He does anything. He does anything we day one Goji to him. Stop right there, criminal scum! What are you doing throwing ki blast at Gojira's game? How's it going everybody? Dato Doy here back with some more online matches for Dragon Ball Fighters. And today I am super excited to be doing something a little special. Normally in these Fighters videos we hop on ranked and just play against random opponents, you know, whoever we queue up against. Whereas in today's video we're going to be doing something that I've actually rarely gotten the chance to do on the channel. And that is playing against some of you guys in the comments or on Twitter and just get some games going within this community. Normally this is something I do during live streams on Twitch, but I've also gotten a fair amount of people that said, Oh, I would love to do this and also have a chance of ending up in a video on your channel. So hopefully over the course of this video you guys can let me know if this is something you want to see more on the channel or just you want to see me have more open ring stuff like that and post about them what I'm doing. But with all of that said, before we jump into the crazy amount of matches we have today, I do want to take a quick moment and talk to you about the sponsor for today's video, Raid Shadow Legends. If you didn't already know, Raid Shadow Legends is a brand new collection RPG game that you can play on your very own mobile device. Within the game you can collect over 400 champions, assemble a team from 16 heroic factions and explore more than a million potential champion builds. The best part about Raid though is that it's entirely free to play. One of my favorite aspects of the game personally is the amazing graphics and details on the champions. It also helps that in Raid you also have the ability to personally customize and choose the artifacts and design a unique mastery build for each and every single one of them. And when you combine that with the fact that more than 10 million players worldwide have already downloaded the game in less than six months and with the fact that with over 300,000 reviews, Raid still has an almost perfect score on the play store. That's a combination for a good mobile game. The game is growing super fast and the highly anticipated new faction wars feature is now live and there's also a new awesome rewards program for new players that allows you to get a new daily login reward for the first 90 days in the game. So don't wait, go to the video description and click on the special links and you will get 50,000 silver and a free epic champion as part of the new player program to start your journey. Thanks again to Raid Shadow Legends for sponsoring this video. Now let's get into some online matches. I did not know Lotus was going to slide through and be the first person here. This is actually while also he's playing 18 who I think you said he doesn't really like in this season.  He's seen the scams as they are. Oh, yep. He's seen that. He's seen that one. That's that's crazy. That's crazy. I didn't expect to snag this man. Let's get him back in the corner. All right, Broly. And we're going to be mixing up teams throughout the video. Let's see. Oh, I got away. Wow. He went straight for the dome. Nice blocks. Oh, never mind. No punish. 6m. We see those overheads. We see those as well. Not going to mix it up. No dragon rush. I know you got one. He did it. Is it going to T.O.D.? No, no, no, no. It's a little it's a little too early in the day. It's only 7 p.m. The night is still young. I think I want 18 now. This might be a bad idea. He does have an air level three. All right. We're good. I'm just going to do a very basic combo here. Trying not to risk too much. Let me see it down on the ground. Oh, big. That's going to be the end of 18. But that 18, I think, you know, I don't know how long your lotus has been playing her again. But probably definitely not the problem character of the team for sure. Oh, that was so close. I panicked. Oh my goodness. All right. Prioritizing the hard knockdown. I can respect it. But do I respect it enough to not spark on wake up? Or just jump? I guess we could just. No, no, I don't want to play that game. All right. You know what? I do want to play though. Trunks does not have an air level three. So let me see Trunks. Yeah, didn't go too hot. Wow. Nice. I always I back dash too often. I got to stay in my ground one of those times. Rolled trunks is the the definitely the problem character. We were talking a little earlier how oh come on stuffed me during my all right coming through understandable. I just wish it wasn't. I just wish it wasn't. We're getting mauled only so tall to be in the corner and my phone's ringing. But there is no time for social social pleasantries. I'm a Broly player. I'm a Broly player. Okay, another good start. Let's Super Saiyan Goku talk to him and you know Broly's got to do some talking of zone. Where's the spark? Oh, okay. That was a good one bad maneuver by me. One too many L's and looking like my old look of like my old sports history just an L after an L. All right. Oh, I hope we don't lose the trunks at this point.  He's got sparking activated. You know, you know he's been waiting for this moment. I'm not going to reflect again the mix up. Oh my God, the damage. This is why sparking trunks is instant S tier material. I'm talking triple S tier here. People wake up for it. What am I doing? Oh, we're beautiful. That was amazing. Oh, I could have won cool. No, the other way as I'm going to lose. I'm going to lose because my my my old son. He just did it to me. He still has three bars left. I thought that was a glitch at first. Oh, we're going to lose. No, it can't happen this way. I'm sorry, Lotus, but I actually have to refuse to lose on this one. Oh, trunks players. He has to mess it up. He did as well. You got to be kidding me, bro. I am just one mess of a play. Oh, you see that wasn't supposed to be a six and what we did that was supposed to be a raw medium. We'll thank this. Oh, Lotus takes it. That's a bad way to start it off. But oh, I have rematch out. I got to turn that off. Oh, I mean, I could get one more match. I'm going to be opposed to it. There might be people waiting, but all right. We got the rematch and let's do one more with Lotus. All right, this is this is good. I mean, obviously we're not starting off good, but I'm just glad we got a chance to rematch going down again. Do we up that? I think we do. Oh, you were close to getting rock. That was close. Oh, Lotus. All right, hold on. Lotus shared a jiff with me. I don't know what it is, but not no beam. Oh, we deserve that one. What a great player we are. All right, we'll take it. Decent hit. Should probably start going for the more meter intensive stuff, but you won't see me complaining. Where's the reflect? No reflect. That's that's interesting. Big protagonist to him. You know how we feel about those. That's a that's a one-way ticket out of here. I am going to enjoy this one. I might have been able to get away with a little something else, but that's fine. Broly's going to come in here. Do the damage we need and we didn't use spark in a panic. Oh, that's the good stuff. Wow. That was ballsy. Notice I swear to God. That was kind of hot. Nice super dash. Nice super dash. Man, that was that the struck the disc really set off the the Broly part of my brain. Um, all right. So here's why that hit me. I thought that was going to go over my head and I was going to I was going to big brain that like I was going to run up and hit him. I forgot it tracks mildly. And let's just let's go ahead and be honest and say I didn't think that was coming down for my skull. I did not think the androids knew where the soft spot was located on my on my skull, but they seemed to clearly be at ease. Now we did lose a lot. Oh, all right. Francis. I mean, yonks is this we got to abuse it while we have it because I feel like yonks is going to be priority target number one. Beautiful. I should start going for 2m's after that jab jab wait to him got me in the corner. I'm a normal tech. Okay. Okay, that's fine. Wow. That was either reactionary or the ballsy as play. Can you level three? No, I can't. Oh, God. Oh, no, you don't got me. Oh, no, you don't got in fact. Let me take this corner back. Oh, you mind if I do this real quick? All right. That was pretty good to age for the ages. Get him down. Oh, he opened me up. I tried to jab in there. Trunks damage. No, okay, drop combos. The correct answer was drop combo. That was hot. That was a little that was a little interesting. What is this Jiffy seven? Let me see. Let's unravel the presence. See if I'm tilted. Oh, no, he was just sweating. All right.  Do we let's just round it off of the print. What is this video? This was supposed to be a little. All right, last one. Then we get back to the purpose of the video. Oh, God. And bounced around gets the full conversion. We're going to go right to the corner. You know, I'm down taking on this one. Well, look at how tall Broly is. You can already tell I'm not having a good time. You could tell you could read a Broly's mind when he's in that situation. It's just not a good look. I try to guard against that's fine. Coming down like a ton of brick. Oh, a second. You give me three chances. No, that's on you, bro. All right. Maybe it's on me.  That's us. Oh, that's not us. That's not us. I'll do it. I'll do it. I'll take it. I'll take it. OK, we're in Broly cook. This is primetime Broly cooking. Oh, no. All right, Lotus. No, you definitely block. So that's that's the facts of the situation. Wow, bro. That did not look natural. Broly was just like, I guess I'm going back. Definitely manipulated his key to do that. I've watched the manga and what kind of big brain jump at me was that? All right, maybe it was a little obvious. Three times was too much. Three times might have been too much for the little mix. What the? I thought that was my turn. We got slapped up this third game. Hardock taking me down to three. Hey, OK, he's cooking. He's cooking. But who's really the chef? He might be cooking. But who's who's kitchen is he working under? I thought it was Yonk just for a second. It might be Trunks.  Definitely ruined a little bit of a reflect. Three bars. This is a hard knockdown for sure. Hard knockdown into mix. No, no mix. Doesn't need a mix. Just a high low. Four bars can kill. Three bars can kill at this rate. But he's extending. Pushing the limits of what it means to combo under mines. Leave it. Ah, we got cooked.  You know what? I'm actually pleasantly surprised to see a lot of good connections here. You guys, uh, y'all really know what you're doing. He also has a witch hat bent. Yo, this guy is like next level thematics. But enough about thematics. Can he withstand the Broly pressure? I know Lotus did. Yamcha already making the name for himself. Broly's going to take it right to the corner. And I got to switch up teams after this. I forgot. Oh, yeah, Broly combo time. I messed it up before and I messed it up again, baby. One is not enough times.  Oh, that was it. That was it. That was very interesting. I know what it's like. I do it as somebody that possesses the sausage finger keke genkai. I can confirm that I know what it's like. We definitely are not the strongest plan. Okay. Oh, back him up. Back him up. I don't want to see it. Not in Broly's dojo. Okay. We are just getting some random hits for you.   All right. Well, shoot him down. Yeah. Do we let Yamcha do the rest of the talking or do I don't know? I feel like Broly's the best bet. Oh, that was sick. That was actually like some movie stuff. The beam upwards to stop him, but only barely through the armor. I going to have to spend a bar to make sure the sliding connects there goes down off. We had Broly up. You know, he'd be coming in right now. We're going to spin take you out. Oh, I was out of there. Lands it. Okay. Oh, Broly players. All right. Hold on. I'm not a Broly player. I've chosen to actually OK. I'm a Broly player. Oh, we didn't clash. Didn't clash in time. Here comes the giant fit. If he dashes at me, we just. Yeah, we're just going to take him down. In fact, let me take him all the way down. I tried to snag him. Why? It doesn't the armor still applies against guard cancel. Oh, I didn't know it worked like that. I thought it took away like vanish and guard cancel. He definitely tried to down heavy there. It was just a little too late. All right. We're making moves now. I'm sure we're fighting. I was I don't know why I like that quote so much. That was brutally honest. All right. First team change of the video. We got beer is coming in at behind Weiss. Man, I really hope they put Weiss in the game. But you also we're going up against pink square get ZL. Sorry if I said that wrong. But oh, that's not y'all. I try to use that like a yoncha assist. Perhaps a poor choice on my part. We got an orb out and we actually land the bunk. The bunk. What should we call it or look at the hitbox on the claps though.  I'm just oh we out. What was I going to say? Oh, yeah. Also, he's playing the theme team of blues. I can respect it. Oh, come on. Oh, we just mashing now. We just mashing. Okay. I'm out. I'm out. Oh, don't bring me back out to Jiren. You know, I switch for a reason. All right. I'm going to send you flying then. Okay. What can I do to maximize that? You know what? I'm just going to go for that was not what I wanted. But I was going for a grab. Perhaps a little too risky. First time post vanish.  My Jiren is pretty wack these days. In fact, there's always been pretty wack. Okay. I got to remember that that does not do anything. I got to start using that more from mix ups. He blast out beer is blocking back it up. You don't got to play his game. Don't have to but we probably should get around to actually doing some damage. Angel isn't as good as I was hoping. Oh, I wanted to be there for that switch. Oh, no, I try to read a beam. Which probably would not have been the best anyway. Oh, Jesus. Ah, nice deep fee. He knew the grab was coming eventually. What does he do here that just swap us out? Yeah. Okay.  No, we don't play by anybody's rules. Oh, one orb got him. Ah, but we'll never get that second orb. Another DP. Why is this a DP game? Okay. Let me send him up this way. Got him. It should just be Beerus. All Beerus. All Beerus, baby. Look at the amount of... All right. I like the stagger game with his death. Beerus's 2L actually feels kind of nice. I don't know what it is. I think it's just because it's so snappy compared to his other moves. We did it again, baby. Just another day in the Beerus office. Oh, that's we definitely blocked that. If it even hit. We missed it. Oh, oh, nice. Look at the go courses. All right. I love this team. It might not be my best team or even like anywhere decent. Oh, come on. DP again. Now Beerus got the best DP in the game. That's actually a fact. There's no opinion there. Oh my God. That was kind of mean. I feel like DPing is pretty obvious now. It choked me. He choked me. Oh. All right. All we have to do is make it through. I'm pretty sure the sparking is almost up too, right? Sparking time should be pretty limited. Let's just make the move. I'm never going to get it. Oh, it's over. It's over. Call it a wrap. You know I didn't have any Jiren combos either. So I had to do what I had to do. On to the next one. What team are we going to use next? All right. We're going up against another Ping Square using Gojira. This is our team, by the way. Android 17 Goku and Trunks. I think this was actually a recommended team that I was going to do a full video on, but you know, there's always two. Oh, no, I didn't really think there. That was supposed to be a three H that that was a big accident. Oh, you know when I when I finally get to use change the future. It's a good day. It's a good good day to use to get up there. Oh, I thought I missed it. Okay. Okay. He chooses to spend it early. Boom. Boom. Okay. That's fine. Get him down. Get him down. Boom. How to run it back. Dude, what was I supposed to do? Okay. Okay. Okay. When do we be a scrub? Boom. When do we get to be a scrub? Because you know, I'm going to do it. It's a natural way of being. I can't hide it for too long. Oh, get him off me. I could have kept punching him. I always forget that. Okay. I'm just going to choose to kind of know where was my I try to get the I don't even remember what they're called. Did the two to S the basically the key words that come out of your hands. I was really looking for it. Our doc is going to sauce us up here. Only going to get a level one. Do we down take buttons? Nah. Okay. That's an easy switch for us. That's one bar coming in. Bardock looking a little weak and meager too. Oh, come on. Lotus give me strength. Oh, no. All right. Well, somebody gave somebody Bardock strength. I don't know who called that one out, but I did the move wrong. Change the future as quarter circle back as nice read nice read. We still got a spark that we could use early here or we save it for maybe two characters left. Now I don't know if I want to lose 17 the heart and soul of the team. I don't know dude. I don't know. Can I really let it go? No, I can't. What is the what is the matter with you 17 on what or on what earth would I ask for those inputs? The game should auto correct.  We're going to use the power of science to say what you really want. I'm just on floor and I'm just my day is ruin. A disappointment is immeasurable and my day is ruin. I cannot believe it. We lost the heart and soul of the team anyway. Okay, I'm glad one bar killed because I was really really thinking I should probably go with heat burning attack, but I'm glad we held off. Come on. Okay, too much too much too much protagonist to him. I definitely got caught sleeping level three inbound. You already know it is. All right. Well, what do we do to secure this win? We at least get one trip back to neutral. So what's the mix low overhead? Okay. You just kind of jump our kids. So that was terrible, actually terrible guard cancel. It doesn't have much you could do here. So he's just going to take that. I was smart. I didn't think about that. Just take that. What are you doing? I knew you. I knew I knew as soon as I open up this ring, y'all would try to drag and rush me. I get it. It's because you think I can't take them and you would be right. This is it.  Kill him. You have to. I'm going full blast damage. It wasn't enough speaking of dragon rush. Get out of here. He was mad. No, got him in the air with the jump in light. Maybe get him down. That one was just for a little extra flair. See, I'm a little tight that the first orange square we go against. I changed to this kind of team, but ah, dude, we got to get this pink square back eventually. I don't know why orange squares are so rare. It feels like I'm the only one other than, you know, my fine, my fine gentlemen right here. Go to GT assist already. Oh my goodness. Get down to the ground. Can you maids rise up? Okay. We're just going to back off there. Okay. You know what? Oh, the range on that normal is crazy. No text. No text. The dragon rush continues this time on dragon ball fighter. I got to pop up. I got to pop up. I got to pop up. I got to pop up. There we go. Further. He can't jump. The pressure is limited. There it is. I cannot believe this. Six bars. I mean, he might be able to kill.  That's that's Bardock we're talking about. Unfortunately, give you, you didn't get them. You didn't get the ball out like I was hoping, dude. May you rest in peace. Dude, I was really hoping I could get a little cheeky body swap in there. Doesn't look like it's going to happen though. Oh, did we wild out too? Did we go too crazy with the team or can Bardock bring? I threw Bardock on here as like a I knew the second I opened this ring. I'd face I'd face hatred. I'd face people who don't want to see a dude succeed. No, where was the spark? I got a spark. I got to save Bardock. The only person I can truly trust out here. Look at the Bardock privilege being thrown around right now. I'm just going to level three. Hold up. Let me keep him in the corner real quick. We use sparking. So it's got to just be mix up city from here on out. I think we go safe jump. I don't try. I haven't seen this guy on wake up yet. Okay. Got him again. I knew there was going to be something tricky coming. I knew there's like a dragger. I thought it was going to be a dragger rushing for being honest, but can we clash with him? Back it up. Where's the comment? I know he's going to throw it out. No comment yet. That just means one's coming soon. Oh, no, we tried to mash out of it. Wasn't the proper choice. Seven bars.  Okay. He's got the spirit bomb, which means he probably can kill with go to GT. Yep. We're definitely dead. Yep. We're definitely dead. That's the power go to GT. It didn't even have to be that fancy. It didn't even have to be that fancy, but he still got Geneva in the back. Come on, Geneva. Oh, you know, we like to keep it fancy. Wait, I completely forget dude. It's been way too long. So definitely Geneva. What do you do with him? Oh, no. Oh, we still make it out. I remember that. Okay. Okay. I remember. I remember. I remember. It just it takes a little while for my brain to click with things like this. I had to see the key blast. I remember now. Oh, I definitely take that one. Yeah, I say that a lot, but that one was tech. That was I'm starting to think he put in the cheat code. It's crazy. People got game sharks out here. No, I went too wild with the team. Game shark. I didn't see it coming. All right. The system is clearly being rigged here. So normally what I do is I back out so I can change teams, but people have caught on and now they're just staying in the ready box. So I have to fight one of them. But I think this guy's a new player anyway. I don't think we faced him before. Hopefully I'm right because this is going to be the grand finale. If I'm wrong, man, I feel it. No, I think I'm right. I don't think I played against Huntsman 73 before. He also was the only red bar in the game. So this might end up being an anti climatic finale. If it is, I'll see what Lotus is doing. But, you know, for the final one, we had to go back to Gojita. Still paying our respects to Gojita all these days later. Oh, oh, D.P. You got to you got to remember Gojita has that because I certainly do not. That's good. One, two, three. Look, gojita power. Yep. Yep. It should be the death of him. But if it's not, all right, you asked for it. I'm going to read the evil, the secondary gojita that can only be one. All right, Huntsman, let's see what you got. I'm going to come at you.  Oh, he came at me with something classic. All right, I'm kind of just going to back up there. Let me hit you with something new. Let me hit you with something new. Okay. Okay. You know, I understand. I understand coming in hot. Beam really stopped my stop myself from moving. All right, getting bounced around. Goku sending me down. We were all level three. Do we is that? Is that the state of the game? Backing up. I'm going to switch out real quick just to get the health pack. Oh, snipe my sniper right back. I tried it. I tried something nice, but turns out I was bad. Okay, keep them low. Keep them low. Oh, I definitely planned to. I thought that bounce was going to be wild. All right, we're going to switch them out here. Get really low. Oh, we pushed it too far, but you could argue that it was a reset. You could make that argument. And rumor has it, you're very handsome if you do. Okay, here comes a big bang coming. I'm switching topics because we're back in it. Let's see if I can't. Let's see if I can't instant overhead this man. Let's see if I can't. Oh, did you? I tried it. I tried it back off of Gojita back off Gojita. Okay, we're keeping it at bay. Oh, he's got a beam of his own. Gojita getting rocked. Don't lose Gojita. Do it raw. Oh, we missed. No, we missed. He does anything. He does anything we day one Gojita him.  What are you doing throwing ki blast at Gojita's game? Ah, we tried it. Oh, he died to that. I was going to DP. I was going to DP on wake up just to keep it nice. But all right, nice. That's how we wrap it up. All right. And with that, we are going to go ahead and call it a ring. I think we play. Let me see. I'm sure there's more people in the lobby. Yeah, there's some people that we didn't get to play, but hopefully you guys enjoyed this. Thanks again to Raid Shadow Legends for sponsoring this video. And we're definitely going to try. Wait. Oh, no, dude. He's perfectly copied me down to the very aspect of the color of Krillin costume. All right. But yeah, like I said, thank you guys so much for watching the video and thank you for coming out. If you were one of the people in the lobby. Good games to everybody I played and let me know down below in the comments if this is something you would want to see again. Other than that, you can check out some more videos on your screen right now if any of them catch your eye. But I have been Dr. Doya. Thanks again so much for watching to the end and I will see you in the next video.


In 2015, she got diagnosed with ALL, which is a formula chemia. She had her first chemo therapy in December and had a massive stroke and she was in intensive care for 10 days, we nearly lost her, but with the help from Lady Saleno Hospital and all that, she's pulled through and over the last two and a half years, she's battled the cancer and beat it and ended up ringing the bell on the 19th of January to say she's chemo-free. When I found out that Alana wanted to be a police officer, Senior Sergeant Stuart Day got on board and said, look, we'd love to bring her down here for a day. I think she'll make a great police officer. She's got that battle, that fight, that determination. To get through this at her age, especially with a chemia and to do what she's doing now, yeah, I believe she'll be a wonderful police officer.


Hello everyone, welcome to tomorrow. So first off, I wanted to acknowledge that Apollo astronaut Edgar Mitchell, the sixth man to walk on the moon, died on February 4th after a short illness, just one day before the 45th anniversary of his landing on the moon with his crewmate Alan Shepard. He was 85 years old when he passed away. So in the spirit of open-mindedness, which Edgar Mitchell was famous for, I wanted to examine some new rocket study contracts that the United States Air Force has been awarding to help replace the RD-180 engine that's used for national security payloads. This is your space pod for February 9th, 2016. So the RD-180 engines are used on the Atlas V launch vehicle. Ever since the United States imposed economic sanctions against Russia, the United States Air Force has been looking at multiple ways to ensure that they have access to space for large, sensitive payloads. Part of that ties in to certifying SpaceX's Falcon 9 rocket for national security payloads, and they still have access to the Delta IV rocket. However, the Delta IV rocket is planned to be retired in the 2020s, and the United States Air Force wants to make sure that they have at least two launch providers. Several small-class launchers exist and are used for this purpose, but for these new contracts, this is specifically focusing on medium-to-heavy-class space launchers. Although the primary focus of these new contracts is a new first-stage engine for the Atlas V rocket, several smaller contracts, worth less than $1 million but more than $500,000, are looking at different subsystems, advanced manufacturing, and other technology advancements. But some of the larger contracts may yield some unexpected results. The first of these contracts were awarded on December 23rd, 2015. Northrop Grumman received a contract worth about $5.5 million, although I don't know what they will specifically be working on. Northrop Grumman does have a long legacy of working on ICBMs, and they recently helped co-design an experimental rocket plane for DARPA, the United States Defense Advanced Research Projects Agency. And if anyone can lead me to more information about this specific contract for this Air Force program and what Northrop Grumman is going to be doing for it, I would highly appreciate it because I was unable to find very good information. Orbital ATK also won a contract worth about $3.2 million, and it draws on the fact that Orbital ATK manufactures composite structures, engine nozzles, propellant tanks, and booster separation motors for United Launch Alliance. And all of these essential technologies would improve the entire rocket systems for the Atlas V, Delta IV, and future Vulcan rockets. Aerojet Rocketdyne also received a contract for about $6 million, and interestingly is going to be used to define the standards to qualify 3D printed rocket parts. 3D printing, or otherwise known as additive manufacturing, can greatly reduce the amount of time and money spent to reduce complex rocket parts, and Aerojet Rocketdyne is working on 3D printed parts of their own. Aerojet Rocketdyne is working on their AR-1 engine as a possible replacement for the RD-180 engine on the Atlas V, and even if not selected because it seems like United Launch Alliance is favoring Blue Origin's BE-4 engine, all the different processes and techniques that they come up with while working towards the development of the AR-1 engine could help them to define the standards for 3D printed parts, and maybe even come up with some unexpected innovations as well. In 2014 the company successfully hot-fired tested an engine made entirely with additive manufacturing that had 5,000 pounds of thrust, and in 2015 the company used additive manufacturing to replicate the injector of the gas generator used on the Apollo F-1 rocket engine to demonstrate that a proven design can be built at a competitive cost without sacrificing performance. Those first contracts, including the smaller ones I didn't talk about, all together were worth about $17 million, and again those were awarded on December 23rd of 2015. Now fast forward to January 13th of 2016, and the Air Force awarded two much larger contracts, one to SpaceX and another one to Orbital ATK, although for different reasons than for the first contract they received, and all together these two contracts were worth about $80 million. Orbital ATK's second contract is worth almost $47 million, and it's for the development of a new solid rocket propulsion system prototype in support of the Evolved Expendable Launch Vehicle Program for National Security Payloads. The Air Force award does look at some additional upgrades as well, valued up to $133 million, and the company Orbital ATK is going to be contributing some of their own funds for this program as well. The main part of the new work under this contract is for something called a common booster segment, main stage, in other words the creation of a new solid rocket first stage core that could compete in the medium to heavy class launch vehicles for the United States Air Force National Security Payloads, in direct competition to United Launch Alliance and SpaceX. Are we seeing some sort of revival of the Ares-1 slash Liberty rocket idea, or is this something else altogether? Part of Orbital ATK's work that will support United Launch Alliance is to help the development of their new Graphite Epoxy Motor 63, or GEM 63 solid rocket strap-on boosters, which in the future is going to replace the AJ-60 Aerojet Rocketdyne solid rocket boosters, which is currently used on the Atlas V. And then the GEM 63 XL, an extended version, is planned to be used as the strap-on motors for United Launch Alliance's Vulcan rocket as well. It makes me wonder if these boosters could be used on the Antares rocket. Interestingly, Orbital ATK is also working on an extendable nozzle for Blue Origin's BE-3 rocket engine. That rocket engine is being considered as an upper stage engine for United Launch Alliance's Vulcan rocket, and on top of that Orbital ATK is also considering using the rocket as an upper stage for their as of yet unnamed rocket that's going to be using that new solid fuel common booster segment. But Blue Origin itself is already planning on using the engine as an upper stage for their Orbital rocket. So no matter which way it's implemented, United Launch Alliance, Orbital ATK and Blue Origin all stand to benefit from that. Now SpaceX's contract is worth about $33.6 million, and is for the development testing of the Raptor upper stage engine. The Raptor, which is a liquid oxygen and liquid methane fueled engine, is designed for use on SpaceX's Super Heavy Lift launch vehicle, the one that will be used for the company's Mars Colonial Transport vehicles. But under this contract, however, SpaceX will develop the Raptor prototype to be used on Falcon 9's and Falcon Heavy's. SpaceX is going to be contributing $67.3 million towards this jointly funded program, but if the Air Force decides to exercise additional options under this contract, they could fund SpaceX up to $61.4 million. However, at this time, I do not know what those additional options might be. Now the United States Air Force will be awarding more contracts throughout the year under this program as they sort out all of the different proposals that they've received for this. But it seems to me like Blue Origin is a likely candidate for a contract and seems like United Launch Alliance itself would be a likely contender as well. Who do you think might be an additional company that would receive a contract for this program? Thank you for watching this video. My name is Michael Clark and please let us know what you think about all of these different rocket contracts. What do you think that Northup Grumman is doing under this contract? Do you think that Orbital ATK will be able to compete with United Launch Alliance and SpaceX? Let us know in the comments section below and on any of our social media. And if you haven't already, please subscribe to our YouTube channel. And if you're able and willing, please consider contributing to our Patreon campaign. This show is entirely crowdfunded and a huge, huge thank you to everyone who has been supporting the show already. Without your guys' contributions, we would not be able to be talking about all of this really cool space stuff, so thank you so much to everyone who is making this possible. Thank you again for watching this video and hopefully you know just a little bit more today than you did yesterday, thanks to tomorrow. Keep moving onwards and upwards everybody and I will see you in the future.


When it comes to entrepreneurial success, what I've learned from 13 years full-time as an entrepreneur is that the external metrics such as how much money we're making, how many followers we have tend to result naturally if you focus on the right internal things and these are the three things internally that I focus on. Mastery of my craft, so improving the skills that I deliver to my clients and students. Number two is the practice of internal values every day in business. I see business work as really a stage for personal growth. And number three, balance between work, rest and hobbies to keep optimizing those. So by focusing on those internals, the externals tend to take care of themselves.


The cast of Dracula, do you know where I can find it? Do not venture there. Do not venture there. It is dangerous as blood and vampires. The ugly wind speaks the truth. No. I must find it. Do you know where I can find it? Look, stay with us overnight. You must not go when the sun goes down. It's too dangerous. No. I must go now. Go in the morning. Ugly wench. I need it now. Do you know where it is? We've got no idea. Hello. Is this cast of Dracula? It is. You can count. I have been expecting you, Mr. Van Vild. Come in. Mr. Van Vild, these are your quarters. I hope they will be to your liking. This is my wife. She is beautiful. She is not really made of stone. In real life, she is made of skin. This is my table. This is an even finer table. Hold on to the rope for your own safety. It's rather narrow. Oh, look. I like these windows very much. But I cannot see through them. They are too short to come. You must be hungry. I have prepared a meal. That's very kind of you. Thank you. It will be here in two minutes.  It is very old. Will you be having some? I don't drink. Good night, Mr. Van Vild.


Well, adapting to lockdown, like everyone else means, I suppose, having arrangements with your family about who does their work in which room. I think the first couple of weeks I had, my daughter was finishing off her work for our hires, so she had one room, so Meldrum Academy was in one room. My husband's a deputy head of Tariff Academy, so Tariff Academy was another room. I actually thought I had symptoms of coronavirus, so I actually wasn't feeling very well, so I didn't have the pick of the rooms, so I was banished to the bedroom, and I did most of my work at that point from my bed with quite bad symptoms. Now that I'm better, I've been relegated to the living rooms, so I think it's one of those things of having to arrange with your family because there's conference calls going on all over the place. So it's almost we've got our own mini timetable, so that's been quite difficult. Pass for working from home, I was actually quite used to working from home anyway because I run my own business before I was elected, so I'm quite used to working from home. And my teams have been very good, my constituency team have been great because we operate a very flexible office anyway, so we would do quite a lot of video conferencing as part of our week, so I think we've adapted quite well. I think it's a case of just recognising that you're going to hear noises off from time to time, you're going to hear a doorbell going and there's going to be a dog barking, but actually I think people are quite enjoying that aspect of things when they see you on television with things going on in the background. It makes us feel more human to people, that's a good thing. Like everyone, we are having to figure out a way to do our work remotely and using social distancing. Of course we've also had to look at our work programme and see what's necessary and pressing at this time and what can wait till after this emergency. Some of that's led by what the government feel is a priority in terms of legislation and some of it is discussions that we're having. So at the moment we're meeting regularly, I'm certainly, you know, I'm in touch with the committee clerks every day about decisions that have to be made, but as far as the committee members go, quite a few of them are actually in the category that they can't come to the parliament, they can't leave home for various reasons. So we have been using, like everyone else, the technology to meet informally to discuss things like decisions about the work programme and what to do, but we've also had a very successful live committee meeting. And I'm very pleased to say that we were the first committee to actually pass SSIs, which effectively were the part of the programme that we were going to be doing around a deposit return scheme. And we have very successful votes. We were the first committee, the first Scottish Parliament vote to actually happen successfully. So we're keeping in touch. We're meeting every week, whether it's informally or formally. And we, as you say, we've got one successful committee meeting under our belt. But of course the parliament are having to look at the capacity as to what they can actually broadcast. So we need to be very careful that we don't be asking for committee slots and things that aren't pressing. So we're prioritising our work and seeing what we absolutely need to do in public. And one of the most difficult things potentially is the stakeholder views. But we're working on that at the moment that hasn't actually come up as been a pressing issue right now, because we did so much scrutiny before lockdown on the things that we actually got in front of us. But going forward, we're working out how we can actually have our stakeholders in our committee meetings. You know, in a strange way, I've quite enjoyed it. That's no face to my colleagues. I come from a background. I used to teach live television. So I actually got quite a kick out of actually working to a script. It felt like old times. It was like directing a programme. What we did actually, just a little bit behind the scenes, the info, is that normally committee convener has a script anyway. So you make sure that you cover everything. What I did is I suggested that every member of the committee also have that script. So they knew what was coming. They knew what I was going to say. That was particularly, I think I actually worked to it, been really useful. So it's been fine. I think because committee meetings are very, very formally structured. No one really talks over one another. Everyone has to go through the chair in order to speak. It actually lends itself quite well to the Microsoft teams or whatever other software people are using for these online meetings. So, yeah, actually, I bizarrely quite enjoyed our first meeting and doing it that way. It was quite funny. I saw Mark Ruskell in the parliament and a socially distanced and had a conversation with him. He was joking that he felt that he had to type something in the text box in order to speak to me. So we're actually getting used to the online meetings more than actual conversations at the moment. Yeah, we have had very intense conversations about what we prioritise. I think because the priority for all committee and the whole parliament right now is the response to COVID-19. And I guess we looked at what are the most pressing things from our portfolio with regard to the response to COVID-19. It may appear to public at large that it's a largely health and economic issue, but of course it's also an environmental issue. A lot of environmental organisations and bodies are having to have their own responses to the pandemic. But we also were very acutely aware of some, well, we had legislation and regulations that were kind of halfway through, I suppose. I mentioned the deposit return scheme. We were taking lots and lots of evidence. We were at the point where we were actually having to have those regulations voted on in our committee. We were getting a lot of stakeholders asking where that was and how that was going to work and if it was going to be stalled as a result of this. So we had to take that into consideration and figure out whether or not we thought it was a priority or not, whether the government thought that was a priority. And of course we were just about to go to stage two of the Animal and Wildlife Bill as well. So we need to work out whether we can actually have our stage two in a remote committee meeting, which I'm confident that we're going to be able to do. Of course, in the backdrop to all of this, we've got the EU exit. EU exit is not changing according to the UK government. So we need to make sure that our work in the scrutiny of the EU exit and the impact on the environment, that still has to happen as well. I mean, we have got this pandemic, we've got this emergency situation, but we also have other situations including, of course, the emergency situation of the global climate change emergency as well. So one of the things that did change in terms of the government's programme, which impacts very solidly on the committee that I convene, is the climate change plan. So the updated climate change plan was due to be in front of us in the middle of April. So now the government has actually changed the focus of the climate change plan. It's going to be a green recovery. So it's very much going to have a kind of like climate change and the economy and the recovery from this pandemic. But with a real focus on reducing their emissions. They have said that that's going to be delivered by the end of the year. So we still have to work out a way to have scrutiny and oversight over that as well. And other things, of course, happened as well. COP26 has been cancelled. So these are all things that I think immediately we thought we still have to look at all of these things. We still have to find a way in order to scrutinise everything, to take evidence and to continue our work. And I think that we've actually been doing quite a good job in that, if I may say so. Because the Parliament and the members have all been up for getting that work done somehow. I'm going to do a double thing because there's two things. Obviously I'm really missing extended members of my family very, very much. Particularly my son, who's at university at the moment. I can't wait to have us all together. I think everybody will be the same. But the first place that I'm going to go is to Nubirah Beach, which is the place that I love the most in this whole world. But it's also where I usually take my dog and because I don't live in the village of Nubirah, I can't drive to go to the beach. I have family members who live in Nubirah and they're driving me insane by photographs of them taking their dogs to the beach. So the first thing I'm going to do when lockdown is lifted or any of the restrictions are lifted on travel to go and exercise is I'm going to go to Nubirah Beach with my dog.


Thank you very much, and thanks to the organizers, the two heroes here for and you, for inviting me to give at this very lovely place a talk about commensurability of hyperbolic coxsetter groups. Let's start. So let's look at coxsetter groups. How can we think about them? We can think about them geometrically. This is what I want to do basically today. So you take a hyperbolic polyhedron, a triangle, a simplex, a cube, a hypercube, a simplex of higher dimensions, and you assume that all dihedral angles between bounding hyperplanes are integers sub-multiples of pi, so of the form pi over m, say. This is a very nice polyhedron in general, so we do not care about the combinatorics, and when you now take the reflections in these bounding hyperplanes, these reflections generate geometrically defined hyperbolic coxsetter group. So we have a group of isometries, but we also have a very nice fundamental domain, a polyhedron. Abstractly seen, it is a subgroup of the isometry group of hyperbolic n-space, real. You could also speak about the complex case, the quaternionic case, or even more generally. Let's greet, and I assume always co-finite. That should mean finite volume of the simplex, or your polyhedron, whatever it is, co-finite. Then the fact that we have reflections in hyperplanes meeting in phases of co-dimension in two, and these phases contain vertices, that means that two reflections in two hyperplanes intersecting one another non-trivially give, when you compose them, a rotation about this ridge. So we have, because of that, very nice, oh, very nice general relations. This is the rotational condition. This is the order of your rotation, m is an integer bigger than or equal to 2, and this means that the two hyperplanes intersect under the angle pi over m. In hyperbolic geometry, we have the nice feature, which you do not have on the sphere. You can ask the same questions, of course, in another space of constant curvature. We can have hyperplanes which intersect at the point at infinity, and then we do not have a finite nice relation order of the composite, and we can also have hyperplanes which do not intersect at all. They are ultra-parallel, or hyperideal, and inside hyperbolic space, the two hyperplanes emit a common perpendicular, that we can also have when we have a cube. So here we are with hyperbolic coxet groups and hyperbolic polyhedra. That we should slowly absorb. Now the problem is, which I would like to address today, we take a particular family of coxet or polyhedra, or reflection groups. A particular family defined through a condition on the combinatorics, and let's take a family which consists, which is not empty, which has a lot of family members. And in hyperbolic geometry, they are quite rare, actually. In fact, the classification problem for co-finite hyperbolic coxet groups is not yet done, and I think during my lifetime I will not leave that. So we do not have a list of all co-finite hyperbolic reflection groups, contrary in contrast to the spherical and euclidean cases. There we have a complete picture. So again, I am too fast. I'll take a family which is particularly rich, also finite, we do not like that too much. I'm looking at coxet or polyhedra, which have at least one vertex at infinity, on the boundary at infinity. And when you are at infinity and you look at the structure close to this vertex, I would like that the structure is a pyramid over this vertex, and when you look down you see close to the vertex at infinity a product of two simplices, of positive dimension, meaning maybe a segment times a segment, so a quadrilateral, then we are in dimension three, or a segment times a triangle. This gives something three-dimensional around the vertex. We are in four dimensions, or two triangle times triangle, or whatsoever. So the structure around this apex at infinity is a product of two euclidean simplices of positive dimension, and that gives us a pyramid or some open part in hyperbolic space, and then we cut by one hyperplane opposite to this apex. So this is a very nice family. They are not co-compact, these groups, polyhedra are not compact, finite volume, at least one vertex is at infinity.  Not so intelligent or smart, right? So Pavel Tu-Marquin classified this family, I can say they are generated by n plus two reflections in hyperplanes with a particular behavior, and he listed them up and it turned out in each dimension we have finitely many, the number decreases with dimension dramatically, but the dimension goes up to dimension 17. And in dimension 17 we have one single example left, and this example I love so much that I like the whole family. Why is this 17-dimensional example so particular? Actually it is a pyramid with 19 facets, it gives us a group, and in that sense it gives us a quotient space, a space of orbits. This orbit fold of dimension 17 is actually arithmetic, it comes with a lattice. It is related to the unimodular lattice of type 2, which is pretty well known. And this orbit fold, arithmetic, has among all arithmeticly defined hyperbolic orbit folds of any dimension, and they exist in each dimension until infinity, the smallest possible volume. That is fantastic. And so this family is a family, these are not simplices anymore, a simplex has a vertex, the boundary, the vertex boundary is a simplex and not the product of two simplices. They exist in higher dimensions even than coxsetter simplices, a coxsetter simplex exists at most up to dimension 9, here we are 17. But okay we have to pay a price, the combinatorics is not so nice, and if you establish the gram matrix, what is the gram matrix? Well, you have n plus 2 hyperplanes, to each hyperplane you have a normal vector in our Lorenz Mikowski vector space model for hyperbolic space, and all these vectors, when you take the bilinear form of signature n1, we get a real symmetric matrix of signature n1. And this gram matrix is not invertible anymore, it is singular. That is also a price we have to pay. So the aim is now to classify them up to commensurability, what does it mean? You take two groups, say in the group of hyperbolic isometries, but this is a general notion. You can take just a bigger group, two subgroups, and they are called commensurable, here we take the wide sense because we are geometers, all of us I hope, yeah, okay. If there exists an element in the ambient group such that group G and this conjugate of H, this group, this intersection group has finite index in both G and it's conjugate in the conjugate of H. What does it mean geometrically? If you pass to a subgroup of finite index of a group of hyperbolic isometries, this means that when you have your orbifold, you pass to a orbifold, a finite sheeted orbifold, cover. You can even, we know by cell back, you can even take a subgroup of finite index which is torsion free by passing even lower. That means you can get the cover by a manifold. So commensurability is something we like to study. However, there are not so many results about commensurability. When you look at low dimensions, what does it mean, PSL2R, PSL2C, we have a beautiful picture coming from Takayushi in dimension two and in dimension three by Colin McLaughlin and Alan Reed and others. There's a whole book about that by McLaughlin and Colin and Alan Reed and there are also quite a few results. In low dimensions because there we have, we understand the isometries of these spaces very well. These are two by two matrices where traces give us a background about the geometry and trace is a conjugacy invariant, so if we are studying such problems when you look at trace fields when we have a group, that is a good, that is a very good tool. But okay, so here is this problem where we make one step further in a more complicated context, non-invertible matrices, isometry groups, which are maybe Clifford matrices or not, things become a bit more complicated. What I tell you today is, I would say, one of the first steps to do that in higher dimensions and it could be the methods which I present to perform this aim can be generalized. They are, the methods are universal in some sense, but one vertex must be at infinity. It does not work so well in the compact case, but okay, maybe you will do it. Let's continue. So what are these? These are coccet graphs and I invest maybe two minutes in order to encode or to let you know how to read or to understand what is coming next. So one node in this graph is a reflection. Sometimes people write also S for generators. So before I had an S, I do not quote it there. Two generators, so two reflections in our group, are joined by an edge with a weight M. If the hyperplanes HS and HS prime intersect and their angle is pi over M, which gives us this nice relation I showed you on the blackboard. So here, there you have an M, N, K and L. M is an integer bigger than or equal to 2. But then there are also these bold faces at these bold edges. That should be bold. I could also write infinity. This means that these two hyperplanes join one another at the point of infinity. So they are parallel in the Euclidean sense. And when you look at these in the upper half space, when you look at these hyperplanes, they do not have a hyperbolic common perpendicular, but a Euclidean common perpendicular on each horror sphere. And the composite of these two reflections gives us a translation by the double distance. So you can read these coaxial diagrams. Each one gives us a group. And what is this node over there? So this node over there. When you take it away, together with the emanating edges, you have here this situation, so something infinite and the translation. And here, if you omit this, you have that connected component of a graph.  And so this is exactly Euclidean times Euclidean. This gives us the vertex structure of the apex at infinity. So you leave away the hyperplane associated to the apex at infinity, and you see the product of two simplices. This is a one-dimensional segment, and this is three-dimensional. So the node always tells you there is the apex at infinity. And we have two Euclidean simplices of positive dimension, giving us the vertex neighborhood. Okay? So this is the classification of two marking in dimension three. This is one group in dimension five. It has one, two, three, four, five, six, seven, n plus two. And what is this here? This is, one could call it the width symbol or the coaxial symbol or whatsoever. This is a graphical, another symbolic notation how to read this coaxial diagram from, not from the British side, from the continental side, from the left to the right. So here is a four. We start with a four. Here is a three and a three, but a bifurcation. That means we write the three, one leg here, two legs here. Then we come to this point, and we have a circuit, a circle, which means we put a parenthesis. It says we have to go three, infinity, four, okay? So now you can exercise on next slides by understanding how we write coaxial diagrams. And I show you now very quickly the full classification of two marking. It consists of 200 graphs. So I do it quickly. That meaning you take one graph from the left side, one graph from the right side, and you glue the circled node together. So we fix the combinatorial structure with a vertex at infinity, and we cut with a hyperplane. That gives already a whole lot of coaxial graphs. Then the next, you glue one of these two to any one of these. This gives higher dimensional ones. N plus two nodes gives us a n-dimensional pyramid. This is one, and finally I end up with the last one. So there are 200 groups up to dimension seven. I call it the queen. I could also call it the king or emperor. It's an important group. And it is this group here with 19 nodes. It's this pyramid group. It is arithmetic. It is the reflection group generating the hyperbolic lattice which has at infinity, which is related to the unimodular group of type two. And so if you look at the set of all arithmetic hyperbolic oriented n or before, it's quotients of hn modulo gamma, gamma discrete in the direct in the group of isometries conserving the orientation. And arithmetic, I will just come back to this word later, among this huge set which exists for each n, no bound, this group here or this group which is the fundamental group of a particular or before it realizes in a unique way the smallest volume. And this is the volume, a certain rational multiple of set of three. And that was shown by Vincent Emry not so long time ago. Now what is arithmeticity? I could now give a talk about not just one hour just to explain what is arithmeticity and to make the bridge. I will not do that. I will give you the working definition which is a deep criterion and an equivalent description for arithmetic which works quickly in our context. So let's look at the group, discrete group, finite core volume in this group of hyperbolic isometries and let's look at the commensurator. So you look at all conjugates of gamma such that these two groups are commensurable. So up to finite number of sheets, they somehow give us the same or before. Now by a very deep theorem of Margoulis which he proved in an even more general context, one can characterize arithmeticity in the following way. This group, it is not difficult to show that this is a group because commensurability has also very nice properties.  I have to hurry up. So the commensurator is a discrete subgroup, is a discrete subgroup of this group and contains gamma of finite index if and only if gamma is non-arithmetic. Or in other words, gamma is arithmetic. You could take this as a definition if you believe Margoulis. Gamma is arithmetic if and only if its commensurator is a dense subgroup of the group of hyperbolic isometries, whatever. But I would like to see once a proof which is not so difficult to understand. Not so good. But okay, we are looking at hyperbolic coxset groups. They are distinguished by having very nice generators and relations. But in that case, Wienberg, the king of hyperbolic coxset groups, provided us with a very efficient and linear algebraic tool with a bit of number theory of course to decide whether such a group is arithmetic or not. It turns up or turns down, boils down to look at the gram matrix of such a group to look at cycles of coefficients and then to check a few things also concerning embeddings of your algebraic number field which you get by looking at the coefficient of your matrix. But I cannot go into this. But I want you to know there is a big separation of discrete groups of isometries, the arithmetic ones and the non-arrismatic ones. Certain tools work only for the arithmetic ones. That's often the case for McLaughlin and Reed's approach. And then there are other problems where arithmetic tools cannot be used anymore. And we are confronted here with both cases. Among our 200 coxset-pyramid groups, quite a considerable part of them are non-arrismatic. Non-arrismatic. Okay. So how to check commensurability? How can we check that a group, a subgroup, is a subgroup of finite index in another one? Don't read this. Look at me first. So if you have a group gamma and you have a group H which is a finite index, it is clear that gamma and H are commensurable. So the first thing we do is among these 200 groups we check which one is a subgroup of the other. This is the first thing one does. And for this, this is already a task. But for coxset groups, George Maxwell from Canada, he provided us with a criterion which seems to be, my God, difficult to read, you get tired, don't read this. What it says is if we have such a group here, I just give you this example. And we start here from the left with a subgroup of type AK. Type AK means linear graph K, K minus 1, 1. This is a famous finite coxset group arising very, very often also as subgroup. So here we have a A2. If we have at one end, for example, here at the left, an AK, like here, or AL minus 1, then there is a procedure to study subgroup relations in other ones. That is the condition. And among our coxset pyramid groups, this happens from time to time. So this helps to decide, ah, yeah, this group is a subgroup of index 3 in here. It is not difficult to prove, but yeah, there are not many general results of that type. The index is also given. Then another one, just a useful criterion is we work on the graphs. Here we see a part of the graph which is of no interest to us. So I call it delta, just some part of the graph could be very long. But here we have one end of the graph which has a high degree of symmetry. And if we have a high degree of symmetry on the level of the fundamental domain of the fundamental coxset polytop, this means we can cut it into two pieces. In different ways even. When the graph is symmetric, we always have some bisecting hyperplanes. It's not difficult to prove. Just to say, aha, let's study symmetry on the graph, and then we take here an associated bisecting hyperplanes, for example, by choosing this reflection or this hyperplane and this one and we cut into, we cut there into two pieces. Okay, a last useful result which I want to mention. Then we go into the dirty work, but this is already the dirty work. So you could see on these different slides there were these cycles, these triangles. And geometrically this means, so here we have such a graph with this cyclic part here, so we have infinity or bold edge, three, three infinity, five. This is the coxset, simple. When you look at this geometrically, actually turns out geometrically, we have, we take this one, one, two, three, four, one, two, three, four nodes, but here we have five nodes. That's actually a pyramid. These are not pyramids, there is lacking one node, but one can complete them. The first one you can complete it like this, infinity, three, three infinity, and the second one in this way. That means when we are here in dimension three, this is a tetrahedron, which is not compact, and we add an additional hyperplane so that it becomes a finite volume. And we do this in this way. These are pyramids. And we could do it in such a way that by adding this additional hyperplane, we can glue them together. And the gluing of these two finite volume pyramids gives us this one. That is something one can see in low dimensions, and then one has to prove it algebraically. This is here very nice to explain. These two groups inject into this coxset group, and actually this coxset group is an amalgamated, it's a free product of A and B, of that group and that group, of A and B, amalgamated along a subgroup, and this subgroup is infinity three. This subgroup here is a well-known group, it is the modular group, which in the upper half plane is given by this coxset diagram. And so it has, this subgroup here gives us this plane along which we glue two pyramids together. So here is an algebraic description, and due to a result of Karas and Solitar, I hope I pronounce correctly Karas and Solitar, the Spanish people, we have a criterion how to decide when do we have a subgroup of finite index. So if we compare two groups, which are both amalgamated free products, and we want to check whether they are commensurable, we have to check a common finite index subgroup has to have a non-travel intersection with this subgroup for each conjugate group of it, and this also a finite index. So we just have to work on this connecting hyperplane. It's a very nice result. Okay, now all these criteria, quickly, quickly, previous slide, okay, so quickly, okay. Now I kind of, okay, now with all these criteria, and there are some others, there is one unpleasant pair, unpleasant. Those are amalgamated free products, but there is not a common subgroup or a plane which they have in common, yeah, here's three, here's infinity three, that group, but here's an infinity four, they are not the same triangles, and we cannot decide whether they are commensurable or not. Maxwell does not work, amalgamation does not work, nothing works, but it is in low dimensions, it has five nodes, it lives in dimension three. In dimension three, we can compute hyperbolic volume. In terms of dialogarism functions, so not just a number, explicit values. In terms of certain well-defined functions, okay, we come back to this unpleasant pair, it's a very nice pair. Now in the arithmetic case, I was speaking about arithmeticity, yeah. In the arithmetic case, and in dimension three, there are two invariants for commensurability, which in fact form a complete set of commensurability invariants. So if we compute them, things are done. And this is the invariant trace field, so you take your group, I call it here G, you take all the elements in your group as two by two matrices, they have a trace, but now you pass to the square of this element. We are working PSL2C, not in SL2C, because of that we have to pass to the square of the element, compute its trace, form the field of all these traces over Q. Turns out that this is a very nice real number field, and we compute the invariant quaternion algebra over KG, which is not so difficult to do, because we are in a non-compact case. In a non-compact case, this quaternion algebra becomes actually isomorphic to the matrix ring over KG, look, everybody can do that. Meaning in dimension three, arithmetic case, the commensurability classification is done due to Reed and McLaughlin, yeah. They form a complete set of commensurability invariants. Now Colin McLaughlin, one of the top experts in this domain, he generalized this approach by using the quaternion algebra to study arithmeticity of higher dimensional discrete groups in isom HN, as long as they are arithmetic. So there's a whole picture with quaternion algebras, which he wrote down in a beautiful paper which was published in 2011, and that was the last paper which he published before he died, unfortunately. So the arithmetic case is done due to McLaughlin. And I will give you now very quickly, quickly, quickly the list, just to show that we worked on that and that, so I give you now the list with all these commensurability classes. How should you read this? Well, it's not so difficult here, in dimension three, I have one, two, three, four classes, four commensurability classes, because commensurability is an equivalence relation, which gives us a distinction. In the first class, I have a number four, that's the cardinality, the number of elements in this group. And these are mainly subgroups of finite index, which we got by Maxwell, etc. In the second class, we have also four elements, this is a representative, this is a representative here also, and in dimension three, we have three classes. In dimension four, we have two classes, one is very rich, has 20 groups. Dimension five, dimension six, seven, eight, nine, ten, eleven, and in dimension seventeen, we have this one group, the isometric queen, okay? So that was easy. Now let's continue. This was the classification for a coaxial of pyramids. The simplex case, which was simpler, was already done with Johnson, Radcliffe, and Chance, we did that some years ago, and used the fact that the gram matrix is invertible, and developed a ratio test with the determinant of the gram matrix, and so the simplex case was done. And what we did here, in these classes I had before, we just tested, when do we have a tetrahedral group? So these are simplices, coaxial to simplex representatives. And this means, you have in dimension eight, in the first class, in this class, you have 16 groups of coaxial to pyramids, and all these 16 groups are commensurable with this simplex group. So apparently there is a common sheet, a common finitely sheeted cover of an orbit fold, which has a combinatorial model, which you can dissect into a finite number of coaxial to simplex, and you can dissect it in 16 different ways into coaxial to pyramids. If you want to do it by hand, this is not possible. Okay, home baby, voilà. Now the non-arithmetic case, I will not speak too much about these technical proofs, but I want to mention what we used. I think that is a new idea. Don't read this, don't read all this. It's a long text. So we have our coaxial to pyramid, and very important is, we have at least one vertex at infinity. That means we have a subgroup, which stabilizes this vertex at infinity. This subgroup is a Beeperbach group. It's a Euclidean co-compact group, a coaxial to group, and by Beeperbach we know it contains a translational lattice of finite index and full rank. That means we have, when we are in dimension four, four translations, which are independent. But that means also that we have four translational lengths. And these translational lengths, which are Euclidean, Euclidean on a horoscope, we can compute them by using hyperbolic geometry, by looking at hyperplanes, which go up to infinity. They are connected, they are bounding hyperplanes of our polyhedron. And we have between vertices on these hyperplanes, a geodesic line of a certain hyperbolic length. And this hyperbolic length is directly related through cosine hyperbolic to the Euclidean distance. In this way, we look at this translational lattice, and we can check whether these translational sublattices, which we have in this group and in that group, whether they are equivalent or not by looking at the supergroup, which is a hyperbolic coaxial to group having this stabilizer. I hope this is somewhat clear what I say. Well, I want to come back to this annoying pair, which is a little bit a different story. But in this way, we can, in a geometric way, this is much more geometric together with Biberbach, but we use translations, decide whether two groups are commensurable or not. So difficult, but I cannot go into details, or you can ask me a question at the end when we have five minutes left, and I'll come back to that with a big pleasure. Okay, let's continue. So that is basically the idea. You have two hyperplanes intersecting at the point at infinity. Here we have vertices on the polyhedron, and this translational length, which is delta. We have half. There is this formula. There is this formula, which maybe many have forgotten or never used. You can read it in Cocksetter's book, but not in all books on hyperbolic geometry. That is a default. So in this way, for example, we can easily also treat amalgamated products, because they are glued copies. When one shares a vertex at infinity, the other one, we also have translational length. So this is not a drama. You don't have to read that. I continue, if I can. What's going on? Didn't I tell you that I have this unpleasant couple? I do not say unpleasant anymore, now I come more with interesting couple. So we have these two here, and this proposition above does not help to decide whether incommensurable or not. So what do we do here? Well, what is funny, I gave you here the theorem 2, which gives you a complete classification in a non-arismatic case. It's complete. So what about this unpleasant couple? Apparently, we could deal with it, of course, we could deal with it, but with another thinking. Do you want to go with me through the list? Yes? No? No. The lists are not so important. Nevertheless, it's some work. Okay. So this unpleasant pair lives in dimension three. Not in even dimensions, not in an even dimensional space. Because you know, in even dimensional space, if you have a polytop which tessellates space, then its volume is a certain rational multiple of a certain power of pi, always. That is due to the Euler characteristic. But in odd dimensional space, Euler characteristic is not there anymore, and what replaces the topological Euler characteristic are highly transcendental functions like dialogarism, Lobacevsky function, or hypergeometric functions. We are dealing now a little bit with analysis and number theory. So let's look at this pair and let's look at the Lobacevsky function. It is one-half of the imaginary part of the classical dialogarism function. Lea to z, sum, r to 1, infinity, z to the r, r squared. So you take this dialogarism function of roaches and others. You put for z a point on a circle, on a unit circle, and then you take here the imaginary part, one-half of it, this is called Lobacevsky function. And you can write it as a simple integral over the log, and that makes that this is a typical three-dimensional volume functional, because length in hyperbolic three-space is given by cosine hyperbolic, sine hyperbolic, tangent hyperbolic, something with hyperbolic, and that is always in coordinate something with logarithm, if you, with logarithm. And so integration over length gives you volume in some sense. So for these two, in our unpleasant pair, we can compute the co-volume in terms of values of the Lobacevsky function and which have arguments which are all rational multiples of pi. Oh, that is not so evident. That is not at all evident. And there are conjectures about rational combinations of Lobacevsky numbers at rational multiples of pi and commensurability of numbers which are unresolved up to today. Our approach has to do with that problem. Why? So if this pair of groups were commensurable, the ratio of their volume is a rational number, right? So if this number is irrational, the two groups are not commensurable. And those who started my list before saw, they live in different commensurability classes. So okay, a numerical check of this explicit quotient gives us a number which never ends here. We continue with high precision, but it still could be a rational number, right? But somehow this suggests this terrible number. Probably this is not a rational number. Probably. Okay, let's assume it were a rational number. Now, this Lobacevsky function, which is related to this dialogarition, satisfies three beautiful functional equations or as properties which John Milner called essential functional properties. First of all, this is an odd function. It's pi periodic. And it satisfies this strange distribution law for which we unfortunately do not have a geometrical interpretation. We don't know what this means for on the level of volume of a polyhedron. We don't know. But it's not difficult to prove. Now Milner, in chapter seven of Thurston's notes, he conjectures the following, first, a very rational linear relation between real numbers with omega, a rational multiple of pi, is a consequence of the three essential functional equations. He writes and he says, he checked that for a small set of rational numbers. Probably this is true, but it's completely open since the 70s. And B is also interesting, fixing some denominator, like 24 before, 24 was appearing as denominator. All these real numbers with k relatively prime to n and because of the other relations, we can take them, k between 0 and n over 2, that they are linearly independent over q. That's conjecture part B. Okay. So let's assume that our number alpha, this quotient volume, is rational. What happens? Nothing. Doch. So alpha is rational, so if we take 2 over 3 times 1 minus alpha, beta is also rational. Yeah? This is just a rational, that is with our expression before, beta minus 1 times the Lopacevsky value at pi over 4 is this combination. That follows from the slide before, before. Now by the distribution law applied to two different cases, once with pi over 8 and once to pi over 12, gives us for these two parts, these two terms, this and this. So if you plug in everything here up, we get this. That means if beta were rational, these two numbers are rationally dependent. But then also we have, then we have a contradiction to Milner's conjecture part B and if Milner is conjecturing something, I'm pretty sure it's true, although it's still open. But okay, then we became confident, let's prove that they are not commensurable. This is not as yet the proof. And what is the proof? Now I come back to this commensurator. These two groups of this interesting pair, both are non-arithmetic. With Winberg's criterion, one can't decide it. Then they are non-arithmetic and they are commensurable. Then they belong to the same commensurator, which is a discrete group in isome HM, right? So they belong to the same commensurator. That is now here. Let's suppose that their volume quotient is irrational. This is what we want to show. This volume quotient is irrational. And we assume the contrary. So since both are non-arithmetic, this commensurator, which contains both of them, they are commensurable as we suppose. This is a discrete subgroup herein and contains both as subgroups of finite index, magullis. Thanks to magullis. C, it contains these two groups, which are both non-co-compact. C is also non-co-compact, but has finite core volume, because we are finite index. So for this hyperbolic group, which is discrete and non-co-compact, we know due to Robert Meyerhoff, we call it atoms in a non-orientable case, we know in Dimension 3, every hyperbolic 3 or before is bounded on the level of its volume by this universal number. This is the volume of this coxate tetrahedron 336, or you take an ideal regular tetrahedron of angle pi over 3, and this one dissects it 24 times. So this bound of Meyerhoff gives us this bound from below. This number is computed with high precision, and this helps now in the following sense. So we take alpha, this is this volume quotient, but this one is the finite index in C, so we just divide both by co-volume of 3. This is the same, but this here we can interpret as a quotient of two indices. This is an integer number, this is an integer number. And now, by the volume computation which I showed you before, we have an explicit value, which is this one, and that gives us this number must be at most 16, at most 16. And that allows, so you multiply alpha by this denominator, alpha times this number can be at most 16. But on the other hand side, it is this integer, which we can also, so now we just check, is it possible that by taking integers here, 1, 2, 3, 4, 5, up to 16, multiplying with alpha, we get our other number, and that is not possible. So they are not commensurable. This allows you to conclude that there is no rational solution alpha 2, this one, number 1, with alpha being of that size. I have two more minutes, but I think I finish here. Thank you for your attention. We should leave the room in time, and if you want to come back to one of the questions I put you in the mouth, please don't hesitate. Thank you very much.