Root exudates can be used to improve the solubility of phosphorus in soils, which can then be taken up by plants more efficiently. This can help to reduce the need for fertilizers and make agriculture more sustainable. This article was authored by Hugo A. Pantigoso, Daniel K. Manter, Stephen J. Font and others.


Hey co-friends, I'm Bianca Renee and you're watching Bianca Renee today and today I'm gonna help you guys solve a problem that I know many of you are struggling with which is frizz on the top of Your head. Yes, it happens to the best of us even me now The short solution to this problem is just don't care like it's a little bit of frizz It's not the end of the world as long as the lengths of your hair nice and defined like that's most important If you look real closely Every curly person struggles with a little bit of frizz on the top or like the exposed parts of our heads But if it's getting a little out of control like this piece right here This is Unacceptable now. This is a really common problem if you have bangs and I'm gonna tell you right now I'm gonna probably make a full video on like The truth about having curly bangs But it's a lot of work and I would not recommend cutting your hair into this bang shape If you are a beginner in your curly hair journey But because we're putting all the hair on the top of our head.  Brianna my Barbie doll her little water bottle Let's use it Shout out to Barbie for making toys that can actually Help us in life. You could use an actual adult squirt bottle, but I'm gonna use Barby's cuz I don't want to get up so Find the problem areas this whole this whole chunk right here Needs to be fixed. Okay, so here's the before Hot mess I'm gonna spritz it Is this actually functional? Yes, Barbie clutch All right Sometimes just adding some water will do the trick because it'll reactivate whatever products already in your hair Well, I'm too tall So you're gonna add some water to smooth out that frizz So already That looks Better now to keep it pretty and curly. I would suggest Using a gel any type of gel you have shall work you even could try your cream stylers But if you use a gel like this curl keeper gel, which is one of my favorites It has more hold now you could just apply the product you could do a little twirl around your finger Let's split this into two So what you're gonna do is you just kind of Twist it around And you're kind of training your curls To be better and you just continue to do this for any frizzy parts on the top of your head More gel Just to smooth the of those flyaways. There you go. Don't look amazing Okay, just kidding. Obviously. This is not a good look It's just a very flat triangle look with no volume nothing that I really desire But now that I use the gel I know that it's going to create a little bit of a cast making it safe to diffuse Now I'm gonna grab my diffuser and we're gonna fix this so now say a closer look Much better see how all these curls up top are now just more defined Look how shiny this one is and because it was a gel It was able to create that cast and not create more frizz. This looks I think so much better It's really easy to do the the more you do it the faster you'll get So this is what I do to make my curls look extra poppin even on the top of my head creating more of that definition Less frizz like I don't think you're ever gonna be able to get rid of every single piece of frizz like I mean Tiny micro things but I dare Someone to stop you and say something now Although I showed you how to do this for the frizz on the top of your head Feel free to use this anywhere you have frizz Maybe have like a trouble spot in the back Maybe you have some little pieces in here that always get frizzy wherever it may be get a little bit of water A little bit of gel wrap it around your finger and let it dry and it'll look so much better So that is my quick easy video on how to refresh those annoying pieces of frizz on the top of your head If this was helpful to you make sure to give me a thumbs up and subscribe I post new videos every week And I have so many curly hair videos on my channel to help you with your curly hair journey So this binge watch my entire curly hair playlist This isn't like Disney plus or Hulu or Netflix where you have like a subscription fee every month. This is free It shouldn't be but it is and what beyond grenade today has to offer that other subscription services don't is Beautiful curls after viewing so make sure you subscribe and hit that notification bell So you don't miss any of my videos and you can follow me on Instagram at miss beyond grenade That way you get daily tips and you're always the first to know what I'm about to be talking about here on YouTube I'll see you guys next time. Thanks for watching beyond grenade today


What's up guys, my name is Michael and welcome to my YouTube channel and today what we're going to do is we're going to do this problem. I saw this problem on the code where we have an array and we need to split it into k sub arrays, such that the maximum of each sub array is as large as possible, right? Now now we're going to do a variation of this problem where we're going to basically just split the array into k sub arrays, such that just all the possible get all possible combinations of it. Okay, so what do I mean by this? Let's say our array is 123456 and we need to split it into k equal to three. So we're going to split into three sub arrays. And what I mean by a sub array is that they're contiguous sub arrays. So I take one and then I split it by one, then I split by two, and then I have 3456. What I'm not doing is I'm not taking one and then taking four. Okay, I'm not taking one and then taking six. Okay, so as you can see here, we have 123456, right? We're one and two are individual sub arrays, and then we have 3456 as the last sub array. And in each of these, the length is only three, right? We only have three sub arrays in each of these because k is equal to three. We're splitting the array into three different sub arrays, three contiguous sub arrays. And the second one, we're also splitting it by three. So you have 123456. As you can see here, we have 23 is three is was in the previous in the previous array. But now it's in the same array as two. So we have 23456. Then we have one, then we have 3234, then we have 56, right? Then we have one, we split by one, and then we have 2345, then we have six. Now the next one, we cannot have one, then 2345, and then add six to it. We cannot do that. And the reason why we cannot do that is because then our k is not going to equal to three anymore, right? Our k would only have, our k would be, would only have two values, right? We need to split this array into three groups. So that's why this is the last possible third group you could have in this situation of one, then 2345, then six. Now in the second, in the third one, we have one, two. One, two is in one group now.  And the reason why we cannot have this five inside this one, two, three, four is because then we won't have three subgroups, okay, we won't have three subgroups, and we'll try to split it by, okay. So I think you guys understand what I mean by this, and by splitting it into three different subgroups, okay, sub arrays. So how do you do this problem? First of all, we need to each row here is going to represent a different combination that you could split it by, right? We have one, two, then three, four, five, six, that's one possible combination. Then we have one, then one, then two, three, then four, five, six, that's another combination. So each of these rows are individual 2d arrays, right? If you could see here, we have each of these rows are 2d arrays. So the first thing we need is we need a global variable called whatever it is, whatever I'm going to call it all combinations, where it's a three dimensional array, like a three dimensional array, and each of the row is going to represent each combination that you could possibly have. And in each row, it's a two dimensional array, right? We have one, two, then two, four, five, six, then we have one, then two, three, then four, five, six, then we have one, two, three, four, then five, six, and so on and so forth. So the first thing you need to do is I'm actually going to create, recode this up as if I've never coded it before, okay? So here I have, yeah, I'll just recode this up since I never, assuming I never coded this up before. So I'll code it up for you guys. So let's say I have my array of one, two, three, four, five, six, it's a vector. Then I need my all combinations. And all combinations, like I told you before, it's just going to be a vector vector vector integer, right? It's going to just be a three dimensional array where each row is going to be a two dimensional array of each sub arrays that we're splitting by. So if I'm going to have this function, I'm going to call, I'll call it solve. I need all combinations, right? I need it. And the reason why is because in each combination that I'm generating, I need to add my new combination into the all combinations, this variable of all combinations every single time. Okay. What else do I need? Well, I need my array, right? If I don't have my array, I can't split it. So I need to pass that in. Okay. What else do I need? Well, I need my K value, right? Because I need my K, like I need to split it into three different sub arrays. And if I don't have my K, then like, I don't know how many sub arrays I'm splitting it by, right? I don't know how many different arrays I could split it by. So that's why I need my K. Now, now it's going to get a little tricky. It's going to, this is going to get a little tricky, okay? As you can see here, we have one, two, three, four, five, six, right? The next one we have one, then two, three, right? So this, the, the location that we're splitting by the index we're splitting by got moved up by one, right? We have two, three. So this little next location was now split this two, three was originally in three, four, five, six. Now it's coming up to two, three. So that's where this is now. So that's split about that. Then the four that we split it by the index where we split it by, we originally split it by three. Now it's four. So now this four is now in two, three, four. So what we need is we need an index that we're going to keep incrementing up by one as we recursively call this function. And that's going to represent the location that we're splitting by. So we need to pass this in. I created an index called zero index, index equal to zero, and I'm just going to start from beginning and yeah, I'll pass them into the pass end to my function. Okay, here's the part where it's going to get a little tricky. So we have each our total combination array, right? That's three dimensional, right? What do I need now? Well, in each row, it's a new combination, right? It's a new combination of different sub arrays that I'm splitting by. So we have, see in each row, we have one as it's individual, then two, then we have three for five, six, then this one is going to be one, then two, three, and four, five, six. Then the next one is another combination of one, two, three, four, then five, six. So in each row, what we need is if we're going to keep modifying our previous previous row, we need to recursively pass in a combination array, a combination array that has each sub array inside of it, right? A combination that has each individual sub arrays. So this row is going to represent how many different sub arrays, and in each of them, it's going to have each sub array. So what is that? That's a two dimensional array, and in which each of these rows are the three sub arrays that we're splitting by. So we need to have this vector, vector into each combination, I'll call each combination. Okay. And then what else do we need? Well, we need to the, since we're recursively adding this new sub array into each row, right? We would also need is that we need each individual sub array, okay? So look at if we have each combination, each of these combinations have individual sub arrays that we split it by one, then two, then three, four, five, six. So because of this, we need to our each sub array, which is going to be just a vector of integers. So we have this vector integers, each sub array. This is going to be the each group that we're splitting by each, these individual groups. So we're going to pass it in. So here I have, I'm going to pass in, have my array, my K, I'm splitting by the index, and then I'm going to have each sub array, then I'm going to have each combination. And then I have all my combinations that I have. Okay. So now let's actually write this method. And then we could do this, we'll be on our way. So I'll just take this out and just record it void, solve, put this here, or whoops, and I'm going to write the method types. So the first array is just going to be vector integer passing in the K is just going to be an integer. Right. The index is just going to be integer also, each sub array is going to be a vector of integers, right? Each sub array is a vector of integers. Each combination is going to be a vector vector combinations. All combinations is a triple vector, a 3d array. So I believe that's right, missing one. Yeah. All combinations. Okay. So let's see which one we need to pass in by value and which one we're modifying. So we're going to pass in by reference for array because, yeah, it's not good to keep a copy of that. And the integer is we could leave that is that each sub array, I'll explain why we're not passing by reference later, later afterwards, separate and then we have each combination. We're going to pass by reference because we need that and then all combinations we're going to pass by reference because we need that. Okay. So what am I going to do in this function? Well, I'm going to recursively call this function called solve. And I'm going to pass in my array. Okay. And how am I going to do this? Well, every time I call this function, I'm going to split the array. I'm literally just going to split the array, right, into its individual sub arrays. So every time I split an array, the number of remaining sub arrays gets decreased by one.  Because if I have three, if I have three, three groups I need to split it by, right? I have K equal to three. And then I've already split it by one. So I've already split it by one. Then I have two remaining groups I need to split, right? Then once I split it by two at this two, split it at this two location, then only I only have one group remaining that I need to split it by. Then once I split it by here at this after this two, then I have three for five, six, then I have zero groups I need to split it by. And then at that point, my function ends. So every time we recursively call this function, I'm going to decrease my K value by one because every time I split it, the number remaining number of arrays I need gets decreased by one. So I'm going to use K minus one. Now, what about our index? Well, what we're going to do is, as you could see, looking at through this pattern, is that we need to increment our index by one every time, okay? So we have two, we have one, and then two, right? In this case, two was split at this part right after the two, okay? Then we have three for five, six. Then here we have one, then two, three. So the location that we split it by is now after the three, and this three is actually in the same sub array as the two. So we split it there, and then we have four, five, six. Then we have one, then we have two, three, four. So we're going to split it at this location at four. Then we have two, three, four, and then we have five, six. So every time I call this function, I'm just going to increase my index by one. Because that's going to be how the location that I'm splitting by gets incremented by one every time. See, four, five, and five, six, and so on and so forth. Then I have my each sub array, and then I just have my each column, each column, and then I have my all combinations. Okay, all right. Okay, so what do I need to do now? Okay, so we have our recursively call our function here. What do we need to do now? Okay, well we need to, every time we recursively call this function, we need to add our current sub array that we're splitting by to each row of the combinations right. So as you can see here, let's say we have one, two, three, four, five, six, and I split it by one. Well, let's say I already have this one here, that I already split it in my new sub array. I need to take this one and add it into my, each combination that I could possibly have. Right, because if I have one, here I split it by one, I need to add this one into my combination array. Right, then if I split it by two, like right after two, then I need to take this two that I just knew sub array that I created of two, then add it into my new combination array. Then if I split it by three, four, five, six, I'm going to take the sub array and add it to this new combination array. So if the K value, and not K, if each sub array is not empty, right, if the area that I'm splitting it by is not empty, then I'm just going to add it to my combination array. Right. Each of these, after you split it by, you need to add this new sub array into your combination array. Okay. Now, now we need to think about our base case. What is our base case? Okay. So we are decreasing the number of sub arrays. We're splitting each sub array, right? We have the original one, two, three, four, five, six. We're splitting it by one, and we split it by two. Let's say I'm on the last sub array. So what is that last sub array, right? Because we're decreasing the sub array every single time we're splitting it, like the number of sub arrays K minus one every time, the last one is going to be when K is equal to one, okay? So like if K is equal to one from here, this location, right? So let's say I split it by one. So now originally I had three sub arrays I needed to split by that got decreased by three. So the three minus one is will be two. Then I split it after this two, then this two gets added to the array and it gets decreased by one. So now then now I have one, when K is equal to one, that's my last sub array. So that's going to be the last sub array that I need to split it by. Okay. So let's say I'm on the last sub array when the case equal to one, what do I need to do? Well, I need to loop through from the index and just add all the remaining values that I need, right? I don't need to like, like pop things or push anything. At this point, all I need to do is start where I'm starting at this index. I'm sorry, I just loop through to the end and add it to my last sub array. And then I need to take that and add it to my combination, right? So that's all I need to do. So here I'm going to create a array called last. And I'm just going to loop through from my index, start from I into my index, I'll loop through to the end. And then from here, what I'm going to do is I'm just going to take last pushback added to my array index I, and then from here, I'm just going to add it to my combination array. So each combination. Okay, you guys understand what you guys following along, we're going to push by added to our last combination. Okay. And then I'm just going to return after this. Because we added to our last combination to your basics. And then once we're done with this, we just return it. Okay, we just return it. Right. Return. And then we need to get our new combination. Now let's look back at into our code. Let's look back into our code. Let me just make sure I follow this along. Okay. Yeah. Yeah. Okay. Yeah. Pushback. Okay. Yeah. Okay. Now, now we're going to add this last array into our combination array. Okay. Now there's some tricky situations where the last, like where we're splitting back could be empty. So let's actually check if it's not empty. It's just like a case of like every time you add something, you should check if it's not empty because otherwise it might give you like an empty thing. Okay. Okay. So now we push back or each combination array, then we need to pop it back. Okay. So each way you got pushed back. Okay. Each combination. Once we pushed this back of our combination, right, one, two, three, four, five, six right? I need to add this to this type of combination into our global array, our total array, right, that we have.  Okay. Here's going to be a little tricky part. It's going to be backtracking. What do you see here? We have one, two, then three, four, five, six. We have each combination, which is one, then two, then three, four, five, six. Let's say we have this first combination already. Look at the second combination. Do I really need to do, what is the difference between the first and the second? The only difference is the location I'm splitting by, right? The first two values are still there, one, two. And the only difference is that now my location I'm swimming by is there. So do I really need to copy all the values from four and put it here? Really I don't. I still have my previous combination, right? My previous combination is one, two, then three, four, five, six that I generated. Now what I could do is I could just, the only difference that we have from here is just the last element, right? This last element is three, four, five, six. What all I have to do is literally just remove this last element, okay? And the reason why is because if I just remove this last array, all I have to do, right, I have the one and the two here already, all I need to do is just take the next index, the value of the next index and just add it to my previous array of three and two and three, right? Do you guys understand what I'm saying? Right? If I, if I'm on the last one, right, let's say I already added this into my new combination array, I don't need to copy all the previous elements. All I have to do is just remove the last element and then add, add it at a new element, change the last array again and then add another combination. So for that, what I'm going to do is I'm going to just do each combination dot pop back. What this does is it removes the last sub array of three, four, five, six, that's gone. And then I could just go to the next one, add the three into my previous two, and then add the four and five, six, then I just remove four, five, six, add four into my previous two, three, then I have five, six, there's removed the five, six, take the five, add it to my previous two, three, four, five, then remove, then that's it. Okay, so that's basically our base case. All right, we have our base case, we have the, what we're adding every single time and so on and so forth. All right, now we need to think about how we're going to pass and solve. Okay, so each of the array that we're having is going to be this, okay. Let's say we're at two, one, two, and then three, four, five, six. We added this already. Then we go to the next one, right, we go to the next one, okay. Now we, how do I get from here one, two, and three, four, five, six, two, one, to this array of one, two, three, four, five, six, okay. So let's say I already popped the last element of three, four, and five, six, right. What I need to do now is that I need to generate the next possible array, right, the next possible array. So the next array I need to split by is going to be four, five, and six, right, I move the index up by one, I added the three into the previous one, and then I need to add four, five, and six. So the next array that we're adding is has to be empty. So what I'm going to do is I'm going to actually clear each sub array. So I'm going to do each sub dot clear, and what this does is it's going to completely erase all the elements in it, and then have our a new sub array that we're adding. Okay. So now what am I going to do? I need to loop through. I'm going to loop through here and just continuously add the next, what am I missing? Oh, so remember, let's say we have two, three, one, two, then three, five, six, what am I going to do is I'm going to go to the next location I'm going to split it by. So we have one, two, then three, four, five, six, right. I'm going to go to three, then split three from five, six. Then after this index, I'm going to go to four, split it, then I'm going to five and split it. Then when it calls go back to the beginning of one, two, I need to loop through from here to the end again, and split it after the two. Then we have three, split it there, then four, go to the loop to the end, split it one, two, then three, four, then I split it for four, then go to the five, six, then split it, so on and so forth. So in this case, I need to actually put this into a for loop.  And yeah, now every single time I call this, I need to add a, every time I add a loop through here, and split it of where I'm increasing my index to split it at, I need to essentially is regenerate a new array, right? So like we have two, then you have three, right? You have two, this two was originally two, and then it becomes two, three, then you have two, three, four, so on and so forth. So before I generate this, I'm going to actually push back the current value at my current index, okay? And then, yeah, then after I push back the current value at the current index, recursively call my solve function, then, then what I need to do is just check if the combination array is empty. Okay, so like let's say I am at, yeah, so let's say I'm at like one, this part one, two, three, four, five, six, then I move to my next index to split at three, then I split at four, then I split at five, and I'm done here. This recursive call is going to get at one, two, okay? So how do I get from one, two? Well, simple, I need to go back to my previous combination. Each combination I need to pop it. So remove the last elements. So before here, we would add like one, then two, three, four, five, then six, right? At the end of this, it already popped the six out to generate the next one, right? And then it tries to generate one, and then two, three, four, five, six, where the six is at the previous one, and it can't add it, right? So then it pops, it goes back to here. So now, now we're going to have just one single element of one. Then in order to get the next one, I'm what I'm going to do is I'm going to pop and remove the two, three, four, five. All right, I'm going to remove this previous, previous array, sub array in our previous combination. So to do that, I'm just going to go do if each sub is not empty. I believe it's each sub. Yeah, your sub is not empty, I'm going to pop it. So yeah, then when it comes back up to the top, it goes to the next one. So it'll generate the next one, and so on and so forth. So yeah, that's basically the gist of the code. I'll just run it real quick for you guys to show you what it does. Make sure this runs properly, actually. Oh yeah, I should show you guys how to print it out. Because now you have like all the possible combinations, but you don't, you can't print it out. Okay, I don't know why it's sticking forever. Hold up, I plus one new sub club. Each sub array is not just pop it. Each sub array. Oh wait, each combination. Oh, my fault. This should be each combination. So yeah, each time you push the sub array back. Yeah, that was supposed to be each combination. All right, so every time you call this, and you finish your combination of one, two, three, or five, right, you need to pop the last combination out because, you know, when you generate the next one, you pop the six, you pop the one, two, three, four, five, then you can go on to one and two. So you need to pop the combination. It's not the sub array. The sub array is just the individual element here, right? So yeah, yeah, I'll show you guys how to print this now. So what we're going to do is we're going to loop through every, our three dimensional combinations. I'll print out a bracket, then I'll loop through from j equals zero to all combinations outside, print out another bracket, I'm going to loop through from K to all combinations outside. So I'm going to go through all my, all my, my two dimensional, three dimensional array, and we go through every single two dimensional, two dimensional array inside of it. And then I'm just going to go through every single individual array and print it out. So go through three dimensional array, print it out, two dimensional array, print this out, and then each individual array, print it out, then just, if it's not, but yeah, if I'm not at the last element, now I'll just print out a comment at the end. Then at the end, I'll just print out another bracket. So yeah, and then in each of these, I print another bracket also. So that's pretty much just the code. Hope you guys enjoy this video. This is a fun video to generate all possible, split it by K combinations. Yeah. Hope you guys enjoyed this video. Great, come subscribe. I'll check you guys later. Peace.


Welcome to the ITU studio in Geneva where we're joined this morning and we've got the great pleasure of having Jean Taudin the studio here Who is the president of the FIA as well as the UN Secretary General Special Envoy for Road Safety Jean welcome to the studio Thank you glad to be here with you We're here because we've got the future network car symposium that's happening today here at ITU Would have been at the Geneva Motor Show but that was cancelled As I mentioned in my introduction as well as being president of the FIA The Federation Internationale d'Automobile You're also the UN Secretary General Special Envoy on Road Safety Now 2011 to 2020 was declared the UN Decade of Action on Road Safety Where do we stand today? Yes that's right I mean 2011-2020 was the UN Decade of Action for Road Safety Incidentally, I like your badge Road Safety is part of the SDG 3.6 with the aim to half the number of victims by 2020 And unfortunately it will not happen And we had a ministerial meeting in Stockholm 10 days ago And of course it was very important to make an assessment of what had happened And what needs to happen in the next decade 2020-2030 So I mean unfortunately I mean due to the increase of number of vehicles around the world It had been one increase of victims on the road And if you take the WHO figure Last year 1.4 million people died on the road Which means over 3700 people dying on the roads every day And you know in a period where we speak about other pandemies I mean can you imagine what it means 3700 people dying every day It's number one cause of mortality for use between 5 to 29 And on top of that we need to have injuries because of road crashes And it's one evaluation about 150,000 injuries with disability So of course it's a pain for the victim but it's also a big burden for the country And it's depending of the country between 3 to 5% of the GDP And you know contrary to other pandemies we have the prescriptions And here it comes to I mean the theme podium It's around education, law enforcement, vehicles, road infrastructures and post-crash care So here with all the new technologies we can be applied as essential And also UNEC is developing the UN Road Safety Conventions Which is absolutely essential to convince the member states to apply those conventions So I mean that was part of the topic which were discussed in TOCALM And I mean I would say that's also a great opportunity to have this sympodium To refresh memories, to make also an assessment about what had happened What are the new technologies and of course we speak a lot about autonomous vehicles We speak a lot about electronic stability control I mean and of course we are talking about two different level of technologies Autonomous vehicles, I mean we speak a lot about it but it will take still years before I mean globally there is access to this kind of vehicles So I do believe that you will have available autonomous vehicle probably for public transportation But I mean we need to add now to develop new technologies Or to make sure that they are applied in developing countries Where over 90% of the victims are occurring And of course the average of age of the vehicles which are four wheels or two wheels Are of course much older So it's very important to see also what can be adapted to those old vehicles with new technologies I would say it's a fascinating time, challenging time as well But a lot has been done and much more can be done Let's talk about the technologies If we had been at the car show, I'm sure we would have seen a lot of innovations there What are the most interesting and the latest technologies in automotive? I will say at the moment we have been talking about road safety But I mean one big topic which rightly we are talking is about environment Climate change So I mean a lot of pressure is on manufacturers for vehicles with zero emission So I mean electric cars So if Geneva Motor Show would have happened I'm convinced that a lot would have been about electric technology Which has a lot of merits but also a lot of downsides So in reality I do believe a lot on hybrid technology which is a mixture And of course new technology are under development And soon coming like fuel cell and hydrogen technology So I would say it's a fascinating period But I mean even more than a period I think we are living in a fascinating part of the society Because when you see, I mean motoring is quite new It's just over a century ago And when you see I mean the development which has occurred over each decade It's absolutely fascinating But of course it does create problem It does create problem because the number of vehicles have increased Number of vehicles increasing traffic, congestion, pollutions are increasing And it's something which needs to be addressed And again a symposium hosted combining UNEC experience, expertise And ITU will allow us to move forward Are automation and connectivity the final answer to road safety in your opinion And how would this answer differ in developed and developing countries You touched upon in your previous answer But I just wanted to just probe a little bit more here I mean of course that would be ideal you know Again if we take statistics 90% of road crashes is due to human errors So and you can see that I mean in the air You know in the air I think last year it was less than 1000 people who died because of air crash And I mean all most of it is automized you know So it would be fantastic if we could transfer that to the road But unfortunately it is not yet possible That's why we must concentrate to the reality And the reality is driver head And it's a lot of cheap driver head And I mean I will take an example I mean anti-brake system, ABS you know Which is quite new Maybe 20 years back or 30 years back it was an invention So I mean it allows you when you brake I mean the car is deciding I mean the pulsion on the pedal Because it's better than human being You know so a lot of those technologies are smarter than human being So we must make the difference of what is possible now What is achievable And what is not yet achievable Which will come later And I mean make sure that everybody will be able to enjoy as soon as possible Of those new technologies Finally let me ask you in your view What are the key barriers to achieving the full potential of new automotive technology I would say finance You know finance regulations amongst the government But I mean you must bear in mind that unfortunately a lot of countries Around in the developing world have not yet access to public transportation Which incidentally is another SDG 11.2 To give access to public transportation to any citizens around the world by 2030 So you know I mean here we are blessed Because we are living in developed countries where there is a lot of good education On road safety it's a good law enforcement Vehicles are good road infrastructure are good Post-cross care is good But it is not the case in the developing countries which are 50 years behind Thank you very much again for joining us this year at the Future Network Carson podium We look forward to catching up with you again very soon And I wish you the very best for the future Thank you and thank you for your engagement to help us to address better road safety Thank you


Let me hit you with something new! I'm staying in Super Saiyan 1! Ugh! I'm going to keep corner! Ha! Let's go! Staying in Super Saiyan 1! Ha! You know, I never thought this would happen, but... It's time to go back to the basics! Let's go! How's it going, everybody? Dahtonoi here, and I am very excited to finally be able to say that we are now entering the next iteration of Fighters with the main team of Super Saiyan Goku, Krillin, and Yamcha. Now, obviously this patch changed so much about the game that it's hard to cover in just this video, but the main character we're focusing on is Goku, and maybe if we're lucky, I'll try to show you that Krillin can come back from the dead now. Uh, wait, we'll get into that later. But we'll also get a...  Yeah, dude, he finally got his leg fixed. You know what? In celebration, I'm gonna start with a 2M. Oh, he knew I was coming. Oh, yeah, key charged. There's so much I want to cover. There's so much I want to cover. I was afraid. I was just afraid. All right, let's see if we can't come. Oh, no, I'm bad at the game. I'm bad at the video game. Why am I bad at the video game? Wait, wait, am I good at the video game now? Nope, still terrible. Oh, he saw me. He saw me coming with that new mechanic. Oh, that was sick. Unfortunately. Man, I actually have like... I don't know what to talk about because there's so much, but I think I'm gonna hit him with the EX Wolfang Fist. EX Wolfang Fist used to just be a move that I never used, but now it's kind of cheap if I live. Hold on. All right. Oh, no, he... Oh, he sparked it. He knew it was a threat. All right, let's try it again. Oh, he's dragging us. No, Yamcha's gone. Yamcha's gone. All right, this works out though. Maybe if Krillin dies, I can show you that he can self-revive, self-res. It's definitely a bug. I don't think they meant to include it, but hey, you know what? It is what it is. Also, they nerfed Krillin, man. I can't show you in this video because I don't want to drop the combo, but Krillin time is gone. You'll see I have to do a different route. You'll notice that I didn't do the standing S, and it's because it would have dropped. Also, I forgot that I didn't use my heavy. That was sick. I'm actually totally fine. I'm totally fine with Krillin dying. I want to try to hit this in the first video. If we can hit self-res Krillin, the legendary Sensubin. Oh, I could have been sick.  You got to piece him up after that. Spirit Sword, Spirit Sword should have respected it. I didn't, but you should have respected my mash. You should have respected my right to mash. Let's go. All right, so the only way this Krillin strat is going to work is if I get low enough that a Keyblast can kill me, but not, I can't die in one combo. Because if I die in one combo, I won't have a Sensubin on the ground. Okay, we're getting close. You should have let that hit us, honestly. Oh, no, we can't. We're not going to be able to self-res unless he drops it. No drops, but that is fine. There's a reason we kept Goku in last place, and it's because I am a 2M God again. I ignore this, ignore this for a second. Give me one second. Give me one second to become the 2M God I was meant to be. Goku Black Clear, just because he got buffed, bro. Hold on. Let me show you that you're still not the best Goku. Come here. Let me show you something real quick. Let me show you something real quick. And also the Sparking Blast, another system mechanic is if you're in Limit Break, your Sparking Blast doesn't go away as you're getting hit. Oh, I almost got him. Oh, you thought I was playing. Let me hit you with something new. I'm staying in Super Saiyan 1. I'm going to keep corner. Ha, let's go. Staying in Super Saiyan 1. So if you didn't know Super Saiyan Goku is a new super, well, it's the same super. We all know that, but he doesn't go Super Saiyan 3. Instead, he stays 1, and for less damage, you get to keep the corner. I mainly did it there to flex and show you guys. It wouldn't have mattered which one I did, but hey, it's still pretty cool, to be honest. Exactly. Hey, Goku, I like the talk. Dude, talk that talk, but there's still so much more we got to show off. Thank you. Thank you. Thank you. Thank you. Goku Black. Dude, Goku Black is so good now. He's so good. Oh, Double Rock. Talk to me about it. Oh, don't even try it. Oh, the anti-reflect is nice too. All right, let me see if I still got it. Nope, I don't got it. Resets though. I got resets. That's what I got. Take this man down to the ground. Let's go. Do I want to see if I still got it? I don't got it, dude. The fact of the matter is I don't got it. That's the real fact of the matter. Come at me, huh? Yeah, I got beams too. Let's go. Beams from up top, the top rope. We love to see. Kick this. All right. We got to respect Goku Black these days. Oh, yeah, let me show you some. Yamcha actually got booked as well. Uh, not in any way you're going to notice now, unfortunately. But if we were mid-screen, I would still be able to do that now. Oh my goodness. All right, let's use the new system mechanic. I just wanted Goku out. I just felt like letting Goku get some... No, get off me. Okay, I respect your wishes and I understand. You know what? I'm bringing in Krillin and I'm going to throw.  These are fast now. Oh, he's rolling, rolling. Oh, yeah, I'm just still got him. Okay, I'm going to throw Krillin's life away. Obviously, we're going to make it look tasteful. We don't want him to know that we're trying to res. All right, just hit me. Hit me in the corner. Hit me. He's respecting me. He's respecting me. I got lucky there. All right, I'm going to try to res. Oh, that was maybe actually perfect. I really want to try to res. Level three. Nice. Unfortunately, now I don't know. Are we close enough? I have to block a mix. The only thing we can allow to hit him with is like a beam or something. All right, let's throw the beam just in case. Oh, no, I healed.  Hey, my lunch. All right, I don't know how I would throw just enough to get this health back. I'm just trying to res, dude. I'm throwing, dude. All right, there we go. This should be perfect. That's perfect. He pushed me. Well, okay, we deserve that one for sure. For sure. I would say we did. Oh, let's go. Talk to me about it then. Oh, God, panicked. I panicked. Can I still kill? Yo, won't bang. This, I was really trying to do the Krillin res, man. It would be so funny. It's so impractical, but it is hilarious when it happens. Oh my goodness. All right, I will pull it off, dude. I promise I will. I don't know if it's just going to be through luck or through skill, but it will happen. All right, we are back and we are playing against somebody that clearly wants to mix, dude. Oh, that felt good. That felt good. Come here. Oh, Goku. Yes. Protagonist 2M, Protagonist 2M, Protagonist 2M, Protagonist 2M. All right, all right. Understandable. Understandable. I was just ha- Oh, what was that lag? Oh my God, the lag no jutsu. Also, I can't believe he forced my Yamcha into that costume. All right, let me see if I still got it. I'm trying to see if I still got it. I don't got it. I got to stop trying. I don't still got it. In fact, I never had it. I just wanted to do two. I just wanted to do two Kamehamehas. I just wanted the double super again to feel alive. Is that so bad? All right, Yamcha. Can we do the mix? Oh, I went behind him. No, the mix. Dude, he's going to mix me with these two. These two are so- The mix with Gotegs and Yamcha with the new system mechanic to switch out. I did hit a button. I must admit. I do have to- Even I must admit that I did hit a button. Oh, the- Oh my God, that was scary. All right, here's the Yamcha buff. Hey, mid-screen. Pretty cool, right? All I wanted to show you is that you could do that mid-screen now. Oh, don't even try that. Come here. Come here. Come here. You know you deserve this. I did the wrong button. Oh, why did I do this version? Why did I do that version in this- In this- In this weather? Nice. All right, good enough. Good enough. Good enough. Yamcha. No, you don't. Let me hit you with Yamcha, though. Let me show you. Let me show you why Yamcha's good. EX will thank this. Mix, mix, mix. Oh, shoot. Well, you know, I didn't get over to him, but the mix was good. It was good until it wasn't. Ooh, flex. She's going for the level three. I dropped it. I felt that in my core. All right, I'm going to do triple Kamehameha. Oh, why did I do that? Hold on. I want to do triple Kamehameha. Stop. Wolf, Fang, Fist. Love it. Wolf, Fang, Fist. Oh, shoot. Wait, how did he get Goku? It doesn't matter. Kick this man in the face. We're just switching in and out. What's that? That game felt a little laggy in some places, but I don't know if it was lagging up a story. I don't know. I'm hitting rematch. If it's too laggy, editor skip it. But if it's not, let's keep it rolling. I am the protagonist. Let me hit 2M. Let me hit my protagonist 2M button. That's all that comes down to is that I'm Goku and you're Gohan. That felt unfortunate for me. All right, but we still just take these for sure. We take these every day of the week. All right, we got to kill.  Why not, dude? Let's try it. Well, I don't want to try to kill him, but we'll fight with Krillin so that. Oh, that was nice. If Krillin dies, I want a self-res. I am buying that self-res kit. Well, I might have mashed. Stop that. Oh, nice. Oh, come on, drops. Nice 2H. Am I dead for that? I'm feeling pretty dead. Oh, the lag. The lag actually saved me. That is unfortunate for him. Somebody ate my bean. I ate my bean. Yo, I am just medpacking up. I am just medpacking up. Yeah, somebody stop me. Somebody stop. I just have my god. That is another buff. Sensei Bean does heal you for a...  Sensei Bean, if he kills me, I will self-res. Oh, no. No, he keeps eating it, dude. Stop eating it. All right. No, we got to put ourselves in front of it.  He just wouldn't. He wouldn't let me get in front of it.  Lag. Game, you need a lag. You need a lag, so he drops this. No, no, no, no. Come on, for Krillin, dude. He never got to resurrect himself. All right. Well, oh, wow, I'm bad at the game. I am so bad. He didn't switch. I like it. I don't know why I ran headfirst into that tornado. Oh, give me that. Oh, you got it. Let me have that one. No, let me do the combo. Dwych. Ah, ah, ah. There's nothing I can do here. You know what? I'm going to try to hit him with the Key Charge Drag Rush. Oh, that would have been sick. That would have been crazy. If you don't know, Key Charge Drag Rush goes pretty much full screen. I'll try to do it here. Ah, no, he's not standing on the ground. I'm just using every, every bit of this patch I can get my hands on. Every system mechanic change I'm doing right now. Dude, I'm playing like it's old Yamcha. I don't need to do that anymore. He also buffed my Key Blast, so it hits crouching. You can't duck it. Ah, look, he was like, what is happening? What's going on? I am small. I should be dodging these. Oh, poor Yamcha is dead, though. Poor Yamcha just, it has died, unfortunately. Unfortunate as it may be. It's tough. Wait, we're living? We're dead. We're living again? Oh, not like that. How do I just let him weasel out of the way of that? You know what? I'm the protagonist. I am the protagonist. Just let it happen. Just let the protagonist take over. You know what? I think I feel like flexing. That's right, Yamcha, you don't. Oh, I dropped that. He could attack the last day I dropped that. Well, Super Saiyan Goku still got the demo. You know what? I feel like we can include that. I'm going into another game, though. Usually we only do two sets. I need another one. Oh, no. We are not going against Genemba, dude. Genemba is another big winner from the patch. But do I care? No, I'm the protagonist. I just be clicking buttons. I just be clicking buttons. He got double teleport and everything.  I'm just saying what people got. Oh, that's bad. Oh, barely tech that. Let me switch out. Oh, he got hit by that. All right. Man, I miss. I miss the way Krillin used to work, dude. All right. We still get corner, though. Still get corner. Yikes. Yikes. Yikes. Unfortunate. Unfortunate. Oh, even with the faster startup, we got the bead. Oh, he countered. Life is pain. Life is nothing but pain. Let's go. Give me that corner back. He kicked us. But he kicked us into the Sensor Bead. Oh, no, that would have been sick. Oh, no, that was so funny, though. Oh, that would have been perfect had we lived. He kicked us into the Sensor Bead. If we would have died, we would have been resurrected. But it's fine. He's building up a lot of Dragon Balls. All for my sake, dude. I don't want Goku to not be the one to do this, though. What? I thought I was sparking for sure. Oh, no, Goku's down. The protagonist is down again. Wait. Drop it. Yes, I knew he was going to get greedy.  You got to be kidding me. All right, we've let him run ramp it for too long. Bringing in Goku, fine. Wait, why couldn't I guard cancel? Oh, because I'm not alone yet. Let me talk to this guy. Do I still got it? I don't got it. Resets. I love that that's happened twice, though. Hey, where you going? Come here.  Don't run from me. Don't run from Goku. Don't run from Goku. He tried it again. Where are you running? Kicked upside the head two different ways. All right, that's fine. We're still alive. Get off me. The funny thing is, if I do get a dragon rush, I mean, I mean, auto combo, shed run is mine to summon. Oh, no way. Let's go. All right, there is seven Dragon Balls, correct? There are seven Dragon Balls. All right, we should just kill him. We should just kill him. Back dash. Oh my goodness. Oh my God. All right, I'll take it, dude. We'll take the W. We'll take the W. Not how I wanted to win.  So that's fine. Do I click to MEN? Not my best opening gambit. Oh, he tried to two-h me. He tried to two-h me. He really tried to do it. We ain't going to let him. You want to reflect? Talk to me about it then. Oh, I tried to do the new super jump. I got to just do what I'm comfortable with. Oh, I'm so bad at the game. I'm so bad at the game. I should be saving this for later anyway. I'm terrible. I'm just straight up terrible. I never knew I was terrible. Deep down, I always had thoughts that I was. That's bad. I'll take it. Obviously, we're not doing the best right now. Two-h that, yeah, talk to me about it. I need to discuss that with you. Also, new buff. Destructed is don't scale this hard, but they still let me down. They still find new and innovative ways to let me down. I'm just getting DP. I'm the king of getting DP. No tech. Oh, no, no tech. I'm dead. Oh, gotcha. Oh, it hurts. Understandable. Yamsha did not get through baby B. ABB still devastating.  We'll have to make another comeback unless he drops it. Nah, he's not, he's not dropping this one. He's not letting this one slip through his fingers. And then Genemba comes in. Now we got to deal with Genemba on point. I already used sparking early, which is highly incentivized against in this patch with the way it works now. So we might have made a critical error, but can we still win? Can we still win? Why am I? I forgot that Yamsha is crazy now. I forgot that Yamsha can watch this, dude. Ridiculous. This is not the Yamsha I used to know. He's a better Yamsha. Oh, he just woke up mash and I missed. I miss timed it. Huge. That is devastating. Oh, whoa, whoa, whoa, whoa, whoa, whoa, whoa, whoa. Don't don't don't. Oh, that's bad. That's really bad.  I don't know. I genuinely don't know. No. I got caught jumping, but he didn't kill me.     Yeah, he wanted to get those stacks so bad. Just ex Wolfgang fisted to him. I was pretty sure that Genemba was going to hit, but I wasn't a hundred percent, and it seems he was thinking the same thing as me. Oh my God. All right, that one was rough.   All right, let's not start the way we have. Sit down.  One, two, three. Oh my God. Okay, we got to stop the wake up mash. The wake up mash is eating us alive right now. All we have to do to stop that is hit buttons at the right time. It's eating us alive and baby B assist. I always forget about baby B assist. Another classic, a staple of the genre. I sparked early again. Oh, I hate it.   Guard cancel out.  Man said, there's no escape at the full rotation. That's still crazy. That's going to take some getting used to. All right, he wants to, he's just hitting wake up buttons. We should probably play around that. I know we said we would, but I'm quick to forget. Oh, dragon rush was brilliant right there. Brilliant. I must say three bars still won't kill though. Get in the corner. One, two. Do I need to do this combo? I'm doing it anyway. I got to stop. Somebody has to put a stop to me. I am a, I am a, I'm a danger to myself. Oh my God. That's so crazy. I got to stop doing that. I said I'd play around the down tech buttons. Why am I not doing that? Or even normal tech buttons? Why am I not? I keep forgetting. All right, we're good. We're good to escape. All right. The reason I switched into Krillin is because I'm just going to heal Yoncha's blue out right away. I think I'm okay with that. All right. Yeah. Let's just heal Yoncha. All right. Yoncha's all good. Uh-oh. Two H didn't go high enough, eh? Classic mistake. Come here. Come get this work. All right. Two H that. Let's go. No more buttons. No more Mr. Nice buttons. All right. That was pretty expensive. All right. And now just Jinemba. Oh, that was nice. I was also risking a lot. He could have dimensional portaled me. He's got to use the stick. Yeah. Goku's down. Goku is down. The protagonist has fallen yet again. Give me that. Just give me that. I'm just going to level three. What would I do here as Krillin though? I can't believe it. I can't believe I called Yoncha into that. Oh, Yoncha. Fancy with the new mechanics. Oh my God. Live it, Krillin. Somehow live it. Live it. Oh my God. I want to get fancy. Oh no. There was no need to call Yoncha there either. That is just unfortunate. But that is where we're going to end it. Hopefully you enjoyed it. If you did, definitely let me know down below. Other than that, I have a dot today. We're going to have a ton more of these videos. So if you liked this, subscribe and check out some of the other videos that should be on your screen right now. Other than that, I'll see you in the next one.


Thank you for the introduction, thank you for the invitation, thanks Tom for existing. Great, so I'm going to tell you that I'm not in number and cabling and this work is joint with Ty Libman and John Long Park. So here we have the truff oil, here we have the truffel. It differs from the truffle by a single crossing change and if you notice that this is in fact the unknot, great so today I want to talk about the unknoted number. Great so the unknoted number of k, well it's the minimum number of crossing changes maybe to unknot k, so one thing that makes a unknoted number particularly difficult to study is that it cannot always be realized in a minimum crossing diagram for your knot, right? So in 1984 Steve Blileau came up with this example of a 10 crossing knot where it has, so in its minimum crossing diagram, well there's no two crossing that you can change to a knot or knot but there's some 14 crossing diagram for the knot where two crossing changes do suffice to a knot or knot. What are some of my favorite knots? Well the torus knots are one of my favorite bosses of knots so the truffel is the 2-3 torus knot, more generally the p-q torus knot is the knot that sits on a torus that winds p times longitudinally and q times linearly and without loss of generality we'll always take p to be strictly bigger than 1, p is 1 then it's the unknot and by doing some symmetry properties you can always assume p is positive. All right, so what do we know about the unknoted number of torus knots? Tom, what do we know about the unknoted number of torus knots? Okay, well Tom doesn't remember but in 1993 he and Peter proved how they resolved the Milner conjecture so they proved that the unknoted number of tpq is p minus 1 times p minus 1 all over 2. So the unknoted number is one measure of complexity or not, what's another measure of complexity or not? Well the four ball genus is another measure of complexity so the four ball genus of a knot k is the minimum genus of a surface s where s is smooth, properly embedded, oriented surface and before his boundary is k sitting in the boundary of your four ball. Great, and so it's an exercise to show that the four ball genus is a lower bound for the unknoted number, basically you can use each crossing change to sort of sweep out a genus, but each crossing change sort of gives you one genus. Great, and so in fact what Peter and Tom actually proved, they actually proved that this is also the same as the four ball genus, the pq torus knot. They proved something more general about algebraic knots and sort of that goes by the name of the, it's called the local tom conjecture. Okay, and so maybe let's keep track of these facts that we're collecting. Great, so we know that the unknoted number of tpq and that's the same as the four ball genus. Any questions so far? Let's talk about some operations on knots. So let's talk about connected sum, and so one question you might ask is well how does a knotting number behave under connected sum? And so well the conjecture is that it's additive, that the knotting number of the connect sum of two knots is the sum of their unknoted numbers. Okay, so well one direction is clear, right, but certainly this direction is clear, and the other direction is open. Great, so what's sort of the state of the art about the unknoted number connected sum? Well the state of the art is due to Charlemagne from a while ago. Okay, so he proved that the unknoted number of a non-trivial connected sum is at least two. But for example, as far as I know it's still open, the unknoted number of the left-handed truffle connects some t25. It's either it's either two or three, but we don't seem to know. And then I guess maybe you'd also wonder well what's the four ball genus of a connected sum? So in some cases the four ball genus is additive, so for example the four ball genus of the connect sum of the right-handed truffle of itself, well that's it's additive, but in general this is sort of wide open. So for example the four ball genus of figure eight knot is one whereas the four ball genus of the figure eight connect sum itself. Well this is slice, right, so this is zero. What's another operation we can do a knot? So we can take the satellite of a knot. So take the satellite, we have a pattern knot p in the solid torus. Great, so maybe you have some pattern p, solid torus, and you have some companion knot. And then you basically tie your solid torus into the knot or more precisely remove the tubular neighborhood of this knot and glue in the solid torus where the longitude of your solid torus goes to a sacred frame longitude of your companion. And so we'll be, one particular number associated to the pattern that will be important to us is the winding number, the algebraic winding number. That plays an important role. So for example, it plays an important role in the formula for the Alexander polynomial of a satellite, which is the Alexander polynomial of your companion evaluated at t to the w. So this is sort of like, I think of this as like stretching your Alexander polynomial by a factor of w times the Alexander polynomial of p of the knot. Studying the knot in number of a satellite. So if you notice by the single crossing in your companion, well that becomes w squared crossings when you satellite. And so maybe a naive guess that the knot in number of your satellite is maybe at least w squared times the knot in number of your companion. Great, so that's maybe a guess. Well, the answer to that is nope. Wait, so let's look at the two one cable of the trough oil. One, two, okay. And so I'll leave it as an exercise for you to check if you're not a number of this is at most three. So there's three crossings you could change to get the unknot. And as far as I know, it's also open what the unknot in number of this knot is. There's bounds, I'd say it's at least two. So maybe by the end of the week someone can tell me what the unknot in number of this is. That would be great. I would be very happy. Maybe that was a bad guess. Maybe the new guess is that the unknot in number of p of k. Maybe it's at least the winding number times the unknot in number of k. All right, and so what's the state of the art in unknot in number of satellites? There's a result of Charlemagne and Thompson from 1988 that the unknot in number of p of k is at least two when the winding number is down zero and k is non-trivial. And then maybe a big open question is, well, the four ball genus of a satellite in terms of p and k. I don't have a good answer for that. So maybe study in general satellites is too difficult. So what if we focus on cables? So what if we focus on satellites where the pattern knot is a torus knot sitting in the solid torus in sort of a standard way? Think about, go back to our chart. Okay, maybe let's talk about the four ball genus of cables first. Maybe that's a difficult question, but I guess maybe I'll make a conjecture. Maybe we'll conjecture that the four ball genus of a cable is zero. Possibly exactly when the four ball genus of k is zero and q is plus or minus one. That seems like a reasonable guess. I guess there's some recent sort of evidence and maybe why you might believe this. So recently Dai Kang, Malik Park and Stoffigan proved that the 2-1 cable of the figure eight knot is not slice. Strong evidence for why you might believe this conjecture. But anyway, so what I want to talk about today is right here. I want to talk about the unnotting number of cables, which is joint with Kai Libman and Zhang Baik. Right, so we'll let kpq, I'll be the pq of kable, pq kable of k, where p denotes the winding number. And then if k is not the unnot, the unnotting number of the pq kable of k is at least three questions. Like most of my theorems, the truth relies on nozzle homology, which the graduate students are still here from last week. I've learned about it. And right, so that's due to Ajvap and Zabo and then independently, Jake Basky-San. And it relies on a cabling formula for inverse curves. So that cabling formula is due to Kanselman and Watson. And it relies on an unnotting bound coming from not-for-homology that's due to Akvamalashahi and Iman Ektifari. So before I talk about the proof of this theorem, I want to talk about some correlaries of the proof. So notice, in general, the unnotting number of a knot doesn't have to equal the four-ball genus of a knot, but for torus knot it does. And so one question you might ask yourself is what about iterated torus knots? So iterated torus knots are cables of cables of cables of torus knots. And so the answer to that is no. So this is the corollary of the proof of that theorem, which is that there exists a family iterated torus knots, K i, such that the gap between the unnotting number and the four-ball genus goes arbitrarily, goes arbitrarily large. And so I can even tell you what that family is. So K i takes the two negative five torus knot and then takes the two five cable of that and then takes the i comma one cable of that. And then it turns out that the four-ball genus of these knots is two i. And the unnotting number is at least four i minus one. And so maybe for the torus knot experts in the audience, maybe you look at this family, maybe something about this family that won't surprise you is that we have a negative parameter in here. So for example, for positive rate closures, the unnotting number always equals the four-ball genus. And for sufficiently large iterated torus knots, so those are torus knots where we're sort of the, sort of the cube parameters are all sufficiently large relative to the cube parameters in some society sense. Those, for that family of knots, the four-ball genus also always agrees with the unnotting number. The four-ball genus agrees with the unnotting number for positive rate closures and hence also sufficiently positive iterated torus knots. Well, what's another quarrel I want to tell you about? First, let me remind you of some things. So recall that j is concordant to k if the four-ball genus of k connects some minus j is zero. So here minus j denotes the reverse of the mirror image of j. I also want to talk about algebraic knot. A knot k is algebraic if k is the link of an isolated singularity of a complex curve.  So k of this form where qi plus 1 is greater than pi qi pi plus 1. And so, well it's the consequence of the of the local tom conjecture. Well, this implies that algebraic knots are in certain senses the simplest knots in the concordance class. So what do I mean by that? Well, this implies that algebraic knots minimize both the sacred genus, the unnotting number, in their concordance class. So in other words, if you have a knot that's concordant to an algebraic knot, well, its genus is at least that of the algebraic knot that it's concordant to. And it's a knot of numbers at least that of the algebraic knot that it's concordant to. And so, well, you might ask, well, are there other notions of complexity to which algebraic knots are sort of the simplest knots in their concordance class? Okay, the unalgebraic knot or an iterated torus knot, basically any iterated torus knot with one maybe small exception.  So this can't be the chuck oil. But besides this, you can have negative numbers here. So that's why this is more general than just being an algebraic knot. Okay, and well, the statement is then that k minimizes both the braid index and the bridge index in this concordance class. So in other words, in other words, if j is concordant to k, then the braid index of j is at least that of k, and the bridge index of j is at least that of k. And this relies on work of U-Haas, Maggie Miller, and Rengi, where they have a bridge index bound coming from novel homology. P2 minus 3 is also not okay. Thanks. This, right, so we expect that it should be true for all iterated torus knots, but sort of they're proofalized on novel homology, and sort of you need the novel homology of the starting knot to sort of be complicated enough, and it is for all torus knots except for either chuck oil. That's sort of what goes on here. Great. I'm impressed by your level of attention. Okay. Well, then I'm honored to be the first speaker. Great. Other questions? Great. So let's talk about, let's talk about the proof to a knot k. We can get it's not for homology, so it comes in a few different flavors. So we have HFK hat, which is, this is the invariant that categorifies the polynomial. Great. And so, well, the flavor of knot flow homology that we'll be using is the minus flavor. So, okay, so the graduate students who are here heard me talk about this last week. This is a finitely generated rated module over the PID F join U, where the degree of U is minus 2. Wait. And then right, so a finitely generated graded module over a PID always can be written as a direct sum of three parts plus torsion parts. And then it's a result of Peter and Zoltan that this is a non S3 was exactly one free part. And since since U has degree minus two, well, the only homogeneous degraded polynomials are monomials. So the torsion pieces look exactly like this. You can define a measure of complexity associated to this called the order of k, which I'll just declare this to be the biggest n i. If you're worried that that depends on your choice of splitting, well, you could just look at the torsion sub module and look for the smallest n that annihilates the torsion sub module and that'll be the same as this. Knot flow homology detects genus, so that's a result of Peter and Zoltan. And I'll phrase that in a particular way that sort of in terms of h of k minus. So the way I'll phrase that is that equal zero exactly when k is bad not. And so well, this this torsion order is exactly what features in all of Shahi and after parties are not in bound. So the car is that the not a number of K is at least the torsion order of K. And this is the same invariant that also features in the U Haas-Miller-Zemke bridge index bound. So for the proof of that corollary, it's also useful to have the following theorem that the bridge index of K is bounded below by the torsion order plus one. Okay, and so what's what's the main idea behind the proof of our theorem? Great. So the main idea is that well, if K is not well by Aschbach and Zauberl, that implies that the order is at least one. And then the cabling formula of Hanselman and Watson helps us that the order of the PQ cable of K is at least P. And so you should think of this here as it's somehow somehow under a cabling things get stretched by a factor of P. And that's analogous to the formula for the Alexander polynomial, the satellite, where you evaluate the Alexander polynomial at T to the P where it sort of stretches Alexander polynomial curves. So these immersed curves are a reformulation of the bordered invariance of Lipschitz-Aldschbach and Thurston, where to a three-manifolders boundary, they associate, say, a type D structure over some complicated algebra. But for three-manifolds of torus boundary, this has been reformulated in terms of immersed curves in a punctured torus. Great. So basically they take the not flow complex, or maybe the experts in the audience, they work over the VIN, F-adjoint, UV, mod, UV. Great. And then, well, to that they associate some immersed curve in a punctured torus. Great. And then, so this curve has lots of nice properties. So for example, if you want to glue together two-manifolds of torus boundary to get a closed three-manifold, but you just sort of intersect these curves in a certain way and then do Lagrangian intersection homology on the result. But what will be important for us is the following cabling formula, which says that if you want to take the cable of K, will you take the immersed curve gamma that's associated to K? And, well, if you want to take the PQ cable, what do you do? You take P copies of gamma each stretched by a factor of P, and then you shift them by a factor of Q, and then you connect them up in some certain way and then push them. So maybe that's best illustrated by an example. Okay. So I said that these curves live in T2 minus a point. Well, we'll draw them in the universal cover of T2, and then the point will just lift to sort of give us a lattice upstairs. Right. So being very unassociated to the figure eight knot looks like this. So sort of this is lattice here, and then this is sort of just sort of the fundamental domain of the lift of the curve. And so what's the relationship between this immersed curve and the knot flow complex? Okay. Well, if you look, if you look at these like, okay, I'll call them like right arcs. So you look at this, you see this arc here? It sort of is on the right of some of these points. And it sort of, it sort of arcs around a single, a single one of these marks points. Great. And this, such arcs are in one-to-one correspondence with torsion summands in Hfk minus. So this arc here, since it sort of is length one, this is, this is sort of in one-to-one correspondence with a effigy and u mod u to the one summand in Hfk minus. And so, well then, then this cabling formula of Hanselman Watson, so it tells us we stretch out, it tells us we stretch out, and then we, we smush everything. So, right, so this is the immersed curve for the figure eight. And then what I'm going to draw here will be the immersed curve for the two one cable of the figure eight, which looks something like this. Okay. And so, so what did we do, right? We, we stretched the picture, and then we had these two copies of it. And then from this picture to get from here to here, we just compressed everything horizontally. And so the key thing to observe, right, well we had one of these arcs here. Now, now let's sort of follow through the image of this arc in this picture. And now, now you see it here. And so, well it got stretched by a factor of p. And so now, well it used to just sort of go across one point. Now it goes across p points, where here p is two, two one cable of the figure eight. Great. And so, well this is going to imply that we have, in this case, an effigyuanu mod u squared sum in an hfk minus. And then using a la shahi and eptakaris, a knotting bound, well that's going to tell us that a knotting number is at least two. That's everything I wanted to say, so thanks for your attention. Oh, what's next? It'd be great if we could just look at all satellites. Sort of the Hanselman Watson formula is sort of very special to cabling. Maybe it could be adapted to what are called one one cable, one one satellite knot. So those are ones where the pattern in which the Hager diagram that's genus one. Oh, well the, a knotting number of a whitehead double is always one. So it's a sort of non-zero winding number is the interesting case to look at. Great. So the question, great. Yeah, the question was, well, there's a knotting bound coming from Kovanov homology. That's sort of the analog of this and it's the same true for the a la shahi eptakari knotting bound where I guess rational and not a number means instead of just changing a single crossing you replace a rational tango with some other rational tango. Yeah. Don't know if there's anyone in the audience. Great. Robert says that he believes that, so great. Yeah, so I guess, I guess, yeah, the rational and not a number of cables. Yeah, I'll tell you, PXPose.


In this video, we will present the solution to question number nine for practice exam number two for math 12 10 We are given the limit as h approaches 0 of 1 plus h to the 10th minus 1 over h And we're told that it represents the derivative of some function f at some number a and we're supposed to identify What the function f and the number a must be now this requires? We know the definition of the derivative which it really comes in two forms So the first form that we're going to use very often the f prime of x is the limit as h approaches 0 of f of x plus h minus f of x all Over h like so and in particular since we're looking for a specific number not the general function necessarily What we can do is we can erase the variable x right and Instead we use the specific number a You'll see that right there That's the first form the other version of the derivative that could come into play right here Is you'll see something like the limit as b approaches a of f of b minus f of a Over b minus a For which that could be a b this looks really like the slope formula But that symbol b could be anything it wants to be right it doesn't actually have to be a b It's just a placeholder just a slider just much like over here doesn't have to be an h Although that's the symbol we use most often you could have like the limit as x approaches a of f of x minus f of a over x minus a Those two versions are both appropriate in this one now looking at the limit as h approaches 0 We have this h in the bottom it tells us that this is the format that we are going to be using here So we should see that then this right here should then be f of of a plus h and Then this right here Should be f of a Well given that this function right here is f of a plus h I see a one plus h that makes me think that hmm Maybe a plus h is equal to one plus h that is to say a equals one Well, if that's the case, what's then the function in play well The a plus h seems to be sitting inside of the power function f of x equals x to the tenth Notice if I take one plus h and I put it inside of this function right here You'll get one plus h to the tenth so that seems to check out here Let's then also test my hypothesis if f is x to the tenth and a is equal to one What is then f of one that'll be one to the tenth? One to the tenth which is then equal to one which is exactly what that number is right here So that that gives us what we're looking for We're looking for x is the tenth power and a equals one which that then gives us as The correct answer B


one ship or one thousand here I also have monkey there with me because she wants to be in my movies all the time so so anyway this is my Christmas stuff I wanted to share with you guys so there's a couple of these things gonna be in some upcoming videos stick around I'll show you what we got monkey got me and her mom these scratch off tickets this one I can went up to half a million on this one I can went up to two million so if I win the two million you see in a lot of videos of like new garages being built and stuff like that and on her channel you see her her mom going on cruises drink a margaritas and I'll be drinking beers with the guys putting up my shops so wish me luck on and she got me a couple shirts here which is pretty cool hold that up for me was anything true timber camo so that's pretty cool shirt mm-hmm it's like a spandex yeah it's breathable yeah and it's long sleeve so that's a nice shirt and this one it's got a fish on it mm-hmm it's pretty cool fish alright this one badass was a guy Harvey shirt same material yeah pretty cool nice stuff that's freaking awesome so can't wait for them I still haven't worn that red one she bought me a red one bought me some shirts one day just cuz I needed some shirts I want it's a spider one of them is a spider mm-hmm I guess it's a company mm-hmm or whatever and it's got big spider on it so I haven't worn out yet but then she got me a new pair of slippers cuz my Papa bear slippers are about done for their over a year old so she got me a new pair of slippers so I can't I wanted to wear these last night but I wanted to make this video for you this is the day after Christmas so I got these nice and pushy and I've seen a scene from my friends and loved ones and family and I got these little spoon in here and these are the how do you pronounce it name I put sriracha on my sriracha so that's right finally got some chopsticks and this bowl which will be in future videos I'm almost sure of that that's pretty cool that was a fun lunch time then Bruno got me a present Bruno got me this a new soldering gun kit it's cool because that and her some of her old dad stuff he had a bunch of different tips that will fit this gun so that's pretty cool awesome and I got this get her done this is get her done on there and I got the hot sauce the wing sauce buffalo style and the steak sauce bold and spicy so I can't wait to try this out and then it's got a little brush there from Larry the cable guy get her done so that's gonna be cool because I built that grill so they seem to be liking stuff I'm cooking on it so and this is cool because I've always wanted one and every time we watch a movie with one of these in it I always tell monkey I want one of them a bonsai tree and she remembered that and she got me not one but there's four bonsai trees in here now I've never grown a bonsai tree I've never actually seen one in person so this should be interesting it's the whole kit it's even got some trimming shears but I'm gonna have to do some googling and heavy YouTube and find out about how I can make these things grow and look cool so garden Republic that's a nice wooden box for wood it's got the burnt look to it I like that that's what I was talking about knowing the big foot box remember that's what I was talking about which I'm gonna make that into a bluetooth speaker that's gonna be cool and I may do this look to her mom got me this and I know you guys watch my videos will know how many times I said I need an ultrasonic cleaner I need an ultrasonic cleaner her mom got me an ultrasonic cleaner two and a half liters so this will be in future videos I'm sure a lot of them would review on it but and then we're gonna do a lot of videos with this thing in it so it's gonna be cool really needed this right now we gotta go outside show you my new truck that monkey got me no but we do have to go outside though yeah okay guys so sorry about the wind noise wanted to show you the Chevy Traverse monkey gave me her since I bought her a brand new truck so this is my now okay I'm kidding again you guys know that because yeah right alright so we're gonna open this up and this is my present that monkey got me me you guys see what that is got me a blast cabinet be blaster pretty cool huh so my desk in the garage has got to go because I gotta make room for this thing so I'm gonna have to get so we'll be doing some a lot of videos with this but we'll do a video of putting it together and then we'll do a review video I got to get my media blast for and like I said I've got to get it get my garage cleaned out so I can fit this in there and then like I said you'll see this in future episodes we'll be using it on different stuff I like to do so I just wanted to share you share with you guys this the stuff that monkey and her mom and Bruno got me and Snickers got me something too right yeah yeah did she give me the gun too the soldering gun too oh Snickers yeah Snickers the cat got me the slippers I know they're just animals they don't know there are family yeah there are family but I and Bruno monkey got him a squeaky toy a plush squeaky toy it shaped like an adjustable wrench he likes it okay so anyway I wanted to share some of this well share my stuff with you guys and show you guys this so if you have videos of your gifts that you got link them down below I'd like to check them out but you know you guys are like our family so I just wanted to share some of this with you and hope you all had a Merry Christmas and hopefully you have a happy new year we got big plans for the new year so don't wait she's just agreeing she's like we do no actually we don't I'm sure we'll do something fun I'm sure we'll make a video maybe we'll go live for a little bit you know so but anyway again guys hope you guys had a Merry Christmas and everyone's safe and hope you enjoyed the light show that display that we put on this year and stay tuned for more videos of this and the ultrasonic cleaner and the bonsai tree I'll be interesting and I'm sure you'll see some of the you know maybe me flipping some burgers or something with the get her done turn her thingy there was a myth to that one because see I get to enjoy the food that you put the spaces on there was a myth hang on a second guys more of a we kind of you're you're using it but I'm getting to eat it I miss oh yeah what's miss you know what I mean no missus you mean an alternative motive right yeah I miss is like like dragons but yeah her mom bought her an electric griddle good size electric ceramic in it get ceramic griddle and so you guys will be seeing that in videos I'm sure it's gonna be cooking more than me because well when she opened it she's asked her mom she says Marty does it cook and why don't you give it to him she said well I wanted to give it to you but it's for Marty to use to cook us breakfast and stuff so so she got it for monkey for me to use so have a good good Christmas yeah that was fun huh yeah you made a good day yeah I didn't get to do my Christmas special this year but next year doesn't matter I'm doing one I don't care what it takes because monkey got into her candy making please go check that out one video I don't think she's gonna upload but I think she should but that's up to her she she she has a little trouble melting chocolate well wasn't just that it was the mic and then the mic yeah well the mic the battery start going on the mic yeah so it wasn't working right so I have to narrate it and I did narrate it and then when I went to put in my other stuff the narration disappears so I have to redo the whole thing so I'm like you didn't save it did you yeah it's saved it's just for some reason when you do something else I had to put in you know your little things that you put in for each yeah and when I did I did that after I did the narration and it kind of just disappears the narration it goes away for some reason so it's gone so you got to save it that's what I'm saying yeah it was saved not the narration yeah it was saved but it goes somewhere you should stay in there and I don't understand you should be able to do the narration and then put in your other stuff I think you have to put in your other stuff first and then the narration I don't know because it messes it up for some reason okay we'll agree to disagree on that because you know I've done a lot of narration never had that and I really don't like doing the narration thing it kind of makes your videos not as good as she was wondering why her white chocolate turned brown I kept nuking it nuking it well I guess I'm like you know why your chocolate turned brown she's like no why I said you burn it well it didn't taste burnt no I didn't know you got all that peppermint in it it tastes pretty good so whatever it worked for a way it worked but it was fun and it took longer than I thought I wanted to be done by noon hour because I had other things to do and it just took like three o'clock in the afternoon and he wanted to do his thing and my special that you wanted to be in yeah and then I didn't get to do that by the time you got done with your stuff I was out here getting ready set up for the live thing so it just was not enough time and my stuff took too long so it is what it is oh wow so anyway it was fun though yeah we had a good time yeah yeah so it was just the three of us but that's a beautiful day it's supposed to be a nice week yeah so maybe next weekend we'll go take the boat out or something that'd be nice I was thinking today but I was too tired and you weren't feeling good so it's a little breezy you know it's got that nip in the air but it's nice enough I can wear shorts it would have been nice on the boat yeah up there on the river yeah it would have been nice sometimes you have to take listen to yourself and take a break but so I gotta get this unloaded it's very heavy that's why it's still in the car I couldn't get it out they lifted it and put it in for me and she won't let me I can't get it out by myself so we're gonna get the dolly and put it on the dolly the best and wheelie then I guess all right let's film that and we're gonna do it that way let's film putting it on the dolly I'll go get the dolly it's in the garage no it's not the green one is silver ones right there is that the one with the good wheels well the green one in a garage the tires are flat so we'll use that one then okay all right it's fine hang tight we'll film this maybe something funny for you guys will happen okay guys that should be we should get that shot there if something wild happens so what we're gonna do is just slide this out and let it slide down onto this okay this ain't heavy down I got it here we go now I'm gonna wheeled over here by the garage so anyway guys thanks for watching appreciate it monkeys titties are in a shot and we'll see you on the next one again if you guys have some video clips of of your your gifts or opening your gifts and you want to share with this please feel free drop a link in the comment section below and we'll see you guys later Shaebert I'm at the main ledge I'm gone for now month so she's out for now all right bye bye guys take care we'll see you in the next one


In a moment asleep towards establishing sustainable agri-food systems, a comprehensive training session was conducted last week for a cohort of 76 participants across various government institutions. This initiative was organized as part of the collaborative effort between the governments of St. Lucia and Argentina under the Triangular Project. The training titled Experiences in Strategic Management of Water and Soil for the Sustainability of Agri-Food Systems sought to empower a diverse range of stakeholders within the agriculture sector with the ultimate goal of revolutionizing water and soil resource management practices in the region. With the aim of enhancing the capacity of local stakeholders, the training session organized by the National Institute of Agricultural Technology, a key partner of the Triangular Project, provided strategic insights and practical approaches to water and soil management. The program aimed to equip participants with the necessary skills and knowledge to implement sustainable practices in their respective fields. Representative of the Secretary of the Agricultural Livestock and Fisheries attached to the Triangular Project, Maria Camelli, says that the training emphasizes the importance of tailored strategies and practical implementation methods that align with the local context of St. Lucia. When we came last year, we noticed that one of the problems that you have is that you have water. In different parts of the year, you have like a dry season and a wet season that makes you some problems with storing water and to have water for the dry season. So what we're going to do at the third college is a flood demonstration to harvest rain water and then to have it for the flood demonstration to have the water that is necessary at the dry season. Chief Extension Officer of the Ministry of Agriculture, Eloy Alexis, believes that the training has equipped stakeholders with the ability to navigate challenges, adapt to change and contribute to St. Lucia's agriculture industry. He expressed hope that it will inspire innovative approaches and enhance sector resilience and growth. The theme of the project is strengthening soil and water management amidst auto for the sustainability of the agri-food system amidst the COVID-19 pandemic. We also saw what happened during the height of the COVID-19 pandemic. A lot of people lost their employment in the service sector and the majority of these people had to gravitate to the agricultural sector not having a proper background in terms of soil and water management and that can be chaos in terms of on these two critical factors of production. Also, we saw the disruption in the supply food chain and that it caused much headache and inflation in the price of goods and services. So the Triangular Project is going to address some of the issues that we have with soil and water conservation and also it will endeavor also to address some of the cross-cutting issues related to climate change. As the Triangular Project continues to unfold, the collaboration between the governments of St Lucia and Argentina promises to yield transformative results. By strengthening the capacity of local stakeholders in water and soil management, the project paves the way for a more sustainable and prosperous future for agriculture in St Lucia. From the communications unit of the Ministry of Agriculture, I am Anicia Antoine reporting.