From theCUBE Studios in Palo Alto in Boston, connecting with thought leaders all around the world, this is a CUBE Conversation. Hi everybody, this is Dave Vellante and welcome to the special CUBE Conversation. I first met Frank Slutman in 2007 when he was the CEO of Data Domain. Back then he was a CEO of a disruptive company and still is. Data Domain, believe it or not, back then was actually replacing tape drives as the primary mechanism for backup. Yes, believe it or not, it used to be tape. Fast forward several years later, I met Frank again at VMworld when he had become the CEO of ServiceNow. At the time, ServiceNow was a small company, about a hundred plus million dollars. Frank and his team took that company to 1.2 billion and Gartner at the time of the IPO said, you know, this doesn't make sense. It's a small market. It's a very narrow help desk market. It's maybe a couple of billion dollars. The vision of Slutman and his team was to really expand the total available market and execute like a laser, which they did. And today, ServiceNow, a very, very successful company. Snowflake first came into my line of sight in 2015 when SiliconANGLE wrote an article why Snowflake is better than Amazon Redshift, reimagining data. Well, last year, Frank Slutman joined Snowflake, another disruptive company. And he's here today to talk about how Snowflake is really participating in this COVID-19 crisis and really want to share some of Frank's insights and leadership principles. Frank, great to see you. Thanks for coming on. Yes, thanks for having us, Dave. So when I first reported earlier this year on Snowflake and shared some data with the community, you reached back out to me and said, Dave, I want to just share with you. I am not a playbook CEO. I am a situational CEO. This is what I learned in the military. So Frank, this COVID-19 situation was thrown at you. It's a kind of, it's a black swan. What was your first move as a leader? Well, my first move is let's not overreact. Take a deep breath. Let's really examine what we know. Let's not jump to conclusions. Let's not try to project things that we're not capable of projecting. That's hard because we tend to have sort of levels of certainty about what's going to happen in the next week in the next month and so on. All of a sudden, that's out of the window. Creates enormous anxiety with people.  What can we do? What do we control? And not let our minds sort of go out of control. So I talk to our people all the time about maintain a sense of normalcy, focus on the work, stay in the moment. And by the way, turn the newsfeed off, right? Because the hysteria you get fed through the media is really not helpful, right? So just cool down and focus on what we still can do. And then I think then everybody takes a deep breath and we just go back to work.  Pretty much what I was doing before, except I'm not traveling right now. Yeah, so it's not that different than what it was before. It sounds very Bill Belichickian, you know, to focus on those things at which you can control. When you were running service now, I really learned from you and of course, Mike Scarpelli, you're then and current CFO about the importance of transparency. And I'm interested in how you're communicating. It sounds like you're doing some very similar things, but have you changed the way in which you've communicated to your team, your internal employees at all? We're communicating much more because we can no longer rely on, you know, sort of the running into people here, there and everywhere. So we have to be much more purposeful about communications.  Just to bring the connection back, the normalcy, you know, it just helps people, you know, get connected back to the mothership and like, well, you know, things are still going on, you know, we're still talking in the way we always used to be. That really helps. And I also, you know, check in with people a lot more. I ask all of our leadership to constantly check in with people because you can't assume that everybody is okay. You can't be out of sight, out of mind. So we need to be more purposeful in reaching out and communicating with people than we were previously. I mean, a lot of people obviously concerned about their jobs. Have you sort of communicated, what have you communicated to employees about layoffs?  You know, I've said, there's no layoffs on our radar. Number one, number two, we are hiring. And number three is, you know, we have a higher level of scrutiny on the hires that we're making. And I am very transparent. In other words, I tell people, look, you know, I prioritize the roles that are closest to the direct train of the business, right? It's kind of common sense, but you know, I wanted to make sure that this is how we're thinking about it. There are some roles that are more postponable than others. I'm hiring and engineering without any reservation because that is the long-term, you know, strategic interest of the company. On the sales side, I want to know that sales leaders, you know, know how to convert to yield, right? That we're not just sort of, you know, bringing capacity online and the leadership, you know, is not convinced or confident that they can convert to yield. So there's a little bit, you know, finer level of scrutiny on the hiring, but by and large, you know, it's not that different. There's this saying out there that, you know, we should suspend all non-essential spending and hiring. I'm like, you should always do that, right? I mean, what's different today? If it's non-essential, why do it, right? So all of this comes back to, you know, this is how we probably should operate anyways. You know. I want to talk a little bit about the tech behind Snowflake. I'm very sensitive when CEOs come on my program to make sure that we're not, you know, I'm not trying to bait CEOs into, you know, ambulance chasing, that's not what this is about. But I do want to share with our community, kind of what's new, what's changed and how companies like Snowflake are participating in this crisis. And in particular, we've been reporting for a while, if you guys bring up that first slide, that, you know, the innovation in the industry is really no longer about Moore's law. You know, it's really shifted. There's a new, what we call innovation cocktail in the business. We've collected all this data over the last 10 years, you know, with Hadoop and other distributed data. Now we have edge data, et cetera. There's this huge trove of data. Now AI is becoming real. It's becoming much more economical. So applying machine intelligence to this data and then the cloud allows us to do this at scale. It allows us to bring in more data sources. It brings an agility in. So I wonder if you could talk about sort of this premise and how you guys fit. Yeah, I would start off by reordering the sequence and saying, you know, clouds number one, that is foundational. That helps us bring scale to data that we never had to number two. It helps us bring computational power to data at levels we've never had before. And that just means that queries and workloads can complete order of magnitude faster than they ever could before. And that introduces concepts like the time value of data. The faster you get it, the more impactful and powerful it is. I do agree. I view AI sort of next generation of analytics. You know, instead of using data to inform people, you know, we're using data to drive processes and business directly, right? So I'm agreeing with obviously with these trends because we're the principal beneficiaries and drivers of these platforms. Well, when we talked about earlier this year about a snowflake, we really brought up the notion that you guys were one of the first, if not the first. And guys, bring back Frank, I got to see him. One of the first to really sort of separate the notion of being able to scale, compute, independent of storage. And that brought not only economics, but it brought flexibility. So you've got this cloud native database. Again, what caught my attention in that redshift article that we wrote is essentially for our audience, Redshift was based on ParXL. Amazon did a great job of really sort of making that a cloud database, but it really wasn't born in the cloud and that's sort of the advantage of snowflake. So that architectural approach is starting to really take hold. So I want to give an example, guys, if you bring up the next chart, this is an example of a system that I've been using since the early January when I saw this, you know, COVID come out, somebody texted me this and it's the John Hopkins data set, it's awesome. It shows you that, you know, around the map, you can follow it, it's pretty close to real time and it's quite good. But the problem is, I thank you guys, the problem is that when I started to look at, I wanted to get into sort of a more granular view of the counties and I couldn't do that. So guys, bring up the next slide, if you would. So what I did was I searched around and I found a New York Times GitHub data instance and you can see it on the top left here. And basically it was a CSV and notice what it says is we can't make this file beautiful and searchable because it's essentially too big. And then I ran into what you guys are doing with Star Schema, Star Schema is a data company and essentially you guys made the notion that, look, the John Hopkins data set is great as it is, it's not ready for analytics, it's got to be cleaned, et cetera. And so I want you to talk about that a little bit, guys, if you could bring Frank back and share with us what you guys have done with Star Schema and how that's helping understand COVID-19 and its progression.  We have taken that concept from private also to public so that data providers like Star Schema, you know, can list their data sets because they're a data company. So obviously it's in their business interest to allow this data to be profiled and to be accessible by the Snowflake community. And this data is what we call analytics ready. It is instantly accessible. It is also continually updated. You have to do nothing. It's augmented with incremental data and then, you know, our Snowflake users can just combine this data with supply chain, with economic data, with internal operating data and so on. We got a very strong reaction from our customer base because they're like, man, you're saving as weeks if not months, just getting prepared to start to do an L, let alone doing it, right? Because the data is analytics ready and they have to do literally nothing. I mean, in other words, if they ask us forward in the morning, in the afternoon, they'll be running workloads against it, right? And combining it with their own data. Yeah, so I just point out that New York Times GitHub data set that I showed you, it's a couple of days behind. We're talking here about near real time or as close as real time as you can get, is that right? Yeah, it's every day gets updated. So the other thing, one of the things we've been reporting, and Frank, I wonder if you can comment on this, is this new emerging workloads in the cloud. We've been reporting on this for a couple of years. You know, the first generation of cloud was IAS. It was really about compute storage, some database infrastructure, but really now what we're seeing is these analytic data stores where the valuable data is sitting and much of it is in the cloud and bringing machine intelligence and data science capabilities to that, to allow for this real time or near real time analysis. And that is a new emerging workload that is really gaining a lot of steam as these companies try to go to this so-called digital transformation. Your comments on that. Yeah, we refer to that as the emergence or the rise of the data cloud. You know, if you look at the cloud landscape, we're all very familiar with the infrastructure clouds, AWS and Azure and GCP and so on. It's just massive storage and servers. And obviously there's data locked into those infrastructure clouds as well. We've been familiar for 10, 20 years now with application clouds, you know, notably Salesforce, but obviously, you know, Workbase Service Now, SAP and so on. They also have data in them, right? But now you're seeing that, you know, people are un-siloing the data. This is super important because as long as the data is locked in these infrastructure cloud, in these application cloud, we can do the things that we need to do with it, right? We have to un-silo it to allow the scale of querying and execution against that data. And you don't see that any more clearly than you do right now during this meltdown that we're experiencing. Okay, so I learned long ago, Frank, not to argue with you, but I want to push you on something. So I'm not trying to be argumentative, but one of those silos is on-prem. I've heard you talk about, look, we're a cloud company. We're cloud-first, we're cloud-only. We're not going to do an on-prem version, but some of that data lives on-prem. There are companies out there that are saying, hey, we separate compute and storage too. We run in the cloud, but we also run on-prem. That's our big differentiator. Your thoughts on that? Yeah, we burned the ship behind us, okay? We're not doing this endless hedging that people have done for 20 years, you know, sort of keeping, you know, a leg in both worlds. Forget it, this will only work in the public cloud because this is how the utility model works, right? I think everybody is coming to this realization, right? I mean, the excuses are running out at this point. You know, we think that people will come to the public cloud a lot sooner than we will ever come to the private cloud. It's not that we can't run a private cloud, it just diminishes the potential and the value that we bring. So as I sort of mentioned in my intro, you have always been at the forefront of disruption and you think about digital transformation. You know, Frank, we go to all these events who used to be physical and now we're doing, you know, the cube digital. And so everybody talks about digital transformation. CEOs get up, they talk about how they're helping their customers, you know, move to digital. But the reality is when you actually talk to businesses, there was a lot of complacency. Hey, you know, this isn't really going to happen in my lifetime or we're doing pretty well or maybe the CEO might be committed but it doesn't necessarily trickle down with the P and L managers who have a nut date. One of the things that we've been talking about is COVID-19 is going to accelerate that digital transformation and make it a mandate. You're sort of, you're seeing it obviously in retail play out and the number of other industries, supply chains are, you know, this is wreak havoc on supply chains. And so there's going to be a rethinking. What are your thoughts on the acceleration of digital transformation? Well, obviously the crisis that Rich Bernstein is obviously an enormous catalyst for digital transformation and everything that that entails and what that means. I think we, you know, as an industry, we're just victims of inertia, right? I mean, I haven't understood for 20 years why education, both K through 12 but also higher at, you know, why you're so brick and mortar bound, you know, on the way they're doing things, right? We could massively scale and drop the cost of education, you know, by going digital and now we're forced into it and everybody's like, wow, this is not bad. You're right, it isn't, right? But we haven't. So the economics, you know, the economic imperative hasn't really set in, but it is now. So these are all great things. You know, having said that, you know, there are also limits to digital transformation and I'm sort of experiencing that right now, you know, being on video calls all day and oftentimes the people that I've never met before, right, there's still a barrier there, right? It's not like digital can replace absolutely everything and that is just not true, right? I mean, there's some level of filter that just doesn't happen when you're digital. So there's still a need for people to be in the same place and I want to sort of over rotate on this concept.  That's not the way it will be. Yeah, be balanced. So earlier you made a comment that hook, we should never be spending on non-essential items.  You've seen, you know, the black swan memos that go out.  You've done a couple of stints in the VC community. What are you seeing in the Valley with regard to investments? You know, will investments continue? Will we continue to feed innovation? What's your sense of that? Well, this is another wake-up call, you know, because in Silicon Valley, there's way too much money. There's certainly a lot of ideas, but there's not a lot of people that can execute on it. So what happens is, you know, a lot of things get funded and the execution is either, you know, no good or it's just not a valid opportunity.  I mean, when the tide is running out, you know, only the strongest players, you know, are going to survive that.  It's not necessarily a bad thing because people get reallocated. I mean, Silicon Valley is basically one giant beehive, right? I mean, we're constantly repurposing money and people and talent and so on. And that's actually good because if an idea is not worth investing in, let's not do it. Let's repurpose those resources in places where it has merit, where it has viability. Well, Frank, I want to thank you for coming on.  You know, you could have retired long, long ago, but having leaders like you in place in these times of crisis, but even when in good times to lead companies inspire people and we really appreciate what you do for companies, for your employees, for your customers and certainly for our community. So thanks again, really appreciate it. I hope you to do it. Thanks, Dave. All right, and thank you for watching everybody. Dave Vellante for theCUBE. We will see you next time.


We've spoken on the last two shows about the disgrace that is the government and BBC following Nigel Farage's lead in pretending that 4,000 people seeking asylum in the UK constitutes a national. Crisis Britain receives a poultry number of refugees compared to our European neighbors not to mention countries in the developing world we talk about 11 on population of 7 million 1.5 million refugees Britain a population of 60 million and we're all freaking out because fall fall only 4,000 people coming over in dingy's the campaign to pretend this is some kind of crisis is nothing other than a big lie which the government are using to distract from the incredibly poor record on coronavirus the highest death toll in Europe. And I think that means the world actually and the deepest recession in the G7 which we're going to talk about next. But I suppose one of the things that's particularly upsetting I suppose when it comes to the discourse on migration is that there's very little pushback against this. So as we spoke about on Monday, all of the major channels are now doing live footage of people coming over in dingy's as if this is some sort of entertainment show reality TV. Not really going into the backgrounds of anyone there but looking at it as some kind of sport and the opposition party are also really not pushing back at all on the narrative that the people who are trying to get here from France coming from war zones. Many of them are a threat. So let's look at some of Labour's statements on this. They haven't been doing broadcasts themselves for a while. But yesterday, Sienna Rogers at Labour List got a statement from the shadow immigration minister, Holly Lynch. So asked for comment on migrant boat crossings. Labour's shadow immigration minister, Holly Lynch said that immigration minister's emergency visit today to France today shows the lack of grip and competence shown by the government on this issue. Today they have announced a new comprehensive action plan but have failed to reveal what that involves or when it will be enacted. We're calling on the government to urgently provide the detail and reveal why it has taken them this long to make any progress on a solution. This line was, I suppose, backed up this morning by deputy leader of the party, Angela Rainer. She tweeted, the UK has been tipped into the largest recession on record. We have the largest death rate of coronavirus in Europe. We have the Chancellor saying, help out the PM saying cut back on eating too much immigration ministers getting our borders muddled up. So here it's being presented that alongside what are genuine failures of the government, yet a failure to control coronavirus and because they spent so long sort of twiddling their fingers while the virus was running for our population, ending up having such a deep recession, they are genuine failures of this government. It's not really a failure of this government that they're letting some desperate people cross the channel to come to this country. I say letting. I mean, they haven't yet prevented it happening. So it does seem like the argument the Labour Party are taking here is completely accepting the premise of the overwhelming narrative would be presented by the mainstream media and by the Conservative Party, that the people who are coming over in these dinghies are a threat. The only problem is that the Tories aren't competent enough to stop them. That seems to be the line here. And one of the reasons this is so dangerous, actually, is because if there is ever an issue that needs to be pushed back on, it's this one. Because sadly, the British public do not have much sympathy towards people who are crossing in dinghies. People who, you know, to my mind, it should be pretty easy to create sympathy for. All you have to talk about is the background. You know, how did these people get here? Why would people risk their lives crossing the channel to get to this country if it weren't because they were fleeing somewhere much worse? Right. Let's look at a YouGov poll because most of the country don't agree with me, unfortunately. So almost half of Britons say they have little to no sympathy for the migrants who have been crossing the channel from France to England, breaking that down. 22% say they have little sympathy, 27% no sympathy, which adds up to 49%. You can see that on the left hand side of the graph. But, frankly, among Labour voters, 73% have a great or a fair amount of sympathy for the migrants crossing the channel. But, as I say, that that sympathy does not seem to have been yet expressed by anyone on the Labour front bench. Disappointing. This is really when we need political leadership, because as I say, there should be a fairly easy argument to win if anyone would dare to make it. This is the culmination of two things. It's the culmination of consistent dehumanisation and toxic narratives around migration in the media. The UN has had to intervene multiple times in order to, you know, the UN Commissioner on Human Rights has at multiple points said that, you know, the British tabloid media in particular, the representation of migration, the approach to reporting on migration is the worst in Europe.  We have a, you know, a deeply unethical approach to reporting on migration on the part of the media. And we can see that not just in I think the tabloid and the right wing press gets a lot of gets a lot of the stick. But the important thing is that we see it across the political spectrum and we see it in the BBC, where, you know, the filming of refugees trying to cross the channel on a dinghy of, you know, emptying their boat. We're using plastic bottles in order to stop it from sinking and you have a BBC presenter sort of showing their faces and just sort of like casually reporting on it. It's deeply unethical and it's deeply troubling. So it's a culmination of this incredibly toxic media that we have when it comes to migration, which is something that has existed for decades and decades. Something that, you know, we often talk about the Windrush generation as people who obviously are examples of, quote unquote, good migrants. But at the time that the Windrush generation came to this country, they were frequently the subject of incredibly toxic representation by the media. They were frequently made to scapegoats, but all of Britain's problems by the media. It's also the culmination of the complete failure for any kind of counter counter narrative. And unfortunately, we are seeing this happening right now in the portrayal of this as simply mismanagement or ineffective management rather than a massive violation of human rights. And I think also what worries me is what's going to come next, you know, at times. And I think I might have said this last last week or maybe the week before. But when the state feels that it is in a precarious position, it's being seen as weak. It's maybe not doing so well electorally. It's not being able to get the handles on the crisis. It will often recourse to the realm of race and migration and sort of stoking up hatred against the racialized minorities and migrants in order to portray itself as strong. And I worry that with, you know, the coronavirus crisis, with the upcoming climate crisis, with this recession and the fact that that is probably all going to be compacted by Brexit, that we are going to see the state and the media repeatedly go back to that well used tool in their toolbox, which is to whip up hatred and to scapegoat migrants for all of the issues that are being shipped, that are being faced and for the state to use migrants and to use racialized minorities as a kind of space where they can show their might as the government when they are failing, failing elsewhere. So and now I guess with with the with the fact that the Labour Party has somewhat abdicated responsibility for protecting racialized minorities or protecting migrants from this kind of crisis. It is going to be up to grassroots migrants rights movements and it's going to be up to the community stepping in to draw the boundaries of what around what can and cannot happen to those in our communities that come from migrant backgrounds or migrants themselves.


You know, you look at what he did throughout the season and how it impacted winning. And I think that's the thing is like, you know, there were what six or seven teams that won 50 games this season and we were one of them. And, you know, obviously a lot of that had to do with Donovan and his performance. The guy had a 71 point game. We were down 20 at half and he figured out a way, you know, to kind of will us to victory. I feel like, you know, he should have been first team. You know, again, I know we all would have traded that in to be continued to play further on, but, you know, he definitely earned it through the regular season proved that he was one of the top five players this year.


So, in the last capsule we were trying to solve the Dirichlet problem for the Laplace's equation on a disc x squared plus y squared less than or equal to 1. The function was given on the boundary x squared plus y squared equal to 1 namely u of cos theta comma sin theta equal to f of theta, but f of theta was a Lipschitz continuous 2 pi periodic function so that the Fourier series converges etc. We also saw that the solution can be obtained as an integral with the Poisson kernel. What is the Poisson kernel? Let us look at the Poisson kernel.  So, we got this integral 2.18. Now, we got to look at the properties of Poisson kernel. One of the properties of Poisson kernel is that it is a non-negative kernel unlike the Dirichlet kernel unlike the Dirichlet kernel the Poisson kernel is a good kernel it is a positive kernel. The Dirichlet kernel is not a positive kernel. It takes both values positive and negative values and it oscillates very rapidly Dirichlet kernel is not convenient because it was not a positive kernel because it oscillated very rapidly we had trouble when we try to prove point wise convergence. Here the situation is much better the Poisson kernel is a non-negative kernel and 1 upon 2 pi integral minus pi 2 pi pi r of theta minus t will be 1. How to prove this 2.19?  Integrate this equation term by term with respect to s from minus pi to pi. You can use uniform convergence because r is strictly less than 1 and that is how you prove equation 2.19. So, what you can do is that you can imitate the same thing that we did for Dirichlet kernel you can put theta minus t equal to s in equation 2.18 and you will get 1 minus r squared upon 1 plus r squared minus 2r cos s except that over here this will become theta minus s theta minus s and the integral will be with respect to ds. So, do this small change of variables. Now, once you do this change of variables then what you do is that you can prove the next exercise show that limit as r tends to 1 minus more u r e to the power i theta minus f of theta equal to 0. Remember we got the solution u of r e to the power i theta. Now, we want to prove that this solution does indeed assume the correct boundary values namely f of theta that is what 2.20 is trying to tell you will carry out this program later on, but here I would like you to think about it. Now, what about the three dimensional Laplace equation? We could try to carry out a similar program for the Laplace's equation for a ball in r 3, but the problem is going to be more complicated because the analog of a Fourier series is going to be more complicated you have to develop the theory of spherical harmonics. The corresponding Fourier analysis in r 3 by writing the Laplace's equation in spherical polar coordinates that will lead you to the Legendre functions, associated Legendre equations and the spherical harmonics. So, the analysis of spherical harmonics is the higher dimensional analog of Fourier series. So, suppose you are given a continuous function f of x on the unit ball b centered at the origin then the solution of the boundary value problem Laplacian of u equal to 0 on b u on the boundaries f is given by this integral. So, you recognize the Poisson kernel for higher dimension and this d capital S x is the area element on the surface. You can derive this by other methods also, but the proof using spherical harmonics you will see in G. B. Folland's book Fourier analysis and its applications and page 180. It is a very delightfully written book and you must definitely consult this book and G. B. Folland is an authority in Fourier analysis. So, now we will come to the heat equation, how to solve the heat equation using Fourier series. This will be developed through a series of exercises. So, let u of xt be a smooth solution of the heat equation in the upper half plane. We are going to work in two variables t and x and the variable t is going to be considered as a time variable and the variable x is going to be considered as a spatial variable. We are going to work in the upper half plane real line Cartesian product with the positive half line. So, time is going to be positive t greater than or equal to 0 and we are interested in 2 pi periodic functions. We are going to postulate that the solution is 2 pi periodic u of x comma t equal to u of x plus 2 pi comma t for all x in R and t greater than or equal to 0. The first exercise is for you to show that the energy integral minus pi to pi u of xt the whole squared dx is a monotone decreasing function of time. Prove that the same result holds if the integral is over any interval of length 2 pi and then deduce that a 2 pi periodic solution of the heat equation with given initial condition if it exists is unique. Let us look at this exercise and let us try to see how to do it call the energy this integral Et energy integrated with respect to x and so the energy is a function of time call it Et. So, let us differentiate with respect to t I am going to differentiate under the integral sign I am going to get twice u of x comma t del u by del t. So, it is twice integral from minus pi to pi u of xt u t of xt dx. But remember u is a solution to the heat equation. So, ut equal to uxx. So, here in the last equation I am going to integrate by parts I am going to throw one of the derivatives on the other factor and I am going to pick up minus sign. So, when you integrate by parts I am going to get minus 2 times integral minus pi to pi u sub x x of t the whole squared. What happens to the boundary terms the boundary terms will cancel out because a function is 2 pi periodic u of xt equal to u of x plus 2 pi t. Remember that if the function is 2 pi periodic its derivative is also going to be 2 pi periodic. So, use the 2 pi periodicity to check that the boundary terms cancel out when I perform the integration by parts. And so, what you are left out is e prime of t equal to minus 2 times integral minus pi to pi a squared factor and everything here is real value. So, it is always less than or equal to 0. So, you have finished proving that the energy is monotone decreasing with respect to time. So, we get energy is obviously non-negative function. So, 0 less than or equal to e of t less than or equal to e of 0. So, the initial value u of x comma 0 were to be 0. So, the initial energy is also 0 and so, e of t will be identically 0 for all t and if e of t is 0 remember that e of t is a integral of a non-negative function. So, u of xt must be identically 0. So, we have now drawn the conclusion that the initial data is 0 and the solution is also going to be 0. So, for 0 data the 2 pi periodic solution is unique namely 0. So, from this uniqueness followed now we prove the existence uniqueness of the 2 pi periodic solution has been established we now derive the existence. Because f of x is continuous 2 pi periodic function the data the initial condition u of x comma 0 is f of x. Further conditions will be imposed as we go along we begin with the assumption that the initial distribution of heat f of x is simply continuous. Now, we are looking for the solution of u t equal to u xx we will now look for a solution of the form u of xt equal to a naught plus summation n from 1 to infinity a n cos nx plus b n sin nx. Remember that the solution that we are looking for is a 2 pi periodic solution because it is a 2 pi periodic solution it will be a Fourier series. But this time the coefficients will depend upon t because u is a function of 2 variables x and t. So, I can think of t as an auxiliary parameter if you like. So, that is how we get equation 2.22. So, let us substitute this ansatz 2.22 into the differential equation u t equal to u xx and see what happens. What are the what are the time derivative del u by del t will be a naught prime plus summation n from 1 to infinity a n prime t cos nx plus b n prime t sin nx and then we have to differentiate twice with respect to x. So, a naught disappears summation n from 1 to infinity a n t cos nx and then n squared will come plus b n t sin nx and n squared will come and this whole thing will have a negative sign in front. So, compare the two expressions u t equal to u xx we got two Fourier series and compare the coefficients we get a naught prime t is 0 a n prime t equal to minus n squared a n t b n prime t equal to minus n squared b n t equation 2.23 in the display. The first equation simply says that a naught of t must be constant. Then the second equation says that a n of t must be an exponential function. There will be some initial conditions a a naught of t is a constant alpha naught a n t is constant times e to the power minus n square t b n t is constant times e to the power minus n square t these constants are alpha n and beta n. Let us put t equal to 0 in equation 2.22 let us put t equal to 0. So, I get a naught of 0 will be alpha naught a n of 0 will be alpha n b n of 0 will be beta n u of x comma 0 will be f of x. So, f of x equal to alpha naught plus summation n from 1 to infinity alpha n cos nx plus beta n sin nx equation 2.24. But f of x is a given function and it is a 2 pi periodic continuous function and I can write out its Fourier series and I have my alpha naught alpha n and beta n with that I have my solution u of x t which assumes the form u of x t equal to alpha naught plus summation n from 1 to infinity e to the power minus n square t alpha n cos nx plus beta n sin nx equation 2.25 in the display. Now comes the task for us to show that as t goes to 0 this equation 2.25 does indeed converge to 2.24 the initial condition f of x is attained as t approaches 0 plus that has to be established the rapid decay of those exponentials help us. So, the problem amounts to a passage to limit under the summation sign what we have to do is you have to show that limit as t tends to 0 plus summation n from 1 to infinity that is equal to summation n from 1 to infinity limit as t tends to 0 plus that is what it really amounts to because when you allow the t to go to 0 you get the thing under the summation. So, to check that this approaches this if the limit as t goes to 0 is the same as checking this. As you know exchange of 2 limiting operations is a tricky business one should do it carefully it is not difficult to carry it out and it has been carried out in this book by Bachman and Naricci on page 203 I already given a reference to this book earlier in this course. So, I will just refer to this book and we will not carry out the analysis in this case. Instead I have given exercise determine the solution of the heat equation with initial condition f of x equal to pi squared minus x squared on the interval mod x less than or equal to pi that is an even function on the interval minus pi to pi extend it as a 2 pi periodic function on the entire real line and for that initial condition solve the heat equation using Fourier series. So, now let us continue with the last part of this chapter Abel's summability. Remember the case of the Poisson kernel I left it as an exercise for you to check that the solution that we obtained u of r e to the power i theta converges to f of theta as r goes to 1 we will discuss that in detail under the broader context of Abel's summability. So, let us recall what Abel's summability means a n is a sequence of complex numbers. Okay now let us recall Abel's limit theorem from elementary analysis for example you can look at Rudin's principles of mathematical analysis and you will see the proof of Abel's limit theorem. So, theorem 25 summation n from 0 to infinity a n converges suppose that the infinite series converges then cook up the function f of t equal to summation n from 0 to infinity a n t to the power n this function is obviously holomorphic on mod t less than 1. But this holomorphic function will satisfy the equation 2.26 limit of f of t as t tends to 1 along the real axis and from inside the disk of convergence that will be exactly equal to summation n from 0 to infinity a n. It is as if you can put t equal to 1 directly in the defining equation but you might wonder is not that obvious simply put t equal to 1 in this equation and then you should get this that is not quite the case the result is non-trivial the theorem is quite non-trivial and let us try to understand why is this theorem of Abel non-trivial because the left hand side of 2.26 what is the left hand side of 2.26 it is limit as t tends to 1 minus f of t that is limit as t tends to 1 minus summation n from 0 to infinity a n t to the power n but what is this infinite series this infinite series is limit as capital n tends to infinity sum from 0 to capital N. Let us look at the right hand side what are the right hand side right hand side is summation n from 0 to infinity a n well shall we write this says limit as capital N tends to infinity summation a n little n from 0 to capital N. But this thing can be written as limit as t tends to 1 minus summation n from 0 to capital N a n t to the power n. So, what we see here is an exchange of two limit operations. So, Abel's limit theorem is non-trivial because it asserts that a certain exchange of limits is valid. Now, let us say that a series summation a n n from 0 to infinity 2.27 is said to be Abel's summable. If the associated series summation n from 0 to infinity a n t to the power n has radius of convergence 1 and further the sum function f of t has a limit as t tends to 1 minus. Again t approaches 1 along the real axis and from within and the limit is then said to be the Abel's sum of the infinite series 2.27. It is important that the limit is taken as t varies along the real axis. I can relax the condition and I can take I can take the circle and I can take the point 1 and I can take a sector I can take a sector with vertex 1 and opening angle and some acute angle as opening angle that is called a Stoll's region. I will not get into the details of that you will see this being discussed in classical books. So, let us discuss Abel's summability of Fourier series. We know that if you take a continuous 2 pi periodic function on the real line the Fourier series may not converge to the given function point twice, but a nice analog of Abel's summability holds. So, let us introduce an r to the power n in the nth terms in the Fourier series. So, we pass from 2.29 to 2.30 exactly in analogy with the Abel's summability that we talked about a few slides ago. The series 2.30 will certainly converge for 0 less than or equal to r less than 1 simply because the a n's and the b n's decay to 0 by Riemann Lebesgue Lemma. So, now that we know that 2.30 converges for this range 0 less than or equal to r less than 1, we denote its sum by U R e to the power i t. It is of interest to know whether for a fixed angle theta limit of U R e to the power i t equals f of theta where R t approaches 1 theta. So, e to the power i theta is a point on the boundary of the disc mod z less than 1 and I want to know whether when R e to the power i t converges to e to the power i theta from inside whether do I recover or capture the boundary data f of theta that is Abel's summability. Let us look into the proof of this Abel's summability. First we begin by recalling the Poisson kernel which we studied a few slides ago.  We already have a formula for the sum of the series 2.30. We simply put in the definition of a n, b n and a naught as integrals from minus pi to pi and we find the sum of the series U R e to the power i t and we get it as an integral 1 upon 2 pi integral minus pi to pi pi r s minus t f of s ds or by the usual change of variables put s minus t equal to U and we get it is 1 upon 2 pi integral minus pi to pi f of t minus s pi r of s ds. Now remember that when I integrate pi r s ds from minus pi to pi I exactly get 2 pi so 1 upon 2 pi integral minus pi to pi pi r s ds is 1. So integral minus pi to pi f of theta pi r s ds is f of theta after dividing by 1 upon 2 pi.  Now we exploit the fact that the Poisson kernel is a positive kernel. So the last expression that we got in the previous slide we have to take the absolute value and we need to take the absolute value under the integral sign nothing will happen to the Poisson kernel because the Poisson kernel is a positive kernel.  Now it is time to bring in the epsilon's and delta's. So let epsilon greater than 0 be arbitrary. First we invoke the uniform continuity of the function f that will give us a delta such that mod f x minus f y less than epsilon whenever mod x minus y is less than 2 delta. Now we break this integral 2.32 into 2 pieces integral from minus delta to delta and on the complementary piece namely mod s greater than or equal to delta. So first let us deal with this piece. Now in this piece what happens mod s is less than delta.  So I can say that this factor mod f of t minus s minus f of theta is less than epsilon by 2 and then I am left with the integral from minus delta to delta pi r s upon 2 pi. But because this Poisson kernel is a positive kernel the integral from minus delta to delta is less than or equal to the integral from minus pi to pi and this last one is 1. So I get epsilon by 2 equation 2.32 prime in the display. Now let us proceed further and look at the other piece where mod s is larger than delta.   Again go to the definition of pi r s the expression for pi r s that I gave you.  The cos s in the expression for pi r s has been replaced by cos delta and I get something larger.  So there exists eta bigger than 0 such that when r is between 1 minus eta and 1 we have that this thing is less than epsilon by 2 into pi upon n. So pi r of s is also less than epsilon by 2 into pi by s. The integral that you see on the right hand side of 2.32 double prime is less than or equal to what m upon pi into pi upon m into epsilon by 2 into the range of the integral is mod s bigger than delta and so that is less than 2 pi. And so you all in all you will get that this contribution from the right hand side of 2.32 double prime is less than epsilon by 2. All this drama is taking place when provided 1 minus eta less than r less than 1 and remember mod t minus theta less than delta.  In other words I have estimated this difference to be less than epsilon and that completes the proof of theorem 26. So suppose f of t is a continuous 2 pi periodic function of the real line with Fourier series given by equation 2.29 then the function u r e to the power i t which is a sum of 2.30 converges to f of theta as r t converges to 1 minus comma theta. So this shows that the Fourier series of a continuous function converges to f in the sense of a bell summability not in the sense of point wise convergence. I think with this we should close this capsule and we shall close this chapter on mean convergence and a bell summability some applications to partial differential equations. In the next chapter we will begin with the fairer's theorem. Cesaro summability that is the next item. Thank you very much.


skies, it is a big story behind all this range but does it perform t'is what we're going to find out very very soon. I'm not going to go straight to drivel day so the idea of this section is that down in the IV Golf Chester and I'm not going to be swayed by what it is seeing in numbers.  The thing I want to talk about is how this club looks at ydw i'r adreth yn gwneud y gweithio i'w ddysgu'r ystod, mae'n gweithio'n oedd yn gweithio'n gweithio'r ystod, ac yn ychydig i'w ddweud o'n ddiddordeb gwahanol yn gyda'r gweithio ystod, felly mae'r maeth yma yn cyflwyno ar y maeth, a'r cyfrwng cyfrwng ar y gweithio'n cyfnodol. A'r gweithio'n gweithio ymlaen yn eu gwahanol yn y gweithio, is that it looks a lot thinner than it probably is.. ..in reality... ..but ad-addrest it's got a thick top line. It's a game improvement iron and its got that little bit of mass. But their certainly getting.. ..every year.. ..the super game improvement iron are getting that thinner.. ..that little bit more compact. And to a level now where I would not have a problem.. ..so the game in this. But let's see how it does in terms of performance Now were expecting a long distance is a 28.5 degree.. 7 iron, ond we're going to look a bit deeper in that. Superstart. I've said on every video, I don't know, I'm at the ball quite so well this morning, but it's a good day. It's a good day for testing because, or is it, should you test on a bad day? Dear me, certainly again, it's both firing out there, it's going long and like I said, it's what we kind of would expect from a 28.5 degree iron, but there's a big notable difference which I'll talk about after this shot. Last one. 3Ds and balls, last one the worst, a little bit heavy and a little bit down the left. Big difference for me is sound and feel. They've changed things, I don't know what it is and again I won't pinpoint exactly what has made the difference, but there is a difference and it's a massive improvement and for me this again is back down where game improvement irons are making this massive leap forward. This is another one where we've seen where they're doing, they're packing it full of power, full of forgiveness, decent launch angles, bigger sweet spots and then they're throwing in a bit of softer feel as well, so really, really impressed so far with that, but once again, as I've said with all this, next stage of this review is going to take out there on the course and I'll give you some feedback in real situation. So summary on the M6 irons, I've just literally done the finishing bit from on the course with the M5s, it's not too dissimilar and the clubs aren't too dissimilar either, there is literally just that little bit of added bulk on the M6 iron for a lot of people that will appeal to them, they'll give them a bit more confidence, I was more than happy at every point in the address throughout the course, it sat very, very nice, it wasn't put off by the size or bulk. Y yardage wise I'm not seeing it being overly long, although it's strong in terms of loft, I'm not seeing that overly long distance wise, but I'm seeing a real high ball flight and launch angle off every club and the two clubs side by side in the M5 and M6 even though there's a degree and a half separating them in terms of yardage is not a great deal of difference to be honest with you, so that again, always this myth about who I'm not myth so much but loft equals distance, yes it does but it depends in terms of launch as well and I think this ball goes high, it's easy to launch, very playable in and around as well greens and things like that, so once again impressed, the whole range of this thing has been impressive out here on the course, it's performed well or have I, who knows? Anyway back in the warmth let's do a bit of a round up on this one. Right so M6 irons or what did I think? Well let's talk about, I'm going to talk about looks first of all on this one, for me not overly keen on the finish that they've put on it and I'll reach for the club now, I don't know, I don't know what it is, it doesn't look, I mean comparing it to the M5 for example, the M5 I think is a really really good looking club, the colours that they've put in this and this little carbon insert in the back of it, I don't know, it doesn't, it doesn't appeal to me, it doesn't look, I suppose it doesn't look classy enough for me, that's the thing I would say and the bulk is there, they've done a real clever thing which I mentioned in terms of the top line, in terms of thinning it down by giving it two different colours but so when it's sat behind a ball it looks nowhere near the size and bulk that it probably actually is, there's a fair bit of offset again which I would not be comfortable with on a personal level but we're going to remember this club is aimed at perhaps the higher handicapper, it has that bulk and mass will appeal to others but it wouldn't necessarily appeal to me to gain this one and much preferred the looks and visuals and compactness of the M5, having said that that's a negative so there are a lot of positives believe me, the first thing that I'll mention is the sound and feel, I mentioned it in the M5, they've greatly improved that element and that's a huge huge step forward once again with a game improvement iron of this sort of size to get the sound and feel that they've got is absolutely bang on the money that's a massive leap forward what they've done here, that's a massive and for me it's a big deal not for everybody but for me it's a big deal, I don't like the sound and noise that has come out of these irons in the past and I'm not talking about tailor made I'm talking about everybody's irons in terms of when you get into this game improvement category but they've really changed that then when I throw up this dry ball data you will also see one other element that is incredibly different 28.5 degrees worth of loft don't forget, 113 ball speed 19.8 so again incredibly high ball flight launch angle rather for the loft but it's a spin number, 5 3 spin once again and we've seen it I've seen it in the M5 iron I can't believe I'm getting that number out of a game improvement iron my spin number would drop off the charts with these kind of things and 5 3 spin on a 28 and a half degree 7 iron don't forget is incredibly good for me 32 pki 161 carry, interesting bit for me was that it wasn't that and I think I've sort of slowed down the old swing a little bit in the last year or so I'm not trying to literally rip these balls in the range so but there was a drop off in yard is probably a little bit shorter than I was expecting I would expect this club to be out there at sort of 170 plus, arguably like you said the swings I was putting on on the day but also the loft difference the and I've had this argument many times before about this is 28.5 degrees the M5 7 iron is 30 degrees and there was virtually little difference in terms of overall average yardage with dry ball data so there are always different elements that go into me loft launch spin a lot of different things go into that equation never as plain as simple as just all this club is lofted strongly at 28.5 degrees and I think you see that in this 7 iron easy to launch got a spin number out of it performed really well on the course too bulky for me personally but for a lot of other people that will be a superb club it's got plenty of help in there plenty of assistance I think it's a real real winner once again a movement on from last year's models not massively maybe not uh not maybe not huge amounts but for me some major differences sound feel and spin they're a big leap forward from last year's models but we'll have a closer look and we'll do some heads heads in the week's head but for the first look massive thumbs up on the M6 irons for the right kind of golfer anyway thank you for watching as ever comments down below and we shall carry on