Johnny Dollar. Dollar, this is Michael J. Kendry at the San Francisco Office of International Life and Casualty. What's the Kendry? No, I'm not in my office. I'm over at Lake Tahoe, you know the place? South Arena? Can I have a telephone? Can I have a Nevada? That's right. I'm staying at a motel called Seven Plains Large at the south end of the lake. On the map, the town's called Al-Tahoe. Seven Plains Large in Al-Tahoe. Yeah, got it. Oh, and I understand it's a pretty great fishing in that lake. So Kendry, I'll grab the first one I can. Fishing, huh? What? Maybe that's what you'll end up doing all right. Great. For the body of a man named Marvin W. Smedley. Huh? Yes, radio brings you far Bailey in the exciting adventures of the man with the action-packed expense account, America's fabulous freelance insurance investigator. Truly, Johnny Dollar. The expense account submitted by a special investigator, Johnny Dollar. For the International Life and Casualty insurance company, San Francisco Office. Following his mention of expenses incurred during my investigations, the Twins of Tahoe matters. Expense account at a 1-182-50 cab to Bradley's Field and an early-money jet flight to San Francisco. Item two, when I got the F-50 dollars deposit on a rental car. I could have taken a ferry plane to Sacramento or even North Carino. But I wanted another look-see at this part of the country. And I'm glad I made the drive. For Route 50, took me through a lot of the romantic old gold rush country. Yeah, through Placerville, now a quiet, busy little city. But once it's one of the most lawless, notorious places in the West, where the corner of Maine and Columbus streets was known only as Hangtown. And at one end of Main Street, you can still see the famous Hangtown Bell, where we used to call out the vigilantes in times of trouble. And then, finally, Lake Tahoe. And my way of thinking is one of the most beautiful spots on Earth. Over 20 miles long, about half as wide, its deep, clear water is the bluest I've ever seen, except in the shallows and on the edges where it takes to an almost brilliant emerald. Trolls still kept peeps stand out above the dense, forever-green forest that surrounds the lake, barely hiding the many resorts that nestle together along the water's edge. And the water's cold out there and very deep. There are parts that have never been fathomed. Now and then I could see the clean, cloudy way of the freeboats, sometimes with the waters here and tall. A group of small steelboats like canvas spread to the wind. Here and there along the shoreline, a few hearty soles and swimming, bathing the chill waters in that breathless high altitude before the early winter snows would come. And then, in contrast, in various spots along the highway, on the Nevada side, as such things are illegal in California, there were huge, gaudy, restless, noisy gambling houses. And out town, though, it was seven pines large, and there were seven tall pines around these little motels. Kendry was waiting there for them. Yeah, that's right. Glad you made it. I'm Mike Kendry. Oh, yeah. How are you, Mike? Good morning to Mike Cabin. I'll tell you what it's all about. Oh, Mrs. Elton, Mr. Dollar's alive. I'll settle him in the cabinet tomorrow. Come on, Johnny. Yeah, yeah. Hey, you mentioned fishing for the body of a man by the name of Smedley. The police are fishing for it now. Marvin W. Smedley. He's a client of... He was a client of ours. Very wealthy, retired. Good morning. Good afternoon. Good morning, sir. Well, on the way in here, you passed a big piece of lakeside property with a small stone house set back away from the beach. Oh, yeah, I know. Smedley, where he lived with a housekeeper, Mrs. Turner, and his twin nephews, Tracy and Alfred Smedley. They're over 25 or six. So what happened? About three o'clock this morning, Mrs. Turner woke up to what she thought was the sound of a shot. She went into the boy's room to see if they'd heard it, but they weren't there. Oh. And she remembered Tracy had said he'd be out gambling all night over on the Nevada side, and that Alfred was at the two or three-day house party at the upper end of the lake. And rather than try to wake an old man Smedley, well, she decided it was only a backfire and went back to bed. Yeah. But when she got up for breakfast, well, she found the library in a real mess and the safe had been rightful. Uh, no sign of Smedley? No. That's the one, Mike, please. Now, she knew that old Smedley had a habit of getting up in the middle of the night and going in there and counting his money. He never did trust the banks. He kept everything he had in that safe. So she knew right away what must have happened. Uh-huh. She knew for sure when she found that Smedley wasn't in his bedroom or anywhere else around the house. Oh, uh, also, she said, she could smell gunpowder there in the library. Well, did she call the police? Yeah. And Sergeant Corder came running, and when he found the footprints leading down to the beach, down to the dock, he found the marks of a body dragged along. And the body itself? No sign. Must have been weighted down and dumped in the lake. Who, Mike? No question about it. They've been living easy often for years, just waiting for them to die so they could get their hands on his money. One of the twins. Only one of them? Only one said the footprints out there in the sand. Had the police been able to find the twins? Sure have. They've got them locked up in the local Huzoff. Then what's the problem? Only one of them did it, Johnny. Okay? But both of them. Both of those boys have absolutely airtight alibi. Crime, delinquency, threats of war. These are the subjects that dominate our news headlines these days. Not very pleasant subjects, are they? You may say that somebody ought to do something about cutting down on crime and delinquency and in promoting peace among nations. But that there's nothing you personally can do about it. Well, that's where you're wrong. You can wage your own fight against crime and delinquency in your own family by taking the family to the church or synagogue of your faith this week. The inspiration and guidance you and they will receive from spiritual contact will strengthen moral background and faith. Regular attendance at religious services will help your family to work out its own problems and give them comfort in facing the tensions of our present day life. Worshiping together brings your family closer together too. And supporting your own religious institutions provides funds to help those individuals and families who, unlike you, are unable to help themselves. Find the strength for your life. We're together this week.  He said roped off a lane from Smedley's house to the beach to the dock where it was obvious from the traction of the sand that somebody had dragged the old man out there, piled him into one of the robots, then dropped him into the lake. It was one of those twins Mr. Dollar had to be. Now, why do you say that, Chief? It started. It started to build cars.  Which one? Either Tracy or Alfred. But you don't know which. Well, that's the trouble, not only because it was a pair of brand new shoes that didn't have any what you might call identifying mark, even where they showed up here in the damp sand, but because those two young punks always dressed exactly alike. You know, so it's a cover up for each other. How do you mean? I mean, not only because they were always... Those boys were always mischief makers, Johnny, and sure were and still are. They, well, they just didn't have anything else to do. One of them was provoked a fight at one of the gaming casinos or borrowed somebody's car without permission, if that's nonsense, and each of them was where it was the other. Oh, yeah. And because they look and talk and act exactly alike nobody can ever tell which is which. Well, the police here never knowing which one they could legally hold and they were always threatening the super-false arrest. And you'd be as mad as we are over how many times they've gotten away with that, darling. Right. And not only because witnesses to whatever they did couldn't tell them apart, but that fool uncle of theirs had always come along and pay a fine or something before we could nail down the right one. And you say they both have alibis for whenever it was this thing happened. Yes, sir. About three o'clock this morning. And you're sure about those flip flops? Sure I am. Found the shoes that made him in the house in the clothes closet. Found the sand on him. In those closets. Tracy's. Aw. Yes, sir. And Alfred's. Aw. Same closet. tragic quarter. Mike and I went on into the house to talk to Mrs. Made-Centers. Nightly's old housekeeper. Tall, raw bone, but erect and strong in spite of years having worked hard all her life. She looked as if nothing in the world could ever face it. But obviously the death of her employer had. If only he'd have listened to me, Mr. Dollar. If only he'd have listened. Because I know something like this would happen to the poor old man. He'd been keeping all of his money here in the house instead of down to the bank where he ought to. How much money, Mr. Center? Goodness knows, Mr. Dollar. But for all he had was a rich man and everybody knows that he kept it here. And that he'd come down in the middle of the night and cut it all out to himself. And look, he didn't even pull down the shade so as anybody and everybody could see him sitting here with all his money. And I told him, I told him. And now, now it's happened. In other words, Sergeant, almost anybody around here could be suspected. If it wasn't for the footprints out there in the sand. Well, what footprints did you say? Well, to be honest, it's not it. That's why I came in here earlier, Miss Turner.   Well, that is... He doesn't need to tell me, Officer. Because I kept telling him about them, too. About the way he was spoiling them to that. Giving them everything they wanted. I told him sooner or later one of them would get so greedy he'd turn on to him. But which one, Miss Turner? Which one? They're the only relatives that he had. So it was ours. Oh. Oh, and it was facing. Oh, yeah. My next move was to go on down to the jail and talk with the twins. Oh, they were identical, all right. I wondered how they themselves knew which was which. But in spite of what seemed to be the general opinion of them, they impressed me. He is just a couple of other nice, normal, high-spirited kids. So what if we did raise a little pain around town now, and then, Mr. Dollar, none of it ever really did any harm. And we had to do something with our time. All right, now look here, Tracy. No, I'm outside. Oh, and what I mean is, it's no reason to pen a lousy thing like this on us. Even though we aren't too sorry that he's gone. After all, what did he have to live for? Oh, I mean, just having to drag on maybe 10 or 15 years more with that lung trouble he had.  Well, not since we got through school. But do you know why, Mr. Dollar? Why, Al? I mean, Tracy? Because Uncle Mar didn't want us to. He'd worked so hard as a prospector and a miner accumulating his pile, and he wanted somebody to enjoy his money. And that old skin-flint ever want somebody else to enjoy it? That's right, Tracy and me. We're his only kin. Yeah. That's why he told us he was leaving us all his insurance. And what about the rest of the estate? We don't own, we don't care. After all, he did plenty for us. I mean, even before this insurance we'll get.  All now, wait, Mr. Dollar.  Proof that? Well, what? I mean, sir, that unless you find his body... Don't you worry about that, Tracy. I mean, Al. My boys will find his body if they have to drag that entirely. But unless you do, you won't really have a case against us or anybody, will you? We'll find it, all right. And why don't you suspect somebody besides us? Yeah? Like who? I'll tell you who, Sergeant. What about all those old prospectors from the old days that he did out of mining claims that have kept coming back to me for handouts? Oh, you're a fine man to talk about handouts. Now, listen, don't tell me you aren't glad that he's gone. Yes. Tracy and me and that old witch Turner and all the other people... All right, I mean, if Miss Turner hadn't taken care of him all these years... Oh, sure, sure. All right, all right. I understand you boys have pretty good alibis for three o'clock this morning. Do we have to go over all that again? What, afraid you might slip up retelling them out? I'm crazy, Mr. Dollar. Okay, okay. Now tell those alibis to me. Okay, Mr. Dollar. Whatever you say. We'll leave Jimmy over there on the blackjack table. When he gets back, you can take over one of the relays. Oh, Mr. Market. Sure, Dollar. One of the medley twins was in here all night. Which one? Came in around midnight and didn't pull out until about 8.30 this morning. All right. Are you sure that he was here at about 3 a.m., huh? Sure, I'm sure. Playing one of the craps. He didn't step out for a few minutes, perhaps? No, sir, and I know. Because I always keep an eye on those medley kids. You don't watch them. They're liable to pull tricks on our dealers. They think it's funny. All right. Which one was it?  It was Alfred Smedley or his brother, Tracy. Oh, nice. Let me give you a dollar. No, no, no, thanks. Just go on with what you were saying. Well, the party got pretty big, and we all had a lot to drink. I didn't watch the clock, and it didn't really break up until about 9 o'clock. This morning? Mm-hmm. And he was with you all the time? Every minute. All right. Now, which one of the twins was it? Who cares?   But, now listen, I have to know if it was Alfred or Tracy. But I haven't believed that. You know, what's this all about, Johnny? Murder. Oh, well. Well, it was Tracy. Maybe it was Alfred. Oh, thanks a lot. The world of doubting Thomas's decades ago never would have been possible for stars to shine in the daytime. But in this day and atomic age, anything's possible. Thus, it's not only possible, but a fact that only on stations of the CBS radio network, stars do indeed shine in the daytime. You'll find on illuminating an entire segment of your daytime listening every Monday through Friday. These stars have all been named by celestial authority vested in their forebears. Their names are Arthur Godfrey, Art Linkletter, Bing Crosby, Rosemary Cluney, Gary Moore, and Derward Kirby. You won't find them in your books of the planets of stars, but no compendium of show business luminaries to be complete without them. Monday through Friday, your radio set is your personal telescope on this star-studded display on your CBS radio station. To list them once again, Arthur Godfrey, Art Linkletter, The House Party Man, Bing Crosby and Rosemary Cluney in tandem, and Gary Moore and Derward Kirby. Enjoy them often.  Lacey and Alfred Smedley had alibis, if for no other reason than that neither of them could have been in two places at once. And those alibis might be pretty hard to bring. Yet, according to what little actual evidence there was, one of them must have killed their uncle for the insurances. Wait a minute. Yeah, there was something in Alfred or Tracy. Anyhow, there was something in one of them that suddenly gave me a hunch. Expensive count, I'd have 370 cents for phone calls to locate Smedley's lawyer on the 16th advantage. Then I dropped into his office and asked him just exactly one question about the old man's will. And his answer gave me a new fresh start on the case. I'd have 410 cents for another call just for the sergeant quarters. The boys found the body all right. Now you know how the water off that beach stays clear and shallow until it's all getting to 150 yards or more straight out from the dock. And it drops off over 300 feet straight down just like that. And the body had been dumped over it over that ledge? It hung up on a rock, not more than 20 foot under. I see. Did you find the gun? Same caliber as one that was issued to those twins a couple of years ago. Did you find it? No, sir. We've never found it in a thousand years. Maybe we don't need you if I can pull off a little bluff. In other words, we're never going to know whose friends are on it. I want to go back to the house. You meet me there on the library. Smedley's house. I'll be there in five minutes. That's right, Mr. This is turning. This is turning. It's a dollar. Well, let's quit fooling around with this thing. I just finished talking to Mr. Kenneth McManick. Mr. McManick? That's right, the lawyer. Yes.  Yes, he mentioned that. Is that all he told you? No, no. He told me that the remainder of the estate goes to... Let me see, I scribble down some of the actual words. The balance of my estate, including my home, I bequeath to one who, although I'm certain her motives have been entirely selfish, that she has simply been looking forward to being provided for in my will, nevertheless as a reward for Mrs. May Turner's faithful service and I have told her it is. He left something to me. You know that he told you he did. And that was your motive, Mrs. Turner. What is? Of course, the gun belonging to the twins was just as available as a pair of shoes that were borrowed to carry the body out there and leave the footprints. The point is, though, that when Sergeant Quarter found not only the body, but the gun out there in the lake. Oh, no, darling, I take your fingerprints and compare them with those we found on the gun. You, uh... You don't mind giving me a set of your fingerprints, do you, Mrs. Turner? Well...  Half of my life I spent taking care of that stinky old man and then giving me nothing. Out all the time, he kept giving his money to them. Them boys. So that by the time he finally would die, that's what they'd be left for me.  Oh, no. Oh, no? I hated those boys, but I know they could never put it on to them. Nobody could ever put anything on to them before. I'm afraid that's beside the point now. Now that we know you did it. But I had to. Because by the way, he'd give them all his money, and when he did die, there was nothing left. Nothing for me. Nothing. Don't you say? What's left for you now, Mrs. Turner? If you're on in and set to the court, then no doubt the whole estate will go to the twin. You know something? I hate this kind of a case. It leaves a bad taste in my mind. I spent the count on putting incidentals in the trip back home, $416 even. Yours truly? No, I don't. There's something people don't talk much about, but it can be a problem for anyone, even doctors. And when constipation occurs, it's interesting to see just what doctors consider important about a laxative they might use or recommend. Now, a majority of the doctors we heard from have this to say, a laxative should be effective, gentle, close to natural acting, a medicine that can be used with complete confidence. Well, pleasant-tasting chocolate x-lax is effective. Overnight, it helps you toward your normal regularity. X-lax is so gentle, so close to natural acting, there's no upset. That's why many doctors and millions of people use x-lax with complete confidence. X-lax, a laxative that helps you toward your normal regularity gently, overnight. Now, here's our star to tell you about next week's story. Next week, one of the most beautiful, most charming girls I've ever known involved with a killer. But, now, don't misunderstand me. I didn't say how she's involved. As a matter of fact, it's to protect her from a killer that I take on the case. And it just so happens that among the people on this man's list are intended victims.  He's also after me. Jonas, won't you? Yours truly, Johnny Dollar. Starring Bob Bailey, originates in Hollywood and is written, produced and directed by Jack Johnson. Heard in our cast were Virginia Gregg, Jean Tatum, Paul DuBois, Boris Lewis, Sam Edwards, Terb Ellis, Tom Hanley, and Bill James. Be sure to join us next week, same time and station, for another exciting story of yours truly, Johnny Dollar. This is John Wall speaking. Manus looms over blistering desert waste. Suspense brings you Inferno, next on the CBS Radio Network.


All right good morning cyber traders happy Friday and hope everybody's getting excited for the fourth of July weekend I am actually this morning I just went outside and I put my nice little flags outside all excited about it actually might even do a post on it that'd be pretty cool. But but let me tell you what a great day we had yesterday so many good moving stocks. Top is moving again that top we're gonna put this on the watch list we're gonna watch it what a beautiful moving stock top. Now a lot of you remember the stock and for some of you maybe put this over here. You remember the stock we trade this back in the end of what day was that that was on May 28. Okay so this stock went from $48 up to 250. And it also you could see right here from six to 24. Now this stock was a crazy stock yesterday right around here. Let me just blow this up and show you what we have going on right here. So we watched this stock in class in the morning and the big thing is it is a very fast moving stock.  It is a very nasty stock. You cannot trade a lot of shares in the stock. You can make a lot of money but you probably blow a lot of it because it is very very volatile. But it had this nice little pop right here right around 1045 and then got halted. It literally ran from a price of 670 all the way up to 870. You could see these candlesticks on here this thing was moving like literally like 50 cents 75 cents a minute. That is a lot of money you can get yourself hurt there.  You are going to trade a lot of it. You know three very very small. It is fun because it is in your portfolio because we talk about it and most importantly we learn from it. So that stock has gone on the watch list. But before we start going and talking about all the stocks on the watch list let us talk about what also we traded yesterday before getting ready for this holiday weekend. There was a couple of other really nice stocks. Fry was also a very nice stock yesterday. I mean here you can see from 9.30 came right out of the gate. This was one of the stocks that I posted. You can see it right here. So basically it is right here. So let me just show my other camera covering the room. There we go. So everybody can see this. This is my legal pad that I have. And right here in the bottom you can see Fry. You can see the IO. It was the Javi. We're going to talk about that one. Not all of them but I did a I don't know if anybody here is following me on Instagram. I took a couple of photos. I'm going to actually do a couple of photos and post them up there. I know we got some videos but this looks pretty cool. You know what time we posted that list. And you know it was on there. It was a nice stock. And you know and right here at 9.30 you could see it. We traded the stock in our morning class stock ran from like 860 all the way up to a whopping nine dollars and 90 cents. Now it was nice about this stock is that like what we teach at Cybertree University is what I was taught being a market maker on Wall Street is you got to follow the money and that stock had a gorgeous resistance levels. We had the game plan and you know it's funny I was asking the students I'm like I did a survey on it was that luck or is that skill. And a lot of them said it was skill and I says you know what it actually was more of a little bit more luck in a way that usually they do a shake. This did not shake. This came right down. So we got lucky we got out of it. It was the skill part of it knowing where the order was and having a game plan and maximize that profit on it. But the skill of the luck part about is like we didn't get out of it.  So anyway another nice little move we had on that fry. Javi also the stock in the last two days had a really big run up. You could see it was at $6 a long term and sure enough the day before the thing ran from like six to nine. And then they had a nice little pop yesterday and I think the party was over on Javi but you could see it ran from like 1050 all the way up to about 12. Same exact thing that happened with Javi happened with the FR-EY stock. So that was also another one. Now a couple of the stocks that are moving that are coming back into play. Now some of you probably know this plug was a stock that we liked for a while. Plug came out some pretty decent news in the last couple of days. It's been doing pretty well. It's turned in not only into a day trade but it also turned into a swing trade. You can see it right here from $9 ran to 10. It had a really good pop right here in the early morning hours. Just ran from like 930 up until about 1013. And then it kind of like just did a, which you could see a channel. It was basically a channeling there. That one did pretty well. This one came back down because it was up big the day before. Revon came up pretty well too. This stock, it was kind of slow and boring. It went up really slow but Revon's been doing pretty well the last couple of days. I mean, you could see here long-term. Now it's on a major resistance levels going back on a June 21st. But you know what? She's up this morning. She farmed from 1580. If she can break that 1620, we're going to, we'll check it out. We'll check it out. We'll put it on our watch list and we'll see how it goes. So you could see how stocks from the day before can dwindle in and now coming from the day of. Now also Carvana, another stock that's been doing pretty well last couple of days. She came back down. She was a good short yesterday, but she's up a little bit this morning. She's up about a dollar. We'll put this on the watch list too. So wow, that's a lot from the day before. We got four stocks that trickled in from the day before into today. Now what else is moving? Well, first of all, let me bring up top really quick. So I want to show you a couple of interesting things here. If you look right here, it had this big resistance levels right around here at around 1030. And a lot of orders came in. You see a lot of green balls, a lot of volume. And then all of a sudden you could see how the stock literally just came right straight down. Now the orders starting to come back. This is called a ladder effect order kept coming down here from early from 1060. He dropped down to 1020. Now he dropped down to 1018. So I'll be a little cautious on that. Remember, the biggest thing that we always teach you and I'll repeat it a thousand times and we'll keep saying it over and over and over again is that you have to follow the money. Okay. And for some of you here that are watching on social media, YouTube or me watch this later.  It just overkill. It really is. And I'd rather follow the future than worry about what happened in the past. So anyway, that's that's one of the stock. Let's talk about what else is in the big percentage gainers. Now this day was this RNLX. She's not really going anywhere right now. She gapped up pretty big and you could see she kind of just kind of flat line. There was really no orders out there. So I'm going to just because we find these stocks in a big percentage gainers like, Oh, look, a 38% traded 3.7 million shares. It doesn't mean it's really going anywhere. Might move in the market opens up. As of right now, it's not doing nothing. Another one, CYT gapped up pretty nicely, not really going anywhere. Tops did make number three on the list though. We already know that one. That one's already back on the watch list. CTMX, this one looks pretty interesting. So you could see it starting to trend up a little bit. She's about 27% very inexpensive stock. She could get up and test these resistance levels right around $1.90, which are not too far. Order-wise, there were a couple of big orders that did show up around $1.78. I'll show you what I'm talking about right around here. And it looks like they got executed, but to have this huge jolt right here from a buck 70 up to a buck 86. I like to see that. That means that the stock has good volatility and good movement. So that one is going on the watch list, CTMX. Okay, that one putting that on there. What else we got going on? This one just made a quick pop right here. 20,000 shares, 29,000.  You all know about that, the three T's, right? Tradeable trend and trap. I don't know if that one is as tradeable. I don't care what people think on that. CIS, new IPO came out. Probably one of the first days are up. It just went from $1.30 to $1.80. Coming back down. I have the list here. It looks like I saw a lot of buyers out getting executed. There's one pretty decent buyer at $1.23. I mean, talking about pretty far away from where it started. I guess that's right there. You can see him right here. I'll show you where he is. See him right there? You have this guy right here and you see the sellers out there. So once again, not super, super excited about it. All right, I'm done with the New York's. Then Azac, any good losers? Don't see anything exciting there. New York Stock Exchange. Ugly spread of 17% flat. APV. This one's been doing pretty well on the swing trade, but day trading. We scratch this out several times. Fry, we already know about. Looks like that finally made it work. We're at 244%. Fourth biggest percentage gainer. We're done there. That's done. All right, tops just made a nice little pop. I know I saw that grant. Really nice pop. 80 cent move. Remember, we started, we started at nine o'clock. Look at that big, big pop right there on that top. So hopefully we're seeing that one. So anyway, it is the day. It's a very, very big weekend. This is probably one of the longest weekends next to a holiday, like a Christmas or something like that. When kids are off from school, but this is actually a very, very big holiday because, you know, when you have a holiday falls on a Tuesday, people like to take off Monday. Monday is a half a day. So everyone be aware of that.  So by the way, happy birthday for everyone. Everyone, it's everyone's birthday. You have two birthdays. I always tell everybody your personal one and living in America. You got a second one. So, so we got a bit. So you have that. So what's going to happen is you could probably spill into like maybe Wednesday might be a little bit slower. But today, I think you probably get a lot of action early, whatever's out there. I usually days like today are a little bit easier to trade. People want to get out of their trades and something's moving. Sometimes they can't. So they got to get stuff out of way before they make it a long weekend. So, but we'll be here on Monday, but let's see where we could go. We could pull out. All right. Anything I'm missing. I did not pull out.   A lot of you showed up this morning, which is pretty awesome. I look great in that. Okay. All right. I'm going to mark it open. We've got about 20 minutes. Uh, cyber traders. You see something. You trade something. You let us know. I know we have a bunch of trial members that joined us. Hopefully all of you had spoken to your education advisor, teach a little bit more about what to expect. Remember, it's not just about this trading room. It's about doing those classes that we have for you those videos and those lessons to kind of let you know how to trade properly. And if you're watching on social media, like I said, uh, if you want to ever join us just to take your phone, scan that little barcode up on the top corner and you'll be happy to join us for a week in the trading room. And in the meantime, we'll see you back here at 2.30 and all the traders, anyone in the room, we'll see you back when the market opens up at 9.30. Good luck, everyone. Happy trading and enjoy your long, long weekend and happy birthday, America.


It's Cruz Pipe. Matt right's calling me Big C. I'm back with you guys today. I got a whole bunch of stuff for you. First thing I want to talk about, we're going to take a look at the latest YouTube statistics. The video stats are out, guys. They're crazy. And I'm going to give them to you. And of course, if you want to verify what I'm telling you isn't just me making up, there's a link in description below. The second thing I want to talk to you about is a Facebook strategy. It's a potential strategy that I've seen big Facebook guys and actually big YouTubers use. I don't know if it's intentional or not. And it just helps them grow their channel. And it could make you more money. So we're going to talk about that second. And the third and final thing we're going to talk about is, well, your skill sets. As a video editor, as somebody that understands social networking, video editing, YouTube and all that stuff that goes with it, I want you to know that you have an amazing skill set that is going to serve you very well in the future. And if you are in a position where, you know, you're not getting the number of views you want to get or you're not getting, you know, maybe the subscribers or anything, you know, that kind of stuff. We've all been there. We're all like, I want more. Guys, the world is your oyster. I promise you I'm going to give you some tips and some encouragement at the end of this. All right, that big mumble jumbled over the weight. Let's get into the statistics. All right, stat one. All right, guys, the first statistics are the YouTube statistics. We've all heard them. YouTube's huge. YouTube's growing.  Well, we've got a new stat and here it is. For every single 24-hour period, YouTube shows 46,000 years of video. So for every day globally, the world watched 46,000 years of video. That is an unbelievable statistic that's almost incalculable and it is massively growing. It is not stopping. In fact, the train is just choo-choo-choo, man. It's just, it's unstoppable. Here's some more stats for you. YouTube videos are currently 11.3 times more popular than Facebook videos at the moment. For every 11, I guess for every 11.3 minutes, a YouTube video is watched. A Facebook video is watched once. So that's a stat that, you know, it's kind of shaky. We'll talk about that later. But also, millennials, people between the ages of 13 to 17 are 18.5 times more likely to watch a YouTube video than a Facebook video. These are just some stats that I wanted to throw at you because, well, here's the thing. For those of us, and I talked about it briefly at the beginning, sometimes we feel like, hey, you know, we're not getting where we want to be. We're not getting the number of views we want or, you know, when is this thing going to pick up? When am I going to start, you know, getting them subbed and getting them views and all that stuff? Well, here's the thing, guys. This train is still going. I mean, it's not slowing down. It's growing crazy fast. And without other way, I want to talk to you about a strategy that might work. Here it is. All right, guys. The first thing I want to talk to you about is Facebook strategy. That's right, guys. I mentioned it briefly at the beginning. YouTube is way bigger than Facebook, and it's very hard to compare them. It's really apples and oranges because they use different measuring systems. But I'll talk about that at the end. But here's the thing. Has any of you or have any of you tried uploading videos to Facebook, uploading them organically? I'm not talking about embedding links onto your Facebook page, even though that does work. So, using Facebook's organic uploader, uploading your video that you made for YouTube, put it on Facebook, and then use Facebook to drive traffic back to YouTube. Let me give you an example. If you create a Let's Play for a new game and you make a quick one, instead of uploading it only to YouTube, upload it to Facebook on your personal page or on your, you know, or maybe you have a game page or something like that. I know Dissector, if you want a good example, Dissector has his own Facebook page. So, upload it there and then say, hey guys, to watch more videos like this, check out my YouTube channel. Put a link in the, in your right up there and try and send traffic back to YouTube. I know big YouTubers like Key and Peele and a couple of those comedy guys, Amy Schumer, they have uploads both on YouTube and Facebook. And I wouldn't have heard about them in many cases if I wouldn't have seen it on Facebook first. I watched it on Facebook and I was like, hey, this is hilarious. And then I went to YouTube and I found all their videos. So this is another strategy that might work for you guys.  Second part of this is have any of you ever tried this? If you've tried it, let me know what kind of results you've had. I've uploaded my own personal stand-up comedy videos to Facebook and I've had great results and I've seen search engine increases going to my YouTube channel. So I can tell you it does work. Let me know what you guys think about it. Secondly guys, the other thing you want to know is that Facebook, they count their videos a little differently. So while this sounds like a great strategy, do keep in mind that Facebook counts a video after three seconds and on Facebook news feeds, they auto-play and they count that. So keep that in mind and they also auto-play without audio. So there are some caveats, there are some things that, you know, ain't quite, like I said, they're apples to oranges but it is a potentially effective strategy. Give it a try and let me know if you've tried it or if you do try it, what kind of results you get. Alright, the last thing we're going to talk about is some words of encouragement. Here we go. Alright guys, the last one. This isn't real fluffier guys. I'm not trying to inflate egos but I'm going to tell you a little bit from my situation as a YouTuber in the past and currently I've had great success and I've also had channels that sort of didn't get off the ground and they didn't get the number of views I wanted. And sometimes it used to get me down or it can get me down but not as much anymore. I'd be like, hey, I spent all this time learning how to video edit, spending, I've uploaded all these videos and nobody's watching them and you guys, if you feel the same way, there's no shame in it. I've been there, we're in this together but you know, I've created all these accounts, I've changed my channel art and I only get like a hundred views or a thousand views or whatever the number it is. It doesn't really matter. Here's the thing you've got to think about though. Like I said, those statistics at the beginning, I told them to you for a reason. Because YouTube is growing, whatever skills you are getting right now, whether it's video editing, social networking skills, all of that stuff guys, these skills are not just good to have right now. They are going to be incredibly, if not infinitely more valuable in the future. Even if you're a young YouTuber, you're like 14 and you're not getting the views. I'm just using an example. Guys in 10 years from now, when you're out there in the job market, cranking it out and of course doing YouTube on this side or maybe you've made it by then, doesn't matter. You are going to have a skill set that will be able to get you jobs at all sorts of companies, whether it's a large corporation that needs professional video or small, medium businesses that don't know how to do YouTube and they want a video because they sell a product or a service. You can go in there, film that, edit it, do the audio, create a Facebook account, create a YouTube account, do the channel art, do the icons, do the search engine optimization, the back linking, et cetera, et cetera. You guys know that stuff or you know most of that stuff and this is going to help you in the future. You're going to be able to make a lot of money off this. So if you're feeling discouraged or you're not getting what you want right now, I just tell you that to try and tell you to keep the faith. You're creating an amazing, invaluable skill set that's going to serve you for the rest of your life and you get to do what you like to do, make it YouTube videos. How awesome is that? Anyways guys, that's my wrap up for today. Thank you for watching this video. Stay the course. The stats are coming. Featured on Freedom is coming. We got all sorts of stuff coming here at Freedom. Alright, stay tuned. This is Big C. Check it out.


OK. So maybe we'll start. So we were talking about products. And then, in particular, about completeness, which, as I said, was supposed to replace compactness for varieties. So the definition was that the variety x is complete if we take the second projection from x times y to y, then this is a closed map for all varieties y. So that means that the image of any closed subset in the product with any y, if I project it down to y, will be closed in y. And as we had seen, so we had shown that one thing that, if x is complete and f from x to y is a morphism, then we find that the image is always closed. So this is also property that you have for compactness. If you have a compact space which has a map to a house of space, then the image is always closed because it is compact. So now we want to give some. And we had, finally, the big theorem was that any projective variety is complete. So this was a somewhat more tricky theorem that we proved. Essentially, one had to show it only in the case of Pn. And then it was finally reduced to some kind of linear algebra statement by using the projective null-stellar sense. OK, so now we come to some corollaries. So this would be new now. So first, if let x be a projective variety, if I take the functions which are regular on the whole thing, then these are just the constants. So any regular function on the whole of x is constant. That's quite different from the affine case where we knew that the regular functions, for instance, on an are all the polynomials in k. But here it's very different. There are very few regular functions. I mean, if I assume that to be regular everywhere. The second statement is every morphism from, so say, P from x to y from a projective variety to an affine variety maps the whole of x to a point. That means there are essentially no morphisms from projective varieties to affine varieties. If I have a morphism from a projective variety to an affine variety, the only possibility is that everything is just mapped to one point. So in particular, if x is affine and projective, so affine means that it's isomorphic to a closed sub variety of affine space, and it's also projective, then it follows x is a point. So the affine and projective varieties are really something very different. So this is quite simple. So and in fact, it all follows from the first. So assume we have a regular function. So let f be regular on the whole of x, where x is a projective variety. So that means that f, a function, can be viewed as a morphism to a one. It's the same thing. So f is a morphism. So by this statement here, as x is complete, it follows that f of x is closed in a one. But so what are the closed subsets of a one? So they are finite sets of points, a one or the empty set. And f of x is irreducible because the image of something irreducible, so it follows that f of x is a point, or f of x is equal to a one. Now we have to exclude that it can be a one. But that's simple. So via the embedding of a one into p one, say a one identified with u zero or something, we get that f gives a morphism, f from x to p one, which however we know maps everything to a one, because the image is the same as before. So the image is still closed. So f of x is also closed in p one. But obviously, a one is not closed in p one. The closure of a one is p one. So it follows that f of x is a point. So this proves that f of x is a point. In other words, the function is constant. So the second statement is kind of clear. So y is isomorphic to a closed sub-variety of some an. So if I compose with this isomorphism, I can assume that y is a closed sub-variety. Well, then we know that a morphism from anything to something in a n is given by an n tuple of regular functions. So thus, p can be written as f one to f n, where the f i are functions which are regular on the whole of x. But we know then they are constant. So each of them is just constant equal to some number a i. So phi maps everything to a one to the point a one to a n. So that's also very simple. And the rest is clear. If x is a fine and projective, then the identity would be a map to which this applies. And the identity map would be supposed to map x to the point. And so x, et cetera, is a point. So as I said the other time, the fact that the image of a morphism starting from a projective variety is always closed. So a sub-variety of whatever you map to gives us a new way to construct variety. So if I have any variety and the morphism from that to some projective space, so any projective variety and morphism from that to some projective space, then the image will also be a projective variety. And so this is a way to construct many projective varieties. We want to look at one example, which is the Veronese embedding, again after some Italian mathematician Veronese. And that's rather simple. It's somewhat similar to the Sege embedding in some way. So we fixed two numbers. Why? We fixed just one number. So we fixed the number, say, d and n, so some positive integers. And we put n to be the number n plus t choose d minus 1. And so we want to construct Veronese embedding, which is new d, from pn to pn, the Veronese embedding of degree d. So how does it go? So we want that the coordinates here corresponds to all the monomials of degree d in the coordinates on the original n. And that is precisely the correct number. And so let mi from, say, of x0 to xn for 0 to i to n be all the monomials of degree d, while in x0 to xn. And then we define w vd, this mu d, I suppose mu d d4, from n to n as given by all these monomials, m0, mn. So just the ith coordinate of an image point is just the, they are somehow numbered in some way. It's the ith monomial of degree d in these coordinates, evaluated on that point. So I claim this is a morphism. So to define a morphism, this should be all homogeneous polynomials of the same degree, that they obviously are, they are polynomials of degree d. And they should have no common zeros on pn. So that's only other condition. So and that's easy to see, because we can just choose a few one of those, which already have no common zeros. Because already x0 to the d, until x1 to the d, xn to the d, which certainly are some of these, belong to these, have no common zeros. Because x0 to the d is 0, x0 is 0 and so on. So these have no common zeros. So this defines us a well-defined morphism. And so we get nu d of pn is a closed sub-variety. So in this case, actually, one can show that this map is an isomorphism onto the image. Maybe I will not claim nu d as a morphism from pn, so its image is an isomorphism. And this is basically because one can define the inverse morphism. I just do it locally. So if we look at, so the open subset ui, so this is the points of all a0 to an. In pn, such that ai is different from 0, they form an open cover of pn. So maybe I will not, the other end.  So say, for any monomial of dvd in p0 of pn, we may be right, so this monomial is called m. Maybe I write zm for the corresponding coordinates on p large n. Now you see that according to this, the coordinates on the large pn correspond and 1, 1 correspondents with the monomials of degree d in x0 to xn. They're just, as a notation, I index them by the monomials. It's just the way, the only thing that I know is that they are large n such coordinates. I can call them whatever I want. I call them zm, where m is the monomial. So then I can form, say, ui tilde, which is the set of all points, which is pn minus the 0 set of z xi to the d. What is it? Yeah, that's a bit so there are two different z's.  So this is the set where the coordinate corresponding to xi to the d is non-zero. So this is obviously an open subset, not pn, nu d, open subset of nu d of pn. And this ui tilde form a cover of nu d of pn. Because we have kind of used this already here, we know that on pn, at least one of the coordinates is non-zero. And if we end the image, then at least one of the xi to the d is non-zero. Because what we started with, one of the coordinates was not zero, the corresponding coordinate xi to the d on the p large n will be non-zero. So this form an open cover. And we see that nu d, if I restrict it to ui, is a map from ui to u tilde i. If the i-th coordinate here was non-zero, then the coordinate with xi to the d will be non-zero there. And so we just want to give always the inverse of this map. So and this nu d, so give inverse of nu d restricted to ui. And this is very simple. This is just, so it's given by, well, I say, you just take, say, zx0 to the d zx0 to the d minus 1x1. So I assume here, for simplicity, assume that i is equal to 0. You can look for any other i, obviously. And then this is given by zx0 to the d minus 1xn. So why is this the inverse? So if we start from ui, we map x0 to xn to x0 to the d, I mean to all these things. And then if we go back, we just have to, so this inverse map. So if I look at this, this would just map, so if we would start with a0 to en, then it maps to all these two quotes. And then, so the cm, and then we project this just to a0 to the d, a0 to the d minus 1 a1, and so on, a0 to the d minus 1 an. But this is the same as a0. So to an, because we just have to divide by a0 to the d minus 1. And you can also check that it works the other round. If I start, that's a bit more what? Yeah, yeah, that's just a misprint. Thank you. OK, so you have to check that the other composition gives you that. OK, so we find this. So in particular, let's just give some. So one thing which occurs often is the case that n is equal to 1. So the simplest example is the map nu d from p1 to pd. If you work it out with the numbers for n equal 1, you get just pd. And the map now is just, if you look at it, to take all monomials of degree d in the variable. So we have, so a point a0 a1 is mapped to a0 to the d, a0 to d minus 1 a1, and so on, until we finally get to a1 to the d. And so the image curve is usually called nu of d of p1 is called a rational normal curve. So this is just a curve which is isomorphic to p1, but somehow it lies in pd. And this is considered because it often gives a counter example to some things for months, might think to be true. And another case is if you take nu2 from p2 to p5, then the image is called the veronizosurface. Again, it is considered because it often gives an example of something one would not expect. OK, so this is this example. There's one kind of application. So we have seen here that in some sense, the coordinates on this large pn correspond to monomials of degree d in the coordinates on the original pn. And this will lead to the fact that if you look at the intersection of p small n with a hyper surface of degree d in the image under the veronizosurface, this will correspond to the intersection with a hyper plane. And that somehow simplifies things sometimes. So let me write this down. So let f equal to sum ai mi. So mii again our monomials, i equals 0 to n, the homogeneous polynomial of degree d in x0 to xn. So now let x subset pn be a closed sub-variety. Then I claim, so if I look at nu d to the image under the veronizomap of x intersected the 0 set of f, then I claim this is equal to the image of x under the veronizomap intersected 0 set of the hyper plane sum i equals 0 to n ai z. If you think of it, this is completely obvious because the image of, so the map is just given by sending by all the monomials of degree d. And so you find precisely this. If I take the intersection of x with a hyper surface of degree d, it's mapped to the intersection of the image of x with a hyper plane. These are just the coordinates, it's a linear combination of the coordinates. So this is essentially directly from the definition, even though it maybe looks strange. So we know that nu d is an isomorphism onto its image. So this allows us sometimes to restrict to reduce questions about intersections with hyper surfaces to intersections with hyper planes. You can use this, reduce questions of about intersections with hyper surfaces to intersections with hyper planes. Because after taking the veroniza embedding, intersection with a hyper surface becomes intersection with a hyper plane. So we give an application, so that's not just theoretical nonsense. So let x in pn be a projective variety. So let f be a non-constant homogenous polynomial of degree, so homogenous polynomial of degree d, degree d bigger than 0. So then first, so f is obviously in x0 kx0 to xn. Then the first statement is that if I take x minus a 0 set of f, this is n a fine variety. And the second statement which actually follows, if x is not a point, then the intersection of x with a 0 set of f is non-empty. So that means if you have any sub-variety and you intersect it, and it doesn't happen to be just a point. Well, then the intersection with any hyper surface will be non-zero. So it's kind of a very strong strengthening of the fact that in projective space, any two lines meet. So we have something much more. Anything which is not a point with a hyper surface will always intersect. And we will later make use of it when we talk about dimension. So let's prove this. This is a relatively straightforward application of the above. So first, so if f is a hyper plane, so degree is 1, we can apply a projective transformation such that the image of f under the projective transformation is just the hyper plane at infinity. I could say maybe whatever, it doesn't matter. So you could write a to the set of f. This is equal to the 0 set of x 0. And then, obviously, we know that x minus 0 set of x 0 is just the open subset u 0, which is isomorphic to an. Or we can view it as an. So pn without and x intersected x without the 0 set of x 0 is equal to x intersected an, which is certainly a fine. And this projective transformation is an isomorphism of pn to itself. So it follows that it's also true before that. So by the projective transformation reduced to the case that the hyper plane was just the hyper plane at infinity and then it's open. So this is the. And now, if it's not a hyper plane, we apply the Veronese map. If f has degree v, so we see that x without the 0 set of, so via the Veronese map, we have that x without the 0 set of f is isomorphic to x minus a hyper plane to mu d of x minus a hyper plane, which is just the hyper plane which was obtained in this way here. And therefore, mu d of x minus a hyper plane is a fine by what we already saw. So this is a fine. So for the second statement, it's just an application of the first. So assume it's not the case. So if x intersected the 0 set of f is empty, we have just seen. So what do we have then? If we take x minus the 0 set of x, this is, after all, just like equal to x, then this is a fine. And it is projective because x is projective. Thus, it is a point. And so this is precisely the statement here by a contraposition. If we say that if the conclusion is false, then the assumption is false. It's the same as saying that the statement is correct. OK, so this proves this. OK, so this is everything I wanted to say for the moment about completeness. So what we, for the moment, say about the product. And now we come to one more topic about, in some sense, morphisms, which is rational maps. So we want to do this relatively fast. Maybe a bit more than one lecture of the lectures over, so we're not expecting to finish today. So rational maps. So I mean, we have these morphisms. We have introduced these morphisms because we want to use them to study varieties. So they are somehow the tool that we use to study them. Under we apply the morphism, we see what happens, and we find out something about the varieties. So the problem that we, in some sense, have with this is that there are not very many morphisms. So for instance, I mean, for instance, we have this. We have just proven that from a projective to a defined variety, there's basically no morphism at all. It maps everything to a point. But in general, it's difficult to construct morphisms. You have this condition. If you have a projective variety, you want to map to another projective variety, you have to find so-and-so many polynomials, homogenous of the same degree with no common zero, and whatever. And that's just not easy to find. And so we want to try to make do with something which is not as good as a morphism, but it's easier to construct. And this is a rational map. And that's something quite simple. It's kind of almost a morphism. Namely, it is a morphism which is not defined everywhere. So there's just an open subset of the source where it is defined, a non-empty open subset. And then the morphism we had before is just a special case where this open subset happens to be the whole of the variety. And it actually, we will not really come to that. But if one is doing more advanced later, the way how one actually usually construct morphisms is to construct them as rational maps and then prove that they are defined everywhere. So it's somehow, it is actually the way how one does it. Well, so a function for us is always a map or a morphism to K to the ground field. And we will prove that a rational map to the ground field is the same as a rational function. But I always call, it's maybe my personal viewpoint, but for me and also in the usual language, now we're talking about the function is always thought as a map to the ground field. And a map is always to another variety or a morphism. But so it means that rational functions are a special case of rational maps. But that we'll have to see. And they're actually quite, we will find out that they're quite closely connected to rational maps and rational functions are closely connected. So lemma. So first before starting, we need, I want to, so that everything is reasonably well defined, we prove a small lemma, which in principle we have already done, but I want to repeat it. So if phi and c are two morphisms from x to y of varieties, and we assume if there's a non-empty open subset in x such that phi restricted to u is equal to psi restricted to u, then it follows that phi is equal to psi. And if you remember, we have in some sense already proven this, we have, namely we have proven that if we have two morphisms from x to y, then the set of points in x where phi of p is equal to psi of p is a closed subset of x. And now we are looking here at this closed subset in terms of the condition that it contains a non-empty open subset. Now we know that any non-empty open subset in a variety is dense. So it follows that the closure of it is the whole of x and that proves this, okay? So this we already have proven it, but it's anyway. So now we want to define a rational map and it's slightly more complicated than what I made you believe. So a rational map. So maybe I can basically say what the idea is. So I said a rational map is a morphism which is defined on an open subset of the target. But we don't want to care about which precisely what open subset that is. So therefore a rational map will be an equivalence class. So where we call two such things equivalent, so where a representative will be an open subset plus a morphism from that open subset and we call it two of them equivalent if on the intersection of the open subsets they are the same map, okay? But this is just, it basically just says it's a morphism on an open subset and we don't care about which open subset. So a rational map f from x and it's usually done with a broken arrow is an equivalence class. So I write u comma phi of pairs. So u and phi where u and x is non-empty and open there's an open subset in x and phi from u to y is a morphism. And as I said, the equivalence relation is that two of them are equivalent if only the section, they are the same map. So here say u comma phi is equivalent to v comma psi if and only if this is restricted to u intersected v is equal to psi restricted to u intersected v, okay? So obviously when I say I'm talking about equivalence class one has to see that this is an equivalent relation to exercise this is an equivalent relation. If you, you know, you have to see symmetry which is trivial, reflexivity which is trivial and transitivity which is not trivial and to prove that the transitivity you use this lemma and then it's actually quite simple then. So I mean I just, so we say that say this map the insertion of that psi is defined by say v psi if this is representative. So if it's one of the things equivalent to it. So now that I have made this complicated thing with the equivalence classes I want to tell you that I could have also done it in a different way which is in some sense easier. Namely instead of saying I say it's a morphism on an open subset and I don't want to say which one I can instead say it's a morphism on an open subset which I make as large as possible on a maximal open subset. And so that cannot be further extended. So there's kind of one best representative always. And so this is as follows. So let v from x to y the original map say defined by you know phi on an open subset then the claim is phi defines a morphism from an open subset. So phi I could still call it phi from the what I call the domain of phi to y. So with where the domain of phi will be the largest open subset on which I can define it. And it's just what? So the domain of phi. So I take the union over all. So I take another one which is equivalent which is in the equivalence class. So we take all v psi which are equivalent to u phi. So that means that psi restricted to intersection is equal to that of v. So it's just, so I take any function which on the intersection of u intersects so which is defined on some open subset such that on the intersection with u it is equal to phi and I just take the union of all these open subsets. This is an open subset. And how is this morphism defined? Well just whenever I am in a v I take the corresponding map. So by, so we define phi of a point p to be equal to be say psi of p if we have that v psi is equivalent to u phi and p is in v. And obviously you have to see to show that this is well defined and the morphism. Well defined morphism. And it's kind of clear because to be a morphism means there is a neighborhood of the you know that for every point where it's defined it must be morphism in the neighborhood but it's given by morphisms. And so we have to see it's well defined but so if we have a point p which lies in two different such open subsets then it means it lies in the intersection of the, so on it lies maybe in v one and v two and we have c one and c two. So then it's a point in this intersection of v one and v two and we know that c one and c two will be equal to each other on the intersection. Now that is will again follow from the transitivity of the equivalence relation. So it is well defined morphism. So we can instead of saying it's an equivalence class we can also always extend the rational map to the largest possible open subset. And if this largest possible open subset happens to be the whole of x then psi is actually morphism. Well yeah so it's somehow glued. Yeah I mean you can say yeah but yeah you could call it we kind of glue it yes but you know in some sense it's just that the equivalence relation in such a way that you just but yeah you sometimes call this gluing in more complicated situations. Okay so let me see. So first then so I have some remarks. So the first one is what you asked in the beginning. So the rational maps from x so f from x for a one are the same as the are the rational functions. K x namely so in O x U for some open subset. So if f is an element in K x then it follows that f is regular on some open subset. So namely you can always write it as a quotient of two elements say in the coordinate ring and then the if where the denominator is non zero it is a non zero. And so as f is a map function or a map and so it's a regular it is a rational map to a one and you also find that the equivalence relation you have on both sides are the same. You know we had seen that if two rational functions coincide on some open subset then you have a non zero function two rational functions coincide on some open subset then they are equal. So it's somehow the same story. Conversely if f from if say U f from x to a one is a rational function or rational map well then why we see you can just reverse steps.  As I said the equivalence relations are the same. So you know the two things are the same as elements in K x if and only if they are the same as rational maps. So it's really that the rational maps are the same as rational functions. So you can use this to look at some other examples. So if x is a variety and you take any elements f one to f n in K x then we get a morphism then say f one to f n from x to p n so maybe a n is a morphism is a rational map and so these f i will be regular on some open subsets which I maybe call the domain of f i so this is the set of all points where you can write f i as a quotient of two regular functions so that the nominate is non-zero and so it is a morphism so from the intersection i equals one to n the domain of the f i which is some open subset of x to a n and we can also do the same in the objective case so let x in n be the quasi-projective variety and we take some say n f zero to f m some rational functions then say i write f zero f m from p m is a rational map which is defined on some open subset namely the open subset will be the points where all of these functions are regular so each of them is regular on some open subset so there will be the intersection of them is still a non-empty open subset and they have no common zeros so therefore I must have that not all as zero but just very different before it was very difficult to for instance to have morphisms but now you can just take any rational functions, anything this will always give you a rational map which is defined somewhere and here it's in some sense even more extreme this could be a projective variety so you know there's no morphism but you have an enormous supply of rational maps there are always lots and lots of rational functions the rational functions are the same as that on any open subset and for any tuple of them you get a morphism for a n whereas you know that there's basically no morphism at all and we can do this also in a different way with four polynomials so we have the same so you have the same assumption that x in p and part of the variety and say f0 to fm the polynomials homogene is of the same degree so and they are not all in the ideal of x so not all of them vanish identically on x then f0 to fm from x to pn m is a rational map so we don't have to worry whether these things have common zeroes as long as they are not all identically zero on x we always get a rational map okay so for instance in just a stupid example so say we take projection from a point so we have p equal to the point zero one in pn then we have the projection from we can look at we had talked about projection from a point if we have a sub variety which does not contain the point but we can instead also define the projection from a point as a map from pn to pn minus one is a rational map so pp so projection from p so this is p equal to x0 to xn minus one this is a map from pn to pn minus one and in this case we had seen this was just the map was just given by taking any given point in pn and taking the line from that which connects that point with this point and intersecting the line with the hyperplane so the map is actually defined everywhere on pn except the point p so the domain of pp okay so now we have these rational maps but usually if you have maps you want to compose them and now the problem with these rational maps is that they cannot always be composed and that's kind of obvious because a rational map is not defined everywhere so if you have two maps and the first map maps everything to the locals where the second map is not defined then you don't have a composition so you want to avoid this and you avoid this by only considering maps where the image is dense in the target so then this problem cannot happen okay so definition a rational map so phi from x to y is dominant well if in a suitable sense its image is dense in y so you can say this if I could call it phi of the domain of phi is dense in y and you can check as an exercise or it's also in the notes equivalent is so if say u phi is representative so u is any open subset on which phi is defined only non-empty open subset then this is equivalent to the fact that the image of the domain is dense is equivalent to the statement that the image of u is this is equivalent to so the statement that the image is dense if it holds for one open subset holds for all open subsets so now I want to define a composition so let phi from x to y be a dominant rational map so assume that it is defined on u an open subset u and psi a rational map from y to z rational map then we can define the composition so say defined by defined on some open subset v of y then it follows that if I take the inverse image of v under this map if I take the map from u to x to y then this is a non-empty open subset of u then u intersected phi to the minus one of phi is a non-empty open subset of u and therefore of x now obviously q to minus one of v if it's the only question is whether it's non-empty it's certainly open because the inverse image of an open set by a morphism I mean the morphism on u so this is a non-empty open subset and we can define our rational map by just this as a representative so if I take the pair u intersected phi to the minus one of v and on this I define psi composed phi this is from x, z is the composition so I get the rational map which I have now defined to be I have given it some open subset on which it is defined we don't know precisely how big the maximum open subset might be might be much larger but there is an open subset on which it is defined and that's what we call the composition and then the domain of this would be the maximum open subset on which I can extend it in many cases this might be the whole of x but even if it's not true for the individual pieces so in particular we can do this if the second one was a rational function so we can use rational maps to pull back rational functions so this allows to define the pull back of rational functions so let f into y be a rational function again and let phi from x to y be a rational map so then we have again the pull back so we have phi star so this here I would call psi composed of phi star of f is defined to be phi f composed with phi which is a rational function so a map from x to a1 so one would do the same thing one finds some open subset which phi is defined one takes the inverse image of that open subset intersects it with the set on which f is defined and then from this to that gives a rational function so we have this pull back and again we can easily see that this behaves nicely so if we or something so easy to see if we take phi star from ky to kx so I should say that it's not a rational map it's a dominant rational map so for dominant rational maps we can define the pull back so phi star is a homomorphism so if we take a constant function it pulls back so actually homomorphism of k algebras if you pull back the product it pulls back the product of the pull backs the sum as the sum okay so I'm a bit faster than I thought so now you know we want to so we had before defined what an isomorphism between varieties is this is a morphism which is an inverse and if two varieties isomorphic then in many ways we kind of view them as more or less the same thing you know I mean they have all the same properties now we have this rational map which is a bit less than a morphism and we have so therefore we can also have something which is a little bit less than an isomorphism which is a bi-rational map so that's a rational map which has a rational map which is inverse to it and in that case we call the two varieties bi-rational and that means equivalent to the statement that they contain open subsets which are isomorphic so we know that in a variety an open subset is dense so if two varieties are bi-rational to each other then they contain open subsets non-empty open subsets which are isomorphic now if you know the point is that in Zariski topology open subsets are very large so it is a non-trivial statement that they contain open subsets which are isomorphic if you are in differential geometry if two manifolds have open subsets which are isomorphic this says that they have the same dimension so that's not very exciting but you know here it is a much more interesting notion I mean it means really that they are almost everywhere isomorphic and so we'll introduce this so definitions so a dominant rational map say phi from x to y is called a bi-rational map so it's again if it has an inverse so if there exists another dominant rational map say if and only if there exists another dominant rational map which I just call phi to the minus one phi to the minus one from y to x such that the composition of them is in one direction identity of x and the other one the identity of y so such that phi to the minus one composed of phi is equal to identity of x and phi composed of phi to the minus one is equal to the identity of y and here so I should say this is they are equal as rational maps so what this means is that wherever this is defined you know this is so when phi is defined and maps into where phi to the minus one is defined the composition will be the identity but that means that if I extend this composition to its maximal domain it will be the identity on x defined everywhere and the same here so it is not necessary that phi is defined everywhere or that phi to the minus one is defined everywhere but the composition is automatically defined everywhere if it's the identity so x and y are called by rational or by rational equivalent if there exists a by rational map phi from x to y okay so let me see what I can do now just 5 minutes the next thing would be some theorem which is a bit tricky so one special case which so in particular a variety will be called rational if it's by rational to an for some n or the same to pn the variety is called rational if it is by rational an for some n and so you can see that example we have that well obviously an is rational and pn is rational no because we have just the embedding of an to pn is a by rational map if I take say the caspital cubic which is the 0 set of y squared minus x to 3 subset a2 this is rational because we had seen that there's an open subset of it so we have an isomorphism from an open subset to another open subset if you remember a1 so because if I take the map from a1 to c which sends so it's what I could say it's given by t squared t to 3 so this is a morphism and the a bijective morphism actually and the inverse is a morphism on c without the point 0 0 so we see that we have an inverse rational map no that we had seen before and finally if you have pn times pm this is also rational because we have an open subset we have seen that it contains open subsets which are isomorphic to a n times a m so at this point you might wonder whether maybe all varieties are rational which would be a little bit disappointing no I mean we cannot be sure but in fact in a very strong sense most of our varieties are not rational it's a kind of a special thing to be rational so I think for instance I cannot prove it because that's actually would actually be difficult but I can just say if I have I cannot now write down the equation but if you have a a general so so if f c in k x0 x1 x2 is general I don't say what general is but somehow if you write down one at random polynomial of degree homogeneous polynomial of degree d bigger equal to 3 then the 0 set of f is not rational I haven't told you what general means it means for instance non-singular we have to see so if we take the so we will not explain this now but anyway so we have one can there is a certain but anyway maybe it's enough to say for any d bigger equal to 3 there is a polynomial in fact many of degree d such that the 0 set of f is not rational and actually much more is true I mean there are very many different kinds how things cannot be rational I mean usually by rational to each other okay so anyway maybe that's enough now and then next time we will say something more what being by rationally equivalent has to do with the function field and then we will talk about particular briefly talk about particular rational morphisms by rational morphisms so these are morphisms such that the inverse is that they have a morphism as a rational map which are is the blow up which is a very important construction I mean in general we are not going to use it very much but I just want to introduce it briefly and then this will finish the story about the rational maps and we will start talking start a completely new topic which is dimension so you know how to define dimension how to compute dimension and then we will find that now we actually are able to say something about it and make some theory because we can use these what we have learned about morphisms to study study the dimension okay so see you on Wednesday


What's up guys? My name is Michael and welcome to my YouTube channel today We are going to go over this problem called dynamic array on hacker rank So basically we declare a the problem says is to declare a 2d array n of empty arrays They're all zero indexed. We're gonna create an integer and set to zero Then there's two queries in a queries array. All right. The first one is if it's a one What we're gonna do is we're gonna create this variable called index It's gonna equal to x x or our The last variable last answer and then we mod it by n then what we're gonna do is we're gonna append y to this answer Append the integer y to this array array at index IDX The second query we could do is to x y which is in this case. We do the same thing Where we get the index which is gonna equal to x x or a last answer mod by n And then we're gonna sign the very the value array at index at y mod size of our Index to last answer and then we're gonna store this new value into our answer array in the end We're just gonna return our answer array So I'll just show you guys how to do this the code. We're just fall literally following these steps So first thing you want to do is create your two-dimensional ray of size n So that's how you do that here in C++ you vector vector our n vector Integer, okay, because it's a integer integer here Next thing you want to do is create your last variable Your variable is equal to called last and they want it to be said to equal to zero So that's what we do here. We set our last variable last is gonna equal to zero Then they say they want to create an answer array Right array of answer and then to literally just return that in the end. So we're gonna create an empty array called answer Here it's an integer now what we're gonna do is we're gonna loop through our queries So remember there's two types of queries in each query, it's We'll loop through our queries array and in each query. There are two types one is a Query with a value of one then x y and the second query is about query of value two and then x y So in this case what we're gonna do is we're actually going to get both of these values so The first Value of x is x is gonna equal to queries at i at one and the reason why it's i at one is because If you look here at the values here Okay, so if you see here each of the query very Queries has this form of one x y which is an array that has values one x and y So the value of x is going to be at index one Okay, because the value of One is at index zero the value of x is at index one and the value of y is at index two Right. So what we first have to do is let's actually get each of the values of x and y if I just indexing them So we're going to index x is going to equal to queries at at i at one and then the Query for y is going to equal to queries at i at two so zero one two So that's what we do here We're going to set the queries at integer x is going to equal the queries at i at one integer y is going to equal the queries at i at two And I already explained why it's At one and two is because each of these values are arrays. So zero one two, okay So, yeah, i is the current query by the way because this query is a 2d 2 dimensional So once we get our values of x and y We need to create an index variable So they said here Your index is going to always equal to x x or last mod by n So that's the same case for both of them.  So I'm going to put it outside of the f statement So that's what I do here. So I create integer index is going to equal to x and this is how you xor To xor you use this uh carrot symbol So I take x xor with last Which is last is our last variable and then take this answer and I mod it by n and that's going to give us our index Okay All right, so that's our index in this line this line of index Index is equal to this. Okay. Now what we need to do is that once we have this Right the index, um, we actually have to do the individual statements for both of them So we need to check, uh, which statement we're going to do So do that we have to actually check if the query is equal to one because if the query is equal to one We just have to do These two statements the first two statements and if the query is equal to two we have to do these three statements Write these three statements So to do the value to check if the query is equal to one We just do queries at i at zero is equal to one And then else if we check if queries at i at zero is equal to two Because the first value at each of these arrays is either one or two Okay, like at each of these queries is either one or two for both of these situations and circumstances So now that we have this we need to do the individual Uh individual thing So in the case of when the query is equal to one We need to append the integer y to array at index So to do that all I did was here is uh array at index We're going to push back the value of y We're just going to push back the value of y into array at index The reason this works is because array at index is a two-dimensional array See right here and you can do a push back at the index Now, um, otherwise if it's not If if it's not the query of one we're going to do the values the the stuff for query or two So otherwise we're going to set last answer is going to equal to this whole thing Which is going to be r at index at y mod by r at index dot size So that's what I do here. I set last is equal to r at index at y mod r at index dot size And this is just going to set this new last variable to equal to this And then it said to push it back to or your answers array So store this new value last answer to your answers array. So I just do a push back So I do answers dot push back last we're just going to store this low value last At the end of our forlet. We're just going to return our answer and that'll be done So yeah, and I pass all the test cases. Hope you guys enjoyed this video rate com subscribe This is a tutorial on how to do dynamic array on hacker rank And yeah, hope you guys enjoyed this video rate com subscribe to check guys later. Peace