Hi there. I'm Faisal Islam. I'm the BBC's economics editor. Thank you to the many thousands of people who are joining in this very important topic of the Jobs Resets Summit, the new vision for social protection. We have an excellent panel and I'm your moderator. I feel like I've been a moderator of sorts during these extraordinary health and economic times, trying to moderate between people's lived experience in this extraordinary situation that we go and film as the BBC and what's actually happening. We're finding that actually going out and seeing people and filming them, you often get better, more up to date information than the data. And you're seeing huge innovations in terms of public policy, huge reactions in real time and also learnings from other parts of the world and other countries that we haven't seen before. This is a time to take stock, but also to think about this moment, whether it's a time to sort of relaunch a different type of protection system going into the future. As I said, we have a stunningly brilliant panel and I think without further ado, let's just get into that. I think I'll introduce them as we go and we'll start with the, we're lucky to have the federal minister for and special assistant to the Prime Minister of Pakistan on poverty alleviation and social protection. And that is Sanya Nishtar. So, Sanya, if you just help us, you know, given everything that's happened this year, how can national social protection systems in emerging markets be redesigned for this new norm? Well, first of all, thank you for having me on your, on this programme and thank you for convening this very important conversation. I think we need to think of social protection as a right now and governments have to be very clear that we need to massively expand social protection and make it very responsive in, you know, in a post COVID context because the need is so immense. I mean, the loss of livelihoods have been so decimating, you know, in countries like mine as a result of COVID that we will really need to draw social protection de novo. And as I said, look at social protection as a right. As governments do that, I think that there are certain important pillars on, on which to, on which to construct a sound multi-secure strategy, a good data backbone, adequate funding, you know, appropriate digital infrastructure, making sure that the right principles is what we draw on and then of course accountability for delivery. So I just want to quickly outline to you what we have done here in Pakistan because we're in a bit of a fortunate position because Pakistan started off with a very expanded social protection and poverty alleviation strategy pre COVID. So we started back in March 20, March 2019, where after a series of deliberations we came up with a whole of government, multi-sectoral poverty reduction and social protection plan with 140 elements and actions. And all of these were linked by theory of change. So there was a completely new strategy with different programs that were mutually reinforcing targeted at poor women, orphans, the homeless, poor laborers, farmers, the disabled, widows, individuals who are at the risk of catastrophic health expenditure. You know, there were interventions for primary school going children for students, they were livelihood promoting programs for the extreme poor and they were also nutrition interventions from a human capital perspective, a number of different condition cash transfers. So once we had the strategy in place, the government committed funding, you know, indigenous funding, not donor driven. And we made investments in the new social, national socioeconomic registry and not just a household survey, but also the necessary investments to make sure that the data analytics architecture is there, which would immigration data, a land ownership data, telephone data, government employment data and so on and so forth. So that was the second bit that we did the data part. And then we made sure that our registry is not just a one-time exercise, it becomes a live registry. So in addition to the strategy, the data piece was the second pillar. Thirdly, we invested resources to develop the digital architecture. So we did the procurement of a new biometric payment system. We developed a SMS seeking request, SMS based request seeking mechanism, which then allowed us to deliver emergency cash to half the country's population. And then in terms of the principles, we made sure that it was a whole of government approach that we were adopting, that the private sector was an equal partner, that technology was very much the mainstay, and that we were using tools such as financial inclusion and the, you know, to enrich whatever we were doing. The focus was very much on women and particularly on girls, and we have now very explicit incentives for girls to be prioritized in the new social protection framework. And in addition to all that, we made sure that accountability for delivery and delivery with integrity is one of the cornerstones. So we got cabinet approval for a governance and integrity policy, made it absolutely binding on all the executing agencies to follow it. We established a delivery unit, started tracking metrics. And lastly, information because during my experience, giving cash to half my country's population, the most important tool that I found was information. So it was not just investments in digital infrastructure and data and the strategy formulation and the delivery architecture. And the principles on which all these things were founded, but the information citizens must hear the truth from governments, the unvarnished truth, and they must have the confidence that governments are there to deliver for you. That they have clean hands, and they have every intention to be very truthful with you. So I think in order to espouse and in order to follow that policy principle of social protection is a right, you know, just to wind down what I had to say. Governments have to of course legislate and formulate policy, but in order to execute it, there is an institutional architecture you have to build. And on top of that, the right information truth is what will really do the trick. Well, Minister, thank you very much for that. And obviously I share all our participants and saying get well soon on all that. I'm quite remarkable to think of a system that distributes cash to a sort of 100 million people in Pakistan and will develop that conversation. We're now going to move on though to Jean Zaino, all the way in the USA, talking specifically I think area of expertise here is that he is the executive chairman and founder of MBO partners and specializes in the area of freelance workers big really important underserved in some of the developed country schemes. Let's see what Jean has to say about these millions of independent workers have been particularly hard by the disruption, what needs to change to better support them in the aftermath of COVID and beyond. Thank you fossil and thanks for including me on this panel and with all these great, great speakers and, and it's a very important issue. So specifically regarding the independent workforce, which goes by other names like the gig economy and gig workers and independent contractors. My personal expertise and passion has been understanding this workforce over the last 20 years at MBO partners and we today and over the last certainly 10 years or so we've been studying this workforce with a report that's been fairly well recognized called the state of independence in America and it's, it's surveyed by an independent research firm of the last 10 years that the growth, the behaviors, the attitudes and the motivations of people doing this type of work and in fact we're about to release our 10th year data so it's interesting to look at the trends over the last 10 years. So, so what I would offer here in my perspective with regard to this segment of the workforce, which, which is a significant segment and growing around the world. I would say there's two things that we need to think about as we reset. And as we enter into this new post COVID world. One is independent work is highly desirable by many people. And that's not only from our research but from all the reports that are out there some that might be more self serving than others but even the most recent United States Bureau of Labor statistics. When, when you survey the people doing this work, they, they, they're happy doing it they actually feel more in control of their world and their life and, and people are in either independent workers or traditional employees. They tend to do this work at different stages in their career. And I think that's important. It's important for retraining and skilling reskilling and, and, and filling gaps between traditional work, and even in this COVID situation that we're currently in. I think it's actually been one source of trying to fill some of the financial needs by by doing some type of app based work. So, so the first thing I would offer is that we need to recognize that this is a significant part of the workforce, and it's a desirable way for many people to work. And, and it's usually about 70 to 75% of the people that feel that way there are certainly people that that don't want to be doing this type of work and they're forced into it, which leads me to my second point. I think the second point is, we need more clarity into understanding the classification of the type of workers. It's very, I believe it's very dangerous for us to consider one broad stroke of a solution. There are, you know, as, as the world becomes more technology driven, more diverse, more segmented, there are different work arrangements that require different apparatus and different policies, and different enablements to make it work well. And in our study, what we've learned is that there are there are what we call occasional independent workers or gig workers, and they tend to supplement full time traditional work during the weekends, moonlighting in between work to supplement income. And then there are the other extreme people that at different points in their career are looking to actually work differently on a on a full time basis, as building their own independent what we would call a micro business. And these micro businesses need infrastructure. They, they need liquidity in the market. And a lot of that is a result of innovation technology things like we're doing right now through through zoom and collaboration and, you know, the the delivery of services is much more digital today. So working from anywhere, in fact, one of the recent aspects of a study that we just released is digital nomads in 2018. In the United States, there were 7 million of these people that consider themselves digital nomads where they would relocate at least three times during the year as recently as July and August of this year we did the study again, and that number is now grown to almost 11 million. So people are realizing they could work from anywhere they could deliver their work remotely. And this is more of a of an experience for them that we want to make sure that we don't disable by having rules and regulations that are overly complex by trying to protect people that do need the protection. So, you know, my second point here is, you know, let's make sure we don't have a one size fits all. So in summary, you know, independent work is very desirable for many people. And we need to think about what are the different structures that are needed to enable the different types of work arrangements, and, and making sure that we both enable the people that that want to do this type of work, as well as protect the people that need to protections. And that's not an easy problem to solve. We've been working with some of the policymakers here in the United States. I certainly have done work with the World Economic Forum which have released last year, a charter of the promise of the platform which I think has a lot of good aspects to it. So anyway, thank you and I hope this, you know, is informative for the listeners. Thank you, Dream. Thank you for shedding light on that really important issue of the of the gig economy in this tough time. Let's move on to Sharon borough who you all know is the General Secretary of the International Trade Union Congress. Sharon, let me ask you, and despite all the hardship brought about by this COVID crisis, there is a sense in scale the need and the relief measures we've seen around the world have transformed the public imagination of what's possible. What would a global social protection fund look like? So the minister from Pakistan, let me say congratulations because they've actually put in place the backbone of a social protection system. Now, the moment it's focused on cash transfer, but at some point on income for those who are unemployed or don't have enough income to live on, plus health access of the two key areas we've been promoting as a first step, but a social protection floor covers a range of areas to give people security. What's what the COVID-19 exposed in huge craters? It was there before and Olivia Distuta knows the history as well as I do, but we've got commitment in the UN system to a universal social protection floor 10 years ago. And yet the progress we've seen is so narrow that indeed 50% of 55% of the world's people have no social protection, no resilience, no foundations for any sense of security. And when you marry that with informal work and you're not making money every day, there's 1.6 billion of our brothers and sisters facing destitution potentially on a daily basis.  So it's an inadequate base of social security. We have to separate indeed the classifications of work. But let me say this, if you look at the ILO declaration, we can put a floor of decent work under every worker irrespective of the contract, the nature of the employment contract. I've very firm views about technology can't be used for exploitation. And indeed last session in this discussion we guy writer and I pointed out that indeed some of the new work looks like work of the last century before it was cleaned up with employment regulation. That doesn't mean everybody works in the same way. And technology is changing that although for the bulk of people not so much. So what do we have to do if we're going to build a just future, we have to have a new social contract that is about recovery and resilience and of course jobs are at the center of that decent jobs, climate friendly jobs if we're going to sustain the planet for future generations but we must have universal social protection. And if we're going to build that social protection in the poorest countries, then the only way we will enable that to happen is indeed to start by kickstarting those systems with a global social protection fund. And, you know, when you think that there's many ways of building up such a fund, but even official ODA going to developed economies only 0.4% of it goes to social protection. That's not acceptable and we need to look at the ways through direct funding through liquidity swaps from special drawing rights through debt relief and conditionality around SDGs. But if we don't care as a world about both universal social protection and that's for everybody. And indeed solidarity for the poorest countries, then we are going to see any kind of disaster, climate disaster, pandemics, further pandemics, whatever, knock people out of again a hope of a secure future. When you know those same people are losing trust in democratic systems, when indeed young people now are saying they won't have pensions, they won't have access to health, then something is very wrong. We are three times richer in just the last 20 years and yet we have not put affordable systems in place universally around the world, not just in the poor countries in some of our wealthier countries, but we'll fight those battles. What we have to do is act together for a global social protection fund for the poorest countries. Well, thank you very much to big idea there from Sharon Borough much appreciated we'll discuss that but let's move on now to the UN special report on extreme poverty, Olivier deschutes. So, Olivier, thanks for joining us. How can we move from the current short term relief measures around the globe to a comprehensive approach to universal social protection in this new normal. Thank you. I think this is indeed the exactly the right question. As a matter of fact, for the United Nations. I recently surveyed the approximately 100, sorry 1400 social protection measures that were adopted after the COVID-19 crisis in the field of social protection, covering some 200 countries. What was really impressive was that although that figure seems very high and very encouraging only 600 billion US dollars were committed to social protection during the crisis, which is a very small portion of the 13 trillion dollars that have been injected in the government's in order to stimulate the economy recovery, but more importantly, many of these 1400 social protection measures were actually leaving many people behind. The findings from the study I made of those measures can be summarized as follows first many of these measures were temporary stop gaps and were improvised in order to respond to the crisis in other terms countries were taken off guard. They were not prepared for this. And many of the support schemes, whether in the form of cash transfers to poor households or in the form of unemployment benefits under temporary unemployment schemes have been phased out after three or four months. Secondly, many people have been left out workers from the informal economy workers in precarious employment positions. Together, these categories represent some 2 billion workers, 61% of the global workforce with weak or no access to social protection, migrant workers, people who are just too poor to fill in the forms, people who do not have access to documentation, people who cannot fill in online information because they have no access to Internet, for example, have been left out from quite a few of these schemes. And the, and the third finding is that very often these schemes that were designed in the midst of the crisis, although of course it's very encouraging to see that governments have reacted were designed without any participation of the social partners. In fact, the, the ITUC, the International Trade Union Confederation has examined some 95 countries, examining how they adopted social protection schemes, and only in 49 of these had unions being consulted. So I think it's very important that we learn the lessons from what happened and what we learn is that we are ill prepared for the next economic climatic, or indeed sanitary shock and we must build much more robust social protection course on the basis of a rights based approach as quite eloquently described by Sanya Nishta at the start of this panel. What this means, building on the rights based approach means we have to provide people with entitlements they may claim in order to be granted the benefits they have a right to that is empowering to them. It means we have to organize social protection in ways that protects everyone and not selectively or in a way that is only targeting the ultra poor people, and it means that we need to ensure the participation of social partners in the management of social protection schemes. And yes, one important part of what we must do is support low income countries, perhaps developing countries as a whole to help them overcome the financing gap today that they experienced, making it difficult for them to invest in social protection. And quite frankly, this is affordable. The latest study we have on the financing gap shows that to cover all low income countries, social protection floors, we would need some 79 billion US dollars per year. That may sound a lot, but it's actually about one half of the total official development assistance already provided by OECD countries by which countries to developing countries, and it's 0.15% of the gross national income of rich countries. In other terms, it's entirely affordable for us in the OECD countries to provide this encouragement to low income countries to invest in social protection for us to cover all the population, all women, children and men throughout their lives. And if we make this a political priority, and that shall be indeed for the next few months. And my main objective is to convince governments and international agencies to join in this worldwide effort to support poor countries efforts to strengthen social protection for us. Great. Thank you very much for that. We'll move on to Mike Mansfield from the private sector from the insurance company, age on Mike is the program director of the Center for Longevity and Retirement. Mike, what is the role of the private sector in this type of world, both in terms of what's happened during the pandemic and what might happen and what changes might occur on a long term basis. Thanks guys and thanks for the opportunity to speak today I think we can we can build on some of the points that have already be made by looking at the private sector as being one of the key partners in the contract that we've created to support social protection for people when they need it during their life and as they move into retirement. I think employers together with other social partners like governments and individuals need to work together to develop more sustainable systems that rebalance responsibility in a way that everyone has the ability to retire with dignity and that nobody has no one is left behind. And for the past three years we've been writing retirement research reports calling for a new social contract, recognizing the fact that our current system is under severe financial strain and responsibility is shifting to the individual. The report we brought out earlier this year is looking at what employers can do to create age friendly workplaces and help people prepare for retirement. And I think there are three areas where the employers and you can private sector can help in that area. And these are our results of the long term relationship that many people have with their employer. And they offer opportunities for people to prepare for the future and enable them to become more resilient and support our social protection systems for the long term. And the first area is in providing access to retirement savings arrangements. And here I think the encouraging news is that over half so 52% of the workers in our survey said that they were offered a retirement plan by their employer. And this creates a great opportunity for workers to save by opening them up to all workers, workers in the gig economy and so forth, and making them portable and people move jobs, employers can make sure that retirement savings becomes the norm. Employers really have a home game advantage here in that 41% of the workers in our survey said that they started to save for retirement based on a nudge from their employer employment related reasons. So employers can really play a role in helping people on the journey to saving for retirement. So the second area I think in making people more resilient is creating an environment that supports workers in making healthier choices easier and helping them take better care of their health. So even small steps like providing healthy food and snacks in the office were found to have great appeal by almost a third of the workers in our survey. And the third area I think builds on something that we've spoken earlier about, and that's offering opportunities for continual learning and development for people throughout their career. And in an increasingly fast paced world, where the job market is becoming increasingly uncertain, many employers may not be able to guarantee employment for people throughout their careers, but they can help them stay employable, which is why it was disappointing to see that just over one in four workers in the survey said that they didn't receive training to help keep their skills up to date. I think this is a great opportunity for companies to be able to help out and it's a women opportunity. So I think that there's three areas just cognizant of time, where I think employers in the private sector can help make things more resilient in the future. Great stuff. Thank you very much, Mike. Much appreciated. I was going to check whether I'm on mute or not with this. Well, that's a great discussion that we had new insights from emerging economy from the minister about how to create a system of handing out cash to half a country's population from gene on the gig economy. The big idea there from Sharon borough about the global social protection fund. A price tag put on something like that from Olivier the shooter of $79 billion a year and support that will be needed for developing countries. And then Mike Mansfield telling us about the role of the private sector in trying to manage some of the change of this extraordinary time and trying to create some sort of opportunity for sustainable and sustained change. So thank you very much to everybody that joined us on the live stream.


to the best layout in general, so it's pretty good at that with our normal reports. You're not usually going to get a report that's going to have like the columns on the right printed on another page because it has the smart page breaks and the report, the smart page breaks in essence are doing that. So within here, you could email it within here, but you probably don't want to email them one at a time. You'd like to email one email with all the attachments, otherwise it's going to get tedious. You can save it as a PDF or you can print it. So clearly if you wanted to print it, then you can manually print it from a printer here. I'm going to just save it as a PDF. So I'm going to say let's save it as a PDF. I can also hit the drop down here and export to a PDF and then just say save as PDF. There it is up top. And so let's go ahead and open up my report. So over here, I have it open twice. I think let's open up my reports and then I'm just going to drag and drop drag and drop. So drag and then I pulled it into the reports and it was scared. But now it likes it there. It's at home with the balance sheet reports. It's 4.0 or just like four. So I'm going to number it. So there's that one. And let's do the next one tabbing to the left. I'm going to right click and duplicate it, pull that to the right. And then we'll open up the income statement, the second income statement report. And let's go ahead. This is going to be our income statement vertical analysis. I will then say that we want to export to PDF, save as PDF, drag it into where it needs to go and drag. And pulls and dropped it into the just dropped it down a thousand miles into my report here. But don't worry. It survived the fall because these reports are tough. This is going to be a 5.0 and our five point. And then I'll say then the next one is going to be one more time. Uno base moss, right click, duplicate, pull it to the right. And this one I'll close up the hand boogie and open up the income statement. And this is going to be income statement by quarter. And let's just do the same thing here. So I'm going to say let's say that we're going to export as a PDF. And we're going to say save as a PDF. And then I'll pull that one over. Boom. And I'll just rename that one. Rename, right click, number six. So you can see now the numbering system having numbers on them might make it more likely that it'll be in the order that you want to present it in. From this point, of course, you could then email it to someone attaching these to an email, but we're getting a lot of reports here. So it would probably be nicer to zip the file if you're doing it by email. Or you can possibly use a cloud drive, provide these on the cloud drive like Amazon OneDrive or Google Cloud or something like that. If you were to email it, you'd have to I would suggest zipping it like we did before. So if you put them all into a folder, right click on the folder, then you can compress to a zipped file. So I cannot attach this folder all as one folder, but I can attach the zipped file, which is nice. So the next thing that we could do if we want them on one PDF, then that would be even cleaner possibly. So if I go to this first page, one way is that we can use the management report that we talked about before. I won't do it in detail here just to give you the idea again. Though if I if I say that I want to duplicate this report so I don't overwrite the default and then I'm going to edit it. Then we can go into where our reports are located, which are which are down here instead of the standard profit and loss and balance sheet. I'm going to add the new reports, which would be the I'm looking for this, not the standard, but the custom area. So there's the custom report number four, and then boom, I'm going to add another one to do it to do it and then it's going to do it. And I'm going to say custom report number five, boom, add another one to do it and then custom report number six, and then I can delete these two, delete, delete. And then I'll edit the name so it doesn't show as month end report. So I'll just say edit and get rid of that first bit. So it doesn't show on the table of contents with the month end report there unless you want it to might not be too much of a problem. Talvez possibly. And so we'll then remove this one and then to do to do it and then we'll remove this one. So there's that. And so then if I was to just to check that out, just what we've done thus far, and we can preview it down here. Preview. Wait, wait a second. It removed it now. What in the world? Let's go ahead and do this again. And then I'll edit it and then get rid of this bit. I think I did it on the wrong spot or something. Okay. So, okay, I see what I did. I was deleting it up here instead of down here. So now I fixed it. I think I have it now. So now let's preview the report.  Look at the look of it is a little bit different. You can see in here versus what we had over here. So this one looks like this versus this one looks like this. There's a little bit of color in it. So it's nice, but it's basically the basic color scheme still. So you can't really do too much else with the color scheme, but you can get them all on one report. It's pretty easy to do that. Once you do that a few times and it gives you a nice header page that you can use. And that's great. If you wanted to further adjust anything in these reports, like the font or to just some of the titles of things, for example, then you could use Excel. The next format would then be to export these reports to Excel, put them on one Excel file in different tabs, and then you can further do a little bit more adjusting of them and customize it even further. And we'll talk about that a little bit more in a future presentation.


Welcome to this course on partial differential equations. In this lecture, we are going to discuss some basic concepts and the nomenclature that is involved in the study of partial differential equations. So, the outline of today's lecture is first we discuss certain preliminaries, then we go on to define what is a PDE and what we mean by solution of the PDE and then we discuss classification of partial differential equations. So, study of partial differential equations involves functions of several variables, theory of differentiation and integration for such functions. Therefore, let us discuss what are the domains of those functions are going to be. So, we consider functions defined on an open subset omega of rd. Since we are studying partial differential equations, functions are of several variables, therefore, d is greater than or equal to 2. We use the notation omega d to remind us that the set is contained in rd because later on we are going to see that in a certain problem we will see many sets one is lying in higher rd, higher dimension and one is lying in lower dimension. Therefore, this will keep track or it will remind us that we are in certain dimension. So, an element of rd, x is written in bold phase, it is not difficult, it is not easy for us to write on paper, but in print it is possible. So, the bold phase x actually stands for d tuple of real numbers x1, x2, xd. We may also use normal x, x in rd and there should be no confusion because the moment you know you are dealing with functions in rd and you have a function let us say u of x, it means x is a d tuple. So, writing x takes more time and effort than writing a very simple x. So, when working with functions defined on subset of r2, elements of r2 will be denoted by xy, that is the standard product it takes and we will not use x1, x2, we stick to the standard notation xy. Similarly, when we are working with the functions defined on r3, we use the notation xyz, the 3 tuple is denoted as xyz instead of x1, x2, x3. And in the context of PDEs, the variables x1, x2, xd these are called independent variables. Now, we have discussed the domains of functions, let us discuss the core domains of the function that we are going to deal with. So, core domains of functions means the function where the values it takes. So, we assume that the functions take values in rp, p greater than or equal to 1. That is we consider functions u defined on omega d, I need not tell you that omega d is a subset of rd again because the notation tells us and takes values in rp. Thus, the function u consists of p real valued functions defined on omega d and we use this notation u equal to u1, u2, up. So, the function u is given in terms of the sometimes what is called coordinate functions u1, u2 etc up to up. In partial differential equations u1, u2, up are called dependent variable or unknown functions. In our course, we are going to deal with PDEs which have only one unknown function. That is only one dependent variable in other words p equal to 1. So, the PDEs that we are going to study in this course are going to be a single equations. There is only one unknown function not systems. Some more notations let u be a function defined on omega d taking real values. Then first order derivatives the notation we use here is du. It denotes all the first order partial derivatives which are described here dou u by dou x1, dou u by dou x2 up to dou u by dou xd. Then the second order derivatives are denoted with d2u. This is a notation d2u stands for all second order partial derivatives of the unknown function u which is dou to u by dou xi dou xj i and j vary from 1 to d. Now in general dku for k greater than or equal to 1 denotes all kth order partial derivatives of u. So, recall that the notation dku stands for the collection of all kth order partial derivatives. However, we need a notation for any specific kth order partial derivative. Therefore, we are going to introduce that notation to do that what we do is let us fix a d tuple of numbers which are either natural numbers or 0. That is alpha is a vector which is equal to alpha 1 up to alpha d. It is a d tuple. Where does it belong? N union 0 power d that means each of them is a whole number either a natural number or 0. For such an alpha of d tuple of numbers whole numbers we denote mod alpha equal to sum of the alpha i's.  So, the notation d alpha u means I am going to differentiate u with respect to x 1 alpha 1 times with respect to x 2 alpha 2 times and so on up to with respect to x d alpha d times.  Let us see an example assume that d equal to 4 that means we are dealing with functions of 4 variables. So, the independent variables are denoted by x 1, x 2, x 3, x 4. Let us take alpha equal to 0, 2, 1, 3 and we want to differentiate you this alpha d alpha u. What is that? It tells us that do not differentiate with respect to x 1 with respect to x 2 differentiate 2 times with respect to x 3 differentiate 1 time and with respect to x 4 differentiate 3 times. So, sum of the entries of alpha is 6, 0 plus 2 plus 1 plus 3 that is 6. That means what it defines is a particular 6th order partial derivative of u. So, d alpha u is dou 6 u by dou x 2 square because the second entry here is 2 that is why second variable differentiate 2 times. Third entry here is 1 therefore differentiate once with respect to the third variable and similarly with respect to x 4 variable differentiate 3 times because there is a 3 here. So, there is a room for confusion because we are using d k u as an at the top of the slide to stand for collection of all kth order partial derivatives and here also d alpha u to denote this particular partial derivative. But there should be no reason to be confused because this d k u stands or makes sense only when k is a natural number. This d alpha u makes sense only if alpha is a d tuple of whole numbers. So, there is no room for confusion here. So, the order in which partial derivatives are taken does not matter. For example, if I want to differentiate with respect to x 3 first in the above example and with respect to x 2 later there is no way to take care of that in this notation. In other words this notation does not care the order in which you take the differentiation which is not okay. We know that partial derivatives do depend the order in which they are taken but for reasonable functions they coincide. So, we are going to consider such functions only where the order in which you differentiate does not matter okay. For example, if you are considering a c 2 function then it does not matter in which order you differentiate that is a theorem in multivariable calculus. Maybe that is the reason why this notation does not respect the order in which you will differentiate. Now let us go to the definition of a partial differential equation and what do we mean by its solution. One normally says PDE is an equation involving partial derivatives. This can be accepted as a definition provided we understand this statement. Not only that everybody understand this statement in the same way. Question what do we understand by the above sentence right. We are saying involving partial derivatives. So question is how many derivatives finitely many or infinitely many is also allowed. There is one question there. Second question is what does the equation involving stand for what does that mean? No idea. However we seem to be knowing examples of PDE is therefore we do not question this kind of a definition if found we do not question because we know what our PDE is. We know okay. So how do we know? We have examples and this English statement seems to be okay but the problem is this English statement might allow more types of equations which you do not think is a PDE. We will see one of them as an example okay. Therefore these are the example that we seems to be knowing dou u by dou x plus dou u by dou y equal to u, u dou u by dou x plus dou u by dou y equal to 0 and another equation dou 2 u by dou x square plus dou 2 by dou y square equal to sin x plus cos phi and so on many such examples we know. And the above sentence PDE is an equation involving partial derivatives seems to be acceptable okay. Therefore there seems to be an answer to the first question. That means how many derivatives should be there in equation because whenever we are writing an example of PDE we are not thinking of writing an equation which involves infinitely many derivatives. Therefore we know that a PDE should have only finitely many derivatives appearing in the equation. What about second questions answer? Still no idea. So we need to capture what we are writing here in terms of some definition which is very rigid okay. So in mathematics as we know definitions are set in stone and vaguely worded sentences are not allowed as definitions such as this PDE is an equation involving partial derivatives. So we reformulate the above sentence or rather we give a mathematical meaning of this sentence or a possible mathematical meaning of the above sentence. And once we write down as a definition we stick to that that is a definition okay. So we understood that PDE will have only finitely many derivatives in it. Therefore find out the order of the derivatives which appear in the equation look at the equation ask which is the derivative appears and let m be the largest having the above property. Being the largest it will be unique and this largest derivative which appears largest order derivative which appears in this that particular order is unique and that is called the order of the partial differential equation. So now we need to define a PDE may be defined as an equation satisfied by all partial derivatives up to order m including the function as well okay. So a relation of this type capital F of x u du d2u dmu equal to 0 recall what is du collection of all first order derivatives d2u collection of all second order derivatives and so on this is collection of all mth order derivatives. So for some m and some function f we should be able to cast our equation in this form now where x is in omega d questions what is m what is the domain of f so m is the order of the PDE okay. So this is how we plan to define we can if you have a PDE which involves only first order you can also think I can write this equation with m equal to 2 but such a thing should not be allowed our definition will not allow because this m we would require that m should be the order of the PDE as identified on the last slide. m is the order of the PDE means some mth order partial derivative appears and no higher order partial derivatives appear in the equation that is why that m is unique. And now once you know this it must be easy to write down the domain of f once you see the equation it is easy to write let us look at some examples examples it is always easy okay. So for this PDE dou u by dou x plus dou u by dou y equal to u m is 1 that means first order PDE. So first order PDE means it has to be expressed as x y u du equal to 0 du means all first order partial derivatives which are dou u by dou x and dou u by dou y. So it should be written in this form what is f? f should be some function of how many variables 1, 2, 3, 4, 5 f should be a function of 5 variables. So this is always a good practice to write any such function like this f is f of x y z p q okay x y is independent variables it means we are working with a PDE in 2 independent variables here z is the location where you are going to substitute u p and q are the places where you are going to substitute for dou u by dou x and dou u by dou y respectively okay it is important to stick to this kind of notation understand this notation.  Now larger domain on which this f is meaningful is r 5 because it is defined for every 5 to pull x y z p q this function is defined. However we may also consider the domain to be any open subset of r 5 that is also okay. Observe something nice about this f it is a linear polynomial in this variable p q and z of course it does not x y does not appear therefore it is still you can say it is a linear polynomial in all the variables okay. Let us look at second example it is dou u by dou x plus dou 2 u by dou y square minus dou 2 u by dou x dou y equal to cos of y by x. What is the m here it is 2 because second order derivative appears and higher order derivatives other than 2 are not appearing only 2 okay 3 4 5 etc nothing appears so m equal to 2.  Therefore f must be written like this I told x y z is a place we are going to put u p q for the first order derivatives or s t for the second order derivatives. I have included s and s tilde just a placeholder for dou 2 u by dou x dou y and dou 2 u by dou y dou x. As I mentioned earlier we are not going to really distinguish between these 2 so after sometime we will not write s tilde we will simply write f of x y z p q or s t s tilde will be forgotten we will not write that because we will maybe we will consider functions for which both are same for reasonable class of functions this mixture partial derivative is coincide and what is the formula for f one can easily write down it is this. Now we will ask what is the domain of f then there is a problem here because x equal to 0 should not be allowed everything else is allowed and what is the domain of f it is which how many tuples tuple of what size 2 here 3 plus 2 first order derivatives and 4 second order derivatives so 2 plus 1 that is 3 plus 2 5 plus 4 9. So domain of f will be in r 9 but it should not contain x equal to 0.  Note that the given p d is meaningful for those x y is in r 2 for which x is not 0. So if we denote the projection of omega 9 to the first two coordinates by omega 2 then omega 2 is an open subset of r 2 which does not intersect the y axis thus we may write the domain omega 9 to be like this f is from omega 2 cross r cross r 2 cross r 4 this omega 2 stands for the independent variables x y vary here this r stands for the unknown function u which we do not place any restriction on what values it must take so therefore it can take any real number r 2 stands for the derivatives the p q variables where dou e by dou x and dou e by dou y are going to come r 4 is all 4 second order partial derivatives. So we can further write omega 9 in this form now dou f by dou s is not 0 in fact dou by dou t is also non-zero it means f is not a constant function in s it means s variable is used by f it is not constant it depends on s it means it is a second order p d this guarantees that some second order partial derivative appears in the equation.  So a system of partial differential equations why system because we are considering q greater than or equal to 1 so there can be more than one equation that is why in general we call it is a system of partial differential equation order m because we have d power m here is defined by this equation okay so d m u and all mth order partial derivatives which are so many okay I am not discounting the symmetries like earlier I have done I am writing full okay so we are at least one of the mth order partial derivatives of u appears in the system of equations only then we are going to say it is a mth order p d e so as I said if q is bigger than 1 it is actually system of p d e is q can be equal to 1 then it is a single p d e or a scalar equation and if p equal to 1 it is a single unknown function if p is bigger than 1 unknown function is vector value. So now we have to define what is the solution of the p d e a solution of the system of p d e which is defined here is a function solution is a function when we say function with some domain so it is a function phi with domain u of course here we have written system for vector variable dependent variable is a vector value therefore this phi is also vector value r p with what property u is a subset of omega d what is omega d omega d is that set where this equation is meaningful for x belongs to omega d the domain of f was omega d cross some things but a solution can be a function which is defined even on a proper subset of omega d it is a function with some properties what should be the properties all partial derivatives of phi up to order m should exist including m because I am thinking of a mth order equation therefore all partial derivatives should exist up to order less than equal to m. Second once it exists look at this tuple I am going to substitute inside this x phi x d phi x d 2 phi x d m phi x therefore I would require that this tuple belongs to the domain of f and I have the right to apply f to that when I apply I get 0 that means this equation is satisfied in this way by this function phi not phi we are demanding that it should be defined on a subset of omega d and we are not demanding that it should be defined on omega d that is very important to be noted. So, we do not insist that a solution must be defined for every x in omega d you may be proper subset of omega d this should be omega d now answer the following questions decide which are the following are PDEs write down capital F as in the definition how do you say somebody is a PDE you have to go to the definition of the PDE that involves a capital F it comes with this domain right you should be able to give such a function f and then this becomes a PDE.  So, this circle stands for the composition of functions now at this point you must stop the video write down you think about the answers write down the answers if you have and then proceed because answers going to come on the next slide. What are the answers?  Think in your leisure time if you do not get this answer do not worry keep thinking about this question motivated by the following example of Arnold in ODE is d y by dx equal to y circle y he says it is not a ODE. So, the vague definition would allow this as a PDE because it just says it should involve unknown function and its derivatives which this equation does involve U x plus U y equal to U circle U. So, there is no point in admitting such equations if you cannot have a general theory for some special class of such equations. So, maybe that is the reason why Arnold called this as not as a ODE. So, if you understand the answer to this the answer of this will be immediate. Now, we move on to classification of PDEs. So, what is classification classification of PDEs means creating banks with labels and associate each PDE to exactly one of the banks. Examples of a few methods of classification we are going to see each one of them based on the number of equations, based on the order of PDEs, based on algebra I will mention this details when we come there. So, question is it important to classify. Recall that a complete theory is available for solving systems of linear equations A x equal to b and the same is not the case with transcendental equations which are also known as non-linear equations. You have to hold each and every equation and analyze for solutions. Now, coming to linear ODE is there is an oscillatory behavior of solutions that is different from first order ODE and second order ODE. Look at simple equation like y dash equal to y and y double dash plus y equal to 0 or y double dash equal to y. Think of the solutions some of them are oscillatory. So, it changes. So, it is important to identify a reasonably interesting or useful class of PDEs for which it is possible to obtain a larger understanding possible is possible may not be complete, but at least a larger understanding good understanding. So, first order PDEs we are going to see that classification leads us to uncover certain aspects of solutions starting from the quasi-linear to the general non-linear equations. We will see that in the next chapter. So, we will come back to this discussion on classification later on also in chapter 3 where we are going to classify second order PDEs we ask the same question and we have some better answers there we will see that. So, now a classification based on the number of equations recall that a PDE is defined through a function of this type. If q is bigger than 1 it is called system, if q is equal to 1 it is called scalars. So, this is an example is a Maxwell system of equations in 3 independent variables x, y, z and a time variable that means there are 4 independent variables here and e is 3 vector even e to e 3 functions b is 3 functions b 1 b to b 3 and dou e by dou t equal curl b dou b by dou t equal to minus curl e divergence of b and divergence of e are 0. So, this is indeed a PDE you must write in this setup and that is left as an exercise to you. As I told you when q equal to 1 that is called a scalar PDE but when q equal to 1 we also take p equal to 1 that means the unknown function also is a single unknown function. For this otherwise what happens is that you have one equation because you said q equal to 1. If you have more than one unknown it means that information is I mean conditions are very less. For example, just recall what happens with one linear equation two variables not so much interesting right I mean x minus y equal to 1 infinitely many solutions that is one of the aspects okay. So, this is an example of a single equation it is a very important example we will be studying a depth later on. Now, second thing is the classification based on the order of the PDE. So, order of PDE as we know is uniquely defined and hence they may be classified based on their orders. This is a first order PDE this is a third order PDE. Now, this is the important classification which uses algebra. So, this classification is based on the manner in which the unknown function and its derivatives appear in the equation. What are the possibilities? First possibility dependent variable and all its derivatives whichever appear in the equation they all appear as a linear combination with coefficients which are functions of only independent variables that equation such equations which feature which has this feature or will be called linear equations. Second option is highest order partial derivatives appear as a linear combination with coefficient which are functions of only independent variables and no conditions on the appearance of low order derivatives and the function u. This may be called semi-linear equations. Now, highest order partial derivatives appear as a linear combination but now the coefficients depend on independent variables and also on the lower order derivatives including u. So, that will be called quasi-linear equations. The highest order partial derivatives do not appear as a linear combination at all. So, such equations will be called non-linear equations. Now, we may ask this question why only highest order derivatives are important for the last three classifications. They play a major role in determining properties of functions is a vague answer. You should be in a position to appreciate this at the end of this course. Now, let us look at these equations these two equations. One is the first order PDE and this is the second order PDE. What is common between them is that u and du in the first equation, u, du and d2u in the second equation they appear linearly. It is a linear combination of those derivatives and the function. X and Y we do not ask any questions. The classification is based on ZPQ variables. In this for the second order PDE it will be ZPQ or SS tilde t. So, we will see very specific examples for first order PDEs, the classifications and we will see most general possible equations that we can write of that type. So, linear first order PDEs means the unknown function and the first order derivative should appear linearly that is coefficients are allowed to depend only on independent variables.  So, this is an example, semi-linear. Here what we ask is dou by dou X, dou by dou Y should appear linearly. Coefficients are allowed to depend only on the independent variables and u may appear in any way but not in the coefficients of dou by dou X and dou by dou Y. So, therefore the most general possible semi-linear equation of first order is like this. This is an example. Now comes the quasi-linear first order PDEs. Here the first order derivatives dou by dou X and dou by dou Y should appear linearly, fine. Coefficients are now allowed to depend on the independent variables as well as the lowest lower order derivatives. If this is the first order derivative what is the lower order derivative that is function itself, they are allowed. So, therefore this is the general possible quasi-linear first order PDE. This is an example. Now a PDE which is not quasi-linear is called a fully non-linear PDE. So, it is basically the complement of quasi-linear equations. This is an example, very important example, it is called Ikanl equation. Now this is the picture or Venn diagram, first is linear. This is the formula how the general linear PDE looks like. Linear is also semi-linear, that is a bigger class. Bigger class is quasi-linear. Out of quasi-linear will be full PDEs, this color in this color, red color but not in green will be fully non-linear. Otherwise these are all general first order PDE will look like this. That was the definition. Now for a linear PDE, first order PDE, F will be of this type. F of X, Y, Z, P, Q will be like this. This is for semi-linear and this is for quasi-linear. Now we can ask the questions like which quasi-linear PDEs are semi-linear. When this A does not depend on Z, B does not depend on Z, a quasi-linear equation becomes semi-linear equation. Similarly, this is another question, please answer yourself. Now we can classify the m-th order PDEs. We have explicitly seen first order PDEs classification but same thing can be done for m-th order PDEs. What is the guiding principle? The 1, 2, 3, 4 that we have written down. How unknown function and its derivatives appear? If all of them appear as a linear combination, it is called linear PDE. If that is not the case, we straight away go for highest order derivative and ask how they appear. Then we had 2, 3, 4. So we will write down here once again. Linear, see here, mod alpha less than or equal to m. So when alpha equal to 0, 0, 0, 0 that is m equal to 0, the 2-pull alpha is 0, 0, 0 that means no derivatives which means it is a function. So here it says function as well as the derivatives up to order m appear as a linear combination. Coefficient depend only on the independent variables. Semi-linear, once again highest order partial derivative that is m-th order appear linear combination. Coefficient are functions of independent variables. Rest however it appears no restriction. Quasi-linear, once again it is on the highest order derivatives. Coefficient are allowed to depend on apart from independent variable x, Udu up to dm-1 that is m-th order derivatives. Then this is called quasi-linear. Outside that it can appear in any way they want. But as coefficients are the highest order partial derivatives, they can only be like this. Now PDE which is not quasi-linear is called fully nonlinear PDE. So Laplace in U, its second order and linear. E power Laplace in U equal to 1. It is second order fully nonlinear. And this equation third order semi-linear. Of course you may also say quasi-linear but when I say semi-linear it has more information because semi-linear much smaller set. This equation second order fully nonlinear. So this is an exercise for you through examples show that the following inclusions are very strict inclusions. That means find a semi-linear PDE which is not linear, find a quasi-linear which is not semi-linear and show that there are PDEs which are not quasi-linear. That means given example which is not quasi-linear PDE. These are the questions for you and thank you.


When AWS introduced the modern cloud in 2006, many people didn't realize the impact that it would have on the industry, but some did see the future of an as-a-service economy coming. I mean, SaaS offerings came out several years before, and the idea of applying some of these concepts to infrastructure and simplifying deployment and management kind of looked enticing to a lot of customers. And a subscription model, or better yet, a consumption model was seen as a valuable proposition by many customers. Why not apply it to infrastructure? And why should the hyperscalers have all the fun? Welcome to At Your Storage Service. My name is Dave Vellante. And as an analyst at the time, I was excited about the as-a-service trend early on. And one of the companies that caught my attention back in the beginning of last decade was Pure Storage. Pure not only was delivering cloud-like simplicity, but it's no forklift approach to infrastructure was ahead of its time. And that's why we're here today, to dig into what's happening with the as-a-service trends that we see popping up all over the world. Today we're going to dig into three sessions with noted experts in the field. First, Prakash Darji is the general manager of the Digital Experience Business Unit at Pure Storage. He's going to join us. And then we bring in Steve McDowell. Steve's a senior analyst for data and storage at More Insights and Strategy, a well-known consultancy, an analyst firm. And finally, we close with Emile Stam. Emile is the chief commercial officer and chief marketing officer at OpenLine. OpenLine's a managed service provider. They serve the mid-market. And Emile's got a very wide observation space. He's going to share what he's seeing with customers. So sit back and enjoy the show. The cloud has popularized many useful concepts in the past decade, working backwards from the customer, two pizza teams, a DevOps mindset, the shared responsibility model and security, of course, the shift from CAPEX to OPEX, and as-a-service consumption models. The last item is what we're here to talk about today. Pay for consumption is attractive because you're not over-provisioning, at least not the way you used to. You'd have to buy for peak capacity events. But there are always two sides to every story. And while pay for use more closely ties IT consumption to business value, procurement teams don't always love the uncertainty of the cloud bill each month. But consumption pricing and as-a-service models are here to stay in software and hardware. Hello, I'm Dave Vellante and welcome to At Your Storage Service, made possible by Pure Storage. And with me is Prakash Darje, who's the general manager of the Digital Experience Business Unit at Pure. Prakash, welcome to the program. Thanks, Dave. Thanks for having me. You bet. Okay, we've seen this shift to as-a-service, the as-a-service economy, subscription models, and this as-a-service movement have gained real momentum. It's clear over the past several years. What's driving this shift? Is it pressure from investors and technology companies that are chasing the all-important ARR, their annual recurring revenue stream? Is it customer-driven? Give us your insights. Well, look, I think we'll do some definitional stuff first. I think we often mix the definition of a subscription and a service. But subscription is, hey, I can go for a pay-up front or pay as I go. Service is more about how do I not buy something just by the outcome? So the concept of delivering storage as a service means, what do you want in storage? Performance, capacity, availability? Like that's what you want. Well, how do you get that without having to worry about the labor of planning, capacity management? Those labor elements are what's driving it. So I think in the world where you have to do more with less and in a world where security becomes increasingly important, where standardization will allow you to secure your landscape against ransomware and those types of things, those trends are driving the classification of storage and the only way to deliver that is storage as a service. So that's good. You may be thinking about it differently than some of the other companies that I talked to. But so you've made inroads here, pretty big inroads actually, and changed the thinking in enterprise data storage with a huge emphasis on simplicity. That's really Puerh's raison d'etre. How does storage as a service fit in to your innovation agenda overall? Well, our innovation agenda started, as you mentioned, with the simplicity a decade ago with the evergreen architecture. That architecture was beyond the box. How do you go ahead and say, I can improve performance or capacity as I need it? Well, that's a foundational element to deliver a service because once you have that technology, you can say, oh, you know what? You've subscribed to this performance level. You want to raise your performance level. And yes, that'll be a higher dollar per gig or dollar per terabyte. But how do you do that without a data migration? How do you do that with a non-disruptive service change? How do you do that with a delivery via a software update? Those elements of non-disruptive updates, when you think SaaS, Salesforce, you don't know when Salesforce does an update. You don't know when they're increasing something, adding a new capability, it just shows up. It's not a disruptive event. So to drive that standardization and classification and service delivery, you need to keep that simplicity of delivery first and foremost, and you can't allow, like, if the goal was, I want to change from this service here to that service here, and a person needed to show up and do a data migration, that's kind of useless. You've broken the experience of flexibility for a customer. Okay, so I like the Salesforce analogy, but I want to jump out and do a little side for a second. So I've got to make some commitment to pure, right? Some baseline commitment. And if I do, then I can dial up and then pay for what I use and I can dial it down. Correct? Okay, I can't do that with Salesforce, right? I could dial up, but then I'm stuck with those licenses. So you have a better model in Salesforce, I would argue. Okay. Yeah, I would agree with that. Okay, so, and I got to pay for everything up front. Anyway, let's go back. I was kind of pushing at you a little bit of my upfront about the ARR model, the all-important financial metric. But let's talk from a customer standpoint. What are the benefits of consuming storage as a service from your customer's perspective? Well, one is when you start your storage journey, do you really know what you need? And I would argue most of the time people are guessing, right? It's like, well, I think I need this. This is the performance I think I need or this is the capacity I think I need.  You find out as you're deploying. So in a storage as a service world, when you have the ability to move up performance levels or move out capacity levels and you have that flexibility, then you have the ability to adjust to meet demand as you deploy. And that's the most important element of meeting business needs today. The applications you deploy are not in your control when you're providing storage to your end consumers. You're gonna want different levels of storage. You're gonna want different performance thresholds. And it's kind of a pay for performance type culture, right? You can use HR analogies for it. You pay for performance. You want top talent, you pay for it. You want top storage performance, you pay for it. You don't, you can pay less and you can actually get lower performance tiers. Not everything is a tier one application and you need the ability to deploy it. But when you start, how do you know the way your end customers are gonna be consuming or do you need to dictate it upfront? Because that's infrastructure, dictating business inflexibility. And you never want to be in that position. I got another analogy for you. It's like, you know, we do a lot of hosting at our home and you know, like Thanksgiving, right? And you go to the liquor store and say, okay, what should I get?   We're gonna get some beers. Should I get bubbles? Yeah, I get some bubbles because you don't know what people are gonna have and so you over provision everything.  So you just over buy. But there's a liquor store that actually will take it back. So I gotta do business with those guys every time because it's way more flexible. I can dial up capacity or I can dial up performance and dial it back down if I don't use it. Or you're gonna be drinking a lot more the next few weeks. Yeah, exactly. Which is not the last thing you want. Okay, so let's talk about how Pure kind of meets this as a service demand. You've touched upon your differentiators from others in the market. You know, love to hear about the momentum. What are you seeing out there? Yeah, look, our business is growing well largely built on what customers need. Specifically where the market is at today is there's a set of folks that are interested in the financial transformation of capex to optics. They're like, that definitely exists in the industry around how do I get a paper use model? The next kind of more advanced customer is interested in how do I go ahead and remove labor to deliver storage? And a service gets you there on top of a subscription. The most sophisticated customer says, how do I separate storage production with consumption and production of storage being a storage producer should be about standardization so I could do policy-based management. Why is that important? You know, coming back to some of the things I said earlier in the world where ransomware attacks are common, you need to standardize security policies. Linux has new vulnerabilities every other day, like find two, three critical vulnerabilities a week. How do you stay on top of it? The complexity of staying on top of it should be, look, let's standardize and make it a vendor problem and assume the vendor is going to deliver this to me. So that standardization allows you to have business policies that allow you to stay current and modern. I would argue in the traditional storage and appliance world, you buy something and the day after you buy it, it's worthless.  The very next day, the car is not worth what it was when you bought it. Storage is the same way. So how do you ensure that your storage stays current? How do you ensure that it gets like a fine line that gets better with age? Well, if you're not buying storage or buying a performance SLA, it's up to the vendor to meet that SLA. So it actually never gets worse over time. This is the way you modernize technology and avoid technology debt as a customer. Yeah, I mean, just even the words you're using and the way you're thinking about this Prakash I think are different. And I love the concept of essentially taking my labor costs and transferring them to Pure's R&D. I mean, that's essentially what you're talking about here. So let's stick with the tech for a minute. What do you see as new or emerging technologies that are helping accelerate this shift toward the as a service economy? Well, the first thing is I always tell people you can't deliver a service without monitoring. Because if you can't monitor something how are you gonna know whether you're meeting your service level obligation? So everything starts with data monitoring. The next step layering on the technology differentiation is if you need to deliver a service level obligation on top of that data monitoring, you need the ability to flexibly meet whatever performance obligations you have in a tight time window. So supply chain and being able to deliver anywhere becomes important. So if you use the analogy today of how Tesla works or a IoT system works, you have a SaaS management that actually provides instructions that pushes those instructions and policies to the edge. In Tesla's case, that happens to be the car. It'll push software updates to the car. It'll push new map updates to the car but the car is running independently.  In the same way, what if you think about storage as something that needs to be wherever your application is? So people think about cloud as a destination. I think that's a fallacy. You have to think about the world in the view of an application. An application needs data and that data needs to sit in storage wherever that application sits. So for us, the storage system is just an edge device. It can be sitting in your data center. It can be sitting in a equinex. It can be sitting hosted and MSP can run it. It can even be sitting in the public cloud. But how do you have central monitoring and central management where you can push policies to update all those devices? Very similar to an IoT system. So the technology advantage of doing that means that you can operate anywhere and ensure you have a consistent set of policies, a consistent set of protection, a consistent set of prevention against ransomware attack regardless of your application, regardless of where it sits, regardless of what content you're on. That approach is very similar to the way the IoT industry has been updating and monitoring edge devices. Nest thermostats, Tesla cars, those types of things. That's the thinking that needs to come to storage and that's the foundation on which we built Peer as a Service. So that implies, or at least I infer that you've obviously got control of the experience on prem but you're extending that into AWS Google Azure which suggests to me that you have to hide the underlying complexity of the primitives and APIs in that world. And then eventually, actually today because you're treating everything like the edge out to the edge, maybe mini pure at some point in time. But so I call that super cloud, that abstraction layer that floats above all the clouds on prem and adds that layer of value. And as a singular experience what you're talking about pushing policy throughout is that the right way to think about it and how does this impact the ability to deliver true storage as a Service? That's absolutely the right way of thinking about it. The things that you think about from an abstraction kind of fall in three buckets. First, you need management. So how do you ensure consistent management experience creating volumes, deleting volumes, creating buckets, creating files, creating directories like management of objects and create a consistent API across the entire landscape. The second one is monitoring how do you measure utilization and performance obligations or capacity obligations or policy violations wherever you're at. And then the third one is more of a business one which is procurement because you can't do it independent of procurement meaning what happens when you run out. Do you need to increase your reserve commits? Do you want to go on demand? How do you integrate it into companies procurement models such that you can say, I can use what I need and it's not like every change order is a request of procurement that's going to break an as a service delivery model. So to get embedded in a customer's landscape where they don't have to worry about storage you have to provide that consistency on management monitoring and procurement across the time. And yes, this is deep technology problems whether it's running our storage on AWS or Azure or running it on Pram or at some point in the future maybe even I'm pure mini at the edge, right? So all of those things are tied to our piracy service delivery. Yeah, technically non-trivial but hey, you guys are on it. We got to leave it there Prakash, thank you. Great stuff, really appreciate your time. All right, thanks for having me man. You're very welcome. Okay, in a moment, Steve McDowell from more insights and strategies is going to give us the analyst perspective on as a service. You're watching theCUBE, the leader in high-tech enterprise coverage. Why are customers making the change to pure as a service? Other vendors offering flexible consumption models will promise you the world. On the surface, it's just what you need but then you notice the asterisk that dreaded fine print that turns just what you need into long-term commitments, disruptive upgrades and unpredictable costs. Pure storage launched pure as a service to provide the flexibility to respond to your ever-changing needs with clear per-unit costs, no large upfront purchases and no asterisks. A usage-based model should be simple, innovative and adapt with the changing market. Unlike other vendors, pure is offering exactly that with options for service tiers and short-term contracts in a single unified subscription that allows you to improve your discounts over time. Pure makes sure you can grow and upgrade without ever taking your environment offline and without the constant worry of hidden costs with complete billing transparency unlike any other you only pay for what you use and pure one helps track and predict demand from day to day making sure you never outgrow your storage. So why are customers making the change to pure as a service? Convenient solutions with unlimited potential without the dreaded fine print. It's as simple as that. We're back with Steve McDowell, the principal analyst for data and storage at More Insights and Strategy. Hey Steve, great to have you on. Tell us a little bit about yourself. You've got a really interesting background and kind of a blend of engineering and strategy and what's your research focus? Yeah, so my research, my focus area is data and storage and all the things around that or whether it's on-prem or cloud or software as a service. My background, as you said, is a blend, right? I grew up as an engineer. I started off as an OS developer at IBM. I came up through the ranks and it shifted over into corporate strategy and product marketing and product management. And I've been doing, working as an industry analyst now for about five years at More Insights and Strategy. Steve, how do you see this playing out in the next three to five years? I mean, cloud got it all started. It's kind of snowballing, you know, however you look at it, your percent of spending on storage that you think is going to land in as a service. How do you see the evolution here? IT buyers are looking at as a service consumption-based as a natural model. It extends the data center, brings all of the flexibility, all of the goodness that I get from public cloud, but without all of the downside and uncertainty around cost and security and things like that, right? That also come with a public cloud. And it's delivered by technology providers that I trust and that I know and that I've worked with, you know, for in some cases, decades. So I don't know that we have hard data on how much adoption there is of the model, but we do know that it's trending up, you know, and every infrastructure provider at this point has some flavor of offering in the space. So it's clearly popular with CIOs and IT practitioners alike. So Steve, organizations are at a, they're different levels of maturity in their transformation journeys. And of course, as a result, they're going to have different storage needs that are aligned with their bottom line business objectives from an IT buyer perspective. You may have data on this, even if it's anecdotal. Where does storage as a service actually fit in and can it be a growth lever? Again, absolutely be a growth leader. It gives me the flexibility as an IT architect to scale my business over time without worrying about how much money I have to invest in storage hardware. Right? So I get kind of, again, that cloud-like flexibility in terms of procurement and deployment, but it gives me that control by oftentimes being on site within my pyramid and I manage it like a storage array that I own. So, you know, it's beautiful for organizations that are scaling and it's equally nice for organizations that just want to manage and control cost over time. So it's a model that makes a lot of sense and fits and it's certainly growing in adoption and popularity. How about from a technology vendor perspective? You've worked in the tech industry for companies. What do you think is going to define the winners and losers in this space? If you were running strategy for a storage company, what would you say? I think the days of a storage administrator managing rate levels and recovery and things of that sort are over, right? What these organizations like Pure delivering with their offerings is simplicity. It's a push-button approach to deploying storage to the applications and workloads that need it, right? It becomes storage as a utility. So it's not just the consumption-based economic model of as a service. It's also the manageability that comes with that or the flexibility of management that comes with that. I can push a button, deploy bytes to a workload that needs it and it just becomes very simple, right? For the storage administrator in a way that kind of old school on-prem storage can't really deliver. You know, I want to ask you, I mean, I've been thinking about this because again, a lot of companies are moving, hopping on as a service bandwagon. I feel like, okay, in and of itself, that's not where the innovation lives. The innovation is going to come from making that singular experience from on-prem to the clouds, across clouds, maybe eventually out to the edge. Do you, where do you see the innovation in as a service? Well, there's two levels of innovation, right? One is business model innovation, right? I now have an organizational flexibility to build the infrastructure to support my digital transformation efforts. But on the product side and the offering side, it really is, as you said, it's about the integration of experience. Every enterprise today touches a cloud in some way, shape, or form, right? I have data spread, not just in my data center, but at the edge, oftentimes in a public cloud, maybe a private cloud. I don't know where my data is. And it really lands on the storage providers to help me manage that and deliver that manageability experience to the IT administrators. So when I look at innovation in this space, it's not just a stored array in a rack that I'm leasing, right? This is not another lease model. It's really fully integrated into end management of my data and all of the things around that. Yeah, so to your point about lease models, if you're doing a lease, yeah, you can shift CAPEX to OPEX, but you're still committed to, you have to over provision, whereas here, and I wanted to ask you about that. It's an interesting model, right? Because you got to read the fine print. Of course, the fine print says you got to commit to some level, typically.  And that's got to be very attractive for folks. I wonder if you will ever see like true cloud-like consumption pricing. That has two edges to it, right? You see consumption-based pricing in some of the software models. And yeah, people like it, the lines of business, maybe because they're paying by the drink, but then procurement hates it because they don't have predictability. How do you see the pricing models? Do you see that maturing or do you think we're sort of locked in on where we're at? No, I do see that maturing, right? And when you work with a company like Pure to understand their consumption based and as a service offerings, it really is sitting down and understanding where your data needs are going to scale, right? You buy in at a certain level, you have capacity planning, you can expand if you need to, you can shrink if you need to. So it really does put more control in the hands of the IT buyer than, well, certainly than traditional CAPEX based on-prem but also more control than you would get, you know, working with an Amazon or an Azure. Okay, thanks, Steve. We'll leave it there for now. I'd love to have you back. Keep it right there at your storage service continues in a moment. Some things are meant to last. Your storage should be one of them. Say hello to the Evergreen Storage Program. Say goodbye to refreshes and rebies. 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And if something on plan comes up, Evergreen Gold provides Upgrade Flex, the leading anytime upgrade feature to upgrade controllers whenever you need. As you expand, Evergreen Gold provides credits to consolidate storage with denser, more modern flash. Evergreen is your subscription to continuous innovation for storage that lasts 10 years or more. Some things are meant to last. Make your storage one of them. A storage service. Emil Stam is here. He's the Chief Commercial Officer and Chief Marketing Officer of OpenLine. Thank you, Emil, for coming on theCUBE. Appreciate your time. Thank you, David. Nice, glad to be here. Yes, so tell us about OpenLine, your managed service provider. What's your focus? Yeah, we're actually a cloud managed service provider. I do put cloud in front of the managed services because it's not just only the service that we manage. We have to manage the clouds as well nowadays. And unfortunately, everybody only thinks it is one cloud but it's always multiple layers in the cloud. So we have a lot of work in integrating it. We're a cloud managed service provider in the Netherlands focusing on companies who ever had office in the Netherlands, mainly in the healthcare, local government, social housing, logistics department and then in the mid-sized companies between say 250 to 10,000 office employees. And that's what we think we provide them with excellent cloud managed services as it should be. Interesting, a lot of early on in the cloud days, highly regulated industries like healthcare or government were somewhat afraid of the cloud. So I'm sure that's one of the ways in which you provide value to your customers is helping them become cloud proficient. Maybe you could talk a little bit more about the value prop to customers. Why do they do business with you? Yeah, I think a number of reasons why they do business with us or choose to choose for a managed services provider. At first they of course are looking for stability and continuity and from a cost perspective, predictable costs. But nowadays you also have a shortage of personnel and knowledge. So, and it's not always very easy for them to access those skill sets because most IT people just want to have a great variety in work, what they are doing. Towards the local government, healthcare, social housing, they're actually a sector that are really in between embracing the public cloud but also have a lot of legacy and bringing it together best of all worlds is what we do. So we also bring them comfort. We do understand what legacy needs from a managed service perspective. We also know how to leverage your benefits in the public cloud. And I'd say from a marketing perspective, actually we focus on using an ideal cloud being a mix of traditional and future based clouds. Thank you. I'd like to get your perspective on this idea of as a service and the as a service economy that we often talk about on theCUBE. I mean, you work with a lot of different companies. We talked about some of the industries and increasingly it seems like organizations are focused more on outcomes, continuous value delivery via suites of services and they're leaning into platforms versus one off product offerings. Do you see that? How do you see your customers reacting to this as a service trend? Yeah, to be honest, sometimes it makes it more complex because services like you look at your Android or iPhone, you can buy apps and download apps the way you want to. So they have a lot of apps. But how would you integrate it into one excellent workflow, something that works for you David or works for me? So the difficulty sometimes lies in the easy accessibility that you have to those solutions but nobody takes into account that they're all part of a chain, a workflow, a supply chain and they're being hyped as well. So what we also have a lot of time in managing our customers is that the tremendous feature push that there is from technology, providers, SaaS providers, whereas if you provide 10 features you only need one or two. But the other eight are very distracting from your prime core business. So there's a natural way in that people are embracing SaaS solutions, embracing cloud solutions but what's not taken into account as much is that we love to see it's the way that you integrate all those solutions to something that's workable for the person that's actually using them. And it's seldomly that somebody is only using one solution. There's always a chain of solutions. So yeah, there are a lot of opportunities but also a lot of challenges for us but also for our customers. You see that trend toward as-a-service continuing or do you actually see based on what you're just saying that pendulum swinging back and forth somebody comes out with a new sort of feature product and that changes the dynamic or do you see the as-a-service really having legs? Ah, I think that's a very, very good question David because that's something that keeps us busy all the time. We do see a trend in as-a-service looking at, I'll talk about pure later on we also use pure as-a-service more or less and it really helps us. But you see that sometimes people make a step too fast, too quick, not well thought of. And then you see what they call the sort of cloud repatriation tendance that people go back to what they are doing and then they stop innovating or stop leveraging the possibilities are actually there. So from our consultancy, our guidance and architecture point of view we try to help them as much as possible to think in as-a-service thought but just don't use the cloud, there's just another data set. And so it's all about managing the maturity on our side but on our customer side as well. So I'm interested in how you're sort of your philosophy and this relates I think in terms of how you work with pure but how do you stay tightly in lockstep with your customers so that you don't over-rotate so that you don't, and send them to over-rotate but then also you don't want to be too late to the game. How do you manage all that? Oh, there's a world of interactions between us and our customers. And so I think a well-known thing that people know is customer intimacy. And that's very important for us to get to know our customers and get to predict which way they are moving. But the thing that we add to it is also the ecosystem intimacy. So know the application and services landscape of our customers, know the primary providers and work with them to create something that really fits the customers. To just not look at from our own silo where our cloud managed service provider. No, we actually work in the ecosystem with the primary providers. And we have, I think with the average customers I think we have in a month we have so much interactions on operational level and technical level, strategic level. We do bring together our customers also and to jointly think about what we can do together what we independently can never reach. But we also involve our customers in defining our own strategy. So we have something we call a customer involvement board where we present a strategy and say, does it make sense? And this is actually what you need also. So we take a lot of our efforts into our customers and we do also understand the significant moments of truth. We are now in this broadcast David. So you can imagine that at this moment nothing can go wrong. If the internet stops, we have a problem now. So we actually know that this broadcast is going on for our customers and we manage that it's always on. Where in the other moments in the week we might have a little less attention but this moment we should be there. And these moments of truth that we really embrace we got them well described. Everybody working on the line knows what the moment of truth is for our customers. So we have a big logistics provider. For instance, he does not have to ask us to have a higher availability on Black Friday or Saturday or Monday. We know that's the most important part in the year for him or her. Does it answer your question David?  You know, the other thing, Emil, about this as a service approach that I really like is it's a lot of it is consumption-based and the data doesn't lie. You can see adoption, you know, daily, weekly, monthly. And so I wonder how you're leveraging pure as a service specifically and what kind of patterns you're seeing in the adoption? Pure as a service for our customers is mainly never visible. We provide storage services, so private storage solutions and storage of its part of a bigger thing, of a server, of an application. So the real benefits, to be honest, of course, towards our customers all flash and they have the fastest storage is available, but for ourself, we use less resources to manage our storage. We're far more, but we have a near to maintenance free storage solution now because we have it as a service and we work closely together with Pure. So actually the way that we treat our customers is the way Pure treats us as well. And that's why there's a huge click.  Now we only have one. And knowing that's a shortage of IT personnel, the other persons can well be involved in other parts of our services or in other parts of an innovation. So that's simply great. You know, my takeaway, Emil, is that you've made infrastructure, at least the storage infrastructure invisible to your customers, which is the way it should be. You don't have to worry about it. And you've also attacked the labor problem. You're not provisioning lungs anymore or tuning the storage with arms and legs. So that's huge. So that gets me into the next topic, which is business transformation. That means that I can now start to attack the operational model. So I've got a different IT model now. I'm not managing infrastructure in ways. So I have to shift those resources. And I'm presuming that it's a business, now becomes a business transformation discussion. How are you seeing your customers shift those resources and focus more on their business as a result of this sort of as a service trend? I do not know if they transform that business. Thanks to us, I think that they can more leverage their own business, have less problems, less maintenance, et cetera, et cetera. But we also add new certainties to it. It's like the latest service we released was immutable storage. Being the first in the Netherlands offering this thanks to the pure technology, but for customers it takes them and give them a good night rest because we have some geopolitical issues in the world. There's a lot of hacking, people have a lot of ransomware attacks and we just give them a good night rest. So from a business transformation, does it transform their business? I think that gives them a comfort in running your business. Knowing that certain things are well arranged. You don't have to worry about that. We will do that. We'll take it out of your hands and you just go ahead and run your business. So to me, it's not a transformation, it's just using the right opportunities at the right moment. The immutable piece is interesting because of course, but speaking of as a service, anybody can go on the dark web and buy ransomware as a service. I mean, it was seen as a service economy hit everywhere, the good and the not so good. And so I presume that your customers are looking at immutability as another capability of the service offering and really rethinking maybe because of the recent ransomware attacks, rethinking how they approach business continuance, business resilience, disaster recovery. Can you see that? Yeah, definitely, definitely. I don't, not all of them yet. Immutable storage is like an insurance as well. I have what you have when you have immutable storage and you have a ransomware attack at least if you part of data, which never if data is corrupted, you cannot restore it. If your hardware is broken, you can order new hardware, but data is corrupted, you cannot order new data. And now we got that safe and well. And so we offered them the possibility to do the forensics and free up their data without a tremendous loss of time. But you also see that you raise the new, how do you say the new baseline for other providers as well. So the security of the corporate information, security of the service, TIO, they're all very happy with that. And they raise the baseline for others as well. So they can look at other security topics and look from say a security operation center because now we can really focus on our prime business risks because from a technical perspective, we got it covered. How can we manage the business risk, which is a combination of people, processes and technology. All right, makes sense. Okay, I'll give you the last word. Talk about your relationship with Pure, where you want to see that going in the future. I hope we'll be working together for a long time. I experienced them as very involved. That's not, we have done the sale and now it's all up to you. Now we're really closely working together. I know if I talk to my prime architect, Marcel Haider is very happy and it looks more or less if we work with Pure, like we're working with colleagues, not with a supplier and a customer. And the whole Pure concept is quite fascinating. I had the opportunity to visit San Francisco head office and they told me the vision of how they launched Pure being, if you want to implement it, they had to be on one credit card. The menu had to be on one credit card. Just a simple thought of, put that as your big, heavy audacious goal to make the simplest implementable storage available. But for us, it gives me the expectation that there will be a lot of more surprises with Pure in the near future. And for us as a provider, what we really look forward to is that for us, these new developments will not be new migrations. It will be a gradual growth of our services, our storage services. So that's what I expect and that was what I, and we look forward to. Yeah, that's great. Thank you so much, Emil, for coming on theCUBE and sharing your thoughts and best of luck to you in the future. Thank you, you're welcome. Thanks for having me. You're very welcome. Okay, in a moment, I'll be back to give you some closing thoughts on at your storage service. You're watching theCUBE, the leader in high-tech enterprise coverage. Welcome to Evergreen, a place where organizations grow and thrive rooted in the modern data experience. In Evergreen, people find a seamless, simple way to leverage data. Through market-leading, sustainable technology, financial flexibility and effortless management, allowing everyone to innovate with data confidently. Welcome to Pure Storage. Now, if you're interested in hearing more about Pure's growing portfolio of technology and services and how they're transforming the enterprise data experience, be sure to register for Pure Accelerate TechFest 22, digital event is also taking place as an in-person event on June 8th. 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