There's always something on the tip of my tongue Something political to spit or just to sit when I'm drunk But the fear is intrinsic and the shit isn't fun The fear of doing nothing or the criticism went down I mean it's kind of pathetic, like why would I stress it I know the right kind of rhymes to make my writing impressive Can rap with violent aggression or all insightful and pensive But time together I get declined in progression Cause life moves faster and there's a lot to observe A lot I got to preserve if it's a song in the words To find honest with myself when I'm concoctin' these words like Mr. Jump to top tier from the Proving Grounds That's old news, cause I'm Mr.       See, against Pedro you said, it's like I really hate golf because I'm not playing around. And obviously there was no reaction. Well, no, it looks all right, carry on. It's atmospheric. It's atmospheric.  And of course there was no reaction.  And the silence the second time was even more painfully profound and that is why you got put on the demo channel. It's not astounding to me. You can claim you aren't playing around but how can that be?  Hey, check me out.  Some bars. Look, your whole thing, I just don't get you. You're not some top professor. Why'd you drop with the conjecture of a condescending lecture?  I ain't having them. Those tiny rounds put me to sleep. That's value. See, I don't need to switch up and start shooting. I'm still here to bring the art to it trying to get some hearts moving. If I did start schooling with the weapons drawer, my gun would be so big it'd have its own stand. That's tech support. That's the main update. But look, I'm not going to do a gun bar. Or am I?   That's some proper in over your head awkward white guy stuff you never understand so why try?  Just saying, here's a quick tip for you writing, dude. Don't spit bars in a battle if they apply to you. It's obvious, isn't it? Think of it like an own-goal in football. And you were well too chuffed when Rob Wilson called you black. I didn't have a punchline in store for that.   Even Joker Star said I'm blacker than you. He speaks for all black people so that's actually true. There's no way you're winning this. Not just a little bit. Listen kid, run your stuff.  I deliver it. Intricate. Standably random and interesting. Stand so bland and deliberate. Four buck instruction from you or else in to cut, set up and pause.   That trait is so blatant. Just face it. That stuff just a standard like jokes in a way, kid.   This guy. Nah, that's fine. So you sometimes do this thing where you start your third by going third round breakdown. Like your closing angles really gonna leave them laid in the earth. Well, let me try that for this round and we'll see if the tables can turn. See, against Mickey, you tried to claim, as many others have, that you invented the nerdy style. And that should have meant your praise, name dropped, Jefferson Price and Mark Brist as you grabbed off this set of names, but they were better than you. What? You attacked the ones getting fame because you were incapable of stepping up to adapt to the better game. You also name dropped, Mixie, most prob, Eek said all that when on center stage and wait, you're responsible for Eek.  But all that style really comes down to is dry British comedy. And I bet you think that stuff's impressive.  But if you think that concept works, then you must be stretching your partridge pitching monkey tennis.  Now back. There's a lot of dry British comedy.  So I didn't just come here with the intention of outwitting you. I came to Al Javais, Al Cougan and Al David Mitchell. And that's why you should have given up when you saw this club I knew were members because you're not properly veteran, maybe an honorable mention, making way for the tougher new contenders. And yes, you did sarcasm five years ago. And at the time that stuff meant you were clever.  Check. I'm off the pills the kid in limitless, hidden his crib. For that hair you have to pay, that's why I'm splitting your wick. You know I'm setting up a sad line and doing it bare-washed. The credit is cool. It wouldn't copy his hairstyle. Look, check it out. That is your exact hair.  It's too small, you can't really see a problem. Look, that is your exact hair. It's weird, isn't it?  You've got that thing in common. It's crazy.  But I'd distract him in the battle and it wouldn't be fair.  I thought this kid's going to be deep, he's going to have a lot to say. And surprise, surprise, guess who didn't have a lot to say. Let me guess. UK is tough, cops and government corrupt. You have to drink and smoke weed to put up with the stuff.  So Klaiber gives you put in some irrelevant literature reference. That shit's about as clever as an FB standard as you're really getting clever mixed up with pretentious.  It truly is a prophecy. What do you do there champ?  And he talks about the Bible in olden times. He's clever, isn't he? Look, no, he's just read a GCSE history. You've ruined, you've ruined, you've ruined quoting stuff sarcastically to break it down. I don't want to even do it because you played it out. But you can have that, put it in your little hemp backpack. And even what I'm mad at. See look, part of my style is your whole style, we're not equals, that's dumb. It's a mass exaggeration to say you're even my son, you're like a copy of part of the steed, I keep him a thumb. See, it doesn't matter how you spend on it, it doesn't matter how long you spend on your setups getting deep or you do full bar constructions and it's boxed up very neat, I can better your whole style, it's not any reach, any line you make is useless on the wrong Tetris piece. See, you've got no idea your software, I see weakness in you. I've got rage, you feel the quake when I speak in the room.  Yo, the pen is legit so you know that Stoes are fu- All right, the pen is legit so you know that Stoes in the function, that's the end of the shit, when you're explosions or something. Now you said in one of your tracks, I'm not in it for the dough but I wouldn't mind the hovis.


Hello everyone, welcome to Channel KIS Academy's current affair MCQs. Let's see today's questions. Question number one, which country's Kabaddi team has clashed the Asian games title in 2023? The options are A. Iran, B. Indonesia, C. India and D. Pakistan. The correct answer is C. India. Indian men's Kabaddi team defeated Iran by 33 by 29 to win the gold medal in the Asian games 2023, which recently concluded. Indian men's team won their eighth gold medal in nine edition of the sport in Asian games. India women's Kabaddi team has defeated Taiwan in the final earlier. Question number two, when is the world hydrogen and fuel cell day celebrated? The options are A. August 8, B. September 8, C. October 8 and D. October 18. The correct answer is C. October 8. World hydrogen and fuel cell day celebrated annually on October 8 is a global initiative aimed at promoting awareness of hydrogen as a clean and sustainable energy source, along with the versatile technology of fuel cells. On the occasion, the R&D roadmap for the National Green Hydrogen Mission was unveiled by the Ministry of New and Renewable Energy. The roadmap, which provides for a budget of rupees 400 crores, seeks to provide guidance for developing a vibrant research and development ecosystem, which can help commercialize green hydrogen. Question number three, red-rumped hawklet, a new dragonfly species, has been discovered in which of the following region? The options are A. Western Ghats, B. Terai region of Bihar, C. Indo-Miyamar border region of Arunachal Pradesh, and D. Eastern Ghats. The correct answer is A. Western Ghats. Let's know about red-rumped hawklet. The scientific name for red-rumped hawklet is epithymis vayana denses. It is named to mark the lush and biodiverse vayana plateau in the Western Ghats. Its habitat is shaded stream-side marsh near vayana forested terrain and parts of the Nilgiri-Kurg landscapes. Its features are darker pigmentation, a restricted red coloration on the abdomen, absence of the yellow ante, humeral stripe, let's know about the growing condition. It is profoundly seasonal, which is observed in during the month of October in skies. It spends the remainder of the year in the aquatic larval stage. It bore a striking resemblance to ruby-tailed hawklet. Let's know its significance. A 2% genetic variance indicates the presence of a new species. In this case, an astonishing 12% genetic difference was observed. This marks the first instance of an Indian dragonfly being documented with genetic evidence substantiating morphological distinctions. Question number four. Ayam palayam netai recently seen in news is a type of the options are a. rice, b. sugarcane, c. turmeric and d. coconut. The correct answer is d. coconut. Ayam palayam netai is a variety of coconut from Tamil Nadu for which the local farmers are trying to get a GI tag. It has high oil content and a sweet kernel. It is dropped and disease resistant and promotes intercropping. Question number five. What is the primary purpose of card on file tokenization in the context of digital payments? The options are a. to store card holder data securely for easy access in future transactions, b. to convert sensitive card information into a unique token for enhanced security, c. to facilitate international money transfers between banks and d. to provide discounts and rewards to card holders. The correct answer is b. to convert sensitive card information into a unique token for enhanced security. Card on file tokenization is a security technique used in digital payments to replace sensitive card holder data with unique tokens. This process enhances security by reducing the risk of data breaches and fraud when storing and transmitting payment information. The primary purpose of tokenization is to improve the security of digital transactions. We conclude today's session here. Tomorrow, we will meet with few more MCQs. Till then, please like, share and subscribe our channel. Thank you.


Big shout out to Warner Brothers for sponsoring this video. Concentrate, punches to the face, to the face. Oh, and then she puts you out over here. What's going on guys, Fumbros here. As you guys may know, self-defense is always such a hot topic. And there are so many different levels for so many different age ranges. There's a physical aspect, a mental aspect, de-escalation, avoidance. There is something for everybody. And in this video today with the help of a professional martial arts coach and self-defense specialist, we're actually gonna be showing you real life self-defense techniques that you can actually use that are inspired by the characters in the new Mortal Kombat movie out in theaters now and on HBO Max. And by the way, thank you Warner Brothers for sponsoring this video. Over 20 years of martial arts experience, we got Jay Lee. Hey, what's up guys? And today we're gonna go over some basic moves inspired from the Mortal Kombat movie. There's a lot of real elements of martial arts that's displayed throughout the whole movie.  I wish I could teach you guys how to throw ice balls, but we're gonna go over the realistic self-defense moves inspired by the Mortal Kombat film. And coming along for the ride and learning with us, we got Stephanie here. Hey guys, my name is Stephanie. I'm a fitness influencer and model. I'm here to learn some super cool moves. I am super excited. Let's go! We're gonna go over the basic level of self-defense and the advanced. The basic level, we wanna create distance. We wanna de-escalate the situation. We wanna diffuse the situation. Basically, we do not wanna engage in the fight. You don't need to be a hero, okay? Just run away from the situation. The heroes we can watch and film. The advanced level is when you have no choice but to engage in the fight. How you can maximize damage with minimal amount of effort. But before we get into any of the characters, we do wanna talk about the number one most basic thing you have to remember when it comes to self-defense, when you're facing an aggressive situation, which is creating distance. So I'm still in my fighting position no matter what. Okay, I keep my hands up. I'm not keeping it close because that's threatening to you. Okay, good thinking that I wanna fight. I don't wanna fight and I'm communicating. Hey, I don't wanna fight. I'm moving, I'm not moving back. That shows that I'm scared. It's gonna give him a predatory sign to attack me. Keep my hands open, just in case he attacks me. Even if he throws a straight punch and notice what I can do, I can parry the punches, I can parry the punches. I can still block. But this means surrender, surrender. I don't want any trouble. I just wanna get out of the situation. And once I see that the coast is clear, I'm getting out of there. All right, so in this real quick demonstration, Stephanie has been followed by this bad guy and she's been trying to get away but now she is cornered. Let's watch. Hey, what's up? What are you up to tonight? Don't worry about it. Oh yeah, where are you going? None of your business. Why is it none of my business? You need to get away from me right now. Oh yeah? Yeah, I'm not trying to fight. I'm not trying to fight. I just wanna get out of here. I just wanna know where you're going. All right, but that's none of your business. I just wanna know where you're going. Hey, don't touch me, all right? Don't touch me. All right. All right. Back up. Okay. Get away from me. Notice she had her hands up. She was ready to parry strikes. She was ready to guard her face and notice how she wasn't moving back. She was confident, she looked strong. She was verbalizing what she wanted. She was communicating efficiently. She was loud and she was circling. Good footwork, good positioning, good communication skills that prevented the fight. Yo, I love that to start off with. That was very, very simple, non-combative, but still got out of the situation. But now, let's talk about what the characters might do. All right, so our first character is Sonya Blade. Everybody's familiar with her. She's special forces, meaning that she's actually trained in probably a mixture of martial arts. She's particularly trained in Campo, Taekwondo, and also Krav Maga, obviously, because that's the military martial arts. All right, so for our first base level move to get out of this situation, Jay, if I'm trying to choke you and I go in for this, what are you gonna be doing? I'm just gonna parry it out. I gotta need more meat on here. Okay, okay, okay. When it comes again, I can parry it outwards, or I can parry it inwards, onto you. So now, what if my hand actually makes it to your throat and I'm trying to choke you like this? Okay, so it makes it, boom. I wanna be balanced. The worst thing that could happen to you is if the guy chokes you and you just fall on the ground. Okay, you wanna be balanced once again in your fighting stance. Keep your hands up, stand up, surrender. Also distracts him, right? Mirror side attacks the wrist. Simultaneously attacking the trachea. C-sharp strike the trachea right over here, boom. Notice I'm not attacking the forearm. Too much muscle, too much bone here. I'm isolating the wrist joint, okay? I wanna slap away the wrist joint and notice how I move my head the other way. This motion. And then I am not letting go of this. And by the way, thank you Warner Brothers for sponsoring this video. Because if I let go of this, what can he do? He's gonna, boom, clock me. I wanna keep it locked and loaded and attack. Actually, I'm really surprised about the C-shaped strike because I would always think that you're supposed to punch someone when you're getting attacked. Yeah, the problem is not a lot of people know how to make a proper fist. You can break your knuckles pretty easily. You're supposed to punch first through strong knuckles and make a really strong fist.  And people don't know how to punch properly. You're gonna end up hurting yourself more than the other guy. Yeah, that totally makes sense. Oh, nice. Case to space it. Oh, nice. Oh, oh. Dave, you wanna get some in? All right, all right. Everybody's doing that. Yeah, get some in. See, this guy's got a pretty thick neck, but just good enough for the... Boom, boom, boom, boom. Boom, boom, boom. All right. Okay, so for the advanced level of the Sonya Blaine Krav Maga self-defense, what do we got? The guy chokes you, the base level doesn't work. And I'm really strong, by the way. Yeah, if this doesn't work, and you're really struggling, if this doesn't work, you have to make sure to transfer that C-shaped strike to take you into an elbow strike, into a knee strike, pull in the back with the neck, and push out over here. And then, boom, stomp the head. That's a little bit more advanced. I was just feeling out how strong your neck was, bro. What are you doing with that? So our second Mortal Kombat character that we can learn something from in real life is Raiden, the God of Thunder. Now, you're probably thinking, thunder, lightning, it comes from the sky, it's very loud, but the real life version is the tactical flashlight and the key chain alarm. Yeah, so the flashlight, a lot of self-defense specialist argument is the most important self-defense tool. A lot of people that try to attack you, they do not want to be exposed. They want the identity hidden, but once you shine a flashlight at them, the show is that, hey, I know who you are. If you mess with me and report to the cops, I know exactly how you look.  That's gonna buy you some time to run away, or if you want to engage, you can attack. Base level situation. The attacker's a little bit further away from you, and all you gotta do is just shine the light. Hey, I know who you are, I see your face, I see your face. You got good lighting on that photo, so you're clearly gonna see my identity. I don't want that to happen. So I personally have a key chain alarm already, and I really love it. Jay, what would be the best way to use this at a baseline level? Most attackers don't want attention. They want to do their crime and get out. They don't want to be exposed. They don't want anyone to know. This, however, tells everyone in the vicinity, there's a problem. Someone's getting attacked, something bad's happening. Notice how as she was flashing the flashlight towards me, she said she knows who I am. She said she can see me. As an attacker, I do not want to be exposed. My identity should not be exposed because there are going to be consequences. And that scares me. That prevents me from doing anything bad from attacking. It gives her control. And at the end of the day, anything that can buy you a few seconds to figure out what you want to do and how you want to exit is really important. So not only can the defender use this as well, but also a bystander, just someone in the crowd seeing an attack and just pull this alarm to just let everyone know that something bad is happening. So our third character is Cole Young. He's actually the new Mortal Kombat character and it's played by Lewis Tan, your friend. Yeah, Lewis Tan is a good friend of mine. Comes from a fighting background of real fighting, real martial arts. And so we're going to break down certain moves that we saw in the trailer. Okay, what is Cole Young style in the movie? So his style is pure MMA, does all his moves on his own because he's classically trained in martial arts. All right, so let's learn some basic MMA moves for self defense. All right, so we got the front push kick or the T. Notice how as soon as he comes in, I'm just going to throw my leg, okay? I'm using the ball on my foot to drive through. Boom, right back. Notice how chambered my knee up and chambered right back. Why? Because I don't want to capture my foot. A lot of people say don't use kicks unless you've practiced them a lot. Are they talking about like just a roundhouse kick?  And once he catches your leg, he could throw you down, he could hit your face, for example. And also if your range is off and you kick the guy with your foot, you might even hurt your foot more than you hurt the guy. A proper roundhouse kick, you want to kick with your full shin so you can blast the ribs. Right, so the easiest kick to do to create distance is the T, aka the push kick. Yes, but your hands are up, same side, same side arm, thrown down to maximize the power. Notice how I'm using my hips to drive through. Using my hips to drive through. I chamber in and I want to chamber right back my original position. Why? Because I don't want to catch my foot.  And once again, thank you Warren brothers for sponsoring this video. Please do a push kick on me. Oh, yo, any lower. Woo, that wouldn't have been a demonstration. All right, so when it comes to the character, Cole Young who specializes in MMA, what's the advanced version for those who are more experienced? The advanced version, if you want to get more momentum, more power, we're going to go for the jump-scissor push kick, okay?  He's focused on this because he doesn't see that coming. And notice in that trailer, what Cole Young did, what Lewis Tan did, he had a jumping start. So now I'm moving and I'm going to drive my knee up and I'm kicking as hard as I can to the face, to the chest. And then from there, you can follow through. From here, you can follow through with the Superman punch, elbows, you can work on knees, roll kicks, do whatever you want. Dude, it's getting crazy in K-Town right now.  Boom! Ah! Finish him! All right. So for our fourth character from Mortal Kombat is we got Hong Lao. He is known for using his hat as a weapon. In the movie, he slices people with it, but how do we use our clothes to help us, whether it's as a weapon or at least a distraction? First thing you can do with the hat, store that thing as a distraction. Throw it at them and then run. Throw it at them, do a kick to the groin and then run. Okay, so that's the basic level, is a distraction to buy you a couple seconds or to just maybe distract him so that you can kick him. All right, so our final character that we're talking about is Liu Kang. He's probably the most famous character of all played by Ludi Lin in this movie. So shout out to Ludi. Liu Kang is known for Kung Fu and then also in the movie, of course, he can make a ball of fire. What are some basic Kung Fu moves? So you can use it for defensive blocks, 360 defense, you can come from the attack from the top. Or attacking from the top. I can block here, right?  I'm here. Everything is firm. You're using your form, you're using the braver form to attack. What would be some more advanced Kung Fu moves? So you wanna go over some counter attacks.  You can do a lot of different things which is based on Chinese martial arts. What about chain punches? How do we get the chain? Chain punches look really cool. Chain punches. Do they work? Chain punches, yes, they work. If you practice them enough, a lot of people with chain punch from here, they just do this. It has to come from the chest. You have to do it fast and you have to use your whole body to progression. Ideally you will do a chain punch to the face though. Constant chain punches to the face, to the face. Yo, that video was so dope. I was watching, I had no idea that Mortal Kombat characters moves could connect so much to real applicable device and self-defense just to reiterate, recap in a very big picture sense. What did you want people to know about self-defense? So guys, most important thing, if you can de-escalate the situation, if you can diffuse the situation and just get out of that dangerous place, that would be the best case scenario. If you don't have to fight, don't fight. If you could run, just run. We don't have to fight, but if you do have to fight, you have to inflict the most amount of damage, efficiently, as fast as possible and get out. All right, Steph, what did you learn from today? I actually learned a lot. I didn't know a lot of these moves and that I could use a lot of these tactics to defend myself, but I'm definitely gonna need more practice to feel more comfortable and safe. Yeah, at the end of the day, guys, we know that watching one YouTube video isn't gonna make you a master at anything, but hopefully just, this just sparks some discussion and we are able to use Warner Brothers' Mortal Kombat movie as a starting point, as a conversation to get into self-defense, you know? So remember, it's something that you have to practice, but also remember that number one, get yourself out of that situation, stop it before you even get there. Shout out to Warner Brothers for sponsoring this video. Mortal Kombat is now out in theaters and on HBO Max. Share this with somebody you know. I just think it's great to get just the mind sparked and sometimes you do gotta use, you know, something dope like pop culture, like Mortal Kombat just to get your mind running. Also shout out to Louis Tan. Thank you for the motivation, brother. Yo, we gotta see you in the next Mortal Kombat 2 and Mortal Kombat 3. I'm sure we gotta see you again. Yeah, I'm gonna do it. Comment, bro, comment. Guys, in the comments down below, let us know what are some other basic and real-life self-defense techniques that you've heard. That's it for the video. Make sure you hit that like button and until next time, we out. Peace. Here, before. Let's put some out over here. Constantly punches to the face, to the face. Superman punch, elbows. They can work on knees or roll kicks. I'm out.


A huge question when it comes to going to college is should you take out student loans? And that is what we're going to be talking about today. But before we get into it, make sure to gently tap the like button, hit the subscribe button, ring the notification bell and let's jump right in. So this is going to be a pretty quick video today. I'm basically going to go over kind of like the origin of student loans, why they've kind of gotten out of control or they've gotten so high and what you should do about it. First of all, let's talk about how much college is going to cost. Now it's going to be slightly different depending on what study you look at or what data you look at. But basically, the total cost of college, if you add in rent and your cost of living is going to be somewhere between $80,000 to $100,000 for a four year bachelor degree. And you're also going to go on average almost $40,000 in student loan debt, right? So that's about $20,000, $25,000 per year for four years of total cost. Now, when I say a four year degree, a lot of the time people are going to end up taking more than four years for a four year degree. Sometimes they end up changing their major in the third or the fourth year, or they just take a lot longer than it should take in order to get that four year degree. And sometimes that's kind of the college's fault.  So the most expensive college in the US at this point anyways, is going to be Harvey Mudd College at $77,589 a year. Right behind them is going to be the University of Chicago and Columbia. But typically for the different types of colleges, what they're going to charge is going to be around, for a private four year college, it's going to be $32,000 a year. That's about $128,000 for the full degree. And that's just for tuition that doesn't add in all the other additional fees. For a public four year college, but you're attending it out of state, it's going to be about $23,000 a year. So for four years, it would be just under $100,000. For a public four year college where you're attending it in state, it's going to be about $9,400 a year. So for four years, it would be a little under $40,000. And for a public two year college or a community college, it's around $3,400 a year. So if you attended it for two years, it would be $6,000 to $7,000. Right. So this data is kind of just giving you a general idea of how much college is going to cost and how much you're going to go into debt for college. If you do decide to take out student loans, this is of course just for the average person. Some people have parents that pay for all of their college, some people don't get any help at all. So this is going to change depending on your specific situation. Now let's go over student loans and talk about the things that pretty much nobody talks about, but are extremely important. So first of all, let's talk about the positives of taking out student loans. The reason that this whole student loan system was created in the first place is there were many people who couldn't afford an education, right? So they couldn't afford to put themselves through college. And so the government made programs to where basically anybody could apply to get into a college and get that paid for. However, they had to take out what's known as student loan debt, right? So this whole system was created with very good intentions. It was designed to make it to where anybody can get an education. This was also back in the day where pretty much the only way for you to get an education was to go to college, whereas today it's not like that at all. There's many different ways for you to get educated. Now by the way, the student loan interest rates on these loans, because every loan has an interest rate, of course, at least last year they were about 4.53% for an undergraduate loan, and they were around 7% for a graduate or grad plus loan.  Now there's actually a lot more to that. I made an entire video where I kind of compared those and all the nuances of them. And basically the loans that you take out for undergraduate are much better in pretty much every way than the ones you take out for grad school. You have more protections, they're less interest rate, there's more things you can do with them, there's more ways to pay them off, etc. And again, you can just search my channel for that video if you want to check it out. I believe I called it like the truth about student loans or something along those lines. And then there was another video where I touched on it where I talked about should you go to grad school. Now let's move to the con side of things. The way that the student loan system was set up, even though it had very good intentions, created some unintended consequences. When the student loan system was first created by the government, only a very small percentage of people had to use it in order to go to college. And those that did use it usually didn't use it to pay for all of college, it was only a small part. Now more than half of Americans who go to college have to take out student loans and on average they're taking out around $37,500 in 2019. And overall, students owe about $1.8 trillion dollars in rising. This is the highest type of debt out of any type of debt out there, except for home mortgage loans. It's higher than credit card loans, it's higher than car loans. And some facts about student loans when you compare it to other types of debt is it has some of the highest default rates. On top of that, it can't be discarded or gotten rid of in bankruptcy like many other types of debt. There's also less consumer protection laws than other types of loans. So for instance, they can take you to court, they can seize your wages, they can garnish your social security checks, disability payments, all without a court order. And like I mentioned before, graduate loans are much, much worse than the undergraduate variety. And I'm not saying all of this to scare you, I just want you to be aware of the reality of what taking out student loans is like. And you knowing the truth is going to allow you to approach this situation with caution, right? So getting a college degree can be a life changing experience. I've talked about this a lot on the channel, but people who get college degrees on average are going to make much more than people who don't get college degrees. There's also a lot of other benefits of going to college that are outside of making money. But with that being said, if you are going to take out student loans in order to get a college degree, you need to be very, very careful.  And so they don't really take it seriously. And it all goes back to basically the way that they set the system up, student loans have basically become a blank check. And this is why student debt growth has grown by 500% since 1999. And it's only going to get worse and worse if things aren't changed. Colleges and the entire student loan system is still back in their old, you know, 1900s way of doing things when in reality, there are so many different ways of getting educated that shouldn't cost nearly as much. But colleges know that getting a college degree is almost like the American dream. People dream of it all throughout their childhood. Parents want their children to get a college degree. They know that people are very emotional and attached to it. And so they know, especially when you combine that with the student loan system that they can basically charge as much as they want, right? So I kind of just wanted to go over the pros and cons and just give you a very realistic idea of whether or not you should take out student loans. And really the answer is going to be take out the least amount of student loans possible for you to be able to attend college. Now, there's many ways of doing this, making sure that you're budgeting, you know, living below your means, working if you can, you know, attending colleges that are not ridiculously expensive, great ideas to attend community college, for instance, for two years before you transfer to an in-state school, that's going to save you a ton of money, getting classes out of the way so you don't actually have to take them in college and therefore saving the total amount of time that you have to spend in college.  And these are all things I go over in my other videos. So if you haven't done it already, hit that subscribe button, ring the notification bell, and comment down below any thoughts, comments, criticisms, etc. that you have on the video and I will see you guys next time.


Okay, welcome to the final lecture of this series. And this lecture will not be on the blackboard there. I want it to be a little more light since I think everybody is exhausted and looking forward to the dinner, those of you who are coming. And so what I want to talk to you this afternoon about is another application of the techniques that we've developed in this course to a problem in discrete mathematics and in particular in combinatorics or graph theory. So as you all know, we now live in a world where connections, social networks, the internet, meta, and these are modeled by graphs. Networks are graphs and here is a picture from 2003 of the internet. Okay, so the thing you realize here is of course that these are huge graphs and things you would like to understand is how well connected are they. You want to find measures that in simple terms characterize if a network is highly connected or not. And so you're looking for simple quantities, a number that describes the connectivity. So here's the mathematical structure of a graph. It consists of vertices, these are the dots, and edges between them. And you can also put a metric on the edges by assigning a length or a way to it. And one of the numbers that describes the connectivity of a network is its diameter. And the diameter is the maximal distance between any two vertices that you can find. And distance defined with respect to whatever metric on the edges you choose. And these days you can just go to your favorite program here, Mathematica, and all these things are built in. So it's really easy to play around with it. So I've just generated a random graph here, the so-called, from the so-called what-stroguards model. And you see the red line is a curve that connects these two vertices. It is the shortest such curve. And it realizes the diameter of this graph. And it's well known and has been observed and known for a long time that the networks you see in the real world satisfy an interesting scaling law, namely that the number, that the diameter grows very slowly in the number of vertices. So typically like the logarithm of the number of vertices. And so the diameter is obviously only one of the kind of quantifiers that you can choose for connectivity, but it's the one that we want to study in this talk. And you see here just a numerical realization of the what-stroguards model that I had in Mathematica. So for each n, I'm generating a random, exactly one random graph. And you see that's the log n function and it looks pretty good. Of course, there are fluctuations around it. There's no question. Now the what-stroguards model is a very, very simple model that doesn't capture all the effects, but it's become extremely popular. So these kind of papers have, I don't know, tens of thousands of citations. And the model is very simple. So you just start with this circular graph. You have n vertices and you connect here the nearest neighbor and the next to nearest neighbor.  So small world means that you have extremely high connectivity. So maybe you've heard of this law that everybody on the planet is connected with anybody else by at most six neighbors or something like six other people. So this would mean that the world has a diameter of six, right? Something along those lines. I don't know if it's true or not, but that's sort of the catchphrase as you see. And so what-stroguards model was one of the first to try to model this behavior, this logarithmic growth of the diameter. And so what you do is with probability p, you remove a connection and you reassign it randomly with equal probability to one of the other nodes in your network. And so as the probability goes from zero to one, you see these networks, these whole family of networks and the numerical observation of what-sand-stroguards was that already for quite small values of p, of the removal probability, you see a logarithmic growth in the diameter. Now there are rigorous results, beautiful papers. In fact, this one here, is very closely related to the what-sand-stroguards model. It's also, you start with an n cycle and then you add random matching. So it's almost the same thing. But of course it was written by some pure mathematicians and so probably it only has a hundred citations or something like that instead of tens of thousands. And they actually proved something. So they proved that the diameter is bounded about below by log n up to a small log log n correction factor. There's also a nice paper by Bollerbosch and Riordan. So you see, there's a big industry here of these papers. You can't really see it, maybe from the back. There are many other papers here which I put in gray because I only wanted to mention these two. So there is a whole industry of people working on this and it's a beautiful problem in combinatorial probability. Now what we want to look at is not none of these sort of random small world network models but a different family of random graphs that are very, very regular. And we want to understand how big the diameter is and for those guys. And these are the so-called circular graphs. Circular graphs appear in particular in computer science where people want to construct highly symmetric networks that for instance are used in parallel computing and that have a certain redundancy. So you want still good connections but you also want to have sort of a very symmetric structure. Okay, and so this is how they're defined. So you want to construct a graph with n vertices and so you pick numbers a1 up to ak that tell you how to connect the vertices and let me just go through one example to make it very clear what's going on here. So let me construct a circular graph with n equal to eight vertices. So let me draw the vertices here and now I'm going to connect them. So let me choose two connectors, a1 will be two and a2 will be three. Some nice colors. So these I'm gonna do in orange and this one in blue. So okay, and let me actually label my guys. So let's start here. One, two, three, four, five, six, seven, eight. Okay, and now you see the way to connect them, the rule to connect them is you connect two vertices if their difference is congruent to two modulo eight. We want to assume that these three numbers here in this case have GCD one because you can convince yourself otherwise you would get a disconnected graph and we don't want that, right? Because then we can always just throw one away. So it's a natural assumption that the GCD is one. Okay, so what we're gonna do, well one is connected to three connected to five, connected to seven connected to one and similarly, okay, two is connected to four, four connected to six, six connected to eight and eight connected to two. Right, so now we've done these guys. You see that gives us two graphs, disconnected ones but now we also have to connect with a2. So one is connected to four, four is connected to seven, seven is connected to, so seven plus three is 10, is two modulo eight, right?  Five, eight, eight plus three is 11, is three modulo eight, six, yeah, thank you. And one, oh good, yeah, thank you very much. So this is our circular graph, yeah? Does everybody get the idea? Okay, so good. And now we want to know what is the diameter of such a graph and here's an example when n is large and we are interested in the case when n is large and when we throw these numbers a1, a2, et cetera, at random. Okay, and here's the result. Okay, so we'll now pick those numbers a1 for some given k and n to be random and how do we do this? Well, we have a choice here. We can take some set D, this could be a ball and then we enlarge it. So we've done this already very often, right? You say I can set D, you enlarge it and you take all the points in that set. And as I said, we want that the GCD of the coefficients of a and n, that that is one because otherwise we get disconnected graphs. And then we construct the circular graph which is Cnl of a. So I glanced over l, l are the length. So I can assign a length function here for all the blue edges. I want the length to be the same and also for the red edges I want the length to be the same. For now just take them to be one. The reason why I bother you with, you know, having lengths that are not necessarily one will become clearer later in the talk. Okay, very good. So this theorem now says that if you look at a random circular graph where the a's and the n is random and you look at its diameter, if you normalize it in exactly this way, then this random variable here has a limit distribution. And I'm gonna show you in a minute how this limit distribution looks like. So what you see already here is that the diameter is not scaled by log n, but by n to the one over k. So we don't see this effect of the small world network so when k is large, you know, it's growing slower and slower, the more connectors you put, but that's the scaling. And here is a picture of this limit distribution in the case when we have two of those connectors, k is equal to two and there is an explicit formula for this, which is again some formula that we've never seen before. Okay, and now what is the limit distribution? It is given by the covering radius of a random lattice in RK with respect to this polytope. So I have to tell you what is the covering radius. You already know what a random lattice is, but I'll remind you. So that's the answer. And there you see, of course, our random lattices come back and that's why I'm talking about this, right? Because what we've done so far all the time is we said we start with some problem directions and things like that and then the limiting distribution, remember this morning we talked about quasi crystals and yesterday evening. We looked at when you rotate a quasi crystal, you ask what's the probability of finding k points in a randomly rotated stretch set, that has a limiting distribution which is given by a random lattice. And so that appears here again. So what is a covering radius? So you take a lattice, so let me start with a lattice. Here's a two dimensional lattice and that's one thing that's given L and the other thing is k. So it's some set k. Let me think of the set k being something like this. The disc, let's say, right? And then what you do is around each lattice point you draw your set k. So this is a very bad picture because obviously this should be a periodic picture. And the covering radius is now, the factor are by which you need to expand or shrink your set k so that the expanded or shrunk copy of k translated by L covers all of our D. Now if you make R large enough then as of course will always happen. So the covering radius is the smallest R, the infimum over all those R so that you just cover all of our two in this case and generally our D. That's the covering radius. So it's a function of the set that you're using and the lattice. And what is a random lattice? Well, you're all experts in what a random lattice is, right? It's a random point in your space of lattices with respect to harm measure. And so the interesting observation is that in this limit theorem, let me go back to the limit theorem. This limit distribution here you see only depends on the choice that we have here, the polytope, which is a fixed thing. So in dimension two and k equals two, it would just be a square and then k equals three, it would be octahedron. And it doesn't depend at all on the choice of D. And that's something you remember that we have seen also before when we were averaging over our spheres and we were averaging over the sphere with respect to some measure lambda, the lambda didn't feature in the limits and this is exactly the same observation here. And of course, the way we're gonna prove this conjecture is exactly in the same way as we did before. Now, also the lengths, the limit distribution will be independent of the length. That's also a funny thing, right? So there's some universality here. Note that we normalize by the product of the length. So the only dependence is in this normalization. And this settles a conjecture of some graph theorist who wanted to understand whether indeed random circular graphs have a limit distribution. The diameters of random circular graphs have a limit distribution. Now, how are we going to deal with this problem? Exactly in the same way as we dealt with all our problems, we have to translate the question into a question of equidistribution on the space of lattices, right? And then pick the right test functions and integrate over the right thing. And this will be done in three steps. So first, we will show that the circular graphs can be identified with such lattice graphs. So think of this as the analog of the plane or RD that we always looked, right? And then we looked at a sub lattice and RD modulo that, not a sub lattice, a lattice in RD. And if we looked at RD modulo lattice, then we get a torus. And here we will do the same thing. We will take this lattice graph where you have Z2 and then you connect all the nearest neighbor vertices in this way by edges. And then you look at a subgroup of Z2 which will identify points and then you'll get a finite discrete torus in this way. Then we'll approximate the discrete tori with continuous tori. And then we show as before that the tori coming from circular graphs are actually uniformly distributed in the space of all tori, which is nothing but, as you've heard in Anton Zorich lectures, the same thing as the space of all lattices. Okay, so let's do it. So first of all, let's start with the lattice graph. That's just the vertices are at the integer lattice points and we are going to define a metric on there. And that's exactly done by simply, it's much simpler than it looks like in the formulas. So let me actually make the picture over there using already the right colors. So here's my lattice graph and I just define a metric by saying the vertical distance between two points is say L2 and the horizontal distance is L1. Yeah, and that defines a metric on this object. So that's the first line. And this lattice graph will then be denoted by Lk. Okay, and then we can sort of lift this lattice graph up to define some SLK plus one action. And then you can forget about all of that I've written there. I don't want to bother you with that. The main observation is this first lemma and that is that we can, if we take the lattice graph and we mod out by the correct discrete subgroup, not the discrete subgroup, I'm sorry, with a correct sub lattice of Zk. And this was what all this was about is just to give you a formula basically for which sub lattice in Zk to choose that gives you the right identification. You just trust me that this works and I'm gonna illustrate this particular example how to get from the circular graph to this finite torus. Is everybody with me? Right, you get a torus, a standard continuous torus by starting with a lattice and then identify opposite sides. And how do you do this? Well, the lattice points give you that identification. So now I wanna do exactly the same thing here. I want to find vectors that identify points on this discrete lattice graph and that will give me then a finite graph with finitely many vertices. And what I'm saying is that that graph will be isomorphic to the circular graph that we're interested in. And it's a little surprising that that wasn't used more in the circular graph literature to us. We found one paper where that surprise was expressed that paper was just a few years old. Okay, but I might be wrong. I mean, there's a huge literature there and I'm pretty sure that this correspondence has been used before. But for us, it's the key. Okay, so let me just illustrate this here how this works in this particular example. So I want to start here. Let's start with a vertex one here. So let's suppose this is vertex one. And then, oh, let's say this one here. Let's suppose this is vertex one. And now I'm just simply going, let's say this way. So I'm going to vertex three. So then this would be vertex three, right? This would be vertex five. And then where do I go next? Vertex five, no, have I done this correctly? Sorry, I'm doing the orange one. So I'm going from one to three, three to five, five to seven, and then I'm back to one. Okay, I need another line up here. And then I'm back at one. So what does it mean? It means that this point and that point have to be identified, right? So I'm going to just draw that line that identifies those two points. With a different color. Green. So this guy and this guy are identified. And then you see the periodicity coming up here. So this one should be vertex seven and so on, okay? So that's if you like one of the vectors that would give me the identification of the torus. Now, let's continue. So let's do the blue guy. So one is connected to, let's go the same way. One to four, seven, one goes to four, then I have seven. And what else have I got here? Three goes to, three goes to, I have to be careful I got the right way. Now let me do the whole thing here. Four, seven, two, okay. And then I should do that one as well here. So four goes, the orange guys, now four goes to six, right? Four goes to six along the orange line and then this will be eight, yes? What will this be? What? Two, very good. Why? Because yeah, we're taking things modulo eight. 10 modulo eight is two. What will we have here, two, right? Ah, so it works actually. Look, this identification also works here. This one is identified with this one, perfect. Okay, now what have we got here? We go along the blue line, three, six, and then, three, six, one, three, six, one, okay. Now we see one and one is identified. So again, I use a blue line, so these guys are identified.  Okay, so this should go, this should be the same vector as here, so there is an image of one here, right? So what you see here is that, what comes up here? What comes up here? Can that, someone tell me? Hmm? Four, exactly, yeah, four. Four, okay, and now what do you see? I have a fundamental domain here of my discrete torus, right? This is sort of the fundamental domain of my discrete torus. No numbers repeat, they're exactly eight numbers. So these eight vertices are the fundamental domain, and that fundamental domain tessellates everything, okay? So I've showed you here that this graph is isomorphic and isometric to this graph, because these lengths are exactly chosen, and they're all the same, the orange guys. The blue guys might be to have different lengths, but I've done it in a way that it does. And this works, this works, right? And full generality. And the proof is completely algebraic, that fact. Okay, so you believe lemma one. And so what we're gonna do then is remember, we are really interested here in the limit of large n. Large n. So when n is very, very large, what you can show here is that the diameter of such a finite discrete torus, now there are many, many fundamental domain, fundamental cell will now contain many. So here's a general fundamental cell of a discrete torus, so there are now many, many points in here. And we are interested in the diameter, right? So the diameter will be the distance, the largest distance that you can get between any two points. Let's say here, and the metric is sort of going along the edges. Now, let us rescale the lengths so that we scale everything to have length one. So we just have to multiply this picture by a diagonal matrix which has the L's in the diagonal. And that'll do it for us. And then we compare the distance that we get in the limit when n is very large with the continuous torus with R2. And you see when you just look at the diameter of a continuous torus, the diameter cannot be much different from the one on the discrete torus because you're just moving by something of the order one over n in each direction, right? And n is large, so you're making a small mistake on it. That's this lemma too, okay? And then the third observation is if you now look at the diameter of a continuous torus, that's the torus we've been seeing today in Anton Zoric's lecture. RK moduloel lattice. That you can show is nothing but the covering radius that appeared before with respect to this particular polytope, symmetric polytope. You just believe me, right? And you can prove it for yourself if you like. So actually the more natural geometric information is not this polytope but it's simply the diameter that appears here of a random lattice. So I could have formulated that in the theorem. The reason why I'm using this is because that's also a very natural fundamental object in the geometry of numbers. So both of them are equally, equally natural. And then, so that was step two. So we have translated the problem of the diameter on the circular graph into a problem of a diameter of a continuous torus. We said that's approximately the same thing. And now all we need to show is that the diameter of the continuous torus that corresponds to this kind of random lattice that we are picking. So the random lattices that come from circular graphs are not all random lattices. They are very, very special. In particular, the vectors that you have here, they're all rational. These are all rational lattices. And so the thing that you now need to show is that when you sum over all the rational lattices, so this is now the analog equidistribution theorem that we considered before where we were rotating our lattices and then stretching them and we were integrating over this rotation, right? You remember, we had an integral over V over the direction in which our lattice was going. We were integrating that and we would say that would become equidistributed. Now, we don't have an integral over rotations anymore, but we rather have a different averaging here over all those lattices L that come from those specific circular graphs. And what one can show is that this average over those lattices again becomes equidistributed in the space of lattices, okay? So now we are in the position of again, applying our trick as before of realizing choosing a particular function F here that will give the covering radius of the particular lattice that corresponds to this circular graph. We plug that in here. We again remember that we can choose a characteristic function that not just continues, bounded continuous functions by the argument involving the Siegel Beach formula or here in this case just the Siegel formula. Okay, so and then we do what we've done before and we get our limit theorem, okay? So that's the basically the main message. Get your equidistribution result, use it to prove the theorem on the randomness of the diameters that you wanted. Now this equidistribution result has been improved by Han Li in a very beautiful paper. And I should say the way I've proved this result is just using the equidistribution of unipotent translates that we had in the lecture. So there is a very nice approximation argument which is similar to this Margulis trick of thickening things that you can use. So I've simply thickened the neighborhoods of these little lattices by a tiny amount and then showed that you can then, once you've thickened it, you can relate it to a continuous integral and reduce it to the questions that we studied earlier in the course. And this theorem has been recently improved. So remember, note here that I'm summing over both A and N and there is a fantastic improvement by Einziedler, Moses, Shah, and Shapira who simply don't have to sum over N but keep N fixed. So in other words, what this means in this scenario here is that I had to also throw the number of vertices at random while they don't. And they use very, very heavy, egotic theoretic techniques that built on retinas theorem. Okay. And now let me come to the other problem in the talk and show you how this connects to the previous question. And I should say, as I've indicated, what was before that was all joint work with Andreas Strimbergsson. Strimbergsson. So now we are looking at also a beautiful problem in discrete mathematics that started over 100 years ago or even more than that on so-called Frobenius numbers. So you start with selecting a primitive lattice vector. You all know what this is. So a primitive lattice point. And then you ask, which linear combinations can I form here with coefficients that are non-negative integers? And what you can show is that from some number onwards for every fixed A, you can represent every integer in this way. And Frobenius asks, what is the largest integer that cannot be written in this way? And so this is just the formula for it. So what's the largest integer that doesn't have a representation of this type? This is called the Frobenius problem. It's also called the coin exchange problem or posted stem problem. Why is it called a coin exchange problem? Well, you think of the A's that you have here as the denomination of the coins that you have in one currency. So two cents, three cents, five cents. And then you have another currency and you want to represent, you want to ask, or let's put it this way, you go into a shop and you want to understand what things can I buy, what values can I buy with those coins, right? And you don't get change. That's because it's non-negative linear combination. So there's no change. It's a little bit like going into the cafeteria here. Okay. And this is a very old problem. Sylvester asked, what's the formula if you just have two coins, okay? And here's the formula, A1, A2 minus A1, A2. And there are no such formulas, believe it or not, if you have more. It's already extremely difficult for three and people to develop sort of continued fraction style approaches. And there's a beautiful paper by Brouwer and Shockley. So this all goes back to Frobenius. And then Frobenius had a student, Shure, who basically understood something and told Brouwer and Shockley about it and then told Brouwer about it. So this is, I think, Alfred Brouwer. There were two brothers, Richard Brouwer and Alfred Brouwer. I think Richard Brouwer was the more famous in representation theory, I think. And anyway, there's a beautiful paper here that goes back to Shure on this problem. And I'll show you in a second what this is about. And then there was lots of work. I got inspired to work on this by a paper by Arnold who called this Arithmetic Turbulent. So what Arnold's question was is, okay, we don't get a formula for F of A, right? But when you do numerical experiments and you put the A's in, you see that it fluctuates wildly. So it's really a random, it looks like a random function of A. And so he said, well, can we describe those fluctuations in some natural way? And Sina and Bourguin then worked on it and characterized those fluctuations. They were up and lower bound. So this is a very popular program in our whole community called Linear Integer Programming. Because certain optimization problems can be phrased in terms of integer-linear combinations that have to satisfy certain bounds and have to have certain solutions. Okay, so, what I proved here is that you can show that indeed the Frobenius numbers have a limiting distribution. Okay, so you take A random just as before. You take any nice domain D, you make it large and then you pick the A's in that big domain random with uniform probability and you get a limit distribution. And the limit distribution, as you will not be surprised, is given again by the probability that a covering radius of a random lattice, yeah, so that's the limiting distribution is given by the covering radius of a random lattice with respect to this thing. So it's no longer this regular polytope, but it's now this simplex here. And as I said, in dimension three, Burgan and Sinei, and also Schur, Sinei and Ostinov proved a version of this. And then everybody said, well, but we don't have high-dimensional continued fractions. And that's when you should say, ah, but we have the space of lattices and high dimensions, which is so much more easy to control, right? And that's how you can solve these problems. Anyway, so here are some images of the limiting density that Andreas Trümberkson produced from the limiting distribution. So these are, we don't have exact formulas here. And so the only thing we can do is we can take this random variable and devise and put it on the computer and make these drawings. Only in D equal to three, there is an explicit formula. In higher dimensions, Andreas Trümberkson proved tail asymptotics for these distributions. So that's very interesting because they don't decay exponentially, they have a heavy tail. Very good. And so now let me just say in one word, I don't wanna go through this whole slide. I just wanna say in one word, why do I talk about these two things in the same talk? Well, A, you've seen that the two limit theorems look sort of very similar, except that in both cases you get the limit distribution of the covering radius of a random lattice, just that the covering radius is taken with respect to two different sets. In one case here, and the Frobenius number, it's a simplex. In the other case, it's another regular polytope. Okay, and the reason is it can be explained in a nice way following this original idea of Brouwer and Shockley that as I said goes back to Schuyl. And so the idea is to first reduce everything modulo, one of the coefficients that you're looking at here. So remember, we're looking at this primitive lattice vector A that has coefficients A1, A2 up to AD. And so you reduce it mod one and you show that the Frobenius number can be written as the maximum over these FRA, which you can think of as the largest number that doesn't have a representation of this form, that doesn't have a representation with MA congruent to R modulo AD. And now what can you say about those numbers? Well, if you define the smallest positive integer that has a representation in R modulo AD, that's NRA, okay? So that is the smallest number positive, strictly positive, that has a representation as a non-negative linear combination of the A's equal to R modulo AD, then you can convince yourself, and this just takes some sitting down and thinking a little bit. I don't expect you to follow every little state. At least I couldn't follow this. Then you see that the largest guy that does not have a representation is exactly the smallest positive that has a representation minus AD. So you have this formula here. And then the key point is that actually you can compute NRA, and this is the formula that Brouwer and Shockley found. And so you've now lost somehow one dimension here, right, in this formula. So now NRA becomes the smallest of the non-negative linear combinations because you've taken that mod AD here that satisfies this equation. Okay, very good.  Now have a close look at how this looks like. It's the maximum of some minimum. Okay, that looks a lot like the diameter of the shortest distance. Think about it a little bit. What is the difference here? The difference is that we're only allowing non-negative coefficients and only non-negative coefficients. In the diameter, we would also allow to go backwards and go this way. Here we're only allowed to go right and up, if you like, in the lattice picture. So that actually looks a lot like the diameter for a directed graph where you're only allowed to go in one direction, okay? And so the difference, oops sorry, the difference between the directed, the undirected graph and the directed graph is that simply now you assign a direction for each edge. So here we would simply say in our example, we would simply say I'm only allowed to go from one to three, from three to five. So I'm really saying A minus J has to be congruent to A1 mod, okay? And before I had absolute values. And so this means I also have to put now directions here on my lattice graph. So I will have directed lattice graph. And similarly for the blue, guys. So now I'm only allowed to go from one to four, et cetera. So I will have directions in this way. So now I'm only allowed to go right and up, okay? But I still can ask for the diameter of these things, okay? And then if you do this, there's a beautiful observation by Newnhoos, which is what I've just explained to you, that you can express the Frobenius number as the diameter of a directed, circulant graph minus N. The minus N is because AD now plays the role of N. Right? Yeah, where is my ND? So you see, this will now be the diameter, right? It's the minimum. That's like the distance, the minimum, the distance is a minimum, the shortest path. And I'll take the biggest of those, that's the diameter. And so I get this formula. Aha, but there is a length. So I have actually weights now. Why do I have weights? Well, because these now in this geometric interpretation, these guys here now play the role of a length, right? As I go through my path, I'm assigning a length here of these guys. And that length is exactly A2. And that length is exactly A1. So the lengths are not all one in this identification. Do you all see that? Yeah, right? Because you're forming these. If you would just look at the standard graph distance where each edge has weight one, then you just have ones here everywhere. So this is different weight. And that's why the weights appear here. This is the length. And this is the labeling of the edges. So they're both the same in this case. And so now there you see. The Frobenius number is exactly the same as the diameter of a circular graph with these particular choice of lengths. And so here you have the two limiting distributions. And now on the left, you see the same distribution you've seen for the Frobenius numbers. But now, just to make things a little more beautiful, here is I haven't chosen the length that come from the Frobenius number. I've chosen length one, which is the more natural thing when you look at your directed graph. I've just given every edge length one. And you see it's exactly the same distribution as for the Frobenius numbers. Because I told you earlier, the limit theorem that we had said that the limit distribution is independent of what length you choose on your graph. And that is a very non-trivial statement and it's reflected in this thing here. Right, so that was the last lecture. And let me just recap what we've done in the last week. We've shown you what measure rigidity and homogeneous dynamics is good for in terms of proving sensational problems like the Oppenheim conjecture, quantum unique, a Godicity, and some very nice observations on randomness and sequences like square root n mod one. And then we've told you some of the basics that we have here. Everything in two dimensions and the SL2R has a beautiful geometric interpretation in terms of hyperbolic geometry, SL2R acting via fractional linear transformations as isometries. The main game played by SL2Z as the stabilizer of the lattice the basic lattice. And so identifying the space of lettuces with SL2R modulo SL2Z. We've told you how to prove equidistribution results that then later can be used in the distribution of problems, in number theory, et cetera, that we were interested in. We didn't talk about how to prove the equidistribution of horocycles via Eisenstein series. Andreas talked about how to prove this doing representation theory and also how to derive it directly from the mixing property using Margules' proof. And we looked at the space of d-dimensional lettuces, how to understand when you go to infinity in a d-dimensional lattice using Mahler's criterion. And then you remember maybe we then use these things to re-prove some classical theorems in number theory on say the statistics of fairy fractions, which have a natural generalization to higher dimensions which people previously couldn't do without the space of lettuces because somehow even though it's natural, it's never been done in this way. People were stuck in the one-dimensional problems. The three-gap theorem for n-mode one interpreted in terms of space of lettuces. So there's a little handout on the web page which you can download. That was one of the tutorial problems. And so the solution is on the web of this thing. And then we discussed other applications, statistics of directions and distribution of free path lengths in the Lorentz gas. The key tool, the Siegel Beach formula. Yesterday we talked about quasi-crystals and this morning, and that was the last lecture. So if you're interested, here's the reading list again. It's I think also somewhere on the web. And I hope you've enjoyed it.


These that I've talked about can be seen through the body's eyes, and they all involve constructs. They're just different shifts in the construct. But there comes a point in the mind where the entire cosmos is, the mind is aware that that is a construct. And therefore it's the duality between me, a little person, and a city, and a country, and a world, and a solar system, and a cosmos. Their perception always requires a perceiver and something that is perceived. And there is a state in which this is no longer the case. That perception is literally laid aside. It no longer serves. It no longer has value. Someone may say, I like the world. I like what these eyes perceive and so forth. It really comes down to peace of mind. And really honestly looking at one's life and one's emotions, one's fluctuating emotions, and heartaches, and whatever the form they take, loneliness, isolation, this insatiable restlessness. I guess the question is, above all else, what do I want? What's it going to be? And if I think above all else what I want is what I see in the world with these eyes, then I'll probably keep going with that until I get to a point where that doesn't do it for me. It's not enough. I'm tired of trying to find something there that isn't, and I come around to, above all else, I want peace now. How do I go about that? There's an openness in the mind. The mind that seemed to be so determined to make something of itself, to make something more of itself, so determined to hang on to individual, unique individuality, separateness. There's a crack or an opening saying there has to be something beyond all of this.  I think just out of a feeling of desperation, there was a willingness to open up to something else.


The Russian military plane IL-76 that crashed in the Belgorod region was carrying 65 Ukrainian prisoners of war, six crew members and three other people, according the Russian Defense Ministry. According to some reports, the plane was shot down by Ukrainian armed forces. Ukrainian sources confirmed that the plane was shot down and said that it was carrying S-300 missiles, according to information. The plane crashed while trying to make an emergency landing at Belgorod Airport Russian Telegram channels suggest that the plane was hit by a Patriot or Iris T-Missile


This is George Gilbert. We're at the San Jose Convention Center at the at scale conference. I'm with Rodrigo Schmidt from Instagram. And Rodrigo has some very big picture and down and dirty architectural details about the design and implementation of the very, very scalable stuff that Instagram's using. So Rodrigo start with telling us for those who don't know the latest experience in Instagram, what are some of the features that your systems have had to support, to support the evolution of the user experience? There are many parts of the product that are powered by our data system. So you can think of the whole explore tab is one thing that is powered by Instagram data systems. The newly launched trending hashtags and trending places, those are products that are again powered by data that we collect in real time, then we process it, we identify the trends, we rank them and then present them to the users. All the way into the analytics itself, like we have a number of analysts that are continuously looking to how Instagram is growing, like the health of our growth and presenting us with statistics on Instagram usage and whatnot. And all that is powered by the systems that we build on Instagram data. Okay, so it sounds like the site data is feeding into, I think what you talked to me earlier about like as a stream and different systems are gonna consume it. So tell me like for the trending feature, where is that the data that feeds that coming from and then what type of analytics do you have to do that and in what timeframe to make that work? Sure, so imagine like somebody's using Instagram on their phone right now. And as they like, let's say upload a photo or like a photo, those requests will go to our servers and our servers are gonna look like, oh, there's something going on, like this person just liked this photo. That generates an action and that action goes into like a stream of actions in real time. We're using a system called scribe for that. So each web server, like the connect to scribe and they say like, this is activity that I'm receiving from multiple users around the world. And that scribe stream is open to any other system within Instagram to plug into read from and start processing that data. So for example, like for our data warehousing, for offline processing, we have some readers that read from the scribe stream and write that data directly into Hive so that our analysts can like look into and process the data afterwards. All the way into, for example, the trending system that plugs to that stream in real time and starts taking that data, processing it, counting how much activity we are getting for each hashtag, for each place, like aggregating that data in real time and servicing it back to other systems that then will rank and surface it to the users. Okay, so let's drill on that last point. If the audience is leading edge IT practitioners and they want that sort of interactivity in real time sort of analytic feedback that trending and to be able to surface that on their sites, what's some of the machinery that they should consider where you talk about consuming, performing the analytics in real time and then surfacing it? Yeah, it depends a lot on the scale of the system that you're building. I think at Instagram, we have a bigger challenge just because we have like hundreds of millions of users like at Instagram on a monthly basis and even on a daily basis. So we need like a much more powerful kind of system. So we have a distributed system that runs in real time and then you have to think about like how you wanna distribute that data, so how to make sure that it's processed in real time in a distributed way. But let's say if you are building something to a much smaller scale, even a simple system with a single machine that aggregates the data like in real time, in memory, that should suffice. We actually try to do something like that in the beginning and it's amazing how much you can do with a single machine these days. Okay, so tell us what are some products that would work in a beefy single machine and then what did you guys have to do that might in the future where volumes are so much greater for mainstream people where they might have to move to? Yeah, so what I would say is like kind of what we did is like we always tried the simplest solution first and I am a strong advocate for that because then you can see, like as you could really fast on your first prototype and you can see where your problems, your pain points are and iterate on top of that. So I would say like one, so you can start with that. I think like if you don't have many users, like if you have like just a few million users or if you're just starting, I think like having a single machine system that just reads from scribe, processes it and outputs the trends should suffice. If you would that be like a stream processor or would it be an in-memory SQL database or a key value store? Just a simple like stream processor, something that like reads from scribe, like it very easy to implement. If you want to get into something like more complicated then you can start using like say an in-memory key value store that might help with a few things. Then if you want to distribute, I would say like instead of doing, we actually ended up building our own system for training. But that was because of our scale and we couldn't find anything that would fit our needs in the market. But today, like these days, there are so many things, like there are so many open source tools that you can use. Like things like storms out there and like while scribe in itself is open source and it's very useful, like it's impressive how much you can do with something as simple as scribe. Okay, we just have two minutes. Tell us, let's switch gears from the real time to the batch analytics. What does that pipeline look like? What are you trying to solve? What analytic problems? So there are many things that we do with batch processing this day. So the same stream that generates these actions that can be accessed in real time that feeds into Hive directly. So that gets stored on a daily basis in our data warehouses systems and that can be used for analysis. So our analysts get information about the activity on how many photos are being uploaded, how many users are accessing Instagram on a daily basis. They can do all sorts of generate second hand statistics on top of that and all the way into other systems that can use that data too. So you can think about the ranking for search or the ranking for suggested accounts on Instagram which are products that we also build on Instagram data. Those come out of batch processing pipelines that run on the data that we accumulate on a daily basis. So those would be fed back into whatever the operational database would be. Exactly. So let's say on the... What do you use for the operational database? So we use some integrated technology, some in-memory database systems that we have at Facebook. And what it does is just on a daily basis, you take the data that comes from the stream, gets into Hive, you process the data in Hive, so there are like many pipelines that run. You generate some data that then we want to get back to the user. So that gets exported into some distributed in-memory databases that we can access from our systems with the data from yesterday. So that goes for a whole day. So the analytics were done sort of batch mode but you want to serve them up in real time? Yeah, like you can think of it that way, exactly. Thanks for watching. This is George Gilbert.


Hi, so it's a cloudy day in New York City, a little bit overcast, a little bit dingy. And a little snowy, yeah, but it's winter here. But inside, where we're talking about cloud, it's a lot brighter in some of the record breakers. So, Bob Olwig, WWT, what's your first take on this EMC announcement? Well, first of all, I want to thank EMC for inviting me up here. Fantastic event, both from the dinner last night to the fun they had this morning during the venue. What got me here, what made me most excited about this particular event is the breadth of products that were announced today, both from the refresh of every one of their product lines, basically. And then they went downstream in terms of the market applicability of their new VNXC product line. So it now hits the small to medium-sized market, which isn't really a place that we play, but I can see for a lot of other partners that makes a lot of sense. And then in terms of the cloud infrastructure announcements that were announced today, very important for us, we're a VCE partner, we're one of Cisco's largest partners in the world, so it's very relevant for us, especially the speed and simplicity of the software that they've introduced in the systems today. Great, so Bob, when we talk about the Converge Network, I would think the new VNX message really helps with the unification, unisphere, and UIM, if you're talking on the V-block, it really helped a lot. Absolutely. You know, really, how have you seen Convergence really changing some of the dynamics in the marketplace today? Well, being a systems integrator, we've been all about Convergence for many, many years, but anytime the manufacturers like us, Cisco, and EMC, and VMware can get together, create the reference architectures, start enhancing their products to make the products simpler to begin with, it makes our job easier. Great, Bob, thanks for joining us. We're here in New York City for EMC's record grade chain-breaking mega-launch.