Hi everyone, I'm Joe for jaspyscasebreaks.com with Jaspys 16 box baseball mixer. It's a pick-a-team mixer. Number three half of a case of tier one Half a case of clearly authentic second half already in the store if you want to get in on it And if you bought at least two teams if you follow the instructions here you get a shot at some break credit We'll do that randomizer at the end big thanks to This crew right here for making it happen if you got some rooftops next to your name That means you won those spots in that filler Diego you ended up with double last-bought mojo cubs and tigers oh Will Smith's daughter will was on SNL People watch SNL. Is SNL started again? I just watched the highlights the next day All right, here's tier one. I'll set that aside for a second here, too And let me pop this open here Ten boxes out of here. Oh, yeah, she did do that with my hair back and forth song She should really throw people for a loop and Then and then like shave her head or that too obvious All right, so there's big stacks of ten right here twenty boxes total Ten on the left side ten on the right side So we're gonna roll the die One two three for the left side four five six for the right side one left side So this will be saved for pick your team four This is pick your team three. This will be for pick your team four. Oh, she is bald now Seems a little too obvious Little too on the nose But I guess I guess the this the Smith family may be not known for their subtlety All right, here's the tier one case Do I with my hair back and forth how I was asking? I've been known Gotta gotta I have to loosen up those muscles though before I do that All right 12 boxes here so six on the left side six on the right side same deal one two three for the left side four five six for the right side One two three left side four five six right side. This will also be for break four. So left side for break three right side for break four And you can pick your teams in break four right now All right, we'll do the tier one last Some bigger hit potential in tier one clearly. I think it's pretty solid too. Let's see what we got in here Good luck everybody. All right one in case car per box and the first one is Conor Wong 80 out of 99 for the Red Sox That's gonna go to Nick and the Boston Red Sox nice Next up is Another Red Sox old school this time It went from a rookie Conor Wong to Red Sox legend Carleton Fisk 89 out of 99 Conor Wong has half the career that Carleton Fisk has I think Red Sox fans will be happy That's right Scott Bad barrel bad knob opportunities in tier one Halfcase from a fresh case. So all is possible. There's Jim Abbott 77 out of 99. Why haven't there been more? I mean, maybe he doesn't want to but I feel like there's there should be like a documentary or the Netflix documentary of the big ESPN 30 for 30 on Jim Abbott that goes to Nick and the Angels spot the Angels straight up And we've got Paul Conerco White Sox that's gonna go to Nick won that spot in the filler Nick S Next is Cal Ripken jr. The Iron Man That's for Chad and the Orioles won that team in the filler Actually one it has two rooftop decks with the same one and one a spot within the filler And then that spot won the Orioles So a nice little parlayer there Next is Salvador Perez 15 out of 15 that tech design That's Chad and the Royals. All right next up Is Hans Kraus rookie Otto for the Phillies The fighting Phillies fighting their way to the next round Steven Steven C with that one and there's Rodolfo Castro rookie Otto for the Pirates That's gonna be for Scott Some cool penmanship there from Rodolfo Castro Scott. You're on the board with the buckos And there's Fernando at the East jr 33 out of 99 That'll be for the Padres. That's for Chad. Maybe work his way back into the good graces of Padres fans And baseball fans next year at some point And the 10th and final box clearly authentic. It's gonna be Juan Gan Gonzalez 56 out of 75 Rangers Diego with the Rangers Excellent mustache for Juan Gonzalez. All right Now Tier one two autographs one relic part inside sometimes we'll get an extra relic. We'll get a fourth hit On all the time But sometimes Remember the second half in the store right now jassescasebrakes.com You can get your team straight up. No filler for now So they didn't see that fourth extra card right there. He's that blank side to hide the hits All right first relic. Nelson Cruz 240 out of 3 99 Ray's left to retool a little bit for next year. That's David and Tampa Bay Yoan Moncada 293 out of 249 game used relic the white sock pinstripe there That is for nick s and chicago A new manager search for the white socks The white socks look great on paper, but Maybe new manager will help next year autos Corbin burns 299 Maybe another team that feels like that maybe under achieved a little bit Brew crew nico picked up the brewers straight up And edgar martinez for the mariners It's the last time The mariners are in the playoffs now they're backing it again won their wild card series and Seattle's at houston, right? I think that's going to be a pretty good matchup It's like logan gilbert taking the mound for the mariners In houston facing justin verlander They've got the astros at a heavy favorite minus 2 4 minus 2 14 If you're into that sort of thing I know I feel like there's always good values on underdogs in uh In baseball Especially with the way the mariners have been able to To fight back at end at the end of games Giancarlo stand in 69 out of 3 99 they're hosting the guardians Garrett Cole on the mound another heavy favorite minus 201 But I feel like the guardians play uh Play the yankees pretty well this goes to nick s bought the yankee straight up And here's brice harper in the fillies 180 out at 1 99 Steven c with the fight in fills Noel would give his left. Well, you know what thanks for keeping it family friendly on this family friendly show Noel especially since your comments are now Are now recorded Um to see the mariners win That'd be pretty cool Steven with the fillies There's yolpozo yolpozo 173 out of 2 99 that'll be for diego and the rangers And we've got Yes, carlia stremsky 30 out of 75 nice one for the red sox nick s The spot that he won Yeah mariners are Plus 195 Fillies are putting out rangers suarez Against uh in atlanta against max freed They're also heavy favorites at minus 190 dodgers also heavy favorites minus 215 julio urias They're hosting the padres Everything flew into la early this morning from new york Somebody must have arrived in the morning and read I Got a chance to relax in la today. We'll be playing Tomorrow night. That's the late game. So it's uh, so tomorrow is going to be a pretty fun baseball day Hopefully we'll be able to do that 21 box um postseason mixer while we're doing this there's blake snell 150 out of 249 that's for the padres chad with the fryers Should be a fun series There's michael brandley astros That will be for nick s another playoff team hidden So fillies at braves is the early game tomorrow Mariners at astros the second game cleveland at new york Uh, the third game in the late game will be padres at dodgers Playoff baseball back There's ernie clement 193 out of 299 for the guardians another playoff team hitting diego cleveland. This is for you And diego Who bought the guardian strato? All right three to go Anyone have any thoughts on the game tomorrow? Who's gonna win? I may just go all You know if i'm if i'm pick i may just go all underdogs tomorrow. I have to do a little more research tonight, but I think all my picks might be the road dogs Especially how heavy those favorites are it's maybe not worth and worth it to lay down that That big number lay that big number. It's tough eight out of 25 logan web Giants nico we got riley adams 260 out of 299 for the nationals craig with naddies We've got a giant 225 out of 299 wan marshall nico san francisco baseball giants last two boxes Are you out? Check out these two boxes. Yeah, let's see what we got. Good luck everybody The second half in the store jaspy's case breaks.com tier one definitely has bat knob bat barrel opportunities one-of-one chances really nice hits Think uh, I think mike got that Part of his hall of fame swag for being a hall of famer Hall of famer mike jaspy. Well rex just cursed the guardians Yeah, rex said it would be kind of cool to see the guardians win First world series after a name-chame and they have the current longest streak for no world series win rex also supported the idea of wainwright and yachty and poohols going out with a world series win and then they got Now rex is like 95 out of 349 tay oscar hernan does Rex's a cubs guy too. Can you believe it? It's disgusting that that'd be that'd be like me saying boy I'd love to see buster pose go out with a world series That's what rex claims, but I don't know if I don't know if that's true. Is he that clever Kenneth the lujay's He's a rich jersey that broken clock being twice. That's true. That's true 24 by 25 max kepler copper Coppering golden golden. It's technically I think it's technically golden. Yeah golden got a graph twins that'll be for Diego San Diego The translation was lost over time, but scholars believe that san diego means the whales It's true. You know that especially area I think it does do 257 out of 299 jake burger, you know the little cheddar cheese on that burger Maybe little iceberg iceberg lettuce a slice of tomato.  Oh jack cheese on the burger. All right. All right No soft cheese no feta No, no, no, that's first that's for salads. I like my lettuce shredded Really a shredded lettuce man with uh interesting and always season your vegetables Even if they're on a sandwich or a burger a little salt and pepper a little almojo goes a long way That's just too much. I feel like the burger itself should be seasoned and that and then the the cool crisp iceberg lettuce That crunch and then the the the season meat onions pickles pickles anymore season obviously, but onions raw onions, you know salt pack Walker Bueller 68 or 249 are the Dodgers chris jaspy in on Jacob de groom because this guy's not there a lot of money coming off the books He's 30. Is he 34? I don't think he's no spring chicken All right, I know you got maybe either 32 or 34. You got called up late. Yeah, he's 34. He turns 35 in june I didn't realize he was that old. I don't think he's a Dodgers guy. No, actually no It's the reason why he didn't sign Max Scherzer, and by the way I was like saying all that trash he talked about not managing his arm, which like it was dead arm How does that start the other day max? I know not great. Not great. Hey 007 Out of 299 guess who this is logan gilbert game one starter tomorrow. How about that? Yeah, he had a heck of a sophomore season I've been kind of high on logan gilbert.  I think he got some he got a cup of coffee this year, right George curvy have a nice rookie season. They have castillo for another year. Robbie ray. He's not going to be 2021 No, but but I mean he's still he's still lefty gobbled up in ink striking out dudes It's a good picture. It's a good time to get this logan gilbert That's for kenneth with the mariners, especially if he has a nice start tomorrow If you sorry, and if you're a houston fan If you're rooting for the astros and not the mariners here if you're not an astros fan What's wrong with you? And the final hit is a bow jackson.  All right, let's flip back over to this screen right here. Let's make that a little bit larger there Let's alphabetize by your first names here And if you got an even number of teams half of that number becomes entries for break credit So chad you won four teams That's two entries in the promo Daniel bought two teams. That's an entry in the promo david one two. That's a spot in the promo. Diego bought eight That's four in the promo. Good odds there. Diego. Good luck. Kenneth won two teams gets an entry Nick s bought a couple one some That's six total. That's three entries Nico got two. That's an entry And steven c got two That's an entry All right, so I got chad Daniel yes, david. Yes, Diego Ken Nico Let's see. All right. Remember second half in the store. We can run this back. Pick your teams All right, let's copy those names new dice new list And top two will win two hundred and fifty dollars of break credit each. It doesn't expire You don't have to use all of it at once. It saves your balance Just keep that in mind and check your emails for it after 10 five into five 10 the hard way top two after 10 Good luck everybody one two three five six nine And fingers crossed 10th and final time top two Nico and steven Congrats you 250 dollars of break credit each going your way I don't think I called your names very often in these breaks either So, uh, so that break credit should help. Thanks everybody. I'm joe for jaspyscasebrakes.com I'll see you for that second half. Bye. Bye


So I think we have 30 minutes to do this so I think we're gonna jump in I got like five questions And maybe more and I don't think I want to waste a lot of time one thing You know when we do these kinds of interviews a lot of times they're saying oh the army is at a critical moment it actually is because a lot's going on and You know, we'll see what a Trump Pentagon will mean for the army, but one particular question. I wanted to ask was The campaign has signaled that it doesn't like some of the their word social engineering in the military including women in combat transgender wondering what your thoughts are on Is it too late to roll back that? Some of those initiatives are they too far along? What are your thoughts first? Let me say I've never liked the word engineering when it comes to this because I think what we're doing is just building off of shared values of opportunity and inclusion And and the path to do this over decades going back to integration of the military Has never been purely linear, but I do think It's very hard to roll things back completely for a number of reasons society is changing very quickly The soldiers we assess just in the active component 60,000 soldiers a year They come out of a different society than I came out of and And so that that's one aspect the other aspect is when you're leaning in or looking into making some of these changes The debate is whether or not you're going to let someone wear a uniform Which is a very different discussion than whether or not you're going to make someone take the uniform off so I do think This this tide is moving forward and is is Difficult to stop that doesn't mean it doesn't have pivots and ups and downs and so forth But I feel pretty confident that that these changes in this direction is here to stay is the is the Transgender initiative More in its nancy and so maybe easier because these guys Potentially come in and say we've got a new new plan, which is to undo the old plan Sure, anything that is new a story You know in the early stages is is easier to unwind or to change But a couple of things one in many ways these policies were coming up with are just catching up with what's already happening out there we already have transgender solar sailors airmen and Marine and Trying to find a way for them to serve with dignity and for commanders to give some help and guidance to commanders Who were being left to figure this out on their own was really the the goal we had in mind here And it is all the science now the medical community treats it as a medical issue Which is how we're treating it and that that's a hard thing to walk back and the other thing is the numbers just aren't what People thought they would be or expected they would be and I think what we see over and over and over again Certainly in this administration starting with don't ask don't tell is these implementations are actually pretty smooth one last one on that is it is it possible and this is kind of above your pay grade, but you know the army and the Navy and the Air Force on women in combat had kind of a general view of it Marines may be a different Can you imagine? Some services going one way and other services emphasizing enough going a different direction Yeah, I think in almost anything you can imagine that that's one of the One of those questions that's always out there. Why do we have four services? Why don't we just consolidate to one? Well, each of those four services is pretty immense The army is the second largest employer in the country behind Walmart And part of the value and the beauty of that is that you have some competition and some different ways of doing things And you can learn from each other and so in all areas you see the services doing a little bit differently, okay? As far as I know Trump campaign where the transition folks have not made an appointment with the Pentagon to talk transition yet Let alone the army, but just kind of generally speaking and as I was saying to you backstage like There might have been an expectation that you might have stayed on a little bit longer You had things you wanted to do things that are probably not yet complete so kind of Two questions in one like what's your message to the transition people when you do talk to them? And what are the things that you had hope to do that you're? Hoping they will continue Gordon. You always have at least two questions in one night So the the I've worked kind of on the incoming side of two transitions Clinton in 93 and then was early in an Obama and and In as is the case in every transition the main message is we're here And we will do whatever is necessary to make this as smooth as possible We have a hundred thousand soldiers outside of the country right now in a hundred and fifty different countries In combat in harm's way there can be no gaps There can be no seams and that is the first and only priority and we're completely dedicated to that and whenever they're ready to talk We will do whatever They need us to do to make the transition as smooth and fulsome as possible And as for me thinking about whether I was going to stay or not stay I've done this long enough to know that what you hear in November isn't necessarily what you see in January And so I wasn't you know I had thoughts about what I would do if there was a longer tenure But I was also approaching the year to make sure that Whatever I was working on at least would leave some modules that were complete and can be built on or had What I what I call escape velocity it would it would continue after I left But I think what I was really focused on and will and we'll talk to whomever replaces me If he or she wants to talk about that Was further defining what the army needs in the future the army in the last 15 years really has been pulled into the day day after day after day after day Stripped a lot of investment out of the future even a lot of the intellectual process of thinking about the future so really with General Milley focused on what the army needs in the future and then Trying to make sure the army was structured to deliver on that We have a lot of creativity a lot of innovation in soldiers in our civilians And there are a lot of things the army does really well Despite the processes that we have in place and the barriers that we put in front of those innovative and creative people And so trying to help structure restructure the army For the way Innovation exists today and technology is fielded today so that we can deliver on that vision for the future We're kind of the two things that were really in the back of my mind and then thirdly because you're always thinking about your people really Wanting to think about behavioral health very differently the the army has done a lot to move behavioral health out into The operational level so that it's there if it's needed But I think we need to fundamentally shift the paradigm here Which is not there is help if you need it but of course you're going to need help and and special operations command is not a lot of work in this actually that that this should be an Expectation based on what we ask a soldier to do day after day after day after day deployment after deployment That it's not there if you need it You're gonna need it and it's a part of your program when you return when that de-stigmatizes it Yeah, nobody has to say like raise their hand and that's what And I had a question about special forces folks, but That community is a smaller community that Where they can kind of road test ideas and be kind of innovative when it comes to resilience and that kind of stuff How can you or how would you hope that maybe somebody would expand that model to the to big army? Well, there's a I think first of all it's expanding the model big army But it's also thinking about the the transition from one-year soldier to becoming a veteran because we are we've been focused on this for years now There's a lot of research underway. There are a lot of new programs in a way There's a lot more to do to understand PTS traumatic brain injury how you should treat it And and the veterans medical community Focuses on for a number of very valid reasons different things And so I think there's some things that we can do that the two medical communities Maybe to overlap on that transition a little bit more but in terms of Institutionalizing I mean special operations is smaller than than the army It's a lot bigger than it used to be and so they've shown you can scale some of these things and how they acquire weapons and what they do on behavioral health But you know the analogy is what what when we when we run a platform hard we bring it back we strip it down and you know put it back together and I think We need to think about our soldiers that way and special operations has been doing this for a while What is you know special operations forces have been kind of the force of choice certainly for this administration Over the years But a guy was telling us the other day that you know when we look at the Operations in Mosul and then soon kind of more expansively in Raca, you know that frankly some of these SF guys are You know stretched a little bit not stretched too thin the person was careful to say but But clearly like there's some where there Can you talk a little bit more about how to address those particularly in that community how I address that stretch Well, we you know army special forces and US special forces are an incredibly lethal power It's amazing what they can do it and what they do Many times a day day after day after day We are working them really hard army special operations command Is putting a plan together to has a plan together to to mitigate this which is a whole series of things behavioral health family programs? Really trying to focus on how we can sustain readiness because that's part of the problem when you're running this hard You're impacting your readiness and we need to keep doing more on that but part of what we've recognized too is as As amazingly capable as our special forces are in partnership with a lot of other countries special forces They can't do everything you need you need your full army and you need the the the joint force To have the total and long-standing effect that you want Do you see other kind of major muscle movements that need to kind of occur to You know assuming the new folks come in and say we want to use Continue to use these guys at this kind of rate like what kind of major muscle movements might you even recommend to the transition folks About what they should consider doing to kind of set those guys up. I think I I mean I I Support the the strategic plan the president has in place now I mean you can use special forces you can send them in night after night after night after night But you need something else to to clear ground and then something else all together to hold ground And so that's why for the second element you need your larger conventional army for the third element You need to be developed these other these countries that we're in need to be developing forces And so that's why a large a large percentage of of our effort and our people are in Afghanistan In Iraq helping build the police build the military because something other than us needs to hold whatever we've gained Um segue to a size question size always matters with the army in particular I Think you're going to 450 450 that for the in the active in the active side. Let's not forget our friends in the reserve and the other group But as you go to 450, I think that the kind of number that I think some army leaders had embraced was more like 490 Talk about, you know, and I should say new demands on the army in particular with the Contribution in NATO that starts I think this year about a thousand folks and then you've got a heavy brigade on rotation six to nine months Going in and out of Europe, but if you do if you count it differently It's really like a division because one's on the staff one's there and one's regenerating right so a lot of demand there So can you address, you know, kind of the size of the army and also? What's the analysis that's been done to kind of? Defend those numbers because I think that's one criticism that sometimes the army just as a let's do more numbers But the capabilities question is left aside. Well, there's a lot more than two questions in that one. I know right I'll come back so I'll start I start by saying when I when I moved to the army last year on my first tour I had a senior general asked me if the Secretary of Defense had sent me to the army with secret instructions to shrink it more because the army is You know sitting at the Navy table its chair sitting at the Air Force chair I was so the Navy the army always feared they were going to be a billpayer for some of these big technological investments of the future and That's not why Carter sent me to the army. He didn't talk about that at all He did say you have to get them away from talking about themselves as a number a and be 450 480 490 520 whatever it is is not the number because that's just the active component and we cannot do what we're asked to do if we Don't think as as fully as possible as a joint force all three of the components are important But you know the number by itself doesn't really tell you a lot There is a lot of analysis behind the number, but you can't there's no Precision to this because it's all about risk and so you can come up with these different numbers In the context of the joint force and then just describe what the risk is in certain scenarios You know, there's no question that that the army is being Worked hard the 450 target for the active component when when that was agreed to because all this takes time to implement into into achieve was before Russia was being so provocative On the on the border with Eastern Europe before China was getting More provocative in the in in the Pacific before we saw what ISIL was doing in Iraq and in Syria and as I said soldiers already in 150 countries a day as it is and so there's a bit of a stretch there but I but I caution people to get away from thinking about it just in terms of four structure numbers because If that's the focus to not be 450 but before 90 or 520 or whatever it is you could end up creating greater imbalance and actually lead to more hollowness because Those people cost something we need to make sure if we have that number that we also have the money to keep them trained and to equip them And then there's tail that goes into the future for a very long way So whatever your future modernization plans are there over a larger force So I think it's important that we We don't oversimplify these things about the number of soldiers the number of fighter squadrons the number of Carrier strike groups and think about the totality of what it means to have them and have them ready And as far as the analysis There is no shortage of analysis on any of these things We could all do a better job of communicating the analysis and what we're thinking in the context around it There's no question about that, but again This all then you can do all the analysis you want and it gets down to judgment calls and and conversations about risk And about balance art not science On yeah, and as a guy was saying to me yesterday You know it's unclear which Trump will show up to govern on one hand He's sent the message that there are inefficiencies even even in DoD on the other hand like probably wants to send You know more money toward toward the building. I There's no question. There are more efficiencies to be found in an organization as large as the Department of Defense That's never that process should never stop But I think We have sort of approached efficiencies over this administration of kind of identifying them and taking the money before we necessarily implement them And we haven't caught up so I I would caution about Banking on those dollars right away, but I absolutely would say don't ever stop looking for it's not just efficiencies in pulling money out of the system It is doing business differently, you know, we used to drive Technological change we think about stealth the internet GPS or what have you and in and we still we do incorporate and feel technology very well But in many ways we cannot keep up with what the private sector is doing how the private sector iterates on technology And so it's not just changing to get more money out of our system It's changing to partner better with places outside the Department of Defense that are doing things. We need to embrace I Had another question, but you've greased this kid so well for my next one that I think a previous I think senator Codman's here I think he's one of several people who have pushed the army on You know buying off-the-shelf stuff commercial stuff that's available and as you're familiar there was a ruling Against the army for essentially shutting out One particular tech firm who wants to do work now Palantir When you work for secretary Carter, certainly you're aware and you know Carter has pushed this idea of bridging the gap between the building and the and Silicon Valley I Know you're not going to comment about the legal issue But can you kind of talk more broadly about is the army open for business with tech firms? And is it willing to embrace off-the-shelf stuff? Which is cheaper typically cheap typically cheaper faster ready to go Yes, there is more work to be done, which was which I intended to be a real focus of mine General Milley's really You know interested in that as well. I believe the faster you get things into the hand of soldiers the better Because we'll they figure out how to learn how to adjust how to use things differently and give us feedback very quickly Some of that's off the shelf some of that stuff. We sort we already have that we can modify It's not just Silicon Valley First of all technology is not just Silicon Valley. It's not just Silicon Valley There's a lot of creativity innovation in our defense industrial base We need to find better ways to partner with them as well We need to open ourselves up to all of this because Technology is being developed differently and in faster and faster and faster cycles right now And we need to change to be able to get that in and feel that getting the hands of soldiers faster So absolutely You know one of the one of the many nice things about having lawyers around as they tell you a whole bunch of things You can't talk about so they shield you a little bit So I can't I cannot talk about that case Right, but I hope we use it and and all other cases as a learning opportunity for how could we do things differently? Because I do think in in some ways. We're speaking different languages now We need to figure out how to collaborate better because it seems you know We can say here and I think anybody would agree with kind of what you're saying But the the military bureaucracy can get in the way and it doesn't always get the memo about what leaders Want to achieve it's not just it's some you know I keep going back to something Secretary Gates said when he had his farewell address and he said bureaucrats and bureaucracies are different things And nobody suffers the bureaucracy more than the bureaucrats have to fight it every day These are rules and procedures Bureaucracies are additive something bad and they're always steering by the way something bad goes wrong You had a process to prevent it from happening again And you layer and you layer and you layer and you layer and you layer and you create a whole bunch of people whose job is to Say no and we've got a lot of people who do want to get to yes And that's one reason we set the rapid capabilities office to try and field some of this stuff faster And it's not meant to be a workaround for the entire acquisition process We need to keep reforming that and making it more agile and responsive My kind of ink, but what I call an ink block question kind of back to where I was going to go is you know because we are Where we are in terms of military strategy overseas and what do you say 150 countries the armies deployed But not in huge numbers right and then people associate the army with huge numbers generally And so my question really is is is there? Does the army suffer from some kind of identity complex? Because we're not doing what people generally associate the army with doing major occupations meet big forces You know in some ways that's that's a question. You could have asked me in any of the jobs I had in any of the military departments I do think that before you work for the army before I work for the army Yeah, any jobs in the army as well the They the services do try to do tend to focus on that tip of the spear kinetic thing Because that's what they identify with and and what I found moving from service to service and no more more so than with the army Are all these unbelievable other capabilities they bring to that are required for the army to be the best army The world is ever known but that they bring to the joint fight and that they bring to the coalition fight as well And it's it's an impressive Group of and I don't even like the word enablers because I feel like it Doesn't give enough power to what that part of the army is and and we have thinned that out a lot to support that tip And so that's that's been a struggle But I don't think they have I don't see any sort of existential issue about being spread thin in small groups over 150 countries it does impact how we're structured We've proposed Setting up training advice and assist brigades that are smaller than a regular brigade because right now when we create those we We take the leadership out and a part of the brigade we separate them and it impacts our readiness to Recognize that we are fighting in many different ways and working in many different ways around the world Are you is there a readiness argument to be made for? I think everybody's kind of lining up to to make a big argument And you it potentially won't be here to make that argument with the new folks and on the hill, but is there a What's what's your kind of a general assessment of army readiness right now? We're on a path to get to our readiness goals really in this in this fight up but You know they say in the military no plan survives first contact with the enemy and the enemy gets a vote And we don't know what we're going to be asked to do in the next five years and and I I'm cognizant of time. I want to just say one thing I get asked all the time What's the biggest threat to the army and there's two ways to think about that? What is the geopolitically?  So that's where is it Russia is it North Korea is it ISIL? but I I think of that question in terms of my 2010 responsibilities as secretary of the army in the Department of the army in many ways I'm a CEO of a business unit and the biggest threat to the army that I that I've been working through and to the Air Force into the Navy Marine Corps has been the budget instability and so all these issues that we're talking about readiness technology modernization force deployment if we had some More stable baseline which to plan we could I think get to a lot of our goals faster Okay, I know we had two seconds left, but I know a guy Just quick. What are your plans assuming that things end for you next day next week? I My my model in all this was was the man who hired me out of college One of the two Rudy de Leon who in the Clinton administration at four jobs and Ended his deputy secretary and I'm like well, that's what I want to in the bomb administration. I want to stay eight years If I can have four jobs, I'd be great. I had nine so I overshot a little bit and Pace myself but be tired at the end so that I I know I've Contributed and so my plan was always to be someplace really warm come January 21st And that's yeah, I we are committed to doing whatever the incoming team needs for us to do on the transition And some people stay and some people don't But the the plan is to take a break and and figure it out later. I this is this is an Amazing job. It's a huge honor I and it's a struggle every day But a but a joy every day and I don't want to waste any ergs Thinking about what's next and getting caught in the mob the transition because I've been through a few of those and they the anxiety grows in A way that's suffocating and so I'm I'm gonna just hold the stick until they drag me out of the building And then I'll go and rest for a while. We may see you again. It's Dr. Fanny. Thanks so much


Hello and welcome to the session. In this session, we discuss the following question which says, find the power set of the following set. Set A is equal to x such that x is equal to y minus 1, where y belongs to w and y is less than 3. First let us define the power set. The set of all possible of a set A is called the power set of A and it is denoted by P of A. And if we have that a set A contains n elements, then its power set contains 2 raised to the power n elements. This is the key idea that we use for this question. Let us proceed with the solution now. We have a set A equal to x such that x is equal to y minus 1, where y belongs to w, that is the set of whole numbers and y is less than 3. Now that we have y belongs to the set of whole numbers and y is less than 3, therefore y takes the value 0, 1 and 2. Now we have x is equal to y minus 1. So now we will find the values of x for the values for y as 0, 1 and 2. So for y equal to 0, 1 and 2, we obtain x as 0 minus 1, that is minus 1, 1 minus 1, that is 0, 2 minus 1, that is 1. So we get for y equal to 0, 1 and 2, we obtain the values of x as minus 1, 0, 1. Thus we get set A is equal to the set containing the elements minus 1, 0, 1. To find the power set of set A, we have to find all the possible subsets of this set and the set of all possible subsets of this set would be the power set of A. Now subsets of A are phi, singleton minus 1, singleton 0, singleton 1, set containing the elements minus 1, 0, set containing the elements minus 1, 1, set containing the elements 0, 1 and the set containing the elements minus 1, 0, 1, that is the set A itself. So these are the subsets of set A. Now the power set of A is denoted by P of A is the set of all possible subsets of set A, that is set containing the elements phi, singleton minus 1, singleton 0, singleton 1, set containing the elements minus 1, 0, set with elements minus 1, 1, set with elements 0, 1, set with elements minus 1, 0, 1. So this is the power set of set A. Now as you can see that the set A has 3 elements, so power set of A would have 2 raised to the power 3, that is 8 elements and these are the 8 elements in the power set of A. Thus given the set A, we have found the power set of A. So this completes the session. Hope you have understood the solution of this question.


Everybody forced tear ripping two boxes of 2022 absolute football hobby for a wansi. Thanks Wanda. Good luck Let's find a kaboom It's been too long since I pulled a kaboom right Robbie Anderson drew lock tools of the trade Do a relic Gabriel Davis 78 of 199 rookie out of Brian Robinson jr. Two of five Two out of five on the Brian Robinson. Nice low number there in a rookie in Hutchinson 72 of 100 Alan and Watson We got magi Sanders 226 of 275 Kenneth Walker the third on the triple relic rookie auto 33 of 299 and draft diamonds rendez-barber numbered 40 of 50 Ryan and Jacobs James and Williams in Hutchinson dual rookie Jersey 64 of 99 a Kalei is Campbell legends auto 4 of 25 and Lamar Jackson numbered 53 of 75 and box to Bosa Hawkinson we got Lewis sign 15 of 275 Tyquan Thornton rookie triple relic auto 93 of 99 and Stargazing Tom Brady 11 of 100 on kaboom. Come on. We got Cooper Cup patch card 39 of 49 Showstopper rookie auto Jelani Woods 24 of 25 and Deandre Hopkins numbered 8 of 75 Eckler Montgomery, we've got tools of the trade Alexander Madison dual relic 43 of 49 Historical duals auto flipper Anderson Henry Ellard 14 of 199 and a Damian Pierce rookie numbered 58 of 100 Appreciate the order one. We'll get these cards right out to you


Welcome back folks, Dao. Dao Industries right now, up to 219, you get the Nasdaq down 30, S&Ps are up three and a half. Our guest today folks is Bill Letera. Bill is the chairman and CEO of iSocrates. Now check this out, this is pretty cool. Bill's a former Wall Street head of research and portfolio manager. In the 1990s folks, bottom line, billstartedart.com. When Jeff Bezos was out there with Amazon and the bottom line, Getty Images bought it. Bill's got a new company, it's iSocrates. Bill, welcome to TFNN. Thank you very much, how are you doing? I'm doing great, man, yourself? Good, a little bit hot and humid here in Florida, but we're doing okay. And we're glad you're in St. Pete, man. We're definitely glad you're in St. Pete. So hey, so tell us exactly what this company is. Yeah, you bet, happy to do so. So iSocrates began life a little over six years ago. We work in the digital media and marketing industry, started out doing some pretty high-end consulting work around strategy and operations. And all about 90 minutes into our first major consulting assignment became pretty clear that companies were having difficulty executing with all these literally thousands of platforms that they have to work with and didn't take long for us to figure out, actually, there's an opportunity here. In addition to the consulting, maybe we should be doing hands-on execution, letting them outsource to us and where we can run mission-critical applications for them in media and marketing. Publishers, marketers, agencies, different data and tech companies. And we're headquartered in St. Pete, but it's a global business today and we've got clients around the world. We've taken those services that we've been providing and learned where the gaps are in the technology. And so like many companies, we are evolving a SaaS solution to go with our services. So like if a large company wants to advertise, would they basically hire in you so that you could figure out what these different platforms are doing and just what type of return on investment they could get? Is that what we're talking? Yeah, and we're hands-on keyboard, actually planning and buying or in the case of media companies selling, doing the monetization for them. Okay. Reporting in analytics, data science, all that. Runs 24 hours a day, seven days a week in our operations. Yeah, you know, this business has come so far. It's really intriguing and large companies, you'd be really surprised folks, large companies spending a huge amount of money that they really don't understand it kind of how this works. And the advertising companies have everything to gain and zero to lose by just saying, okay, you got this many hits. I mean, years ago, Bill, I used to own radio stations and Abitran used to drive me crazy because I'd be saying to myself, this is amazing that a large company would just buy this buy based on a certain amount. But I've seen it and where we are right now. I still see it. I actually got a spreadsheet because we do a lot with a couple of big advertising firms and they sent me a spreadsheet one day that they never should have sent me and it was like, I thought I was seeing things, man. I mean, and what it was, it was supposedly the number of hits that these large companies were getting. But if anyone ever drilled down to it, they'd realize that this is not what they need and more than likely that's what you're digging into. Yeah, I mean, the reality is that not only is there a lot of waste in the system, but increasingly, particularly in this economy, it's hard to find the staff that sophisticated understands that's affordable and available. Yes. We've been able to fill that hole fortunately. So businesses has been very good and it has allowed us to be able to reinvest in building our own technology. So after having been a professional services company for so long, we have begun to build Madtech BI, which is our SaaS platform and very fortunately for us, all of our services clients have taken the platform and now other new clients who've never been services clients are taking it. So we're benefiting from a lot of learning here and on our way to building a nice growth company. It's a great name, man, Madtech. That's definitely a great name. So what level company-wise, like with gross numbers, would be your clients? Is it 5 million, 10 million, 20 million, 1 million? Where are we at in that? Typically when they're on the buy side, it's gonna be companies that have over a million dollars a year or over $100,000 a month to spend on digital media and marketing. But since you're from the radio business, you might remember Enercom. Oh yeah. Enercom, CBS News, they are a long-time customer for us. Gosh, probably going on almost five years already. Yeah. Do work for them and a lot of other big businesses that you know. But it can be a smaller company. I think the big thing is if you're looking for full transparency, lower costs, very significant accountability, we're the right people to go with. If you're looking for a full service agency and you really don't wanna know more about what's going on, we're probably not the right fit. All right. Where do you feel the most amount of advertising fraud is in the business? That's it. I know. I'm not putting you on the spot because I know it's not close to what you do, okay? But we both know that it's like, oh my God, it's like sometimes I'm trying to figure it out myself. Do you know what I mean? Absolutely. So first of all, there's fraud in many ways. I'm a graduate school professor in this subject matter and I definitely share this with my students every semester. I love it. This is definitely a business where caveat emptor. You wanna trust but verify every step along the way throughout the ecosystem because there's plenty of fraud. I think that probably the biggest fraud from a dollar standpoint is gonna be in the ad inventory itself. What advertisers believe they're buying but actually what is not being delivered. But it can also happen where the publishers are getting short change themselves. Overall, what I would suggest is I'm not familiar with another commodity industry where fraud is a bigger factor. And I'm gonna call it flat out. This is a commodity industry. These are companies that are being bought and sold and increasingly in auction-based transactions just like financial services evolve to basically from May 1st of 1975 with the onset of the negotiated commissions of the New York Stock Exchange. And that was really the beginning of loosening up of the market. In the same way, electronic real-time auctions have come to advertising inventory and we're seeing the same kinds of challenges but in a very short, much more compressed period of time. And in that activity, with hundreds of billions of dollars being transacted in auctions going forward, there's plenty of fraud. So this is not a place for the timid and it requires constant observation and clients, they need to know what's going on. And you know what's so cool when clients do know what's going on, folks, okay? It's awesome because if they're educated, I can tell you from me trying to sell them advertising it's much easier. When they're not, it's a real problem, man. I mean, because they can say to me, I've got this many, what I can't say to them, hey, man, you better go check it again because I know you don't have this many. Meaning, and that conversation is very hard when I'm on the selling end of it, do you know what I mean? Yeah, yeah. I don't think just so it's what it's worth, I don't think it's just that it's gonna be the unification of all those things because there's a lot of platforms as well. This is a complex industry. No, there's no doubt. Well, listen, this has been a pleasure. Appreciate the update, appreciate the education. I'd love to have you on again, Bill. Great, thanks so much. Thank you. Have a great one, have a safe one. Yeah, good place. Stay right there, folks, come right back.


Hi, I'm Garth Russell. Over the years I've had the pleasure of telling extraordinary stories about Nova Casterians. None more inspiring than this life long, if unlikely, friendship between a young Australian family and an ambitious Chinese boy. Ken and Judy Morley raised their young family in Newcastle in the 1970s and 80s. Ken was concerned about matters such as Aboriginal rights, peace and disarmament, environmental protection and renewable energy. Issues that were yet to have their day, but now feature prominently in national and international debate. In 1980, when the People's Republic of China was just opening up to the West, Ken took his family on an Australia-China friendship society tour across China, where they met Jack Ma, who was then an enterprising teenager who would ride his bike around the West Lake to practice speaking English with tourists. Jack first struck up a conversation with Ken's older son, David. A long friendship grew between Jack and the Morley family. Jack would write letters to David and Ken, and Ken would correct the letters to improve Jack's English. Before Jack's first visit to Australia in 1985, Ken jumped through many diplomatic hoops to host a Chinese national. Jack now considers that visit to Newcastle to be a turning point in his life. He discovered that there was an entire world beyond China. The Morley and Ma families became one, with Ken and his family visiting China frequently. Ken challenged Jack with rigorous debate on values and ideals. In 1999, Jack founded Alibaba.com, which grew into China's largest e-commerce company, with hundreds of millions of users. Jack is now recognised internationally as both a leading entrepreneur and philanthropist. The profound and enduring friendship between Jack and Ken remains strong, up until Ken's death in 2004. The bond, now spanning multiple generations, remains incredibly special to both the Morley and Ma families. In 2014, Jack established the Jack Ma Foundation. Its mission is to work towards a world of bluer skies, cleaner water, healthier communities and more open thinking. The Jack Ma Foundation and the University of Newcastle are embarking on a lasting partnership to honour this extraordinary friendship by creating a unique and impactful scholarship program that will inspire, educate and cultivate the next generation of globally aware and socially conscious Australian leaders for many years to come.


So, we have been discussing iterative methods for solving linear algebraic equations and then after discussing algorithms, we started looking at convergence of these methods. We also derived necessary and sufficient condition for convergence. Necessary and sufficient condition for convergence was spectral radius of matrix S inverse T should be inside unit circle and we said that well it definitely gives us lot of insight into what happens, but we need something even simpler to calculate and we have started looking at matrix norms to come up with something that is easier to compute in terms of assessing whether a matrix whether A matrix which we used to formulate the iterations, you can apply some quick test to come up with convergence analysis. So, induced matrix norm was defined as norm of A, let us talk about norm in which the norm induced by identical norm on both range and domain space. So, this is max x not equal to 0 norm A x by norm x and the way I try to explain this was this is something like amplification power of a matrix other way of writing the same thing. So, it is 2 norm here and 2 norm if I want 1 norm it will be A 1 norm 1 norm and 1 norm and so on. This is called induced norm because this is induced by the norm defined on the range and a domain space that is why it is called induced norm. Other way of writing this is this is a generic thing. So, I could actually I need not write 2, but we were discussing 2 norm at the end of the last class. So, this inequality can also be written like this norm norm A x divided by norm x is less than or equal to. So, norm A is in some sense a bound upper bound on this ratio of course, this is always greater than 0 for x is not equal to 0 this is always greater than 0 or it could be always greater than or equal to 0. Suppose A is a matrix which is rank deficient and A x is in the null space then A x will be 0, but denominator is not 0. So, this can be greater than or equal to 0 and this is an upper bound norm is an upper bound maximum of this ratio or maximum of the gain maximum of the amplifying power of the matrix whatever way you want to view it. Now, in particular we started discussing 2 norm. So, in 2 norm I just started by putting A x 2 square divided by x 2 square which I wrote as x transpose A transpose A x divided by x transpose x.  So, this is ratio of 2 scalars.  So, this one we showed that we made additional assumption that matrix A, matrix A transpose A this is a symmetric matrix positive definite matrix. We made one more assumption that A transpose A has linearly independent Eigen values. So, we wrote this matrix A transpose A as psi lambda psi inverse actually what I argued was that since this is a symmetric positive definite matrix the Eigen vectors are orthogonal. So, which is same as psi lambda psi transpose where lambda is a diagonal matrix with Eigen values of A transpose A appearing on diagonal on main diagonal and this matrix psi is nothing, but matrix formed by keeping Eigen vectors of A transpose A next to each other. So, I am just keeping Eigen vectors of A next to each other this is a n cross n matrix not only invertible it is a orthonormal matrix which means psi transpose psi is equal to psi transpose is equal to I special property of this matrix and then I transform this ratio using this I transform this ratio to something very interesting. So, I wrote this x transpose A transpose A x as x transpose psi lambda psi transpose x psi psi transpose x and defining y to be psi transpose x we wrote this ratio defining this new vector we defined y or we defined z we defined z. We wrote this ratio we wrote this ratio to be lambda 1 z 1 square plus lambda 2 z 2 square plus lambda n z n square. So, you can see that this is a ratio of 2 positive quantities is a ratio of 2 positive quantities because Eigen values of A transpose A are always positive.  So, this is ratio of 2 positive quantities. Now, if we order the Eigen values as all the Eigen values are real positive for positive definite matrix and then if I say that lambda 1 is greater than or equal to lambda 2 is greater than 0. If this holds if this holds well there is an implicit assumption when we started all this analysis that A is full rank that because we are assuming that when we are solving this we are talking about a full rank matrix. So, that is why you get minimum Eigen value greater than 0 otherwise you may have possibility some Eigen some Eigen values will be equal to 0. Now, if I number my Eigen values such that lambda 1 corresponds to the largest numbering is depends upon me what I call 1 and what I call 2 matrix as Eigen values they do not come numbered we number them. So, I am numbering lambda 1 to be the largest ok. So, it is very easy to see that this ratio will always be less than or equal to lambda 1 z 1 square plus lambda 1 z 2 square plus lambda 1 z n square upon z 1 square up to z n square is everyone with me on this I am just replacing lambda 2 by lambda 1 lambda 3 by lambda 1 lambda 4 by lambda 1 ok lambda 1 is the largest magnitude Eigen value of A transpose A ok. So, I am allowed to do this denominator is same numerator I am replacing by a larger value for every term by term ok very easy to see that you can take lambda common ok. So, this ratio is independent of this is lambda 1 ok. So, this this ratio can never exceed lambda 1 this ratio can never exceed lambda 1 very very nice property ok what did we start with we started with we started with this we wanted to find out maximum of this ratio what is the maximum of this ratio lambda 1 ok this ratio can never exceed Eigen value of maximum magnitude Eigen value of A transpose A ok. So, what is 2 norm what is 2 norm of matrix A lambda 1 this is the upper bound and you can show that this upper bound is attained when Eigen vector is aligned in a particular direction ok what happens when x is equal to V 1 when will when will this be an equality for which direction V 1 transpose A transpose A V 1 upon V 1 transpose V 1 ok, but V 1 can be chosen to be orthonormal. So, V 1 transpose V 1 will be unity ok. So, this will be V 1 transpose lambda 1 V 1 upon V 1 transpose even if it is not chosen orthogonal even then this ratio will be equal to lambda 1 ok. So, when x is aligned along x is aligned along the direction which corresponds to the Eigen vector associated with lambda 1 Eigen vector of A transpose A not Eigen vector of A Eigen vector is defined for a square matrix ok Eigen vector is defined for a square matrix A transpose A is a square matrix A in general need not be a square matrix, but A transpose is always a square matrix. So, of course, right now we are dealing with square matrices we are dealing with square matrices. So, there is no question of non-square matrices we are dealing with solving A x equal to B where A is square. In fact, we also are have a problem where A is invertible otherwise. So, this ratio is becomes equality when x is aligned along Eigen vector of A transpose A first Eigen vector first in the sense that correspond to the maximum magnitude Eigen value. So, this is equality, but this ratio for any other x is not equality for any other x this ratio will not be equality this ratio will be smaller that is why the maximum amplification power of a matrix using two norm is given by this ok. So, I think I have this picture somewhere drawn for a two dimensional case actually when you have defined this z is equal to psi transpose psi transpose x you have actually defined a transformation which is rotation ok. So, suppose this is your this is your x y then this is your z. So, this is so this is x 1 x 2 this is let us say this is x 1 x 2 and this is this will be your z 1 z 2 coordinate space. So, this is the invertible transformation you can go from one to other ok. So, actually you are just rotating your coordinate axis when you are multiplying ok and what happens in the rotated coordinate axis in the rotated coordinate axis this x transpose if you draw this x transpose A transpose A into x that is which is same as if you draw this inside a region where norm x is less than 1 we also had a other interpretation of norm right. Remember we did one more interpretation of norm max of A x cap where x cap is a straight to unit circle ok. If you draw that you know locus of points then you will see here that this actually corresponds to a ellipse actually corresponds to an ellipse ok. And this coordinates will be nothing, but this will be v 1 and this will sorry this will be v 1 have drawn I think wrong the ellipse will be like this the ellipse will be like this this z 1 will be actually aligned along your direction v 1 z 2 will be aligned along direction v 2 ok. It will be an ellipse this ellipse will be major axis will be along the along the Eigen vector corresponds to maximum magnitude Eigen value minor axis will be along the direction which is smallest Eigen value all other axis are in between. So, that you can. So, it will be an ellipsoid in three dimension it will be a ellipsoid in n dimension depending upon what kind of matrix we are looking at. So, I have drawn a picture in somewhere in the you can have a look at it, but again the problem with this two norm is you have to find out Eigen value of A transpose A if A is large does not help us ok. So, we could actually we could actually come up with a criteria which says that convergence will occur if two norm with two norm is nothing but Eigen value of A transpose A which is strictly less than 1, but as I said does not really help. So, actually this what we have found here is ratio of the squares it is square of this by square of this. So, what is the what is the two norm? So, two norm it turns out is that is square root of lambda 1 that is square root of lambda max of A transpose A that ratio that ratio is for square of see this ratio was found for A 2 square ok A 2 square is equal to lambda 1. So, what is A 2 square root of lambda 1 and this will always be positive because A transpose A will always have positive Eigen values ok. So, this number will be positive ok. In fact, Eigen values of A transpose A are called as singular values of A and square root of the maximum magnitude singular value that is what is what do you expect when A is symmetric there is a problem in the problem sheet. If A is symmetric then it will be A transpose is equal to A. So, A transpose A will be equal to A square. What is the relationship of Eigen value of A and A square? Square it is very easy to show that if lambda is Eigen value of A lambda square is Eigen value of A square ok. So, if A is symmetric A transpose A will be and if A is symmetric positive definite then it is much easier A is symmetric positive definite then it is lambda max of A square ok lambda of max of A square is lambda square of A and you can deduce for a symmetric positive definite matrix just look at it is Eigen value and then you can maximum Eigen value will directly give you the norm, but then this norm is again inconvenient because you have to compute Eigen value ok. Now, I am going to state two other norms without deriving the derivations to some extent required for these norms are there in the as a part of your exercises. So, you should look at you should try to work out and then you can see whether you are able to derive that or you end up into some difficulties. I am just going to write these final statements for other norms ok. So, now even two norm though it has some nice geometric interpretation it is not quite convenient for me for computing. So, I am going to talk about one norm. So, one norm is nothing but is one norm is nothing but max over this one norm is nothing but max over this ok well one small correction I realize I had made a small mistake here when I wrote the earlier expression I will just correct it. So, it is not norm 2 square is equal to this equal to max x not equal to 0. So, earlier when I started I had forgotten this max operator is there without max operator you cannot proceed and then I have simplified this quantity ok. So, moving on to one norm. So, this one norm is induced by one norm on the range space and one norm on the domain space ok one norm on the domain space maximum of this ratio. Now, you can show that one norm. So, what is one norm? How do you interpret this? What is this? What is this summation? Is it a column sum or is it a row sum? It is a column sum. So, you take all the elements in one column a 1 1 a 2 1 ok. So, 1 is 1 is summation over 1. So, I am taking summation of more of each row each column ok. So, and max over that. So, I find out first of all I take a matrix which is consisting of only absolute values only positive numbers ok. Then I find the column sums ok I find the column sums max over the column sum is nothing but this ratio max over this ratio that you can show not it is not very difficult to show this. So, max over column sums absolute of column sums that is one norm and what you can show is infinite norm. So, infinite norm is nothing but max over absolute of row sums ok. So, this these norms you can see here computationally is much more easy to see what you have to do when you want to compute one norm or infinite norm.  If you take all column sums find max over it you will get one norm take all row sums ok find max of the row sums that will give you infinite norm. This is much much easier to compute one norm or infinite norm are much much easier to compute let us close the. See this one norm and infinite norm are much much easier to compute then computing a transpose a and it is Eigen values. So, much more complex business than doing this this is very very easy ok. So, I wanted some easy way of computing norms now where am I going to use this ok why am I computing norms because our condition necessary and sufficient condition was spectral radius. Spectral radius is nothing but Eigen value ok. So, what is the relationship between norm and Eigen value. So, that is the next part of the puzzle is this clear now there are we have three different ways of computing norms two norm one norm and infinite norm among these one norm and infinite norm are computationally preferable ok. And now comes the point is why do why am I talking about norms yeah lambda one was nothing but Eigen value of lambda i we have this a transpose a v i is equal to lambda i v i. So, lambda is are Eigen values of a transpose a ok. Now, I said that I have numbered the Eigen values such that lambda 1 is greater than or equal to lambda 2 is greater than or equal to lambda 3 I have numbered them ok. So, lambda 1 is nothing but another way of saying lambda 1 is lambda max instead of giving a number and remembering it is easier to remember this as a formula lambda max of a transpose a that is why I called it ok. So, this theorem which I am going to state is now the corrects of the matter for any matrix b any square matrix b ok spectral radius sorry any matrix a for any matrix a its spectral radius is always less than or equal to any induced norm ok. The spectral radius is always less than or equal to induced norm. So, can you prove this how will you prove this what is spectral radius spectral radius is spectral radius of a is max over i or let us use the new notation that we have lambda max spectral radius is nothing but max over this. Now, what is what is induced norm of a matrix any induced norm for any induced norm is this is true this is the definition right is the induced norm definition ok. So, this thing also holds for this thing also holds when x corresponds to Eigen vector of a let us say v i is Eigen vector of ok v i is equal to lambda i v i right. So, I am going to write this is equal to norm. So, a v i and if I substitute v i here if I substitute v i here this is nothing but lambda i v i right, but what is this quantity when lambda comes out of the numerator what happens mod lambda mod lambda i norm v i divided by norm v i right mod lambda i norm v i by norm v i. So, this cancels ok what remains is mod lambda i this is hold this holds for every for every Eigen vector. So, this also holds for maximum magnitude Eigen vector right. So, this holds this holds for every Eigen vector ok which means it also holds for that Eigen vector which has a maximum magnitude ok, but what is the maximum magnitude Eigen vector Eigen value maximum magnitude Eigen value is nothing but the spectral radius ok. So, this inequality that is mod lambda i is less than or equal to norm a this holds for all i 1 to n and this implies that spectral radius of a is less than or equal to norm of a. Everyone with me on this ok so far so good.  So, computing using a software these days for a any huge matrix is just very very simple of course, computationally for a large scale matrix this is much much easier you know it just has to take absolute absolute sums of rows and find the max very very easy as compared to doing this. But so and we have a very nice relationship here ok. Now, how I am going to exploit this I am going to use this relationship to come up with sufficient conditions for conversions ok. Is this alright we have matrix norms we know how to compute them and now we know what is the relationship between the spectral radius and the matrix a its eigen value. We were at one point analyzing behavior of systems of the time z k plus 1 just about 2 lectures back be linear difference equations we are analyzing behavior of this and we said that if spectral radius of a is strictly less than 1 then what happens then norm z k goes to 0 as k tends to infinity ok as k tends to infinity but spectral radius for a large matrix is difficult to compute you have to compute eigen values ok. But from this inequality what I know is that spectral radius is always less than induced norm ok. Now, suppose I take a matrix ok compute its induced norm I compute its infinite norm ok or not a matrix here we are talking about sorry this should be b matrix here I am really sorry just I stand corrected this should be b matrix here b should be strictly less than 1 I take my b matrix ok if I take my b matrix compute its norm say 1 norm or infinite norm and that norm turns out to be less than 1 what can I say about the spectral radius right. So, if norm of b say infinite norm is less than 1 or I am not going to compute 2 norm I am going to compute only 1 norm or infinite norm. So, this infinite norm or norm of b 1 norm if this is strictly if this is strictly less than 1 either of them are strictly less than 1 ok from this theorem from this theorem what I know is that spectral radius of b should be strictly less than 1 ok. So, which means a sufficient condition a sufficient condition for conversions of this z k sequence to 0 norm of z k sequence to 0 is that take b matrix find its 1 norm or find its infinite norm if that norm turns out to be less than 1 strictly less than 1 I am done ok I know that I know that z k is going to go to 0 irrespective of irrespective of what happens to what is your initial condition it does not depend upon what is your z 0 z 0 can be large z 0 can be small ok z 0 can be arbitrary I know that if this condition holds then the spectral radius is always less than 1 because induced norm gives a upper bound on the spectral radius induced norm gives a upper bound on the spectral radius and if upper bound is smaller than 1 obviously spectral radius is less than 1 and then ok. So, I think we had lot of side stories now let me go back to solving a x equal to b so far so good is this a line of arguments clear ok. So, now let us go back and look at what we were doing finally we are back to what we wanted I wanted to solve a x equal to b using a iteration method x k plus 1 is equal to s inverse t x k plus s inverse b ok we have we have this iteration method and we said that the error behaves according to e k plus 1 is equal to s inverse t e k ok s inverse t e k and then we had this condition that spectral radius of s inverse t now s and t are different depending upon whether it is a Jacobi method or Gauss-Siedel method or relaxation method and so on. Now typically s is a simple matrix s is a diagonal matrix or s is a lower triangular matrix and inverting that matrix is not that difficult well we will come up with conditions which even do not require inversion further, but right now even if you want to do it by brute force by actually inverting s matrix even though it is not difficult, but now we have this condition spectral radius less than 1 for error convergence this is necessary and sufficient condition is necessary and sufficient condition I am now giving you a sufficient condition that if if induced norm of s inverse t is strictly less than 1 then spectral radius is less than 1 and why this is why this is just sufficient why this is not necessary it can happen it can happen that this is greater than 1 and this is less than 1 it can happen see this inequality says it is like saying 0.1 is less than 5 or 0.1 is less than 0.9 ok. So, if you get norm of a to be 5 quite likely that spectral radius of a could be 0.1 you do not know, but if if norm of a is 0.9 I surely know that spectral radius of a is less than 1 ok. If if a norm of a comes out to be 1.1 I cannot say anything about this being less than 1 or this being. Sorry if if induced norm typically will be 1 norm or infinite norm 1 norm if this is less than 1 definitely this is less than 1 ok, but if this is greater than 1 we cannot say anything about what is this ok. Inequality just says that this quantity is you know is less than this quantity we are particularly interested in this number 1 whether this is less than 1. So, this is less than 1 we are sure that this is less than 1 that is the model of this quantity ok. So, I can actually use the norm computation 1 norm or infinite norm computations to to come up with you know a sufficient addition for convergence ok. So, now let us actually apply this I know with more practical conditions because you know I will be talking about 1000 matrix how do I know how do I compute S inverse if it is lower than to the matrix computing S inverse is again not a big idea ok I still want a simpler conditions ok. So, what is this going to be now I am going to define a special class of matrices called as diagonally dominant matrices ok. I am going to define a special class of matrices called as diagonally dominant matrices or strictly diagonally dominant matrices and finding the summation I am taking a matrix ok. I am taking summation from z is equal to 1 to n. So, I am taking rho sum I am excluding one element from the rho sum which is that element. Now, if one of the diagonal element ok if one of the diagonal element is strictly greater than some of the remaining elements absolute some of the remaining elements ok. Then I am taking a matrix and this should hold for every I ok this is for I equal to 1 2 n. So, if this if these inequalities hold for each I then such a matrix is called as diagonally dominant matrix such a matrix is called as diagonally dominant matrix. Now, where is where will it help me diagonally dominant matrix I will go on here I will just I hope you have all these in your group book. Let us go back and see in the Jacobi method what was we wrote a as s minus t ok for Jacobi method for Jacobi method we had s is equal to I will then we also wrote this that a is equal to n plus d plus u then for Jacobi method s was d and t was n plus u minus of minus of n plus u ok minus of n plus u.  Now, if matrix a if matrix a is diagonally dominant if matrix a is diagonally dominant ok then what happens a i i a i i are all greater than the summation of all the row elements absolute number of elements what is this matrix can you this can you this right now what is this matrix for Jacobi method s inverse t will be what will be this matrix 0 minus a 1 2 by a 1 1 minus a 1 3 by a 1 1 minus a 1 n by a 1 1 a minus a 2 1 by a 2 2 0 minus a 2 3 by a 2 2 minus a 2 n by a 2 2 right. So, s inverse t will be a matrix which has 0 on the diagonal elements will be like this infinite norm no sum will be nothing, but what will be each row sum summation a i j j going from 1 to n divided by a i i a i i is divided in each row just look at here ok. But, this is the diagonal the other matrix which means this sum is strictly less than right this sum is strictly less than this if this sum is strictly less than this what does it mean that all these ratios are strictly less than 1 what does it mean infinite norm is strictly less than 1 if infinite norm is strictly less than 1 what can you say about the spectral radius of this matrix ok. So, now I have reduced checking whether J2B method will converge or not just to see whether a matrix whether it is right or not if matrix a is right or not ok my iterations will converge irrespective of where I start from ok irrespective of where I start from my iterations will converge if my matrix a is diagonal element. So, after doing all these jacquery in lot of arguments learning about you know matrix norms and then in spectral radius all that we have come up with a very simple criteria for finding out whether jacquery iterations are converged or not I will give you some more theorems either I can do the proofs of each one of them but I will give you some more theorems which are very elegant and from which you can ascertain whether your iterations will converge or not or you can modify your problem such that the iterations are guaranteed to converge ok. So, that is what we will see next class briefly and then move on to some other analysis, but this is where you can see you know I can value spectral radius and norms everything is action where you take it to analyze the behavior without actually having to solve it just looking at diagonal arguments I can come to the conclusion for any initial guess 5 iterations will converge ok very very powerful