The following is a presentation of TFNN, the Tiger Technician Hour with your host, Basil Chapman. Call now. Call free at 1-877-927-6648. Good morning, everyone, Basil Chapman on this Monday, October the 9th, and we're looking at the Dow down 60 at 33,349, I must say. For my subscribers in my opening call every weekend, I do about an hour long overview, go through our stock positions, what we're looking at, what I'd like to buy, what we didn't buy, what moved without us, what we have in terms of new positions, etc., etc. And what we're looking to buy in just in relationship to maybe adding to positions that we already have. And it had just been often news about the Israel attack on Israel. And I cheated with respect in the sense that it was something that, for market-wise, because that's really what I was doing, market-wise, but I want you to look at the market as if what would happen if the selling pressure that we've seen, a pretty intense selling pressure for the S&P was July, July, for the Dow it was August the 1st at 35,679. What if the selling pressure just lightened up a little bit? What would happen? So with that context, look what we've got. I'll do this in the daily chart right here. I discussed the symmetry between the August the 1st high in the Dow and the low of May the 25th at 32,586. And basically what I'd said is there was a one-to-one upside, etc., but we've got that turnaround and based on really a lot of emphasis on one particular technique, although I used others with the RSI, the Red Group Strength is gray line declining before we got to the top and August the 1st. But within that context, there was a very good sign that on a short-term basis the Dow would turn down, but that nine-beard exponential moving average was still strong, but there were hints that it could turn around based on some of the technical indicators that I follow. So within that movement to the arch right here where you're always looking at patterns of an arch and a cup, there's the arch, he has a V or cup formation, there's another V or in this case a beautiful cup formation and there's an arch. Within that context, what we were looking at is that the sell signal generated later on with that big 500-point spike, I think it was back about the 5th or so of August and then it failed the same day and then went lower and then suit afterwards on the 15th of August, that's when it says 15 days, it's 15, not 15 sessions, but 15 days later, counted the days, that's where we turned down and that's where the nine-beard moving average turned down and what I was suggesting is that within that context, the arch formation, this H that goes to it, lower case M pattern that creates a larger arch formation, should see a break of the 34,000 level, which it did and if that was broken and that orange 200-period moving average, the pink orangey line, just taken out decisively, there's a chance that we could have symmetry between May the 25th, the lower 32,500, the higher 35,600, August the 1st, that's the day we actually went short and this low coming in right here on the 6th of June, that was Friday. Well, within that context, what we were looking at is there was a low that was made at 32,846, was just a little bit above the low that was made on May the, no, June the 1st of 32,704, so that was a low that was the first low, well, so far it's held and the inside wedge dash pink support line hasn't been taken out, so this move right now is possibly the start of something and we're getting to that as soon as I can go to Garrow in Newport Beach, Garrow, how are you? How are you, sir? I'm very good. You'd like to look at? I'm looking for a square SQ, if you have the time, I need your opinion regarding the daily chart. This has the second dot from the bottom and it's showing some strength here. Is this your idea, just your idea? Is this going all the way up to 21 day to around $47 or this is, I'm just hovering down here, it's going to hover down and again. Yeah, I think this is still in the basing area. It's had a very nice move, it's down 68 cents today and 43.16, we were looking for some kind of test of the 40 maybe 39 area, what did it actually get to? It got to 40.77, nice green ball on Thursday, a very nice green ball on Friday and today it's stalling a little bit. If I look at the weekly chart, this weekly chart says is preparing an attempt at a rally to the 45, 44, 45 area. But I think you have any position at this particular point? No, sir, I'm short. Today I shorted it. And I'm just looking at it. I'm in profit. I may get out a little bit later or so on. But no, nothing at this time. My positions are very short in the longest that I wait is about 15, 20 minutes, sir. Okay, with that said, let's just go to the 10 minute chart right now because I agree with you. That's why I was stalling to say I don't see much on the upside. So let's go to SQ, which is square, which is actually blocked. They changed their name. So you are short. If you are short, then what you want to see is training at 43.24 right now. You can see that the 10 minute went right to the 200 period exponential moving average. And that was peak. Let me just check. Yeah, this is a brand new buy mode. So this is A, B, C, D, E, F. Yeah, so this made a top at about eight o'clock this morning, then across negative. And oh, that's not eight o'clock this morning that was on Friday. So now what we're looking at, I just be a little careful because it's trying to show a little bit of strength and the selling's been intense. My thinking is on the very short term, and that's the only reason why I saw it, because I see in the very short term there's an attempt to move higher, but I don't see it as a big move up. I just think it's an attempt to move higher. So I'd be a little careful because right now it's trading at 43.28. The stochastic is very good at 89%. And that just says that there's good support and it's got 43.21 as support. I kind of like what I'm seeing right now for a bounce. And I think 43.57 is a 200 period moving average in the one minute chart of square. And I don't know if it's going to get there, but it will get there if it's able to close in any three minutes, three minutes of a one minute bar holding over 43.40. And then all of a sudden I think that's in play. But right now it's got a little bit more strength and weakness. But I don't think this is the big move to the upside and the longer term. But yeah, I hope that helps you. Absolutely. Thank you, Shane. Thank you so much. Thank you. If you're looking for potential trading setups in the stock market, then Rocket Equities and Options Report is a newsletter you should try. Tommy O'Brien delivers options and equity trades when the markets present them using a combination of fundamentals and technicals. Sign up for Rocket Equities and Options Report today with a 30-day money back guarantee so you have nothing to risk. 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So on the way to our overview, I also want to just tell you that I'm going to go to a couple of questions and do that on the way because I might forget them and it's my, I did this on Friday, I written it down because I keep getting asked by a number of people, is it time to buy Eli Lilly and on Friday during my show, I wasn't going to do it for my subscribers because we were looking at some other things completely in different areas. I forgot to say, this is the time for a trade, not the big long term buy, but at least for a trade that could allow you to hold much longer than normal to buy it. And it would have been run about the $5.52 because it already just moved out because it gaped up higher by the 10 o'clock time I got into my show. And I completely forgot about it until I was looking at it. Last night, I think I said, Oh, no, did you forget to try? I'm almost sure I forgot to talk about it. I might have, but I got a feeling I forgot it. So now it's got an even bigger move than I expected today. It's up 11 at $5.76. And it was at $5. I believe at about my show, it was maybe at $5.50. This is 25 points. I mean, 25 points about it. It's, you know, what about 23%. So from the low, so yes, I like this. And it's my apologies. I know that if you had, if I'd seen you put something in the dead, because it was a really important session, because I was expecting that there could be a turnaround. We got the turnaround. I was able to add for subscribers to some of our long positions. So I was really busy. But wow, that was silly. And why was it silly? Because at the five, at about even 552 level, what I was saying is I would have a five point stop, which is a 1%. That's nothing for a $552 stock. Because if this was the move that was the oversold move that I was anticipating could happen in this environment, Eli Lilly, we're looking at large farmer with really fantastic products online, especially I guess a diet one or weight loss, I should call it. And here it is. So now I'm sorry, because if you get in now, you've lost that risk reward that I was really looking forward to for some way for the people that were asking me. So it's a little difference. I'm going to say, you've got to use less. If you're getting in now, because it's a $574 stock. This is not like when I have these streamers under $10 and all that you can hop in and hop out. I don't think you really want to be hopping in and hopping out of this. So this is what I'm going to recommend. We don't know if the turnaround on Friday and I have to go into them and I'll do that. But I want to spend a moment in this because it was asked enough of me by quite a few people. I'm going to say you've lost the opportunity because I would have said start a small trading position. Now I'm going to have to say make it a half or even a little bit less than the trading position. But I would start the position here. But now you've got that risk reward because if the general market, let's just say this is real. This is absolutely obscene optimism right now, because you've got second war potential right here with almost everybody else involved. What are you thinking about going along? Well, you know, the market sometimes, if you look historically, markets sometimes fights its own battle and actually moves up during a period of strife when it comes to war. That's more the bigger picture of what happens in war. But in the shorter term, if there's a sell off the end of the day and instead of being looking at the down 50, let's call it 60 and there's to be down 13, we double that to the downside. That's going to be a problem and that will drag Eli Lilly down. So I I've lost for you. I've lost that impetus that we would have had if we were in early on Friday, where I could be looking at and say, OK, now just raise your stuff. How easy would have that been? Well, it's not there. So now it's a little difficult. So I'm going to say out of the position you would have started, I'd only put a third to work right now and let's give it two or three days because if it holds the day's lower, 563, that's 10 points lower, by Wednesday, then it's saying, hey, fund managers are looking at this as counter to anything else that's going on in the market. So start your position, but it has to be small. In fact, I'm not going to tell you how much to do. Think it through and say you've lost your biggest your littlest risk reward. Now it's somewhat bigger, but it does look really good. The fact that for two days, it's actually for four days, it's got green candles, that money's going in says that this is what people want, even though it's extremely, I wouldn't say overbought on the dating because it's had a bit of a pullback. And even the weekly, but the monthly chart says it's acting extremely well, but this is pretty much a vertical move, at least for the last five, six months. Hope we got that out the way. Next question came in. Did I just hear a thing? Yep, I did. I've got Bill in Montana. Bill, how are you? Great, Bezel. Good. I got a general question for you. And as an example, I'm looking at Bank of America. Yes, BAC. OK. And I, for example, it's in when looking at the downward move, it went from 50 to 30. Yes. And then did a more or less three eight to up to 38. Correct. And so if you're looking for a support level, perhaps down below now in an ABCD, do you use the $20 from 50 to 30 or do you use the 40 percent and would look at 40 percent down from 38 or $20 down from in general in in terms of your trying to, you know, gauge where they now come in in a in a chart like that. That's a that's a very good question because I spent quite some time over the last I'd say five or six marker days really looking at Bank of America. I've spoken about it before. I've said this is one that we over the years we've had it and we've held it for months and months.  Get it and do it again. Haven't done it for a long time. But what I did, I just want to give you a little perspective of why so far I'm avoiding it. I'll get you on to try to answer your question. One is that and we got a more online bank on Friday. Not quite at the level that I want it, but lower than the high of the day, which is good. And I'm watching it closely today and it's acting way better. Look, Bank of America is down one percent down twenty seven cents at twenty five point eight. And the one that we got is down point twelve. But it is within a penny of what we paid for it on Friday. So I'm I wanted to put it into perspective that I think in the financial area, and this is really important in the phase that we're in right now, the XLF and really needs to move. And I think that they are kind of stuck. We'll see what JP Morgan does on Friday and some of the other banks. But you can see this 200 period exponential moving average in the weekly chart of the XLF. It's like a magnet. It refuses to let the price go much higher because it keeps coming back. And it even doesn't let it go much lower because it keeps coming back to the 32 83 level Bank of America. Let's go to. So what I like to look at, and I'm sure you've heard before, I look at a pattern that I call the dreaded age. It's the lowercase H. You just have to move something away. There it is. So it looks like a lowercase H. And you can see a whole bunch of them here is Bank of America from the 50 point one, one high or back in I think it was February, January, February in the weekly chart. And it ran. It came straight down for 50 11 down to the slow of about 39 and then ran up and then failed at the H pattern and took it out and it keeps doing that. Now it's making the H to M pattern. I'll be back because I think it's a pretty good question. Bank of America finding makes the student to the upside. I think all the financials will be happy. I'll be back. Commodities, commodities and bond markets are as important as ever right now with how they're driving the volatility in equity markets across the globe, which is why it's a great time to try out Teddy Kegstad's Tiger Forex report. Teddy Kegstad breaks down the forex markets every Monday using his 30 plus years of experience as a trading veteran of futures, forex, stocks and options. 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Am I correct? Yeah, you know, percentages or points, I guess I sort of felt like on serious declines, points can become not feasible even, you know, so kind of I kind of drift or percentages on declines and points going on going up. OK, you see, if you use that and it works for you, I say just stay with it because it's a methodology that you consistent in. But this is what I'm really looking at. So the pattern that I was talking about is both a pattern, but within the chambering methodology back in the practice days when he was talking about Elliot Wave and everywhere you went, CNBC, whatever medium you were looking at, people started to talk about Elliot Wave. It's a it's a very, for people to understand it, it's very helpful, but there are a lot of alternate counts. I said I don't want anything to do with Elliot Wave. I've got something very different. Although I know a couple of people use Elliot Wave and say, wow, in the Chamber Wave, it very often turns out that you get the results in a much simpler way. But anyway, with that said, I look at just counting each success of the high peak and label them ABCD on the way down and it can go E, F and G, but on the way down I use lowercase and it's the same thing. So it's a very simple idea. I want to mix it up with Elliot Wave, which uses numbers and numerals. So mine's alphabet. Okay, now we've got that out of the way. What I am looking at is if you look at the weekly chart of Bank of America, there's a down channel. And that's why this particular week is so important because when you get the arch formation and there was an arch yet another arch formation in the weekly chart, the acceleration down of the last few bars that takes out the left side low. In this case, the 27.68 low of end of March, beginning of February, April, we got that.  So today it's kind of important to establish some kind of a base, but the measured move for me, and I'm going to blow this up right. I shouldn't use the term blow this up in this particular environment. So I'm just going to, I'm going to expand the chart right here.  First of all, there's the big rectangle, the long, long narrow rectangle that just lasts a lot longer than your patients, and it's been in there. Now, I don't know if you can see my chart. I'm going to do this, I'll have to move away the background of the rectangle. I'll just grab it and move it to the side. Move it down. And you can see I've got a pattern yet that looks like an oval.  It's using somewhat different technique, but the result should be more or less the same. I do it as conservatively as possible. So I go the straight line move. In this case, I've gone with one, two, three, four bars, four weekly bars. I go click and I got this measured move and it takes me to the high. And then I see where does it go to? And you can see that it goes down. It's this particular, the only time I'm using it right now is for this move that went from the candle, the doji candle of the week of the 28th of July in Bank of America, down to the low of the 25th of August at 2818, then to the high, not this high. I want to be as conservative as possible. So I'll go to the high over here on the 15th of September, 2944. And the measured move takes me to just about 24.89. So it's a dollar lower, but that also says, who if it does break that level this week and it needs to be this week, there's a chance that this declining and I'm going to do this. I'll make this green. I like to always have these colors. I came to this because I'm a very visual person. There it is.  And this would be my target. If Bank of America doesn't ready in the next two days from 2580 to about 2635 in the next day or two and takes our Fridays low, then I'm looking at sometime this week, the green line will start the support level and that'll be at 2442. And if it takes that out, then I'm looking at this whole channel, which has been, I could even do a symmetry, bar symmetry. Yeah, but I don't really want to interfere with this. I like it as it is. And it says, if Bank of America is weak in the next two days, you got to be careful because this whole area between just over 24 and just above 23 is going to be absolutely imperative to hold. That's the move that I like. So that's the one-to-one, a little different to your usual, what you hear from most of the hosts here at TFNN who use their own technique very favorably. They use the lightning bolt, you know, that pattern. I've got a slightly different. So this is the way I'm looking at it. If it's, if it, certainly if it closes last week's low of 2547, then I'm looking at this particular little mini channel right here. I don't know if that helps you, but that's kind of the way I'd be looking at this right now. And the reason, only reason why I decided to avoid Bank of America for quite some time is something's wrong with the chart. It's got, it's got the meral part of it, which is the brokerage. And if, you know, if you look at Schwab, their chart doesn't look anywhere as bad as this. So there's something quite, not quite right with Bank of America, and that's why I've avoided it. But I don't know if you're looking at it to buy, but if you are, these are the techniques that I would use. Well, no, I'm, I'm, it's, to me, it could, could even possibly get to 18. I'm not disagreeing with you at all. I don't know about 18 in this particular move, but certainly if that whole 23 area fails. No, no, over many months, but. Yeah, because it's not, it's making lower highs. 23, 20, about to, to be intermediate support. Oh, I can't disagree with you. It's making lower lows and lower highs in the monthly chart, and that's what you look at for a trend. So yeah, so we're on the same page there, but you asked me about the one-to-one. I'd first do that. Yeah, no, that's crazy stuff, Basil. Thank you. You have a great day. Good question. Thank you very much, Bill. I appreciate it. Call again. So folks, now you've got the Dow down 91, the SMB's down 14. This early morning search to the upside is the reason why I said to my subscribers, we're going to raise stops, we're going to take tight stops, we're going to take off. Oh, so Garo, Garo, that price did go high. We're right to the tune. I think I mentioned the 200-period moving average of 43.54. It hit it exactly that square. Now, if you are still short, and you said 20 minutes, so you're probably out, now's the time that you could start to see a retest. If the general market stays weak, square could go all the way back to 42.98.  Okay, so this is what I want you to do. I didn't go through all the different industries. Let's just run this again. There's the Dow. Oh, it's a break. Oh, dude, really quickly. Oh, I can't do the Dow in the one-minute chart. I want you to have a nice, big look at this. So we've got a break coming up, and I'm going to go, as we go out, I'm showing you the Dow, so I can give back some of the early gains. Got repels of the pink nine-period moving average. I'll be back in a moment. Baselchap and Tiger technicians, I'll see you in a few. Gold report. As a precious metal, gold is still king. It continues to hold the most effective safe haven and hedging properties across the global major trading hubs of the London OTC market, the US futures market, and the Shanghai Gold Exchange. The Gold Report. 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It's trading at 786 as UNG United States Natural Gas Fund. I haven't done, drawn in the, I think I did it in natural gas, continuous contract. Oh, typed it in the wrong place, sorry. Let me put it over here. NG. Oh, I didn't even do that. So yes, so UNG, this is a very nice move. I said it's going towards this candle. I wasn't sure it was gonna fill the gap. Well, not only did it fill the gap, it went to 793 today. The high that we're looking at is 815. What was the eight? 808, that's right, 808 on the 9th of August. So it's getting towards in this beautiful cup formation. Remember, I'm all about cups and arches, cups and all straight line, cups, arches, is a straight line mixed with the cup. Now, what we're looking at is, how does it hold going into Wednesday, Thursday? That's gonna be important because I haven't been saying for a long time, the rectangle formation can last a lot longer than your patient. Remember, I've had webinars based on narrow rectangles and large rectangles. And what I was looking at is, this particular pattern with a trading band says, if there is a push above 808 and a close above that, you've broken this resistance and you can start to the left side, which takes you all the way to 10. And that seasonality wise, this is kind of where I would expect that natural gas. So this is very nice. I missed it for subscribers, so subscribers. You spoke about it. I just didn't feel I wanted to chase it. And you had to be in it to enjoy it without any risk. That is, gap down or gap up risk in the morning. It's doing very well. If this week on a weekly basis, it can close above 808, that's gonna be really important. A question came in here. Mike and Orman Beach, Mike, how are you? Haven't heard from you for a little while, are you good? Yeah, I've been pretty busy, Basil. Basil, I've got the ultimate question for you. Do you think the low that we had last Wednesday at 4235 and the futures, do you think that is A low or D low? And what would make you make the determination either way? So that's a question, of course, not only you, but a lot of people have been asking themselves as well as me and other hosts. I think that this is going to be very important. For me, it's just A low. I don't think it's D low, not at this particular time. So I'm treating it as if it's perhaps an internal low that needs a residual low to confirm it. But in the meantime, I'll tell you what I'm looking at. The fact that the S&P right now is only down 14 after that, I mean, the news over the weekend, the fact that crude oil did spike immediately, gold spiked immediately, and then they kind of stalled is telling me that this is a process and the process itself is the most important thing. So when I get questions about, once we have Kevin over the months and months has been talking about starting position and then adding to the position on a dip and then waiting, waiting. I've been saying for this other position, which I'd also treat as a split position, I wouldn't go all in on the next move down. I'd still split it. I'm waiting for, I've got eight series of lows that have been trying to form. But the, I want to see to get the next big load like we did in October or just a really decent low. I want to see the VIX close at least one day up in the 23 to 24 area. And intraday, I would like to see 20. I wouldn't even be surprised if the news is so bad that it goes to 38 to 37. And then I'm starting to look at it and say, wow, I'm getting really close to such an oversold condition that we're looking at a rally that if it turns around, doesn't last a week or two, we're talking months. I haven't got that yet. I think that's your question. That's my answer. Yeah. So basically we need to see another panic low. Yes. Big spike in the VIX. And I prefer- Before you're totally convinced. And I prefer that it didn't happen under the conditions that we saw last week where we saw decent bounces almost every day and then a sharp sell-off. I prefer just a week, week, week until everybody just- Even people expecting a low and wanting a low say, I can't stand this anymore. If it doesn't happen now, I'm done. And that's when you're going to get your loan. That's where the VIX has to be. I prefer a Friday or a Monday. You remember, I said, down on a Friday, we've done that before we did that back in March of 2000. We went along in 2009, I'm sorry. But then it was Monday for the S&P. This is the reverse. This is Wednesday on the S&P, Friday on the Dow. I think we are attempting to form some kind of a tradeable rally. But I don't have this. This isn't for me, not the big one. No. Yeah. And Friday's bounce. I mean, in the past, I have seen huge bear market bounces also. So I'm kind of agreeing with your thinking.  Well, if we didn't have the news on Saturday, I think we would have followed through on the upside. And that would have meant many of the conditions. You remember, I also have the Dow chart where on the 6th of October was exactly the time sequence and bar symmetry from the left side to the right. And that said, whoa, that's a good day to have a really good turnaround. And we got that turnaround. So all I'm saying is that many of the conditions were met. But I'm also at the same time kind of impressed the day is young. But I'm impressed that we aren't down 500 points right now, based on all the news and that crude oil isn't up $5 and gold isn't up at least $38 to $42. It could still happen the day is young. But so far it says what's going on now is in the not in the economic purview that deals with the market in the United States at this particular moment. I think that's the only way I can assess it. It is emotionally and in every other way something you just say, wow, how can the how can the market run against its backdrop? But only reason why it's holding is it got very oversold and it needs to work that off in some way. So I hope that answers your question. Yes, it did, Basil. Thanks for everything you do for us. Thank you very much for calling. Always appreciated, Mike. So folks, let me just do this because I know we're going to run out of time. I've already got a bunch of things. Maybe I've got the daily chart up. Look at the gold. And as I say, gold is I mean, the days the week is young. We just barely begun. So it's gapped up and it's holding. It's not holding. It's up 18. That's not nothing to say. Well, what's the big deal? It's a big deal because we've covered one, two, three, four, five. We've gone to the fifth trading day from last week in one fell swoop. So I'm thinking, look, GDX, these are the gold stocks. We spoke about this the other day. I said, I like the fact that the GDX, the gold stocks are running and gold in a way is lagging and that the move on Friday. You remember, we talked about a sort of a bill. Here's another way that I look at the one to one. To the downside, I use this pattern that I call the chamois forwarding exclamation. And when it turns down, that's when you can get it. Well, it went almost to the panning of the GDX and now it's running quite nicely. So I'm not saying gold is not acting well. I'm just saying I'm a little surprised it isn't as high as it should be. And that even the GDX only have 50 cents. Should be up about 78 cents to 92. And it's not. The day he's young could still do that. I'll be right back. Giles down 140. You might think that if you want to be successful at trading in the stock market, you're going to need a crystal ball. After all, it's impossible to predict the future, right? Like any endeavor in life, before you decide it's impossible, get some advice from the experts. You might find that it's not so impossible. After all, for daily market overviews that give you direction on the key indices, selective stocks and commodities, subscribe to the opening call newsletter at TFNN.com. The opening call newsletter is written by Basil Chapman, creator of the trading methodology known as the Chapman Wave. The Chapman Wave up-down sequence gives you an edge in identifying price turns, finding the peaks and valleys and stock prices.  First time subscribers also get a 30 day money back guarantee. If you're not satisfied, let us know and you'll get a full refund within 30 days of signing up. TFNN.com Educating Investors Everything in the universe is governed by the Fibonacci sequence. This mathematical principle is responsible for everything from the most aesthetically pleasing artwork to patterns in the stock market. To stay on top of stock patterns you can take advantage of, sign up for the Fibonacci 24-7 newsletter at TFNN.com. When you subscribe, you'll get a weekly report from Veteran Day Trader Larry Pezzavento on stocks you need to pay attention to and you can trust Larry's analysis. After all, he's got 45 years experience as a day trader. Larry will also provide daily charts, videos and data on the key markets that he's tracking. Expect notifications from Larry on market movement you need to act on at any time. First time subscribers also get a 30 day money back guarantee. If you're not satisfied, let us know and you'll get a full refund within 30 days of signing up. Subscribe to the Fibonacci 24-7 newsletter today. TFNN.com, educating investors. TFNN has just launched their new trading room, the Tiger's Den. Hosted at Discord, TFNN has been educating traders for more than 20 years with live programming hosted by a variety of professional traders during market hours and now they are expanding the reach with the Tiger's Den available to all Tigers and Tigresses for just $1 for the year. There's no catch or added costs when you join our community of traders. In the Tiger's Den, you can look over the shoulders of Tom O'Brien and the other TFNN hosts while they analyze charts during their live Tiger TV programs and join an interactive trading community with hundreds of members exchanging ideas, interact with other Tigers and Tigresses as they share trading ideas, news analysis and discuss the market action all trading day even at night and on the weekends. The Tiger's Den at Discord is accessible on mobile or tablets as well. So it's always at your reach. To sign up today and become a part of this educational community of traders, just visit the front page of TFNN.com. Don't forget, you can listen to TFNN live on your mobile device 24 hours per day. Go to TFNN.com and hit watch Tiger TV. That's TFNN.com and hit watch Tiger TV. Okay folks, hold on to your stick, Stratocypels, we're gonna go through this quickly. So GTX, what would make me more bullish on the GTX? And I do, I like it. In fact, we considered getting it for subscribers even on Friday morning, haven't got it. I like, well I wanna see how we close because if GTX is holding well, then I think it's gonna tell us that gold will follow then maybe at some point gold will lead and then silver will be doing the same thing. So it's not out of the question. I'm just saying I'm a little surprised it isn't even higher at this particular time. Rig, this is a TransOceanRIG, there we go. I just think it's stuck and I've been saying that for about a month or two and it's just kind of stuck. It's fabulous, but it's kind of stuck. It's having a high level consolidation. So when I look, it mustn't close under 70 and if it can start to close for three days above 842, then 860 and then 880, that's gonna be very positive and say the TransOcean, the offshore drilling, oil and gas is in favor again. TLT, let's just wait until tomorrow. I need to see TLT if it's able to hit 85 to 80, sorry, 8650 to 80, 8790 in the next couple of days. That'll be really good. But right now it's just, it's a nice move up. The next question was that PLTR, Palantir, I spoke about this other day, I said, I like the pattern. I like it, it's having a nice move up to 65 cents, 60 cents in 1725. It's going to the area that I said it needs to get into that's this candle right here, the Chapman Wave Roman candle of August. I would say between 1790 and 18.30, it'll have, it should have some resistance but it's really doing very nicely. The next question was CS. CS is Z Scaler, very strong move today. This is a leg E in the week chart. This is what you want to be looking at. Up 170 CS as a symbol. Nice action. Keep supporting us, keep supporting us. Thank you for Steve Rose, check out most of the competition newsletter and listen to your big, big programming here. See you tomorrow.


Hi everyone and welcome to today's webinar. My name is Karina Barquette. I am a senior research fellow at the headquarters of the Stockholm Environment Institute. I will be talking to you today about islands of innovation. So a few words about myself before we move on. I am leading a team of multidisciplinary researchers working on water, coast and ocean issues, including islands. Our work covered aspects related to, for example, systems and infrastructures, disaster risk reduction and adaptation, blue economy and marine spatial planning issues, both in Europe but also globally. Perhaps something a bit more personal about myself is that while growing up in Mexico, I was constantly in contact with disaster relief work through my dad. I would say that this experience has probably got me interested in what I do today. These experiences made me appreciate water in all its forms, but they also made me a little bit scared of it. So I actually hate swimming, and though I love the sea, I'm always terrified when jumping in, especially when you live in the cold waters on the Baltic Sea. They're never warm. Anyways, today I will be talking a little bit about islands, climate and innovation. But before that, I would really like to know who's behind these screens. I can't see you, but hopefully I can find out a few things about you. My colleague Brenda is going to help me paste the link to a mentee on the chat. If you don't see it or if you cannot access it, just go to mentee and provide the code and tell us a bit about who are you. Where do you come from? What do you do? It would be great to know who's behind these screens. And I'm going to change my screen just to see how the answers start floating in and I'll give you a few minutes. That's super nice. Creative, kind, somebody from Slovenia. Maybe you hear my pain with the cold weather. That's great. That's super nice to see how you all define yourselves. I should also say that I found out a few minutes ago we're registering for this webinar that two years ago, since two years ago I stopped being young. So today I'll be the granny reading you the bedtime story, I guess. Nice to see. Welcome everybody. It's super nice to have you on board today. I hope we can have a conversation despite the limitations. So now I'm going to continue with my presentation. So island systems are among the most vulnerable to climate change. And we know that climate change is predicted to reduce shifts in temperature rainfall and sea levels. And while these changes are exacerbating challenges and putting people infrastructure and biodiversity at risk in unprecedented ways. I would say that these challenges are also leading to innovation. This session is proposing seeing islands, not just the spaces of vulnerability, but also spaces for innovation. That can offer an opportunity to think beyond what we are usually used to thinking beyond the established paradigms and beyond the ways we usually organize society and build our cities. And I will be sharing with you some examples of innovation, but I also want to hear from you your experiences and thoughts about ways in which islands could innovate. But before we dig into the core of this presentation, maybe we should take a step back and sort of think about what is an island. But we all know that an island is a piece of land surrounded by water, and essentially every landmass, no matter how big is surrounded by ocean waters. So how many islands do you think there are in the world? And this is our second mentee. Again, my colleague will paste the link on the chat, please click on it. Alternatively, log into mentee and type in the code. And I will again share a new screen just to see your answers. You have three choices. I hope you can access the link. Right. There's some competition here between options going on. Any more answers before I change screen. Great. One more for 169,000. Let's settle for that. Let's see what the right answer is. So many of you actually were much wiser than I was when I first found out what the right answer was. There are over 340,000 islands globally. I was personally shocked to see that there were so many islands. I think most of us know about the bigger islands, which is roughly about over 21,000. Big islands are considered islands that cover over one square kilometer. But there are many, many more smaller islands that are below one square kilometer. And actually we didn't know the exact number of islands until 2018 when a group from the US Geological Survey was tasked to map a global map of islands, which is quite interesting. It took us so long time to actually know how many islands we have in the world. Some of these islands have been intensively developed and become the local or even the regional center of economic activities. But there are many islands that are not even populated or not developed at all. I want to tell you a little bit about my island story. I mentioned I grew up in Mexico, but I've lived in Sweden for about 14 years. And the last years I have lived in Stockholm. You can see Stockholm in red square way up north. Maybe some of you know that Stockholm is an archipelago and archipelago is a cluster of islands. And Stockholm archipelago is in fact the world's largest archipelago, consisting of 30,000 islands and cobs extending over 1,680 square kilometers in the Baltic Sea. And these islands are huge tourist attraction. It is also the everyday life for many people. But they're also increasingly hosting second homes that could be summer houses, summer cottages or even all year round kind of weekend houses. And if you look at the image on the right, you can see that this is one of the places in Sweden with the highest concentration of second homes. Now why is this interesting? So Cliffhanger, I'll get back to that. What I can say for now is that all of these islands are very, very different. Some are larger and very urbanized. As you can see in the far left, one of the biggest islands, Sandham. Some others are much smaller. Some of them are even, you know, one island, one house. Most of them have lots of boats activity, increasingly so as well. If you ever come to Stockholm, I highly recommend the boat ride to any of these islands. It is a rather inexpensive ride, partly because it is subsidized by the government. It is also for many, especially the islands that are closest to the city, it's the everyday commuting. Yeah, it's an everyday commute. And so because they are relatively accessible and relatively cheap to get to, they're also quite popular, particularly during the spring and summer months. And so this creates a lot of opportunities, especially for the local economy, which largely relies on tourism. But it also creates many challenges. Archipelago, like many other parts of the world, is risking to run out of clean freshwater. And this is partly due to the risk of saline intrusion, but also partly due to that groundwater is extracted at a faster pace and it is replenished. And this extraction has increased in recent years due to demographic changes. In the Swedish Archipelago demographic changes, or to be more precise, demographic fluctuations are extremely challenging to manage. Places and landscapes are usually planned in relation to projections of activities where people go, where people buy, where people live and work and so on. But in a setting with constant fluctuations, planning and financing investments, for example, from large infrastructures like water, energy, wastewater becomes very, very difficult. These types of infrastructures usually go hand in hand with the taxpayer base, which means that the larger the population, the more needs, but also the more taxes available for covering large and expensive infrastructures. But in the case of the Swedish Archipelago, the taxpayer base doesn't correspond to the needs on a yearly basis. So there are around 10,000 local inhabitants and about 50,000 temporary homes. Together, they pay for services that are used by themselves, as well as the two million visitors per year. So obviously the systems are not sufficient, but the taxpayer base is not large enough to cover for an expansion. And this creates obviously other types of challenges, which leads to insufficient services, but also not the best environmental services or most environmentally designed types of services. The overly polluted Baltic Sea suffers from severe eutrophication and hypoxia. And this means that insufficient structures, for example, for cleaning water before it flushes before it's being flushed out in the sea is not enough. I will come back to this in the coming slides, but before I do that, I want to hear from you once again. Do you have an island story? Do you live in an island? Do you come from an island? Or do you have any experience or memories from an island? Or simply you're just interested in having something to share with us when it comes to an island. Once again, please follow the link, I think, on the chat. Otherwise, login to Menti with code that you see on the screen. I will switch screens so that I can see your responses coming in. And I will give you a few minutes. We have somebody from Sri Lanka. Is there anyone else from elsewhere? Great. This is great. A lot of you have visited islands or would like to live in one. Some of you have actually grown up in an island. A compliment island. We have somebody from Comoros, from Thailand, seashells. This is great. And for those of you who live or work or have experience islands, you're probably a lot more knowledgeable than even I am. So, I hope you can share your insights towards the end of this presentation. I will go back to my screen again. Thank you for sharing your island story. We come from different islands and I shared my island story, which is it's got its challenges, perhaps not the immediate thought like if I would mention Sweden you wouldn't immediately think about vulnerability or even islands necessarily. But I think all islands have commonalities and the challenges that I described in the Stockholm area are probably faced by many other islands. But there are differences. There are islands that are way more vulnerable than others. If we were to look into the whole equation, then studies point out at smaller island development states or developing island nations so-called seeds as being the most vulnerable islands globally. Especially in the Pacific, you can find some of the world's lowest islands. If one would take into consideration two things, one, exposure factors, which means the people, the landscape and the infrastructures that are exposed to risks. Plus a second factor, which is vulnerability and vulnerability is considered a lack of capacity to manage effects in the short term and to adapt to change climate in the long term. Then we can see that not all island nations are equally vulnerable or prepared to deal with the effects of climate change. And amongst the islands in the Pacific Ocean, Papua New Guinea is considered the most vulnerable. On the other hand, Fiji is less vulnerable and more ready to invest in adaptation measures. So even amongst the Pacific Islands, there is differences between them. What we see is that many island nations are taking measures to reduce risks instead of trying to prevent the effects of sea level rise. You can't prevent the effects to sea level rise.  For example, Fiji and Tonga are already working to expand their economy by investing in public infrastructure. They are adjusting taxes and they are reorienting the economy away from agriculture to our services and tourism.  And you probably wouldn't want to have your crops close to a potential risk area for saline intrusion. Now it's expanding its network of marine protected areas to reduce the strain on its reefs and fisheries. The Kiribati government has bought land in Fiji to grow crops and perhaps even evacuate the entire population of the country if worse comes to terms. And parallel to this, there's a lot of international pressure, for example, the World Bank pushing neighboring countries to help out. So for example, Australia and New Zealand are being pushed to allow for migration of people displaced by climate change from some of the most vulnerable islands like Kiribati and the other Pacific islands. Until now, New Zealand has allowed 70 people to migrate from New Zealand to New Zealand per year. So we'll see what happens in the coming years, but I think it's important also to recognize that pressure on islands is also pressure on the neighboring countries to take action. And how and how that will happen and how that will be financed is still a question that's not really solved. When it comes to finance, we do see an increased stream of finance globally and this is what this figure is showing. Between 2010 and 2014, a total of 748 million in finance, $748 million in finance, was principally targeting climate change, was committed to the Pacific Island countries. And this was, this corresponds to about 6% of the total flows for the Pacific that were reported for those years. The largest recipients of climate finance are Timor-Leste, Samoa, Tonga, Banuatu, Papua New Guinea and Fiji. If you want to read more about finance, you can find it in this publication that my colleagues Aaron and Nella published a few years ago. Really interesting to see and really desegregate finance, what goes where, who gives who and what finance is useful. Within the Pacific Islands, the Caribbean is also facing real threats and within the Caribbean we also see differences. And it sort of highlights, these differences highlight a fundamental difference between hazards and disasters. They are natural, they are a result of climate and weather events, but their aftermaths are human made. So disasters are human made. And nowhere, this is clearer than in the contrasting reality between Haiti and the Dominican Republic. Both of them are located in the same island. Both of them are exposed in various ways to storms, to storms, hurricanes and climate change. But there are fundamental differences in both their landscape and societal organization that defines when a hazard turns into a disaster and whether a hazard turns into a disaster. On the left side, you see probably a line, a dividing line, which is a border between Haiti on the left side and Dominican Republic on the right side. And you can see the degree of deforestation on the Haitian side versus the green coverage on the Dominican Republic. And a lot of research has pointed out, while there are many variables involved in creating a disaster, deforestation being a very key thing for enlarging the impacts from storms and hurricanes. Partly, it basically, there's no hinder for the storm or there's no hinder for the waves or there's no hinder for the wind to destroy everything in its path. But they've actually found that deforestation can trigger earthquakes through the rapid erosion, which is in turn made worse through hurricane activity. So that was an interesting finding and Haiti and Dominican Republic are both exposed to storms and earthquakes. If we look at what's causing deforestation, we go back to the issue of systems and infrastructure because the main reason for deforestation in Haiti is the demand for charcoal, which causes then people to chop down trees essentially to be able to get fuel for cooking. And so we are back to having the systems and infrastructure for world living but also for adapting. And that we see everywhere, but in islands, there's basically no escape so this is so clear in an island setting that it has brought to light for researchers and even policy of the importance of resilient disaster resilient infrastructure and how this is going to define whether the communities are able to reduce vulnerability and adapt. This is an image of a house in Puerto Rico, nine months after Hurricane Maria. That caused the largest blackout in the US history because Puerto Rico is part of the US as you know, it's the second largest power outage in the world on record. It filled a housing shortage, suicide crisis, the spike in the murder rates, and about 4,600 deaths at least was recorded. Nine months after the hurricane, which is when this picture was taken, the power grid was weaker than ever. And so, once again, this brings light to the importance of systems and infrastructure, because very often we see that the systems that are worse heat are also those that are most critical. Very often, these are provided through centralized weights, centralized wires and pipes. This way of building of designing cities is very much following one way of thinking one way of development. But what we can learn from islands is that there is a need to think differently about the way we build cities and about the way we organize societies.  How do you think islands can innovate? How do you think islands can think differently? And what can islands teach the world when it comes to adaptation and building resilience? Once again, please follow the link that my colleague is posting on the chat. If you have any problems, just log into mentee and this is a code that you need to access the question. I will give you a few minutes to think and I will switch my screen to see your incoming responses. I will give you a few more minutes. Your answers are looking great. I'll give you half a minute more in case there's anything incoming in before I switch back. So we have suggestions for off grid technology, ecotourism, there's a need for more funding, awareness campaigns and capacity building, electric resources from natural resources, I guess renewable sources of energy, mariculture, floating structures, shipping industry. And I think there's a couple of answers that perhaps belong to the previous question, sustainability conservation. Thank you so much for your answers. I'm going to go back to presentation. So I want to give you some examples and in fact some of my examples are really well aligned with some of your comments. It would be great if we could discuss this, of course, this webinar doesn't allow for discussion, but perhaps towards the end of this presentation, you can tell me a bit more about what you know, which relates to some of the examples that I have or perhaps even completely new things. And what we see in our research, but also in the work we have with the private sector is that islands are coming up not just as places of vulnerability, they're also triggering innovation. So, one example is solar panels, we all know and have witnessed the extreme pace at which solar panels have developed, the prices have become much more accessible. It basically has taken over in a much faster rate than anyone could have imagined just a few years ago. However, panels have also been designed for stable conditions. Solar panels have been installed in conditions of extreme hazard events like the Caribbean or many of the islands could not withstand extreme weather. So based on these conditions, we have private companies and researchers developing a new type of solar panels that can withstand wind intensities at the level of hurricanes. And many of these have now also been able to be adjusted to extreme weather conditions, extreme cold conditions, for example, ice and snow that usually breaks the panels have now been also inspired in the development of solar panels in islands being adapted to other contexts beyond islands. So this is a very interesting and important development when it comes to solar panels. Another, let's say less techie development, but super important is how the limit of space availability in an island forces us to think multifunctional. And once again, I'm going to use the example of Sweden, even if even if it's, if it doesn't present so extreme and extreme case as many of the other islands. As I said, the threat or the risk of running out of water is real and in fact we've had extreme droughts for several years in a row. And this has led to building nature based solutions that can be used for multiple purposes. What you see in this picture is a water retention pond that can be used for irrigating agriculture. It can also be used by cattle, while it can also capture nutrients running from agricultural fields. A third type of innovation which was actually mentioned by one or a couple of you in your comments is the use of offshore platforms. And essentially we know that the level of the sea is rising. Islands know that they are going to lose land. So we need to use the ocean space more actively. And one way of using it is through these types of platforms. We've seen offshore platforms normally through oil platforms and gas platforms, but platforms have potential beyond these dirty types of energy. And what we're exploring and what we're seeing is the use of this space for many different purposes, whether energy in the same space where you have aquaculture activities or energy to power for example, wastewater treatment plants offshore, or even desalination plants. And another example which also one or a couple of you mentioned was a floating solutions. So why not convert ships into mobile, multi-purpose mobile modular marine solutions, what we call in our gridless initiative as for solutions, as M4 solutions, mobile multifunctional modular and marine. And these are essentially solutions that can travel wherever there is need, and it wouldn't take space in land, and it would be a much more demand based type of solution. Now, I've given you lots of or let's say for examples, all of them are interesting, all of them require more exploration, especially because the marine space is an unregulated space in contrast to land based activities. So how will all of these different types of activities be managed and governed in the marine space is not really clear. This is some of the stuff that we are engaging with and hopefully we'll try to figure out in the coming years. But for now I think that the main message of this presentation is that there are exciting opportunities. Islands can provide, can provide, can be spaces for innovation, and we can learn a lot from them on the types of solutions that can be applicable beyond the spaces of islands. An island in the Pacific Island shares, in the Pacific area shares a lot in common with the Caribbean but also with islands in Sweden and elsewhere in Europe. And this is sort of the main message of my presentation today. Before we wrap up and I hand the word back to Hannah, I am wondering whether there are any questions or comments in the chat or in the Q&A function. My colleague Brenda will help me keep an eye on the chat on the Q&A. Feel free to just pop out anything you want to comment or question. Not sure I can respond to everything, but I will do my best. Brenda, I will ask you to open your mic if there is anything that is coming through the chat with the Q&A. Absolutely, I'm keeping an eye on the chat. Thank you. In the meantime, I would like to thank you for joining us today and invite you to get in touch if you want to learn more about our work. And visit the SAI website, we've got lots of resources there for you. And of course the presentation will be made available. It's being recorded as well I think. I will, I think hand over to Hannah. I have a comment saying it's very informative. Thank you very much from Sri Lanka. Thank you so much from Sri Lanka for joining us today. Hi everyone. I can turn on my video but maybe, maybe Mario if you can help me. It says that I need permission from the host. There we go. Hi everyone. My name is Hannah Begweit. I am the project manager of this summit and it's really, really great to have you all here. Karina, thank you so much for a brilliant webinar. It was really interesting. The material, the slides that she used today are available in the chat in the Google Drive link that my colleague, Mario just sent. And I thank you very much for joining us for these last, for these pre-summit webinars. This was our last one, but I do encourage you to have a look at the recordings that are available on our website. You can see dash youth.org slash y40. That is also in the, in the chat box. We're uploading, we will be uploading Karina's webinar today or tomorrow the latest. And, but we have two out of four uploaded already that we encourage you to have a look at. For those who have signed up for the summit. In two days. And for everyone just explore the recordings, explore the material. And I wish you a wonderful rest of the day or afternoon or evening wherever you are. Thank you so much.


Last time, we have defined divided difference and proved some of the properties of it. Today we are going to prove some more properties of the divided difference, but before that I want to consider map from C A B space of continuous functions, which will associate f with a interpolating polynomial. And we are going to show that this map is linear, when you consider the three functions 1, x and x square, then you are going to have convergence. So, we look at a continuous function defined on C A B, x 0, x 1, x n, these are n plus 1 distinct points in the interval A B. We know that there is a unique interpolating polynomial which interpolates the given function at these points x 0, x 1, x n. In fact, that interpolating polynomial is given by summation i goes from 0 to n f x i l i x. So, this is the Lagrange form. Now, consider map i n from C A B to C A B, which will associate our function f to the interpolating polynomial. Now, it is easy to see that this map is linear and by that we mean that i n of alpha f plus g, where f and g are continuous functions belong to C A B, alpha is a real number, is equal to alpha times i n f plus i n g. P n approximates, P n interpolates the given function at n plus 1 points, it is a polynomial of degree less than or equal to n. Q n interpolates g, then when you look at alpha P n plus Q n, it is value at x j that is going to be equal to alpha f plus g at x j.    Now, look at the three functions, f x is equal to 1. In fact, the interpolating polynomial they reproduce the polynomials. So, when f x is equal to 1 interpolating polynomial will be equal to f for n bigger than or equal to 0. For the function f x is equal to x, i n f will be equal to f for n bigger than or equal to 1 and f x is equal to x square that means, i n f will be equal to f for n bigger than or equal to 2. Hence, for these three functions norm of i n f minus f it is infinity norm will tend to 0 as n tends to infinity. Now, recall Karoukin theorem. In the Karoukin theorem it was that if you have a sequence of maps which are linear and which are positive such that there is convergence for three functions 1 x and x square. In that case there is going to be convergence for all continuous functions. In fact, when we wanted to show that the Bernstein polynomials converge to f in the infinity norm it is Karoukin theorem which we have used. Now, we have a new i n new map i n which is linear which converges in for three functions 1 x x square, but it is not going to be positive. So, we have we can look at a simple example. Look at f x is equal to x square x belonging to interval 0 to 2. So, clearly f x is bigger than or equal to 0 on this interval. Now, we have to look at interpolating polynomial. So, the graph of function will be something like this. So, here you have got at 2 it is going to take value 4. If I take my interpolating points to be 2 and 1. So, I choose x 0 is equal to 1 and x 1 is equal to 2. So, I consider the linear polynomial. So, I look at p 1 a polynomial of degree less than or equal to 1 such that p 1 at x 0 is equal to f x 0 and p 1 at x 1 is equal to f x 1. We know that it is going to be the straight line passing through the point x f x corresponding to value 2 and value 1. In fact, we can write p 1 x explicitly it is going to be equal to f at 1 is equal to 1 f at 2 is going to be equal to 4. So, p 1 x will be given by 4 into x minus 1 plus p 1 x is equal to 4. So, p 1 x will be given by 4 into x minus 1 plus 1 into 2 minus x. So, that when x is equal to 2 it is 1 when x is equal to 2 it is going to be equal to 4 and when x is equal to 1 then it is going to be equal to. So, it should be 1 minus x you want that p 1 at 2 should be equal to 4. So, it will be 1 minus x at 2 it is going to be 4 at 1 it will be 1. So, it is 4 into x minus 1 plus 1 into 2 minus x. So, that is going to be equal to 3 x minus 2. So, our p 1 x is 3 x minus 2 p 1 x will be less than 0 if x is less than 2 by 3. So, even if f x is equal to 4 it will be much bigger than or equal to 0 p 1 x need not be bigger than or equal to 0 and hence you have the Karoukin theorem it needs the function to be the map to be positive which is not the case and hence it is not applicable. In fact, I had said that for the interpolating polynomial there is going to be problem about the convergence that for not for all continuous functions you are interpolating polynomials they are going to converge. So, the map is it reproduces polynomials and hence it we are going to have convergence for all polynomials it will be a linear map, but it will not be positive map. So, now let us get back to the properties of divided difference. So, the divided difference is defined as look at the coefficient of x raise to n in the interpolating polynomial p n x there is a unique polynomial p n such that p n x j is equal to f x j. So, whatever is the coefficient of x raise to n that is going to be our divided difference based on points x 0 x 1 x n. Now, last time we saw that if f is a polynomial of degree m which is less than n then the divided difference is going to be 0 because if f is a polynomial of degree m which is less than or equal to n then in the interpolating polynomial is function itself f being a polynomial of degree m it will be of this form. So, a 0 plus a 1 x plus a m x raise to m to that we add 0 into x raise to m plus 1 plus 0 into x raise to n. So, the coefficient of x raise to n is going to be 0. So, this was the first property second property is it does not depend on the order of x 0 x 1 x n and the third is important property which is the recurrence formula that the divided difference based on n plus 1 points is expressed as difference of two divided differences based on n points divided by x n minus x 0. Then the value of divided difference in terms of function values is given by this formula where w x is function x minus x 0 x minus x 1 x minus x n. So, these were the four properties which we proved last time. Now, today we are going to prove another property and that is the linearity of the divided difference. So, if you consider f and g to be two functions and alpha to be a real number then the divided difference of alpha f plus g will be alpha times divided difference of f plus divided difference of g. Let p n interpolate function f at x j q n interpolate g at x j we have already seen that alpha p n plus q n interpolates alpha f plus g. Now, divided difference of alpha f plus g will be coefficient of x raise to n in alpha p n plus q n and that will be alpha times coefficient of x raise to n in p n plus coefficient of x raise to n in q n. So, here this beta actually should be 1. So, it is linearity of the divided difference. Now, the next property is suppose your function f is sufficiently differentiable. Suppose f is n times differentiable then the divided difference of f based on x 0 x 1 x n that will be equal to nth derivative of c divided nth derivative of f evaluated at c divided by n factorial for some point c in the interval a b. If you take special case to be n is equal to 1 then this theorem says that divided difference of f based on x 0 x 1 which is equal to f dash c. Now, that is nothing but the mean value theorem. So, let us prove this result and then once we prove this we will define divided difference when the points are not distinct, but they coincide. So, our f is sufficiently differentiable. So, f sufficiently differentiable x 0 x 1 x n these are distinct points in the interval a b p n at x j is equal to f at x j j is equal to 0 1 up to n. So, let me define e x to be difference of f minus p n. We assume that f is n times differentiable p n being a polynomial it is infinitely many times differentiable because of the interpolation condition e of x j will be 0 for j is equal to 0 1 up to n. So, now look at e x j is equal to e of x j plus 1 is equal to 0 and e is differentiable. So, by Rohl's theorem e dash y j will be equal to 0 for j is equal to 0 1 up to n minus 1. So, that means e has at least n plus 1 0s then e dash it is going to have at least n 0s which will mean that e double dash will have at least n minus 1 0s e has at least n plus 1 0s e dash has at least n 0s. So, e double dash will have at least n minus 1 0s and when you continue the argument you will have an argument e n this will have at least 1 0. So, suppose e n at c is equal to 0 for some c now we will look at the nth derivative of e e n x will be equal to nth derivative of f minus nth derivative of p n p n is a polynomial of degree less than or equal to n. So, its nth derivative is going to be constant and then so we have e x is equal to f x minus p n x e n at c is equal to 0 p n x is a polynomial of the form a 0 plus a 1 x plus a n x raise to n. When you consider nth derivative of p n that is going to be n factorial into a n and by definition of divided difference a n is nothing but divided difference based on x 0 x 1 up to 0. So, this is the definition of nth derivative to x n and thus we get f n c minus p n it is nth derivative at c to be equal to 0 which will give us f of x 0 x 1 up to x n to be equal to nth derivative of f evaluated at c divided by n factorial. So, this is the generalization of mean value theorem in order to define the divided difference we do not need function to be differentiable we do not even need it to be continuous, but if it happens to be n times differentiable then the divided difference is going to be nth derivative evaluated at c divided by n factorial. Now, we said that there exist c in the interval a b in fact c will be in the smallest interval which contains our interpolation points x 0 x 1 x n because when you look at the interpolating polynomial what comes into picture is our interpolation points x 0 x 1 x n and value of function f at those points. So, now the divided difference is equal to f n c divided by n factorial where c lies in the smallest interval which contains x 0 x 1 x n. If your points x 0 x 1 x n these are ordered then that interval is going to be closed interval x 0 to x n. If your x 0 x 1 x n they all coincide they are all equal then our interval is a single point and then it seems logical to define the divided difference based on x 0 repeated n plus 1 times to be f n x 0 divided by n factorial. So, f of x 0 x 1 x n is equal to f n c by n factorial c belonging to smallest interval containing x 0 x 1 x n and hence we define f of x 0 x 0 x 0 repeated n plus 1 times to be equal to f n x 0 divided by n factorial. We want to show another property of divided difference and that is the continuity of the divided difference. Our divided difference f of x 0 x this is going to be equal to f x minus f x 0 divided by x minus x 0 if x is not equal to x 0. If x is not equal to x 0 then f dash x 0 if x is equal to x 0 our function f is defined on interval a b. Let us assume f to be differentiable so that we can talk of f dash x 0 now you fix x 0 then f of x 0 x this becomes a function of x. Our claim is that if you consider g x to be equal to f of x 0 x x 0 fixed this g is going to be continuous because when you consider value of x not at x 0 then f x minus f x 0 divided by x minus x 0 f being a continuous function so we have got quotient of two continuous functions. And at x 0 we have limit x tending to x 0 and at x 0 we have limit x tending to x 0. f x minus f x 0 divided by x minus x 0 that is f dash x 0 so that proves continuity of f x 0 x and now consider the higher order divided difference. If x 0 and x 1 are fixed the divided difference based on x 0 x 1 x it is same as divided difference based on x 0 x x 1 because it is independent of order. Now this will be equal to by recurrence relation f of x 1 x or x x 1 minus f of x 0 x is equal to f of x 1 x or x x 1 minus f of x 0 x divided by x 1 minus x 0. We have already seen that this divided difference is continuous this divided difference is continuous you are dividing by a constant. So, this divided difference will be continuous and in general by induction if you look at divided difference based on x 0 x 1 x n x that is going to be a continuous function. Now, we want to look at building of polynomials when we looked at the interpolating polynomial we when we proved its existence and uniqueness then we already had the Lagrange form explicitly we had written p n x using Lagrange polynomial. Now that time we said that the disadvantage of this Lagrange form is suppose you have found interpolating polynomial interpolating x 0 x 1 x n and then you add just one more interpolating point then you have to do all the work again. So, that is why what we want to do is if I have found already p n x and I add one more interpolating point then I should be able to do able to do it so that I just add one more extra term and that is why we defined the divided difference. Now, let us see how if you have found p n minus 1 x how to get p n x. So, p n minus 1 is the interpolating polynomial interpolating given function at x 0 x 1 x n minus 1 p n is polynomial of degree less than or equal to n which interpolates the given function not only at x 0 x 1 x n minus 1, but also at x n and our claim is that p n x will be equal to p n minus 1 x plus this extra term. So, now, we are going to prove this and then from that we can write the Newton form for the Lagrange polynomial. So, we have got p n minus 1 x j to be equal to f x j for j is equal to 0 1 up to n minus 1 p n minus 1 x n minus 1. So, p n at x j is equal to f x j for j is equal to 0 1 up to n if I look at p n x minus p n minus 1 x this is going to vanish at x 0 x 1 x n minus 1 at the common 0s p n at x j will be equal to f x j is equal to p n minus 1 x j for j is equal to 0 1 up to n minus 1. So, we are going to have factors x minus x 0 up to x minus x n minus 1. Now, these are n brackets. So, right hand side this will be a polynomial of degree n. So, this will be a polynomial of degree n p n is a polynomial of degree less than or equal to n p n minus 1 is a polynomial of degree less than or equal to n minus 1. So, here what we will have will be a constant alpha a constant. So, p n x minus p n minus 1 x it has got this form. Now, let us calculate what is alpha. So, alpha is coefficient of n minus 1 minus 1 of x raise to n on the right hand side coefficient of x raise to n on the left hand side will be p n minus 1 is a polynomial of degree less than or equal to n minus 1. So, coefficient of x raise to n on the left hand side will be coefficient of x raise to n on the left hand side will be p n in p n x because there will be no contribution from p n minus 1 x. But coefficient of x raise to n in p n x is nothing but f of x 0 x 1 x n the divided different. So, thus you get p n x is equal to p n minus 1 x plus this divided difference multiplied by x minus x 0 x minus x n minus 1. So, thus we have shown that p n x is equal to p n minus 1 x plus this term. Now, you can replace n by n minus 1 and get a formula for p n minus 1 in terms of p n minus 2. So, p n x is p n minus 1 x plus this divided difference into x minus x 0 x minus x 1 x minus x n minus 1 replace n by n minus 1. So, you will get p n minus 1 x is equal to p n minus 2 x plus you are replacing n by n minus 1. So, you have got this divided difference into x minus x 0 x minus x n minus 2 which gives us p n x to be equal to p n minus 2 x plus these two terms. So, you can continue this process and then you can have p n x to be equal to f x 0 plus f of x 0 x 1 x minus x 0 plus this term. So, we have got the power form which is p n x is equal to h n 0 plus a 1 x plus a n x raise to n. Now, we have got another form.  So, it is of the form alpha 0 plus alpha 1 into x minus x 0 plus alpha n into x minus x 0 x minus x n minus 1. So, this is known as Newton form. This is the power form. Our alpha n is divided difference based on x 0 x 1 x n. Suppose, your points are all 0. So, if you have got x 0 is equal to x 1 is equal to x n is equal to 0. Then the divided difference f of x 0 x 1 x n this is going to be equal to nth derivative at 0 divided by n factorial and hence your p n x will be alpha 0 plus alpha 1 x plus alpha n x raise to n where alpha n is f n 0 by n factorial. So, that is nothing but the Taylor's truncated series. So, here the center is 0. Here you have got different centers. You have got x minus x 0 then x minus x 0 x minus x 1 and so on. So, this is the Newton form. This is the power form and we have looked at the Lagrange form. Look at the divided difference table. When we want to construct a interpolating polynomial then I am first going to look at only three points x 0 x 1 x 2. Look at the function values.  The first thing we do is we evaluate this divided difference. So, it will be f x 1 minus f x 0 divided by x 1 minus x 0. So, that will give us this term then divided difference based on x 1 and x 2. So, you calculate that and then you calculate the divided difference based on x 0 x 1 x 2 which will be this entry minus this entry divided by x 2 minus x 0. So, you prepare divided difference table. Then when we want to write the p 3 x you have x 0 x 1 x 2. It should be p 2 x. Actually it is a second degree polynomial. So, this will be f x 0 plus f of x 0 x 1 into x minus x 0 plus this divided difference multiplied by x minus x 0 x minus x 1. So, you are looking at these three entries and then you are writing your polynomial. Now, instead of taking the interpolation points as x 0 x 1 x 2 in that order. If you take in the order x 2 x 1 x 0 then you will have f x 2 plus this divided difference into x minus x 2 plus this divided difference into x minus x 2 x minus x 1. It is going to be the same polynomial. It is just that this will be forward formula and this will be backward formula, but the interpolating polynomial it is going to be the same. So, now let us look at an example. So, suppose x is taking values 0 1 2. So, this is x 0 x 1 x 2 the corresponding values of f x they are given to be 1 4 and 15. So, we will first calculate the divided difference 4 minus 1 divided by 1 minus 0 that will give us 3 15 minus 4 divided by 2 minus 1. So, that will give us 11 then 11 minus 3 divided by 2 minus 0. So, that is going to give us 4 then our p 2 x is going to be equal to 1 f x 0 plus 3 into x minus 0. This is f of x 0 x 1 into x minus x 0 plus 4 into x minus 1 into x minus 0. So, this is our p 2 x that is f x 0 plus f of x 0 x 1 into x minus x 0 plus f of x 0 x 1 x 2 into x minus x 0 x minus x 1. It is also going to be equal to if I write the backward formula it will be 15 that is value of f at x 2 plus 11 into x minus x 2 that is x minus 2 plus 4 into x minus 2 and x minus 1. So, x minus 2 x minus 2 x minus 2 x minus 1. You can check that both these they give you the same polynomial. It is the same interpolating polynomial only once I take my interpolation points in the order x 0 x 1 x 2 and in other I take as x 2 x 1 x 0. So, we have calculated p 2 x 0 plus 4 into x minus 1. Now, suppose you are also given the value at 3 to be 40. So, now what we will do is we will complete the divided difference table. So, now first the divided difference based on 2 and 3. So, that will be 40 minus 15 that is going to be 25 divided by 3 minus 2. So, this I am adding extra then 20 minus 5 minus 11 that is going to be 14 divided by 2. So, that will be 7 and then 7 minus 4 divided by 3 minus 0. So, that is going to be 1. So, we added one extra interpolation points and these are the entities which we calculated. And then our p 3 x will be equal to p 2 x and then p 2 x will be equal to p 2 x plus this nu divided difference 1 and then multiplied by x minus x 0 x minus x 1 and x minus x 2.  If you were using Lagrange form then you calculate the Lagrange polynomials corresponding to 0 1 2 and then you have to calculate completely again the Lagrange polynomials corresponding to 0 1 2 3. So, this is how the building of the polynomial it helps. Now, we are going to look at a general this thing I for the sake of simplicity I looked at 3 points, but if you have got x 0 x 1 x n n plus 1 points then these are the function values given to us. The first column contains the divided differences based on two points then divided difference based on three points and the last column will contain the divided difference based on n plus 1 point. And then p n x will be equal to f x 0 plus f of x 0 x 1 into x minus x 0 plus f of x 0 x 1 x n into x minus x 0 x minus x n minus 1. Or you can take the points in the other order and then you will have f x n plus f of x n minus 1 x n into x minus x n plus f of x 0 x 1 x n and your points are in the order x n x n minus 1 up to x 1 you have to go. So, that is going to be the background formula and in compact form you can write it as summation r goes from 0 to n x minus x 0 x minus r minus 1 and the divided difference based on x 0 x 1 x r.   So, this is a forward formula and this is a backward formula they give you the same polynomial but then the reason we write this is because we want to prove next the Leibniz formula for the divided difference. So, that is why I wanted to write it in this form. So, now let me recall the Leibniz rule for the derivatives f into g see we have we are considering various properties of the divided difference. We saw what is the divided difference of alpha f plus g. So, that is the linearity.  What can one say about the product divided difference of the product? There is a relation between divided difference and the derivatives. If your points are identical then the divided difference based on x 0 repeated n plus 1 times that is nothing but f n x 0 divided by n factorial. And for the derivative of the derivatives we have got product rule. So, whether such a product rule exists for the divided difference and the answer is yes and that is where we will need the forward formula and the backward formula. So, let me recall the divided Leibniz rule for derivatives if you consider two functions f into g it is nth derivative at point x it is given by summation r going from 0 to n n factorial divided by r factorial n minus r factorial r th derivative of f at x plus multiplied by n minus r th derivative at x.  So, now, let me fix x to be equal to x 0 and the same formula I write as f n minus r x. So, f into g n th derivative at x 0 divided by n factorial this n factorial I take it on the left hand side.   So, Lebanese rule for the derivatives I am just rewriting in a different manner that it is f into g the divided difference based on x 0 repeated n plus 1 times which is equal to summation r goes from 0 to n divided difference of f when x 0 is repeated r plus 1 times and divided difference of g when x 0 is repeated n minus r plus 1 time. Now, what we want to do is instead of x 0 repeated n plus 1 times. Suppose, I have got n plus 1 distinct points x 0 x 1 x n then whether we have this divided difference is equal to f and then here x 0 x 1 x r and then here it will be x r x r plus 1 x n. So, that is 1 wants to see whether such a thing is possible the question is whether f into g it is divided difference based on x 0 x 1 up to x n whether it is equal to summation r goes from 0 to n f of x 0 x 1 up to x r and g x r x r plus 1 up to x n these are r plus 1 points these are n minus r plus 1 points. So, instead of identical points whether the formula is true for the distinct points. So, this is question such a formula for the divided difference is valid. Now, the answer is yes. So, once again what we will do is our divided difference is coefficient of x raise to n in the interpolating polynomial. We have got f and g to be two functions. So, we will look at their interpolating polynomials. So, suppose p n is interpolating polynomial of f interpolating at x 0 x 1 x n for g suppose you have got another polynomial q n which interpolates the given function at x 0 x 1 x n. Now, if I look at product f into g. So, similarly I will look at the product p n x into q n x. Now, p n at x j is equal to f x j q n at x j is equal to g of x j. So, this polynomial p n into q n it will interpolate our function f into g. So, we have got product f into g at x j is equal to f x j into g x j f x j is equal to p n x j g of x j is equal to q n of x j. So, this is the result of the interpolation after this p n into q n will be a polynomial which will interpolate the product function, but then we do not want just a polynomial which interpolates, but we are interested in the degree p n is a polynomial of degree less than or equal to n g n q n is a polynomial of degree less than or equal to n. When I take its product it will be a polynomial of degree less than or equal to 2 n whereas, for the divided difference of f into g I will need a polynomial of degree less than or equal to n. So, we will see next time how to do this. So, the next time we are going to first prove the Lebanese rule for the divided difference and then we will consider the error in the interpolating polynomial. I had said earlier that the divided difference is also important in getting the error formula. This we are going to do next time. Thank you.