With Donald Trump's positive COVID-19 diagnosis, you know, my thinking was that, okay, well, at least at a minimum, this is going to pressure him to take COVID-19 more seriously, because if you have it, if you experience it firsthand, then you are going to know this is no joke. However, that's not the case either, because after announcing via Twitter that he would be leaving Walter Reed Medical Center today, he also took some time to downplay the virus, saying via Twitter, I will be leaving the great Walter Reed Medical Center today at 6.30 p.m. feeling really good. Don't be afraid of COVID. Don't let it dominate your life. We have developed under the Trump administration some really great drugs and knowledge. I feel better than I did 20 years ago. Yeah. So I mean, there you have it. If you contract COVID-19, just do what the president did. You know, have a helicopter escort you to a hospital for precautionary purposes just to make sure that you receive the best medical care in the country with some experimental drugs that are probably costly.  You could just do what he did. He's downplaying it as he leaves the hospital after contracting COVID-19. So there is going to literally be nothing that will get him to take it seriously. Now, a lot of people thought, well, this is karma after saying that this affects virtually nobody. You know, now he's going to realize how serious it is. No, that's not the case. Because even if he felt like death, he's not going to tell us the way he truly feels. And over the weekend, we have been getting mixed messages. How serious is the president's COVID-19 diagnosis? Is he actually doing okay? Is he requiring one or two, you know, measures of supplemental oxygen? We just don't know. We're kind of left out in the dark. And a lot of people may be wondering, Mike, why didn't you talk about this over the weekend? And I'll be honest, I was just exhausted and admittedly a little bit lazy. Like dealing with Donald Trump's shenanigans, it really weighs you down. And I mean, I would have talked about it if he died, of course. But like this is something that really it's shocking to hear that the president got COVID-19. But at the same time, it's not really that shocking because he hasn't been taking it seriously. And it seems like most Americans agree that this is kind of his own fault. So as Justine Coleman of The Hill reports, an ABC news Ipsos poll found that 72% of adults said the president did not take the appropriate precautions when it came to his personal health. The same percentage of people also said Trump did not take the risk of contracting the virus seriously enough. Among Republicans, 43% said Trump didn't take the appropriate precautions nor the risk of catching the virus seriously enough. For Democrats, 94% said he didn't take needed precautions. And 95% said he wasn't serious enough about possibly contracting COVID-19. And I agree with most Americans. And I feel like the events over the weekend kind of confirmed that we are living in a simulation, at least to me, because it feels like we're watching a parody of 2020 while experiencing it for the first time. Because this is just, this is bizarre, like the set of circumstances, you know, leading to us learning about his COVID-19 diagnosis was extremely strange. And when you have an administration that won't be straight up with the American people, it makes matters even worse. And I can't not mention how insufferable liberals have been throughout this process as well, because Joe Biden decided to suspend all negative advertisements about Donald Trump for whatever reason, because Trump wouldn't do the same thing. On top of that, you have some liberals saying, well, you know, I don't wish the president ill will. In fact, Mr. President, we're praying for you. We're wishing you a speedy recovery because we want you to see yourself lose in a landslide. And we can't wait for you to go to jail. Oh, please, as if that's ever going to happen. And perhaps even more insufferable than smug liberals like Rachel Maddow, who wished him well and said she was praying for him, or the Republicans who are now clutching their pearls talking about how negative the left has gone, you know, to Donald Trump, when, as these tweets point out, Ben Shapiro in no way has ever held back wishing others harm such as Trayvon Martin and whatnot. So, you know, this is just, this whole situation is exhausting.  Like I am forced to keep up with this stuff because this is my job. But this is exhausting. Now, I have to share something with you. So, in case you aren't necessarily sure what's happening to the president, this animation that Dr. Oz brought to Fox News definitely will clear things up. I think I sent you an animation if you're able to show it might be beneficial here, but the virus itself isn't the main culprit oftentimes. The virus, here you are seeing little virus particles, those little red things are spike proteins. And they're looking for areas there. They're going into the president's nose. We don't know what had happened, but a couple days before he tested positive down through his throat, into his lungs, those little virus particles set up shop and begin to cause irritation. They attached to the little green receptors that like a key opening a door, they step right in there, they take over the cell and they hijack the cell to release lots of virus, which is exploding through here. That was really helpful because if he didn't confirm that the virus was in fact going into the president's nose, then we wouldn't have known for sure.  We definitely needed that commentary from Dr. Oz. Thank you so much, Dr. Now we're going to kind of recap what took place starting with the genesis of this entire event, but all of it has culminated in a dispute with the Trump family where somehow Trump Jr. is the one who has the most common sense in Trump's inner circle. Yeah, so it started out when we learned on Friday evening that Hope Hicks, senior advisor to the president, tested positive for COVID-19 after going on Air Force One with the president and other individuals within Trump's circle. Now she reportedly self-quarantined on Air Force One, but still that wasn't enough because we learned that the president after taking a test, tested positive for COVID-19. We learned about this at approximately 1 a.m. Eastern time on Friday, but turns out the president knew that he was positive the day after the debate and knowing he tested positive for COVID-19, he went to an in-person meeting with donors, exposed all of them more than a dozen people, and he pretended as if everything was peachy keen and there was nothing to worry about. Now on Saturday, he was escorted to Walter Reed Medical Center via helicopter after we received reports that his condition had worsened. He was being treated with remdesivir and reportedly received supplemental oxygen. The White House then tweeted out quote-unquote proof of how well he's doing because he was actually still working in the hospital. As you can see here, he was hard at work signing his name on a blank sheet of paper. Then things got even more weird when he was escorted around with his motorcade, all while positive for COVID-19, might you, because he wanted to wave at all of his sick of fans that gathered outside of the hospital and that of course was dangerous because as one physician at Walter Reed Medical Center put it, every single person in the vehicle during that completely unnecessary presidential drive by just now has to be quarantined for 14 days. They might get sick. They may die for political theater commanded by Trump to put their lives at risk for theater. This is insanity and members of the Secret Service whose lives were put at risk because of Donald Trump's antics also spoke out. But I've already gone ahead of myself because it seems as if we learned that the nomination announcement of Amy Coney Barrett, that event, that seemingly was the super spreader event that led to 12 people at the time I record this video testing positive for COVID-19. That includes of course the president himself, his wife, Hope Hicks, Kellyanne Conway, RNC Chairwoman, Ronna McDaniel, Senator Mike Lee, Tom Tillis. We have Chris Christie, who is in Donald Trump's inner circle, who also attended a debate panel on Tuesday evening without a mask on ABC. We have Kayla McEnany, Nick Luna, all of these people have COVID-19. And after we were learning about more and more people within Trump's inner circle catching COVID-19, they still decided we're not going to institute a mandatory mask requirement at the White House. It's personal. What? You're leaving that up to personal choice when we don't know how many people were exposed? Now thankfully, they reversed that because it is affecting them personally. So, you know, they quickly learned, okay, a lot of people have it or have been exposed to it. We have to do something to stop the spread and contain it. So we're all wearing masks. So that's actually surprising to me because I didn't think they'd actually budge. But another weird aspect about this story is that Trump's behavior as it becomes even more bizarre than it usually is, is actually worrying people in his own family, not just like in his inner circle, but members of the Trump family are getting worried because what he's doing is bizarre. So on Monday, he went on a really long and bizarre tweet storm in all caps, tweeting about why you have to vote for him. And it's gotten to the point where even Trump Jr. wants to stage an intervention, literally. His family is fed up with the tweets and Trump Jr. wants to step in and try to get everyone close to Trump to rein in his behavior. In an article for Vanity Fair, Gabriel Sherman explains, Donald Trump's erratic and reckless behavior in the last 24 hours has opened a rift in the Trump family over how to rein in the out of control president according to two Republicans briefed on the family conversations. Sources said Trump Jr. is deeply upset by his father's decision to drive around Walter Reed National Military Medical Center last night with members of the Secret Service while he was infected with COVID-19. Don Jr. thinks Trump is acting crazy, one of the sources told me. The stunt outraged medical experts including an attending physician at Walter Reed. According to sources, Don Jr. has told friends that he tried lobbying Ivanka Trump, Eric Trump and Jared Kushner to convince the president that he needs to stop acting unstable. Don Jr. has said he wants to stage an intervention, but Jared and Ivanka keep telling Trump how great he's doing, a source said. Don Jr. is said to be reluctant to confront his father alone. Don said, I'm not going to be the only one to tell him he's acting crazy, the source added. One area where the family seems united is over the president's manic tweeting early Monday morning. After Trump sent out more than a dozen all caps tweets, the Trump children told people they want Trump to stop. They're all worried, they've tried to get him to stop tweeting, a source close to the family told me. So when members of Trump's own family are concerned that his behavior is more strange than it usually is, you know something is going on. Now as to the severity of Trump's COVID-19 diagnosis, it's really difficult to say because we're not getting a straight answer from people. Mark Meadows basically admitted to lying and downplaying the severity of, you know, his sickness. We just don't know. He could still be very sick as he's being released.  He apparently said he met with veterans while he was at the hospital. Was he wearing a mask? Should they be quarantined now? We just don't know. There's so many questions that we don't have answers to. And all we can do is really just step back and think about how weird this moment is and try to appreciate this because this really is something that we will look back on. And even though we now know how weird it is, but like when we look back on this, this is going to be one of the strangest moments in American history, where the president catches a very contagious deadly virus after months of downplaying it and not taking it seriously and leaving the hospital. He encourages people to not let it control their lives and downplays it more. And he may still be very sick. He may possibly be meeting with more people, exposing them.  This is just so weird.  So that's all that I've got. I'll try to keep you all updated, but you know, this story is changing rapidly. And I'm sure it's gonna get weirder. I don't know if I'm ready for that, but we better be ready because it's gonna get even more weird. And there's less than a month until the next election. All of this happening before a major election. Strange times. I'm unsubscribing.


One of the applications of number theory is something called the Chinese remainder problem, and this emerges as follows. The reason it's called the Chinese remainder problem is that it originated sometime probably around the 4th century with a mathematician by the name of Sun Xie, and he posed and solved the following problem. So I have some number of objects, and if I count these objects by 3s, I have 2 left over. If I count them by 5s, I have 3 left over, and if I count them by 7, I have 2 things left over. And so the question is, how many objects do I have? And this is actually, in modern terms, equivalent to trying to solve a system of simultaneous congruences. So what is that? Well, if I count them by 3 and I have 2 left over, that says however many objects I have is going to be congruent to 2 mod 3. Likewise, if I count by 5 and I have 3 left over, that tells me that however many objects I have is congruent to 3 mod 5. And then finally, if I count by 7s and I have 2 left over, that tells me that this number is going to be congruent to 2 mod 7. So now I have this system of simultaneous congruences, and I can solve this as follows. This appears to be Sun Xie's approach, although he gives the solution without too much commentary, but the reasoning seems to be the following. What I want to do is I'm going to find a number that's congruent to 1 mod 3, but also congruent to 0 mod 5 and 0 mod 7. And what that tells me, in order for it to be congruent to 0 mod 5, the number has to be a multiple of 5, and in order for the number to be congruent to 0 mod 7, it has to be a multiple of 7, which means that we should take a look at multiples of 5 times 7, multiples of 35. And so what I'm going to do is I'm just going to go through the list of multiples of 35 and see what they're congruent to mod 3. So 35 is congruent to 2 mod 3, 70, on the other hand, is congruent to 1 mod 3. I'm going to do the same thing for the other two congruences. So I want to look for something that's congruent to 1 mod 5, but also 0 mod 3, 0 mod 7. And what does that mean? Well, that tells me that I have to look among the multiples of 3 times 7, 21. So anything that's a multiple of 21 is going to be congruent to 0 mod 3 and also to 0 mod 7. So again, I'll just go through my list of multiples of 21, and in this case, the first one has it. So 21 is congruent to 1 mod 5. And then finally, I'll take a look at that last congruent. So I want to find a number that's congruent to 1 mod 7, but congruent to 0 mod 3 and 0 mod 5. So that tells me I want to look for multiples of 3 times 5, multiples of 15. And that's going to give me 15 turns out to be congruent to 1 mod 7. Now, here's why we did that. The thing to notice here, because 70 is congruent to 0 mod 5 and 0 mod 7, then any multiple of 70 will still be congruent to 0 mod 5, 0 mod 7. So I can take a look at multiples of 70 and see what happens to the congruences here. Now, I want this to be congruent to 2 mod 3, so I can look at 140 as congruent to 2 mod 3. Likewise, any multiple of 21 is still going to be congruent to 0 mod 3, 0 mod 7. And since I want my number to be congruent to 3 mod 5, I'm going to multiply by 3. And likewise for 7, I want something congruent to 2 mod 7, so I end up with this, 30. And here's where all of this got us. Because 63 is congruent to 0 mod 3, if I add 63 to this, I still have something that's congruent to 2 mod 3. Likewise, because 30 is congruent to 0 mod 3, if I add 30 to this, I still have something that's congruent to 2 mod 3. So if I add 63 and 30, it's still congruent to 2 mod 3. And I can do that for this row mod 5. I can add 140, which is the same as adding 0. I can add 30, again, same as adding 0. So I'll add that to the second congruence. And then likewise for the third congruence, I can add 140 and I can add 63. And that won't change the congruence of 7. So there I now have three numbers that satisfy my three congruences separately. This number is congruent to 2 mod 3. This number is congruent to 3 mod 5. This number is congruent to 2 mod 7. And those three numbers separately satisfy the congruence.  All three of those numbers are actually the same number. In fact, they're all 233. And so there I have it. I have a number that's congruent to 2 mod 3 and also congruent to 3 mod 5 and also congruent to 2 mod 7. Now I can go one step farther here. Note that the product 3 times 5 times 7, 105, if I add or subtract 105, because this is congruent to 0 mod 3, 0 mod 5, 0 mod 7, adding or subtracting 105 won't change any of the congruences. So given one solution, I can find other solutions. And the solution that I'll be interested in, I'll subtract 105 repeatedly and find the smallest number that satisfies all three congruences simultaneously. Now let's take a look at another solution. So I'm going to look for the smallest positive solution to a number congruent to 3 mod 11, 5 mod 20, and 2 mod 9. So again, I'm going to look for numbers that are congruent to 1 mod 11 and 0 mod 20, 0 mod 9. So I'm going to look among multiples of 20 times 9, multiples of 180. And 180 is congruent to 4 mod 11, 360, 540, there we go. Likewise, I'm going to look for a number that's congruent to 1 mod 20, but 0 mod 11, 0 mod 9. So I'm going to look for multiples of 9 times 11, 99, and 99 isn't, but 1,881 is. We can find that by using the Euclidean algorithm or the multiplicative inverse. And finally, I want something that's congruent to 1 mod 9, but not 3 or 5 mod 11 and 20. And so that's going to be looking among the multiples of 220 and again 220, 4 mod 9, 1540, 1 mod 9. And again, I want a number that's congruent to 3 mod 11. So I'll multiply by 3. I want a number that's congruent to 5 mod 20. So I'll multiply by 5, 2 mod 9, I'll multiply by 2. And there I have my three numbers. And again, these two are both congruent to 0 mod 11. So if I add these two to the first number, I still have a number congruent to 3 mod 11. And likewise for the others. And again, all three numbers are actually the same number. And I can find the smallest positive solution by subtracting 11 times 20 times 9, 1,980 repeatedly.


Hi everyone, Shawn with jaspyscasebreaks.com here doing 2001 upper deck SP authentic golf one box break random pack number nine where we're going to open up a sealed box of 2001 SP authentic golf. We're going to number every single one of the packs then we're going to come over here and with our list of customers we're going to randomize the customer names and the pack numbers from one through 24 then we're going to open up those packs so that everybody can see what they got in those packs and that's obviously going to you so way this works like it says 24 spots will first number all packs one through 24 with them randomized list of numbers and names whatever number you get is the pack will open for you only true numbered and autograph rookie cards of Tiger Woods Sergio Garcia and dozens of other superstars average of two autographs and one tournament worn shirt card per box looks like one of these rookie autos numbered to is that the same one was this numbered out of a hundred PSA 10 sold for a hundred forty thousand dollars or is currently bidding at a hundred forty thousand dollars so I guess they're pretty they're worth the pretty good money so before we do any randomizers we're going to pop over here and pop up in this box so you can see here's our numbers one through 24 on the labels and our sealed 2001 SP authentic box so let's pop this open there's our empty box oh it's gonna be special Greg it's gonna be special 16 so 9 through 16 there back 23 Jordan and 24 So packs are all numbered one through 24. Now, I'm gonna keep the NIC cam on, so that you guys can see. No funny business with the packs right there on top. Pop over here to the randomizers. Once again, names for customers, Sean Matic back down to Gretchen Ternin, Last Spot Mojo, and the numbers from one through 24. Dice World Randomize, both lists, and pair them up five into three, eight times. Eighth and final time from Dimitri, down to Danny Giantsopoulos. Eight times on the randomizer, eight times on the dice, and now eight times on the pack numbers. Eighth and final time from number three, down to number 19, eight times, and eight on the dice again. There you go. So Dimitri with pack three, Andy's in with pack 20, Dimitri with 13, Eric with four, Gale Abloh with 14 and two, Danny with 23, Gale with 15, Gretchen with 18, Sean with 21, Danny with 11, Dimitri with nine, Gretchen with seven, Dimitri with eight, Brian with 24, Danny with 10, Gale with six, Christopher with 12, Eric with one, Danny with 16, Dimitri with five, Gale with 17, Gretchen with 22, and Danny with 19. Okay, so that's the order. But we're gonna do it alphabetically. So, print this out, we'll get going. So Andy has pack 20. So we're gonna go alphabetically. My apologies to Sean Maddock. So there's our final list here. So Andy with pack 20. Good luck, sir. That means selling everything. Ian Woodsman, Aaron Palmer, or no, sorry, Arnold Palmer, and Mark Calcovecchia, I remember him. There you go, no tiger in there. Right, Andy's like, first pack mojo. Thanks for giving it a shot, Andy. Nice Arnold Palmer there for you. I remember Mark Calcovecchia. Brian Frank with pack 24. You got Tom Kite, Fred Couples, Bernard Langer, and Tiger Woods, 2001 Masters Tournament Champions. Focus on a champion, Tiger Woods. I love Arnold Palmer, those are my favorite. There you go, not bad. Not the rookie, but is it great? Nice, I like that. There you go, Brian, congrats, man. Greg says that is still worth a lot. My Arnold Palmer ratio is like 80% ice tea, 20% Arnold Palmer, and like, or 70, 30 at most. Christopher with pack 12. Tom Weisskopf, Chichi Rodriguez. He was in Happy Gilmore. Chris Perry and Lee Janssen on a roll. There you go, Christopher. Oh, John Daly's are much better, much better. Especially with Sweet Tea Vodka, Jimmy. Ooh, that's a nice brunch drink right there. Danny has 10, 11, 16, 19, and 23. Yeah, it's with Sweet Tea Vodka. 16, 19, and 23. It was Lee Trevino. I thought that was Chichi. So pack 10 to start. Lauren Roberts, Raymond Floyd. Oh, we got an auto. Ben Hogan and Michael Kirk is our first auto. What's up, Sam, how are you? Pack 11 for Danny. Corey Pavin, John Daly, there we go. There's our guy. Craig Stadler and second auto. Frank Licklitter, the second. Nice auto there, Frank. Always good, Ben. So there are an average of two autos per box. Maybe we have an extra average, over average, above average box. Let's see. Pack 16. Hale Irwin, Jack Nicholas. Got a Payne Stewart, sweet pants tucked into the socks. Look. And Tom Kite. Tin Cup is a great golf movie. What was the one with Will Smith? Or not Will Smith, what's the one with Matt Damon? That's a really good one. Pack 19 for Danny. Bag of Ants. Is that the one with Matt Damon? That's a great one. Marco Mira, Gary Player. Raymond Floyd and Byron Nelson. He has a course named after him. So it's both of them, okay. Caddyshack is a timeless movie that you can watch anytime. Brad Faxen, Colin Montgomery, Tom Wieskopf. And Tiger Woods again. This is the same, Masters Tournament Champions. Rookie Insert, Tiger. There you go, Danny. Congrats. I think that's the same one, right? That we hit? Are they different poses? Okay, Dimitri is up next. Packs 3589 and 13. I think so. If he's able to play, he will. Right? Like, there's not a chance that they're gonna be like, yeah, no. Pack nine and pack 13. Pack three to start. Good luck, Dimitri. Dudley Hart, Ben Hogan, Ken Venturi, and Shopmakers Fred Couples. Pack number five, Jack Nicklaus, Tom Lehman, Craig Stadler, and Arnold Palmer. Pack eight, Ben Hogan, Nick Faldo, Ian Baker Finch, and David Duvall. Anybody else excited for the re-release of SP Golf? Thank you, upper deck. Big ol' John Daly. He's got a full-on mullet there. That is so nice. It's literally like bangs in the front, spiky hair in the back, and then a full-on mullet. Yeah, SP's coming back. They've had four releases this year for golf. Duffy Waldorf, Lee Janssen, and Gary Player. Club Joe, and pack 13. Last pack for you, Dimitri. Good luck. Well, it was exactly like his career, Craig. The beginning of it was talked about his upcoming and how good he was and everything, and it was positive, and then the downfall. Ken Venturi, Peter Jacobson, Tom Weisskopf, and Corey Pavin. There you go, Dimitri. Thank you very much, man. But it's literally like it mimicked his life. It was like the huge buildup, untouchable, and then, my God, what a fall. Eric with packs one and four. Ken Venturi, Lee Janssen, and Shotmakers. Tom Lehmann, and pack four. Good luck, Eric. Fred Couples, Andy Bean, Mark Calcovecchia, Shotmakers, David Duvall. How can you not like the guy? Gale, two, six, 14, 15, 17. Two, six, 14, 15, and 17. Mark O'Meara, Billy Ray Brown. Tour Swatches, Stuart Sink, 373 out of 500, and Tom Weisskopf. After this, Joe, we've got a Bowman, two box mixer, and then orders, which I've got about 15 of them to go through to figure out what's coming up after that. There's never enough times watching Tin Cup. Just so good. Lee Janssen, Gene Sarazen, Gary McCord, and Ratif Goosen. Remember that guy? Loved him. It was part of his shirt, Billy. I believe it's part of the polo. Match worn. Pack 14 for you, Gale. Payne Stewart, with the pants tucked into the socks. Grant Job, Mark Calcovecchia, and Curtis Strange. Can you imagine a tiger one in this with a red polo? So popular, Sam. So popular. Pack 15 for you, Gale. Two inserts. Tom Kite, Gene Sarazen, Jack Nicholas, and a third one. 2001 Masters, again, Tiger Woods with his Sunday red polo. Sunday reds, classic tiger look. There you go, Gale. And one more pack. Pack 17. Gary Player, Bobby Jones, Jeff Maggart, and Payne Stewart. Or no, sorry, Stewart Appleby, 764 out of 29.99. Creator of Appleby's, strangely enough. You know? I think we all knew that, though, right? Stewart Appleby, creator of Appleby's, the number one restaurant in America. Gretchen, packs 7, 18, and 22. 18 and 22. Gene Sarazen, Tom Wieskopf, Ian Woosman, and Brad Faxon, 18. Ian Woosden, Bobby Jones, Ben Hogan, and Shigeki Maruyama.  Larry Nielsen, Arnold Palmer, Gene Sarazen, and Nick Price. Finish this it off for you. Thank you very much, Gretchen. 100% the second one, Rex. 100%, not even close.  Fred Couples, Marco Mira, Frank Dully, and Tom Kite to 500 to finish it off. There you go, Sean. There you go, guys, that was the box. Couple nice tigers in there, none of the regular rookie or autos, obviously, but we did get three 2001 Masters inserts in there. All very nice looking ones. And a little flash from the past. So that was 2001 Upperedeck SP Authentic Golf, one box break, random pack number nine from jaspyscasebreaks.com. I'm Sean, jaspyscasebreaks.com. Thanks for hanging out. We'll see you next time.


Hey freedom, what's up? Have you been using your analytics page? Well, if you haven't today's your lucky day because on this episode we will be talking about that But before we do you know what to do like subscribe and gong that bell today Let's talk about your analytics page and know where how and who are watching your videos on your channel This video will be the first analytics video that we will be releasing on this show So without further ado guys freedom quick tips Before we dive into the analytics page Let's ask these questions to ourselves first so that we can have a better grasp on why we are on that page and also find out The things that we can improve on the analytics page number one question should be how is your channel doing? Is it increasing in views subscribers or likes? You can check the watch time view and subscribers to see if your channel is improving or not You will be able to see if your audiences are watching your videos if your channel is growing in subscribers and the numbers of views You get per video number two is who are watching my videos and are they really interested in it? You can find out more in your video analytics the parts where your audience is p and what parts they are most interested in number three Is how's it engaged with with my audiences? Are they sharing my videos or not? You will see how much your viewers share your videos as we have said in our previous episodes Sharing is caring and it's what makes your videos viral number four is how much money do I generate in my videos? In your analytics page you will see the amount of money that you will be earning on your videos The next part of the video is we're going to be answering those questions that I have mentioned a while ago To answer all of those questions you will need to look at your watch time your views and subscribers But you may be wondering what impact does it do to your channel? There's a lot of things that you can do if you know your audiences and in order for you to do that You will have to look at your demographics page. So what is a demographics report? Well to put it simply the demographics report tells you what kind of people are watching your videos How old are they what part of the world they are watching from what gender? They are and a lot more if you get a grasp of this you can tailor fit your content that will greatly help your growth So now do you see why the demographics page is so important? Well, if you haven't will give you a few more reasons on why you need to visit your demographics report number one Reason is you can tailor fit your content to your audiences very clearly number two You will be able to narrow down your topics number three is you will be able to know your niche immediately The next report will be the most interesting report that you will find in your analytics page Well, I really find it interesting when I look at my watch time report on my videos So the watch time report shows if your videos are being watched by your audiences You will know with this report how much time they are spending to watch your content So what's the impact if you know your watch time report? You will be able to discern what parts your audience is like about your content without being said you will be able to know Exactly what your audience is like you will be able to understand clearly what your niche really wants and what they want to see In your videos, you will be able to get a firm grasp about your niche and start creating content similar to those what they like The next report is the traffic report in the playback location in this report You will be able to pinpoint the location where your audiences is watching your videos made beyond their mobile phones their PCs or on a website So you might be wondering now Why do I need to look at my traffic sources or my playback location? Well, if you know where your audiences are watching your videos You will have a valuable piece of information if you know where your audiences are watching Because it will give you enough information to tailor fit the video quality of your content For example, if your content is being consumed by mobile users You know that people that use mobiles are on the go and maybe they have a busy lifestyle Maybe with that piece of information you can make your videos more brief so that people on mobile will appreciate you more There's a lot of ways you can do with the data on these reports And it will give you a more firm grasp on how you can approach your niche or the audiences that you are catering to Basically guys if you have these reports you are proficient in Number one niche targeting number two audience retention number three audience behavior number four knowing your audience's interest and number five The content quality of your videos So guys, what are you waiting for go to your analytics page so that you can know all of these Reports and that's episode gamers Let us know down below if you have missed parts of the analytics page Also, if you have a friend who doesn't know anything about the analytics page You can share this video to them so that you can help them out guys Please if you have time click that like button subscribe and gong that bell My name is Noel guys bringing you tips on how to grow your channel always remember freedom quick tips I'm gonna be seeing you guys on the next episode. Goodbye


This talk will be about representations of abelian groups, mostly finite abelian groups acting on complex vector spaces. So the problem we want to look at is, we've got a group G, let's say finite abelian, and we want to know how can it act on a vector space V, which is finite dimensional over the complex numbers. So let's just look at an example. Let's take G to have order three, so it's just three elements, one of G and G squared with G cubed equals one. And then there are three obvious irreducible representations where G can act on the one dimensional complex vector space C, because G can act as one or omega or omega squared. Here, omega is e to the two pi i over three, is a cube root of one. And the question is, can every complex vector space be split up into a sum of these sorts of irreducible? And the answer is yes, because we can take any vector V in a complex vector space and write it like this. We have V plus VG plus, sorry, GV plus G squared V over three, and you can add to that V plus omega GV plus omega squared G squared V over three, and you can add to that V plus omega squared GV plus omega G squared V over three. And you notice that this one G acts as one, this one G acts as omega minus one, unless I've got a sign error, and this one G acts as omega to the minus two. So the vector space splits up as a sum of subspaces where G acts as one or omega to the minus one or omega to the minus two. And of course, each of these subspaces can then be split up into one dimensional pieces. So we've seen that every complex representation of G can be split up into a sum of these three special one dimensional ones. So we can now write down the character table of G. So you remember the character table of G has a column for each entry of G and a row for each character. So if we call these characters chi one, chi two and chi three, then we get one, one, one, one, omega, omega squared and one, omega squared, omega. And it's pretty obvious that something very similar is going to happen if you replace three by some other integer. The character table looked like one G, G squared up to G to the minus one. And the first column will be just ones. The next one will be one zeta and zeta squared up to zeta to the minus one. Next will be one zeta squared, zeta to the four, zeta to the six and so on. The last one will go on until you get zeta to the n minus one and so on, where zeta equals e to the two pi i over n is a primitive n through two of one. You notice by the way that the rows and columns of these tables are all orthogonal in the following sense. Here by orthogonality, you need to use a Hermitian inner product. So we define the inner product of two vectors v and w, the sum of the i times wi complex conjugated, where the i are over the components. And then you can see that the rows and columns of these matrices are orthogonal. This is a very general phenomenon that will also hold for non-Abelian groups. So what makes this work? Well, in order for this to work, we need two things. First of all, we need an nth root of unity. And secondly, we need to be able to divide by n. So first for one over the order of g is in the complex numbers. And secondly, the number zeta equals e to the two pi i over n is in the complex numbers. And of course, if we replace the complex number by any field with these two properties, then the theory is very similar. A little bit later, I'll discuss what happens if one of these two properties fails. So that's more or less done cyclic groups. What about general Abelian groups? Well, that's not very difficult because a general Abelian group g can be written as a product of cyclic groups. So general finite Abelian group. And if we've got a representation of g1 and g2 and so on, so suppose g1 is generated by little g1 and g2 by little g2, then a typical representation of g will be a one-dimensional complex vector space in which g1 is omega one to the n1, g2 is omega two to the n2 and so on, where omega one is a root of unity of order, the order of g1 and so on. You can then see that you can split any vector space into eigenspaces of g1 and these will all be acted on by g2, so you can then split those into eigenspaces of g2 and so on. So any representation of this finite Abelian group of the complex numbers splits as a sum of these one-dimensional representations. You notice, by the way, that the one-dimensional representations of g form a group. So if we've got two representations, chi1, which maps g to the non-zero complex numbers and chi2, which maps g to the non-zero complex numbers, we can simply form the product chi1 and chi2 defined by chi1, chi2 of g is chi1 of g times chi2 of g. And it's easy to check. This makes the set of one-dimensional representations of g into a group. This is called the dual group. And you can easily check that the dual group of a cyclic group is isomorphic to a cyclic group of the same order and the, let's call this dual group g star. And if you've got g1 times g2, then the dual of that is just g1 star times g2 star. So this shows you that if you've got a finite group, then g is actually isomorphic to its dual. But this isomorphism is actually not a very good one to use because there isn't actually a natural isomorphism between these. It depends on the choice of a generator of the group. So this isomorphism is a bit iffy. It's not canonical or anything. However, there is a canonical isomorphism g to the dual of its dual. So this is a bit like vector spaces. You know finite-dimensional vector space is isomorphic to its dual, but not in a canonical way because you have to choose a base. Whereas it's canonically isomorphic to the dual of its dual. And you can see this because if you've got a character acting on an element of g, this is a non-zero complex number. And if you fix the character, this is a homomorphism from g to the complex numbers. But if you fix g, it's a homomorphism from characters to the complex numbers. You can think of an element g as being an element of the dual of its dual group. This is a special case of Pontriagin duality for finite groups. So Pontriagin duality for finite groups is actually kind of trivial as we've seen. Pontriagin duality really becomes rather more interesting when g is allowed to be an infinite locally compact group. Anyway, so let's see an example of a representation of a non-cyclic group. Here we're going to take the simplest possible example. Let's just take the group z modulo 2z times z modulo 2z and look at its character table. Well, its character table looks like this. It's got four elements, nought nought, nought one, one nought and one one. And it's got four characters. And the first character is just one one one one. And the second character looks like one one, minus one, minus one. And third like one minus one one minus one. And the fourth like one minus one minus one one. And now you can see the characters of these two groups inside this as follows. So if I call this chi one, chi two, chi three, then this is chi two times chi three. Then you can see the character table of the first z modulo two z in sitting in here like this. So you remember the character table of z modulo two z just looks like one, one, one minus one. And the character table of the second one is sort of sitting inside like this. And in some sense, you just sort of multiply these two character tables together to get the character table of the product group. Again, you can see the rows and the columns are orthogonal. You can show this either by just sort of brute force checking since we have explicit formulas for everything, or you can just prove it directly. So what we want to do for an Abelian group is to prove the sum over g of chi i g times chi j g complex conjugate is equal to naught for i not equal to j. And we may as well multiply chi i by chi, by the chi j to the minus one. So this is just equal to sum over g of chi i chi j to the minus one, g times one. So we just want to show that the sum over g of chi of g is equal to naught if chi is not equal to the identity character. And if chi is not equal to the identity character, we can just choose some element h with chi of h not equal to one. And then we can change g to g h. So sum over g of chi of g is just sum over g of chi of g h. That's just a change of variable. And this is the sum over g of chi of g times chi of h because chi is multiplicative. So you see this expression is equal to this expression times chi of h. So sum over g, chi of g is equal to naught as the chi of h is not equal to one. So this shows orthogonality of characters for any Abellion group. In particular, you see the characters, you can think of a character chi as being a function from g to the complex numbers and these form a base for the vector space of functions from g to the complex numbers. That means you can expand any function of g to the complex numbers as in terms of the characters as sum over f i i bar. I guess I better divide by the order of g of chi i g. So this is called the Fourier series of the function f. And the reason for that is that it's more or less the same as the usual Fourier series for a periodic function. So suppose I take g to be the circle group which is the reals quotient out by two pi times z. So okay, this isn't actually a finite group but it's compact which isn't too far from being finite. And it's got characters, the characters are chi i which just take a number x to e to the i n x. I guess if I'm using i for a complex number I better call this J instead of i. Sorry, let's call that n. So for each integer n and z, I've got a character taking x in s one to this periodic function. And these are orthogonal because the integral from zero to one of e to the i m x e to the i n x bar the x is equal to naught for m not equal to n. So that's just like the orthogonality relation we had here except I'm instead of summing I'm integrating over the group. And we have a Fourier series which says that if f is a periodic function on the group G then f is equal to the sum over or n of the integral of f of x e to the minus n i x dx divided by two pi times e to the i n x. And if you look at this expression here it's almost identical to this expression here except we've replaced the summation by an integral we've replaced the order of G by the measure of G and so on. So the Fourier series expansion for finite abelian groups is and the Fourier series for a periodic function of both really special cases of a Fourier series for abelian groups. In particular, we notice I guess I better just remark the dual of the circle group s one is the group of integers z. And more generally, as I mentioned there's a dual for any locally compact abelian group and an analog of this.  So you remember in order for this to work well we needed to use the fact that the complex numbers had lots of roots of unity and also a characteristic zero. So first let's look at what happens if there are no roots of unity or it's not enough roots of unity but let's assume the characteristic is still zero. Well, what happens then is things get slightly more complicated but they're still fairly easy to handle. So for example, for G being of order that's suppose G is cyclic of order three then we find there are now two representations over the reals. One representation is dimension one and if this is generated by G we have the element G acts as one. And secondly, there's an irreducible representation of dimension two where G acts as rotation by pi over three radians. So it's just a rotation by third of a revolution. And in some sense this representation dimension two has got by gluing together the two one dimensional complex representations. So if you take this representation let's call this V and to complexify it it turns out to be the sum of V one and V two wherein V one G is equal to omega and V two G is equal to omega squared. So it's not too difficult to deal with the case when the field doesn't have roots of unity. The only difference is you find some representations some irreducible representations might have dimension greater than one. However, the second case is if the characteristic is greater than naught can get a lot more complicated. Suppose the characteristic P is greater than naught if P doesn't divide the order of the group then you're fine you can just continue as before. So if P does not divide the order of the group G this is easy. If P divides the order of the group G things get complicated. First of all, there are lots of indeed composable representations but need not be irreducible. So you can't divide everything up into a sum of irreducible representations. And suppose for example that G is a generation of G and G is ordered P to the N. Well, we notice that G minus one to the P to the N is equal to zero because we're working in characteristic P the binomial coefficients P to the N choose I for all equal to zero for not less than I less than P to the N. So this is nil potent. Well, if you've got a nil potent matrix over a field you know it has a sort of block form. It can be written as a lot of blocks each of which has zeros down the diagonal ones just above it and so on. So G minus one might look something like this. And you see each of these blocks corresponds to an indeed composable representation but it's reducible because it's got a one dimensional invariant subspace. Well, that's for cyclic groups. Cyclic groups aren't too bad because we can sort of classify nil potent endomorphisms and pushing them in Jordan normal form. However, if G is non-cyclic this gets very complicated. This runs into the problem of trying to classify two commuting nil potent matrices over a field. And this turns out to be a complete nightmare if the field has reasonably large dimension. As far as I know, nobody really knows a good way to classify all indeed composable representations of even an abelian group in characteristic greater than zero. Finally, I'll just say what happens? Suppose G is infinite and acts on V with the dimension V being infinite. Well, here you've got to be rather careful about your definitions otherwise you run into some rather horrible examples. So you might say, if V is infinite dimensional obviously we should give it a topology and the action should be continuous. So let's take V to be say a Banach space and let's take G to be the integers. And the question is can we find an irreducible representation of G which is the integers on a Banach space of dimension greater than one. And this representation should be continuous. So it should be by bounded operators. And this is the notorious invariant subspace problem. So it asks, can you fight if you've got an operator on a Banach, bounded operator on a Banach space in infinite dimension doesn't necessarily have an invariant subspace. And the answer is it doesn't necessarily have an invariant subspace. So the answer to this question is yes. And the first example was found by Enflow and his example was nightmarishly complicated. It took about 10 or 12 years for people to actually verify that his example was correct. It was so difficult. I mean, several people tried to verify it and simply gave up. So incidentally, this is still open for Hilbert spaces. So what this example shows is not that the integers has an irreducible representation in infinite dimensions, but you've got to be rather careful how you define irreducible representations in infinite dimensions because you really, really do not want to count this thing as a representation of Z. So you have to define infinite dimensional representations in ways it excludes these rather pathological examples. Okay, so next talk, I'll be giving some more examples of representations of Abelian groups and showing how to use them to prove Dirichlet's theorem about primes in arithmetic progression.