Yo, what's up, YouTube? Realistic Gaming here, and here we are here with Mikey from Bulleig Heroes. What up, what up? Guys, hey, Mikey, you ready to get scared today, bro? Scared, bro? When have you ever seen me scared, bro? I don't know, man. He's gonna have nightmares tonight, and I'm pretty sure you guys will too. I just twitch sometimes, bro. I just twitch sometimes. That's what they are. Let's do it. Alright, let's get it. Let's get it. Yo, what is that? Dude, that does seem like a nightmare. Bro, that would creep me out so much, bro. Bro, just close the blinds in there. Bro, just go under the blanket. Oh, that's Nerfball from Jared Henderson. Oh, yeah, I have to do a video with him, huh? Oh, dude, I would be scared. I would just take off right there. Oh, no, he's gone, bro. Cartoon dog? Cartoon dog? Okay, let's see Cartoon dog. Oh, oh. Good, bro. You guys are amazing.  Oh, my God. Oh, dude. I don't know this Trevor. I'm pretty sure this is Trevor Henderson, but I don't know the name of this. It's like Siren Head without a head. Bro, they're trying to tell us something. Bro, isn't gonna happen in real life, bro. The aliens are coming. The aliens are coming. Like Logan Paul said in June. Did you hear that? Yeah. Oh, yeah, he did, huh? They're telling us. No, he's gone. Oh, he's right there. 17. Oh, you can't even see what's over there. Oh, dude, what would you do if you pulled out to your house? Bro, I'd get the water hose out and spray it. Is that cat in the hat? Yeah, it's cat in the hat. This guy is good. Oh, it's because they all run at the camera. Bro, this guy can make a movie. Like a scary movie, bro. He can make a movie. With that little swerve. With that swerve, it could be OP. Ooh, kicked straight to the forehead. That's cool. It looks like the Doctor Strange thing he does when he's kind of does. I guess this is like a self-made monster or something. I'm not sure. Dope. But it looks, yeah, it does look pretty cool. The birds are like... I mean, if I walk out of my house and I see that in the morning, I'm passing out. I'm going back inside. Going back to sleep under my blanket. Ooh. Like, bro, just turn around. Bro, it's super creepy, bro. What is that? He went on a nice little hike and then he saw a little demon. A homeless dude under the cave, bro. He's a homeless guy. Good. He got lost out there. Oh, Slender Man. Dude, is that the rake or SQ96? I think that's the rake. Dude, there's a lot of them. What you doing, my boy?  I probably hit like Po or somebody. This video is going to be sweating. Things. Dude, this one creeps me out, bro. I think this was like his newest video. This thing creeps me out. Yo, what the heck? Oh, my God. Yo, that was terrifying, man. He had some heat, bro. What's your favorite one? My favorite one is probably the Teletubby one. Teletubby? That was good. I'd say for the Trevor Henderson culture, maybe the Cartoon Cat or Siren Head because you know they'd be pumping, man. They'd be pumping. For the culture. For the culture right there, man. All right, well, that seems like that is it. If you guys liked the video, let us know in the comments down below. We'll look for more of these scary videos on TikTok. We'll try to find more pages. And of course, we credit these awesome creators. Shout out to them, bro. Shout out to Lights Are Off. Dude, amazing videos. And dude. Fire. You could do. Yeah, we're a lot of potential, man. If you see this, you got a lot of potential. You got to make a movie. Oh, dude, do some, like, some Thomas the train ones, bro. Oh, please. Do some Thomas the train. The spider one, dude. Some Peppa Pig. Peppa Pig. SpongeBobBXE. Dude, that would be insane. Of course. My little pony. My little pony. Cracking. Dude, that would be nuts if he did. I don't know, man. If he sees this, hopefully, you know, he's like, man, I'm going to do it for these guys. I'm going to do it for everybody that likes scary stuff. For the scary culture. For the culture. All right, that was pretty awesome. Wow. Anything to say, Mikey? If you guys can, go and follow me at Bulle Heroes on TikTok and on YouTube and on Instagram. These are my channels. And on MySpace and on... MySpace. Wait, what? You don't have MySpace? Dude, MySpace is for boomers. Boomers don't even use MySpace anymore. Bro, MySpace is coming back. I'm just saying. To send everybody back in 2018, 2017. I don't have MySpace. All right, you guys, well, thank you guys for watching. Thank you all for the amazing support and we will see you on the next one. Peace. Yo, what's up? YouTube Realistic Gaming here. I'm here with Mikey. Yo, what? Mikey, where are you at, man? Bro, it's the Coast Clear, bro. Come on, bro. We're gonna be reacting to some scary TikToks. I think he's still afraid from last time. Mikey, you ready to get scared today, bro? Scared, bro? When have you ever seen me scared, bro? I don't know, man. I'm tired, man. He's gonna have nightmares tonight. And I'm pretty sure you guys will, too. I just twitch sometimes, bro. I just twitched sometimes. That's what they are. It's a little bit dramatized. He's dramatized, I told you. Dramatized, I said dramatized. Dramatized. What? Dramatized even a word? It is now. Okay, look it up. Look it up. All right, so we are back on the Light Hour Off channel and we did find another channel. What was its name? It's called Short Blockbusters. Short Blockbusters. RIP Blockbuster, by the way. You will always be remembered. All right, so let's get ready to react to this. And guys, if you're new to the channel, make sure to subscribe, smash the like button and turn that notification bell on when you do so. All right, so let's get it. Oh, what is this? Oh, a cat? Jack the Killer? Dude, that is creepy. We're down the surfboard where it looks like sharks. Bro, I'm scared of sharks. I would never do this right here. Like the ocean has too many. Why is this chilling, though? Why is this chilling? Who's that? You're going to get hugged to death. It's like dodgeball. Why did they stop? There's no reason for you to be standing in the street dressed as Pennywise. Part 2? See, this is what would happen if you stopped. He would lose his bullet. Why would you even stop? Why would you get out? I'll cover that with toilet paper all over my mouth. It's supposed to toilet one time. Oh, you see the head? What is that? Who's monster is that? I don't know. He's just chilling, though. He's just walking around. He's not doing like that. Elmo? Hey. Oh, did he move? Who's that? He's outside there. Shortest blockbuster. Shortest blockbuster. This is the other channel we just found. What is that? Like some alien? Go and follow them on TikTok, too. What is that? The heck? Trust in the street, though. Like, honestly, if you think about it, they're just living life. What is this? That's a walkman, by the way. Youngsters don't know what that is. I was born in the 90s, bro. I know what a walkman is. It's like that guy from the dance squad. So it's a nice squad. They're donuts, bro. That's how you do donuts. He's just doing tricks. He's just doing tricks, bro. Dude, they're gonna crash? I don't know. Oh, Jesus. Dude, I'm really scared. I wanted to see a crash, low-key. I thought a monster was gonna come out and eat it. It's an ocean. Wait, I didn't see anything. I think it was just, like, scary whispers. Why did the monster cross the road? Let's find out. We're not gonna find out, okay? Dude, this guy has, like, a lot of alien-type monsters. These were, like, his first videos, too. Oh. Oh. Dude, I would be scared. Oh, look. You see, look. Like, they're just enjoying the weather. It's probably cold weather from... I feel like aliens like the cold weather. I don't know why. Probably. Dude, it's scary. Oh. Okay, that one looks a little intimidating. Yeah, yeah. He sounds scary. Oh! Oh! No way they made it out. You might have the other alien from Science. From Science? Oh, I remember that. Wait, that was a good scary movie, bro. It was a good alien movie. Oh, there it is again. Oh, it's the parking attendant. But no. Bro, I don't think... Oh, he's running to get the car. He forgot something, bro. Is that a spider?  Heck no, bro. I would move countries, bro. I'm out of there. They're playing hide and seek, bro. They're playing hide and seek. I love these because there's like jump scares in there, you know? Bro, put some clothes on. What is wrong with you, man? He's looking for his pants. Dude, how is he running so fast? What is that?  His homie was just standing there. Oh, man. Oh. This is like living at the dorms and it's like... Everybody's drunk. Everybody's drunk. He's looking for the restroom. Yo, what's this restroom? He still doesn't have any clothes on, brother. I gotta go pee. Oh, man. Shortest. Oh, they're supposed to be... Yeah, attack on Titan once, man. He's been making... Attack on Titan, great as anime ever. The top three. Oh, he just got stomped on. Normal sometimes. Do you want to say jack-o-lantern? That's cool. From FNF. They want to find love. I wouldn't say no, but... Oh, she said yes. It's scary at all, but it was... I don't know, I like it. Sometimes. You got a happy ending at the end, I guess. And then there was a happy ending at the end. All right, you guys, well, that was a scary TikToks part two. We will definitely try to make more. Yeah, you gotta find some more. So if you guys know of any channels or any videos on TikTok, let us know. Let us know in the comments. Let us know what your favorite clip was, of course. I don't know which one was my favorite clip this time. Oh, man. I like the Barney ones, but the one that's scary was the first one, the Jeff the Killer one. Oh, yeah, I gotta go with Jeff the Killer, bro. That was scary. I'm telling you, my guy saw it last night at 12 p.m. and I was in my room. It was all dark. I was on TikTok. I saw that video. I said, heck no.  I went to sleep. I couldn't be in, bro. I just went to sleep right away because I didn't want to get into nightmares. Yeah, it's creepy, bro. It was terrifying. All right, you guys. Well, thank you guys for watching. Thank you guys for the amazing support. As always, we really appreciate it because you guys are helping us make a living off of this. Somehow, we're managing. Well... We're managing. We're managing. Why don't you say anything, Mikey? And see, go and subscribe to my other channels, too, because we also got other YouTube channels. You like boomer toys or stuff like that. Boomer toys. You know, we got the boomer games. It's called collectibles. It sounds more fancy, you know. More attractive. Collectibles. Big brain energy right there. So, we got Bule Heroes and Pixel Game Squad. So, go and check them out, also. We're making bio. Check them out. They're on my bio channel, too, if you want to find them easier. Well, that is it. Thank you guys for watching. And we'll see you guys in the next one. Peace. Six months for the world's more dangerous spider. I finally found one. Bro, why are you so close? See, it looks like it's going to jump.  But I put the screen, bro. But I put the screen, bro. A weapon. How did you just love it? Kind of slightly, bro. That was a bit... Yo, what's up? You told Willis to get me here with my boy, Mike. Mike. What up? Oh, man. You good, bro? Oh, my God. You know, sometimes, you know, I get stuff in my throat, man. I'm good, bro. Kind of stuff, bro. And nothing, all right? All right. So we are back with some more scary TikTok shot of two shortest blockbusters and lights are off for the amazing content, as always. Why did you just slam Baby Yoda? He needed to be put somewhere. Yeah, sometimes he'd be acting up, man. You guys see how hard he'd be treating him. You guys don't know how he acts behind the camera. When the camera's off, he's totally different, man. That's crazy, man. Hey, man, sit down, bro. Hey, relax, all right? Damn, bro, you're crazy. Oh, whoa, whoa, whoa. It won't be to see all right, all right? All right, all right, all right, all right. I want you to cancel it. Let's begin. Shortest blockbusters. One of our favorite TikTokers right here. That's six months, like, for the World War Danger Spider. I finally found one. Oh, man. Bro, why are you so close? See, it looks like he's going to jump.  Oh!  Uh-oh. Oh, okay. They're filming the TikTok, obviously. Oh, no. Dude, what was that? The monster was already closer. She was all filming. I know. Oh, he's taking a little hike. Oh, these alien freaks. They're back. Yeah, they're back. But look, he's hiking. He's hiking. Nothing crazy, you know. Mother nature. Yes, just mother nature. He doesn't love mother nature. Oh, he's back again. Now, instead of hiking, he's going to go tag up the walls. Just tagging the walls. Obviously. Like, you know, why are you snitching? Oh! Oh, man. And the home just popped up out of nowhere. Oh, see, now he's going to go cook dinner. He's fixing the roof.  I don't think he's fixing the roof, bro. Oh, so they keep running at the camera. They hate cameras, though. Oh. It sounded like a cat. It's the sound that they use on TikTok. It's so random. Oh, all right. What are these? Like a vlog. Oh, it's like, dude, what are they called, cicadas? I don't know if you've seen those.  Oh, yeah. Dude, just get the bug spray. Come on. A flame thrower. It's obvious. What the? They're popping out. That's how they're telling us. Shortest blockbusters knows. What's up? Yo, that's pretty cool. That would be crazy, though. That's how it was. They just spun in like that. Oh, now you got babies. Even the babies are crazy. Little family outing, you know.  He's like, oh, look at the camera. You cannot film them, bro. You can't film them. Yeah, they just hate cameras. You better get out the middle, bro. What would you do, bro? Run towards the alien or the spider? I would go around the alien. I would jump over the spider or something. If the spider has fur, I don't want the fur to touch me. Oh! Oh! That was a video to that 25 years ago. 25 years ago? I'm lurking. Yeah, it's the creeper after the bar. Oh, it's this. Oh, it's like the attack on Titan once he does. Oh, yeah. They're acting like things are real. Oh, bro, it's over for everybody. Yeah, man. The monster among us. Oh, I think it's like the little little series he's doing. Oh, the alien. I think he's going to do a feature film with these aliens, bro. Maybe. Because I don't think those aliens are from a movie, are they? Or is it like his own thing? I think they're his own. Oh, no. There's a new Halo trailer. Halo. Oh, he's behind them. There's going to be inside of him. That was a look like he is. He's wearing the suit. Oh, these are like the shapeshifters, bro. They turn into humans, too. Oh, that's it. Come on. Oh, two Titans fighting. Oh, and another one coming. That's a boss. Yeah, that's what it's like. Like he owns a spot. What are you guys doing over there? Oh. Why did the aliens come after us, man? We just filmed it. Don't let them see the camera. You know he's going to chase the camera. He was chasing someone. Dude, I don't understand why he hasn't made a movie. Is that UFO? Oh, it's a UFO. It's the mothership. Oh, I thought it was going to do some crazy. Oh, another Titan. Oh, it's fat.  It's like he has a Titan right there.  Dude, they have all sorts of Titan. Oh, okay. It sounds freakishly licking. Yo, that looks real, bro. It looks like Mr. Clean. Stank breath alert.  Deep blue sea.  Uh-uh. Oh. That looks like Markiplier. Oh, bro. It does. It looks like Markiplier. That's what he's doing, though. Oh, not a dog. He's a dog out of this. He's a dog. He disappeared. What happened? Oh, this guy again. Dude, he looks like a god just standing there. That is huge. He moves pretty fast. That's a good idea. That's a good honor. Oh, not the Eiffel Tower. Imagine the people up there. Okay, but is that the one that's over there in France or the one that's in Vegas? I think that's a really good one. Oh. Of course. A camera ray blast to the face. Dude, that's pretty cool. Oh, bro. It didn't attack us. Hunter Road Nightmare 1. What is that? That's what you get for that reckless driving. Oh, it's part two? Yeah. Oh, nope. Nope. Dude, there's zombies. Keep going. I love zombies. You got to keep going on those, bro. If we ever have zombies, I hope they're the slow ones. What's this? It's like a Titan with no skin. That looks pretty cool. Yeah, it's like Titans with no skin or something. Yeah, but this is not human, bro. It's like a Centipede Titan. I don't know. They look pretty cool.  Our other favorite channel right here for scary stuff. Dude, that looks so dope. Where does that happen to even come from? That looks so dope, bro. Come on. Are you going to see the camera? This is bro. Yeah, you can't point the camera at them. I'm still alive, huh? Come on. I thought we made it out of there. Oh, what is that? It looks like stone. That's a pretty dope weapon. Kermit. Hey, what's up, little frog? You look so cool.  Hello, bro. Dude, it kind of looked cute at first. Dude, that dude was freaky, man. Yeah, that was the last one. Oh, man. It seems like we have reached the end of the video. Hopefully, we find more for next time. I don't know if we'll have a part four. I'm pretty sure there won't be a part four. I'll keep looking for that, bro. If you guys find any, let us know in the comments. Let us know on TikTok. Or if you have any videos you guys want us to check out. Let us know. Comments. The hardest part is finding the videos. That is the hardest part. If you guys have any, let us know. Or if you know any TikTok channel or page, whatever you call it, let us know in the comments below. We'll check it out because we need more content. All right, you guys. Well, thank you guys for watching. Thank you for the amazing support as always. I'm saying the thing. Yeah, if you guys want to follow us on Boulay Heroes, which is our other channel where I do like focus pop stuff and toy stuff. And then our other channel, Pixel Game Squad, which is for older people, you know, if you like retro gaming and all that. Go check it out. Link in the comments. Take a mile, y'all. Go check it out. All right, you guys. Well, we'll see you in the next one. Peace.


The Nigerian Custom Service is set to take off its E-Custom Service by 2022 to improve trading and provide end-to-end automation aimed at eliminating physical contact. The NCS said it had integrated and automated over 90 percent of its activities by deploying tools in order to eliminate clearing and forwarding difficulty at the ports. Meanwhile, the controller of our Papa Command has warned against making Nigeria a dumping ground for illegal goods. Our correspondent Destiny Momo reports that NCS declared a revenue generation of 2.3 trillion naira for 2021, so far beyond the federal government's set target of 1.6 trillion naira. Nigeria is not a dumping ground for used items. That is why it has even said clearly that used clothes is a contraband. So other items, used fridge is a contraband. Used television is also a contraband, but you dismantle a lot of things and say they are used, or used spare parts because, you know, in Nigeria we are very used to using used cars, used cars. When the government in Islam magnitude allows that, then why do you have to declare falsely on that? So, my brother, we are not discouraging the issue of used items, particularly the one the government has approved, but we are discouraging you to make a false declaration on your item. So you get notified about fresh news updates.


Our cell phone has become our video camera where we record fun moments of our lives. But did you know that the first ever video camera recorder was made in 1956 and was the size of a piano? Good thing we don't have to carry one of those to record good times with friends, right? But did you know?


Hi, we are learning polynomial interpolation of a given data set. In this we have learnt that for a given data set we can always find a polynomial interpolation and such a polynomial interpolation will always be unique for a given data set. This is what we have learnt so far. In this class we will learn two ways to construct such a polynomial interpolation for a given data. One is the Lagrange interpolation and another one is the Newton's interpolation. Let us quickly recall what is an interpolating polynomial for a given data. We are given a data set it means we are given a set of nodes and the corresponding set of values at these nodes. These values may be coming from a function or these values may be generated through some source like experiments right. Now, the question is can we find a polynomial say P n of x such that P n of x i is equal to y i. So, that is what is called the polynomial interpolation more precisely a polynomial P n of x is said to be an interpolating polynomial for the given set of data that is more important. If its degree is less than or equal to n remember from where we are catching this n it is coming from the number of nodes that are given in our data. If you are given n plus 1 nodes starting from the index 0 and goes till n then the degree of the polynomial should be n that is what we have to remember. So, the polynomial P n of x should be of degree less than or equal to n and it also satisfies the interpolation conditions given by this ok. So, if these two conditions are satisfied by a polynomial then that polynomial will be called as the interpolating polynomial for this set of data that is what we have seen in the last class. We have seen that for a given data set there always exist an interpolating polynomial and not only that such an interpolating polynomial is also unique for a given data set that is what we have proved in the last class, but we never constructed such a polynomial in our last class. We only proved the existence by formulating a Wanderman system and showing that the Wanderman matrix is invertible right. Today, we will give two different formulas to construct this polynomials remember although we give two different formulas finally, those two formulas will lead to the same interpolating polynomial for a given set of data why because we have proved that such an interpolating polynomial is unique. It is only that the methods or the form in which we construct the polynomial is different, but they both will lead to same polynomial. Let us start with the first method called Lagrange form of interpolating polynomial well we are given a data set like this. The Lagrange form of interpolating polynomial is given by this formula you can observe that this is the linear combination of some polynomials. How this polynomials are given these are given by L k of x and its expression is given like this, this is nothing but the product. So, product of these terms where you have i here i runs from 0 to n, but it excludes i equal to k that k is sitting here. And this is called the kth Lagrange polynomial note the difference between Lagrange form of interpolating polynomial that is given by this expression and the Lagrange polynomial means this polynomial. You can see that it is a polynomial of degree n why because in this product we have n plus 1 terms, but we are excluding one term here which is nothing but the kth term and that is called the kth Lagrange polynomial. Therefore, the product runs with n plus 1 index, but in between 1 index is removed. So, it has n terms in the product and each term contributes to x therefore, its degree is n and this polynomial is the linear combination of nth degree polynomials. There are n plus 1 such polynomials therefore, this is a polynomial of degree less than or equal to n. Now, we have to prove that it also satisfies the interpolating condition. Once you do that then the uniqueness says that your polynomial interpolation is nothing but this form. Let us try to prove this again I will tell you what we have to prove. Lagrange suggests that the interpolating polynomial will look like this where all this L k's are the Lagrange polynomials. Remember the space of all polynomials of degree less than or equal to n forms a vector space right and we are trying to get our interpolating polynomial as a member from this vector space and you can also show that the Lagrange polynomials form a basis for this space. And therefore, our polynomial P n of x in fact, can be written as the linear combination of this basis elements that is the overall idea behind this. So, as a proof what we will do is first we will show that this expression is nothing but a polynomial of degree less than or equal to n that is not very difficult to show. And next is that we have to show that this q of x which is now a polynomial of degree less than or equal to n satisfies all the interpolation conditions. There by q of x will also be an interpolating polynomial to the given data set. Already we have P n of x as our interpolation polynomial we do not know how it looks like, but that is our assumption that we always have this polynomial for a given set of data. Now we constructed another polynomial like this thanks to Lagrange. Now the uniqueness says that these two should be equal right that is how we will prove this theorem. As the first step we have to prove that q of x is a polynomial of degree less than or equal to n and that I have already told you how to show that you take each Lagrange polynomial you can see that each Lagrange polynomial is actually a polynomial of degree n right. So, that is not very difficult for us to observe from here you can easily observe. And then q is written as a linear combination of all this polynomials of degree n therefore, q itself is a polynomial of degree less than or equal to n. So, that proves the first property of an interpolating polynomial. Let us now try to prove the second property that is q satisfies all the interpolation conditions. How to do that? Again take this Lagrange polynomial L k of x equal to this this is the kth Lagrange polynomial. Let us closely look at this polynomial and try to understand how it looks like. Well I am just writing this expression by just putting this product explicitly instead of using this notation that is all I am doing I am not doing anything. Therefore, you can see that I runs from 0 to n therefore, when I is equal to 0 you have x minus x naught that is coming from here divided by x k minus x naught that is coming from the denominator right into when I equal to 1 the numerator is x minus x 1 and the denominator is x k minus x 1 like that it goes till x k minus x 1. Then you see that we are just excluding the term I is equal to k right we do not want to include that term into our product. Therefore, there is supposed to be one more term x minus x k and similarly here x k minus x k and that is actually removed in a way that is our product in this definition right. Of course, that is quite nice that we have to remove it because otherwise this term would have become 0 it would have made the entire term to be not defined at all right. Therefore, that kth term is actually removed from this product and then it jumps from k minus 1 to k plus 1 and then it goes just like how it went here it goes till x minus x n to x k minus x n. So, this is how the kth Lagrange polynomial looks like. Now, let us take x is equal to x k can you see what happens it means you are putting x is equal to x k here that will get cancelled with this term right. Again here x is equal to x k if you put this term also get cancelled similarly all the terms will get cancelled right and you will get this term equal to 1. It means l k of x k is equal to 1 this is what we are seeing similarly what happens is equal to 1 this is what we are seeing. Similarly, what happens if we take x is equal to some x j for j not equal to k just see what happens if you take x is equal to x j for j not equal to k then see you already excluded k from this term. Therefore, x is equal to 1 therefore, this j will be one of these terms it this jth term will be sitting in this product at some stage right and that will make x j minus x j in the numerator and that will make the entire thing to become 0 right. So, therefore, you can see that l k of x j is equal to 0 when j is not equal to k. So, that is what I am writing here l k of x j equal to 1 if j is equal to k and 0 if j is not equal to k. Now, come back to our polynomial that Lagrange suggested as well this can be written as y not l k of x j is equal to 1. So, l not of x right plus y 1 l 1 of x plus y 2 l 2 of x plus dash dash dash up to y n l n of x right. Now, let us take x is equal to say x 1 what happens here. Then on the left hand side you have q of x 1 and that is equal to l not of x 1. If you see when k is not equal to j then l k of x j is 0 right. Therefore, this term will become 0 and l 1 x 1 will come again when k is equal to j l k of x j is 1. Therefore, this will give us the value 1 and all other will give us the value 0 right and therefore, you will be left out with y 1 right. So, that shows that q of x 1 is equal to y 1 this is just an example I have taken. You can similarly show that q of x not equal to y not and so on. In general you can show that q of x j is equal to y j for each j equal to 0 to n. This is precisely the interpolation condition right. Therefore, you can see that q is a polynomial of degree less than or equal to n and also it satisfies the interpolation conditions. That shows that q of x is an interpolating polynomial. We have p n of x as the interpolating polynomial on one hand. Now, Lagrange suggested a formula and we called it as q of x and we have shown that that formula is also giving an interpolating polynomial. Now, by uniqueness that formula that we got should be equal to p n of x which is the interpolating polynomial. Now, from this theorem what we get is a nice formula to get an interpolating polynomial for the given data set right and it is of course, unique. Therefore, that is the interpolating polynomial for the given data set that is what we understand. Let us take an example. Let us take this nice function if you recall we have taken an example in the last class where interpolating polynomial was not successful in giving a good approximation right. Now, we are taking a nice function f of x equal to e power x. Let us see how the interpolating polynomial works. Now, I can construct and show you because we know one formula to construct the interpolating polynomial. Therefore, we will go to construct the polynomial explicitly to keep our construction very simple I will take only two nodes if I take two nodes I will have to construct the polynomial that is the interpolating polynomial of degree 1 that is linear polynomial right and my aim is to find the value of the exponential function at the point 0.826 which is something in between these two points right. So, I want to find the value of the exponential function at 0.826. The idea is to first construct the linear interpolating polynomial and then construct the linear interpolating polynomial and then find the value of that linear interpolating polynomial at the point 0.826 and just take that as the approximation to the corresponding exponential value that is the idea. Let us now use the Lagrange form of the interpolating polynomial to get the expression for p 1 of x. For that what we have to do is we have to first find the zeroth Lagrange polynomial and then first Lagrange polynomial and then write this linear combination where f of x naught is this and f of x 1 is this value right. So, let us write this what is l naught of x well l naught of x should be x minus x 1 divided by x naught minus x 1. You just see the general formula of l k of x and try to understand why in this case l naught of x is given like this. Similarly, l 1 of x is given like this.  So, that is very simple. So, this is the Lagrange form of interpolating polynomial I just simplified it in general you do not need to simplify you can just leave it like this, but I have just simplified it and this is precisely the linear polynomial interpolating the given data set. Now, we are interested in finding e to the power of 0.86 that is what the problem that we post for our self. Now, we will use the interpolating polynomial to get the value and this is a nice story a successful story you can see the relative error in p 1 of x which is the approximate value when compared to the exact value. The relative error is pretty small even for the linear polynomial you see that is the good news for us of course, the function is very good and also the nodes are very close to each other you can see that the nodes are very close to each other that is one good thing and second thing is this function is a very nice function you can see that it is a real analytic function therefore, often it goes well with such approximations that is the idea. Just to have a feeling let us go and insert one more point into our data set previously you had this and this. Now, we are having this x 1 as this and x 2 as this. Now, we have this data set and thereby we will have a quadratic polynomial as our interpolating polynomial for this data set remember this will be different from the previous one because the data set itself is different now. Now, you can construct the quadratic polynomial how will you do that you first have to find l naught of x now it will be x minus x 1 see x naught term you have to leave therefore, x minus x 1 will be there x minus x 2 will be there divided by x naught minus x 1 into x naught minus x 2. Similarly, l 1 of x will have all the terms removing x minus x 1 term that is x minus x naught into x minus x 2 divided by x 1 minus x naught into x 1 minus x naught into x 1 minus x 2. Similarly, l 2 of x is equal to x minus x naught into x minus x 1 divided by x 2 minus x naught into x 2 minus x 1. Once you have this then p 2 of x is nothing, but f of x naught that is this one into l naught of x plus f of x 1 into l 1 plus f of x 2 into l 2 of x. Once you have p 2 of x 1 plus f of x 1 plus f of x 2 into x 1 plus x you plug in x equal to 0.26 and that will give you this value you can see that the relative error is much much smaller than what we have obtained from the linear interpolating polynomial. So, this is a successful story for us in applying the interpolating polynomial to approximate a function at some given nodes. So, this is the discussion about the Lagrange form of interpolating polynomial. Let us now pause on to Newton's form of interpolating polynomial and see how Newton proposes a formula for the interpolating polynomial. Remember whether you apply Lagrange's formula or Newton's formula when you finally simplify that polynomial both these formulas will lead to the same polynomial that is what the uniqueness says. Now, given Lagrange polynomial why are we actually worried about another formula that is the question. If you see Lagrange form of interpolating polynomial is given like this. This is simply the linear combination of the basis element of the space of all polynomials of degree less than equal to n. In that way Lagrange form is very nice especially to use it in doing any theoretical study. On the other hand Lagrange form of interpolating polynomial is not that preferred for computational purpose mainly for one reason that is often we come across a situation where we considered a data set and constructed a polynomial p n of x and then saw that the approximation that we get out of p n of x is not enough for our problem. So, we wanted to have a better approximation therefore, we will include more data into our data set and try to get a higher degree interpolating polynomial. Such situations occur quite often in applications. In such case you see even if you put one point into your data set all the Lagrange polynomials have to be constructed right from the scratch ok. You cannot make use of your effort in computing p n of x in order to compute p n plus 1 of x. All this computational efforts have to be trashed and p n plus 1 of x have to be constructed right from the scratch because all the Lagrange polynomials have to be reconstructed even if you put one node point more into your data set. That is a big disadvantage of writing the interpolating polynomial in the form of the Lagrange. Now that is where Newton's form is quite handy because in the Newton's form what we can do is we can write p n of x equal to p n of x remember this is when you already computed p n of x therefore, this is already known to you plus you will simply put some more extra terms to get the higher degree polynomial p n plus 1 of x. In that way most of your effort is now saved from your previous computation and you have to put new effort only to obtain this extra term. Now the Newton's form of interpolating polynomial is to just see what is this extra term. Well we propose this extra term in this form and the question is why should I put this form? Well you can see that if you propose this form for the extra term then if you put x equal to x naught then this term went to 0 because of this term and then you will be left out with p n of x naught and you know that p n is already an interpolating polynomial for these data and therefore, this p n of x naught will give you y naught and in that way the interpolation condition for p n plus 1 at x naught is readily achieved through p n. Similarly, you can achieve the interpolating condition for x 1, x 2 up to x n if your extra term is like this. Now the only problem is you have to make sure that p n plus 1 of x satisfies the lost interpolation condition that is p n plus 1 of x n plus 1 is equal to y n plus 1. This is the only remaining interpolation condition that we have to ensure, but for that we are including a free variable shear you see that is why we have inserted a free variable C here. Now what you can do is you put x equal to x n plus 1 in this expression remember that may not give any known value to this it will come as its own value when you put p n of x n plus 1 it will not give y n plus 1 because p n is not going to interpolate the node point x n plus 1 it will only interpolate the points x naught to x n only therefore, this will be some value, but then you can choose your C such that p n plus 1 of x n plus 1 is equal to y n plus 1. How will you do that? Well you have to choose your C such that the lost interpolation condition is satisfied that is not very difficult what you will do is you will put instead of p n plus 1 you will put y n plus 1 then you bring this to the other side and all this can then be brought to the other side to get your value for or expression for the unknown C. So, in that way you also obtained the interpolation condition for the lost node. So, that gives us the Newton's form of interpolating polynomial the advantage is that in case if you go to add one more point to an existing data for which you have already computed the interpolating polynomial then Newton's form will help us to just do an extra computation and keep the previously computed interpolating polynomial and to get the higher degree polynomial that is the advantage in the Newton's form. Computationally in this sense Newton's form is little efficient then Lagrange form the Newton's form of interpolating polynomial will look like this where all this a i's are constants that has to be obtained using the interpolation conditions. Let us see how this constants a i are looking like well the Newton's form is proposed in this form now if you put x equal to x naught you can see that all this terms will vanish because all this terms involve x minus x naught right. So, therefore, all this terms will become 0 and you will be left out with a naught and you put x naught and that you force it to be f of x naught or y naught and that is how you will get the value of a naught. So, you can see that a naught depends on the function f and its value at x naught similarly to get a 1 you put x equal to x naught x 1. So, all these terms will go to 0 because all these terms involve x minus x 1 other than these two terms you already know that this is f of x naught right from here you can get a 1 as f of x 1 minus p naught of x 1 see p naught of x 1 is nothing, but f of x naught that is nothing, but a naught that is how it is coming. Therefore, a naught depends only on f of x naught and a 1 depends on f of x 1 and it also depends on the lower degree polynomial. So, this will go on with all the other coefficients a i's a 2 will look like this which will depend on the lower degree polynomial and in general a n is given like this this is what precisely we called as c in our previous slide where a n depends on the immediate lower degree polynomial right. So, this is how the coefficients of the Newton's form of interpolating polynomial is obtained. Therefore, to compute Newton's form of the interpolating polynomial p n for a given data set it is enough for us to compute the coefficients a naught a 1 up to a n right where each coefficients expression is given like this. You can observe that it depends on the immediate lower degree polynomial this is actually again not computationally efficient because in order to compute all this coefficient you have to remember all the lower degree polynomials again that will lead to computationally very expensive algorithm that is again a bad news from the way we have posed the Newton's form of interpolating polynomial. Again I will tell you we need to know all the successive interpolating polynomials in order to construct the coefficients of the Newton's form of the interpolating polynomials. Also we need to evaluate this polynomial remember evaluating polynomial is not that easy it is a very expensive computational cost is involved in that and therefore finally, if you just see the total cost involved in the Newton's form in the way we have posted is no way going to be better than the Lagrange form. Now that seems to be little depressing because we have put effort to construct another form of interpolating polynomial exactly putting what is needed for us and made our polynomial in that way, but still we see that the coefficients involve the lower degree polynomials and that again increased our computational cost right. So, now the question is can we somehow rewrite these coefficients in such a way that we can reduce the computational cost that is the question the answer is s. So, there is another interesting concept called Newton's divided differences. We can see that these a k's are precisely the Newton's divided differences and we can derive a formula for all these divided differences that is all these a k's which does not involve the evaluation of the polynomials of lower degrees that is the good part of the Newton's form. We will discuss the Newton's divided difference formulas in the next class. Thank you for your attention.


Part of what made the Kyle Renhouse story so outrageous aside from the fact that he literally murdered people is The fact that he was walking alongside police officers. This was a 17 year old child with a gun Who showed up to maintain order and police officers gave him water police officers? thanked him and the other vigilantes for being there rather than saying no you don't have the training We're the ones with the training. You shouldn't be carrying a weapon. You can hurt someone They embraced this individual who went on to commit a terrorist attack This is why communities of color don't trust police departments now There's been other stories of police officers working with the Proud Boys I mean we learned that police in Kenosha notified the vigilantes and let them know that they were pushing protesters Towards them, which is when Kyle ran house ended up shooting two of them But this phenomenon of police officers getting cozy with the far right It's way more prevalent than people want to admit and there's a report from the Guardian that talks about this and how it's a Little bit alarming that nobody's sounding the alarms about this and talking about this in the mainstream media So in an op-ed for the Guardian Mike German writes for decades the Federal Bureau of Investigation has routinely warned its agents that the white supremacist and far-right militant groups and Investigates often have links to law enforcement Yet the Justice Department has no national strategy designated to protect the communities policed by these dangerously compromised law enforcers as Our nation grapples with how to reimagine public safety in the wake of the protests following the police killing of George Floyd It is time to confront and resolve the persistent problem of explicit racism and law enforcement I know about these routine warnings because I received them as a young FBI agent preparing to accept Recover assignment against neo-nazi groups in Los Angeles, California in 1992 But you don't have to take my word for it a redacted version of a 2006 FBI intelligence assessment white supremacist infiltration of law enforcement Alerted agents to both strategic infiltration by organized groups and self-initiated infiltration by law enforcement personnel Sympathetic to white supremacist causes a leaked 2015 counterterrorism policy guide made the case more directly warning agents that FBI Domestic terrorism investigations focused on militia extremists white supremacist extremists and sovereign citizen extremists often have Identified active links to law enforcement officers if the government knew that al-Qaeda or ISIS had infiltrated American law enforcement agencies It would undoubtedly Initiate a nationwide effort to investigate them and neutralize the threat they posed yet white supremacists and Far-right militants have committed far more attacks and killed more people in the US over the last 10 years than any foreign terrorist movement The FBI regards them as the most lethal domestic terror threat The need for national action is even more critical in recent years white supremacists have engaged in deadly rampages in Charleston, South Carolina, Pittsburgh, Pennsylvania and El Paso, Texas more ominously neo-nazis obtained Radiological materials to manufacture dirty bombs in separate cases in Maine in 2009 and Florida in 2017 Which were only avoided through chance But in June 2019 when congressman William Lacey clay asked the FBI counterterrorism chief Michael McGarady whether the bureau remained concerned about white supremacist's infiltration of law enforcement since the publication of its 2006 assessment McGarady indicated he had not read it asked more generally about this infiltration McGarady said he would be suspect of white supremacist police officers But that their ideology was a first amendment protected right more importantly the FBI's 2015 counterterrorism policy Which McGarady was responsible for executing indicates not just that members of law enforcement might hold white supremacist views But that FBI domestic terrorism Investigations have often identified quote-unquote Active links between the subjects of these investigations and law enforcement officials So this is disturbing to any sane person Of course, we don't want white supremacists infiltrating police forces They already are biased enough. They already exist in this culture Where they dehumanized people and feel as if they're always under attack And even if you want to be a good cop and do the right thing and report police misconduct You're dissuaded from doing that. You'll be outcasted. You'll lose your job possibly. So, you know, they are complicit because they are bullied into silence and that's if you have the intent to do good It's a culture that needs to change We need systemic reform and this story proves it because it's such an issue that the FBI is Pointing it out saying hey, are we gonna do anything about this? You have white supremacists with active links to law enforcement. This is not okay But when we see police officers Coordinating with or working with proud boys. We kind of just like chalk that up to being an isolated incident and We, you know, rightfully call it out for being the scandal that it is But the news just moves on when they don't talk about this being a systemic issue that really needs to be looked into further It needs to be investigated with urgency because if there are white supremacists in law enforcement That's not just them like exercising their first amendment, right? In fact, McGearty was wrong to say that because you know to hold a belief isn't exercising your first amendment, right? But you should not have Those beliefs is if you're in law enforcement those types of individuals who are white supremacists and aligned with right wing Militant groups should be rooted out of law enforcement, of course But the fact that we even have to say that even have to talk about this as an issue shows How deep the problem goes? With policing in this country, which is why people are saying we have to defund the police because police can't respond to all of these issues They just can't like throwing more police at the scene isn't going to reduce crime So this is terrifying The issue is more widespread than we thought and the fact that there's no action surrounding this is It's frustrating


Alright, he doesn't reflect, he just kinda stands there and takes it, Piccolo with the conversion! Aww! Back up on me! What you doing? Yeah! Oh no! No, no, no, no, no, no! Oh God! How's it going everybody? Dato Dui here, back with another rank session for Dragon Ball Fighters, and this time we're playing a team of Piccolo, Tien and Vegeta. The team really doesn't have a theme to it, these are just some characters that individually have been getting some requests to play. And I'm always down to play some Vegeta, let me tell you, I do not get to use Super Saiyan Vegeta's assist enough. Tien, you know, we've had a falling out over the... Aww, you see, this is... This is why we ain't too big of a fan of Tien, ah, the dodon ray is just so good. Two lights, but you gotta watch that head, I'm coming for that, dude! Man, Vegeta is just such a fun character, base Vegeta as well, their normals are just so satisfying. Oh my goodness, we probably should have died there. Or at least drop the combo like that. Alright, so Sparking's down, we can go ahead and have a blast, aka run into a down-heavy. That was not part of my scheme. Alright, quick tag, we out. Oh, I could feel the change coming, but it was just too soon, so we dropped the down-heavy. Alright, we got it, he's jumping all over the place here, but we get a free knockdown in the corner. There we go. That's about gonna do it. Oh no, how did I mess that up so badly? And I get caught by the overhead, I just wasn't paying attention at the wheel there. Darn it, I could have actually switched them out. Ooh, okay, sword clean overhead. Oh, let's not forget, we do have a DP, we can threaten that option, we can go ahead and pop out if we want to. But we're just gonna keep it to the classic, there we go, threaten it. No! Ooh, baby, just cause you're invisible doesn't mean I'm not. Oh, god. Alright, let's get a do it on Ray in here. She just squared it up, dude. The final hit was enough to connect. Oh, that's pretty good. Alright, DHE out again. Big lag. Alright, that's fine, we're back to normal, get deflected, but that's nothing to even worry about. Do we get the snapback pressure? I think we do, why not? Vegeta, get out here. There we go. Pressure it, get him in the corner. Down-heavy. Oh, how many down-heavies have we missed by an inch? Too much. He's had seven bars, by the way. Ooh, let's go. Ah, I'm probably gonna regret this, but let's go ahead and get TN in here.  We got a time that better. Ah, Super Dash to the assist. Piccolo, help me out. Okay, okay, that's fine, that's fine. We'll take that hit every day. Okay, we're good. He takes the vanish, that was actually probably pretty smart. I don't know if he meant to do that, but definitely not the worst option. Ooh, more lag. There we go. I could have probably done a 2M there, but I was not about, ah, I was gonna say I was not about to miss that, but that's fine. Oh, God. Okay, Super Dash nerf. We take those. Oh, just when I air-dashed in, I probably shouldn't have done it from that range. This man is holding bar. This might be the greediest player I've ever played. He wants, okay, I get the last two down heavies, but that one game, come on, you gotta give me that one. Oh, that was supposed to be a standing medium. I mean, up a vanish. I got standing medium. All right, here we go. Here's the bar. Into TN. Yeah, that'll do it. That'll get it done. There goes all the bar. All right, not the worst situation to be in, by the way. We still have Sparking. We got two characters left.  Oh, no, you hate to see it. Oh, God, you hate to see it. All right, defense. We don't got any bar, so this is the all-natural. Oh, just kick him in the face, Piccolo. All right, Vegeta, you could have stayed inside, but that's all good.  Oh, that was a switch and a half. Piccolo's light just whiffed. Jodan Ray in the air. Oh, Eyebeams? I'll fully admit that was why Auto Combo. I would have dropped anything else from that range. And here we go. One last hit to seal the deal. Oh, I can't believe we didn't block that. Oh, get that off me. Because I was standing. Because I was standing. All right, we out. We out. Go! No, I can't believe it wasn't fast enough. It never is. I don't know. It's like, that's consistent, but yet I always rush headfirst into death. We might die. And not just Vegeta. Vegeta is, of course, dead, but we might lose. I was not planning on losing a character without touching him. Uh-oh.  That was, that was a super dash of any super death. No. How do you play neutral? Oh, we live. Because I was standing. Because I was standing.  Oh, my brain, my P-brain can't comprehend that. It looks, oh. That looks like a move you have to block overhead. Oh, so I stand up every time consistently. Oh, and that time I got beat out. All right, let's go with that. Okay. Can we not lose this time? Can we not lose this time? That would be excellent, dude. Oh, you know what? I will take that. You won't see me complaining about that, though. Yeah, okay. Now we got to get him out, though. That's the real tricky part. We got to land a hit on Videl. Okay. Okay.  I super dashed, too. All right, we take the hip check. That's not too bad. Back dash on that. That's fine. Great save, man. Just not letting me get in there. I'll give it up, dude. Beautiful dodge. Level one. Bring out Tien. Oh, God.  All right. Well, not the cleanest, but we'll definitely take it. Don't go jumping away from me. He did it. Oh, no. Too low. Too low. Too low. Too low. Not low enough. Not low enough, though. Oh, guard cancel. This is a different man than the one we saw before, boys. Oh, no. How to... What? I guess because we weren't high enough. This like... Uh... This does put a smile on my face, to be honest. DP, we keep taking the DP. I forgot. Okay. Kind of forgot if Adele had one of the... Down heavy doesn't connect again. That's fine. Spin out. We good. We good. Ain't nothing on deck that will scare us. All right. Throw out one of these bad boys. Wait for the super dash. We got that. What am I doing? All right. We'll take sliding here.  That was a bad back dash by me. Whoa, whoa, whoa, whoa. I kind of thought that was me switching out for a second because I called him Vegeta assist. You could probably see it there in the corner. Get down to the ground. Get that hard knockdown. Pick a low. What's the mix up? There is none. We just put a ball out. I don't know what that was, but we'll take it. Down heavy once again misses. Vanish, he should probably level one here. He lets it go. Don't on raise being fired left and right. This is a war of attrition. Oh, I'll give it up. Beautiful dragon rush. I'll go ahead and say it. I'll say the catch phrase. Beautiful. Didn't expect it. That was the last thing I expected. All right. We can't let him just play the offense. We got to go ahead and mash them buttons here too.  I don't think so. This is probably going to kill. Say goodbye. Yup. That about wraps up game two. We take it. That one significantly less close to the first one. Man, we did get super lucky with that TN happy birthday, double Sunday. Sorry. Let's use the correct terminology. Oh, don't on Ray. Why are you so oppressive sometimes? Damn, he just jumped away. That knee. I'm liking a lot of this. Unfortunately, it's not good things happening for me. But once again, Vegeta and then down heavy. Let's go. Unfortunately, we don't get the follow up I was hoping for. Sick with it though.  Let him go. Oh, no, that was not the guard cancel. I don't know what I was doing. Okay. Okay. I respect that. I respect that. Oh, I respect that too, actually. I thought that would be so random that no human being would see it coming. All right. Couple of lows, play some deep. That was the move I was going for in the corner to think I'd see it coming. There we go. Boom. Exactly what you want to have happen. Oh, keep that same pressure. Oh, my God. That was some lag affecting my thought process. Don't worry, they clash. War of attrition between TNs. Let's go. It was obvious. It was a little too obvious and a little too slow. Level three. Does he use it? No. We out. Sliding knockdown. We take those. This is a man that likes to lay on the ground sometimes. We've all been there before. Sliding. Yep. This is a good reminder. This is like a Videl move set rundown. I love that move, though. That was spacing 101. Missed it by an inch. Spacing 101. Get some core values on display here. Oh, no. Oh, no. Oh, no. Darn, dude. Darn it. This could be death if he wants it to be. Yeah. I don't know about this one. I don't know about that one. Do we got an extra jump? We do. We take that. Man, I'm trying to bait out this super. I'll be honest with you. DP, though. DP, though. Go ahead. Eat a little bit of this and a little bit of that. Damn. Full course meal. We got to meet in everything. Oh, my goodness. Super Vegeta. No. There we go. There we go. I don't have the bar. I don't have the bar. I thought we'd get it. Come on. Oh, no. To be fair, I didn't place myself in a position where I could punish, except with DP.  Switch. Please. I wasn't close enough. Oh, what are you? Oh, clicking buttons on my time. When I'm on the clock, you can't be clicking buttons. Very good, dude. All right. Game two is off to a successful start. We got it. I respect that. Behind the back. Mix up. Classic. All right. That didn't work out like I was hoping. I wanted to catch him not. I wanted to catch him off the controller. Neutral. Neutral. Oh, better neutral. The better beam went out in the end. He doesn't want to spend the cash, though. Respectable. Oh, but unfortunately it all comes crashing down one hit from Vegeta. I think we've been saving Sparking too long. I'll definitely say that about my play. We haven't been using it. We either save it for too long and never use it, or even when we do use it, it's not that useful. So let's try working on that. Game three. Can we take it? Oh, no. This is game four, actually. Two to one. Get knocked down. Not the greatest start in the world. Not the great. We're tagging out. Anything unsafe? I'm out of here. Okay, that was actually... Okay, I might have said anything unsafe, but please understand that that was not what I was expecting. Let's play some games. Ooh. Just because I hate this character, when he goes against me doesn't mean I didn't made him at one point. Oh, Tien grabbed this man. Tien, you have to... Oh, look at the buttons. Look at the... Oh, Vegeta assist backs out on us. Last second back out. Flake. Counter. Ooh, let's go. Switch. Tien's back. All right, go with the classic overhead. No, he doesn't have to stay. That's why you got to pair that up with an assist, so they actually kind of stick around and have to take the mix. Oh, no. Ooh, Tien. You know what? Let's keep this slider. Let's go ahead and get this Tien out of here. Chow-Tzu? Ah, Chow-Tzu. Chow-Tzu? Nah. No meter spent. We got a dragon rushing more as well. Play the neutral. Where's the super dash? Clutch, actually. Ah, we could have back dashed, but again, I froze in fear. Oh, beautiful. Beautiful chase down off that hit. In order to confirm it, gets me sliding. I'd level three here. Wants to level one. Into level one. Into level three. That's what we do see, but will it kill? Find out next time on Dragon Ball Fighters. That's unfortunate to say the least, but it's what happens. Oh, get back. Get back. Probably should have. Yeah. Yeah, I definitely should have gone with the spacing. Spacing 201. We're in the next class. All right, spark out of the blocks then. Let's go. Let's set up some pressure of our own. All right, he saw it coming. Let's go. Piccolo? Not going to lie. That's not what I wanted you to do. That's fine though. That is exactly what I wanted you to do right there. Keep him kicking, Piccolo. All right. What do we do now? I think we got to do something feisty. Couple of air lights. That's definitely feisty, but the result was... Yeah, that's what I call feisty. Please kill. Okay, nice. Couple of sound effects later and Piccolo takes another knockout. I'm coming for it, dude. I really am out here. There we go. Another clean hit, and we might build up some bar here as well. Yep, we build up some bar. No, it doesn't work out how we want, but does he... All right, he doesn't reflect. He just kind of stands there and takes it. Piccolo with the conversion. Back up on me. What you doing? Oh, no. No, no, no, no, no. Oh, God, I was unfortunate. That was the definition of unfortunate. Switch out. Free. Why did you... Why did you air dash back? Why did you air dash back? We can't let Videl come back on us again. Oh, no, no, no, no, no, no, no, no, no, no. No, I can't believe that happened again. Let's get a little feisty with that. I knew you were going to Super Dash, baby. So I got to do it first. If I'm not going to down heavy, I got to beat you there first. Oh, that's going to do it for today's video. Hopefully you guys enjoyed it. If you did, let me know your favorite moment down below if you have one while you're down there. If you like the channel and the videos, make sure to hit the like button and subscribe. We got more Dragon Ball Fighters coming. We're going to be covering all of Season 2. So if you want to hear any of that, when the news comes out, make sure you subscribe to Stick Around.  So make sure to check those out if any of them interest you. Other than that, I'm Dr. Doha. Thank you for watching this video, and I will see you in the next one.


To understand the remaining essential collection functions you must first understand sequences. What Clojure calls a sequence is not any particular collection type, but rather an interface, a set of operations implemented by some collections. The first function returns the first element of the collection, calling first on a sequence of zero elements returns nil. The rest function returns a copy of the sequence with the first element removed. Calling rest on a sequence with no elements or just one element returns an empty sequence. Strictly speaking, only types which implement the Java interface Clojure.lang.iceq are sequences. However, many collection types which do not implement this interface are considered sequenceable or as we say, seekable, because they have one or more operations which produce sequences. For many seekable types, we can use the seek function, Seq, to get a sequence of their elements. For example, the map and vector types do not themselves implement ISeq, but calling seek on a map or vector returns a sequence of its elements. This extra step may seem silly, especially for vectors. If we can produce a sequence from a vector, why shouldn't the vector just be its own sequence? To my understanding, the rationale is that the rest operation is intended to return a sequence of its own type, but for efficiency reasons, rest called on a sequence produced from a vector does not return another vector. Instead, rest returns a sequence object that references the original vector, but with an offset such that the sequence starts at an index of a vector other than zero. These sequence objects are cheaper to produce than new vectors, and a core goal of the sequence operations is to allow for quick iteration through all the elements of the sequence. In contrast, list objects are themselves sequences because their implementation allows for very cheap first and rest operations. Rest on a list returns a new list, whose head is the second node of the original list. For a map, it's more natural that we don't consider them sequences. When you call the seek function on a map, it returns a sequence of its key value pairs as two element vectors. If we want a sequence of just the keys, we use the keys function, and if we want a sequence of just the values, we use the vowels function. Be clear that items in a hash map have no defined order, so you shouldn't expect the items in the return sequences to be in any particular order. So that's why some collections are themselves sequences, but others are merely seekable. In practice though, the distinction is usually really not that important, because most functions that operate on sequences, including the first and rest functions, actually call seek on their argument. So here, when we use first and rest on a map and vector, we can pass the map and vector directly rather than call seek on them ourselves. A core purpose of the sequence interface is to enable iteration over collections. Here for example, we use the special forms loop and recur with the first and rest functions to iterate through every element of a vector. Note that, unlike a conventional iterator, a sequence has no mutable state. Each called rest returns a new sequence, and every sequence instance is immutable. The name cons comes from old dialects of lisp, where it was short for construct, as in construct list. Over time, cons became a noun denoting a list-like data structure. In closure, a cons is an object that holds a value and references another sequence. The cons function creates a cons, though if the second argument is nil, cons returns a list instead. Here, this first called cons returns a list with one element three, but the second called cons returns a cons, which has the value four, and references a list seven, eight, nine. Calling first on a cons returns its held value, and calling rest on a cons returns its referenced sequence. Because a cons itself is a kind of sequence, conses can reference other conses and thus form a chain. Here we deft to z, a cons with the value five, which references another cons with the value six, which references a list with one element three. This is very similar to the chain of nodes that make up a persistent list, but the difference is that list nodes may only link to other list nodes, not sequences. In a chain of conses, the last cons might reference some other kind of sequence, such as a list. As we'll see in a moment, this makes conses useful in connection with what are called lazy sequences. Transcribed by ESO, Translated by —


Hello admins! As a security precaution, it's a good idea to remind your users not to open attachments or click on links from unknown senders. One easy way to implement this in Office 365 is by setting up a mail flow rule. In this tutorial, we'll show you how to set up your own warning message for all external emails sent to your users inside your organization. After logging into the admin center at portal.office.com, click the ellipses and then click the admin icon to bring up the list of admin portals. Select exchange at the top of the list. Once in the exchange admin center, click mail flow from the left hand menu and click the plus symbol under the rules tab. Click create new rule to bring up the wizard. Give your rule a name and apply the rule if the sender is located outside the organization. Scroll to the bottom of the window and select more options. Click add a condition and from the drop down menu, select the recipient is external slash internal, then choose inside the organization. Under do the following, select apply a disclaimer to the message. Pre-pen the disclaimer. Then click enter text. Then we enter the message using htmi. After entering the text, you'll need to specify the fallback action. Click select one and choose wrap. Then okay. If you wish, scroll down to priority to set the priority level. If this is the only rule you have, you can set it to zero. Click save and you're done. Now when users receive an email from an external sender, they'll be notified with your message. I'm Steve with the Protected Trust Train Department and we'll see you next time.