Good morning, y'all, and welcome back to Chicago. We're here for KubeCon Cloud NativeCon, and it has been an already thrilling show so far. Day one was absolutely packed. John Furrier, my co-host, we've got our analysts here. John, I'm going to go to you first. What's your take on day one? What are you looking forward to, day two? First of all, day one was great. First of all, the crowd's perfect, and the place has spread out, so it looks like it's not as many people here as it seems, but it's definitely packed. They've got good numbers. The booths are great. The companies are representing well. You can see it's thriving from an ecosystem standpoint, so check the box on that, and that's a good sign, especially in this market, where there's no IPO there and there's growth VC funding kind of gap challenges, but just overall, great ecosystem. You know, I think day two is going to be a great lineup, and I think the trend is platform engineering in practice starting to show more things that are next level, security challenges, observability, best practices, things on how to recover and roll back, DataDog was on the keynote this morning. I saw their best practice. So seeing people who have the experience with Kubernetes put into action, and I think that's the theme you want to start to see more of, is okay, when it becomes invisible or under the covers, the other stuff will be enabled, so I think we're going to see a lot more of that today. Yeah, I think so, and speaking of expertise and very smart people, we have a brand new guest analyst with us, Joe Peterson, welcome. Joe, how you doing today? I'm doing great. Thank you so much for having me. We are so lucky to have you. We feel very lucky. We've got two women on the stage, folks, and two Petersons. This is a world record for you. Woo, woo, woo, very excited to be repping here in Chicago. Joe, give us a little bit of your background. You've got a lot of security experience. Yeah, so my day job is Vice President of Cloud and Security for Clarify 360, and we predominantly work with the holdings of two large private equities, Post-M&A, and we go in and we do a benchmarking and a baselining of what their current footprint looks like, so lots of great experience across a number of verticals. Yeah. Super fun. And then in my spare time, I am the Chief Analyst at ClearTech Research. So, yeah. Do you sleep? Wasn't wild, a little bit, a little bit, a little bit. Yeah, occasional nap. Well, we're lucky to have your brain and your insights, and Dustin, day two here as an analyst with us officially. How are you doing this morning? Doing well, just came out of the keynote. I would think, highlight from the keynote today, the Datadog session was incredible. Really transparent post-mortem on the March 8th, 48-hour Datadog outage, which was- Yeah, 60% of their cluster, right? I saw something like that, or their nodes, yeah. Now, I'm going to confess, I missed it the first time around because on March 8th of this year, I was in the middle of hiking the Camino de Santiago across Spain. Casual flex. Casual flex. Yeah, Cindy. Whoa, got a good life there. That was so much fun. But yes, I came back to, it's a good one. We should talk about that sometime. But yeah, it was great to actually get the deep dive into the post-mortem, what happened, why it happened. I will tell you this, I would attend an entire conference agenda, all day agenda of post-mortems. Real companies talking about what they've seen in the fire, in the furnace, when the stuff hits the fan. What do you do about that? And look at that in like, how do we fix that? That was a highlight, it was a 10-minute highlight of a hour and a half. Take us through the use case for the hooks that didn't see the keynote, it's online, YouTube, you'll get to watch it later. But what was the net net, what was the highlight, what was the thing that they learned, and how does it speak to the Kubernetes journey? Yeah, so basically, Datadog admins woke up one morning and had this tremendous outage. They weren't getting responses from a number of their, this is Kubernetes, from a number of their Kubernetes clusters. They said within minutes, there were 493, was the note, I took 493 people on a Zoom meeting as they're trying to root cause and figure this out. And they walked us through step by step what they did to identify, they first, they thought it was a network problem, turned out to be in the network area. They found that if they were, that it was only certain nodes on reboot, that when they would reboot those nodes, those nodes would come out without internet access. They were able to narrow it down to system D and specifically system D networking. And then further root cause analysis showed that it was the unattended upgrade function that there was an upgrade that runs every night and kudos to Datadog, I would say, for running upgrades, for doing their updates on a daily basis, for using an unattended upgrade function. This is on Ubuntu, I worked on that by the way. There was a bad update that went in there around system D. And the real root cause was that system D upgrade, that system D network upgrade removed any other network configuration that wasn't coming from that package. And Cilium, which is part of the Kubernetes Network Stack, actually installed some of their own IP tables, rules as part of the Cilium upgrade, and system D ignored those, deleted those, and that brought us to a state where these machines would come out, or these nodes would come up without networking access. So once they'd identified that, they were able to walk this back, reinstate their rules, disable that reboot, and they were back up and running full scale within a day. They also went through some of the cascading effects of then trying to immediately stand up 50% in scale, massively to respond to this. And there were some second order lessons learned. Fascinating talk. I think to me, I think that speaks to what Savannah, we were saying earlier about how architecting the future, which is the theme of the show, the platform engineering conversation is about systems architecture. Consequences, rollbacks, if something fails, it's going to be collateral damage. That's a whole discipline. That's not just a dev thing. The application development is going to do their thing. AI will be a great thing for them, but I think this is like the nerd infrastructure, like dream scenario. Kubernetes is getting better. Still needs work to get done. You got security challenges cropping up. So it's kind of like a builder mode, kind of architect mode. How do you build that operating system? And I think that's a good sign for Kubernetes because that means it's going to the next level. Right, and you bring up a really good point because we're starting to see as it matures that the platform, the platform ubiquity across the BU's is pretty important. So everybody was kind of doing their own thing and silos now, right? And they're realizing that this is an important component of their architecture. So they need to sort of get on the same page. Joe, what's your research point to when you look at the market customers as you've done benchmarking. This is a big conversation around how the transformation journeys are. What does a customer go through when you look at a customer environment, when you get to look at that holistic view and then set the table for platform engineering or an environment that has set up for governance, set up for automation, set up for good network functionality. What are some of the things that you see here emerging from the show that points to maturity? Well, so first of all, adoption. According to markets and markets, we're going to triple the spend in Kubernetes in the next five years. Almost triple. So it's going to go up to about 7.5 billion in spend. What I'm seeing, and I'd be curious to hear what your take is as well, is that, you know, we're still doing lift and shift, right? So older environments are moving as is to public cloud and where we're seeing innovation occur is the net new environments being built and those are being built in containers. Absolutely, and look, lift and shift, everything that's easy has been lifted and shifted, right? So we're down to, you know, the hard stuff, the long tail, there are some solutions out there. There are, there's plenty of technology here that helps you run a great Kubernetes on your own on-prem infrastructure, right? So I think that era of the easy lift and shift has been shifted. Net new, you're absolutely right, is cloud native from the start. And in fact, many that I've seen at least are architected such that they can go to any cloud. You know, you just came from the super cloud event, right? The idea that you can treat almost any cloud infrastructure, you know, take advantage of some advantages here or there, but to architect an application such that it can run in any of the hyperscalers or on-prem, you know, that's one of the huge advantages you get from net new development when you adopt cloud native methodologies. And you know, the conversation around abstraction layers too is interesting because if you take that multi-cloud or super cloud approach, it's not easy, right? And it's like, no vendor can own that, right? So the question is, is that, are we going to see a platform of platforms? It reminds me of the old mini computer days when you had network operating systems, SNA was IBM, DECNet was digital, and then you had proprietary NOSs, network operating systems for the environment. So like, what I worry about with cloud is, you have Amazon stack, you got Azure. Who's going to own the control plane? Who's going to control the semantic layer? Who's going to control the abstraction across clouds? Or do the companies build them? So I mean, this is like a kind of a middleware kind of vibe going on here. Who's going to control the middleware between clouds if you assume the clouds are the hardware? I mean, not to oversimplify the cloud, but they're kind of the hardware. I mean, this is a big conversation that no one's really kind of settled on yet. Do the ISVs, like the snowflakes of the world, build their own end-to-end across clouds? Or is it some merging standard emerges? Good question. What do you guys think? Yeah, I think snowflakes super interesting and that they're coming at it from a purely software perspective, obviously depend on lots of hardware, but coming at it from a purely software perspective, that's quite different than IBM right behind you, Dell right behind you, Dell EMC and so forth. There is plenty of incentive for the hardware vendors to invest in Kubernetes in that control plane. Call it middleware if you want, but that helps ensure that the best of that hardware gets deployed. Snowflake, though, I think is interesting in that they said bring us any of your hardware, you know? What's interesting, I watched watching the OpenAI first developer day happened on Tuesday or Monday, I should say, and OpenAI got a major kudos what they're doing with the chat GPT, but one interesting thing on Dustin on that point is maybe it's a feature, not a bug, maybe this idea of a plugin or some AI code that could be low code, no code to handle switching between environments intelligently because what chat GPT did this week as interesting was our OpenAI is they're adding plugins to the interface, so you don't have to change context. If you want to tap a data source or Canva, for instance, you can hit Canva and it'll do images for you inline into the user interface. So question is, does the interfaces dictate what comes in? We had people on theCUBE yesterday saying press a button and infrastructure just develops. Like that's the dream scenario with AI and generative capabilities on the infrastructure. It kind of opens up the kind of a new way of thinking. This is kind of interesting concept when you think startup opportunities. I think so, yeah, I mean, I think I think what we're going to see is a lot of collaboration in this ecosystem to your question, I don't think we know the answer yet. I think it's going to be a lot of different players that are going to build the little pieces and whether or not there's a grand monolith that owns that ecosystem. I do think the players in the space are going to be a lot of these rising startups and a lot of the projects that we see here at CNCF that are going to help build the ecosystem. Well, I got a question I want to ask you guys, you know, mine, because I got a text from a friend. He said, John, ask people where the white space is. This is entrepreneurs, so the startup culture. Where's the white space for opportunities for entrepreneurs in this market? And we kind of talked about that yesterday. What is this market, right? So if you're an entrepreneur, where are the opportunities to get a position and build value? Oh my gosh, I'm going to go first because I know you're going to have a richer answer given your background. So I think cloud security and container security is still a space to play. You know, I did a primer on cloud security for a bunch of cloud people and they were asking me to explain what the acronyms were, right? So cloud security is still evolving and you're starting to see platforms evolve that are making siloed tooling become a thing of the past, right? So that's the trend in that space. Reading this really interesting report from Red Hat on the Kubernetes state of security and they were saying that lack of security investment is slowing down development in the space that is underfunded, you know? Underfunded and one of the recommendations meaning enterprise-wise and one of the recommendations was to look more holistically at your environment which I thought made sense, like look not only at the security of your CICD pipelines but look at the security of your infrastructure side and see if you could get a tool that sort of bridges both sides. But I'd love to hear what you have to say. I'm glad you covered the security space because I think that would have been my number one. Absolutely, lots of opportunity to innovation around security, money to be spent which I think is super important if you're going into a VC pitch you want to show that path to revenue. I think you covered that really well. You can't not say AI. I think the next step for AI we have now laid the infrastructure for artificial intelligence through Kubernetes as that scheduler call it the large language models and all the open source on top of that. Now bringing that to other industry verticals I think there is a ton of hundred million dollar to billion dollar businesses to be built around taking those nuggets of AI and that insight and applying it to ed tech to health tech, you know, elsewhere. Okay, I think that one. Now if you really want the next frontier the thing that we'll be talking about in the next five years, at the end of the decade it's quantum computing. I was just going to bring up quantum, I love the use of that. And as that sounds, yeah, yeah, yeah. So did AI sound science fiction yeah, yeah. 10 years ago. Well actually open AI is only seven years old. Yeah. And they were nothing on the radar. We are still in such an early stage. I mean people are talking about being experts in this, we are, it's the landscape is still being created by the earthquakes and the movement of the tectonic plates in the technology sector. Jeepers that was an analogy. I wanted to ask you a question that you just said that was really interesting. So one of the, in our morning moment yesterday we were talking about the how Priyanka was saying Kubernetes is maybe having its Linux moment right now. Do you think that AI is the reason or part of the catalyst for why that Linux moment may be happening for Kubernetes right now? Not necessarily. I wouldn't draw that as a direct correlation. Look AI is a, it's almost a predetermined outcome of 20, 30, 40 years of computing. I went to, I got a computer science degree 25 years ago, right? And I took AI classes in the 90s, okay? So that technology has been around. What hasn't been around is the incredible GPU and hardware. The compute baby, yep. What hasn't been around are the exabytes, petabytes and exabytes of data that we need to solve some of these, in the marriage of those two, it's now practical. Some of the algorithms that we're using for AI, 30 years old, you know, that's 30 year old. And Kubernetes helps with the dynamic scale out. I wouldn't draw that as like, you know, the exclusive driver of Kubernetes Linux moment. Yeah, and I think the Kubernetes growing up is a proof point that it's going to be part of the machinery, right? Under the hood, behind the curtain. The antel inside is what I was saying. And I think the opportunity for startups, again, I've been through multiple innovation ways. I've seen kind of these major inflection points and there's a common theme that crops out of all of them. When you have an interface and a user expectation shift, like we're seeing with AI right now, the innovation is the interface, not so much the magic of degenerative content so much. PC was interface shift. Accessibility, I think. The web was an interface shift. Bubble, you could argue, and then Dave and I have already done podcasts, I would say is an interface shift, although that's less relevant now. In all those markets, there was three things that made people successful. Make it simpler, faster and easy to use, don't understand. So if you look at anything, whether it's Kubernetes or AI, if you just make it simpler, faster, and easier to intuit or use, then that could be the single pane of glass model for observability. It could be a security integration play. It could be setting up automated data pipeline. I mean, there's opportunities everywhere. So I think it's going to be a renaissance. I think that would be a great answer to your friend that texted you about where is the clear space to innovate? The first, I would ask a clarifying question. Are we talking about innovation on the order of tens of millions of dollars, hundreds of millions, billions or more? The quantum computing revolution, that's not a small problem that a couple of kids in a garage, I don't think, are going to crack in the near term. There's going to have to be a whole lot of it. If they are, that's some garage. And they've got some serious hardware in there. The quantum disruption of the quantum. That's Dave Brown's garage. I would like to go visit that garage if that's the case. The quantum disruption is going to be radical, what's scary about quantum in a good way is the upside, but the other side of that coin is, will it scare people to death on crypto, encryption, cracking, does it become too good of technology that, remember the old OS2 DOS days, right? Better operating system, but no one ever used it. Because the standard was inferior. But to me, I think that's going to be an interesting challenge. The point is that their interface point, I think, is the really salient here, which is there's lots of opportunities to innovate by just making things a little bit easier to use, a little bit more accessible, a little bit more available. De-increasing complexity. There's always a market for it. Remove complexity. It's embryonic. I mean, it's going to grow faster. As more people use AI, they're going to figure it out. That's why I was honed in on this plug-in concept that the open AI guys introduced. Essentially, it means you can just plug in a GPT model into their system and be the go-to resource for that versus switching out of it. It just completely game-changes it. I think we're going to see a lot of those little bridges. No, I think you touched on something there. So my answer to your friend's question, now that you asked me, John, but I'm going to go ahead and give it anyway, is I think you just hit it there. It's going to be on that ease of use, that education side. So to your point, we're tripling down on investment in the Kubernetes space. We know that adoption's been slow here. A few years ago, Gartner was saying only 15% of companies had even established their strategy for container management. So I think there's going to be an opportunity for that zero to 10 moment for new users, whether they be new to the space or wanting to learn how to integrate Kubernetes into their system and make it operate better. If you can be the red pill that you can take that all of a sudden you can get up and go with Kubernetes, I think that's actually a big opportunity. We see Kastin by Beam doing that. They've got an educational program. We've had Cassandra on the show with her dad at JFrog who teaches teenagers how to use Kubernetes. These are tiny little examples, but if you can do that level of education at scale for more people to understand how to leverage this technology and leverage AI, I think there's a massive opportunity that will increase the diversity of players in that tech space as well. Yeah, it's awesome, it's awesome. And I think this event's going to be bigger. I did a tweet last night. It's getting bigger every year. Joe, I'd love to get your take since your first time as an analyst attending. Yeah. What's your take of KubeCon? I love it. I think it's, I love the energy. I would agree with some of what you said about this being a great big space, but there's lots of people here. There's 200 sessions. Wow, I didn't realize it was that many. 200 sessions, and what I thought was interesting is that almost 100 of them fall into four buckets, right? AIML, security, observability, and darn it, I forgot the last one. Four buckets. Puffer engineer. I know. That's a great analysis. Yeah. You must be an industry analyst. Thanks for coming on theCUBE. What are you working on? Give a quick plug of some of your work because we're big fans of you, what you do, and we appreciate you coming on. Give a quick plug of what you do. Oh, thank you so much. I'm going to say it here, and I really would like to get your take on it. To your friend's question, I would say that AI security, hashtag AI security and or governance could be the next bacon in AI. Next bacon. All right. I love bacon. That's a great call. Great call. Sizzle, sizzle. Now I want some bacon. On that note, folks, I think it's time we wrap this morning segment and get to our guests for the rest of the day. Justin, Joe, thank you both so much for being here. Your expert analysis is priceless. John, always a pleasure to sit and giggle next to you all day long. And thank you for tuning in, fellow cloud native community friends. My name's Savannah Peterson. We're here in Chicago. You're watching theCUBE, the leading source for emerging tech news.


Subscribe? Welcome or welcome back to my channel. I've been making youtube videos for a little over a year now and every single time I post a video I always end up getting comments about how my videos have like these scammy ads all over them.  It's so so obnoxious. If you're new here my whole channel is about exposing scammers and shady people and shady businesses and business practices. So for some reason the youtube algorithm thinks oh she talks about scams let's show her audience scammy ads. Makes sense. And for a while I didn't really mind that much because the ads are playing to an audience who already knows that these ads are for scams. At least most of the people that watch my channel I would hope at least know that it's probably not a good idea to buy and to get rich quick schemes or MLMs. So the ad is kind of speaking to the complete opposite audience than the one that they want to target. And it's sort of hilarious and also funny because they're definitely wasting a lot of ad money. So for a while I was unbothered by it all but lately I've been seeing some of the most ridiculous outrageous ads I've ever seen in my life on my own videos. And this is one of those ads that I'm going to show you guys today. I found it on my own video. This ad is an ad for a supplement. The supplement industry is an industry that has very very little regulations.  And it's super annoying as a consumer who wants to be educated and make smart purchases because I'm sure that not all supplements are bad. But there are so so many bad, ridiculous and even at times dangerous supplements that it's hard to tell what is legit and what's not.  Cue the fitties. All the random pills that are supposed to cure you of any imaginable disease, etc. Basically there is so so much snake oil out there when it comes to supplements unfortunately. But apart from letting you guys know that this is an ad for a supplement, I thought we might as well just react to the ad. Watch it without giving any further information away so that you can experience what I experienced the other day. Here's the ad that for some reason, somehow YouTube allowed on their platform. Did you know that the average adult has 5 to 20 pounds of toxic poop in their body at any given moment? That's what the ad opens up with. The average adult has 5 to 10 pounds. 5 to 20 pounds of toxic poop in their body at any given moment. First off, toxic poop. It's only been one sentence and there's already so much to unpack in this one sentence. Toxic poop. Between as compared to non-toxic poop? What? What? I'm sorry for people that don't like to poop. I should probably put a disclaimer. This ad deals with the subject of poop. Let's just look that up real quick. And the first thing that comes up is from this supplement company. And then on their blog post for this company, it says, scientists confirm the average person is carrying around between 5 to 20 pounds of poop in their body at any time. This blog post goes on to have a ton of health claims. So many ridiculous health claims. I don't even have time to unpack it all. Apparently this supplement can help with health concerns like obesity, diabetes, insomnia, mood swings, depression, arthritis, inflammation, joint pain, heart disease, autism, memory loss, increased aging and much, much more. Oh my goodness. But it doesn't even go into the stats of how many pounds of poop are in my body. I want to know. You said scientists confirm that there's 5 to 20 pounds of poop in my body at any given time. I would like a confirmation of that. Where are the studies? Please give me more information, but they don't. There's an article by McGill University that explains this a little bit further.  So say the promoters of various colon cleansers. The message is clear. Our colons are loaded with a repulsive, noxious sludge and the result of an improper diet in a toxic environment. This putrid goo sticks to the wall of our colon, boosting our body weight. The result?  One product claims that we have anywhere from 6 to 40 pounds of waste, feces and undigested food stuck in our body. Another one compares the weight of the waste to carrying a bowling ball in our gut. Now let's get real here. Have pathologists who have carried out thousands of autopsies seen pounds of goo encrusted in our intestines? No.  No. Have radiologists who have pursued thousands of x-rays of the colon noted the buildup of mucoid plaque? No. Why? Because it doesn't exist. The term itself was the invention of naturopath Richard Anderson, who created a rise and shine, a popular colon cleanser. So he didn't have any motivation behind the invention of this insane amount of poo in our gut, right? It's not like he wanted to sell anything in particular. Seems crazy, right?  Is it any wonder that colon disease is on the rise with all this waste just sitting inside your body? Literally none of it is true. Why is this allowed? Why is it allowed that someone can get on an ad and literally spew absolute doo doo? Click through to read the hilarious reviews about how consistently effective zoo poo truly is. The supplement is called zoo poo. Who is buying this? You can search the scientific literature high and low and you will not find any proper controlled trial of colon cleansers showing they have any health benefit. Any health benefit. But for some reason, woo woo culture is obsessed with the colon. For whatever reason, there's colonics, which have been proven to be very dangerous. And now there's colon cleansers. Although the specific ingredients in these products vary, they all contain some sort of laxative. So if you're feeling sluggish bloated around the waist or even just plain backed up, and give zoo poo a try for 14 days, it will clean you out guaranteed. It will clean you out guaranteed. It contains the perfect dosages of all natural ingredients that will get things moving and flush out that extra waste you're carrying around. You'll feel amazing when it's gone. This supplement has bentonite clay in it. Clay that you're ingesting. And I love how right on their website it says digestion, cleanse, weight loss. So not only are they marketing this as a way to avoid apparently every disease ever, but in the process also lose weight. I literally hate this. As this toxic waste builds up, your body becomes a walking vessel of fecal toxicity. Once this happens, you begin to experience excess fatigue, weight gain, digestive issues like IBS, leaky gut syndrome, constipation and diarrhea, skin conditions like eczema, psoriasis, rosacea and acne, brain fog, focus and attention issues, lute swings, anxiety and depression. There's very little studies on all of this. When researching for this video, I did find there are a lot of newer studies coming out on gut health and how that impacts the rest of your body and the rest of your health. But it's still a developing area. And so these set in stone claims are just not backed by extensive research. It might be a bit embarrassing to talk about your poop, but it has a huge impact on your health. And it's something that you have to understand if you want to look and feel your best. Even this guy looks like he's so over it. He just looks like he's like, what am I saying right now? Understanding the health of your digestive system begins with understanding your poop. If you're not producing well formed stools after each meal, then you're most likely suffering from a toxic buildup inside your digestive system that's secretly causing all of your health concerns. New research suggests that most people have up to eight undigested meals in their gut at any given moment or possibly even more. No, it doesn't. There is no research. Immune system and overall health and well being. As the quality of modern food is deteriorated with the invention of unnatural, hard digest, heavily processed foods loaded with artificial flavors, preservatives, sweeteners and GMOs, we're unknowingly putting a massive strain on our digestive system that's leading up to an epidemic of illnesses. It's just so frustrating because it's absolute fear mongering. First off, if you suffer from digestive issues, constipation, weight gain, gas, bloating, indigestion, skin issues, fatigue, or a slew of other health issues, your bad gut may be slowly and secretly talking about how inevitably our environment and our food quality is getting worse and worse and so your body can't handle it and therefore you need this supplement to fix all of your problems while also preying on all of your insecurities saying you're overweight or you're tired all the time and not productive. All of your issues that you could possibly be struggling with are due to this poop problem and you need their supplement to fix it. Why is this allowed on YouTube? Literally just in this ad so many health claims were made, it's insane and just so wrong. So first off, I will be leaving a link to the FTC page to report this ad. If you, like me, believe that they're making a lot of unsubstantiated health claims. This product is being sold by umzu, umzu, and on their site they have a ton of different supplements that they sell. Zupu itself is $30. So it turns out this extremely charming dude is actually the founder of umzu. His story on his website is a little concerning because it sort of insinuates that he cured his own tumor through naturopathy and his knowledge of supplements, which is very concerning. Lots of red flags. I just have no words for this ad. No words. And so many questions. After watching that ad the first time, I kept thinking this has to be a joke. There's no way that this is real. But the ad kept showing up, it kept following me around, and I ended up finding out, yeah no this is legit. They are literally trying to sell us a poo-be-ridding supplement. Also it's a special kind of irony considering lately some of the videos that I've been posting on YouTube have gotten demonetized or limited ads, and the notification that YouTube gives you when you get demonetized or you get limited ads is that this content has not been found to be suitable for all advertisers. If I swear or I talk about a subject that's a little bit taboo or a little bit controversial, YouTube is like yeah no this is not suitable for all advertisers. Meanwhile there are advertisers. It might be a bit embarrassing to talk about your poop. Meanwhile these guys are YouTube advertisers.  These are your advertisers YouTube that my content does not sue. We're in the first sentence they talk about how pounds of toxic poop live in your body. Did you know that the average adult has five to 20 pounds of toxic poop in their body at any given moment? My mind is blown. So anyways that's all for today's video. I just really wanted to share this ridiculous ad that I found because what is this? There's definitely been a lot of ridiculous ads on YouTube so if you have any other ads that you'd like me to review then let me know down below. And if you made it to the end don't forget to subscribe and I'll catch you in the next one. It will clean you out guarantee.


Should we switch? Ladies and gentlemen, it's my pleasure to introduce the promoter who will speak on relaxation, oscillation, limit cycles, and astronomy on your unit. Thank you. I'm very happy to be here to talk to the audience on the Poincaré. So I'm going to divide my talk into two parts. In the first part, I'm going to say a few words about... More or less, I will try to put what I'm going to do in the first, second part in perspective with what was done by Poincaré and others at the time of Poincaré a little bit later. And I'm going to explain what is written on the board. And afterwards, I'll switch to more recent things and to some work we have done with Martin, who is here. So this has been... Oh, she's not... I'm sorry for my poor handwriting. So everything basically will be concerned with limit cycles. And as probably everybody knows, the theory and the word limit cycle goes back to the second memoir of Poincaré sur les courbes définies par des équations différentielles. And this is a place where I introduce... This is chapter six, published in the Journal du Mathématique pure des appliqués in 1882. And in the same article, it defines the theory of limit cycles and what we call now Poincaré application. And the others have explained that before me. And what I'm going to say now is that this series of memoirs were about a purely mathematical problem. But afterwards, for reasons I'm going to explain, at the beginning of the 20th century, Poincaré became interested in a problem of modern science, modern science at the side. And even the most modern science of this time, which was the propagation of electromagnetic waves, what he called Hertzian waves, in a long distance. So the first, what triggered the interest of Poincaré in this was the success by Marconi in 1901. And Marconi managed to transmit Hertzian waves across the Atlantic. So this was between a place called Poldu, Poldu, I don't know to pronounce this, in the western part of the United Kingdom, to Canada, to Newfoundland. And this was a big success because given the method used at the time for emission of waves and for the reception of waves, I'm going to say a few words on that. So Poincaré became interested in what Marconi had done. And he wrote a paper in 1910, I'm going to comment a little bit afterward, about this paper of 1910. But for my topic, what is important is that in 1908 and 1909, Poincaré delivered a few lectures called the Telecommunication in France. And this lecture has been published in Revue d'Electricité, and this was discovered only recently in this publication by Giroud. And in this series of lectures, he solved, he considered two problems. One problem, which is the one connected with this paper in 1910, was the transmission of electromagnetic waves on long distances. And it was not the first one to be interested in that, but the big problem they had for the transatlantic crossing, but you had the emission in the UK here, the reception in Canada on the other side of the ocean, and of course we know that the earth is not flat, is round, is spherical, and so because of that, the inline transmission, direct transmission by ray, is impossible. So there is no link, you would have to cross the entire of the earth, which is not possible because the earth is a too good conductor. So the only possibility, supposing that outside of the earth, you have a uniform medium, which is not true. So the only possibility was to look for the transmission at the diffraction, the diffraction across perpendicular to the straight line to the geometrical optics. And what Poincare did compute in this paper published in 1910, was the amplitude of the diffracted wave. And this amplitude, this is a complicated problem, a difficult problem by the way, which was considered again by Fock, the Russian mathematician Fock in the 1940s. And what he found, Poincare, was that the amplitude of the diffraction of the diffracted wave was too small to explain the reception by Marconi in Newfoundland. So which was nevertheless, Marconi was successful. And at the end of his paper, Poincare suggests that the reception was made possible because there is a conducting layer in the upper atmosphere, and instead of having a straight propagation, actually the propagation is in between two conductors, the conductor in the upper atmosphere and the conductor on the ocean, and which is actually what happens. And because of this bouncing of the rays, so the direct, not so direct, but the emission and reception of the wave is possible on long distances. So this was the first part of this conference and on the paper in 1910. In the second part, which is very short by the way, it was considered the other problem posed by the radio waves, which is the generation of oscillations. You have to have electrical oscillation because by the equation of Maxwell, with a time-dependent current in an antenna, you radiate some energy outside of the antenna, which is basically the energy which is received at the other end in Canada. And to have an oscillating circuit, at this time it was in 1908. So the only way to have oscillating circuit was to have RLC circuit. This is a capacitor, an inductance, and the resistance R or R, by the way. And in between you need to have some amplifying system, a system that feeds energy into the oscillation. And the system used by Marconi and many others was an arc, electric arc. This electric arc has many instabilities. It's not well understood even yet. And because of the instabilities, you have an amplification. So the equation for this problem are written here. You have the RLC, which is actually RL, which is L, and C is H. I'm using the same notation as Poincare. And the non-linear element and the amplifying element is contained in this theta function, which is non-linear, this theta function as a linear and non-linear part. Actually it's not absolutely, Poincare probably mixed two papers, two things in his article in the conference, Téphonétien Field, because he writes this equation and then he writes another equation which is closer to the truth by connecting, coupling two equations, one equation for X, which is the charge of the capacitor and an equation for the temperature in the arc. So he has a system of two couplets equation and he claims without proof that this, which is, by the way, but this is true, that this set of two equations has actually stable oscillating solutions, which comes by, which it's not a system in a plane, for a plane flow, because there is a second derivative here. It's for three-dimensional system. And Poincare also gives a constraint on theta to have sustained oscillation. So, and this paper is apparently the first one, for the first paper where there is a connection made, this was pointed out by Giroud, there is a connection made between the limit cycle and the periodic solution, periodic in time, periodic solution of differential equation, nonlinear differential equation. Stable nonlinear. So, the same year, 1908, was the beginning of the modern era of radio waves, which is the year of the invention of triode by an American inventor called Leder Forest. And this triode is the element that replaces the arc in the Marconi and other experiments. So you have an amplifying element, which is much easier to use than the arc lamp. And the triode of Leder Forest was much improved by someone called George Abram, who was the director of the Laboratory of Physics of Eco-Normal, 100 meters from here. And so we are in 1908. So after 1908, before the war, because the war was getting close, basically all the research on all sides was becoming completely secret. So there is almost no publication on this topic between 1908 and after the war, after World War I. So after World War I, there is a first paper by Blondel, I think, André Blondel, who worked on this electrical circuit, the one I have written here, with the arc lamp substituted by a triode. And Blondel wrote an equation, I'm going to show again and again, which is called now a Van der Poel equation. So Van der Poel equation, a year later, wrote a paper on the theory of triode oscillations. So it's not very much a theoretical paper, it's only a paper where Van der Poel only writes the equation, and he says that this is a good equation for the oscillation, but it does not bring too much argument to prove his point. And his point was only proved in 1926, that the equation, which is called now the Van der Poel equation, has a self-associating solution, and this is a paper on relaxation oscillation. This is, as far as I'm aware, the first paper where you have this notion of relaxation oscillation that I'm going to talk about. And this was published in the London Journal of Science. And Leonhardt, in 1928, made all the theory of the Van der Poel equation in the two limits which are convenient, the limit of small amplitude and the limit of large amplitude. Although the paper by Balthazar, how to pronounce, Balthazar Van der Poel is a paper where the equation, which is a set of two coupled equations of first order, he solved the equation by a graphical method, geometrical method. It was not, he did not solve the equation by analytical method, you know, he used only this graphical method, which has also a long history and which is basically the kind of method which is used nowadays, you know, because when you solve differential equations with computers, you are following the same idea as Van der Poel. But only with some much big improvement in the technical method, but the idea is the same. And Leonhardt, you know, again, he solved analytically the problem that was considered by Van der Poel. So, this is basically the history and afterwards, you know, of course, you know, I should need 100 blackboards to say everything that's been done on this subject afterwards. But this is basically sets the stage of the early part of the history of the theory of relaxation oscillation in a simple dynamical system. So, let me try to do it well. So, I, oh, before, sorry, before we continue, I should say that this was also found by Giroud in 1933 in this building, this building where we are now, probably in this conference room. There was a conference on non-linear phenomena organized by Van der Poel and others by Russian. And the topic of this conference, it was years ago, the topics of this conference was precisely the relaxation oscillations, 1933. And one of the attendees of this conference was Yvrocar. And some of us, you know, probably have, we have, please, Martin and myself and others in this audience. I've heard talks by Yvrocar, you know, who wrote a paper on the dynamic, the vibration, which is still well known at least in France. By the way, I should say that this is something which is also pointed out by Giroud and others. You know, the paper after Poincaré, Nublondel, Van der Poel, etc., never referred to, in Yvrocar in particular, never referred to Poincaré. It was only after the Russian came in, Van der Sam, and his students that they began to refer back to Poincaré. You know, the French, or they had completely forgotten or they didn't want for some reason to refer to Poincaré. And from what I remember, you know, classes by Yvrocar, you know, he never mentioned the name of Poincaré, if I remember well. So, now, this is a quotation of a, well, there are many pure mathematicians here. So, this is a very good, very interesting piece by André Valle, l'avenir des mathématiques. And he says that the study of the equation of Van der Poel and of the oscillation called de relaxation is one of the very few questions, and a few interesting questions asked or posed by the contemporary physics because the study of nature, which was, which used to be one of the main source of big problems, big mathematical problems, seems lately to have been more borrowed than it has given, which was very much in the style of, so for those of you who had a chance to meet André Valle, you know, it was a bit critical, you know, a bit ironical, very much in the spirit. But it's true, you know, because this was written in 1946. And if you have a chance to find this piece of work by Valle, you know, it's very interesting because, yeah, the part devoted to the application is very small, you know, and he has a long development on the Riemann hypothesis, and he sees the Riemann hypothesis as the main, the biggest problem by far in mathematics, you know. And I think at this time it wasn't maybe a way which was original. Of course, nowadays, from what I understand, everybody believes that this is the biggest challenge for mathematics today, you know, the Riemann conjecture, but at this time, maybe it was not. So let's follow the advice of André Valle and try to study the Van der Poel equation. So by the way, this was in 1946 and the mathematician of the Van der Poel equation in the two interesting limits, small non-NIT, large non-NIT was done just a little bit later by Doronitsyn in Soviet Union. So I come back in the history much, much earlier. So the problem of limit cycle, if you wish, is following one. You want to describe mathematically an oscillating phenomena. So an oscillating phenomena, a time-periodic phenomena in nature, are very well known. They have been known since the Babylonian Western Hemisphere and Mesoamerican civilization in the Western Hemisphere and Babylonian in the Eastern Hemisphere. And it is known that you have regular trajectories and also not so regular trajectories which are due to the planets. But of course, mostly, and this was explained by Jacques Lascar yesterday, so mostly there is no dissipation in astronomical phenomena. Although if you want to make a clock, I mean a device which oscillates regularly in time, you have to bring energy in the system to balance dissipation. And surprisingly, after all, you know, from the very far away point, from the theoretical point of view, you know, oscillating circuits are just an example of an artificial oscillating circuit, but you have also many other naturally, not naturally, artificial oscillating systems. You have the clocks which have been made since middle age and even before, which are oscillating regularly in time and which might require a little bit of a mathematician. Actually, some great scientists were interested in the making of clocks. Classically, you have organs in the 7th century and Gahan was a British scientist less well known than organs. So what is the problem if you want to make a clock? To make a clock, in general, you have to have what is called an escape mechanism. You have to feed energy in the system and when feeding energy, you have to keep the system oscillating at a given frequency, which is not a trivial problem. So if I... So this is an escape mechanism invented by, I can say, perfected by Graham. So you have a pendulum oscillating at the end of this wire regularly, but you know that if you leave this pendulum oscillate by itself without doing anything else, in the plane of Poincare, vx, v being the velocity in x, the position, the angular position of the pendulum, you know that instead of having a closed curve, you will have a spiral going down to the center because of the friction. So what is the escape mechanism? So the escape mechanism is a way to give to the system some energy to balance this decay of energy. And this is done in the following way. I hope my explanation will be clear enough. So you have this piece is called the anchor. And there is a tooth wheel. You have this tooth wheel which is rotating because there is a weight going down. So the tooth wheel has some energy input coming from the motion of this weight in the gravity. So this anchor oscillates at the same frequency as the frequency of the pendulum. It is linked rigidly to the pendulum. And at each time these little pins hit the tooth wheel, the pendulum gets a kick from the tooth wheel because the tooth wheel is the one that receives the power. And when it does that, it does that at the end when the pendulum is at the extreme end of its motion, when it has no velocity. So what happens is that you see that the motion is the following one. It gets spiral inward and when the pendulum has no velocity, when it is here, it gets a kick. By getting this kick, it moves to the same, you know, it's an instantaneous, more or less instantaneous kick, you know. It moves from here to there and it gets back to the spiral. So, of course, you know, during the motion it will not visit any more this part of the spiral. So it will go from here to there, get back up again here and get another kick, etc. So this is the way you have a limit cycle which was more or less the invention of Agnes. It's a smart idea. So you have only a kick, you know, it's not something which is as regular as a non-linear term in the equation, but this is also a non-linear effect, but you have a kick only at a certain time. And so, let me continue. Clocks. There has been, by the way, very little interest in the year. In the general theory of clocks, you know, and even now atomic clocks, etc. And for instance, there is no general theory telling what is the ultimate accuracy, ultimate precision one can hope for a given clock, you know, as far as I'm aware. It's an interesting problem because it is connected with thermal fluctuations, quantum fluctuations, etc., which are certainly an incidence on the ultimate accuracy of a clock. So this is to recall what Poincare said, you know, it describes oscillating system with losses and energy input, even though this is seen with our eyes nowadays, you know, but basically what Poincare describes are oscillating systems from the point of view of physical system with losses and energy input by a theory of the energy system. So you have the limit cycle, Poincare map, this is something I already said. So this is to recall the Marconi transmitter, you know, he used again as an amplifier the arc lamp, which is not convenient at all. And see what is, this is also called the singing arc because, you know, it is oscillating, you know, including at the frequencies, audible frequencies. And this is the big device which was made by Marconi to generate the oscillation in the antenna, electric oscillation in the antenna. So now I'll switch to mathematics, but maybe not for the rest of the talk. So this is the equation which was derived by, actually there was a first, it was written for the first time by Rayleigh in another form, and as I said by Blondel and it is known now by Van der Poel, Van der Poel equation. So you have an equation of this form which is not exactly of the form of the equation by Poincare Ray, because in the nonlinear term you have a mixture of x dot and a function f of x, which is x square minus one. This equation is called the Van der Poel equation. So what is known that if eta is negative, so all the solution tends to rest, you know, to x equal x dot equals zero. And for eta positive, the only stable solution of this equation is a periodic solution, periodic in time. And there are two situations. One can do an analysis, detailed analysis of the behavior of the solution of this equation, which is a limit eta small, where we have this calculation of this, this is the limit of what is called the Poincare-Andronov bifurcation. And I'm not going to say anything on this. And what I'll be interested in is the limit eta large, which was the limit studied by Van der Poel, numerically again. So what you observe, what you observe without explanation, by the way, what you observe is that when eta is large, there are two stages in the solution of this equation. There is a first stage of decay of x as a function of time, and then a very quick jump for a positive to a negative value. This is, I think, if I remember, this is for eta equal 10, something like that. 100. For eta equal 100, for eta equal 100, so you have a very fast decay, then somehow a slow recovery of x, a big jump to a positive value, so decay, et cetera. So the oscillations are periodic, but with two very well-definite different stages, a stage of slow motion and a stage of very fast motion. In this image, eta large. So with Martin, we got interested into this problem because, to come back to, there is a, this, at least, this Van der Poel equation is very interesting because it is a simple model where you have a transition from, in the same equation, from something which is very slow, very slow dynamics, and a very fast dynamics. And there are very natural, very many natural phenomena where you have also this kind of transition. So the most remarkable one, which I've been studying recently, is, it's not periodic in time. It is an explosion of supernovae. In the explosion of supernovae, you have an explosion that lasts about 10 seconds, and the time of evolution, the slow time scale, is of the order of 1 billion years. So the ratio of time scale is 10 to the 14. So you have really a very large or a very small parameter depending on where you put the numerator and the numerator. And in this, this is probably the most extreme case where in the same physics, because there is no, you know, the physics remains the same, you have, in the same physical problem, you have a very, very different time scale. And you have also the other example where you have a very different time scale, and I will come back to that, is the case of the earthquake. In the earthquake, an earthquake lasts less than a minute, you know, it's about 10 seconds or 20 seconds, and the time scale, the time interval between major earthquakes is in the range of 100 or 200 years. There, the ratio is about 10 to 9, but it is also a very, very large time. So we were interested with Martin in this, the study of this problem, because the idea we had is that by understanding how do you, how does the transition from the slow phenomenon to the fast phenomenon occurs, my help in some cases, in some specific cases, particularly for earthquakes, to manage to predict the occurrence of the very fast phenomenon, the long time scale. Maybe there is something in between an acceleration that could be a way of predicting how does one do the transition. So maybe I'll jump to there. So this is the Lienard, the Lienard representation of the Van der Poel equation. So you have a slow, a stable manifold, a stable, no, a slow manifold, sorry it's not stable, a slow manifold which is this manifold here. Basically, this is the, y is an integral of, it doesn't matter too much, y is found by integration, by a first integration if you wish of this equation. So in the equation, the x derivative is very large. It is eta times, it is eta times the derivative of y, y dot. So you have a slow drift, and again in the limit eta large, you have a slow drift of the representative point where, again, in the realm of Poincare, you know, we are drawing everything in the phase place, in phase plane, y and x, y you can see as x dot. It is not exactly x dot, but it is the same quantity. So you have a slow drift along this curve. You see that this cubic here, which is a slow manifold, is attracting everywhere. This is attracting because the x derivative is much larger than the y derivative, in the limit eta large. So every point starting away from the cubic will be attracted to the cubic. But what happens is that you have a slow, the cubic is not a line of fixed point. It is. There is a small drift in the y direction because the y dot is not zero. There is a slow drift along the slow manifold. This slow drift is going upward on the right part of the plane and downward there. So you see that starting from this point, you drift on the slow manifold along this slow manifold here. But here you reach the extreme point on the slow manifold and jump out of the slow manifold because the drifting continues. And then the point is taken by the very large velocity in the horizontal direction. And it goes along this red arrow, the velocity in the x direction and you reach this point there and drift again along the slow manifold to this point and then to the very fast drift there from this point to that point and there again. So you have basically this explain this relaxation oscillation. The relaxation oscillation, you have the slow part which is a drift along the slow manifold, which is slow because the velocity in the x direction is much bigger. The velocity perpendicular or more or less perpendicular to the slow manifold is much larger than the velocity along the slow manifold. This is this part of the drift. But you reach at this point, the extreme point and jump from this extreme point to the other extreme point. This point is this point over there. And then you have the drift there from here to there, et cetera, et cetera. So this story was understood in terms of qualitatively understood by Liena in 1928 and there is other paper by Liena also. But the details of what happens when the point, representative point takes off from the slow manifold there and lands on the slow manifold here was understood by, as I said, in 1947 by Doronitsyn. This is a tricky problem to understand how things happen in detail. But to come back to this picture. So what you have here, and this is what we're very interested in. You have basically from the point of view of the stability in the transverse direction, here you have a point which is unstable because you know that the blue arrows point outward from the slow manifold. And here at this point, the black, the blue arrow point towards the slow manifold. And at this point, when you look at the dynamics in the x direction, you have the merging of an unstable fixed point and of a stable fixed point at this point, which is you have a kind of dynamical saddle node bifurcation.  When b, I think, when b is negative, you have two fixed points. You have this fixed point, which is the stable fixed point. You have the unstable fixed point. At b equals zero, you have a cubic. So there is a meta-stable point. And when b becomes slowly positive, you have no stable point anymore. So you have a merging of the two fixed points, stable and stable, and a disappearance of the fixed point, of any stable point. So by analyzing what happens, which was done by Dorotnitsyn, what Dorotnitsyn showed is that for the problem of transition of the slow-to-fast transition, in this equation you have to have a slowly-dependent, time-dependent b. So you have b depends on time, but very slowly so that you sweep again, you sweep across this picture from here to there. And by rescaling the equation, so near the transition, you have this universal equation, x dot equals x squared plus t. x squared is there because this is the derivative of the cubic force. And you look at the solution of this equation where at minus infinity you are in the stable fixed point. So x is like minus, I think there should be a minus square root here. Minus square root of minus t. And this equation can fortunately be solved by using the equation of every. So you have x as a function of time, which is x is what happens in the direction of x, in the original direction of the function of time, and there is a singularity at 2338, which is a time where the approximation of the cubic potential by, the approximation by the cubic potential there is not valid anymore, so you have to have higher order terms. So this is done by x dot, by adding to the potential a one for x to the fourth, and now you know the rolling down stops when the particle reaches the bottom of the potential. So with this, with this modellization with the cubic term now in the equation, we have something which is much, which has no singularity after a finite time. And so what is interesting from the initial point of view is that in this framework there is a long time scale, which as I said at the beginning for the application is very long time scale, which is the ripening time for the maturation, and there is a very quick time, oh sorry, this is a long time scale because A is large, and because A is small, there is a time scale, which is a short time scale, which is one, and there is an intermediate time scale, which was found by the way by Doronitsyn, which is A to the power minus one-third. So you have an intermediate in this phenomenon, at least the result of this analysis, that before the catastrophe you have a long time scale, which is intermediate between the short and long time. So what we did with Martin was, and maybe I won't tell too much on that, because you can detect, and this was done in an experiment which was done in the US, and this theory at least is successful in one experiment where you have, we observe this time, this kind of intermediate time scale. How much time do we have? No time? Five minutes. Now, maybe the most interesting part of this work for this audience is a stick slip model. So one could ask the following question. So we have this phenomenon of transition from slow to fast phenomenon, and which is well explained by the Van der Poel equation, the Van der Poel equation. So this is what we call the saddle node bifurcation. So saddle node bifurcation with a dynamical, it is dynamical because there is a slow sweeping from across the transition. So is this only possibility for having, in a differential equation, a transition from slow to fast? And the answer is no. We studied with Martin this model what is called stick slip behavior. So you have a mass which is drawn on a solid surface. So there is some friction, and the mass is drawn by a point moving with velocity v node and a spring of strength k. So this is a classical model for earthquake. And in this model, there is a parameter, there are many parameters, and here is a parameter gamma. The parameter gamma for earthquake is very large. This represents more or less the strength of the friction compared to the strength of the spring here. And what happens in this model? So this model is, as you may have noticed, is a system of three, a set of three ordinary differential equations which are written here. And in this, maybe, so what happens in this model is interesting. It is not at all what happens in this three-dimensional model. This is not at all what you have in the saddle node bifurcation of the van der Poel model. In this model, by restricting the equation to the slow manifold, because there is also a slow manifold, what one has is that the trajectory remains on the slow manifold, but on the slow manifold, the equation, the dynamical equation, have a finite time singularity. So after some time, the slow manifold is the point, representative point, get out of the slow manifold just because it accelerates too much. In some sense, the assumption of having something slow doesn't work anymore. So there is a transition to another kind of dynamics. And you see that this is what happens there. So you have this, the trajectory, which is very easy to draw, because it is in three dimensions.  So in some sense, this is a second way for a system to have this relaxation oscillation. And so somehow it is kind of a point of mathematics, but it is not completely indifferent, because if... Or maybe I'm not going to explain that. But if one wants to, you know, the possible precursory phenomena for the transition are completely different in this case than they are in the case of the transition by the cellular node bifurcation. So it is a complete and another class, or another class of transition from slow to fast that you find in this model of friction. So I think I will stop here. Thank you for your attention. Thank you for the very nice talk. Other questions? Parle Four. Parle Four. Speak loud. Yeah, you didn't mention many people who studied this kind of phenomena using non-standard analysis. Could you comment on this? Which phenomena? The approach to the slow manifold and the times of exit and things like that. I mentioned the ordnance. I think I mentioned the ordnance. I have not used any. Yeah, I know that, of course, there are many followers, but what we are doing is used mostly only the ordnance. But this is the original part. As far as I am aware, these new Parle Four, let's go. Yes, thank you very much. You mentioned that Poincaré was very much interested in DSF wireless telegraphy following Marconi's exploit in 1902. That's true, but of course his interest predated that and there is his correspondence with Hertz from 1890 already. Yeah, but Hertz didn't make any transmission. Sure. The point I'm making is that you talk about long distance telegraphy, but certainly that's not really related to the work that he did on generating the waves from a mathematical standpoint. These are quite different, aren't they? You've got one problem generating the waves themselves and the other part is studying their transmission over long distances. So my question is, is there any relationship between these two problems other than the fact that they're both tangentially related to wireless? No, no, my point was to say more or less to describe what is in this 1908-1909 paper. The first part is about the working of an antenna. It computes various geometrical configurations for antenna and how does it define the geometry of the antenna, how does it define the geometry of the field at large distances. And in the second part, without, with no transition, he switched to this kind of equation. That's it.