There it is, Canada. Maybe they'll let us in another day. Not now. USA is that way. I'll go that way. Suited up. I'm ready to go. The bike is all packed. This bike is my dream bike and I developed it in conjunction with priority bicycles. If you want to learn more about it, click the link in the description of this video. I don't know if it's nervous energy or excited energy or a mix of both, but I I'm just kind of walking in circles making sure I have everything ready and everything set and I do and I should just be out of my Bike right now. I just called my mom to tell her I love her one last time. She's like you called me already This is a dream come true. I've wanted to do the divide ride for about 10 years now Just in case some of you don't know what the great divide is. It's the world's longest off-road bike route at 2700 miles. It starts in Banff, Canada and goes to the border of Mexico Right now Canada won't let us dirty Americans in their country. So I'm starting in northern Montana You might be thinking oh Ryan can just get up and go and it doesn't faze him He just rides and he sings Olay all the time and he's all good. Well, the truth is, you know, I get nervous like anybody else This is this is big. I think my jitters mean that my body is getting ready for something Amazing at least that's how I want to think about it And I look back on the first day of my Honduras ride my first ever bike packing adventure and how Excited and nervous I was I feel good. I've said my goodbyes I've shed at my tears. I got my goodbyes good my hugs everything. I'm ready to roll I'm ready to start the adventure and then the first day of my first cross-country ride across USA and I had the same feelings like oh my god. I'm here. This is happening. This is really what's gonna propel me Not my legs. It's this thing. It's gonna take me across America Whoo, I mean there's so many months that go into preparing and Thinking and buying gear and getting everything set and staring at maps and then boom here. I am Now it's go time. I'm not in the comfort of my home and speaking of my comfortable home You know, I've been in quarantine since early March Essentially quarantine was very comfortable. I eat good food and watch Netflix every night, but it's time to get Uncomfortable it's time to push it. It's time to explore. It's time to be on the edge of comfort And I like that that's when I feel most alive and that's I think why I'm feeling a little bit Nervous and scared and excited. You know, it's not gonna be easy.  I'm in grizzly country. There's bears out there. I have bear spray with me Just in case that's kind of scary, you know, I know that this is kind of a boring way to start a Video for a really big ride, but this is how it happens I sit in a hotel and I pace back and forth before I finally get the courage to walk out the door And I thank you all so much for your love and support in watching my channel And I'm so excited to share this with you It means the world to me that not only do I get to experience this But then I get to share it with you and hopefully Inspire you to challenge yourself with whatever adventure you choose in life All right, I think it's time to leave now Or I could just lay back and take a nap. No, I'm not gonna take a nap. Let's go All right, I've made progress. I am outside of the hotel About to get on my bike and I already feel a lot better. Oh It's gonna be a beautiful day, and you know how we start these things in dozer land. No crashes. No flaties No whammies, baby Right now I'm feeling awesome All it takes is a few miles of riding and then I'm like, oh, yeah This is why I do this. Look at this beautiful road No cars One of the reasons why I'm so attracted to this route is that it's mostly off-road Which means not many cars So I can breathe easy and just enjoy the riding and not worry about getting run over by a semi What a beautiful day. Whoo. Yes It's happening Feeling hard and making big miles But I also like stopping and reading these historical markers Just to get a sense of the history of the area and where I'm riding through and this one's pretty interesting It says three times a year the Kootenai Knutaksa Indians departed their Tobacco Valley homeland and Crossed into the drainage of the North Fork of the Flathead River The Native Americans rode and packed as many as 700 horses for the spring hunts when the whole village made the journey The meat and all supplies were carried on their backs for the return trips Here I am thinking I have a lot of stuff on my bike. I can't imagine Carrying all of that meat on the back of horses. Wow All right, I finally hit dirt about 20 miles south of Eureka. This is what I came for dirt roads solitude simplicity remote beauty It all starts now All I got to do is pedal And that's really all I have to do I have to wake up every day and pedal my bike And I'm incredibly Grateful for that Doesn't get much more simple than that I also have to find water and food And film all of this for you. But for the most part I am just pedaling every day That is a dream come true. Oh, and I get asked a lot about how I know where to go Well, it's quite simple really I've downloaded the gps file of the entire route Onto my wahoo computer and it tells me exactly where to go It's like using google maps on your phone If I go off-route by as little as 15 feet, it'll let me know Which is very nice piece of mine. This is this is some new technology for me I used to use maps like everybody else used to use But it's nice just Having this little computer tell me what to do as long as I keep it charged If it goes dead, then we're gonna have problems It is time to use my magic wand and filter some water You just dip it in there And it magically filters and kills all virus and bacteria actually not magically it shoots UV Light into it. It's how they filter water at big You know water filtration projects for cities It's also nice just to get off the bike and sit next to a river. The sound is very soothing It's not even noon yet and I'm having the time of my life Oh, I love it As I head further and further into the outback I'm starting to see signs that tell you to be beware of bears grizzly bears And black bears. I've only ridden my bike once through bear country in Wyoming for only a couple days But a lot of this route is going to be through bear country and that is why I have a canister Of bear spray right here pepper spray I hope I don't have to use it Um, but definitely it's nice to have it as peace of mind and at night time I'm gonna have to get my food up high. I know bears love refried beans Everybody does not just bears Yeah, here we go into the wilds of montana You know back in colorado the past few weeks. It's been High 90s almost 100 every single day Which is hot but out here It's perfect. It's like 75 It couldn't be better I found some of that uphill that this route is so famous for 200,000 feet of uphill during the entire route That also means there's probably 200,000 feet of downhill, which is awesome So yeah, you you know you put your head down and your pedal hard You go up And then you get to enjoy Screaming fast downhill. I say that's a fair deal. You know Really nice stretch of road right here I feel like the trees are hugging me Whoa, these mountains are different. We don't have these kind of mountains in colorado I think I'm looking at the southern end of the canadian rockies.  I just realized I am very very clean right now It's like I'm wearing my first day of school clothes All spick and span. Look how white the shorts I'm not muddy and dusted up yet That'll happen And I think it's kind of a badge of honor when it does happen And by the end of this tour, I'll be looking A little rougher Which is cool Uh-oh, we've got one of those steep uphill swings And I've already gone up 5 000 feet today I got to the top of the hill Started looking at this mountain that was distracting me and didn't even notice there was this beautiful alpine lake Right here. Oh my god. It is It is a beautiful lake. Wow You want to know what one of the added bonuses to riding in the north is in the summertime? It stays light out really late. It's been a beautiful beautiful day Was to make it to a lake and camp and I've done it I didn't think I was going to get this far though. I'm in whitefish So I rode like 95 miles today and I got a pretty late start But I just kept cruising. I was extra inspired today Not only by it being the first day of the ride, but just by the landscapes and the surroundings and Just so beautiful. So if this first day is any sign of what's to come, I think I'm in good shape This is uh, this is going to be good. This is going to be really good Surprisingly, my legs are doing pretty good and I climbed like almost 6 000 feet total throughout the day But you know what hurts right here? It hurts because I've been smiling so much It was one of those days where I was just like, whoo, this is great


The following is a presentation of TFNN The Tiger Technician Hour with your host Basil Chapman Call now toll free at 1-877-927-6648 Internationally at<PHONE_NUMBER> Now Basil Chapman Hello everyone Basil Chapman, Tiger Technician Hour and let me tell my subscribers The webinar is on the page, you can check it out I'm very pleased, I've got everything I wanted to cover right there, right to the minute, hour and a half And if anyone is interested in subscribing, we've had some really nice trades Hopefully this could be a great help to you And go to the front page of TFNN, there's still time you can check it out because it was archived So you'll be in the same position as those people that were unable to make the webinar two days ago And we're now going to start looking at the archive And everything that was on it is pertinent to what I believe is coming up over the next couple of months Very important So I did this just to show the Chapman wave technique, we made that PD and I always look for the fourth highest peak In the two-minute chart, pullback from the 31 or 350 area, pulls back down to the 3097s And what I draw in right here was a Chapman wave inside wedge, remember I discussed that the other night Inside wedge, target resistance line, dash green line And I did it to what I chose to be left side, right side by time match And let's see where we are, you've got another two minutes to go These are two-minute charts, so perhaps another bar or two We've got right to that dashed line, that's the resistance line When I typed this in, we were right over there And I showed a peak A, I drew the cup formation I don't know, let's see if it goes to the 3102, 310250 area That's kind of a target over the next three to four minutes And here is a peak E in the Chapman wave in the 10-minute chart of the E-mini And it plopped all the way from the 314.50 high of the day down to the low of three Well, this particular low here of 3094 And it's trying to really, we'll see what happens Alright, let's get to our nitty gritties here The Dow, oh, I've got the dollar, let me do the dollar Holding well, it made it, remember, fourth highest peak D On the 1st of October, pulls back from 99.67, comes all the way down to 97.11 Rallys up to 98.45, still got a long way to go It hasn't made that peak D yet But if we can have it like a restart here of another move to the upside, that's going to be positive If it starts to drop under 97.40, it says this weekly peak D Is going to make it probably the lowercase H pattern That says be careful, we could retest the lows of 97.14 Monthly chart is, what, a week and a little bit to go Looks like it's going to be a peak D in the monthly chart Well, let's get to the nitty gritties, here we go We've got, what is today? Thursday, so Tuesday morning Right at the open, I want you to, I want you to short for my subscribers But we had shorted twice, one was an option The other one was the DOG, one to one, short to Dow We took a loss there, a fairly small loss in just one something percent in the DOG But it was about a 20% loss in the option Options always said, hey, if you're not prepared to see a 20% to 30% loss in your option Don't even think about trading options, because in a second they can use 50% In a second they can gain 50%, that's just the way options work Now what's really important is that the reason why I want you to use that I was giving a leg F and the chart wave F with the technicals deteriorating is a sure sign You're getting a pretty steep self, but I wasn't 100% sure that it would happen right then Only the on balance volume was giving me the clear at my notation And we're saying that the strength, let me just do this again The strength of this combination of the green line going up and pink going down Going over or below the black moving average It says to actually and I've been talking about this for the Dow to actually plunge Enough to get the green line under the black line to make a pink line It's going to take 700 points and look what happened twice yesterday and today Went right to the 14th period moving average and it's bouncing above So it's going to be time, so I didn't want to fight it I said we're going to do it short because I'm suspecting that we're in at least a sideways trading range Probably lower highs and lower lows I don't think we're in this formation just yet where it's a really sharp sell-off I don't see any reason why you can't expect a 4% to maybe an 8% decline consolidation Maybe December we stop, but my rule of thumb is that if the Dow, most of the indices But I like to talk about the Dow, if the Dow can survive The August, September, October declines if that's the case you only had one of the decline If it can come back at the end of October the first week of November to almost close to the yearly high There's a chance you're going to close almost at the yearly high That's just my way of looking at the market So keep that in mind I wouldn't get terribly excited because this is such a rotational market I'm looking at some of our stocks up one is up over 1% today The other one is down a little bit, yes down a little bit Another one is up 0.60% This is a mixed market isn't like 90% like in December we're in 1993% of all stocks are going down This is rotational correction So let me show you what I'm looking at here let me just go back to this for a second There's the email there it is so that's the technique that's what I was talking about on my show on my webinar These are the techniques we use I don't believe I think we should actually close pretty reasonably today If there's no bad news further bad news and even with bad news We're down 40 points in the down down 5 points and this is not a real bad news sell-off where you're getting 15 points on the downside every day in the S&P 130 to 117 the Dow there's residual strength here So if you haven't done anything on the short side I think you're going to get an opportunity but I'm going to still say be very selective I'm actually looking for more buys and sells at this particular point That's just the way the market is Alright so we did that exactly that means the 10-minute chart hasn't made a leg bead won't make a leg bead until it goes Above 3103.25 if it goes 0.75 if it goes to 3 I can't read it let me just get it in a different place 3103.75 if it goes to 3104.00 you start your leg bead that's going to be a good sign And every day you know in my opening call newsletter I discuss these things in great deal look he has the chart there's 120-minute chart What it said is today if the Dow I hold a bunch of stuff going on but today if the Dow is able to hold a plus 40s after 1.30pm That will be good action but a rally failure or failure to rally shows weakness that's at that point Especially after so much selling has already taken place So we'll see this is that I do this and that's 120-minute chart I have the daily right here Now you see the daily up until this morning where we retested the 14 EMA and the MACD hadn't crossed negative And the stochastic was at 92% that's very good look how much it drops since then So this is the start of some kind of a rollover that's all I'm calling it right now Could get worse but I'm calling it initially a rollover I'll be right back Bezel Chapman type admissions hour the Dow is down 46 recipes down 5.8 I'll be right back If you're not currently using the TAS Profile Scanner when looking at setting up your trading opportunities Then your arsenal is short a mighty weapon The TAS Profile Scanner is a standalone piece of software that instantly filters over 2,500 global financial markets such as stocks, ETFs, commodity futures and forex Heated by Steve Dahl TAS understands that in today's 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a new and improved homepage with a much simpler navigation Whether you're watching Tiger TV live in high definition or just accessing your newsletter subscriptions We even have new pricing in six months and yearly options Check out the new TFNN.com now and experience all the upgrades TFNN.com educating investors 927-6648 internationally at<PHONE_NUMBER> Hi folks, I'm Dan. We just someone Zia mentioned something about KC which is this is coffee This is a continuous contract Made a peak D, the MACD turned down, did cross negative but very slightly and then a surrogate show support yesterday The stochastic did pull back all the way to 61. So this is this is real. This is a surge of buying And it must be a story behind it because it should have popped maybe to the D level which is at 114 Or this is the continuous contract 113 30 said should have popped maybe to 138 114 10 and then kind of pulled back But it was just going to be a V shape formation something is going on Yeah, this is a stupendous move because it's really it's a leg B now in the weekly chart And it's a leg A a brand new leg A in the monthly with the stochastic Okay at 18% but the MACD has actually started to turn up So this could say that the whole area of one to 95 In In the continuous contract of coffee that could become a base here This is very interesting because everything about you always looking for those these what was this this is a D right here look The reason why the prices are moving around I mean the notation is this is a continuous contract to get smoothed out So it went to a trough D right there on the 11th of October at 95 point 60 This is a continuous contract. So your price might change with the the pattern and the Time absolutely does not just the price gets smoothed out. So there it is. This is a very strong Is this a leg E or brand new is this an old leg E or brand new A for now? I'm just gonna have to say I'm calling it E But my eye suggests that from the monthly chart from the weekly chart There could be a pullback to where D was at the hundred and thirteen hundred high hundred and thirteen Four hundred and fourteen level and then another move up to confirm that this is actually a new and new milk to the outside and coffee Wow, okay, so a couple of things going on here Above in the dances, I think Casey by far the most difficult commodity trade Like Larry says when you have the urge to trade Casey stop and consider vacation So the urge goes away. Well, it's just if you're looking at it purely as a technical analysis Look, it's done. I couldn't complain. It's done everything that you asked for in the Chapman wave I couldn't care if this was Z Z Z Z Z If there ever was such a thing or if it was a water ETF or Coffee, I don't care what it is. It went to a peak D in the Chapman wave We did everything it should had an arch failure This is that this is the dreaded H that we always talk about in the Chapman wave methodology right here And then right left left side Lower right below the previous one and didn't come up for air at all Only differences that I should have put it. I haven't checked this for the one should have put my down arrow there And this should have been an up arrow peak A peak B peak C peak D peak D is where you get a little A cautious you lift your foot off the accelerator just to see what's going to happen A hover over the brake in case you need it yellow light goes on and if everything's safe There's no through traffic. You can just keep going and there it is goes to D I would have put a down arrow in if I've seen it, but I also hold off sometimes and I keep the plus sign Because if this is an M shape formation like it was there, you can only go You might see the price go below the 14 period moving average in this case the 200 period moving average And the magnetic and the stochastic goes under 80% But it could be brief as there's another attempt to try to heal a dreaded age pattern right here That went under and within two bars kaboom it goes back yet. Hey, I you know, I always say this I remember my father once said to me I can't trade I don't remember which country this country stocks. He says they all they know those bankers are crooks I would never tell so then I got a chart and I see Tim. Okay. Yes, some stocks from that country And yes, some stocks from the United States. You tell me which ones are from the foreign country You tell me which ones are the crooked ones from America or the good ones from America or the crooked one No way you could tell could you tell from here that this is not a high tech or this could have been Target this could have been target. Look at the chart target Okay, another peak deep pulls back. Yeah, the MACD was very poor. See, I would never have said from the chart Formation other than with earnings that there could be a spiral from the hundred and four teams in target All the way to today's high of 127.96 in just two days So yeah, so I'm just going to say to you trade what you see There is the guy that teaches trade what you see and this is what you're looking at. Let me just go back to the fuses I'm kind of intrigued Yeah, so we went into the futures. We were right above that because stored at the 200 period moving average in the two minute chart We're at peak B and the technicals all holding well. Let's see if we can start legs see above 3103.00 All right, and I love my one three rule. So this is already one two three four This is the fifth bar from that peak. You got maybe one bar to get close to it one three six rule to see what happens All right, there we go. So now I did the doubt. Let's just go S&P S&P It's kind of holding okay up here. This is a high level consolidation So we'll see what happens. I can just tell you that if the S&P starts to trade a 3082 3082 That's going to go from a cell signal to a cell mode very quickly But so far hasn't even gone to a cell signal not yet. If you close it might sedate each other The IWM I never did in the update the IWM allows it a see this is what I said about the IWM Don't be fooled just because when the down the S&P a week and the QQQ is weak index 100 and IWM The Russell 2000 is holding well that it's great. No It's a very lagging indicator index So you've just got to be very careful and I think you're going to get a wonderful buy on the IWM at some point Not just yet Okay QQQ, 1, 2, 3, there it is Just digesting the gains. Let's get the XLK. I don't know if I've even updated the chart. Yes So there was a let me just, is that a peak over there? 12, 12, 12, 2, 12, okay So you had a big D that it goes to E S&A, F&G Sorry F&B And G&C So the XLK is a little different to the others in that just didn't make a peak right there It was a double top and the next was Bob was a higher high So I'm calling this G&C You might even have to see a D in this one before it's all over and then it pulls back But everything about it says we're in a digestive phase in the S&P select spider fund Tech spider fund had a question about shorting the SMHs the other day You know we were short the SMHs if I had still held that short It really just barely got stopped out, it did get stopped out but barely by a fraction If I just held it, that would be very good I think this is going to be a little more complex than just a very simple smash to the downside Oh, SMHs and SMHs are down to $91 And $130.67 $135.26 just stopped us out the other day Otherwise we would have been short the SMHs And I still think short is probably the way to go because it had a chaff from where you saw the information There's the leg, there's the body, there's the neck, now we're into the beak And the beak is now gone below the body and that's usually very negative So the next balance is where you're going to maybe see an H pattern in the SMHs And then if we start a deeper correction from the next week or two And watch it very closely, if you hide it's $175.27 Since 1984, Basil Chapman has been using the Chapman Wave methodology to advise traders of his expert market opinion While originally hand drawing charts from the late 1970s into the 1980s Basil noticed that prices under most circumstances virtually always had a certain number of legs to the upside before declining sharply Later Basil found that computer software which included the standard market technical indicators Enhanced the degree of accuracy in calling price turns as well as market trend calls Thus was born the Chapman Wave sequence Using the Chapman Wave methodology along with other indicators Basil Chapman advises his subscribers of his expert market opinion Each market day with his opening call newsletter Right now you can get a two week free trial to the opening call Basil's daily trading newsletter by visiting the front page of TFNN.com Cancel at any time during that trial and pay absolutely nothing Get your two week free trial to Basil's newsletter of the opening call today by visiting TFNN.com The path of least resistance is David White's daily trading newsletter And if you're looking for active trading ideas Then now's a perfect time for a 30 day free trial to this powerful daily trading advisory service David uses his years of trading experience to offer his subscribers his trading ideas Each morning in his path of least resistance newsletter Using a combination of equity trades along with options David keeps his subscribers up to date with all pertinent market information With intraday afternoon updates when warranted Don't miss out on this great chance to get a 30 day free trial to David's daily newsletter The path of least resistance with no obligation to pay anything David has been delivering solid recommendations for his subscribers recently And if you'd like to see the type of newsletter he delivers every morning Then visit the front page of TFNN And you'll find the path of least resistance under trading newsletters For all the details and to start your 30 day free trial today Log on to TFNN.com now TFNN is excited about our new software charting program The Art of Timing the Trade Chart In collaboration with Tom O'Brien and using his best selling book The Art of Timing the Trade Your Ultimate Trading Mastery System David White has programmed an outstanding piece of software That will complement any trader's methodology Using this first of its kind program The Art of Timing the Trade Chart allows you to scan thousands of stocks For Fibonacci formation setups including guardleys, ABCs, butterflies and much more The Art of Timing the Trade Chart is designed to help you When scouring the markets for stocks Just beginning to form the trading patterns That many investors spend days, weeks or even months searching to find And right now we're offering licenses available at only $79 a month We are so confident that you're going to love this new charting software That will even give you a 30 day unconditional money back guarantee Don't miss out on this incredible new piece of software Get your copy of the Art of Timing the Trade Chart today By visiting TFNN.com I'm Ron Robach And yes, we made that leg C in the two minute chart That's what we're looking for Why are we looking for leg C? Because we want to go all the way to a leg D And then a peak D Remember Chapman, we've looked for the lowest low bar Most obvious low bar We count each higher peak Alphabetized Upper case on the way up Lower case on the way down You can go to E, F and G But you're ready D is where you're expecting that something can happen It doesn't have to It can recycle to the upside And start a whole brand new buy mode to another four peaks It's an incredible indicator Or you can say just be a little careful here And then you've got the patterns The arch and the cup You can mix them up straight line down And then you try to around it And you fail We just saw that A red because if you have bad technicals It can go much deeper And green because the Y The reverse Y If you take out the left side high You can go much higher In fact, look, all of these are the Y pattern Look, there's the leg up There's the Y And it broke to the upside There's the leg down And broke in the Y breaks So just, you know, these are fractals The fractal is really a small It doesn't have to be small But it is a pattern That repeats identically In all time frames So that's what we're looking at All right, so now let's do a couple of things here Crude oil Crude oil This is the V-shaped pattern Went to a D and pulls back holes They've got beautiful That 14-bit moving average is the black line And then it pops up It walks the 9 and the 14 And then pops to an E Plunges below And it goes right back up again To a new height today And a new recovery height today A 58-28 of the continuous contract There's a peak D Remember what I say about Ds Be careful on 16-14 16th of September Plunges down to 50.99 Oh, let me check the price This is a continuous contract Everything's correct Except the price might be smoothed out So the low was 51.20 I've got 50.99 51.20 And here it is 58.08 I said that I think it's stuck In a rectangle formation Maybe the high 58 59.25 If it goes above that That's something else We'll see what happens here It's been stuck And in the weekly charge You've got about 61 As the rectangle resistance And you've got really important Supporters in the 53 area But it does go all the way to 50 TLT Lehman 20 A Treasury Bond Fund 139.42 Oh, this is going to be so interesting Because look, if you look at this As the price It's been stuck in a range Of lower lows and lower highs And I'd say to go into the 141s Oh, you've got to be careful Because that would mean that money might come out Of stocks into the volatility Of stocks if they go down into the So-called safety of bonds Here we are in the lower register We did the dreaded H It's still acting purely after the H pattern But the monthly chart is still good That was the weekly There's the H pattern I drew in right there And the monthly Is still holding very well In the upper range Of this month's candle And that's good All right Just a review I had a couple of quick questions Come on if they're quick But there were questions that came in So let me just honestly one at a time Yes, this morning I apologize I just between last night A board meeting and stuff And this morning I had to go back into town Back here again I just ran out of time to do all the work I used to do for my subscribers To an opening call So let me just do a couple of things I say to them I do it on the show today Our stock that One of the stocks we've got BDSI The buy delivery Holding really well It's down 0.07 and 624 We're in at 517 We've taken two little Little bits off As part of money management You had got to PDA at 645 That's a big move 517 to 645 It's one of the reasons why I said We're not going to buy A $238 stock like now This is a service now Because we could do the same percentages Without putting that much money to work Because I was suspecting That we were coming into a period of Some volatility And I wanted to have some cash ready For when the really good buys come in But look what happened to service now I said I don't want to go through it again I went through a whole thing in my webinar Please if you're interested In really learning a whole bunch of techniques That I'm sure you don't hear about Almost anywhere else Because these are techniques That I've worked on myself Most of them are proprietary to the extent I'm really stressed if everyone's work But I know that I don't hear about it Discussed So it's my volume, climax, reversal A whole bunch of things And I'd said that now should not After the smash to the downside Of $213.99 on the 23rd of October It was around about year 238, 242 I said We're not going to buy it But follow it Because I think it's going to go For about 28 bars And if it holds 28 bars Without even coming close to the low It can even go for 56 bars It was doing fantastically It went all the way to 267 30 points high And then S&P said It's going to be included I think S&P 500 maybe It was going to be included And it spiked up huge It hit 284 yesterday But on a percentage basis Going from the 24 level to 28 We've got something even better With the BDSI And we've got something Even better with the other ones I said I'd talk about it There's the rectangle formation I've discussed this the other day I said that you could get An inside buy mode That can go to a peak D In a smaller timeframe In this case, a 120-minute chart Going towards Right on or just above the previous Side and then it comes back Into the rectangle And look what happened It went to peak A, B, C Today's a leg D And it's not yet close to the 645 level It's at a height of 636 So we'll watch this Because you don't want to turn around And make an arch formation An arch formation says Oops! You can go and retest the six Areas at 625 right now So that's one of the techniques C, Y, V, R It's another one I didn't have time today to show What I discussed on my In my webinar Was I love the stock We had it at 104 It just came out It popped to 199 It did pull back way sharper Than anticipated Which is the only reason Why I said We had taken two small bits Of one at 100 And in the 116 And one in the 109 area And then I said I didn't get us back in I said We've got a core position I'd like to add back The little bit that we took off The two little small chunks we took off I haven't done that And yet it is today It went to 120.61 To a new height And what did I say? This rectangle formation Had a whole part of my show on On my webinar Had a whole part of it Talking about this pattern Called the inside by mode That you should have A smaller timeframe Taking it to a leg D In certain conditions We met Bam! Look at this We went right just above Now I have to call this Not a B That's something else Now I call it an E Slash B Saying Hey, we've reached our E That's the old notation Are the technicals Strong enough to say You know what? I think it's going higher So that we can go to A new C and a D And there it is Leg B And the reason why Cyber arc Software I love the stock It's in the It's in the It's in one of the leaders In the Internet security And at a whopper of a fall It plunged from 148 Down to 96 94 I mean That's a huge move down So I'm hoping this is One of our intermediate Term buys Like we've had the dollar Like we've got Bank of America We'll see I mean a day's young We'll see what Oh, a day's young Let's see what's happening To a little short Leg D Leg E We're in leg E In the two-minute chart And now new leg B In the ten-minute chart I think that's why It's quite a nice day As a market in there You never know If you're in the CD market And looking for a secure investment The Tiger First Mortgage Program May work for you The security for these First Mortgages Are building lots In the tax opportunity zone In St. Petersburg, Florida The tax act of 2018 Set up tax-free zones Across the country Where you can build And hold for ten years And pay no tax on the profits Which makes these lots valuable The investment is anywhere From $30,000 to $75,000 The interest paid Is 7% yearly paid On a monthly basis According to bankrate.com The best rate for a four-year CD In the country As of February 20th Is 3.1% A $50,000 investment At a normal four-year CD Rate of 3.1% Would give you income Of $1,550 per year Or $6,200 over the four-year period That same $50,000 investment In the Tiger First Mortgage Program Would give you $3,500 per year Or $14,000 over the four years What should you prefer? $6,200 or $14,000 Of interest on your investment If you'd like more information About the Tiger First Mortgage Program You can call me at<PHONE_NUMBER> That's<PHONE_NUMBER> If you're a trader in the market Looking for exposure To gold or gold mining equities Then now is a perfect time To sign up for Tom O'Brien's Gold Report The summer is over Gold is trading back Above $1,500 And the 10-year treasury Is hovering at around 1.5% Tom O'Brien has been writing His weekly gold report For almost 18 years There's no one that knows More about how the Gold market trades And how gold mining equities React New subscribers get a 30-day money-back guarantee So you have nothing to lose Every Monday morning Tom publishes His weekly gold report With coverage of gold, silver, Bonds, the XAU, HUI, GDX, the dollar, As well as more than 30 different mining equities. 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Trade C-H-A-U or C-H-A-D Directions daily CSI 300, China A share Bull and Bear ETFs China A shares in either direction Visit directioninvestments.com Today An investor should consider The investment objectives, risks, Charges and expenses Of the direction shares Carefully before investing The prospectus And summary prospectus Contain this And other information About direction shares To obtain a prospectus Or summary prospectus Please contact The direction shares At<PHONE_NUMBER> The prospectus Or summary prospectus Should be read carefully Before investing An investment in the funds Is subject to risk Including the possible Loss of principal The funds are designed To be utilized only By sophisticated investors Such as traders And active investors Distributor Four-side Fund Services LLC So let's just get to a couple Of questions that came in. I love these technical questions Because this is what we're all About here. So, Greg wants to know Does TRUF-D have the same Importance going down As a peak D? Does on the way up? Thanks, Greg. So, you know, I've spent a lot of time Over the years. Look at this cyber. It went to a TRUF-D In the weekly chart. So, you know, I've spent a lot of time Over the years. Look at this cyber. It went to a TRUF-D In the weekly chart. But I have a rule of thumb That suggests a massive move Down, like... Let me just show you Something here. Mmm... There it is. A massive move down In... This is the dating of the Down from 27,306 On the 12th of September In TRUF-A, TRUF-B And then a huge TRUF-C And that was a temporary 5-day road trip. So, you know, I've spent a lot of time Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years Over the years that historically have worked way better than on the way down so on the way up I use the letters plus the technicals on the way down I use the technicals the letters are just a guide I don't say oh we're at a C we have to go to a D on the way down trough C we have to go to a leg D to a trough D a low low I found too many times you can have bought a down move very quickly at A, B, C, D, E, F it the letters are not as important the only important in the sense that there are times when I get trough A trough B trough C trough D trough E and just keeps going down and that says be careful you're in a continuation pattern but when I get it with a very long leg A to the downside and a short B and then there's a sharp rally to the upside that rally sometimes like here like this is the reason why I use this flagpole pattern for cyber C Y B R trading 119 70 Apodot 25 because the speed to trough A was such that once you especially after a gap once you do anything filling the gap or not completely filling the gap you reverse to the upside if the technicals are still strong I'm done I'm only looking at the upside arm out of whatever position is so that's that so the answer is no it doesn't have the same importance it is important only as a measure and a gauge of time that's used or the speed of the down move but on the way up I absolutely use letters all the time and I mean that's the reason I got that cell signal the reason why I wanted a short Tuesday morning because everything was telling me that there was some kind of a pullback coming of importance at that point it was yes the letter F will know let me do this one at a time let me just show you in the Dow it was an F in the S&P and F is where some of the deepest declines occur in the in the S&P was F in the QQQ that was a leg F perhaps going to get a peak F there it is in the IWM we had already made the sixth highest peak F at a hundred and sixty point forty six way back in the beginning of November yeah we are in a trading major the downside holding quite nicely but in a trading band so yes that's the way I look at it but look at the troughs on the way down trough E you know I look for the technicals more than anything else so the answer is no what do I look for I look for the technicals on the way down I look for notation on the way up and then put the technicals together the other question I had so I hope that answers the question the question I had was I think I'm not sure if I'm getting it right here let me just see I think it's from L yeah we go some what is the question oh twenty five point forty can I buy TBT here let's go to TBT off because I want to look at TLT let's see TBT can I buy here well that made a peak E and then it pulled back sharply and then made this is exactly what I call a rogue wave everything about the technicals were coming down went out of the blue it spirals up to that peak right then that's a peak F and I can't put it down arrow because I've already used it up on the E so it's an accidental new high and I have to use inverted V and I always color that red so yes that's a that's a reversal now the question is is the TBT about to rally and my answer is based on what I'm looking at L I would say no it could bounce a little bit but I'm suspecting that there's a little more weakness to come in in the TLT TLT's at 131 down from the 130 at 139 29 and that should probably try for legs see I don't know if it even go to a deep but I think it should try a little higher so in a timing model I would say to you probably not yet in the TBT which is the inverse of the TLT I think the TLT has a little way to go to the upside just right now then I think it can't solidize and only later does it start to see maybe in January I don't know a really strong move to 149 which would be above the peak C of August and that'll start leg D and 148.91 so just if I talk about a timing model I just don't think it's ready right now I do think you're gonna have a chance to do that but at 25 40 now it's 20 25 40 let's get the exact price TBT 25 48 right now I I don't think you're talking about a bounce I think you're talking about a real position that's gonna just knock the data it's off the wall no I I don't think so maybe even on a short term basis there could be a bit of a bounce in the TBT as the TLT pulls back because I do think there's gonna be an attempt to get close to the recent highs and then fail in the market and then come down to lower lows and that would say to me money or go from stocks into bonds again hey I could be wrong I'm sorry I know you're very good at timing but I just I don't want to I don't want to change from day to day I'm just saying I don't think right now the TBT is the way to go Mike at Almond Beach how are you? Hey Basil, I sold my CGC trade today and my MJ and I was wondering if you think where if first of all if we could get a some kind of a pullback and where you think it may get down to so I can reload again. So first of all congratulations you're ready that was well very well planned. Secondly I'm looking at some of the others like at GWPH which is GW Pharmaceuticals, Medical Marijuana, whole cannabis section but on medical terms and it's just been pummeled from 196 down to today's low I mean 196 to 96 196 to 96 if you take away 100 that's the hundred points that have dropped and that's one of the circle better I think you did the right thing but I'd like to look at it so can you hold on we'll be right back yeah I can vote for you I look at CGC and it's canopy I believe canopy growth does not turn poison right I'm certain you are or strive to be one of the best of the best at everything you do in life it's the most common trait that we tigers and tigers is share if you're looking to become the best of the best when it comes to managing your money let me teach you to do what most wealth managers tell you can't be done which is how to time the markets I'm Steve Rhodes author of Mastering Probability and for the last 12 months timer digest has been tracking my newsletter signals which have earned me the ranking as their number one market timer in the nation for the S&P 500 for the last 12 6 and 3 months timer digest also ranks me as the number one market timer for gold as well the fact is markets can be timed and I'll teach you the exact set of tools that I use that has transformed me into one of the best and what I do sign up for Mastering Probability today by clicking on the newsletter tab on the homepage of TFNN.com and get immediate access to workshops where I take you step by step how to use an extraordinary set of tools as well as provide great market calls to sign up today if you haven't checked out the newsletters page of TFNN.com what are you waiting for all of the TFNN newsletters are informative up to date affordable and a must have for every trader looking to gain a competitive informational edge in today's markets TFNN newsletters cover every aspect of the markets to offer you the very latest in market news plus new subscribers get to test try our newsletters risk free for 30 days from all aspects of the markets including stocks bonds metals commodities and tech there's a newsletter to fit your needs exclusively from TFNN stay informed each day you trade and get the competitive edge that will help you stay ahead of the game visit our newsletters page by going to TFNN.com and click the newsletters button near the top of the page TFNN.com educating investors you know what's cool taking something that's good for you something specifically formulated to help with weight loss better sleep stress reduction and the need to detox nico our hunter and gather ancestors found all their nutritional requirements for health in their wild environment but today our food sources no longer contain the vitamins minerals and nutrients our bodies need to stay healthy and strong that's why we need primal edge daily nutrition it includes a special blend of ionic soil based vitamins minerals fatty and amino acids and an easy to use liquid form primal edge is powered by highly concentrated phobic and humic acids nature's preferred delivery system they've been called miracle molecules because like sunlight air and water life cannot exist without them that's right page they ensure we receive all the nutrition we need to be healthy and thrive we take it every morning primal edge formulated and approved by nico and page of living a primal lifestyle buy it today for just eighty nine dollars click on the primal edge banner on the front page of TFNN.com Hi folks this is Steve Rhodes stay tuned for another great hour of the trader's edge heard here at TFNN.com Hi folks we're back and just that two minute chart made a peak E with two little dojo candles and now it's pulled back so now we'll see how the next thing the phase goes we're looking at CGC which is canopy growth canopy growth in the medical marijuana segment and as I say GWPH one of the one of the real big winners dropped a hundred points so there's a big problem in that area so I suspect so you kept nothing you just got it and you got out you took your money in your hand is that correct you know so I want to thank Tom O'Brien you Steve Rhodes and there's two other technicians that I learned from and for all of the great education I was able to pull off this trade from from what you and everybody else taught me so I want to thank you guys first for educating me and what I did I had see I work full time it's so hard to trade the way I really want to trade and what I did was I had I looked at the chart and I saw we were getting kind of close to the 50 EMA and I saw where that and the 34 EMA has acted as resistance in the past so I put my sell order in that area and it went on you know higher but you know it hit the 50 moving average but you know what I think you did a wonderful trade you did it that was perfect now the big thing is this how would you get re re kind of re-enter and I'm just going to say to you the profits that you've made and you know I'm just going to tell you from my own experience of what I over the thousands of people I've been involved with over the years that I've done this you've told me how they've done and where they've gone wrong etc you cannot making a real nice chunk of money puts two responsibilities on you one is you kind of pat yourself in the back you've lifted your hands off the off the wheel and that's when you hit the hit the pole you got to be careful not to do that number two is where do you get back in and what amount of money and I'm just going to say to you in the CG you've got your way to your patient keep your job do what you're doing that be happy that you're able to do this it's like a game it's like a pleasure it's like a it's like a sport see your next entry and I would just say your next entry should be a little less and take your time then if it pulls back to the twenty point twenty or nineteen point seventy area let's talk again maybe that's where you want to add a little more game have a great day


BSU men's basketball, beginning their final homestand of the year against U-Marie. In the first half, Beavers piling it on. Jakob Hoffman from three. That made it a 19-4 game. A minute later, Beavers keep raining it down as Logan Bader connects. They're up 18 early. Shortly after, John Morgan, driving, avoids the trees in the paint, gets the land. A balanced night tonight as seven Beavers score at least seven points. As BSU wins it big. Taking a tax-deductible contribution to Lakeland PBS.


What's good y'all Chris from Josh's case brace calm. Happy Monday first break of the day going to be a 2023 tops Chrome Jumbo pack random team 5 this fills out the last 16 to tops Chrome Jumbo 8 box case PYT number 20 With nine free spots being given away to start off the break at the very very end will randomize are the names top six We get these six teams in order our big four D-backs Yankees red Sox Rangers and the Marlins and Giants to round it out Well, let's give away those nine free spots first. Thanks again y'all for getting it in Kevin down to Nicholas with the last three spots for the last spot mojo. Let's grab those nine names. Good luck three and two five times one two three four and five fifth and final time Top nine Gonna be Nicholas Craig Jose Jr. Nicholas with the last one module all three of Nicholas's spots get in wow Michael Michael Steven David and that was Nicholas five nice the top nine get in fresh dice and are Now full and fresh list with the arrow next to your name. You see that arrow next your name means you won the spot And the free spot give away So 30 names for our 30 teams from the D-backs down to the Nationals Which roll this dice and randomize for the pack. Good luck y'all three and six nine times one two three four five six seven eight and nine ninth and final time Jose Jr. down to Steven Oh, yeah, I agree.  He just is nine times for the teams one two three four five six seven eight and nine ninth and final time for just the teams in the pack Mets down to blue jays Since this is just a filler No trade window, but if you'd like to take a look and see how everything shook out you absolutely can And I'll put ties by team name so just for the pack Jose Jr. With the D-backs on the Michael with the Nets Let's switch screens real quick. I got hangman is a Personal favorite mind some of the bias towards him, but like he is like an AEW guy he just did like yeah He was in ring of honor and yeah, he was in New Japan, but like he was never anything really there He is a tried-and-true AEW All right For the pack filler Jose Jr. With the D-backs is not all cards should break as well But random team five was a junior with the D-backs All the way down to Michael with the Nets Nice focuses up nice and easy Michael Harris rookie in there With some gold nice. We'll take it from a jumbo pack filler gold Gonna be Manny Machado nice 22 out of 50 for the San Diego Padres.  All right, that's the pack. It's the fun stuff though. Let's switch one more time Let's grab Jose Jr. And Down to Michael our last fresh dies a last fresh list top six We'll get in we'll grab those teams in order. Let's do it. We'll do dice and randomized six times for six teams Good luck one more time one two three four five and six So seven out of 30 sad times like to give it a shot. I think you filled this one out Michael with the free spot down a house a junior with the free spot. Thank you for shot y'all, but congrats to Michael s Michael s David Matthew Ryan Michael and Nicholas one of his last spot Moja spots all straight up spots 20 other place The dice roll likes to be funny What can I say? But this is the way it all shook out So Nicholas has the D backs Michael with the Yankees Ryan with the Red Sox Matthew with the Rangers David with the Marlins and Michael with the Giants. All right. Thanks again y'all And we come back fresh case the tops chrome jumbo number 20 getting broke just in case breaks.com. Bye. Bye


First, we have to select the language, after selecting the language, we will have a console page. Go to YouTube, log in with the admin ID of the course. Once you log in with the admin ID of the course, go to my channel. You will be seeing all the videos which should be uploaded in the course. Click the video which you want to download. Now we are going to see how to submit the assignments. First, you have to boost and add the add-ons to the Firefox. YouTube video can be downloaded only from the Firefox browser. In the Firefox, you go to add-ons. Then in the add-ons page, give one click YouTube downloader. The first four questions are MCQ multiple choice questions. And the fifth question is MSQ multiple select. You will be seeing this one click YouTube video download. You will be displayed as a radio button. You can see the checkbox in the MSQ file. Secondly, there will be one or more answers. Mp3 podcaster. You will select the answers and submit the answers. Podcaster for downloading Mp3 videos. You will be getting a message. Thank you for taking the assignment. If it is a multiple submission assignment, you will be seeing the certain to the assessment. Here, clicking this, I have already inserted, so it is not doing. If you want to add the add-ons, you can be submitted only once. You will be seeing this symbol and download option here. Click this, download the option. Here, the videos will be downloaded in FLB and EMP4 format. And you can see, already we will be having the assignment before. So you have to download the FLB. Within the duration of the assignment, you will be seeing FLB, 3GP and EMP4. In this, and download the video in FLB and 3GP formats. For Mp3 format, you can go to this Mp3 podcaster icon on the top right corner of the window. You can also download the videos here. Also, Mp3, Mp3 programming assignments and FLB format. First, you have to select the language. Just a way you can download all the videos in all the formats. After selecting the language, you will just put these files in the FLB part of your issue. So that we will be taking that video from the FTP path and uploading it on our Mp3 website. So that it will be easy for the students to download from Mp3 website.


Professor Frank Ferris, who teaches at Santa Clara University. I've known Frank longer than almost any other mathematician. I met him when I was a freshman in college. He was considerably older at that time. And he's near, we were just down the road for me in Stanford. So he's done many things in his life, but in the past several years, he's done some very beautiful work in symmetry, using mathematics to generate symmetry. And he published a book, which is called Creating Symmetry, the Art of Mathematical Wall, Paper Patterns, which has received wonderful reviews and is a terrific book. And I'm very pleased to introduce Frank Ferris. Thanks, Ray. Can you hear me all right? When I think about the problems ahead for our planet and for the United States of America, I'm sometimes overcome with grief and fear. But I take refuge in the beauty and constancy of mathematics. And when I look out at this audience and see people who are going to spend the next three weeks learning something new, gathering their energies, to take those energies out to their communities, then I'm filled with hope. So let's all, at all levels, embrace teaching as a radical activity for social change. Thank you for being here and thank you for coming to my talk. I wanted to say that I will be here through Thursday night. I would love to talk to you individually or in small groups. If there's anything that I could do to help you create a mathematical visualization, make something beautiful for a talk you're going to give, or something wonderful to take home and share with your students, I would be happy to help. Before I begin the mathematical part of my talk, and there really is quite a lot of mathematics in this talk, I want to give you an overview of my mathematical process. I make patterns. Oh, we're going to have these lights off, these first row of the house lights. I would much rather you look at the screen than at me. Thank you. So my overview is that I make patterns out of wave functions. And this is a meeting about harmonic analysis. So the waves fit right in. And the way I say it to a lay audience is that I start with a photograph like this, this beautiful peach on a rock. That's my leg and the stream that I climbed up to eat this peach for lunch. And I use mathematical waveforms to turn it into a pattern. For mathematicians, this is a little bit backward. For mathematicians, I would rather say that this is the domain of the function. The pattern is coloring the domain of a function. And that function takes values in that photograph. How do you find functions like this that make patterns? Well, the key is harmonic analysis. But first, it's kind of fun to look at a real close-up of a pattern like this, where you can see the little nub of the peach and a little piece of its stem have become beautiful decorative features of this pattern, including the texture of the denim of my jeans and even the little laces of my shoes. They become an important part of this pattern. Now, yes, it takes certain software to compute these patterns. These are made in C++. But there's a software package that is very easy to use. It's called Symmetry Works. You could play too. So one hope is that some of you will go home and become makers of wallpaper patterns, an activity that I enjoy very much. This is public domain software that was written two summers ago by students at Bowdoin College based on a user interface written by a Santa Clara student, and I have a student at home now working to improve it further. So I want to quickly go over the three major categories of playing patterns, but I need some vocabulary. I need the word symmetry. For me, a symmetry is a transformation of the plane so that when you transform the plane, the pattern remains unchanged. That is a symmetry of this pattern. This pattern is called a rosette, because every symmetry of this pattern leaves one point unchanged, the point right at the center. And I made this one out of a photograph of carrots and bell peppers on a cutting board. Cut them up, thought they were beautiful, took a picture of them. I'm going to try to soft pedal the algebra side. This is an analysis group, but I do have to say that the symmetries of any pattern form a group. Now, if you don't know what a group is, this is the first best example for you to think about. A group is a set that is closed under a certain operation. Here, the set is these transformations that leave the pattern the same. What is the operation? It's composition of those. So this pattern stays the same when you rotate it by 60 degrees. What if you rotate it by 60 degrees again? Well, it stayed the same the first time. It stays the same the second time. So under the composition of those two rotations, it stayed the same. It turns out that there are exactly six things you can do that leave this pattern the same, a group that many people know as C6. So I'm not going to say too much about groups, but I wanted to just give an example of the correct use of that word. The second category of plane patterns is freezes. A freeze is where all of the symmetry, let's see, the translational symmetries all proceed along one axis. So here, left and right. And yes, I'm inviting you to use your wonderful visual cortex to realize that beyond here, there is another cycle of this pattern, and another, and another, and another. And likewise, back this way, too. So this was made from a photograph of my living room before a party with a bowl of cherries and strawberries on the table. This and then wallpaper is sort of my main gig wallpaper. A wallpaper pattern is one with translational symmetry in two linearly independent directions. So I can tell you a little bit of my origin story as a mathematical artist by telling you about a teaching experience I had in the mid-90s. It was a grating definition of a pattern. I was teaching geometry, and there was a little section on freeze symmetries. And this book said, a freeze pattern is a set of points that is invariant under blah, blah, under the translations under a certain group. And their illustration was something like this. Now, this is a good illustration in that you can see the intent to have translational symmetry of one unit, two units, three units, and so on. There's an additional interesting symmetry that I wonder if it's familiar to you. You can flip this one that does not preserve the pattern, but if you flip and slide half a unit, the pattern is preserved. This is called glide symmetry. So I look at this, and I say, but wait, what about the beautiful artistic freezes that have been created by artists for many centuries? I think that patterns should be functions. So it was that teaching moment that set me, I think 25 years ago, onto this long exploration to say, no, you should make your freeze like this. And that is a function on a strip of the plane whose values are in that photograph of an Indian paintbrush from the Sierra Nevada. Another grading definition of pattern, as I learned more about it, I read some of the great books about patterns, and this definition said, a pattern is obtained by taking a motif and repeating copies of it without overlap to fill the plane. If you want a pattern in the plane or a rosette, you could take a slice of pie and repeat it around. So I said, okay, fine. If you want to make that kind of pattern, you can go stamp, stamp, stamp, stamp, stamp, fine. That's your pattern using the strategy of a motif. But I think a pattern should look like this. That's my idea of a rosette pattern with five-fold rotational symmetry. And sure, it's technically true that you can cut a slice of the pie, really any slice of that pie that's 72 degrees at the point will, you can make the whole pattern by repeating that slice of pie. So that's where I come from in this idea about using functions is that for me, they just offer so much more artistic potential. I'm gonna say a lot of things today that are gonna go very quickly. You can read the details in my book. The title is Creating Symmetry, the Artful Mathematics of Wallpaper Patterns. Came out from Princeton University Press in 2015. I've been delighted by the reception of this book. So that's available to you. So here's my list of topics that we're gonna go over. I'm going to tell you about the vibrating string and circle as a first way to get involved with the theory of waves. Then we'll talk about wallpaper waves, which requires me to tell you about something I call the domain coloring algorithm. And then we're gonna talk about color symmetry in wallpaper. And then I can't resist showing you some of my work since the book came out with things that are beyond plain patterns. Quite a lot has happened. And really there are a lot of things for you to engage with as well. As I talk about these topics, I'll be mentioning a few to things that some of the undergraduates who may be looking for short term research projects may wish to pick up on. So yeah, the vibrating string. Imagine a string like a violin string that is tied down at both ends. It's free to vibrate up and down. For technical reasons, I'm going to assume that it only vibrates up and down, not side to side. This is most realistic if the vertical dimension that's shown here is really quite smaller than is shown here. So that's the formula for it. This is the simple arch of a sine curve, just like that. And here's my first space of functions. I hope the mathematical notation is not too intimidating. This says the set of all functions that live on the interval from zero to pi with real values to indicate how high up or how high down with the property that they are tied down at the ends. That's a space of functions in which I may look for my string patterns. And the thing about this string is if you, it's been hanging there at rest for quite a while, but I can release it from rest and it will naturally vibrate like this. Up and down, up and down. Now, I have exaggerated the vertical displacement. The mathematical model is not really realistic for such a large displacement. But what do you think? I think this is one of the platonically beautiful things that the human mind just sort of adores, the motion of a simple sine wave going up and down. The thing, in order to make motions like this, I should try to solve what's called the linear wave equation. And if partial differential equations are too much, well, this means the vertical acceleration of the position of the string. And this represents the concavity of the string. So if you think about it, if the string is concave down, it should be accelerating down. If the string is concave up, it should be accelerating up. So that's a very simple interpretation of the linear wave equation. And here is my solution. I have taken my simple spatial sine function and multiplied it by the cosine of t. And people who know the technicalities by this will see that for this and all of the motions I'm going to show you, I'm beginning a motion from rest. And that's why I use the cosine here instead of the sine. But let's just pause. I bet you can differentiate this twice with respect to x. Sine goes to cosine, cosine goes to minus sine. That kicked out of minus sine. Start here, cosine goes to minus sine, minus sine goes to minus cosine. That kicked out of minus sine too. So this really does, both sides of that wave equation are true. So then I'm going to ask you, what other functions can you think of that would make this happen? So yeah, there they are. You could scale these by integers. And I don't have it on my slide, but n should be an integer one, two, three. Why is that so? Because I required the string to be tied down at both ends. Here are the first three of them. You can see the placid red one and then the somewhat faster blue one going twice as fast and the green one going three times as fast. And then there's this magic moment there where they all go back to their starting positions. It turns out this is unusual in the theory of the vibrations of things to have them periodic, that there's a moment where they all return to their starting point. But this is a very special situation. Sometimes I think that this is why the counting numbers exist. It's not because we have five fingers, but it's because there are modes of vibration of a vibrating string that are in correspondence with the numbers one, two, three, four, five, and so on. That's why we have counting numbers. And I think the next thing that's supposed to happen is to play that little icon so we've got somebody on the sound. That sequence is the basis of at least all western music and really indeed quite a lot of all human music. So I think that this is an iconic thing. This is part of our intellectual heritage, the existence of this sequence of string vibrations. In order to generally solve the wave equation, you wanna superimpose these with coefficients. And that's the thing about linearity. It's hard to get across the emotional impact of linearity. It's just so beautiful that once you have some solutions, you can superimpose them like this, meaning add up a sum with coefficients in front. And so I did some adding, oh yeah, no, I wanted vocabulary. These are called eigenfunctions of the Laplacian. In this context, the Laplacian is just a fancy name for the second derivative. And what does eigenfunction mean? It means you apply that operator, meaning differentiate twice, and you get a multiple of the function, German eigen to the self. These are functions that go to themselves with a little multiple here. And the eigenvalues here are the negatives of the squares of these counting numbers, one, two, three. When we think about frequencies, we're gonna think about the square roots of the negatives of these eigenvalues. So when we think about frequencies, we're thinking about one, two, three, and so on. So I just made a little pattern here. In making art, it turns out that it's best to include smaller amounts of higher frequencies that makes, to my taste, prettier patterns. So I made one amount of the first building block, half amount of the second, third amount of the third, and I made this shape over here. So, and you could think of this as the starting point in a linear space of functions. That's my function space. And say, here's where I wanted to start. Well, each of these waves knows individually how to move forward in time. And when they move forward in time, they look like this. And again, this has this graceful dancing flow. It's, this equation is somehow right because it is showing us something that we interpret as, oh, that's graceful, that's a beautiful dancing motion. So not to spend too much time on this base case, but it turns out that you can make pretty much any starting function from base waves. Experts will correct me. But for instance, I continue, do you see that this is a continuation of that pattern where I had one, a half, a third, a quarter of sine 4x, and so on? It turns out that if you sum those all, it really does add up to that linear function, which is tall at the left and zero at the right. And that, oh, when I take 10 terms of this Fourier series, it makes a pattern like this. And right there, it's trying to make that linear function sloping down. Right there, it's trying to make that slope down linear function. There's something weird at the left-hand endpoint. Think about the slopes of these at zero. The slope of this, each one of these building blocks is one at zero. So you're adding up a lot of very high slopes there. Here it is with 50 terms, and you see something weird going on. Yeah, and experts will be able to tell you about the propagation of singularities observable here. Notice that I'm asking these simple sine waves to do something very unfair. I'm asking them to jump from zero up to pi over two, actually, right in an instant there. It can't be done. But these waves, they do so well to try and do it. Now, you'll notice this little overshoot when they jump here, they're overshooting. I know that one of the projects proposed for the undergraduate research projects is to study that overshoot, it's called the Gibbs Phenomenon. And then just to take delight in the connectedness of all mathematics, if you plug in right at the middle, plug in pi over two there, it turns into the Leibniz series for the number pi over four. So, who makes this stuff up kind of moments, the connectedness of all things? So the summary of the string is that, and we're gonna see this in many situations, is that you discover some shapes, which are eigenfunctions of the Laplacian. You create your desired starting point from those shapes, and then each individual shape knows how to move forward in time. Let's take a leap to the circle. So, I hope people might recognize these as the parametric equations for the circle, cosine t, sine t. As t progresses, this starts at one there, and just goes round and round and round the circle. Well, I wanna take you into the world of complex numbers by saying, you know, complex numbers are just a repackaging of Cartesian coordinates. You just have this other little slot i there, and that's the real part, that's the imaginary part, that's the x coordinate, that's the y coordinate, and we're gonna do algebra on these with the requirement that i squared is negative one. And then you're gonna take another leap by saying, you know what, there's a shorthand for this, which is e to the i t. And this is the beautiful Euler formula that I think many of you know all too well. Anyone could write quite a long article about how you might think about going from there to there. It's the easiest, non-trivial example of a map from a Lie algebra to a Lie group. So, anyway, I'm gonna be using that shorthand. Another thing about it is that that obeys the usual rules for exponentiation that you can exponentiate a sum by multiplying the two individual exponentiations. And that also is an eigenfunction of Laplacian. So I'm gonna ask you to believe that you can do a differentiation rule where you pull down that i, and if you pull down that i twice, the eigenvalue is minus one there. So this too is an eigenfunction of the Laplacian. If I think of Laplacian as being simply second derivative with respect to t here. Well, and then I'm gonna do this trick with periodic time rescalings. That's what I did when I went from sine x to sine two x to sine three x. Here, I'm going to scale these by going once around, twice around, three times around. And yes, negative one times around, negative two times around, that means going around backward. So this now is my space of functions to play in. And my discrete question mark here is this really the same as all periodic functions? Can any periodic function from the plane, from the real line into the complex plane, can it be written as one of these exponential sums? Well, go ask a harmonic analyst to reveal to you the amazing subtlety of that question. There are, it depends on what you mean by is, right? So here's another part of my origin story is that long ago in the 90s, I was using some simple software called DRIVE to show parametric equations to my students. And I wrote out what amounted, if you put it in complex notation, it's this. And I thought I was just making up a random example. And Jerry Alexandersen, who's been a lifelong friend and mentor and who has recently retired after 60 years at Santa Clara, said, well, Frank, what did you do to create the symmetry? They said, I don't know. When you rotate from there to there by 72 degrees, the pattern falls into coincidence with itself. Indeed, the trace of this curve does have five-fold rotational symmetry. Here's the functional equation for that symmetry. The left-hand side says, let's advance time by a fifth of a full period, two pi over five. When you do that, you should get back the original thing you get, but multiplied by this rotating factor. So that multiplication factor is a rotation through 72 degrees. So you start at the red arrow, you advance time, and that is the same as rotating that point to the second red arrow there. How do you solve that functional equation? Well, this was my first article in Mathematics Magazine, and which led me to actually later be editor of that magazine for many years. And so you can read the story. I'll tell you the punchline though, which is, let's see, there's the functional equation we're trying to satisfy. These are the kinds of functions we can use, and this is an example of something that works. Well, when you think about kicking out a factor of two pi over five in the context of putting that six there, well, five of those are gonna give one full rotation which you throw away, and there's one left over. So throw away by fives and have one left over, that means that your coefficient should be nothing unless n is congruent to one mod five. And what about this negative 14? That's one more than a multiple of negative 15. Oh, that does fit the recipe. And then I just have to tell you that, you know, once you've got waves like this, each one knows how to move forward in time, and I'm gonna leave that one in place. That is the circle. I'm gonna think of the other terms as deformations of the circle, and I'm gonna let them rotate forward in time. Oops, I need that word. This is called a recipe. This is a prescription about coefficients in a series that will guarantee a kind of symmetry. That's what I'm gonna mean by a recipe. So these waves know how to move forward in time, and it looks like, oh, that beautiful bounce. Now, I'm totally violating anything about the wave equation, but nonetheless, there's something platonically beautiful about this in that moment where it turns around, you can just,


All right, so let's talk a little bit about how to actually obtain the value for an equilibrium constant. We have seen that we can obtain the value for an equilibrium constant.  So if we're talking about a gas phase reaction with relatively small molecules where we can write down the partition functions, then we can calculate the equilibrium constant using this equation from scratch. And that works pretty well, but it's a lot of work. It's fairly hard work. And it doesn't apply in cases where we don't have a good way to write down the partition function. The other thing that we can do to calculate partition functions is we can take a step back. This is an equilibrium condition. This is the...actually, that's not true. The equilibrium condition is that product is equal to this product of molecules raised to their stoichiometric coefficients. The other equilibrium condition that we've seen, if we think back a few steps before we obtain this expression, is that when a system is in equilibrium, the sum of the stoichiometric coefficients times the chemical potentials had to add up to zero. When the reaction proceeds forwards or proceeds backwards, destroying reactants to make products or destroying products to make reactants in neither direction does it end up changing the net chemical potential of the system. So we can use this expression as it turns out to learn something about the equilibrium constant, and specifically it's going to tell us something about how the equilibrium constant is related to the free energy of the reaction. So let's explore this equation a little more. The chemical potential of the system for each species we've seen can be written as the standard state chemical potential plus RT log of activity. Activity, and that's certainly true under all circumstances, so that we can make this about some specific examples that we can make progress with. For gas phase reactions, I can write that chemical potential, I'm sorry, that activity. So first let's break this up into a sum of nu times mu naught. So I'm going to expand this sum, and I've also got nu times RT natural log. So I'm going to break this up into two separate sums instead of writing natural log of activity. If it's a gas phase reaction, and if I can trust the ideal gas law, then the activity is pressure, partial pressure of the gas divided by the standard pressure. So I've made an assumption or two in going from this equation to this equation, but this is true for gas phase reactions under relatively low pressure conditions. So this is the one that can turn out to tell us something about equilibrium constants. So let's take this second sum and move it over to the left side of the equation. So when I do that, it's going to become negative. As I do that, I'm going to pull the RT out of the sum. So I'm going to write RT times the sum of stoichiometric coefficients times the log of PI over P naught. That is equal to what's left on the right side is this sum of stoichiometric coefficients times standard state chemical potentials. So there's a few manipulations we can do here. Let's start, I guess, with the right side and remind ourselves that the chemical potential is the partial molar Gibbs free energy. So if I'm adding up stoichiometric coefficients times chemical potentials in the standard state, that's chemical potentials of the products minus chemical potential of the reactants all with their appropriate stoichiometric coefficients. So in other words, it's free energies of the products minus free energy of the reactants. This ends up being the free energy of the reaction under standard state conditions, the molar free energy of the reaction under standard state conditions. So that's just another name for some of the stoichiometric coefficients times chemical potentials. On the left side, inside the sum, I've got a constant times a log. So I can write that as log of this stuff raised to that constant. The RT, I'm going to prefer to have that over on the left side. So instead of putting a minus RT out front, I'm going to say minus 1 over RT on the right side. Next, what I'll do, since I have a sum, since I've gotten rid of all the things other than these logs that I'm adding up, I'm adding up log of this for i equals 1 and for i equals 2 and for i equals 3. The sum of these logarithms is the logarithm of the product. So I can rewrite that as log of, again, Pi over P0 raised to the new sub i multiplied together. Again, the sum of some logs is equal to the log of the product. And right side is still the same, minus free energy of the reaction over RT. Now that this is just the log of one quantity, I can undo that natural log by expeditiating both sides. So the log will disappear. On the right side, I'm going to have e to the minus delta g over RT. So I'll just rewrite the term in brackets, product of these pressures raised to the new i's over P0 raised to the new i's. I'll go ahead and pull the pressures out of the product like we've done several times before. They don't depend on which species we're talking about. So in fact, I pull out not just pressure 1 over P raised to the new, but since I pull that out for every term, I pull out 1 over P raised to the delta new, the change in the stoichiometric coefficient. And now that's enough for us to recognize that this term, product of pressures raised to their stoichiometric coefficients, that's an equilibrium constant. It's not molecules raised to their stoichiometric coefficient, so it's not K sub n. It's pressures raised to the stoichiometric coefficients. So this equation is going to become, that term I've underlined here becomes Kp. That's the pressure-based equilibrium constant. On the right side, I have e to the minus delta G over RT. And then instead of leaving 1 over P0 to the delta new on the left, I'll write P0 to the delta new on the right. So that's the main result we were after. That tells me how to obtain the pressure-based equilibrium constant if all I know is the Gibbs free energy of that reaction. So if I do know the change in Gibbs free energy of a reaction, I don't have to go to all the effort of calculating partition functions. I can use this expression instead. If I remember to include this term, standard pressure raised to the difference in stoichiometric coefficient. That turns out to be there largely to make sure our Kp has the units that it's supposed to, as we'll see when we work an example. So let's take a reaction like the formation reaction for ammonia. Nitrogen gas and hydrogen gas combine to form ammonia. If I do that reaction in the gas phase, I can ask myself, what is Kp for that reaction? To calculate it using this expression, I need to know the Gibbs free energy of the reaction. So the other piece of information I have to tell you is the Gibbs free energy of formation of NH3. If we go look that up in a table of Gibbs free energies under standard conditions at 298 Kelvin, that's negative 16.4 kJ per mole. So that's enough information now to ask the question, what is the equilibrium constant for that reaction? So first step is to calculate delta G of the reaction. We were given the delta G of formation for forming one mole of ammonia out of standard state products and reactants in their pure elements in their standard state. We need to know the delta G for this particular reaction, which is similar to the formation reaction, but not quite. So that's going to be products minus reactants. So add up Gibbs free energies of products times the stoichiometric coefficients minus the Gibbs free energy of reactants multiplied by their own stoichiometric coefficients. So the product we have is NH3. Its Gibbs free energy is negative 16.4 kJ per mole. It shows up twice. I subtract from that Gibbs free energy of formation of N2. N2 is an element in its standard state, so it has no Gibbs free energy formation. Subtract Gibbs free energy of formation of H2 three times. It's also an element in its standard state, so it has no Gibbs free energy. So this difference between Gibbs free energy of products minus Gibbs free energy of reactants works out to be twice negative 16.4 is negative 32.8. So that's the Gibbs free energy of reaction that we need to plug into this expression. So kp e to the minus delta g over RT. Now we just need to times standard pressure to the delta nu. We just need to plug this number in here. So I've got e to the, on the top we've got negative 32.8 kJ per mole or negative 32,800 Joules per mole. I'm going to divide that by R and by T. So R in Joules per mole kelvin is 8.314. This, I didn't write it down, but I mentioned as we were writing down the problem that this Gibbs free energy of formation is correct at 298 kelvin. So the temperature we need is 298 kelvin. So that's the exponential. This whole thing we're going to multiply by standard state pressure. So if we're doing this with a standard state of one atmosphere, now we need to figure out what delta nu is when we convert four molecules of reactants, one of N2 and three of H2. Four molecules of reactants are turning into two molecules of NH3. So I'm losing two molecules every time the reaction goes once. So the delta nu is negative 2. And now what our calculator tells me for this particular exponential, that works out to be a fairly large number, 5.6 times 10 to the fifth. The units are atmospheres to the negative 2. So what that's told us is all we need to know is the Gibbs free energy in order to calculate the pressure-based equilibrium constant for a reaction in the gas phase. If we didn't want a Kp, if we wanted a Kn or Kln, we could then convert to a different type of equilibrium constant if we're interested. Notice, as I said earlier, the purpose of this P0 to the delta nu, if you're using a standard pressure of one atmosphere, essentially all that does is introduce the correct units into the problem. Kp needs to have units of one over atmosphere squared in order for when we solve equilibrium problems that look like pressure of NH3 squared over pressure of N2, pressure of H2 cubed. The units on that Kp are not going to work out correctly unless it's got units of one over atmosphere squared because there's more pressures in the denominator than in the numerator. So that's told us something we didn't know before about how to calculate a value for Kp. The next thing to talk about is there is some temperature dependence to the equilibrium constant. Not only the temperature that appears explicitly inside this one over RT, but also the Gibbs free energy has some temperature dependence to it as well. So that's the next thing we'll talk about is how to understand what happens to the value of K as the temperature changes. And so that's coming up next.