Well, in general, of course, the market is weakening, you know, most of them still say order files are still out and very strong, but whether the pipeline continues, you know, that's the other question. Welcome to the Smarter Building Materials Marketing Podcast, helping you find better ways to grow leads, sales, and outperform your competition. All right, everybody, welcome to Smarter Building Materials Marketing, where we believe your online presence should be your best salesperson. I am Zach Williams, alongside my co-host, Beth Popnikov, and we've got a guest on the show today who I have tried to bring on the show, gosh, a handful of times over the last month. Super excited about today's guests on the show with us today. We are really excited to welcome Greg Brooks. He is the moderator for Executive Council on Construction Supply, and in case you're not in the know, that is a think tank for leading independent LBM dealers all over the U.S. and Canada. He's got incredible insight, incredible perspective. We're really excited to have him. Greg, thanks so much for your time. Well, thank you. Thank you for having me. I have been in this business a long time. I'm a steering committee member at the Harvard Joint Center for Housing Studies. I was one of the founding editors of a magazine called ProSales back in the 1990s, and prior to that, I spent 20 years in the business. Basically, I got a job at a lumberyard after high school working for the summer, and I wound up putting myself through school working in lumberyards. I happened to have been an English major, so when I got out, I was qualified to work in lumberyards. So, I've just been watching and studying the industry for a long time, and I have no particular training that equates to being an economist or anything. What we try and do is just kind of look at the common sense what happens day-to-day on the ground, because a lot of times what the experts say happens, they have much more detailed knowledge than I do, but once you get down to the day-to-day impact of that, it's often different, and so that's what we try and look at. Well, I'm going to push you on that, Greg, because I mentioned this to you, but your last couple of newsletters, I feel like have been absolutely on fire. For listeners, if you don't subscribe to Greg's newsletter, you absolutely should. It's, I think, some of the best insight out there about what's happening in the housing space, the dealer space, building construction, and I know you say you're not an economist, but man, some of the insights you share, it wasn't the last one you did on Canada, the Canadian one, but the one right before that about what's happening with housing supply, it is awesome, so good. I know I'm really raving about it, but Beth, I sent it to you and a couple of people- It did, absolutely. It was phenomenal. It was so good. I was like, how is this content free? I'm like, am I on a paid newsletter that I shouldn't, that I'm not paying for? But no pressure on the next edition, right? I know. Now you've got all these people listening. They're going to now sign up. Yeah, that sounds good, but yeah, I just try and figure this out from the day-to-day perspective of a dealer, what are they going to be looking at there? And of course, people are worried about all the talk of a recession. And it was interesting, who was it? It was Fortune magazine did an interview just a few weeks ago with Robert Schiller, the Yale economist of the Case Schiller Index and Nobel Laureate and so on and so forth. His opinion is that we're likely to have a recession because we're also depressed about it, that we're talking ourselves into it, which is to say, why are we doing this? But at any rate, yes, that's that's kind of the issue is just, you know, what does it mean for somebody on the ground? We record these podcasts and then like something will happen in less than a span of a week from when we actually released them. And if I look at it today, so we're recording that the Fed hasn't come out with their rate hike yet, they're projected to do another 75 basis points, you know, this next week. What is your take? Like just macro, if we go even further out, Greg, we look at like the housing market construction as a whole. If you're talking to a manufacturer or dealer, what are you telling them about what's happening today and how they should be thinking about their business? Well, you know, actually for all the angst and all the complaining things are going at this point, just about like they should. The whole purpose of this is to slow the housing market down. When you look at the things that go into the consumer price index, something like 40 percent of it, if you're looking to, this starts to get, they kind of get off in the weeds here. But something like 40 percent of it is housing costs. And so if housing costs are going up, that's inflation. And so if you're going to do anything about inflation, the number one place you got to look is the housing market. And slowing things down. I mean, right now the Harvard Joint Center for Housing Studies puts out an annual report called the State of the Nations Housing. And one of the things that they have been tracking in recent years is this kind of real basic measure of how affordable housing is, which is the ratio between the median home price and median household income.  In other words, if you made $50,000 a year, you knew that a home was going to cost about 155 or thereabouts. It spiked up during the housing bubble of 2006 to 2007. I got to almost five and then it fell back down. Of course, your member price is falling in the at the end of the bubble. And it didn't quite fall all the way back. It fell to about four and it just sort of floated around the low four to four point two for most of the 2010s. And now all of a sudden covid came. And when covid hit in a two year span, it jumped from four point one to a record now five point three. That's the highest ever in the United States.  And so obviously that's what you got to attack. And, you know, it never feels good to be on the receiving end of that attack. But but that's, you know, that's what has to be done in order to get things back into balance. Yeah, I've seen a similar chart. But if we look at what happened during covid, a lot of people weren't necessarily concerned about that because of the lower rates, which meant lower mortgage payments. Right. Now that mortgage payments are going up like I saw a stat recently. And forgive me if we've listed this before on the podcast. But it said that the average home, you know, that cost $450,000 a year ago to get the same mortgage payment would cost like right above 300,000 now to get the same exact mortgage payment today. And like, and that's what's causing this adjustment. But if you listen, if you look at like what realtors are saying, if you look at what homeowners are saying, it's it's more it's moving more towards and I would love to hear your take on this, Greg. But from what a lot of people are saying, is it moving more towards a normal state of a housing market? Like I listen to this one, I like to listen to lots of different podcasts when it comes to real estate, just to hear different perspectives. And there's this one real estate investment podcast that focuses on like multifamily and things like that. And I was listening to them and they're like, oh, you know, we're moving more towards a normal state and people are adding the the inspection back into the offer. Like that got inspections, inspections got removed during covid, which like they were kind of chuckling about because they're like, can you imagine like, like five years ago, we're moving an inspection like that's absurd. But like now we're like, I didn't see the house. I'm offering a hundred K over asking and there's no inspection, no contingency, nothing like that was what we were in. And like now we're like, oh, my gosh, people are adding inspections back into this. Well, like, of course, you know, we're moving back to to a more normal state, a more normal environment where it's not a hundred percent in the favor of the seller. That is correct. Yeah. No, it was and I mean, if you look at what happened during covid, you know, first of all, everybody wanted to get away from everybody else, you know, not social distancing, but get me out of this apartment in the city and get me into a suburban single family home where I can get away from all those sick people. I so you had that. Plus, during the time when we were doing those shutdowns in 2020, everybody who kept their job was had no place to spend the money. You weren't going out to restaurants. There was no Starbucks and all that sort of thing. And we built up something like three point seven trillion dollars in excess savings, in other words, more than more than they projected we would normally have. And so now all of a sudden there's been all kinds of money. And then there's also, like you mentioned, the interest rate thing, which interest rates have normally, you know, they've never been as low as they were following the Great Recession and held that way along. So it was kind of this artificial thing that distorted the market and it distorted it in a hundred different ways. It isn't just, you know, housing was was able to inflate because because mortgages were so cheap. But the the mergers and acquisitions frenzy that is going out there that has been going on was caused by low interest rates that forced pension funds insurers to, you know, look elsewhere for higher returns. So, I mean, you know, those low interest rates have distorted a lot of different things and housing is definitely one of them. So, Greg, let me ask you if you could just like grab your crystal ball really quickly, because we've seen incredible increase in prices in the building materials industry. So, you know, lumber has been a hot topic for 24 months or so. I've kind of lost track at this point. Zach and I joke that like we've never been more popular at cocktail parties because like suddenly building materials and lumber is interesting.  But right now people love to talk to us, right? OK, so lumber prices hot topic for 24 months. Everybody's talking about housing starts, housing costs.  I can't remember the last time I was with a group of people and the cost of housing didn't come up. Let me say two things. Zach, you were at a builder panel virtually. This was about a year and a half ago listening to builders and lumber dealers talk about the price of lumber. Do you remember this? Yeah, they were talking about how lumber prices were coming down. And basically the question was asked, well, are you guys going to start reducing prices? And they were like, no way. We've retrained everybody that lumber now costs 15 times more or, you know, whatever costs 30 percent more.  And then, Greg, in your most recent newsletter, which Zach was referencing, you have. Zach did send it to us. And my response to him was that last line where you said, prices need to come down speaking about housing prices. And you said, I'm quoting you. Builders probably have some room to make that happen, too. Versus 2019, D.R. Horton's 2021 net profit was up nearly 60 percent. While in our it was up 94 percent. That's not revenue. That's profit. That's profit. Man, yes. So how do you like when I ask you to take out your crystal ball? Sorry for my preamble. But when I asked about your crystal ball, my question here is, where are we going from here? Everybody is making a windfall in a industry that typically has pretty tight margins. So what's the where do we go from here? What's the plan? It, you know, it settles out eventually. It takes some time for it to settle out. But, you know, when prices are that high and builders profits are that high, that encourages to builders to build as much as they can. That creates, you know, more supply and which creates more competition, which tends to bring the price down. And that's kind of always been the way that this industry has worked. That's our famous boom and bust cycles, which last six to eight years.  And then we're back on board again. Well, that cycle appears to be broken. We had a long, long run up in the in the 1990s and the 2000s until we got to the bubble, a crash. And then we've been kind of behind the ball ever since then. So so it's kind of, you know, it's kind of weird that way. But now that people have stopped by, you know, housing prices are going to come down. Well, they already are. They already are like they already are. Now, I don't think we'll see them go back to the way they were pre covid. Like we saw some markets are 40, 50 percent up from 2019. But you're seeing 10 percent, 15 percent reductions. And that's on homes from, you know, existing owners as well as builders as well. You know, yeah, we're seeing across the U.S. I, yes, and particularly in the most overheated markets, you know, if everybody's got to live in Boise and Sarasota, then those places are going to be real expensive to live in, because everybody wants to live there.  And you'll find some reasonable prices. Not in Blacksburg. Not in Blacksburg. I can't speak for Louisville. I'm not a wise man. Yeah, I take that back. I take that back. But, you know, the other thing to remember about that in my expectation is that that, you know, well, two things to keep in mind. First of all, if we're talking about the price to income ratio, it doesn't matter if home prices come down all that far, if wages were to go up, you know, if that changes the ratio too. So if that should happen, then everything comes back into balance as far as the price to income ratio. The other thing to keep in mind is that even at five point three times income, the United States is still the single most affordable developed country in which to buy a house. Canada is closer to eight. The UK is eight. France is like 12. Japan is like 12. They're all very much more expensive than the United States for people. So so it's in particularly if you're willing to go to Iowa City, which is a beautiful place, by the way, you know, it's, you know, you can get a home and it's and it's pretty affordable. So I mean, the interesting part to me now is how far is this remote work trend going to go because that's what enables Iowa City? Yeah, it's a great question. I think the other thing I think about too, Greg, and I want to hear what you're hearing from lumber dealers, because now you've got a great pulse in your communication with them. But there was a lot of talk over the last couple of years that like, hey, millennials are not going to want bigger homes or they're not going to want a home. And now we're seeing millennials who are moving into the peak season of their lives where they want to purchase. Covid has made them want bigger homes than what people anticipated. Right. And so I'm curious to get your take on what you're hearing from lumber dealers specifically, because they sell to the local builder. They sell to the big builder. Like you're, you know, and I want to hear from you. What are they telling you? Because a lot of them, they come to you for insight, but you've also probably probably also hearing from them like, Hey, this is what we're seeing in the marketplace and it probably differs from market to market. But can you give you some give us some macro insights about what you're hearing from them? Well, in general, of course, the market is weakening. You know, the I most of them still say order files are still out and very strong. But whether the pipeline continues, you know, that's the other question. We're in great shape. You know, my builders, they're booked for another year and a half. But, you know, and I've had people tell me this, I don't think it's going to affect me because my builders are booked for a year and a half. And by the time we get a year and a half down the road, then the pipeline is going to start up again with it. And so they haven't really been hit by it hard yet, you know, and probably with a few exceptions to that. But I would describe their mood is more caution than anything.  And nobody's saying it's going to be anywhere near that bad. So so I think it's, yeah, you know, we'd rather have a boom. But but we know how to deal with this. And so I don't think they're particularly worried, at least not at this point.  The landing is where it starts to hurt. And so we don't know where we're going to land yet. So I listen to this really interesting Malcolm Gladwell podcast recently. I don't know if you're a fan of his, I love his stuff. It's awesome. He has this podcast called revisionist history, but he was talking about how and it had nothing to do with the economy, but he was talking about how people don't ever really fix things until there's something really wrong. Right. And so, I mean, if you look at the great recession, for example, there's a lot of things really wrong with our financial system that got quote by fixed, you know, through regulation, whatever, because of that situation. And you're seeing a lot of these people like Elon Musk and other well known executives saying, Hey, like, Hey, like a dip in the economy isn't necessarily a bad thing because it forces you to fix the problems that exist in your business. That hot economy covers up because money's flowing and people are just buying, you know, there's issues in every single business that a recession causes you to go, okay, should we really be operating that way? Should we really be doing it this way? And if I think about lumber dealers and manufacturers struggle with this too, is like, is labor is one of the biggest issues. Yeah, right. And so like, you know, and I think that this is an opportunity for regardless of who you sell to or how you sell, to be thinking about how you can improve those types of problems in the industry. Right. That is absolutely true. And that happens at all levels. I mean, you know, the first thought there is that no, a recession is not a problem for you for Elon Musk. And that's, you know, that part's true too. But for the average lumber dealer, yeah, it's an opportunity to change things, make things more efficient. I mean, look at what we're doing right now on Zoom.  They just, we don't do that. Now it's absolutely totally routine. Everybody has changed it and they haven't really scratched the surface of what you can do with Zoom yet. You know, how can this change the way that I sell building materials? You know, I meet with customers. Can I do that on Zoom? And we're still getting acclimated to that thing. But, but yeah, all that stuff, you know, it forces you to start thinking about things in a different way. The issue with labor, and this is, well, this is partly for builders, but, but, you know, a lot for dealers too. You know, there was, it has been a long, long time since anybody has said they wanted to be anywhere near the construction business. Um, NHB has done studies. I used to quote one back and done by a magazine called American Demographics back in the 90s, where they asked high school students to rank their favored up and coming profession. Where, where, where did they want to spend their life? And there were like 254 choices and Carpenter came in number 253 just ahead of Cowboy. So we have not been popular for a long, long time. That was a totally different situation when I was coming up in the 1970s. It was cool to be a carpenter. But at the time, the average carpenter was making like 35 to $40 per hour in today's, today's wages in today adjusted for inflation. Now it's like 25, 28, $28. And so, you know, money makes a big difference in all that. And, um, you know, if there continues to be wage pressure, what it does for dealers and probably for builders too is it pushes them to figure out ways to be more efficient because you can't find people cheap. And, um, you know, in our, in our think tank meetings, we talk about the, the 80, 50 rule. Everybody knows the 80, 20 rule, you know, and, um, the 80, 50 rule is that you can get as much done with 50 people averaging $80,000 per year as you can with 80 people averaging $50,000 per year if you're managing properly. And so that's kind of the next big challenge, or at least the thing we're talking about a lot is how do you get the people who can be, you know, you know, full scale entrepreneurs or intrapreneurs and, you know, unleash them so they can do all that stuff and pay them properly for the contribution they make and reduce your labor costs or make your labor costs flexible to go with your, your business. So labor is it. I love, I love this angle, Greg. So obviously we talk a ton about labor on the podcast and we had someone on within the last six months and they were started talking about, you know, there's a stigma that we've just never really been able to overcome around labor. It's like, you know, it solves it's really hard to make fun of somebody who's who makes $175,000 a year. And I was like, yeah, totally.  Boom, stigma solved. Like, let's just cut out all of the, the razz, all of the like, we don't have to do job fairs, we're not to convince you, we're not to do trainings and like cool events. It's like, you make $175,000 a year out of high school. Are you interested? Yeah, totally.  And there's like 15, 20 lumber dealers and even manufacturers around this table. And like, I'm trying to be like politically correct and listen to people and about them. I mean, a lot of them, they were complaining about the cost and issue of labor.  That's just where they are. And I was like, I was like, you're right. Like that's what that's what solves. I mean, that solves the problem. It's not going to solve, help your P&L. But what would you rather have? Like, would you rather have labor that you can count on? Or would you rather be spending that money on, on turnover and training and recruiting and all of that? Yeah. Yeah. And the other side of that is that you may have a lot of people if you don't have a very attractive job, chances are you've got a lot of people who are willing to put up with a not very attractive job and not working very hard at it as a result. And so to go along with that, you need to figure out ways to, to be able to measure what they're doing and pay them accordingly for what they're doing. I know one dealer, I've known a number of dealers who've done this, but one in particular worked out a system for his load builders, which are the people who package up those framing packages in the yard on a forklift and get ready and his delivery drivers, where he started paying them based on the weight that they had handled.  And so you got to put that all in manually. And then you got a benchmarked for a while. He got them to that point where they went through that process. It was about it was over a year, really. But when he started to switch people to that off their hourly wage and onto this, this whole thing, some of them, some of them didn't want anything to do with it. They left, they left, they quit. Others embraced the whole thing. And next thing, you know, he's got one guy who wants to out compete everybody else. And he's all of a sudden making 75 to $80,000 a year building framing packages, but he's building twice as many as the next guy's too. He's hustling. That's awesome. Yeah. He's hustling, which all of a sudden kills. Who do you want? Yeah, who do you want? Yeah, you want that guy. And it also tells everybody else, here's what's possible.  And so now he's got this super high productive operation, you know, and he can measure the productivity and it's worth it to pay 75 or $80,000 to a blue collar worker.  And I don't know, say, you'll have to tell me how true that is, because you're there and I'm not. But I, but I've never really quite believed it. And I definitely don't believe it about Gen Z or is behind it. Because they're flat out saying we don't mind working hard. And, and frankly, the physical, you know, the physical work of being out in the yard or building a house and stuff like that is actually fun when you're told, oh, yeah, it feels good.  For listeners, for those of you don't know, I don't tan very well by very well, I get an off shade of white in the sun. Greg, this has been awesome. I can't thank you enough for coming on the show. If someone wants to subscribe to your newsletter and or connect with you, what's the best way for them to do that? If they go if they want the newsletter, it is lbmexec.com. My email address is Greg greg at lbmexec.com. And so you can, you can catch me easily that way.  I probably just want to pick your brain and find out what's going on in your part of the industry. So feel free, please. That's awesome. Greg, thanks so much again. And I encourage anybody to make sure you subscribe to Greg's newsletter. It is awesome. No affiliate benefit on my part by plugging. I just think I just think it's, I just think it's great. I think it's really, really great. I mean, Greg, again, thank you for coming on the show and making some time, your contents just awesome and your insight is well received. And for listeners, make sure you reach out to Greg, follow him. And if you enjoyed this episode, enjoyed our content, you can check us out at venue.com slash podcast to subscribe and get more. Until next time, I'm Zach Williams alongside Beth Puppyglove. Thanks everybody.


It's wide up. And I'm always jumping crazy. David Dills, David Dills. Oh. Masks, Masks, Masks, Masks.


So, good morning, and welcome to this talk. Today I'll present joint work with John Finley, Sven Schäver, and Lijun Sang. Before we actually come to continuous time Markov games, I want to start with a more simple model, the continuous time Markov chains. Well, you all know discrete time Markov chains, the standard notion of Markov chain. In continuous time Markov chain, we simply add a notion of time, of probabilistic time. The time when the next transition happens is not any more of a unit distance, but of course there's probability distribution, the exponential probability distribution. And the rate from one side to another indicates the parameter according to which the distribution is scaled. We have several transitions like here, in this small example, we let both experiments run in parallel and take the first transition at fives. This is the same as saying we add the two rates and let one experiment run, and when this fires we jump to one or the other depending on their ratio. So if we now consider games, the only thing that comes into play are the actions. So in each location, we offer several actions from which we can choose. And the rate matrix also takes this into account. So for every action, we might offer different rates to our successor states. The problem we consider is the time-bounded reachability problem, or time-bounded reachability probability problem. And here also our two players come to play. We divide the space of locations into the sets of minimizing locations and maximizing locations, or the location of the minimizing player and the maximizing player respectively. And so these players try to maximize or minimize the time-bounded reachability probability. This is that we have a time-bound, and we want to be in a certain state, or in a certain set of states, when the time runs out. The good news is that an optimal solution to this exists, so there exists optimal strategies for the players. And in this talk, we'll now cover the efficient approximation of these probabilities. If we now take a look at the simple model of continuous time mark of chains, well, this kind of corresponds to the transient probability. So there, of course, we have no optimization problem. We just want, well, it is just one value we can compute. And we know that the transient probabilities in continuous time mark of chains, according to the Kolmogorov equations, are also to be more precise to the Kolmogorov backward equations. So we know the probability values at one certain point in time, that is, if the time has run out, if we have no time left, of course, we either are in the go region or we are not. So there, the probabilities are fixed. And then with the Kolmogorov backward equations, we can develop these probabilities backwards in time until we have our time-bound. So if you have not understood what the model is all about, you can also just look at these equations. This is all we are going to approximate, of course, the control version of this. For a fixed schedule, we can easily extend these equations to continuous time mark of games. The only thing that changes is that we ask the schedule which action he wants to choose. And he, of course, may choose a different action for every location and for every point in time. The control problem, I was talking about, the maximal reachability probability problem now looks like this. Well, it's not surprising that the point-wise optimization of the optimal choice of action leads to the optimal reachability probabilities. This is also a special case of the Bellman equations, if you know them. A small example. Here, let's consider this small continuous time mark of game, which has actually only one place. We could call it also a continuous time mark of decision process. If time has run out, so if our time-bound of one has passed, we know that we either are in the winning region or not. And if we consider here, as here in this plot, the reachability probabilities for location A, then, of course, there's a probability of zero to reach it. And from this point on backwards, we can develop these probabilities. Here, I've done it for three schedulers to either always choose action B, to either always choose action A, or the optimal strategy would be to switch at this point in time, and then, well, in this region, I think, to take action B, and when we have more time left, take action A. A brief overview of the results of our paper is that, so we developed an approximation algorithm that hasn't improved worst case complexity. In the maximal error we want to allow for, previous methods had a worst case complexity of t over pi. So that is quite bad if we want to have a really precise result. And we improved it to the kth root of t over pi, which, well, of course, we cannot use just any k. There's no free lunch, but I'll come to that later. Let's for now just say we can take small constants, like 3, 4, 5, maybe 10. Let's look at different approximation methods. For CTMCs, there's this well-known approximation technique of uniformization, which works amazingly fast. However, it does not extend to control problems. So it's difficult to extend this. I know that there is some relative work on this. We will later compare to that. But it is not applicable in the direct sense. Kut and similar numerical methods could be, in principle, applied. However, they assume, or for their analysis of precision, relies on the fact that all the derivatives exist, at least to the fourth derivative. And here we see that the second derivative of our reachability probabilities are not continuous. Therefore, the result, the Ruh and Kutter algorithms could give us, is not essentially better than the Euler method. Speaking of the Euler method, we will take a closer look at the Euler method now. The Euler method, well, you all know it, is to partition the total available time into small intervals of length epsilon, and we, well, assume that we know the reachability probabilities for the right fringe of the interval and want to compute the reachability probabilities of the left fringe of the interval and then iterate over the state space. In our case, this means that we replace the latter part of the equation just by the constants, that is, by the optimal reachability probabilities of the right fringe. So we assume this to be constant throughout this interval, and then we can, well, then we just have the linear combination over a constant. We have, we maximize over different constants. That's easy. We integrate, and we have our reachability probabilities for the left fringe. Assuming a constant estimate introduces a linear error. That is, an error of order epsilon in our, in the derivative of the reachability probability here. If we integrate over that, that becomes a square error, which is rather small, but it forces, or it leads to the fact that we have to choose, so if we want to achieve a global error of pi, we have to choose t over pi in many intervals, so we have to iterate t over pi many times over our state space. If you now imagine we want a precision of, say, 10 to the power of minus 9, that is quite bad. We have to iterate a lot of times. So we might want something, something, something better here. Now comes the clue. And we, so assuming, or assuming this to be constant, or replacing the reachability probabilities and the equation by constant is, one could argue, essentially the same as assuming, or at least one assumption that is behind this, that it works, is that at most one step may happen during the interval. Just one jump in our process may happen within the interval. And how can we justify that? For that, we take a look at the Poisson distributions. The Poisson distributions are the probability distribution that indicate how probably it is to have 0, 1, 2, and n steps within a certain time interval. That is, if we execute repeated probability experiments with an exponential distribution, we get the Poisson probabilities. Of course, we need a uniform rate for these experiments. But here, we can just take the maximal rate that is in our system to have an upper estimation. OK, of course, for small time intervals, like in a small epsilon interval, it is very probable that no step happens. But for this case, we don't have to optimize it all as our actions don't have any effect. Our choices don't have any effect. So we only have to care for the last three cases here. We see that these probabilities decrease really, really fast, even if we have a moderate epsilon. And in fact, for a small epsilon, the Poisson probabilities to take more than k steps are this. It's the lambda times epsilon to the power of k plus 1. The divided by k plus 1 to factorial. So the denominator decreases really fast. And the denominator increases really fast. This really converges amazingly fast to zero. And that was our motivation. So this, first of all, justifies that we just consider that once it happens and just make epsilon small enough, then we know that we only omit a really small proportion of the probability mass. But we thought that it might be an advantage to consider just more than one step in one interval. Well, for example, one could allow two steps per interval. And let's consider what happens if we allow for two steps to happen in an interval. We have these equations again. We want to approximate for the case that two or less steps may happen. That is, if once it happens, we end up in one of the successor states. And so we need an estimate for the successor states that considers the case that another step may happen. But we have computed this already. That was the Euler approximation. So we just make one pass of the state space, compute the Euler approximations. And then we make a second pass, like here, and replace the reachability probabilities by the linear estimations we got from the Euler approximation. What happens in this case? We have then a linear combination over linear functions and maximize over these. So we need the point-wise maximum over a set of linear functions, which is easy to compute. And we can also easily integrate over this function. And this results in an improved error bound. Well, mainly because of the pass-on probabilities, as I showed you before, per interval, the error bound improves from an epsilon-square error to an epsilon-cube error. So this allows us to take far less intervals. That is, the square root of t over pi instead of t over pi. Now, of course, the natural question is, can't we allow more? And indeed, we can. And this leads then to the improvement to the k-th root of t over pi, many intervals which we have to consider. However, the cost of the calculations per interval increases dramatically once we pass a certain threshold. But we'll come to that later. First of all, I want to show you the actual impact this has on a real example or a small example. Here we have one of the standard examples for city NDPs in the literature. We compare our tool, which allows for up to two steps happen within an interval to the state of the R tool, Markov reward model checker. This actually implements some rather, well, some really nice, heuristical speedups. And it is already much faster than the standard Euler approximation. We see that even for allowing up to two steps per interval, we improve over this. However, not so dramatically, one would argue. If we allow for three steps, the picture looks quite different. For example, for a precision of 10 to the minus 7, and we see that speedup is already in the order of 100. For 10 to the minus 8, the picture looks even better for our tool. But I'm not sure whether this is not an artifact or something. So what actually happens if we allow for up to three steps per interval, we have to make another pass over the state space and replace the reachability probabilities by this quadratic estimate, which we computed before. This means that we have to find intersections between small polynomials of degree 2. But this still works fine. This also works fine for polynomials of degree 3. So if we now want to make another pass, another level, this would work as well fine. We could still find the solution analytically. However, if we go to higher degrees, we would have to employ approximation methods for finding the polynomial degrees, for finding the intersections between the polynomials, which would then probably cost a significant, so would lead not to an improved performance, I think. Additionally, it is the case that the less intervals we have, the more intersections per interval we will find. So this will also decrease the performance. And it's even the case that the number of intersections per interval could increase, in theory, dramatically. In practical examples, this has to be found rather irrelevant. However, one could probably construct some of these examples where this is dramatic. So we argue that a small constant like 2, 3, 4, 5, or so should do. OK, that's it. To summarize, we propose a parametric approximation algorithm that improves the worst case complexity. And we have shown that it, on reasonable examples, is practical and can be implemented. So thanks.


Welcome to module 60 of point set topology part 1. We shall continue the study of topological vector spaces today. As earlier throughout this section we shall fix V to be a topological vector space. I will not use this notation for anything else. We have results parallel to the fundamental lemmas we had for a topological group obtained merely by changing the multiplicative notation to additive 1. Because if you have topological vector space V plus and the 0 element there will give you a topological group which is abelian. So, we shall denote them by additive notation plus instead of the multiplication dot. The multiplication will be there only not from V cross V to V, but only from k cross V to V now. So, all the statements which were multiplicative for the topological group that we have seen will now be obtained by just changing them to additive notation. In addition because of the scalar multiplication and the continuity of that we will have many other important results. So, all those things the new time I will give you a hint or a full proof of those things, but those which we have already got for a topological group as such those I will only restate it at all. So, with this the first theorem here is again a list of statements ab, c, d, e, f which are all easy consequences of the statement that we say topological vector space. You start with any abc subsets of V, then all these statements are true. So, let u be a neighborhood system of 0, then for any subset a, a bar will be equal to intersection of all a plus u, u inside u. Remember there was exactly similar result even stronger intersection of a u, multiplicative notation in the case of topological groups a u or you can take u a and so on ok. And you do not have to take all the neighborhoods just a neighborhood system will do ok. So, that is also an easy consequence anyway ok. So, the next one is a bar plus b bar is contained inside a plus b bar. Once again the corresponding thing was for topological groups we had a bar into b bar contained inside ab bar. So, same thing is true for this one I do not have to elaborate this one. Next thing comes this is new to topological vector spaces only because now I am using convexity. This convexity notion was not there in a topological group. Let c be a convex act then both c bar and interior of c are convex. b is balanced implies b bar is balanced further if b is 0 is in a b interior which is same thing as this b is a neighborhood of 0 then interior of b is also balanced. See unless you have 0 here you cannot claim that one ok. The second thing is this is Eth 1, 5th one if b is bounded then b bar is bounded. So, I have taken convexity, balance in this and boundedness ok. If a is a vector subspace then this closure is also vector subspace ok. So, on c, d, e, f are new to topological vector spaces a and b are old one only right. Let me let me just go through them again. In the context of topological groups you can check lemma 5.4 ok. We have proved that a bar is equal to intersection of a plus u such that u is a neighborhood of 0. If you take all the neighborhood then this is just in the multiplicative notion you have done. So, it is true for this one also. But then you can just restrict it to a neighborhood system. So, that will be a smaller family of same members of this family itself that is smaller family. So, intersection has to be larger ok. But here the smaller family has the property that each member is contained in some member here therefore the equality occurs.  Now, come to statement c. If c is convex I must show that c bar is convex. For any t belong with i ok we have t times c bar from part that you do not have to worry about that also. This is just multiplication by t right which is a translation if t is 0 both sides are 0 if t is not 0 then it is a homeomorphism. So, t c bar is t of c bar. Therefore, in particular what happens is t c bar plus 1 minus t c bar 1 minus t will be also inside 0 1 right. So, I can take these two which is nothing but t c bar plus 1 minus t c bar right from the first observation. But now this is a bar plus b bar. So, it is contained inside a plus b whole bar right. So, there is t c plus 1 minus t c. But for each point here t c plus 1 minus c is a point of c. So, this whole thing is just c bar it is contained inside c bar no problem ok.  The proving this statement here is in here convex it ok. Now, prove the convexity of the interior observe that t c interior plus 1 minus t c interior is already contained inside c because c is convex some point here t times that 1 minus t times some other point here is already inside c by convexity of c. But this one t c is now t of c interior that is an open set 1 minus t times this one is also an open set maybe one of them may collapse to single point that is the caution. If it is a single point when t is 0 or 1 minus t is 0 it is not an open set. But it is already inside the inside the interior of c bar. Therefore, this part is always an open subset of v and hence it is contained inside the maximum open subset that is interior ok. First is contained inside c, but this is open there were it is contained inside c interior. So, c interior is also convex. Now, let b be balanced then for any alpha in scalar with mod alpha 1 we have alpha b is contained inside b. So, it is contained inside b bar also ok. So, b bar the closure it is a closed set right. Therefore, if you take alpha b bar which is same thing as alpha b bar ok. Alpha b bar is the smallest closed subset contains alpha b must be contained inside b bar ok. So, that already assumed that always proves that b bar is balanced alpha b bar is less than contained b bar means b bar is balanced. Further now we assume 0 is in the interior of b.  If alpha is not 0 this will be an open subset. Then alpha b interior is open subset and hence it is contained inside interior of b. On the other hand if alpha is 0 then this will reduce to single point ok. 0 is there otherwise there will be problem. So, alpha b is 0. So, 0 is already there in the interior. So, you are done. So, this proves that b interior is also balanced ok. Alpha b interior in both the cases is contained inside inside b, but then it is contained inside b interior. So, b interior also balanced.  Statement e here is if b is bounded then b bar is bounded ok. What is boundedness? Given a neighborhood v of identity ok. I must produce I must produce as m such that s bigger than m imply b is contained as s of b. That is what I have to do. That is the meaning of that for b bar, b bar is contained as s 1 that is what I have to do ok. b bar is bounded means that ok. So, first I apply regularity of the vector space, topological vector space which we have told ok.  So, I am taking u bar as u now ok. u I am calling I am not calling it as open subset, but it is a neighborhood which is closed. Given any neighborhood there is a closer neighborhood contained inside that to take this one to a close neighborhood ok.   It is contained as s u ok.  So, sorry s v here, s u is contained as s v for every s greater than or equal to m. So, that is what my aim was. So, the proof of that b bar is parents it is done ok. Now, the last one it is a statement that if something whatever notation I have taken same notation. So, a is a vector subspace then a bar is a vector subspace the similar proof we had for the pliable groups also ok. But here, here this you can look at this one this is only for the scalars between 0 and 1. But entire discussion you know can be applied to any t any any square area ok. Because now I am assuming that instead of c I am assuming a is a vector space already. So, lambda times a will be already inside a. Therefore, the same proof will work here for to give you f. If a is closed a is a closed under addition and scalar multiplication that is a vector space sub space then a bar is also closed under scalar multiplication and addition that is what we mean ok.  So, proof is the same thing there with the extra assumption that instead of c arbitrary convex set. So, c a is now vector subspace let us go ahead let o be a neighborhood of 0 in v. Now, the first claim is there exists a balanced open neighborhood b of 0 such that b is contained inside o. So, a arbitrary neighborhood we can improve it to become balanced open neighborhood. Openness is obvious anyway balanced neighborhood that is the whole idea. Further if this o is convex then inside a in the statement a we can choose b to v convex as well that means it is balanced and convex ok. That is every topological vector space v as a local base consisting of balanced under the neighborhood that is the statement a. If v is locally convex that means it has convex neighborhood like this then it has a local base consisting of balanced convex neighborhood. Here the important thing is you just assume one convex neighborhood then there is a whole system of convex neighborhoods ok that is the whole idea here. Sir this balanced open neighborhood this concept is it something similar to symmetric neighborhoods in topological groups? It is just a symmetric it is just one single thing right inverse, inverse corresponds to minus 1 here right. So, that is that is my that is much weaker there even in topological groups you could not do anything the scalar is only minus 1 just comes there it is all right inverse becomes minus right. So, here is a stronger symmetry right you may say only for unit scalars also that is also symmetry but this is much more stronger ok does imply that but it is much much stronger ok.  So, these things will be played the whole of open balls in a metric space. So, we are bringing them in a back door entry it is like that ok. So, they will play the role of open balls that is much more stronger than just symmetry ok. So, by continuity of scalar multiplication at 0 moment you have some neighborhood there exist an open neighborhood V of k such that if delta is positive such that alpha w is contained inside O for all alpha with mod alpha less than equal to delta ok. O is a neighborhood 0 goes to 0 under scalar multiplication ok. So, you must you can control the scalar that is the whole idea there is a positive delta say mod alpha as a delta would imply the whole image alpha w is contained inside O. So, this is by continuity of scalar multiplication all right. So, all that we are doing is the use of scalar multiplication here. Now, let B equal to union of alpha w where mod alpha is less than equal to delta. The first part says that this family is non empty that is at least 1 w. Now, you take all such w which satisfies this mod alpha is less than equal to delta. There may not be any right. So, you have taken all such n where is 1 such that alpha w is contained inside O ok. Now, you take union of all of them then B is an open neighborhood of V ok. So, B is contained inside O and B is balanced. You must put that mod alpha is less than equal to delta I would like to have this one contained inside O, but that is the condition here on alpha ok. So, so I would like all these alpha w's also contained inside O. So, in any case union of all open subsets alpha w therefore, B is open B itself will be now balanced why because if multiple say beta mod beta is less than equal to alpha beta times B will be contained inside union of alpha into beta times B, but if multiply alpha by beta where beta is less than mod alpha will be automatically mod alpha beta will be less than equal to delta. So, those things will be contained inside this one again that is the water. So, so beta is balanced and it is an open subset of contained inside O and containing 0. Now, in the second part I want it to be what second part what is it I want it to be convex as well right. So, suppose one of them is convex that O is convex you have already got. Now, you take A to be intersection of all alpha O where mod alpha is except equal to 1 ok. So, in the case of if the vector space is real vector space this just means O and minus O that is the way you have done to get a symmetric neighbor would remember that. So, here you have to take mod alpha equal to 1 only to saturate it intersection of all of them ok. Now, check that A is convex because O is convex to begin with. So, this you have to check that this will be also convex ok. Hence interior of A is convex that is what we have seen right. If C is convex interior of C is convex was one of the part of the previous theorem. Now, let B contain itself will be a balanced neighborhood of E. Inside this one by part A we can take a balanced neighborhood of E. Then if mod alpha is equal to 1 then you can write B as alpha inverse of alpha of B alpha and alpha cancelled out. But alpha B being the balance is contained inside B. So, it is contained alpha inverse of B, but alpha inverse is also mod 1. So, it is contained inside B. So, all these things must be equal right right. Therefore, B itself is contained inside this intersection because for all alpha is contained inside B. So, intersection B itself will be contained inside all of them ok for all each alpha ok. Therefore, interior is a neighborhood of E ok because B is a neighborhood and 0 belongs to E. So, if you take the whole of interior that will be a neighborhood of E because that is one such. It remains to show that interior is balanced alright. Interior is always convex. Now, interior balanced is what I would show. For this it is for suffice to show that A itself is balanced. Again we have shown that if A is balanced A interior is balanced ok. So, to show that A is balanced take mod beta less than equal to 1 and beta is equal to t times alpha where mod alpha is equal to 1 ok. Any beta can be written as e power 2 pi i t right times t. So, that is why you can write t where mod alpha is equal to 1 and t will be between 0 and 1 positive, non-negative ok. Once you write like that beta A is t times alpha A, but alpha A is equal to A by this property its intersection that is why t alpha A is equal to A that is equal to now t alpha A equal to t times A. Now, what is A? A is intersection of all I can write you do not write alpha it may confusing. Now, write another variable gamma O where mod gamma is 1 ok. This is just A just t times that no, but now you can push the t inside to intersection of all gamma mod gamma equal to 1 gamma of t O ok. But O is a convex set to begin with ok assumption is that one containing 0 and then t times O is contained inside O. t times O consists of all the line segments from 0 to points of points of O t times that one. So, all the t times O will be also inside. So, therefore, what if its whole thing is inside O? So, beta A is contained inside instead of this t I can just write O right. Once I write O, this whole thing will be nothing but A right beta A is contained inside. Thus what we have shown is that in general balanced neighborhood form is fundamental system. If it is locally convex, then convex and balanced neighborhood form is fundamental system ok. Now, we will slowly these are fundamental results. Now, we will slowly you know arrive at some concrete results. Every compact subset of K is bounded. Every compact subset K of V is bounded ok.  So, now we can talk about compact subset being bounded just like in a metric space ok. Of course, our notion of boundedness is also different here. This boundedness is stronger. That is what we have already remarked in the last time. What is the proof? Proof is very easy. Given a neighborhood O of 0 inside V, we have to we have to find some m positive such that S is bigger than m implies K is contained instead S O. For any neighborhood you find an m which is this property that will mean that K is bounded. This is the definition of boundedness ok. Now, the previous theorem says that part A says that I may assume O is balanced in neighborhood. Suppose I do it for a smaller neighborhood balanced in neighborhood smaller than O. The same statement it will be true for O also ok. So, I can assume that O itself is balanced, but I am replacing it by B. B is a neighborhood of 0 which is balanced and contained inside O.  If we prove that then this B is contained inside O it will be contained S O also ok. But since this is a balanced neighborhood now B will be contained inside twice B contained inside 3 times B and so on.  See in R n what you have done? You would have taken a ball with radius 1, you will be content is concentric ball right. So, contained inside twice of that contains 3 times that and so on. This is precisely the property that I was hinting at. And from lemma 5.241 whatever we get K which is you see increasing union of this one ok. So, remember if we have an increasing union of any neighborhood where these numbers go to infinity then this whole thing is covering the entire vector space just like in the case of R n and so on anyway. So, this is what is happening in a topological vector space also that is 5.24. K is a subset of V anyway, but V is contained in the union of N this. So, that is 5.24. But K is compared and these are open subset. So, we will get one of them covering the whole thing because it is increasing union ok. So, we get N such that K is contained in N B. Now, you take M equal to this N if S is bigger than that again using balance these balance of neighborhood K will be already insert N B, K will be also insert S times that one ok because S is bigger than M here right. So, this M can be chosen as N. So, one way increasing union you have seen. Now there is another parallel decreasing union, decreasing intersections balls of smaller and smaller radius right. So, similar thing here if B is a bounded neighborhood of 0 then 1 by 2 power N B are N range over all the natural numbers forms a neighborhood system for 0. It is just like the ball B epsilon where epsilon times 0 is a neighborhood system. Instead of that you can write 1 by B 1 by N also or then you can B 1 by 2 power N also right. So, similarity this is what we are going to prove 1 by N etcetera epsilon times 0 etcetera are more difficult just this is nice easy thing and it is enough because this sequence converges to 0 ok. So, let U be any neighborhood recall by definition of boundedness what implies boundedness of B implies that there is another way of looking at it instead of there exist M such as bigger than M etcetera you can invert the whole thing there exist epsilon positive such that delta less than epsilon implies delta B is contained inside U ok. If B is contained inside S U where S is bigger than M is one definition equivalent to delta less than epsilon implies delta B is contained inside U this epsilon can be chosen like this. Once you have that all that you have to do is 2 power minus N less than epsilon ok. Then 1 by 2 power N B for some N large they namely this much that will be contained inside U for every neighborhood some member is here is inside U means that this is the neighborhood system over ok. So, we shall use all these things in a neat way to you know derive some nice results. Another easy and fundamental result here is take any 2 vectors ok take any 2 vectors let X be any topological space and alpha beta from X to K be any 2 continuous functions ok. So, alpha and beta continuous functions into the scalar the case is scalar field on which V is a vector space ok.  These are elementary things which we have observed inside R n and so on right. So, same thing we are observing in any any topological vector space what is the proof proof is precisely this one namely scalar multiplication and additions are continuous that is all. Now, we have an easy corollary here every linear map K n to V is continuous where n is any positive integer is a special case now ok. X was an arbitrary space ok we produce alpha and beta continuous functions we assume then the linear combination is continuous. Now, I am going to say every linear map from K n to V is continuous. So, for to talk about linear maps you have to have the domain also as a topological vector space, but you cannot repress this one by arbitrary topological vector space. An arbitrary topological vector space to arbitrary topological vector space linear maps may not be continuous ok that is why this theorem this corollary is important. It says that K power n which is nothing but a finite dimensional vector space over n then every linear map is continuous. Understand the importance of this one. If you replace this one by arbitrary where topological vector space then it will not be it makes sense, but it will not be true. Proof is easy similar to what we do for R n and so on ok. Once we have that lemma ok, let F be any linear map alright. Any linear map into vector space from vector space to vector space is determined by its values on a basis ok. So, let E 1 e to E n be standard basis you could you could have chosen any base no problem. Then I will look at the image of E 1 e to E n under F. So, put U k equal to F of E k ok. Now, every element F z belonging to k n can be written as z 1 E 1 etcetera z 10 E n where the dyes are scalars inside k right.  Not only that the coordinate function inside k n to k z going to z k this is a k th coordinate of z right. They are also continuous function. Therefore, all that I have to do is iterate you know the previous lemma for two of them ok. z going to z 1 is a continuous function times E 1 plus z going to z 2 is continuous times E 2. So, add them. So, first two addition is continuous, next third one is continuous and so on. So, what you get is z going to z 1 E 1 is what? F of E 1 etcetera z n into E n is what? F of E 1. So, that is continuous, but that is precisely the value of F that right. So, F is continuous, F is expressed as sum of n functions each of them is continuous alright. So, easily proved, but a non trivial result, an important result that finite dimensional vector space is linear maps are continuous. You know as a student for me because we were all the time studying finite dimensional vector space is linear maps are continuous linear maps are continuous. But suddenly when it goes in infinite dimension and linear maps may not be continuous that was a realization of ok. So, there is no way. So, it took some time I really thought that I can prove that every linear map is continuous, but that is not the case ok. So, I won't emphasize that fact here. So, let us stop here today. So, next time we will reap a good harvest and prove three important theorems and that will be the end of the course. So, today we will stop here. Thank you.


Hello and welcome to the rugged rockhound today I am back out at the location near Jericho where I found some nice aggots especially that plume aggott and here right off the bat I found one just sitting on the ground in the road just a nice little colorless kind of aggott slash calcidney the thing is is Some of it has bad some of these have banding in them some of them don't so some of them are more of an Aggott some are more calcidney, but mostly they're just colorless sometimes some white banding in them So those are pretty neat. They're worth coming out here to rockhound for but the big thing is that plume aggott We're trying to find so Tim's already out there searching and Hopefully today we're gonna try and spend as much time out here as we can because we want to find Where the plume aggott is coming from so we're down in the wash We weren't finding much on the hillside, so it's possible that there's better stuff further down or that this is really where it's coming out of Down some nice colorless. We're gonna picked up several pieces of the colorless Yeah, I'm seeing a couple pieces of it here and there So at least we've got that Across our fingers we can actually find the plume aggott Another decent looking piece. Looks like it's got some calcium in it as well Nice Here's This one cut open Nice Some banding in it some inclusions Nice looks pretty cool. So Tim just found this It looks like it might be some opal. It's kind of light and it's got that look of opal But right here there seems to be some texture This could be a piece of opalized wood. It's really hard to tell that texture could be something else, but Either way just the fact that he found a piece that's opal is Exciting There's a couple pieces right next to each other Nice I still have the macro lens on but Tim said he found something.  It's got yellow on the edge White in the middle. Oh This is gonna look beautiful. We're gonna cut this thing open. Oh Excited for this piece Here's this one. I've been looking forward to this one Unfortunately, it did have a fracture on one side that broke a little bit, but oh, yes Look at that Look at that on the inside That's nice So I saw that one that was pretty obvious Nice And I saw another one over here. Yes Just kind of poking out the wall right there Not bad the other one was better, but there you go. Not bad Sounds like Tim's got one Let's see what Tim's got Tim just found that one in the area. I was just searching. I completely missed that Yeah And a little one little one and here's this one cut open That looks real nice. Look at the banding in that So now that's a good aggott. I just saw that up on the side right there Look at that thing Nice big monster Hmm looks like it's mostly just Cuss Sydney. I'm not really seeing any bands in it. Got this one broke up a bit When I was trying to cut it Hey, it's got some stuff in there A little bit over there Mostly That side All right, Tim found something really nice in the stream right here. Look at the size of that one Uh, that is basically a big chunk of cuss Sydney So we've got cuss Sydney and there's a little pocket in here that's got some quartz crystals It's tiny but there's a yellow one chunk of cuss Sydney. Oh, we found a little yellow one again Okay, a little tiny bit that's got some yellow in it. Yep And here's this one cut open So it looks like on that side There you go. Yeah, a lot of these are just cuss Sydney With maybe that hint of banding in different places Occasionally some inclusions So I just found that piece It's kind of just like a a layer of it. I don't know if there's any banding but it's a decent size So Tim told me to turn it back on and he's being quiet, which usually means he found something good What did you find? Oh That is a nice layer of agate And we're gonna cut that thing down the middle And see what it offers because I love that yellow. You got a little yellow there. You got some yellow along the edges Yeah, it can make a cool design like an actual you could get some moss in there and Maybe have a cool like ocean seascape look. Yeah, so this is uh This is definitely Well, I'm gonna say most likely the best piece we've found so far today Here's this one I've been waiting to see how this one looks and here it is cut Oh, nice. Okay. We've got some stuff on the side here. It looked kind of like some Some plume going into there a little bit Oh, but I can see so much potential in there I think I think eventually I want to slowly grind this one down to try and expose Yeah I'm gonna have to do some stuff with this. This looks nice though So I picked this piece up going.  It's got a nice pocket right here. That's got some Quartz on it got a little bit of yellow going through it White right on the edge. That's a decent decent little agate there Just pulled that one out That's nice. It's got a cool pocket right there I don't really see any banding so Unless they just cut Sydney, but There might be some little bands in it. No banding So I just found this I think it's a little shard of petrified wood So just like Tim found that other piece that might be I think there is petrified wood around here Just saw this on the edge. It looks like it might have a hint of blue in it I don't know. Can you wash that off Tim? No, let the young buck wash stuff off Exactly It's got green on the outside. It's got some green That might just be from the light growing on it, but Sometimes You don't see the blue Okay, it was probably just the way it was sitting in there then Still it's a nice piece. That'll be a great one for cutting Okay, so Tim just uncovered this one right there It looks like it's nice and big so he's gonna pop that out Not bad. That's a good size Ooh, look at all the crystals in there. Yeah, and it's got banding on the ends. So this will be a good one to cut Nice, Tim Just found this one. It's got some nice botched riddle Custed me on the top and you got a little bit of banding in the agate Not bad So Tim just found the biggest one of the hunt. It was right there Tim what did you find? Dang got a pocket there. It's got a little bit of crystal in it. You can see the crystal quartz forming in the center here That's a chunk. Okay. That's the biggest piece we found either today or the last time I was here Well done Tim Tim just found this big one over there and right over here. I found this one. Look at that Nice Cute Another nice one to cut. You need two of those to make mine. No, I probably actually need about three of these to make yours Cut open I was worried that if I cut it this way I'd lose the banding. So I cut it this way nice Looks pretty good Right after finding that piece. I found this nice little piece of jasper Look at that It'd be nice to find a big one like that, wouldn't it? Just found this it's little layers of agate all through it This is crazy. I'm gonna definitely have to cut this And so this one that I found I'm excited for Let's see. Ooh. Oh, okay Yes, looks like it's a bunch of broken pieces of volcanic rock With agate going all through it That is a really cool Just found this piece.  It's pretty nice. Got some banding in there. It's a decent looking agate Just saw this looks like some agate with a little bit of red rain to it Ooh, nice. It's got some nice banding in it Yeah, that should be real nice. We'll cut that up Cut open Oh, very nice. Oh, look at that Oh, that's some nice banding Some I guess some plume maybe up there. I don't know pretty cool though. So here I am This is basically as far as I came the other day and I found this right Probably right after where I finished Cut open Not too bad. We've got some stuff going on here But if I grind down into this further we'll get some more cool things going on So here's the little road that comes down. That's where I had started last time So this is all in searched ground this and I have right here. That's a nice looks like a decent piece I just found that little thing.  We've got some nice agate right here Nice I just found this Nice piece of red jasper A cow died here sometime ago Here's the skull So we've been searching quite a bit actually we Been quite a bit down here as we've come further down. It's kind of gotten less and less So I'm feeling like we're getting away from the actual zone. So it was up there So what I think we'll do now is we're going to go up and over the road And back down the other side and check the other wash So I just found this little piece. Look how red that is It's like a carnelian almost Could be Just another thing to add to the list of things we've found here very small Can we find a big one? That's the question As we're heading up to the road tim found this nice piece. Look at that. It's a nice agate Nice stuff looks like it's even got a little bit of moss going on at the bottom Good find I decided to take this piece and try and grind one side Unfortunately piece of it broke off, but that's what it looks like inside Very nice We're up on the road now And look at that. Maybe we just need to search the road That's beautiful It's a good piece So when I was looking forward to Here we go Oh, that's nice Banding A little bit of stuff going on there small pocket in it Nice piece Back in the main wash. We're gonna head up this way toward the road But look at this that I just came across. Oh, that's beautiful That might have plumes in it I'm gonna have to really think about how to cut that one That is beautiful though. That's gonna be great And this one's too big to cut right now. I'll have to cut it later when I have a bigger saw, but look at it It's amazing. It's got a blue tint to it So if you get good enough pieces out there, you can get a blue tint Just found that one We decided on the way back to the car We'd go up from the riverbed and just kind of follow the edge of the hill And yeah, this stuff just kind of seemed to be coming out here and there All around I ended up cutting it twice because Look at that Isn't that awesome? That's how I cut it again and Look at that Oh, isn't that beautiful? This is a great agate Just found this one with some really nice banding in it That's a good piece Maybe there's more around here made it back to the vehicle We got all our stuff packed up turned out to be a pretty good day We found some really good agates and the yellow ones I'm excited about and some cool potentials You've probably you have already seen everything cut. I haven't yet, but I'm excited to see it And it was a good day. You guys know where it's at now And If I haven't yet, I'll put the GPS coordinates here where we parked But you also have the GPS coordinates from the previous time I went out came out here Anyway We're a little tired. We've been hiking all over the place Remember there is treasure everywhere


welcome to the McLean County 4-H horse day. This camp was designed with the horseless horsemen in mind. A horse sees differently than we do. Let's pretend you are a horse. One of the things we're going to do here is talk about or to have you actually go through a little demonstration on what you can see and then maybe get a perception of what horses can see. So shadow here is actually going to have, remember that the horses eyes are going to be out in the corners a little bit so this is going to be somewhat different. So now what we're going to do is we're going to have you pretend that we're going to sit on the horse. So you get to straddle a horse. This is actually a horse that actually people that are short can get their legs over without getting saddle sores. This is pretty cool.  You can see her back there. You've got great vision. You've got a bigger, you've got a better vision. That's very good. Alright, nice job. So, I don't know. Alright and then underneath is a lot of go in the front part of the saddle. What was it you said? Thanks, Josh. Grab main, grab main. That's a girl. Is that making this horse go right now? So how are we going to get this thing to move? This big horse underneath us. Okay. Can you make noises? Yes. Yes. And without us knowing we're usually using our whole body to get the horse to move. So, we lean forward, use our legs. So everybody move your horse forward. And I would let all you just go, your horse moves. Have you ever been on a stubborn horse that you go, use our body lean forward? So then we use our body again. Yes? Okay. So now let's think of this. If you want to go through a door and the door is closed, what do you do? Open the door? Okay. So, we want our horse to go to the right. Where's our right side? Right hands up. Right hands up. Okay, we're going to go to the right. Okay. So, we have to open that door. We're going to open our rein. This is our rein hand. And we're going to open our leg. So, can we go to do that? Let's... Okay. Tell me what we do to go to the left now. Can you put your same thing? Yes. Open that door. So rein. We open that door. Let's go to... Show me how to go forward again. Let's see it. Yes, there we go. So, we all just learned how to ride. But that's like rein. It's almost a program. We almost need a little printer to put everything in. The time we get through. Because if we have two of them big saw horses. Yeah. And go everybody in. Yeah.


All right, we're back. It's still the breakfast and plus if you have come Monday, Thomas joins us to to look at the World Cup. Of course, very interesting games that have held so far and guitar and miss a lot of complaints, especially from the Western media. But the biggest stories are the upsets that have held occurred in that World Cup so far. We're talking about games featuring Argentina and Germany. You have sorry, Argentina losing 2-1 to Saudi Arabia, Germany losing 2-1 to Japan of all countries. The Canadians should have won 2-1 yesterday, but FIFA did their thing, and they ended up losing 3-2. Monday, Thomas is a sports analyst, joins us live from Aqaibam State. It's great to have you out of said happy Friday, but let me say happy Qatar 2022, happy World Cup. All right, of course, it's a great World Cup and I mean the World Cup where we are seeing not just one of the greatest upsets in World Football, but two of them. I mean, first of all, we started against Argentina. And before that game, Argentina did not lose their past 36 games. I mean, they were on beaten. They were looking to equal the record of 37 on beaten games, which is currently held by the Italian side. And guess what? Saudi Arabia side that are always getting whooped in their first opening matchday of a World Cup game. I can remember in 2002, they were beaten eight goals to kneel by Germany. In 2018, Russia scored six goals past them, and it is 2022. And they pulled up one of the greatest upsets in World Football by beating two goals to one. Germany, of course, thought they had it against Japan. Japan scored two quick goals to, of course, overturn that particular one and beat Germany by two goals to one. And you talking about upsets, we thought we were going to see that from the Blacks as of Ghana actually had an open mind to that particular game. They had a game play in the first 45 minutes or two or two. Was it so brilliant? I don't think so because football, they say the best way to get things done, the best form of defense is attacking. All right, but he was able to defend throughout the first 45 minutes. The Blacks stars were just sitting deep and goodness. They were able to stop the Portuguese from scoring in the second 45 minutes, 60 something minutes on the clock. They are doing everything right. And Cristiano Ronaldo goes down in the penalty box and a referee points to the sport. Sally Sue was complaining. The man that was comfortable that I didn't touch Cristiano Ronaldo. I didn't touch Cristiano Ronaldo. They didn't even take a look at the VRR. And I thought, why didn't they take a look at the VRR? I mean, this guy inside they are complaining that that is not a fault. You should take a look at the VRR. I mean, that's what has been happening, right? But they did not. Maybe because it was Cristiano Ronaldo, and they were just waiting for him to break that record. And when he comes to record, Cristiano Ronaldo loves to chase them. And it's the first player in world football to score in five consecutive World Cups. Congratulations to him. But I don't think that penalty was a penalty completely, although we've seen it given. I can take your mind still in this tournament where Iran commuted a very great blunder in the box against the English national team. Actually, it was Harry Maguire that was held in the box, but that was not given as a penalty. The same thing happened a day later. And that was Sara Riviera against Argentina. Sara Riviera held Argentina. I played it the same way Harry Maguire was held some days ago, and it was pointed to the sport. So we don't get to see it. So it wasn't even as bad as the one of Harry Maguire, where he was being held and talked.  It's not even as bad as that. So I think FIFA are not doing the right thing when it comes to, especially, I think that is one of the most controversial issues right now in this World Cup. But for the rest of the action, I am enjoying myself, and I'm also enjoying African countries playing Ko-fi. I tell you for sure, Cameroon, they put up a great performance against Switzerland, if not for a Cameroonian born, Brille and Bolo, scoring against his home nation. He refused to celebrate. He apologized. And I'm here in Ramos. I know this cannot be true, but people are already peddling this false information that the father of Brille and Bolo has been arrested in Cameroon. You know, for me, I can just make a call from Switzerland, where we hear he normally spends a lot of his time, you see. I arrest him. As far as things in Cameroon, as far as recites in Cameroon. So it's very easy for people to check him out and say, why did your son do that? You know what happened? You know what happened after the, I think it was 2006 or 2004 Nation's Cup, where General Robert Gaye of Côte d'Ivoire, the then military leader of Côte d'Ivoire, you know, arrested the entire Lezalophon, the elephants of Côte d'Ivoire. After they performed a Bismarck at the Nation's Cup, he told them, you know, just take this guy's insight. You know, it's They just come to us, right? It happens. African football for you, you know, African football. Okay. So the upsets, let's start with Saudi Arabia, Argentina.  In the mix is certain Harvey Reynard. Let's talk about that. Let's talk about the tactical acumen of Harvey Reynard. All right. I listened to Harvey Reynard post-match conference, and he was still giving credit to the Argentine, but one thing he was discrediting them was because they did not come with the mentality that they were coming to play a tough game. They thought it was just going to be a walk in the park. But you know, football, it can be very easily played on the papers. But when it comes to the picture of play, it all boils down to the mental fortune cheat. It all boils down on how much you can match your opponent. And I think Saudi Arabia were not lucky in that game. They deserve a victory. They deserve all three points. They were able to match Argentine half of power, pace for pace, skill for skill, tactic for tactics. Although Argentine, they had the most possession of the ball, but the Saudi Arabians, of course, what do you need to do? I mean, football is a game where Fibo is only granted to those with impeccable execution. They had just two shots on targets and two of them was in the back of the net. That's how you get brutally clinical. And that's what gives you the victory in a stage like this, which is the World Cup. At Gentiles, they have so many touches on the ball, but no penetration. They were not able to penetrate against the Saudi Arabia. And I think the Gentiles starting these tournaments as favorites, by the way, they play against Saudi Arabia. I'm not thinking they are favorites anymore because the next game is against the next two games against Mexico and Poland. So if you can't beat Saudi Arabia, I'm sure you can't beat Mexico, who are very familiar with Argentina. If you can't beat Poland, who of course, have got some considerable amount of stars in that particular sport. So I'm tipping at Gentiles to be the biggest flop, although this was not my thought when the World Cup was about to start. But right now, the way that Gentiles play, they have the ball, they have the position, the worst position without penetration. It's quite embarrassing. But for the Saudi Arabians, prolific performance. I mean, compared to the Japanese, who of course, just had maybe the last just had a two-shot of target as well, and they were able to make it to the about four-day Gentile, it is quite a bigger win for the Saudi Arabians because Argentina had gone 36 games on beaten. But let's also talk about the Japanese side. Germany heading to this World Cup, they've not been really good. They don't have the strike force. We used to call them German machines, the effective German machines. But in 2022, it all started in 2018. That's when they lost their machineries. That's when they lost their arsenal, getting knocked out by South Korea. And now they are on the rift of getting knocked out again because losing your first game to Japan. And leading to this competition, Germany played against Oman in the friendly game, the last friendly game. And guess what? Oman. I mean, not even a football country by any means, but they were just able to score once crappy goal past Oman. And you are going to the World Cup where you're going to face stronger positions in people in countries like Japan and the rest.  Right? We should wait for more upset. And for the African teams, the other ones that will pull up another upset today, it's not going to be an upset if sending a big Qatar, but I'm thinking Africa will get it to the semi-finals. At least two African countries will make it to the quarterfinals. And at least one will make it to the semi-finals in this competition. I love the determination, I love the greet. I mean, look at how they're able to stop the teams from scoring the first 45 minutes. Tunisia, Morocco, no African team considered in the first 45 minutes. Yes, I'm right for that. That is brilliant. But it's also true that before the Black Stars scored, no African team has scored any single goal, you know, apart from those two goals, we're not scoring. That's what it is.  I've got to celebrate the Black Stars have gone at 12th school, West Africa just like Nigeria. And they did greatly representing the African region, scoring not just one goal, but two goals. But is there no worrying for you that, you know, Senegal hasn't scored, Cameroon hasn't scored, Morocco hasn't scored, Tunisia hasn't scored? It's worrying, but when you take a look at some teams, you've got to call them some slacks. The likes of Senegal, I mean, it was a crushing blow when Sadio Mane was ruled out of the tournament. Yes, because I think if Sadio Mane was available against the Netherlands, Senegal would have been more dangerous. The Lions would have come out playing. So the Senegalese are still playing as they should. They just lost concentration. You saw that game, Edouard Mendy was comparable for the two goals. I mean, a goalkeeper who hasn't been informed in his club for Chelsea went to the World Cup and it was the same story. Even though for Edouard Mendy's mistake, I don't think the Dutch side would have again gotten two goals past the Senegalese side. So this is what you've said now, lapsing concentration. You look at Senegal, you just said they played well against the Netherlands. They were in control, actually, of parts of that game. You look at Switzerland versus Cameroon, the Cameronians played well. They had chances. Chippa Motting had a chance to score, and then the guy who played for Brentford also had a chance. He could have squareded Chippa Motting. They didn't play badly at all. You look at Morocco versus Croatia. They didn't play badly at all. Do you think African football has a problem? Because the Asian countries, Japan, they've won three points. Saudi Arabia, they've won three points. But the African countries that seem to have the players in Europe can't win. What is the problem? Is it lack of concentration? Because I don't see, I see teams play beautiful football, pass ball around, have skillful players, but there is no end product. All right, what's the problem of Africa? I think I've come with mentality, and I'm going to go back to yesterday's game, the Black Stars of Ghana. They had, in the first 45 minutes, I know it's a game plan of the coach sometimes to sit back and defend, but that's not how you win games. I mean, imagine if they had thrown everything at the Portuguese side from the first blast of the whistle. I know the Portuguese are the better side on the paper. But like I said, if you've got the right mentality, if you've got the ability to think that you are going to last with this players, you're going to match them power for power, toe for toe. I think the Black Stars of Ghana should have just pulled out against the Portuguese side. They did that in the second 45 minutes, and it bore fruits. They got two goals. They attacked against, especially after they conceded a goal, which for me is still very controversial. Cristiano Ronaldo converted from the penalties for, they went all out, a data argue with the captain leading from the front, getting the equalizer, and yes, they were not able to even defend for at least the next five minutes. I just stepped outside, and the next thing I'm here, Portugal, I've scored again. I'm like, from where? Even data argue did not finish celebrating when it was substituted. He was just maybe congratulating, okay, his teammates were congratulating him, and he turns back and he sees that Juan Felix have made him two goals to kneel. So what matters right now is I do not be scared of these people, refuse not to be bullied, because that's their problem. They just let this European country bully them. And the same thing happened against Switzerland. Cameroon were looking really good. Chippo Mote scored in his last seven games. He scored six goals for the Bayern side. He was the atop man, the atalist man. He had a chance where he should have just taken it, but I think it didn't even have enough confidence. For me, he should have held out because the player defender was very close to him. He should have just maybe caught back or something. The defender would have just maybe went away, but he just went on for a show, which was very easy for the goalkeeper Jan Soma. So I think it's a matter of confidence or a confidence issue for the African side. They need to believe in themselves, especially the Cameroonians. I mean, they have what it takes. They have what it takes. They have a very solid precedent. Fager put President Samuel E. Toow, who's talking tough. He has already predicted Cameroon to play in the final. That is what they should build on. The mentality needs to be right. The Senegalese side, you would say without money, they wouldn't go any further. But I think that's a lie. They've got Ismael Saar, who's doing tremendously well. They've played against the Dust side, who were the top best opponent. And now they are up against Ecuador and Qatar. Today they take on Qatar. So that should be a straight win for them. So Monday, Thomas, I will just ask you two double barrel questions. First, which team do you tip to really create the next upset in the World Cup, the next upset in the World Cup? And then, which teams do you predict to lead the trophy?  I saw flashes of a team that can win that cup. France have been on fire. Spain have been on fire. Who gave me the teams to predict to lift the trophy? Top three teams, top three contenders of this year's World Cup, number one, Brazil, number two, England, number three, France, Argentina were on my list, but now they have no way to be found. Another upset that we are going to get is an African team making it to the semi-final. That would be an upset, right? Because I'm thinking Senegal with the way they played against the Dutch side, they already threw to the round of 16. And if they do the right thing, they'll make it to the semi-finals. Cameroon, I've still got Sabia and Brazil to play. I don't see them getting past the Brazilian side. Like you said, they were on fire yesterday. Retali Sen, who scored the goal of the tournament already, in my opinion, that was a great strike. And for the offsets, I don't think we are going to see any offsets anymore because the teams will not be very careful. It was the first round game, and maybe bigger countries were on the rating the smaller ones, but I'm pretty sure in the second round, in the third round, no one is going to give a chance for that. So no more offsets, but in the knockout stages, it's going to be an exciting one. And I'm tipping Africa, Senegal, like I said, to make it to the semi-finals, and maybe another African country, Ghana. They've got what it takes to, of course, get past that particular group and see them in the round of 16. Because by the way, Ghana played in the second 45 minutes. They need to stop playing like that in the both halves of their next game. All right. Monday Thomas is also through having you. I see your shits. I see your shits. Keep up the good job.  I don't know. We need to give him a call and talk to him. But I mean, we were telling him playing so well. I don't know if Gabriel... That team is star studded. I mean, the group name I was substituted, name I was sent to. Yeah, name I got injured, you know. Name I got injured. So they are still taking a look. Sodri will be done today, and before the end of the day, we'll know if Namah has been ruled out. Hopefully, we pray he's not. All right. Thank you so much, and I hope to have you again on the program sometime soon. Have a wonderful weekend and enjoy the rest of the walk-up. We wish Ghana the best, because I'm supporting Ghana somehow, because of you. All right. Right. Let's see how we go. Thank you, Monday Thomas, for listening. That's the size of our package right here on the breakfast on Plus TV Africa. Please follow us on our social media platforms at Plus TV Africa, Facebook, Twitter, Instagram, YouTube. And we have a second YouTube account, Plus TV Africa Lifestyle. On behalf of everyone on our team, technical staff, the production staff, program staff, my name is Kofi Bartel, so return on Monday. Good morning.


It is geometry and that's our new unit. We're going right into geometry. So if we take a minute to talk to your neighbor about what does geometry mean to you? What is geometry? What do you think about? All of them are like shapes and angles. Angles, shapes, corners, sides. Basically, I don't think to do with shapes. Yes. I have no clue how that goes with math, but OK. Adding sides. Yeah. That's how. Right? Yeah. OK, guys. Five, four, three, two, one. Lila, what did your partner think about geometry? I think that it's like all the vertexes and it's like shapes and angles. OK. James, what did your partner say that geometry meant to them? Geometry means like squares. OK. Angles, what the degrees of the angles are, and that's all. All right. Jack, what did your partner say? It's mostly like the shapes and all the angles and sides and corners and things like that. All right. And Andrew? We were saying that they were shapes, angles, lines, parallel lines. All right. Let me get Logan. They were lines, angles, shapes, corners. Does anybody think anything different? Any group say anything different that we didn't catch? All right. That's great, guys. You all have a really good understanding kind of sort of of geometry. But let's start. You ready? We're going into space. Space. What's space? Space. Oh, tell your neighbor, what's space? What do you think space is? Where is space? Talk to your neighbor, where is space? Is it only up high? No, it's down low. Talk to your neighbor, where is space? Space is anywhere that there's not a planet. It's like around Earth, the other planet's been Earth. Yeah. Five, Earth, three, two, one. If I was going to mark a spot in space on my smart board, if I wanted to mark a space on here, could I do that? What would I do? You would find like an open area of white. And what would I do? You would circle it. Circle it? OK, maybe I could find an open area of white like this and circle it. Does anybody have another idea of how I can mark a spot? Leah? You think I could do what? You could put a dot on it, maybe. I could put a dot on it in a certain spot or anywhere. Well, where the space is? Well, space is on the spot. So yeah, I guess anywhere. Maybe in the, you think it would have to go here? Or maybe it could go here? What do you think? I guess, and probably the center makes the most sense. The center makes the most sense. The most sense. Can everybody mark a spot on your white board? Just mark a spot and make it with a dot. Everybody mark a spot. Look at your neighbors. Do you all have it marked in the same spot? No, no. Kind of sort of, you do. Jack, do you guys have it in the same spot? This is like more close to the middle, and I was like out here. But is your neighbors wrong? Why not? Brianna, why isn't it wrong that they're all, even though they're all different, why is that okay? Because all around us is the space that like, if we were to walk across the room, we're in space. So really a point in space could be? Anywhere. Anywhere. You're absolutely correct. So when we mark a point in space in geometry, we make a dot kind of like Leah said, but we call it a point. So everybody right now has a point on their paper somewhere in their space of their whiteboard, correct? We did that, we did that, we did that. Can you make another spot? Mark another spot in your whiteboard. Find a spot. No, do not erase anything. We're gonna keep everything today for almost the whole time, sorry guys, my mistake. Make another point. And again, a point is just a dot, but it's a point. It's an exact location in space. So for us, it's an exact location on our whiteboard. Everybody have one? How many points do you think you could make on your smart board and your little whiteboard? Talk to your neighbor. How many do you think you could make? Because it depends on the space of the whiteboard. It depends on how big or small we make the dots.