As welcome back to the independent investor channel, it's probably be one of the most important Tyleon videos that I put out. I've put out a few half a dozen, maybe a dozen or so videos over the course of the history of the channel. It certainly does not speak to the amount of due diligence that I do offline. As I am a stock owner in the company and a long term bowl of the company, I've always been intrigued at the technology. But up until today, when I was able to view the first stop in the Innovation Council Roadshow in Rochester, New York, it was the first time really that I had saw not only the Hypertruck ERX in action, but also the third party validation that I found was the key takeaway for me to actually hear some of the perspectives from some of the drivers, as well as some of the executives and potentially how the Hyleon solution could tie into some of the governance objectives from Wegmans specifically. Now, this is just one of the 10 total out of the 11 highly unincluded in the Innovation Council that was formulated really to help validate this technology. So from my perspective, I thought when I was watching the five minute video, there was a few key takeaways and I think there could be a lot of different takeaways I had to freeze frame when they showed between the chassis. That is the first time that I've actually got to see the components underneath the electrical systems as the driver was talking about spending more time in his truck than he did actually at home and how important it is to make sure that the rigors that are incurred on the body as they're over there driving over the road are minimized as well as the stress and the noise that are created from being a long haul driver. But before we get into the Wegmans video, I want to touch on a few housekeeping. Items, there's been an overflow of information that has really come to a head here, I think, for me, and that's why I speak about how important this video is at the time that I put it out there. We've been down 10 out of 11 days and I really have never seen anything like this in the stock market in a stock that I've covered owned or covered and not owned. It really just takes the cake on what I consider to be one of the greatest disconnects between what I was able to see with my own eyes today through that product validation, which is leaps and bounds ahead of anybody else out there on the market. There's no doubt about it hands down. These folks have got all the right stuff in all the right places and they've left alone the aesthetics of the truck in all the places that did not need to be improved upon. And if you hear the executives talk about their importance of reducing fossil fuel in their application, this really just plays into it. And also the chairman that was actually on the video that was being interviewed talked about actually partnering with Hylion and speaking about Hylion as if they were going to lead the way in this charge toward decarbonization and lowering the NOx emissions that each of these companies are responsible for putting out through their carbon footprint. I just thought that that was really telling there was a lot through the five minutes that was just absolutely jam packed. I think it was easy to digress after watching five and a half minutes and be hungry for more certainly with the lack of availability of information out on the marketplace right now. I think it's actually been supplemented quite nicely by Drive Mix Game. Dexter's a good friend of the family now is doing a good job providing content that I cannot provide with regard to specific validation, product validation that's going on within Depmar Logistics, which is a Fraxans company down in Texas that has been a staunch supporter of the Hylion team, the Hylion product both from the hybrid CNG perspective as well as the hyper truck ERX. Now we have not got the ERX validation through Depmar. We have got the EX1 hybrid validation through Depmar and this just marks the second company in what I consider to be a template going forward for what to expect. I left a comment in my Facebook group. It was too lengthy to leave to the actual Wegmans video. It actually got held for review because I think the length of the comment was over what they would allow and so it pinged on their filter, which is totally fine. If you want to read my comment in its entirety, I would encourage you to kick into my Facebook group and check that out. All right, but at the time of filming this video in the way I want to kind of express how important this is, the disconnect between what I viewed today in how close they are in getting the product validated and they only have to validate the prototype and then that prototype can be amplified tens and hundreds of thousands of times over again and that's why these initial prototype validations through third party validations from the corporate governance perspective and how the solution can fill those niches is so important and then the driver perspective, which I think is something that has been gleaned over a lot of the times and I think that's where Dexter provides a lot of specific insight from his perspective as a driver on how cutting edge the solution actually is and could potentially be for drivers out there in the marketplace. Okay, but the stock itself, interesting enough, I do want to give you some of my perspectives a little bit and some things that really may help put this thing in perspective for you guys. There's a lot of different holders out there, whether it be from hundreds of shares to thousands of shares. Okay, I typically don't compare myself to anybody out there that chooses to hold a little bit of shares or a lot of shares. Furthermore, I certainly don't want my message. I feel like I'm one of the advocates. I always have been on the stock. I say it like I see it. It is my opinion, but there might be some misconception going on that there is an attempt to hype. I cover stocks all the time, whether it be dividend stocks, whether it be growth stocks, whether it be ETFs that I like to invest in. And I think it really just speaks to my passion for investing. Furthermore, if I can impart to you this, the discipline that it takes to own a company like this, especially with what's going on right now with it, it takes a lot more tolerance to own an asset like this than it does an ETF or a dividend, a risk to crad or a king in the market. Okay, let me talk about that a little bit. I think a lot of people want to apply their investing strategy across the board the same, no matter what they hold. And I think when you do that and you get involved in a specific asset, I think you underestimate sometimes what the opportunity that lies somewhere down the line. And when that opportunity doesn't make itself readily available within the short term, we get frustrated and we start to think about how we're missing out on somehow that long term by focusing on the future. And I think it's really easy to get caught up in the day to day. The stock's down 12 cents, the stock is down 28 cents, the stock is down nine cents, the stock is down six cents, the stock is down, you know, three cents from day to day to day to day to day. And it really, it draws on you with regard to really becoming way too laser focused, which in turn creates hypersensitivity around the stock, the stock, the stock. Now what I'd like to do is draw a conclusion between that hypersensitivity and what is actually going on at the company. Okay, I'm usually somewhat of a neutral broker when I sit back and I look at the application. I have the ability to do that. I'm really proud of my ability as a stock market investor and I do share that perspective openly on YouTube. I could just as easily be a successful investor on my own without the compliments of YouTube. I don't do YouTube for any other reason other than sharing my perspective on how I approach and hopefully give myself the best chance of success on the market. And that's all I would ever hope for anybody out there that here's my message and doesn't take it as hyping the stock because I don't think that really does any good, but I think there's a fine line between hyping a stock for some sort of personal gain, some sort of personal initiative, or sharing a story passionately about a company that's looking to make the planet a safer place, aligning those solutions with other companies out there that are looking to integrate these solutions into their own fleets and are aligned with those same goals. And when I get excited about looking at the specific opportunity, the total cost of ownership proposition, when we look at the cost of renewable natural gas, the initial onset that Hylian is going to be serving up to these companies by way of proposals, when the product has gone through its full and final validation, when they get to pose this, my investment thesis is one to be excited about. And the reason for that is because I believe that Hylian is the most most better appropriate or better positioned to make that proposal to industry than any of the other companies out there. In other words, the nickel proposition has a cost to hydrogen right now to the diesel fuel equivalent of about $50,000 more over the life of the truck. And I've talked about this before about eliminating fossil fuels is not the standalone benefit. If eliminating fossil fuels was the standalone benefit as admirable of a mission as that is, it has to be cost effective to the fleets. So it's something to really take into consideration when you're looking at the bull bear thesis in Hylian. The second point I want to make is about the stock itself. Ultimately, after all the noise and after all the gray matter is separated and after everybody's done with the name calling and everybody is talking about where the stock is going to go in the short term. After all that, you're left with two decisions. And I think this is something that we can all rest on common ground. I've even had some people in my very, very tight community actually hint at this being somewhat of a bad move because it doesn't make sense for them. It's a pre revenue company. Therefore, because the stock is where it is, it somehow justifies the bearish thesis or the reason why that this wasn't a good investment in the short term. The fallacy in all that is selling a stock or not buying a stock in the first place is one of your two options. Outside of all the noise, outside of all of your hypothesis, outside all of your conviction or lack of conviction, whatever it is that you drum up in your head as far as focusing on this or any other stock in the stock market to give you the validation or the hesitation to either invest or not to invest in a said company is really the bottom line in your decision making. And I think where we really get caught up in a lot of introducing the infinite amount of outcomes outside of those two specific outcomes or decision points that you can make is when you start to live between the two. When you make a decision to buy, you stick with that decision to buy and you don't get talked off of that decision to buy unless your decision to sell is completely 100% unequivocally owned by you. That's the only way that that concession can be made. And trust me, you can make that at any time. Okay, you can't handle this, you got overlevered, you have too much capital to risk, you're losing your long term vision, whatever it may be, either it's good enough to fight through right now and stay with your conviction to buy or you go ahead and sell. Okay, but those at the end of the day are really the only two deliberations. Give you some examples on things that don't matter. Okay. Remember and keep in the back of your mind buying and selling a stock are the only two options with this or any other stock. But when you see comments that say the stock is going to two, or the stock is going to 10, or they don't share enough information with us. Okay, it's all that is amplifying to try to get to an end on either one of those two decision points. In other words, if you think the stock is going to two, and you want to become bullish on it, then you wait till it hits two and you buy the stock. Okay, so that's simple. But I find all too often that people get caught up in this really vicious cycle of becoming caught up in between those two options. And if you're going to be caught in between the two options, go do something else with your time. Because at the end of the day, if you don't have conviction to sell or not to buy, or you don't have conviction enough to buy and hold for as long as you need to see that investment workout, then I would rather see you not do it at all. Okay, very, very important when we're discussing the philosophy and the discussion point of owning a stock long term. Okay, whether or not to buy or hold the stock. Okay. A couple of things that I mentioned from the video that I picked up. Number one was the validation by the third party. This is key. I heard the validation from the driver's perspective. And I heard the validation from the executive. And I was also very interested in understanding the technical application and the vision that shared by the Hylian team. I thought that they were very stand up. I thought they delivered on point. And the very end of it said that they thought on their objectives of going up there and really validating the technology was a slam dunk. And I think they were very well received. I think the interviews were genuine. I think the one gentleman that said that he had over four million over the road miles of a 30 year trucker with 15, 16 years standalone with Wegmans alone was key to that validation stage. I thought that was incredible. The driver validation specifically when Roy, one of the drivers talked about how important it was to be quiet and the safety element that is existed in the cab while driving. Just think about a huge diesel engine and becoming accustomed to having that loud noise and how that could potentially drum out somebody honking that is caught in a blind spot as you're driving the truck down the road. Think of the safety concerns that that could alleviate now driving these trucks and having them be significantly quieter while they drive them down the road. The other thing was the how smooth the truck drove. And the third thing is how well the regenerative braking worked when it was tested up there with both drivers. I found that to be extremely telling. One of the executives with Wegmans talked about the customer demands when Greg, who was the fleet specialist with Wegmans talked about how important it was to their customers. When Greg mentioned to the executive Mr. Wegman himself that the hyper truck ERX had the ability by nature of pushing a button to shut down the onboard generator that powers the batteries and run specifically BEV off batteries for up to 75 miles. He caught that right away and he said that's amazing that can really, really cater to some of our customers who demand that. And then the manager with Wegmans talked about one of many of their sustainability goals, some of their packaging, reducing the landfill waste and also eliminating fossil fuels from the fleet itself. I thought that was key. So my closing remarks and my takeaway from the video is this. I expect that in the next coming months we are going to see a similar pattern of visiting each of the innovation councils. And I think this is going to not only solidify their spot for me, for a lot of them, Ruan comes to mind. I haven't really seen anything other than the initial unveil on how Ruan can really take this solution and integrate them. I got that answer for Wegmans. I didn't know Wegmans that well. I knew what they did, but it was really nice to actually see some of the execs, some of the driver feedback, some of the interviews. I thought that was absolutely critical. So I do expect to see some of that stuff. What's going to be unexpected as an investment or a dividend that comes from these meetings is going to be the follow on order. Right now, Wegmans is not on the books for any type of pledge to order with Hylian. Right now, it's at zero. And I think the question becomes, if after this validation, Wegmans steps in and offers themselves to what I would say on the low end is a non-binding reservation order and or they skip that reservation stage, which I do think is the most appropriate right now because Hylian is ill-prepared to step into mass production now anyway with both the supply chain issues and the supply chain hangups at the OEMs themselves. Their inability to take mass orders anyway right now could actually be a blessing in disguise. So when we're talking about a stock at six or $10, is that really reason enough to compromise your long-term thesis about the product that we were able to validate today? I'll let you answer that. But as we look at unfolding over the next couple months, I would expect to see some of these orders trickle in from the Innovation Council themselves. And I'm going to be paying particular attention. Remember, we have 1,590 on the books right now as we speak in non-binding reservations for the HyperTruck ERX. I would love to see them solidify some of those orders with LOIs. I would love to see that fleets are able to validate enough in what they see and feel comfortable enough that those orders will inevitably be able to be rolled off of the OEM hubs and be integrated in their fleets right away. Something that I picked out of the video is that Wegmans was bothering Hylian, not the other way around. Wegmans was saying we're ready, bring the technology up when it's ready, and it just wasn't ready. And I don't think this is any fault of any party at all. I just think it is a reflection of where we currently are in the product development cycle in that product validation stage, having gone through the internal verification stage within Hylian with the assistance of FEV up in Michigan. So I just think that we're always quick to say too fast, too fast, too fast, or too slow, too slow, too slow. We need orders and we needed orders now. There's nobody that has been more staunch with needing some sort of catalyst, and the orders will do that. They're not going to be able to start to solidify an order book by anybody on the Innovation Council or off. I mean, if we have a rogue order from Pepsi or Amazon, it's game over. It really is. Even a non-binding reservation of 10,000 ERX trucks from Amazon, it's game over. And I believe the technology is number one in the industry. There is not a competitor right now based on what I saw in the product validation, the testimonials that were provided, and the specifications that the ERX can provide for a large fleet like Amazon. I just think that right now we're in a waiting game until those catalysts happen, and it's our ability to forecast the probability of those catalysts happening in the future based on what we saw today. If you were really reading between the lines in the Wegmans video on what we are going to expect going forward, we just came off earnings call three weeks ago. The Monet order of 40 hyper truck ERX that came out of nowhere, there's no doubt about that. One thing that Monet and Wegmans have in common is and Denmark is that they are all existing customers having used the hybrid system. Thomas Healy talked about the critical link between the hybrid system and the hyper truck ERX. I think this is going to play dividends in providing that industry validation as a whole. They're only going to be able to keep this thing a secret for so long. And it seems like to me the stock market is assuming that that secret that they have is going to remain a secret. And industries are not going to be excited about incorporating that. I did not hear that from either corporate governance in the company or the drivers. Who else does it need to matter about? Is it people who don't really have any vested interest in seeing Healy on succeed to say that they're not going to sell product to anybody? I just didn't see that when I watched the Wegmans video. And again, I don't think they made that stuff up. I think that was very genuine. I think it was very, very well done. I think there was so much value to extrapolate from that. And we didn't have to infer so much by the one vision that's coming to my head that I remember in my deep dive of Healy on over the last 18 months. And that's the ERX doing peeling out in the Healy on parking lot. That's really all we had up until this point, as well as a few schematics of the rudimentary drawings of where the components go on the truck and what the main makeup is. We got to actually see that in real life today. And I don't think that that need can be understated. I think on the market down day today, I think it was unfortunate. I think Healy on is really stuck in a rut. I really do. I think it's just sell it now and don't ask any questions now on challenging the idea that perhaps maybe we are looking at a bottom. And I don't call bottoms. I don't believe it's possible. There's a lot of people that I view on a lot of the different forums that I visit hungry for information that actually do believe that they actually can forecast the bottom. And let me let me draw it to you this way. If at the stock, if when the stock was $9.50, somebody came out and said, the stock is going to $6. They would have been proven right. But by no other reason than they got lucky. It's just that simple. And where I find the fallacy is that people look at that transaction or the time that exists between the spew of the mouth of where they think a stock is going from an educated guest perspective. And it ends up turning turning out right. The fallacy exists where we start to look at that and we start to put credence in those types of speculative assessments or calculated guesses on where a stock is going. Ultimately, going forward is what is important. It is not where the stock has been is where it is going into the future. And I think that's something that you really need to take into a holistic account. When you're looking at a company like this, validating through third party validation, what it is that we saw today through the Wegmans video, and I've had a chance to think about it. And I think that this is a real game changer. I really do. Now, I'm not looking at the disposition of the stock to change tomorrow. I'm not looking at it changing next week. As a matter of fact, the way I feel right now is that it will never change. It will never change. Five years from now, we're going to be going through the same deliberation. Highly on will have no sales. Highly on will have gone through its roadshow and people say it's the greatest technology ever. It's very comfortable, but they won't have any continued integration into the fleets. There will be an increase in competition in the landscape. There will be an incurred cash burn with highly on. Maybe they've done a share issuance and they've diluted shares and they've done a capital raise with the company. Here's the thing. All of those things that are possible in real life, I think are highly, highly unrealistic. If you took a realistic look at this company and looked at it through the prism of opportunity, can this system perform as prescribed? Yes, that's been validated today. Can they actually draw fuel from the infrastructure grid to actually move goods tomorrow, like right now, if they put this thing into service? The answer is yes. Could there be a potential for the supply chain issues to shake out sooner than later? Certainly. Certainly. I think we're taking very conservative estimates or maybe even just drumming up some conclusions that the supply chain issues will never shake loose. Guys, there's pressure building behind the dams to get these supply chains moving. I think there's more chance of the dam breaking free than the supply chain issues persisting for longer than forecasted. We'll highly on not secure additional orders in 2022. I think that takes more imagination to think about based on the track record over last year and this year of securing reservations to the order book. I'd like to see a lot more variety in the reservation order book. What I mean by that is if I get a five-order reoccurring order because Pepsi is looking to integrate their fleet, I'll be doing back flips because that's the difference in I think they should be at 5,000 non-binding reservations right now. I think they should be on the phone every single day with Amazon. I've actually thought about doing it for them and securing that bid for them and then just handing it on to them with a silver platter on a silver platter because that's the type of things that it takes. Focusing on the Innovation Council is great for the time being but my angst a little bit is what have they been doing over the last 21 days since earnings? There could have been some internal reporting going on on other fronts of the company. There's always different things that can be reported upon the new building, the battery technology, all the 17 million where it went in R&D, all these different things that kind of move away from and give more of a holistic perspective on the goings on at Hylion can absolutely help supplement some of this downtime that we're going to incur because we just cannot progress forward because of the supply chain issues. Hylion is going to have to figure out how to embolden that order book going forward, solidify some of those reservations through letters of intent to purchase and make some of those binding pre-orders. It sounded like to me that Wegmans would be a perfect candidate to come to the table with a 15 or 20 ERX order, not through a non-binding reservation but an actual letter of intent to purchase based on their product validation through what was shared today. Mark my word, expect that that could be something that we see in a short time here into the future. Guys, I really appreciate you tuning into this update on Hylion. I think one of the strategies that I've always looked at as being a win with this company and I sleep easy with this application and that is to buy the stock and hold it. It's just that simple. Don't be stuck being wishy washy between the two. I can respect if you sell. I can respect if you buy. At the end of the day, I don't really care what you do with your money. I've been very, very transparent in giving you the perspective from a shareholder that is bullish on the company and absolutely has 100% conviction. When reports like this come through of industry validation, it only just confirms my existing validation and confirmation that we are headed somewhere extremely special in the future as the Hylion solution is shared with more and more and more fleets and eventually the domino effect will take hold and other companies are going to talk amongst themselves and say, you need to give this company a look.  It looks like they've got the solution that's going to be the one going forward that's going to be the closest to knocking on all three of the value propositions and not just one of them. You want to leave your comments at the bottom of the video, guys, subscribe to the message, share the message with anybody out there that you know are interested in these Hylion videos, man. I post these pretty frequently because I think right now is the time where we have the absolute most disconnect between the goings on of what this company is trying to do to revolutionize the Class 8 space and a stock price that is that absolutely seems like it cannot go up. It just cannot go up. I've never seen anything like it. I'm absolutely dumbfounded. I have no other explanation other than the stock market has it wrong. It gets it wrong all the time. The history of getting it wrong with this and any other company in the past does not and I will repeat it does not guarantee where a stock is going to go into the future as we have not encountered the future yet, but we are marching toward that end. The question that you have to ask yourself is does the scenario that I drew up of the stock being $6.75 in five years make any sense to you now? If you have any conviction at all about the company and you're looking at it here at $6.35 at the time of filming this video, mark my word. I don't think within the short and medium term that the stock is going to be able to ignore the value proposition and the disconnect between what we saw today through third party validation and a stock price that is not reflective of the value that exists on a fundamental level with this company. Guys, thank you so much for tuning into the video and good luck in your investment future.


Section 10.4 is all about inscribed angles, and 10.5 deals with tangents. Make sure you get this in-deer table of contents, and then we'll get started. To begin, we need to understand what an inscribed angle is. An inscribed angle is an angle that has its vertex on the circle, and the sides of the angle are contained in chords of the circle. So if you recall, a chord is a line segment that has its endpoints on the circle. So here, angle ABC is an inscribed angle. That red arc, arc AC, is an intercepted arc. In other words, the inscribed angle cuts off a portion of the circle. It intercepts it, and that portion of the circle in the interior of the inscribed angle is arc AC. And so that brings us to our first theorem, which states if we have an inscribed angle, then the angle's measure is half the measure of its intercepted arc. So if we have our circle in an inscribed angle, let's say it's 50 degrees, that tells me that the arc that is intercepted is 100 degrees. There's this relationship that the angle is always half as big, or the arc is always twice as big. Here's another theorem for us. If two inscribed angles intercept congruent arcs, then the angles are congruent too. So if we look back at this original drawing, let's say there's another inscribed angle that intercepts the same amount of circle. So the black angle cuts off 100 degrees. The green angle also cuts off 100 degrees. That means both angles are congruent. Now similarly, if we have these two angles and they intercept congruent arcs, maybe not the same arc, but the same amount of arc, that means those two angles are congruent. So let's take a look at an example here. Let's say we have a circle, and we have an inscribed angle, and we also have a central angle. Let's say the central angle is 40 degrees. I know from previous lessons that the central angle and the intercepted arc are congruent. From this lesson now, the inscribed angle is going to be half as big. Let's take a look at this next example. Here we have arc AD is given as 4x degrees. We see ABD is 36 degrees. If you look at angle ACD, it is also an inscribed angle, and it intercepts the same amount of circle as angle ABD. So therefore, both those angles are 36 degrees, and arc AD must be 72 degrees. So therefore, we would get a value of x is 18. For another inscribed angle theorem, this states if an inscribed angle intercepts a semicircle, then the angle is a right angle. So if we have, let's take a look at that kind of bottom corner angle, we see it is an inscribed angle, and it intercepts a full semicircle. Well, semicircle is 180 degrees, and so therefore the angle must be half as big. Then our last theorem dealing with inscribed angles, if we have a quadrilateral inscribed in a circle, then opposite angles are supplementary. So inscribed means the quadrilateral is drawn entirely in the circle, and its vertices are on the circle. So let's call this LMNP. If you look at angle L, I see angle L intercepts arc MNP. The angle opposite, L, is M, and angle N intercepts PLM. Well, those two arcs, the red and the blue, make one full circle. And so therefore, angle L and angle N, their measures have to sum up to 180, and similarly angle P and angle M also add up to 180. Next example, let's say we have circle P, we have a diameter, Tu, and we've got these points, V and S. And let's say we're given that arc VU is congruent to Su, angle UTS is X plus 9, and angle SUT is 2X plus 16. Let's say we want to solve for all of the missing angles. Well, we know we've got these congruent arcs. And those congruent arcs, both of those arcs are intercepted by either STU or UTV. So therefore, those two angles are congruent. We have a semi-circle up the middle, and so that means angle S must be 90 degrees because it intercepts a semi-circle. Similarly, angle V is 90 degrees. And so if you just look at triangle TUS, we know all three angles. So therefore, we can set them equal to 180 and then solve for X. So now that we have got a value of X, we can sub that in and find the rest of the angles in the triangles. You know this is 34, TUS is 56, VTU is 34. And then looking at triangle VTU, I know angle VUT must also be 56. One last example for this part. We've got quadrilateral ABCD, and we want to solve for the unknown variables. Well, we know with an inscribed quadrilateral opposite angles are congruent. So B and D add up to 180. And likewise A and C add up to 180 as well. Next, a little bit more vocabulary from section 10.5. So quick review, we all know that segment AD is a radius. Segment BD is a chord. Remember a chord is any line segment that has its endpoints on the circle. So now look at line BC. Line BC is called a secant. Now a secant is a line that intersects a circle in two points. Secants and chords are very similar ideas. A chord, however, is a line segment that intersects a circle in two points. Whereas a secant is a line that intersects a circle in two points. Now look at line DE, that's called a tangent line. And a tangent is a line that intersects the circle in only one point. A secant is two points, and a tangent is one point. So that point is called the point of tangency. One thing about the point of tangency is that the tangent and the radius are always going to be perpendicular, which means angle ADE is 90 degrees. So let's look at this example problem. Let's say circle A has a radius of x units long, and B is a tangent point, and we want to solve for x. Well we know if we have a tangent point, that means angle ABC is a 90 degree angle. Then take a look, AB is a radius, AD is also a radius. And so that means the entire length AC is x plus 8. And so therefore triangle ABC is a right triangle. So we can solve it using the good old Pythagorean theorem. Now make sure you foil out x plus 8. Then I'll subtract x squared from both sides. Subtract 64, and finally divide by 16. Next we have a theorem about tangents. And it states if two segments from the same exterior point are tangent to a circle, then those two segments are congruent. So first let's figure out what this means. We have a circle, and we have a point exterior to the circle. If we draw the two segments so that they're tangent to the circle, what this theorem says is that those two lines, those two line segments are congruent. This is actually a pretty easy theorem to prove. All you have to do is let's say draw in the radii that connect to the points of tangency. Remember tangents and radii are always perpendicular. And then if you draw the distance or the segment between the center of the circle and point A, let's name the rest of those. Right now we can prove that the two triangles ABD and ACD are congruent, right, by HL. And therefore, segments AB and AC must also be congruent by the CPCTC. So there we go, we've got that tangents theorem. I sincerely apologize for the quality of this drawing. Let's say that we have these two circles and these segments and we've got tangents. Sorry about that circle, but let's pretend. And then let's say we know these segment lengths are given. We've got y minus 5, 10, y, and x plus 4. And so if you just look at that small circle, that small horribly drawn circle, we know that these two segments are tangent from the same exterior point. And so that means right off the bat y is 10. Then if you ignore that small poorly drawn circle and just look at these two tangent lines, they're also tangent from the same exterior point. So the blue lines must also be congruent in length. That top side, we know what y is, y is 10, and so therefore the entire length there is 15. And so likewise the bottom length must equal 15. Let's take a look at one more example here. Let's say we have a circle inscribed in a triangle, and the triangle's perimeter is 30 units, and we have to find x. First off, if we have a circle that's inscribed in a triangle, that means we create these three points of tangency. And so since we have exterior points, we know that if that little segment is 7, so is the one right below it. Likewise we have these two segments that are 5 units long, and these two segments that are x units long. And so since the perimeter is 30, we can just add up all of the line segments. 5 plus 7 plus x plus 5 plus x plus 7 must equal 30. Then we'll go through a cleanup session. We see 2x plus 24 is 30, which means x is 3. Thanks for watching.


This time so we have to talk a little bit about discovery or development. Here's the good news about development and plus I hope you don't overreact. From a research scientist perspective, right, from a discovery scientist perspective development is a highly regimented and relatively prescribed process. It's well regulated by the FDA and so there are specific hurdles and specific groups and they're arranged in a specific pattern and you have to do. Okay, you either make it or you don't. It's that simple. Okay, regardless of the type of drug the steps and processes are nearly identical and they're basically broken into four types of questions. First, toxicity testing. Is the drug toxic in non-humans? Can't test it in humans. I have to show you that it's not toxic in non-humans and the FDA requires two species. We have some other rules. One of them may be a rodent, the other one cannot be a rodent and so there's some rules about what you can and can't do but you need two species. It would be a really good idea if those were species that are well understood to the tox community like rats and rabbits and dogs. It's not a good idea to do the testing in a species like a very rare animal where toxicology is not understood like a ferret. Okay, that's not, it's allowed by the FDA but it's not going to be accepted by the community. Phase one, the first time, I know you've been at this for five to seven years. First time you get to put your drug in a human. The only question you're asking in phase one, how do humans metabolize the drug? Is it different than what I've been studying for ten years and rats and rabbits and dogs? Where is it the same? What are the form of poconetics? How long does it hang around? How long will it be there when I give a dose? Okay, phase two, what dose of the drug is likely to be effective? And then the last, phase three. By the time you're at phase three, it is all downhill. Eighty percent of the drugs that make it through phase two make it into market. Okay, phase three, there should be, and this is a rule of thumb guys, if you ever see it just point fingers and laugh, there should be no failures in phase three. Okay, if anything fails in phase three, you're allowed to point and laugh because everything should be locked down by them. It's because people make bad choices or push through bad data. There should be no failures in phase three. And that question is, is that new drug more effective than the existing therapies? Am I going to actually make headway in treating that last five percent of the patients who have heart attack?


Contestant number one we have Laura. Nice smile. Contestant number two Thomas. Nice mustache. Contestant number three Christopher. Nice smile. Nice tie.  T and E, R, H. So some great leg work by Laura there to get basically done. Another feather in your, what is it? People listen at home that haven't seen this clip already. What could it possibly be? Okay. Another feather. Brutal. Mostly that's wrong because that's not the saying and also T's are already on the board because T. You guessed them. So that's a little bit of a momentary lapse and it's a bummer because surely it's not going to come all the way back around to Laura and she did all the leg work. Christopher. G. Ah. No G. Oh. G. No G. Okay. Sorry about it. I shouldn't even have asked. What did he think it was? Another feather in your bag? I'm guessing he thought it was bag. I don't know. So now it goes to. Thomas. But he goes bankrupt so he doesn't even get a chance which means it comes all the way back around to Laura who has another shot at it. I'll solve. Okay. Another. Listen to how Sagex says no. No. And then Vanna White. I mean she was like, you know, one step there. Surely she's gonna get it. She did not. Uh-oh. So it comes to nice tie, nice smile, good glasses. Christopher again and he still isn't choosing to solve. He's spinning, risking going bankrupt and he's gonna guess another letter. D. No. Thomas. D. No. Thomas.  So Thomas again cannot guess. He loses a turn so he's just in a hellhole not being able to participate and watching these two just have no idea what's going on. Brings us back to Laura again and she's like, oh my God. I have no idea.  I'm gonna embarrass myself. I'm gonna go ahead and spin. Yeah, go ahead and spin. Don't take a random guess. A P? Yes, there's a P. Ooh, okay. You can solve. Okay. Another. Oh, no. Oh, that's not a saying. Another feather in your map.  Christopher, it's your turn. And Christopher, he goes bankrupt, but secretly he's like, dude, I'd rather go bankrupt than embarrass myself again. I got no idea what this is, which brings us back to Thomas who finally gets a chance to guess a C. There you go. Another feather in your cap. Yeah, there you go. Yep.  I think I have heard that. Yes. Oh, our bad. Good job. Good job. Good job, Thomas. You have a really good job. I thought it was lap or map. I had no idea. That round is over, but it was fun in a sort of perverse way. We'll be right back. In the sort of perverse way, it was fun. I do agree. Another fact. Wrong. Another. Wrong. Wrong. And if anyone is like, well, what does another feather in your cap mean? I think Native Americans just put feathers in their headbands when they like killed people and it was like honorable and they were like cool. So it's another feather in your cap. No feathers in these people's caps at all.


YouTube team keep it clean. What's going on? It's engraving here with another video and in his video Whoo We're gonna talk about the Browns the Cleveland Brownies And the Browns are making a move at the coordinator position at the office of coordinator position And they are getting ready to hire Todd Haley Todd Haley, so he's gonna be staying in the AFC North Todd Haley was the Pittsburgh Steelers their former offensive coordinator a guy who I mean he made a lot of stuff happen With the Pittsburgh Steelers as we all know their offense was something that could be a thing of beauty Um, but the thing about Todd Haley, you know what before we get into that. Let me not be rude Let me introduce myself me. I mean Raven vision. This is my youtube channel engraving vision I'm a Ravens fan and even though I'm a Ravens fan on his youtube channel We talk about the Ravens, but we also talk about the Steelers. We talk about the Browns We talk about the Bengals we talk about the whole AFC North and we just talk about the whole NFL period So if you're a Ravens fan, if you're a Browns fan, if you're a Todd Haley fan If you just a fan of football period Make sure you subscribe to the youtube channel turn your notifications on And like my boy Ryan mallard says for every single video because it's free and like my boy djp says for every single video If you're breathing make sure you leave a like on the video click the thumbs up But Click the thumbs up button Click it leave a like on the video and um also kids and I'm in such a great mood today because For the next three weeks and I hope hopefully Going forward. I got a new schedule at work to where I'm going to be getting off at 4 30 eastern time Eastern time usually I get off at 6 p.m. Eastern time and it's dark by the time It's dark when I'm driving home, but now getting off at 4 30 It's actually sunlight as you can see and I can see y'all y'all can see me We ain't got to do this like like it's a radio station anymore. Y'all gonna actually be able to see me while i'm talking So this is cool. Anyway, appreciate y'all coming through so fast too, man Appreciate y'all. Thank you. You know, I love y'all. I really love y'all and appreciate all the support And just everything that y'all do for this channel, man. Like I said it man Everything y'all do for this channel. I appreciate it. But anyway Carter's Pat's all dgp. You should be banned just for saying that that's disgusting.  We know They obviously they had one of the best quarterbacks in the game though They had one of the best receivers in the game. They had one of the best running backs in the game Um, but and something that still is also like I always say that i'm very jealous of they are so great at drafting and developing Wide receivers. They're great at that or the browns great at that Or the browns great at drafting anything um Yeah, but the thing about Todd heli the thing about Todd heli They had a nice offense But the thing about Todd heli that is his biggest downfall And we've seen it time after time after time after time after time His biggest downfall as an offensive coordinator Is his situation Is his poor situational play calling his poor situational play calling His poor situational play calling it is the absolute worst in the world It is the worst in the world Todd heli is the he's the offensive version of dean peas Overall, yeah, they're pretty decent coordinators But when it comes down to crunch time when you need them to call a play That's really going to take your team comes to closing the game out They fail fail They fail They fail they are bad. He's a bad.  Yeah, they were up I believe they were up by Uh a field goal They could have The pissberg stealers could have run a clock out Drained a lot of time off the clock, but I believe all either on first or second down um Ben Rothenberger They call the play for him to drop back and actually pass the ball In the past, I mean, I can't blame the the actual Pass that Ben Rothenberger threw on Todd Hailey because it's not Todd Hailey just throwing it Um, but the fact that you have a pass called in that situation It was just terrible It was terrible I couldn't believe that he did that but I was glad because it let it let us save a time out So that was nice, um So a big shout out to Todd Hailey for Trying to help the ravens win that game. Maybe he just knew he was on the way out I think that's what it was. He might have just been mad at pissberg that day and say, you know what? Forget this Throw it throw it um, but anyway I For the browns This could work out. Um a couple of ways it just depends on Their own situation Juan Lopez said big Ben can call better plays No huddle than Todd Hailey can Hey you as a Steelers fan, um I I really I believe you I believe you I do not discredit anything that you're saying because I I definitely believe you because from what I've seen From this and I'm sure pissberg fans are super happy that he will not be returning Did you all get a new office of coordinator yet or are they doing something from within are they doing something from within? I believe they are i'm not 100% sure though, but um, I know Steelers fans are happy about this Because still Steelers man that so jealous of their offense so jealous of their offense But they Yeah, and with uh with Juan saying that It just seems like it's so correct because Big Ben He does his best work When when he's sort of like freestyling that thing When he's sort of just going out there and and even though it sounds so cheesy when they say it When he goes out there and plays like it like he's playing, uh, like he's playing backyard football, man Like he's playing street football. That's when ben rosberg is at his most dangerous All the organized plays and all that that's fine and whatnot and he's still dangerous with those but When that dude drives back and he starts Uh, doing his little hot rounds telling a go deep go deep and put pointing his finger out And when when the play lasts more than six seconds. Oh, it's dangerous. It's dangerous Rosberg is dangerous Dude if that play last more than six seconds I mean if he's holding the ball for more than six seconds, then it's dangerous And it's it's such a high risk high reward type of thing Because we've been rosberg Um, if he's holding that ball for a long time. Oh, you best believe that he can put the ball on the money He's going to put that ball on the money and he's going to get a huge play Out of whoever the wide receiver is or running back to because they got levy on bail He's going to get a huge play for the stealers out of whoever the offensive playmaker is now With it being high risk high reward the high risk part Is that he can end up getting whacked fumble the ball Uh, something like that happened. He doesn't usually turn the ball over though when he's doing a super long play He does not usually turn the ball over like as far as throwing an interception He usually does not do that when he does his backyard plays his backyard football plays, but Anyway, um browns This new offensive coordinator, is he going to get a chance to work with the shan kaiser? I think so. I think they'll give kaiser another shot Because kaiser looked a lot better towards the end of the year than he did In the beginning of the year, obviously he's more A little more mature. I mean he was at the The first the beginning of the year was obviously his first time playing as an nfl quarterback So he was a fresh brand new rookie fresh meat but now He started to get more accustomed to the game started to get used to stuff and he got better As the season went on. Oh, look at this nice car. That's nice. What is that? A fan of this? I don't know what that is. Oh, y'all can't even really see it that much Luen said kaiser's the worst QB in the league Yeah, I was gonna say I think they're gonna give him a shot based off of his play But at the same time, you never know what the browns because they are not Another nice car But they are not consistent at anything Um, so brown they they are a very quick the only thing they're consistent at is keeping Hugh Jackson Because I was about to say the browns They they are so quick to give up on stuff, but they really stick with their coach They really stick with their coach Um, I don't know how I don't know why it's guys with far better Obviously guys with far better records than him that got look at uh, was it was it malarkey? Whatever the Titans Q not the qb code whatever the Titans head coach was The dude got into the playoffs and he even won a game in the playoffs Granted, it wasn't against a really good team. I mean it was against the chief. So they I mean they but He got his team into the playoffs and even won a game in the playoffs and he got fired But a browns coach still sitting there with his head held high Keeping a job after going 0 and 16 after winning what one game in the last 31 games, I think like Does that make any sense to anybody in here? But anyway, um That's that man these browns the browns organization is just they are the browns organization um Stability is one thing that they lack Stability is one thing that they lack Um, except at their head coach position except that head coach position. That's it Except that head coach position Head coach position is the only position there was stability and of course joe thomas too And shout out to joe thomas because this dude could ask shout out to alice mac too because alice mac was there for a long time too But he finally got up out of there and got to enjoy some success with the falcons um But joe thomas is a guy He's been there for forever now. He's obviously experienced nothing but lost nothing but losing um and Joe thomas he never you never ever hear this guy talk about he never throws anybody under the bus Never not one time. You never hear about this guy complaining about the browns.  He plays football and shuts up He played football and shuts up and that's it The the dude He could have all the reason in the world and all the right in the world to complain over the team over the organization because He's been there for forever and he's been doing his thing for a long time but And the browns have also been browning for a long time, but he could He could easily complain over but he doesn't he doesn't So anyway, appreciate y'all. Thank you. Um, I'm looking forward to this schedule because we can get more videos like these And that might add to the consistency a little bit too. We'll see. I mean we'll see. I appreciate y'all though, man This was a lot of fun. Um And again, if you just joined us the browns are going to be hiring Todd Hailey The former Steelers offensive coordinator to be their Offensive coordinator. I wish we had the cap space that those guys have Ravens could really use some cap space. Anyway, appreciate y'all. I will see you guys Later on maybe I might I might see you all later on tonight. I might I feel like I will um So, yeah, we'll see man. We still got to talk about uh the patriots and the eagles And like I said, like I told y'all yesterday, we are not going to beat that thing We're not going to be the dead horse. Um We're going to talk about it maybe one time two times at the most Not going to keep talking about it over and over and over but we got other things to talk about. Anyway, I appreciate y'all Thank you This was fun as it always is It's funny. I'm actually I'm pulling up to my house like within the next 20 seconds So I might as well wait to go ahead and shut it down then somebody just came through. What's up, Garrett? Appreciate you coming through man Appreciate it. Appreciate it. All y'all up in here right now, man Thank you. Thank y'all for coming through Thank y'all for all the support like always say man. Thank y'all for the support on the channel Y'all look great. I want to get a new camera, man. I want to get a new camera I need to start like a go fund me or something because it's a camera that I want to get um, and it's a streaming camera Uh, and it's like So Yeah I want to work on that But anyway, um Thank you all I love you all. What's up guy bear? Appreciate you buddy for stopping by So, yeah, I see you all Oh, you got two dollars on that. I'll appreciate that Or a go fund me I don't know. I had to find out how to set one of those things up But yeah, I just I do want to get that better camera because it'll make the the stream and the quality of it That much better because y'all know that I'm doing it from the phone right now Um, so I mean we'll see I might do a go fund me and just throw it out there. We'll see Um, but anyway, I appreciate y'all man. Thank you all for watching. What am I going to do during the offseason? Oh, we got plenty of stuff to talk about. I got a lot of questions from subscribers that I never uh Answered too. So I'm definitely um Yeah, we're definitely going to do this man. We're definitely going to uh We're going to have fun doing the offseason. It's about five hundred dollars So, oh wow logan. I appreciate that. Wow Thank you Wow y'all uh, um That's very humbling That's very uh That is very humbling I appreciate that a lot man Wow So um I'll talk man. Y'all better make me cry on the stream. That like uh, that made me feel really good. That was really uh That really meant a lot man. I appreciate y'all. Thank you, man. So I had to think about doing that go fund me to see and uh Yeah, man.  It did. Um, all right I'm gonna go ahead and get in here now. Uh, see what carter and my wife are up to and uh, I'll see y'all in a little while We out man


was heading magnetized so the Black Friday is back and ready to react and after I did the 80 miles an hour thing you know there that reaction few y'all said to check out the one with the wife with the dude with his wife because that one's funny so I guess we gonna see if it is funny or because I found it and I'm ready to do this all right so let's do it all right we're traveling from Logan to Boise and I just proposed a math question to my beautiful wife Chelsea and the question very difficult to calculate is if you are traveling 80 miles per hour how long does it take you to go 80 think Chum let's go through the process what do you think you're talking about you think that affects it it's pretty close do you know what the answer is think about 80 miles per hour so how long does it take me to go 80 miles if I'm traveling 80 just whack it in the half it affects it yeah that's true hour and pH so if I say I'm driving 80 miles per hour that means I'm going to go 80 miles in an hour if I'm going 60 miles per hour this is a long discussion on such a simple thing okay child the answer maybe I'd say about 400 times per mile okay I'm going to tell you the answer you're ready for the answer you're ready for the answer the answer to you don't know the answer I do know the answer you guessed me just like I guessed the meaning I'm using math science I'm using my own term the answer to the question how far you know how long does it take me to travel 80 miles if I'm driving 80 miles per hour the answer is one hour your guess was close so say you drive 60 miles per hour at 60 miles per hour you want to go 60 miles at 60 miles per hour is that an hour I'm going to have to subscribe to you good sir because of this is something I would have loved to do to a girlfriend that I had it was really fun she probably really got it though I forgot math before asking the answer asking her the question watch the comments below like and subscribe


All right, now may I invite on stage Mr. Rajneesh Berl, founder of Great Paribas also our esteemed panelists and joining me on stage. Do you remember from the recently concluded news broadcasting awards? Sudhir Mishra, founder and managing partner, trust legal advocates to please come up on stage. We're going to be talking about entertainment and media. Let's find out in the shortlisted nominees.


So, good morning everybody, I am Manjul and Samin is my thesis advisor and this work we are doing in collaboration with Manikandan Mathur. So, the topic is Lagrangian Coherent Structures resulting from 3 dimensional axial water stress axial vortex breakdown. So, first I will define not define basically I will see the examples of axial vortices. So, they are very common feature of and have a wide range of applications in fluid flows as you see is one of the very common example bath of vortices and when it comes to aerospace applications you see these axial vortices resulting in delta wing, flow pass delta wing and as well as in swirl combustion chamber you see that there will be a axial vortex. So, they have a wide applications. So, I will define what is breakdown in axial vortices. So, it is defined as an abrupt change in the in abrupt change in this core of the axial vortex when we go beyond certain Reynolds numbers which we will define later on. So, based on these structure that they result in they are classified as bubble type breakdown you see a bubble here and a spiral type and as well as the double helix type. So, what happens is stagnation point appears and after that a recirculatory region is there. So, here this recirculatory region is resulting in a bubble shape, here it is a spiral shape and here is a double helix type. So, we study this problem in a closed domain. So, this is the two dimensional schematic of the problem that we are studying. So, this is a closed circular cylinder in which the top lid is being rotated with a rotational speed omega the dimensions are as shown and the relevant parameters of this study are Reynolds number and aspect ratio which is the height to the radius. So, as you see when this omega is beyond certain values you will see this first I will talk about the how the dynamics is here when this plate is rotated what happens is that this fluid at the top which is pushed to the side walls and then it is bent towards the non-rotating stationary plate when it reaches here it is again bent upward at the axis and then you will have a recirculatory region. In three dimensional cases you will have actually a spiraling down flow so this has simplified 2D schematic. So, when we increase the omega beyond certain speed this nice axial vortex actually breaks down something like this which we call at the as the bubble type vortex breakdown. So, you have a stagnation point here you have a stagnation point here actually four stagnation points are here and you say that these are the two vortex breakdown bubbles. So, based on the combinations of Reynolds number and gamma you can have different numbers of breakdown bubbles as we will see along. So, to simulate this problem in three dimensional we are solving the incompressible Navier-Stokes equations in cylindrical polar coordinates in this form. So, you can notice here that for the radial momentum equation the qr is not the conventional radial velocity ur instead we are using a variable r times ur. So, this is basically done to achieve the to avoid the singularity at the axis and the grid that we are using to solve this problem is staggered grid. So, you do not have to solve q theta and qz at the axis so that is how we avoid the singularity at the axis and again the Reynolds number is omega r square by nu. So, these equations they are solved using the fractional step algorithm based on the Orlandian, Versailles and Camus's works. So, this is the main part of the algorithm where you discretize the Navier-Stokes equation and then you solve for some approximate velocity which is with the wrong pressure and then there is a Poisson equation which you solve with that approximate velocity and then you enforce the continuity equation to get actually this equation and then after you solve for the correct velocity and then correct pressure using these relations. So, I am not going to detail of this fractional step algorithm. This is actually the Scudius map which he obtained in 1984 when he conducted extensive experiments on this problem and he presented that the number of breakdowns they lie in this map of Reynolds number versus aspect ratio as such the solid lines are obtained by him. This dashed line it separates the region of steady and unsteady flow regimes. The dots are the cases that we have simulated numerically in this presentation I am talking about the aspect ratio 2.5 and more or like precisely I will confine myself for this Reynolds number. So, you can see in this map that for some aspect ratios and Reynolds number you have only a single breakdown bubble for some you have 2 and for a very small region you have here 3 breakdown bubbles. So, first we look at the Eulerian picture of the flow how the flow looks like. So, when you have aspect ratio 1.75 if you look in this picture 1.75 somewhere lie here 1850. So, you have a single breakdown bubble here which is shown here these are the iso surfaces of 0 axial velocity and these are the contours of axial velocity in a plane and this stream lines actually look like this this is a recirculatory region. Similarly for this combination of parameters you have 2 distinct breakdown bubbles and again the corresponding stream lines are like this. So, then I will talk about this aspect ratio so how the transition happens in aspect ratio as we are increasing the Reynolds number. So, as you have seen previously Reynolds number 2200 it is steady you have 2 distinct bubbles and this is a horizontal slice which shows the contours of axial velocity you see it is perfectly axis symmetric case. As you are increasing the Reynolds number further you see both the breakdown bubbles merge and instead of 4 stagnation points at the axis you have actually 2 stagnation points now when you still the flow is axis symmetric then again you increase the Reynolds number and you see that they merge as in this shape and start moving along the axis still the flow is unsteady, but it is perfectly axis symmetric. When you increase the Reynolds number further 23500 rotating azimuthal wave appears which in the flow which is rotating near the side wall and the wave number we have determined of this wave is to be 5 which we will see in the next slide. So, as we are actually increasing the Reynolds number you will have a non axis symmetry and unsteadiness. So, sorry yeah so that is how these both the cases look like we will not play. So if you see that this breakdown region is actually moving then it disappears then appear again, but the flow is axis symmetric, but in this case you can clearly see an azimuthal wave which is rotating and you can see clearly the wave number is 5. Now we but not for the much detail you can get from this flow so I go to the Lagrangian picture of the flow. So for the Lagrangian picture first I will talk about the motivation for this study. The first motivation comes from a paper Stevens et al. in 1999. They have done the dive visualization of this flow in the unsteady regime for aspect ratio 2.5 and their dive visualization clearly shows some so many rib like structures at the axis. The physical significance of these structures is still not understood very well and not much have been talked about these structures. This is the first motivation that we have and the second motivation come from the Sotheopolis paper in 2001 where he has shown that if you have a small perturbation to the breakdown bubble then actually a fluid can exchange across the breakdown bubble with the surrounding flow. You can see here he has distributed some particles in the breakdown region and then after a long time you will see the busting events and particles will escape the bubble this called the emptying and filling mechanism of the bubble which I am not going in the detail. So to study these two problems this is the topology of the breakdown which we have recreated from the Wekin's book. So in the two dimensional you can see the bubble axisymmetric bubble is actually an invariant surface. So this line actually will not allow you to penetrate the bubbles across the surface. So whatever is inside the bubble will be inside and whatever is outside that cannot go inside it will be just moving around the bubble. So bubble actually act as a bluff body to the flow. In the three dimension also it is a sphere, it is an invariant surface. So just to visualize this thing what we have done we have actually advected some particles inside the breakdown region, advected some particles outside the breakdown region and see what happens to the flow whether they really get exchanged or not. So this is the movie, so if you can see here the green particles which are inside the breakdown bubble they are actually just moving up and down and rotating but they do not cross and similarly the particles which are outside the breakdown bubble they do not penetrate inside the breakdown bubble. So they just move around with the flow no particle actually crosses and then how the topology will be in the perturbed case. So you can see here this is plane one, so this point is actually these are two saddle points and this is the unsteady manifold, this is the unstable manifold and this is the stable manifold. So you can see when the perturbation is present there are some trajectories which might actually escape the bubble through this unsteady manifold of this point which we will show with these two movies, so sorry this should be 3400. So 3400 we have seen this is the axisymmetric case and unsteady so these bubble do not actually cross the surface while in this case they will eventually actually go out of the breakdown bubble and mixing will be there because of the non axisymmetry that is present in the flow. So now we move on to the first motivation of the Lagrangian study that we had. So we want to compute the Lagrangian coherent structures of the flow. So Lagrangian coherent structures as defined by Haller they are the shearing material surfaces which are independent of the reference frame. These are some of the few examples that you find that are given in this paper. Then how do we calculate LCS is first we compute the finite time Lyapunov exponent of the flow. So how it is computed? We have a parcel of fluid which is advecting in time at the later time the fluid parcel becomes something like in this shape. So we construct a matrix which shows the relative separation between two adjacent particles with respect to the at some time t0 plus t with respect to the initial time. So we construct a matrix like this then a Cauchy green matrix is defined something like this here t is the transpose of matrix m. Then we find the Eigen values of this Cauchy green matrix and then the FTLE field FTLE field is defined as the maximum using this maximum Eigen value of this matrix is defined given by this formula. So the FTLE is classified as forward FTLE and the backward FTLE. So forward FTLE is obtained when the particles are advected forward in time and similarly when the particles are advected backward in time it is called backward FTLE. So this is the comparison of this FTLE field with the Eulerian picture that we have. So this is the Reynolds number 3500 you see this is the this plane shows the iso contours of axial velocity and this is the FTLE field with different integration time with respect to the first initial time when the particles are advected. You can see that the FTLE field show a rib like structures which is populated in this axial area and it is the whole region actually is populated which is analogous to the to those pictures that we have seen in the dive visualizations of Stevens et al. And similarly so what is the meaning of these rib like structures so forward FTLE is basically gives you the manifolds which are most unstable that is where actually the maximum stretching of the material is happening. So these are the candidates for the unstable maximum stretching manifolds when we actually can compare the two for the FTLE field for two different Reynolds numbers which we have seen 3400. So this one is without the non rotating azimuthal wave this is with the rotating azimuthal wave but in both the cases we can actually clearly see both the cases they are populated with these rib like structures. So what is the significance of these structure we have to further study this but what we can say now is these are the candidates for the where maximum stretching is actually happening for the material lines these red structures. So with this I want to conclude so we have seen that axisymmetric bubbles are actually invariant surfaces but material cannot exchange through these surfaces and unsteadiness actually breaks that symmetry. So symmetry is actually barriers to the mixing and the finite time Lyapunov exponent actually captures the flow characteristics which are absent in the Eulerian field and we can actually conclude those rib like structures they are quite meaningful where the stretching is happening we further have to conduct the study for that and yeah that is the questions.