so our goal is to reduce adverse effect from disaster and improve services for individuals and grow capacity with effective organizing and access to resources at the community level, and to improve efficacy with better informed decisions and enhance collaboration at the agency and management system level. And because of the complexity we cannot only be tracking our impact when a disaster happened so in the in the disaster humanitarian aid world we look at both the blue sky time meaning the recovery and preparation mitigation time, and also the gray sky time meaning the response time, and which we all have to be in compliance with humanitarian aid principles and standards. We are partners because a huge component of what we do is to help them measure their impact. So a lot of this, it is indeed in collaboration to look at what kind of metrics make sense for them and how do we leverage and also look at our internal system. So to give some examples of the blue sky measures that we would look at is. So what huge component is that we are mobilizing community members to be part of community level disaster relief work. So we do measure engagement level, and how much information is crowdsourced and the level of increasing social capital and connectedness as that as a proxy to increasing resilience during blue sky time. And during gray sky time say when a disaster do happen, we use a combination of software technology, analytics and, and also dedicated surveys and also partner data sources to track. If any of the task is completed if there's a response reduction in the response time and if there's reduction in physical and mental health threats. And a lot of this, we are very carefully designing as part of our engagement and governance structure. Our code development process and piloting and simulation and because, again, as I said before humanitarian aid work is very sensitive and we definitely want to do it right and not to risk lives. Thank you. Isaac for SESI Technologies. How do you measure your impact and why is that measurement important in your sector, you know how does that inform a more technical and we've been talking about the commercial imperative and a lot of this so can you talk to us a little bit about what you do to measure and why. Alright, thank you everyone. So when you're when we're looking about at impact we're looking at it from multiple angles we're looking at it from what impacts are we making in the lives of the farmers that we're working with what is important to them and when you look at the farmers, the number one thing that is important to them is more money in their pockets. Now more money in their pockets means that they can pay school fees, they can pay hospital bills and their general standard of living is improved. So, for any group of small order farmers we are working with, we're taking baseline data, we are bringing in our interventions, and then after the season is over, we're looking at what has been improvement in the metrics that are important to them and to us. So to them, what how much extra income did they make, and with extra income translating to a better quality of life for them. Now for us we're looking at how much, how much of grains were we able to save, how much of food waste were we able to reduce, because when you look at it at a higher level from a food security perspective, you realize that 821 million people are going hungry every day and we're losing a third of the food we're producing, which is more than enough to feed all of these 821 million people four times over. So being able to reduce these losses, we're able to quantify how much losses we're able to reduce and that lets us know, even from a high level, how many less people are going to bed hungry. Now, when you look at it from even the perspective of climate, we know that if food waste was a country, it would be the third largest emitter of greenhouse gases. So we're counting how many kilograms of waste we're reducing and estimating what's the benefit even to climate change. So we're tracking a variety of metrics and looking at what is important for the smallholder farmer, what's important for the economy, the local economy, and what is important for, say, global food security. And we are looking at how our work is contributing to improvement in all of these metrics. Thanks, Isaac. You just mentioned security and I think a nice counterpoint to that is sustainability, which is a major component of impact engineered engineering for change, and all of your technology. And so we only have a few minutes left, but I'd just like to ask each of you in a word or two, what you're doing specifically to make sure that your technology is sustainable. And so that when you're going into different areas that they're able to then harness this technology and be able to use it themselves instead of, you know, having to go back all the time to get that expertise. So I'll start with Sagan. Can you have a word or two about how your focus on sustainability? Yeah, sure. So I think the key thing there is to really work through local partners. So for example, you know, for the cooking solution that we provide, it's really supplied to our customer base through local shopkeepers. Right. So again, I mentioned in Kenya, we have 600 shopkeepers in Nairobi, who are all our partners who sell the stoves that we sell, they sell the fuel through these ATM machines. And we're really equipping them to be the point of service and support to customers so that once our solutions are introduced to the market, they can ensure that it continues to deliver for, you know, for the customers. That's a really important part of it because this is an essential solution for people, a stove that they use every day to safely cook their food. Even a day or two without it is causing real harm. And so it's important that we have good service and support. Thank you. Zahin, I'm sure in your realm in terms of water security, this is a major influence. So just a couple words how sustainability applies for hydroco plus. Ultimately, the, I think the total value lies in the aggregate data that we store, rather than the hardware that's being used to store the data so I mean like to kind of extract the data so I guess sustainability comes from being able to see the past and then being able to act on the past for future decisions. Using the existing data that you have, I guess, I guess, and then that would act as a way of more of sustainability of some sort to answer your question. Thank you. Isaac, you talked about farmers wanting more money in their pockets and, you know, because sustainability is a way that makes that process one that can last. How are you thinking about it in the context of SESI technologies. So in sustainability, we're looking at how do we ensure that we're building a system that doesn't necessarily mean that we have to be everywhere. And just to build on what Sargun said, we're also working with local partners in different countries to to make sure that the solutions are made available to them. We're working with other bigger partners like the World Food Program, other multinational companies that play a very key role in the lives of these smallholder farmers and basically are working with partners to ensure that they have access to these solutions in a way that is sustainable to them. Thank you. And Jasmine, I think in the natural disaster context, you know, a major criticism is usually groups go in and they have a response and then there's a lack of follow up or the responses is not sustainable. So, can you talk a little bit about your your role in this context. Absolutely. The core inspiration for what we're doing is what we see what's happening in communities and exactly because governments and external partners go in and not without understanding what's going on on the ground and that's really causing the issue so we're really flipping that whole dynamic and trying to help the communities voices trying to make their stories and and what they see and what they experience count. And so and we do so everyone on the team is very well versed in practice and human and humanity centered design and we care about social justice everyone as a team and we work very very very closely from the very start of the first conversation with our community here in the agency, everything that they are part of both the design and the governance of our solution and we see the, the, and we understand from everywhere around the world that the social connectedness, and the, the self sustaining process of, of community engagement is really both the core of disaster resilience and the core of our work. Thank you, Steven. Same question about sustainability. How is that informing how you would like to change the agricultural realm. That's to me right. My good for us just now, Steve, please, if you can hear us about sustainability. First of all, pricing is important. We want to ensure that the price is one that the average person can afford. So they can sustain it. Secondly, we are going to be set up a very strong service arm so that person who needs to replenish plants and do repairs and so on regular checkups ensure that persons are, they are motivated to sustain their systems. We also work along with NGOs and the government and other organizations to give support to communities politically. And, and, and that is how we plan to ensure that this project is sustainable. Excellent. Thank you so much. So we're wrapping up here and I would like to know that everyone can check out more about our panelists and their technologies, their videos were made available earlier and now is time to vote. You can, you can go into the link which has been put into the chat by ASME staff and be sure to vote for your favorite choice. The winner will be announced tomorrow at 1pm Eastern Standard Time, and this will be during the Impact engineered award ceremony. You can also visit the Rising Stars virtual booth, which is in our exhibit hall, and you can learn more and connect with Steven, Zahin, Sagan, Jasmine and Isaac. And I would really like to express my gratitude and ASME's gratitude. And I guess with one minute left, I would ask Sagan, I see you, are you, what drew you to ASME, how did you hear about us because this is something we'd certainly like to keep going and expand. No, so we've been aware of different programs that have been going on, particularly in Kenya. And so, so for us, it's a network of, you know, just connecting with experts and identifying talent that we can work with and so definitely really appreciative of the efforts that ASME is doing. Thank you. I don't think I would be effective ASME employee if I didn't put in an ASME plug, but I do want to say that we look forward to working with all of you in the future. We've all talked about these partnerships being an important means of continuing these goals so we'd love to be in touch and thank you for joining us. Thank you to our audience members and enjoy the rest of your impact engineered experience. Thanks, everyone.


Hey guys, this is Rich from Rich TV Live. The cannabis sector is on the rise and we've got the hottest cannabis stocks Right here right now. All you need to do is subscribe to Rich TV Live on YouTube. Get access right now. Click the link down below


This is stock number 10870. We are here at Summit Automotive in Fond du Lac, Wisconsin. You're new and use sports car headquarters. Today we are checking out this super clean 2010 Chevy Camaro 2SS. This vehicle has the 6.2 liter V8 motor which pumps out 426 horsepower. It has been fully safety and inspected by our service shop. It has a fresh oil and filter change. All the fluids have been checked and topped off. The car has been gone through mechanically 100% and is 100% ready to go. Inferno orange metallic is the color. We shoot all of our videos in 1080p 60 frames per second. So if you have HD capabilities on your computer tablet or smartphone device, turn them on right now because it is definitely your best way to check out the quality and condition of the vehicle before seeing it in person. If you want to check out all the photos on this car in the upper right hand part of your screen is a link right to our website. Click that and check us out there. Comes with the 20 inch painted alloy rims. No scuffs or scrapes on there. And it has Pirelli P0 245 45 ZR20 tires. And these tires have a really good amount of tread left. I'd say probably 60, 70% of the tread left on there. Comes with the Brembo brakes. Front fender is an excellent condition. I didn't see any dents or dings on there. Comes with factory fog lights. Front bumper and lower valance are in nice shape here. Or bottom of the bumper I should say. Comes with this cool billet grille. You can see it does have the HID headlamps and the LED running lights. Looks really good. And the hood is an excellent shape. I didn't see any dents or dings on there. Going down the passenger side. Front fender looks really good. The rim is an excellent shape as well. And as you go down this side of the car take note of how clean the body is. How reflective and mirror like that paint is. We take these HD videos so if you are far away or even if you're close by and just cannot make the trip down but you're still interested in purchasing the vehicle. You can see the vehicle, hear the vehicle, and have confidence in what you were looking at before you even get here. So that when you do get here there's absolutely no surprises. The roof looks good, does have that sunroof. And that's why we offer the Summit Auto Fast Pass option in the upper right hand part of your screen. A link right to our website where you can apply for financing. Get pre-approved. See a payment quote. Even appraise a trade all from the comfort of your own home. So when you get here all you gotta do is drive the vehicle, love it, and you'll be all set to go. Back rim, no scuffs or scrapes on there. And the back tires have just as much tread as those front tires. Maybe just a hair bit less. But you get Brembo brakes back here as well. And when we come around to the back take note of the metal flake in that Inferno Orange. It's a really cool color, it's got a lot of shimmer to it. If you've got your HD capabilities on, you can check that out. Has a backup parking sensors, the big Chevy chrome tipped exhaust, deck lid spoiler and deck lid all look really good. No dents or dings on there. But as we go down this side of the car just as clean as that passenger side. The back side here looks good, no rock chips. Sometimes this gets all road rash, but these are in really nice shape. Back rim is in nice shape as well, no scuffs or scrapes. And down the rest of this side of the car looks really, really good. I didn't see any dents, I didn't see any dings. Very, very nice. Power heated mirrors. And we'll take a quick look in the trunk. Back storage area is very nice and clean. Those seats do fold down for extra storage in the coupes. If you ever wanted to know where your RPO codes are, they're right back here. So you can definitely see that it is a 2SS. And feel free to pause that and just make sure this car's got all the options you're looking for. And that shuts nice and solidly. Inside you get the Inferno Orange and Black Leather Interior. You get the SS stitched into the backrests. There are no rips, there are no tears on the seats. They all look really good. Power driver seat, both of these front seats are heated as well. And you do get factory formats throughout. The back seats are in great shape too. No rips or tears back there and it does come with the latch child safety system. If you want to try and get car seats back there. Also side curtain airbags, auto headlamps, power windows, power locks, power mirrors. Notice you get the Inferno Orange door sill trim. And to get the interior trim like this is a, I believe it was a $500 option on these cars. So we hop inside here, you can see that this one only has 10,931 miles. Instrument cluster is very nice and clean. Comes with the leather wrapped black steering wheel, Bluetooth and audio controls on the right, cruise controls on the left. Get the SS emblem on the bottom of the steering wheel. Red stitching. This one comes with the AM, FM and Sirius XM radio capabilities on the radio. And they do have your heated seats and climate control buttons there. Comes with the oil temp, oil pressure, volts and trans temp gauges. Also your stability control. The six speed manual transmission with the orange stitching. Take note that that orange dash trim goes all the way around, wraps around to the passenger side door. I always like the styling on these Camaros, passenger side seat is in excellent shape. I don't think this car has ever been smoked in. Comes with that power sunroof. Home link buttons for your garage door security systems and lighting systems as well as your sunroof controls. Then you do get on-star capabilities in the mirror. Dash all looks really good. And we'll take a quick look under the hood. Get the Chevrolet nameplate on the door sill there. Door shuts nice and solidly. I'd personally like to thank you for checking out the video today and hopefully from this HD video. You will have been able to tell just how clean this car is all the way around inside and out. Under the hood we have the 6.2 liter V8 motor. When paired up with that manual transmission 426 horsepower. See the shocks doing a nice job holding the hood up. Engine based very clean, runs very smooth. This car has been fully safe and inspected by our service shop. Has a fresh oil and filter change. All the fluids have been checked and topped off. And this car is 100% ready to go. To see more pictures of this car, one of our other 450 new and used cars, trucks, SUVs, minivans, Wranglers, halftons, three quarter tons, one tons, Camaros, Corvettes, Challengers, Chargers, Mustangs, you name it we got it go to our website www.SummitAuto.com. And if you'd like to check out more HD videos you can go to www.youtube.com. Remember to like, subscribe and share on this video and all the videos that you see there. In fact in a second you will see a link to subscribe to our YouTube channel on your left. It links more Camaro videos like this one on your right if you have not been to our website. On the bottom link to this vehicle on our website click those check us out and we really look forward to helping you with this super clean 2010 Chevy Camaro and Inferno Orange Metallic. Thanks again for checking out the video.


SCP-224 is a wooden grandfather clock, accented with a black lacquer and gold leaf. Though internal examination of its components has been done, the density and construction of its gears make method of function impossible to interpret. The hands of the clock do not move with any known consistency, though the tendency seems to be generally clockwise. Chimes occur at non-regular intervals, ranging from approximately one minute to several months. To further complicate matters, the numbers on the clock also have a tendency to move and shift, though they generally retain ascending order. Each chime has an anomalous acoustic signature that causes a drastic localized temporal acceleration. People and objects in range of the sound begin to age.


This is the last lecture of this module. So, let us complete in the. So, today we have two parts. We consolidate all the results we have discussed so far in the last four lectures. And in the second part of this particular lecture, we will try to discuss quickly about the non-homogeneous equation. So far, we were discussing the homogeneous system, but at the end of it, we will also discuss the non-homogeneous system today. So, to introduce that, we need a concept called to consolidate results and write it in the form of theorems. We are going to write whatever discussed in the form of theorem. We need to have the concept of generalized eigenvectors. So, generalized eigenvector. As you see, our whole trouble is that when there is an eigenvalue, which is repeated and is an eigenvalue is repeated with order m and we may or may not get m eigenvectors. If we get that many number of eigenvectors corresponding to its algebraic multiplicity, then there is no problem. We can always diagonalize the matrix and we can write down the solution. So, the problem comes for any particular eigenvalue. If its geometric multiplicity is strictly less than the algebraic multiplicity, then there will be a deficiency of eigenvectors to find a basis of R n, so that the matrix can be diagonalized. So, in this scenario, we introduce the concept of generalized eigenvector. Let lambda be an eigenvalue with algebraic multiplicity of eigenvalue of a, which is an n by n matrix with algebraic multiplicity. So, we say a vector v not equal to 0 is in R n a generalized eigenvector of lambda. Rather generalized eigenvector for lambda, if a minus lambda i power k of v equal to 0 for some k equal to say 2 to n. Of course, when k equal to 1, it is an eigenvector. So, in generalized eigenvector, we include even the usual eigenvectors. So, for some k, this should happen. When it happens for k equal to 1, it is an eigenvector. When it happens for higher k, it is called a generalized eigenvector. So, we generalized we use it for both the normal eigenvectors and more generalized eigenvectors. So, there is some interesting results, which you can say. For example, we start with an example before going to that. For example, most of the parts are exercises. Look at this matrix a is equal to 2 0 1 2. So, lambda 1 equal to lambda 2 is an eigenvalue with algebraic multiplicity 2. So, the exercise is that there exists only one eigenvector and that eigenvector can be chosen to be independent. When I say that one eigenvector means one independent eigenvector. Of course, if you have an eigenvector, every multiple of that is also an eigenvector, but you do not have a thing and can be taken to be say v 1 equal to 0 1. You can prove that if v is an eigenvector, every eigenvector will be multiple of that one. Now, prove that that is easy. Any vector is a generalized eigenvector. Prove that any vector v is a generalized eigenvector generalized eigenvector. This is because you can actually see that because you can see that a minus 2 i whole square is nothing but, 0 matrix. You see once it is a 0 matrix for every vector it is satisfied. So, you can therefore, one can choose v 2 is equal to 1 0 as another as an generalized eigenvector independent of v 1. So, you have got a basis consisting of generalized eigenvectors and this is the theme we are using. So, I will give one more exercise here. Here it is a let me write. It is a similar thing. Solve the problem. Solve the initial value problem with a is equal to 1 0 0 1 minus 1 maybe 2 0 1 1 2 is the same thing. So, in fact, we are going to write this. Of course, this exercise is valid after the description. This you can write it as of the form s plus n. We are going to say that what is s and what is the. So, you have to complete the problem. I will recall this problem again as an exercise later. So, there is an interesting lemma can be proved which is trivial which is not difficult. So, I will skip it. I will not prove it here. We do not have that kind of time. That much of time let e be a generalized eigen space. A generalized eigen space means an eigen space corresponding to generalized eigenvectors of a particular lemma be a generalized eigen space. Just like it is an eigen space, an eigen space is the space spanned by the all the eigenvectors corresponding to a particular eigen value. You know like a eigen space of a, then this is already we are discussing corresponding to n eigen value lambda. Then e is invariant under a, e is invariant under a. That is, it is a quickly you can prove that is a of e. See a will act on the elements. So, whenever you have something if you take any point in e, it will remain there. So, this kind of things which we know about the eigen space will also work for generalized eigen space. So, I will not prove the results here because I will not have time to prove it. So, because we want to complete something. With this we also yesterday wrote that every r n can be decomposed into stable and stable and all that which are invariant subspaces under the flow not under a. The flow is e power t a. So, the spaces e s, e u and e c which are invariant under the flow e power t a. That is what we have discussed yesterday. So, we will now let me complete all the theorems, but this is whatever I am writing is essentially we have discussed in the in detail 2 by 2 in more detail. So, let me complete for the sake of let me write down in the for the sake of completeness. So, I will do first I will write. So, in the form of theorem. So, I will write 3 theorems. The first two theorems are special cases of actually the second final theorem. So, it is a special case theorem 1. Suppose, a has only a real eigen value. I am not talking about it is a simplicity it can have multiple eigen values, but I am saying that assume a has only real eigen values, lambda 1 etcetera lambda n, lambda 1 etcetera lambda n according to its multiplicity. So, lambda 1 can be lambda 2 equal to lambda 3 there is a multiplicity according to its multiplicity taken according to its multiplicity. Then the result what we have discussed so far of course, which requires a proof we did not prove it everything which is nothing, but the part of Jordan decomposition. Then there exists a basis V 1 etcetera V n of generalized eigen vectors that is it generalized eigen vectors of then of R n. That means, a basis consisting of R n. Consisting of generalized eigen vectors of R n. So, you take the each V 1 is a vector put that vector as a column vector. So, V 1. So, this is the first column V 2 is the second column etcetera V n which is a matrix and it is a matrix formed by the independent vectors it will be invertible is invertible. So, you have a basis of generalized eigen vectors a has the form that is what we are so far discussing that given matrix will not be diagonalizable if you want to have diagonalizable you need a basis consisting of eigen vector, but what we are getting is a basis consisting of generalized eigen vectors. So, what it says that you can decompose a into 2 matrices in which s is diagonalizable and n is nil potent. So, you see so you have the problem of computing e power a, but a very matrix can be written as a sum of 2 matrices 1 s which is diagonalizable and for once it is diagonalizable you can compute it and n is nil potent and again computing e power n is easy and not a s plus n is nil bit. So, it is diagonalizable mean with this diagonal matrix. So, it is diagonal you see you have diagonal lambda 1 etcetera lambda n. Now, the and is not only that s and n commute that is very important n s n nil potent. So, you have all the properties nil potent and hence you can compute e power a therefore, e power a t a is nothing, but e power t s into e power t a. So, you can compute this one and t s can be t s is of the form p inverse of a. So, you can write down this is p inverse of diagonal of e power t lambda j. So, diagonal n p and e power t n is n will be nil potent of order sum k order k. So, in that case you will have e power t n will be of the form this we computed yesterday t power n etcetera e power k minus 1 by k minus 1 factorial into n power k minus 1 of course. So, if you want to find a solution. So, solution is x t is equal to e power t a of x naught. So, you see you have the complete solution you can directly write down using this one if this is a matrix it is x 1 x naught you get that one. So, you have to compute that one. So, this is the case when a has only real eigen value. You can also write a theorem 2 suppose assume a has a assume a is of order 2 n by 2 n instead of n I am taking even order that is all to assume that because complex roots occur in pairs. So, if I want to assume that a has only complex eigen values it order has to be even and has complex eigen values as all complex eigen values. So, it is a this need not be the case I told you as this is a special case complex eigen values lambda j is of the form a j plus i b j lambda j bar is equal to a j plus i b j minus i b j complex eigen value b j naught equal to 0. Then there x is generalized eigen vectors complex generalized eigen vectors complex vectors w 1 j is equal to u j plus i b j w j bar is equal to u j minus i b j this is that one.  So, you are working with 2 n is a basis of r 2 n.  So, it is a 2 n by 2 n thing is invertible again further you have the same representation the only thing is that is a complex further you have a is equal to s plus n s is diagonalizable that means p inverse of s p is equal to you get diagonal of diagonal of lambda 1. So, it is a diagonal is diagonal. So, let me write is diagonal of course, diagonal entries with lambda 1 lambda 1 bar lambda 2 lambda 2 bar etcetera is diagonal n nilpotent. Yeah there is something to be something is wrong not exactly what I said it is diagonal is block diagonal it is block diagonal. What is the meaning of block diagonal I am telling this of the form you will have diagonal of each block block will have 2 by 2 entry a j minus p j b j a j sorry. So, that is the block diagonal. So, that is the correct way of writing p inverse of s t is block diagonal. So, the first d which we have seen yesterday. So, the first entry is first block 2 by 2 and then 2 by 2 then 2 by 2 like that along the diagonal. So, it is 2 by n n nilpotent of course, of some order nilpotent. Yeah that is it s n equal to s n s this always you need it to commute then only you can compute. So, you can again write and we can write the solution as again you see we can write the solution as complete solution as x t is equal to p diagonal because you can compute the e power of this one easily that is nothing, but e power this we have already done for each one cos b j t minus sin b j t sin b j t cos b j t this is the diagonal entry this is the e power of that one into p and then you will have your e power t n e power t n is nothing, but i plus t n plus etcetera t power k minus 1 n power k minus 1 by k minus 1 factorial of x naught. So, you have the complete solution. So, you see of course, this involves the interesting point to be remarked here is that to write down the solution even though this is diagonalization this block diagonalization is done why are the finding the generalized eigenvectors at the n you do not need the generalized eigenvectors to be computed to write down the solution is enough to find the eigenvectors all the eigenvectors and its algebraic motive you do not need anything further. So, you are if this is the situation you do not need to find the generalized eigenvector in these special cases. So, see that is the interesting point. So, now, we will write the last part of it before going to the n only thing. So, the general case which you are already seen this is nothing, but your Jordan canonical form which we have already written yesterday today we have written the special cases separately that is all what it says that. So, you start with I do not write it in the form. So, this is a theorem again you can write it if you want it in the form of the theorem let a b of order to separate of order k plus 2 a. Now, this does not reduce any generality this is just to separate your real eigenvalues and complex eigenvalues. Here I am going to assume that it was lambda 1 etcetera lambda k real eigenvalues. So, you can have any number any n you can decompose in the form k plus 2 n real eigenvalues lambda k plus j is something like or lambda j itself you can write it lambda j equal to a j plus I b j lambda j bar is equal to a j minus I b j b j not equal to 0 it is a complex eigenvalue and j runs from k plus 1 to k plus a. So, you see so you have all the eigenvalues and then there exists basis that is all it is there exists basis we do not write it of generalized eigenvectors generalized eigenvectors you see you can write your complete theorem and you can write down your p.  So, you have the first k eigenvectors generalized eigenvectors corresponding to k this is not here corresponding to the real eigenvalues.  So, you are considering what the most general case and a and with this p you cannot diagonalize you can you can write down the diagonal with the blocks. Now, you will have the problem earlier you had a very particular format in the specific cases, but you will have diagonal of v 1 etcetera for some r we do not know that are all depend on its multiplicity and algebraic multiplicity geometric various factors where each b i is a block where each b i is a block. Block may be can be a 1 by 1 block also. So, if you have only one symbol eigenvalue with one vector it will be a single one lambda it will come and it takes one specific form and takes the form that is the and takes the form and this we have computed already yesterday we have elaborate way computed and any b i j will be of the form lambda lambda etcetera lambda and the upper diagonal elements 1 etcetera 1 the rest of the elements are 0 and the order again depends on the multiplicity there are some estimate product we will not discuss here, but any typical b j will look like this particular form and the order of b j may vary depending on the lambda j this is for the case if lambda real or you can have the block or. So, you can have the block b j will be of the form of the form again we have discussed how to compute this one this will be d i 2 d on the diagonal i 2 here d i 2 these are all 2 by 2 matrices inside where d is equal to of the form d will be of the form a minus b. If lambda is equal to a plus i b b not equal to 0 and i 2 will be of the form is the identity matrix 2 by 2 identity. So, this is the typical eigenvalue. So, using this this will immediately write down your solution as x t because. So, you have decoupled system a x equal to a x it is enough to solve the system for y dot is equal to b j of y for each b j you solve it separately and each b j will have one of either this form or this form. So, with that you can immediately write your solution x of t is equal to. So, let me finish in this page itself you will have p since this is of the diagonal form even for the block the diagonal form works. So, it will be of the form diagonal e power t b j d b 1 etcetera e power t b r at p inverse p inverse of x naught. So, you see.  So, you have that. So, how to compute e t power b j each t power b j is a matrix set matrix which is the same order as the b j and how to compute e power d j is what we had discussed yesterday. So, this gives you the complete analysis of the Jordan composing and representation of the solution for the linear system. So, with this we now move on to the last section of this talk namely for the non homogeneous system. So, we will move on to the next section non homogeneous equation homogeneous again autonomous we are in autonomous only because all these facilities are available. So, what would be your equation you have your equation x dot t is equal to a which is independent of t. So, x t plus g t where the elements g t. So, you can have your g t and your initial condition x at t naught is equal to x naught ok. So, we are going to introduce a concept to concepts called fundamental matrix and transition matrix. This is crucial even in understanding I mean you can write down the solution in a much without introducing this notions which I am going to describe soon, but that is the thing used to understand non-autonomous systems because this represents. So, we are going to represent the solution of this system in a using the fundamental matrices which can be generalized to the non-autonomous system. So, let me introduce this notation which is crucial fundamental matrix and transition matrix. So, let me solve this is system is the system to let me call it and we call it the same notation I am using I am not using in a different notation I am using this notation a for even for the homogeneous system. So, this let me call it one this is my one and. So, let us I want to solve this equation with x at 0 is equal to say x naught I can solve for various initial condition. So, choose x naught equal to the basis vector e n where e 1 to e n canonical basis canonical that is 1 0 except of basis of r. So, for each of this initial condition I will have a solution and let phi i t be the solution of 2 be the solution solution of 2 solution of 1 I am talking about the homogeneous equation that I that means phi i t satisfy the equation 1 with that is that is phi i t at 0 will satisfies the equation e i that is all. So, phi i t phi i dot is equal to a phi i and phi i t 0 is equal to e i. So, it is I can introduce phi i phi t a matrix because phi i t is a vector it is a vector solution. So, each vector I will put it in the first component phi 1 t etcetera phi n t you know that these are all independent solutions and invertibility now you can think you can immediately understand this equation and what is this phi t will satisfy. So, phi t will satisfy each phi i t will satisfy an equation with 1 0 0 as the initial condition phi t will satisfy with the initial condition 0 0 etcetera. So, phi t which is a matrix which is a matrix and satisfy the matrix differential equation satisfy the matrix differential equation max differential equation phi t phi dot of t phi is a matrix now. So, you can still that one it. So, you are arranging your each row properly and what is your phi h 0 phi h 0 is nothing but identity because phi i t phi i t 0 is 1 0 0 etcetera phi 2 h 0 is equal to 0 1 0 etcetera it will satisfy the matrix differential equation. So, you can do it instead of the initial condition I can do the initial condition we can one can take the initial condition at t naught.  So, phi t is nothing in this particular case. So, I can take your phi t t naught. So, I introduce p at t t naught is nothing but phi at these are all very special k works for the autonomous system this is nothing but e power p minus t naught of a and this will again keep. So, this phi t t naught again will satisfy your matrix differential equation, but with the initial condition at t naught that is all the difference we just translated satisfy the matrix differential equation matrix differential equation d by d t that to avoid that t naught are there which we are going to vary phi t t naught. So, nothing but a of phi t t naught t naught is fixed t is varying with the initial condition at t naught phi t naught t naught is identity you see that is the thing. So, you have your solution essentially I am translating a t naught you can write out this is called the transition matrix the matrix definition phi the matrix phi t t naught is called the transition matrix because is the name is suits transition this is where what is it is happening is that it takes the point at t naught if the position is at t naught is at x naught it takes to phi t t naught of x naught at time t. So, that is what happening. So, phi t t naught of x naught is equal to you look at that phi t t naught of x naught is equal to x naught and then at time t the solution will be at. So, x at t is equal to phi t t naught of x naught. So, you take the trajectory. So, if you have the trajectory at the initial point this takes to the trajectory translate. So, the phi t t naught takes the trajectory from x naught to x t at time t. So, the name is correct the transition matrix is correct this also have the other properties which we may probably tell a little later we will we will try to say something later at the end of it. So, there is also a notion of fundamental matrix fundamental matrix it is more or less like a transition matrix, but we do not put a condition on the initial value. So, any matrix any matrix matrix C of t that means, consisting of functions. So, the entries of psi t are functions matrix satisfying the matrix differential equation satisfying the matrix differential equation psi dot of t is equal to a psi t is called a fundamental matrix. So, the trivial fundamental matrices are transition matrices. So, you see. So, every transition matrix is a fundamental matrix every transition matrix is a fundamental matrix. So, hence fundamental matrix in this particular situation is of the form e power t minus t naught into a that is a your thing every transition, but there is also another interesting thing which you can easily prove. Suppose, C is invertible there is no suppose C is a constant matrix which is C is a invertible is an invertible of course, constant matrix invertible constant matrix. Then psi t you define psi t as psi t you post multiply C then consider this one consider psi t. So, this implies psi dot of t with respect to t phi t means phi t t naught if you want to do that one. So, I suppressed t naught here psi dot t is equal to phi dot of t and C is independent of t, but phi dot of t is nothing, but a phi t and C and this is a associative law is true. So, substitute phi t that you will get a psi t. That mean if you take any invertible matrix C and then define phi t is equal to psi t is equal to phi t C then is also that imply psi t is a fundamental matrix. But the interesting result which is trivial to prove. So, I will leave it as an exercise every fundamental matrix is of this form every fundamental matrix given any fundamental matrix given any fundamental matrix C C there exist an invertible matrix matrix C such that C t is of the form phi t into C. In fact, if you observe you can compute C for given phi given C you can given psi you can compute C. In fact, psi t naught is equal to phi t naught psi, but phi t phi t naught is nothing, but phi t t naught at only already the other one is suppressed phi t t naught C is the initial value phi t I represented for phi t t naught suppressed, but phi t t naught is identity. So, this is nothing, but C. So, your C has to be of the form psi t naught. So, you have that this is a trivial exercise with this you can compute and do that one psi is of the form. You just you need the uniqueness of the solution that is all. Now, with this we will go to the non homogeneous equation I use this notation as I said you do not need all this, but it is introduced to understand the non autonomous system which we may not do that one. So, now non homogeneous system we apply the variation of parameters which we have done in first and second order equation we vary as you know parameters vary as you know parameters. So, note that if you see. So, I use this notation phi. So, let me work with that initial condition t equal to 0 that is enough you can work with any t naught also. So, choose t naught equal to 0. So, to avoid, but you can it does not matter there is no loss of generality no loss of generality only you have to work. So, note that phi t is equal to e power t a of x naught for v satisfy the homogeneous equation satisfy the homogeneous equation. These are same idea we have used homogeneous equation with phi at 0 is equal to v. So, the idea which we have used in variation of parameters if you take your flow e power t a and act on any constant vector v which will only satisfy the homogeneous equation. So, you can never expect to have a solution e power t a v for a non homogeneous equation. So, the idea which we have used is to vary v idea is to vary v and view it as a function of t view it as a function of t that may give you a hope. Because if you do not do that one you are never going to get a homogeneous non homogeneous solution this might give. So, assume phi t assume x t is e power t a into v t is a solution of non homogeneous equation solution of x dot of t is equal to a x t plus g t. Now, let us compute here compute the x dot of t. So, it is a product formula once you apply the differentiation to the first term you get a here a e power t a v t and if you apply that to there you get e power t a into v dot of t. This is nothing, but your e power t a into v t is nothing, but your x t plus e power t a into v dot of t. So, if you want x dot is t is a solution to this non homogeneous equation 2. So, if x t is a solution to 2 will immediately imply we need to choose the this term is g t we need to choose e power t a v dot of t is equal to g t. Or otherwise you have the first order equation for v t this is an integral calculus problem you do have nothing to do it is a e power t a into g t.  So, you have v at t naught v at t naught v at t naught if you integrate t naught if you do that one plus integral t naught to t of course you can work with t equal to 0 or not you integrate that one e power minus s a into g of s this is what you want to do. So, you get that. So, now you substitute this formula you substitute in what do you substitute you substitute here. So, if you get if you substitute v t there what you will get is a substitute what you get your x of t will be. So, e power. So, you can do a small computation which I will skip here t minus t naught of a at x naught of course, when you choose t equal to t naught it will be just x naught do the computation substitute and do the computation you get plus integral e power t naught to t e power t minus s is a just a small computation absolutely no difficulty g of s t s. So, you have that formula. So, this is your formula for your solution using the variation of parameters you see. So, when there is no g when there is no g this term would not be there and you know that this is the solution to your homogeneous equation. So, the first one is exactly getting what you have seen homogeneous equation you have a if you recall your second order equation you have a complementary function which is a solution to your homogeneous equation we have seen that and then you have a kind of particular integral and that is what we have done here. So, let me write it in this form. So, this is of the form phi t this is nothing but phi of t minus t naught x naught and symbol form plus integral t naught to t phi of t minus s g of s t f this kind of representations are very important in applications. For example, the non homogeneous term will appear in the control form. So, plenty of applications which you know plenty of applications for example, g t may be a control g t may act as a control control. So, and it may say for example, linear control say linear control we will not get into that one of the form this is a g t is equal to some b of u t. So, you have that control form that format. So, we will not do that one, but it gives you a important thing and another thing interesting thing is that here you can also define this kind of solutions in the weak form and other kinds of thing in ODE it is called the mild form. And which is also important in control theory because you do not look for controls which are continuous and when you do not have the continuity this kind of weaker formulations of the representation of the solutions you can write it as thing. So, we will not get into that, but what we have seen is that you can write down your solution consisting of two parts. One part is the solution corresponding to your homogeneous system and in the last four lectures we were actually studying the homogeneous system completely. You are trying to understand this phi t minus t naught of x naught completely according to the various form how to compute that one. Once you compute you also have a second term that corresponds to the non homogeneous part. With this we more or less endeavor thing, but we make a slide or a just one or two comments about the non-autonomous system and we will stop here non-autonomous system. You know that non-autonomous system works completely different even in the second order you have seen that one. When you have a equation with a constant coefficients in the second order equation linear constant coefficient equation you have a complete analysis how to represent the solution, but when you have an equation x double dot plus p t x t plus q t x dot and q t x x t equal to 0 or y t or g t then you have no there are some methods, but in general finding two independent solutions is difficult. You have the solution structure and here also we are going to give you representation what you have seen even in the n by n system you have a complete knowledge of the homogeneous system in terms of the exponential that exponential representation is not possible. So, you have you have a solution here of this form x dot t. So, the variation of parameters with non-homogeneous is fine, but even with a t x t. So, we are not going to do much on this x t naught is equal to x t, but what I want to or the homogeneous counter part there will be a g t there, but the interesting point is that existence and uniqueness available existence and uniqueness available by the same general theory which you have studied in our earlier module. So, the if of course, the elements in x and the component elements in a t are continuous that minimum assumption of the continuity of the matrix a t is required, but existence and uniqueness available from the general theory from the general theory. Now, the concept of fundamental matrix and transition matrix can be introduced that is important concept of you do not have no e power t f and finding a solution at t naught you cannot translate. So, that is also an important thing you it is not possible you find the solution at x equal to 0 and then translate because of the non-autonomous system which you will see in more elaborate way in non-linear analysis. You cannot just translate find the solution at t naught equal to 0 and then translate like phi t minus t naught and that is not available in this case. So, you have to study if you want to understand at t naught or to study directly at t naught the facility of this kind of translations are not available which you will see in non-linear analysis, but the concept of fundamental and transition matrix can be introduced that is the whole thing transition matrix can be matrix as can be introduced that is the important thing. So, now you have to introduce. So, let phi now phi this is important now you have to write separately you cannot write this is of the form phi t minus t naught that is not the thing. So, t naught and it is works differently be the transition matrix transition matrix transition matrix of 1 means it satisfies this matrix differential equation satisfies satisfies the satisfies phi dot that is d by d t this is a matrix phi dot at t t naught is equal to a t phi t t naught and x at t naught is equal to identity this can be introduced that thing and you can also introduce the fundamental matrix. What is the mean this is a matrix psi t which satisfies the differential equation and you have the same formula you can also write your psi t will be the form psi t has the form all that results are 2 here psi t has the form psi t is equal to phi t is equal to phi t c phi t c you can do all that and your solution will be all that can be introduced exactly this way, but never write phi t t naught is equal to phi of t minus t naught equal to e power t minus t naught t into a and that portion is not available to you the rest of the system you can introduce by using the uniqueness phi i t and then put it is a column wise and you can introduce everything phi t t naught form and you can write down here and your solution also can be written as x t you may think that it is solved, but it is nothing is solved actually we are only saying that phi t t naught is a solution of this matrix differential equation of phi i t each column is a solution of your differential equation with a thing, but how to solve that differential equation and how to find that is not given to you in the earlier case you have a representation of e power t. So, you have your phi of t t naught of x naught u c. So, you have your solution that is all you can do it for the homogeneous equation and the this is called the transition matrix and for the homogeneous non homogeneous system you have your solution x t is of the form phi of t you have the component y all this can be a thing x naught plus integral t naught to t phi of t t naught you have to write this way not phi t t naught of phi inverse of I think this probably may be s if I this may be s there may be some s phi inverse of no this is t naught only. So, you have a representation of that form. So, you have the thing. So, the last thing phi has the group structure phi has the last slide is as the group structure because this is due to the invertibility of the irreversibility of your system. When you know the solution at t naught you can also solve this for the past, but then there are when you go to PDE's and other things you may not be able to do it and you will only get a semi group structure. So, what are the things you have phi t naught t naught this property is identity that you have it and the other interesting property this is the crucial property phi t s and phi s t naught is nothing, but phi of t t naught this is the most crucial property of the semi group structure. So, it means that your solution is going from s to t and s to t naught and. So, you have you want to go from t naught to t first you go from solution t naught to s and then you can take an initial condition at s and then go to t. So, that means you can go from x t naught at time t naught here you reach the time t s here and then you go there instead of you can go directly here that is what you say this semi group property so this immediately tells you that this is invertible the implication is that phi t s if I put phi t naught in particular if I put s equal to t I get s itself and if I put s is equal to t I will get phi t t phi t or may be I have to do it somehow. t naught properly and you get identity so that you will get it as inverse of phi t s is equal to phi s t you can prove all that. So, same thing if you want to see that s is equal to t naught identity I think that is correct. So, we will skip this. So, you have to all that property. So, as far as the non autonomous system is concerned you have all the representation of the solution using transition matrix, but there is no way in general to determine transition matrix as in the autonomous system, but then there are some conditions under which this transition matrix can be computed in some form of the exponential using certain commutativity of that one. So, with this we will complete the module on linear systems and you will see the use of this thing in the when you we study the non-linear systems and this stability analysis. Thank you.