Hi guys, my name is Liz Jackson and we are with SPM Buzz at the Alchemist. So we are here with the one and only Papa Jones, Habariyako. Mzuri sana, Nikofi tika bisa. Now first of all we have to commend you on the good work that you've done on the album that we've had a chance to listen to. So one of the questions that maybe our fans could be asking, what inspired the title Invisible Currency? The title was inspired by, I mean there's a, I was talking to my people having a conversation and that term came about. And you know I was sitting down thinking of a suitable title for the album and then when they explained to me what the title actually meant I was like perfect. This is what I'm going to take for the title of the album so that's why we came up with the title Invisible Currency. For those who don't know what that means, Invisible Currency is the value that you bring to the table that is not visible to the naked eye. So basically it's just the strength of your brand, the strength of you as a person, yes. So when we listen to the album, ojitunaju rawenim to hardcore, to emotions, but we can feel the rawness in the album. How was that transition for you and what inspired you to be so real in the album? It all comes down to growth, ojitunaju and the growth, I've said this, I've done a bunch of interviews right now, I feel like I'm just repeating myself, but I am 32 years old right now and I cannot be moving the way I was moving when I was 25 years old. So the album I felt like I needed to display myself in that manner so that then people also understand that Kali Graf Jones is a human being. But at the same time we balance and being the ojitunaju that I am, so yeah, that's that, yeah. So one of the tracks that you've released is Hiroshima and it's currently trending at number 14 in Tanzania. So how do you feel for that kind of vibe to be trending in Aswahili Nation? How is that for you? I saw that and I didn't even believe it because one of my guys from Tanzania sent me that screenshot. It's like how? Hiroshima doesn't even have a chorus, it doesn't even have a hook, it's all bars from the beginning all the way to the end and for me to have a song like that trend in a country where people don't even speak English, that speaks volumes, right? So I was excited and I was having a conversation with my manager, we're actually planning on doing a tour in Tanzania, media tour in Tanzania and I mean that is something that is part of the plan that we have because we've seen the reception and the guys love it so it's only right that we give them a proper visit and we let them know what's up. One of the artists that you featured is Ali Kiba. What made him perfect for you? Were you trying to diversify your sound in Aswahili Nation or what made you choose Ali Kiba? Ali Kiba came to my studio and he was working on his album, he had his album, he wanted to feature me on a song so it was like some sort of a trade, he actually featured me on a song and it was like yo I give him a verse, he gives me a verse. So it was and the way I worked with Ali Kiba on this particular song, you will notice that it's very different because it would have been expected that Ali Kiba comes in and does the hook, the chorus and then calligraph does the verses but I was doing the hook, I was actually singing on the hook and I sang on the first verse, Ali Kiba only came to on the second verse so I did it differently and he was feeling the vibe you know because he also thought that I was going to call him to come and do a chorus and that was not the case so in Liyonatu from that aspect he would be the best perfect match for this song and that's why we made it happen. One of the people that you've mentioned is Bien. What was Bien's impact on this album? Bien a kuchini. Bien lakini, he is my friend Yanni of course, he is one of, he's a very good friend of mine, he's a good friend of mine in the studio and even outside the studio so we were having studio sessions with the producer and he happened to pop up, he came up, he came to the studio while we were recording the song with Prince Inda and he came with, he actually came that day, came with the guy Rimbui, Aaron Rimbui, they have a joint EP and I mean he gave the album a sense of direction, there are musical terms that I can't even mention right now because people are not going to be able to understand what I'm talking about but just know that Bien is the one who made the album take that direction you know and he, Yanni, Bien imi noma. Thank you so much for speaking to SPM Buzz, good work on the album and we hope that our viewers are going to interact with it, thank you so much. Hey what up calligraph Jones Papa Jones, you know what it is, we out here just dropped the new album Invisible Currency, go stream it, it's on boom play, we're doing massive numbers, God bless you if you have listened to the album, if you haven't God bless you too, go and listen to the album right now, got a new track out together with Invisuals, Calligraph Jones feature in DAX, it's on YouTube, go and check it out and remember keep it SPM Buzz because this is where it happens, respect peace out.


What's up, everybody? Jason for Jasper's case breaks.com. This sells out the hip parade VIP break And we're giving away six spots with this 2014 limited football one box break round teams number two The way this works guys is first we do the break round my sister names and teams strip of the hits out of the box break Then at the end of the break we'll take all 32 customer names randomize them and the top six get a spot in The VIP break which I'm coming up next Let's start roll the dice and it's a four and a six about ten times. Good luck one two three four five six seven eight nine and ten Eric you just have to adjust your Your quality of your video make sure it's not on auto and you are on the highest level which is 720 60 Um It's most likely your your side my side looks perfect. I have four to six ten times William down to Eric one two three four five six seven eight nine ten Falcons down to the Arizona Cardinals Alright, so there you go stick to what you got guys remember this is 2014. I'll give you guys a quick minute to look at your teams All right guys, here you go Trent Richardson Doug Martin Charles Sims It's a dual relic for the Bucks Tom with that one Angela Williams We got coated latimore 56 out of 99 That is for the Denver Broncos When a Tom and we end off with a Reggie Bush, how about that? 13 out of 15 for the Detroit Lions Mixed over. All right, so there you go Now let's switch scenes and give away the six spots Good luck top six getting us Roll it two and a three five times one two three four And five five times Top six get in Christopher Wesley Eric Nick and Tom and Barry right top six Five times boom. Congratulations guys. You guys are now in the break coming up next is the break itself


Well, it's official. Nancy Pelosi has announced that the House will formally be opening an impeachment inquiry into Donald Trump. Now, I want to talk a little bit about the implications of a potential Donald Trump impeachment because this is a conversation that has been dominating the left sphere of Twitter, and I think it's fascinating and I think it's important. Now, before I get started here, let me just say that there's a lot of moving parts to this story. There's a lot of aspects that I won't be addressing in this particular video. There is the whole issue of Biden and his son. I think that's important. Not going to talk about that here. There is the issue of the scope of this impeachment inquiry. There's inklings that Nancy Pelosi may just limit the scope of this investigation into Ukraine and the phone call between Donald Trump and the Ukrainian Prime Minister. That, I think, would be a disaster and a waste of time. Why not just open up his business interests, you know, all the conflicts of interest? But those are different conversations. What I want to do now is just talk about the potential implications of impeaching Donald Trump. What does this mean politically? So let's start by talking about what impeachment is. Impeachment is a process. As of right now, we are only at the inquiry stage where the House investigates. And if they conclude that he's guilty after conducting an investigation, they then vote on it. Now, a simple majority vote in the House sends it to the Senate, where a public trial is then held. Now, it's obvious Republicans are in control of the Senate, so they will most certainly vote not guilty. So this will ultimately not end in the removal of Donald Trump. That's the most likely outcome. And that's especially true when you consider that it requires a two-thirds majority to convict. So the question is, what's the point? Why even bother? Why would we bother to spend political capital and time on this if it's not going to result in Donald Trump's removal from office? Well, it's because, like I said, impeachment is a process. And it's a certain type of process. It is a legal process. It is not a political process. Now, that doesn't mean that it occurs in a vacuum. Of course, that's not the case. There are political ramifications that are attached to impeachment proceedings. But whether we do or don't open an impeachment inquiry should hinge on one question. Did the president break the law? If the answer is yes, then you should open an impeachment inquiry irrespective of what may or may not be politically expedient. That being said, I do acknowledge that people on the left have a point in saying this is risky. There is a risk. Of course, there's a risk with every action comes a reaction. So whatever we do now could potentially impact our chances of defeating Donald Trump in 2020. Doesn't necessarily mean the effect will be negative only. It could be positive, but it is a risk nonetheless. So I think it's important that we talk about this because there are people who don't want to pursue impeachment because they don't want us to end up in a worse political predicament and I hear you. But ultimately, for me, I don't find these arguments against impeachment persuasive. And I do support impeachment. And the reason why I support impeachment is because I don't necessarily believe that anyone can predict with certainty what is going to happen. The best we can do is look at history and see what happened then. But that still doesn't necessarily mean that that will indicate what is going to happen in the future. But I do think it's important that we have a historical understanding of impeachment and what has happened. The problem is that the sample size for previous impeachments is incredibly small because impeachment proceedings have only been initiated against three presidents, Andrew Johnson, Richard Nixon and Bill Clinton. So it's difficult to definitively say that Donald Trump's impeachment will be similar to these presidents, but I think it still is worth looking at what happened when we tried to remove presidents before. And this is all laid out thoroughly in a political article published in January by David Greenberg, who explains when impeachment did and didn't work. And what I want you to take into consideration is the fact that this historical context is incredibly important because throughout the course of American history, there have been certain paradigm shifts with regard to the way that we view impeachment in nonpartisan terms, meaning that both parties viewed impeachment as something that was not politically expedient, overly risky and politically toxic. And then that changed. Sometimes people felt confident that impeachment was a way to remove their political opponents. We're going to get to all of that. But the first thing we should do is talk about the start of it all. We're going back to the 18 hundreds when Andrew Johnson was president. And I get that that seems odd to go that far back when we're in 2019. But I think the context is incredibly important. So Andrew Johnson was Abraham Lincoln's successor. And like Donald Trump, he was a belligerent, petulant in Brazil. He was a buffoon. He was a racist. And he was actively trying to obstruct and undermine the Republican party's reconstruction efforts. Now, this was essentially an incredibly polarizing time. You can imagine it was after the Civil War, there was an endless political battle between Johnson and the Republican Party. But after they widened their lead in Congress, they tried to rein him in, rein in his craziness, if you will. And they passed a law that prohibited a president from firing a member of the cabinet until the Senate voted to confirm his successor. Now, of course, being the petulant child that he was, Andrew Jackson brazenly violated this law and articles of impeachment were subsequently filed. So what ended up happening in that instance? Well, the House ultimately voted to impeach. However, there were some Republicans in the Senate. They were a little bit iffy about impeachment, namely due to the political risks. And also they felt like his VP was worse. Sound familiar? So they were a little bit reluctant to impeach. Now they had the vote and he ended up surviving impeachment by one vote and he survived. He was acquitted. So at that time, impeachment became something that was largely viewed as politically toxic because Andrew Johnson did not get impeached. Now, this was important because it established a political standard. And this standard has essentially carried on until this very day. What's that standard? That impeachment proceedings are not like votes of no confidence in a parliamentary system, right? You can't just vote to remove a president if you disagree with him politically or disagree with his policy positions. You have to have a very high standard. And that high standard has become incredibly relevant today. Constitutional issues have to be at stake.  That's what the Jackson impeachment proceedings told us. And everyone in the country after that, they essentially viewed impeachment as something that wasn't worth the risk. So people were so afraid to try impeachment again that it wouldn't happen until a hundred years later with Richard Nixon. During the 1960s, Richard Nixon was also facing his own non-hashtag resistance movement. People were looking for any and all reasons to impeach him and rightfully so. Anti-war activists wanted him impeached over Vietnam and Cambodia. He was also brazenly corrupt and he broke the law by wiretapping journalists and members of government, political opponents.  They were just overly alarmist, overly reactionary. And they were seemingly acting as if the sky was falling. Sound familiar? But it wasn't until Watergate that we'd actually start the process of impeachment where Nixon's goons were busted, breaking into the DNC headquarters in an attempt to bug the place. Because remember, he really liked to wiretap and spy on his political opponents. But here's what's remarkable. He was still reelected, reelected. He was busted, breaking into the DNC and trying to literally bug it. And voters said, you know what? You get another term, Nixon. How crazy is that? And at that time, when we were talking about impeachment, can you guess the approval for impeachment even after that scandal? Nineteen percent, I repeat, nineteen percent. So after Richard Nixon did that, he committed a crime and was busted. Americans still didn't want to impeach. Only one in five Americans wanted to impeach. But months later, impeachment proceedings were still initiated, regardless of what public opinion was at that time. And after Americans watched the Watergate hearings play out on national television, well, something began to change. Public support started to shift. Support for Nixon's impeachment started to increase until eventually a solid majority favored impeachment because they were presumably unaware of just how corrupt and lawless Nixon actually was. Now, because Nixon was a belligerent buffoon, much like Donald Trump, he couldn't help himself. So he made matters worse. He tried to obstruct justice in the most brazen way imaginable. He ordered two Justice Department officials to fire the Watergate prosecutor. They did not agree to do that. So they resigned. And this happened, mind you, in October of 1973, and the House Judiciary Committee didn't actually take up impeachment until 1974. And as Greenberg explains, at that point, bipartisan support for Nixon's ouster had grown considerably by the summer when the committee voted on the question several Republicans crossed the aisle to vote A, giving the committee's verdict a moral authority that the campaign against Johnson never possessed. It was congressional Republicans, too, who told Nixon he couldn't survive a Senate trial persuading him to resign. So he ended up resigning in order to save himself the humiliation of being the only president in American history to be convicted and removed from office. But the question is, what was the catalyst? What ultimately were the grounds that they impeached him on? Because remember, there are numerous reasons why people wanted him impeached. Wars, tax dodging, brazen corruption. But the straw that broke the camel's back, ultimately. Well, as Greenberg explains, Congress was in effect building on the Johnson era criteria for presidential impeachment, affirming that constitutional issues had to be at stake. The articles of impeachment that passed centered on Nixon's obstruction of justice, such as using the CIA to try to thwart the Watergate investigation and paying hush money to the Watergate burglar's abuse of power, such as using the IRS and FBI for political vendettas and defiance of congressional subpoenas. So understand what they did here. They built on the precedent that was set, the non-legal precedent that was set, the political precedent that was set after the Johnson impeachment failed miserably. Johnson taught us that we can't just try to remove a president because we disagree with him. So what they tried to do was actually build upon that and impeach him, not for political reasons, but for reasons related to the Constitution obstruction of justice, abuse of power. Now, after the Nixon impeachment, this essentially created a paradigm shift where, remember after Johnson, we were all afraid to talk about impeachment because it was too risky. It was too politically toxic. But after Nixon, we learned that if we do it right and we proceed in a way that is, you know, adhering to a particular set of norms constitutionally speaking, where we're not trying to remove a political opponent because he's passing policies we don't like, but instead for very, very serious crimes such as obstruction and abuse of power, then impeachment can actually work. Maybe it's not so politically toxic. So we went from thinking we should never impeach. It's a waste of time. It's going to backfire to thinking, you know what, maybe impeachment can actually be useful if we use it correctly. But we just have to be careful. So that's a paradigm shift. Impeachment too risky. Impeachment is possible. We just have to use it correctly. Fast forward to the Clinton era when Republicans impeached him essentially because he lied about getting a blow job. Now, of course, that's an oversimplification, but it kind of just goes to show you how confident we were with the notion of impeachment again after Nixon. We were kind of imbued with this new fangled confidence that, you know what, it's not too politically toxic to impeach a president. This is no longer the post Johnson era. This is the post Nixon era. And if we want to impeach a president, we will do that. So Republicans at the time were also accused of being too obstructionist and they tried to bog down Clinton with bogus investigations, witch hunts, if you will, sound familiar. And of course, they were obstructionist. They are still the obstructionist shitheads today that they were back then, except now they're just exponentially worse. But I digress. What was evident is that this was incredibly partisan. They wanted to get him for something. It didn't matter that Clinton was capitulating to Republicans left and right, that he was a new Democrat who was literally implementing their neoliberal policies. They just wanted to get someone who was a political opponent. And the problem with the way they went about this is like the Benghazi hearings and investigations. This was super obvious. It was incredibly transparent. They weren't motivated by a good faith desire to defend the Constitution from a rogue president. They were motivated by a desire to get a political opponent. Now, it wasn't necessarily the affair itself that did it. That wasn't the straw that broke the camel's back. It's the fact that he lied about it under oath and committed perjury that led to his impeachment. But here's what happened. Even after Clinton admitted the affair and apologized, Republicans misread the public mood and forged a head with impeachment proceedings, resulting in a historically rare loss of congressional seats in the November midterm elections for the party not controlling the White House. They held onto enough seats to impeach him on two counts, perjury and obstruction of justice in December. But as they did so, his popularity soared to 73 percent. Surprising to no one, Clinton was acquitted in the Senate with considerable bipartisan support. And let's look at that vote count because you have some Republicans even voting not guilty against Bill Clinton. That, ladies and gentlemen, is what I'd like to call a spectacular failure. I mean, it's easy to see why this backfired tremendously and why people are a little bit hesitant that the same could be true in the case of Donald Trump. But I want you to note how this led to another paradigm shift to where now we're afraid of impeachment again. Maybe it's too politically toxic. Maybe it's not the politically expedient thing to do. So I want you to think about this in a historical context and the way that our views culturally speaking on impeachment have fluctuated in the post Johnson era. We were all terrified of impeachment after Nixon. We got confident, but maybe a little bit too confident, which then led to us seeing the Republicans fail and now thinking, all right, maybe impeachment isn't actually the best course of action after all. So do you understand? This is why people view the issue of impeachment really risky. If Donald Trump were president after Nixon and before Bill Clinton, there wouldn't be as much discussion about impeachment because we all were very confident that it wasn't as politically toxic or wouldn't be as politically toxic as it was during the Jackson era. But now we're in a time where impeachment actually is a little bit more controversial. Now, here's the question that we should be asking ourselves as we open this inquiry into Donald Trump. Will Donald Trump's impeachment end up being like Nixon's or Clinton's? I don't think anybody has a definitive answer to that because, you know, we can't see into the future. But let's examine just some of the details here. There's currently a relatively low level of support for impeaching Donald Trump. It's at 36 percent. Now, is that low? Yes. But it's still not as low as public approval was for impeachment during the Nixon era. So what Nixon taught us is that that can change with time. Now, on top of that, when it comes to whether or not constitutional issues are at stake and if we can actually credibly build on that Johnson era precedent that we set back in the 1800s and the Nixon precedent, well, Trump's own memo demonstrates that he did, in fact, ask the Ukrainian president for dirt on a political opponent, which is illegal. But what I think is more pressing is his obstruction of justice. There were more than 10 instances of obstruction during the Mueller report put aside the Russian investigation. You don't have to agree with that investigation, right? I never thought that collusion would be a possibility. But during that investigation, there were numerous instances of him obstructing justice, even if I were innocent and I were being investigated for selling drugs by the DEA. If I tried to obstruct justice, if I tried to intimidate witnesses during a trial, I would be put away. So the fact that Donald Trump is able to get away with this possibly should be maddening to people who don't believe we should live in a two tiered justice system. Now, on top of that, there's the hush money payments that he paid to Stormy Daniels, which amounts to campaign finance violations. There is his refusal to place his businesses in a blind trust, which puts him in violation of the emoluments clause of the Constitution. There are literally thousands of conflicts of interests that plague Donald Trump's presidency. I mean, the fact that he's still profiting from foreign diplomats and U.S. government officials staying at his properties in and of itself, I think, is impeachable. In my Pence's recent visit to Ireland, he stayed three hours away from Dublin, which is where he knew he needed to be just so he could stay at Donald Trump's hotel. Now, in the event this impeachment inquiry leads to Donald Trump's crimes being televised for months on end, could that potentially have a Nixon type effect and drive down public support for Donald Trump and drive up public approval for impeachment? That is entirely possible. But it's also important to consider the fact that he most likely won't be convicted in a Senate, which means Donald Trump could be another Clinton and his approval could potentially soar if it seems as if he was exonerated. It's also entirely possible, mind you, that something entirely different happens. Donald Trump's corruption could be exposed and that could hurt him in spite of a failure to convict his reputation comes out more tarnished. And we're back to being pro-impeachment again. We don't know how this is going to play out. The one thing that I know for sure is that none of us know for sure. The only certainty is uncertainty. But here's what I do know and the reason why I support impeachment. There's a lot of variables. There's a lot of moving parts. But the reason why we should support impeachment is because of principle. Nobody can say definitively that this will help or hurt the left. But what we do know is that Donald Trump broke the law. And if we choose to not open an impeachment inquiry, if we choose to not pursue impeachment and take it as far as we possibly can, I don't think we're right to keep complaining about the lawlessness and corruption of the ruling class. We should shut up and stop complaining about the two tiered justice system that we have where the poor get locked in cages and elites receive a get out of jail free card when they break the law. Because look, here's the thing. If we're not going to hold elected officials accountable when they are brazenly breaking the law, then we shouldn't be complaining about this corrupt system that we have, because if we're not going to hold elected officials accountable, we are only helping to perpetuate our corrupt system. And these elites will continue to misbehave if they know that we're going to be too afraid to hold them accountable whenever they break the law. So we shouldn't give Donald Trump a pass when he breaks the law just because he's in power and he's an elite. We shouldn't not impeach Donald Trump because we're afraid of what might happen. Nobody knows what will happen. This is a process, so it's important that we don't be fearful and we actually stand our ground and be brave for once. I mean, I've been yelling at Democrats to actually be strong and hold Republicans accountable. So why would I now say, no, no, no, don't actually hold Republicans accountable when it seems like they're inching closer towards holding Republicans accountable for the first time since ever? I mean, why it seems antithetical to our views as progressives? Without being said, that doesn't mean that I believe we should not proceed with caution because this could very well be risky. And the naysayers who were against impeachment on the left, I think that they bring up some valuable things for us to consider. I do. But all of these counter arguments, when you take them into consideration, I don't believe that they are persuasive enough. Now, some are more persuasive than others. Some of them, frankly, are laughable. But with that being said, I do think it's important that we evaluate these counter arguments because you never know what will happen. So I think it's important that we are scrutinizing the situation. I'm in favor of impeachment. But let me tell you my view on some of these counter arguments. I've got a couple here that I've seen. So the first kind of argument is that it will only further divide the country. This one makes no sense to me. It's not even persuasive in the slightest bit. We're already divided. Impeaching Donald Trump isn't going to further the divide. Polarization is a political reality. This is about holding people in power accountable. So I'm not worried about further polarization. We will continue to grow further apart. So long as capitalism takes full control of these parties and Republicans continue to become more and more extreme and fascistic. Second counter argument will get Mike Pence. Now, two years ago, this argument would have resonated with me more. But now when we're approaching 2020, we're what, 13 months away? Let's say, hypothetically speaking, impeachment is successful and Donald Trump is removed from office. Mike Pence is sworn in. Here's the thing about Mike Pence. Is he more politically savvy? Could he potentially do more damage from a policy standpoint? Without question. But here's one advantage that people aren't considering. Mike Pence has one year to do that damage and then we kick him out. Mike Pence is far less electable than Donald Trump. If Mike Pence is the president, I think even Joe Biden could beat Mike Pence. So this argument at this time is no longer persuasive. Second or third, actually, this is going to galvanize Trump's base and help him win in 2020. Donald Trump's base is already fired up. They're already galvanized. Here's the thing, Republicans base of support is very loyal. They're always going to come out and vote. So it's not like they're going to be more mobilized than they already are. What will determine the outcome of this election is how mobilized Democrats will be. So you can make the opposite argument that maybe impeachment will galvanize the Democratic base, because we all know that when turned out as high, Democrats win and Republicans lose. Now, I'm of the belief that you're most likely going to get people excited to vote for you if you present them with policies. But there are some resistance Democrats that might feel more excited. So I don't actually believe that this is a very persuasive counter argument. But there are some that I believe are persuasive. So this could make Trump more popular and a not guilty vote in the Senate might embolden him. This is entirely possible, entirely possible. But it could also make him less popular. The point is we don't know. I think that, you know, it's risky. This is a relatively persuasive argument. But I think it's also equally persuasive to consider whether or not this drives down his support if his corruption is aired. If it's put front and center for all of America to see, we could see another Nixon effect. Now, another issue is, you know, maybe this could be a distraction. The media will get bogged down by this. Democrats will use this as an excuse to not talk about policy. I mean, the media already has a sensationalist bias. They're not talking about the issues that they should be talking about. Could this potentially make that worse? Sure. And I get the desire. I understand I empathize with the desire to want to stay focused on policy. But I also believe that we have to do what's morally right. And I don't believe that turning a blind eye, you know, in spite of these possible negative ramifications is the right course of action here. Now, finally, here's the biggest and the most persuasive counter argument. And this is the hardest for me to argue against. What if Democrats bungle this? Very, very likely. I don't know if today's Democrats will be as competent as the Democrats were in the Nixon era, if you want to argue that they were even competent. But this is a very persuasive counter argument that it's going to be done poorly. Democrats will bungle this. Nancy Pelosi won't know what she's doing. And we're already getting indications that that is in fact the case because there are sources saying that she had a meeting with other Democrats in the house and they want to limit the scope to Ukraine.  The biggest political blunder, perhaps in the history of Democratic Party politics. So if you say this, if you think, you know, maybe impeachment isn't the best course of action because Democrats could bungle this possibly. But here's the thing and the reason why none of those counter arguments dissuade me from supporting impeachment, because we don't know. Again, the only certainty is all of our uncertainty. Nobody can see the future. We don't have a crystal ball. We don't know how this is going to play out. It could backfire. There's a chance this could actually hurt Donald Trump more. We don't know. So what we should do is be strong. Not make decisions based on fear and be moral. Hold Trump accountable. Don't turn a blind eye when we know he's broke the law. So that's why I say those arguments, they are persuasive, but they're not enough to lead me to the conclusion that impeachment isn't the right answer. So I absolutely believe that impeachment is the right course of action. Does it need to be done correctly? Yes, does it need to be effective in the sense that we look at all of Donald Trump's criminality? Of course it does. Could it backfire? Yes, but we don't know. So I say let's impeach the motherfucker, because this is a moral issue. This is a matter of principle when people in power break the law, when the ruling class break the law, if we don't hold them accountable, when we when we have the opportunity to hold them accountable, then we have no right to complain about their lawlessness and corruption because they will keep getting worse if they know that we are too afraid to hold them accountable and act when they commit crimes.


Coming up on today's episode of the Pumpkin Patch Show. The PSVR hits another massive sales milestone. We've got loads of firewall, zero hour news. We've got Dreams VR news updates. We've got No Man's Sky coming to PSVR. We've got Iron Man exclusively coming to PSVR. We've got Five Nights at Freddy's coming to PSVR. We have all of the games coming to PSVR. We've got dates for games that we already knew were coming but that we didn't know when they were coming. Now we've got the solid release dates. For example Blood and Truth. That was a big one. We've got a release date for that. Now all that and more on the Pumpkin Patch Show. So hey there lads and ladies welcome to episode number nine of the Pumpkin Patch Show. A PSVR focused show that also just so happens to be the number one most popular show ever made. Now that is a fact. Just don't google that okay. Don't. Don't fact check what I'm saying. Just accept this okay. I'm Petrifying Pumpkins and as always I am joined by a special guest host. Now today's guest host is none other than the shitheads who are very angry that Sony are still supporting VR shitheads. How's it going? Pleasure to have you here. I hope you're well.  Suffering from success lads. PSVR has sold 4.2 million units as of the third of March this year. This is according to Sid Schumann over at the PS blog. So back in August we were told that the PSVR had sold 3 million so that means it's taken 7 months to sell 1.2 million units. And that's not bad for a gimmick. A shithead. Now why should you care about the PSVR selling 4.2 million units? Well the more that the PSVR sells and the more that it increases its lead over the competition that means the more games we'll see come into PSVR as more and more developers will look at PSVR and say that's where the money is and they'll bring their games there so that's good news for us PSVR owners. So well done to the PSVR and if you happen to be watching this and you don't already have a PSVR then now is the perfect time to jump in and these next few things I'm going to talk about are some great reasons why you should jump in. So those of you who keep their ears to the ground when it comes to Firewall Zero Hour will probably already know that something is coming on the 1st of April. We just didn't know exactly what. Well that supposedly has changed thanks to a community or developer livestream from First Contact Entertainment's Frank which took place last night which was Wednesday. Now I did restream that stream on this channel so that we could catch all the new details live but if you didn't happen to see that stream I'm just going to briefly bullet point the new shit that we saw. So first up in the lobby menu we saw a new contractor name. Now we didn't actually see the contractor herself but we do know her name is Ruby. She's from the Unizing Kingdom and she has a skill called Thief. Now Thief allows her to hack dead enemies and steal their crypto basically. It's an interesting skill but keep in mind we didn't actually see her we just saw her name and we know where her skill is. Now another interesting change that we saw in Firewall was the removal of the large red names that you would see floating over enemies heads. Instead those have been replaced with red arrows or red chevrons if you want to be correct about what those shapes are. Now some people might be wondering what's the benefit to this well I think it makes it more fair as people who had really long playstation names their names would sometimes just like stick else behind objects and stuff whereas people with the smaller names they had a bit of an advantage in that way. Now everyone's got the same same length chevron over their head so it's a more even playing field we think. Now interestingly during the stream we saw the CEO of first contact entertainment Hess Berber himself he met a cameo appearance in the corner of the screen. Now what he was wearing was some kind of Oculus rift headset I think or some kind of PC viewer headset. He was going around with the motion controllers. Some people are speculating that means maybe Firewall is coming to PC viewer now we have to keep in mind that Sony do own the firewall zero hour IP so that means the chances of firewall coming to PC aren't exactly very high. But recently Sony have allowed titles that they've already owned that they own the IP to I should say like beyond two souls and Detroit become human they're coming to the epic game store so maybe Sony are loosening their policies maybe we will see firewall on the pc viewer or it could just be that Hess was just messing around having the lords having the bands we'll have to wait and see on that one. So also worth noting is that we've seen changes to spawn locations on both the compound map and the bunker map now those are two of the maps where Russian can be brutal especially when you're out you have a dirty spawn location so the fact that they've actually gone in there changed some things around that could give like a new twist on some of the older maps and speaking of maps most importantly of what we seen perhaps was that first contact entertainment teased not one but two brand new maps now one of them they actually showed in action the other one was just a blink and you miss a kind of screenshot so the one that we did see was called hangar now this takes place in an airport terminal hanger at an airport something like that it takes place during the daytime which is like the first time we've seen a map do that it has it looks fairly large it has interiors it has exteriors and it might be worth noting but if you read the signage around the map but there's exit signs and stuff like that that's all in English so there's a chance that this map is says in an English speaking country that might just be placeholder signage I don't know map might not be fully ready yes we'll have to wait and see what it looks like it could be in English speaking country so then the other map that we saw which was on the map selection screen we didn't even see the picture we just saw the title and that could be a working title it might not be the final title it was fob or fob now if you played metal gear solid 5 you'll probably be familiar with the word fob or the anagram or the acronym fob which stands for forward operating base this is basically just a military base that is forward in the in the battle lines or you know just out there somewhere the name doesn't exactly give us any hints of where this could be it could be anywhere in the world really but it's mostly just nice to see that we're getting two maps confirmed concrete now it's hard to say if all of the stuff that they've shown and teased will be coming on the April 1st kind of patch that we're expecting but it's probably safe to say that at least some of us will be i had a dream so if you've been looking forward to medium molecules dreams finally entering early access then i have good news for you because dreams is coming to early access on the 16th of april which is very close now if you're one of the people like me who was just super excited for dreams solely for the fact that it has psv or support i've got some slightly bad news in that the early access will not support the or will not support psv or and that they will add viewer support to the game later on and they didn't go into any further details about us now dreams early access is going to be somewhat limited you can buy us on the 16th of april provided you're in one of the supported countries you can buy it for a price of 30 dollars if you're in america 40 dollars in canada that's 24.99 sterling and 29.99 euro now if you're like me and you're just really excited for the viewer aspect then i'd still recommend trying to grab a copy of the early access because i'm sure that the price will go up once dreams leaves the early access phase and may become a full price game so right now it's kind of half price some it could be a good investment to get it now while it's cheap and then by the time it's released it'll have viewer support and you'll have it for half price but if this is something you want to do then i would recommend that you act quickly as there is a limited quantity of codes that's media molecule will be supplying to the psn so be quick to avoid disappointments speaking of disappointments are you hyped for dreams shitheads what what do you think about dreams and viewer the no man's sky is the limit no man's sky is coming to viewer this summer as part of the no man's sky beyond updates which will be free to all existing owners of no man's sky so that's right sean murray who only a couple of years ago was one of the most hated game developers of all time has had an epic redemption arc and has pretty much become the jesus of video games lately thanks to the constant support post-launch support that hello games have put into no man's sky all for free since 2016 or whatever year it was again those and i think we can all agree that the viewer patch is going to be the coolest now the best thing about this updates is that it's not just going to be a separate viewer mode the entire no man's sky experience will be available in viewer including multiplayer including exploration flying your ship around dog fighting piracy whatever it is you do in that game all of it can be done in viewer and not only that it's going to be coming to viewer on pc2 and it will be cross play with the playstation so you can play with your oculus and your rift buddies in viewer or just the flat versions as well i will assume now no man's sky is fairly cheap right now if you go looking in places like gamestop so i would recommend that you snatch up a copy while you can because people have been snatching them up quick since the news of this update bro now it's important to note that hello games didn't just want to port it to viewer and just put the viewer vision only they wanted to make it a true viewer experience that's why you're going to be using both move controllers you're going to be interacting with things just like you would if the game was built from the ground up with viewer in mind so that's really cool you can look forward to that virtual downy reality junior so iron man viewer is going to be ps viewer exclusive coming sometime to ps viewer in 2019 now this is going to be developed by camouflage studios the same team behind republic and they've even got members of like metalgear solid x staff on their school on their team you know so they have a talented team now we don't know an awful lot about the game seen as most of the reveal trailer was a cg cutscene but the brief glimpses that we did guess showed us that we'll be using both move controllers using tony stark's hand blasters and his chest blaster to you know shoot down shit in the sky assuming there's definitely flies we don't know if there's going to be ground stuff too yes but there's definitely going to be flies now a lot of people have been concerned of what they've seen they're wondering is this going to be on rails is this going to be just a 20 minutes experience now the one thing i have seen that maybe points otherwise is during the gameplay you can see a compass at the top of the screen now if this game was really just on rails or whatever you wouldn't really need a compass now of course it could just be there for show or for set dressing you know whatever but i think it's it's a small hint at least that this game is going to be some was maybe maybe something like ace combat seven we'll have to wait and see for more details but definitely keep an eye on this one iron man is definitely a huge name for sony to grab as a ps viewer exclusive it could have the potential to actually move units five nights at freddy's is better than one night at mj's five nights at freddy's help wanted vr is officially confirmed i believe i talked about this either a video or maybe two episodes ago after pumpkin patch show now we have confirmation basically it's going to bring back the levels from the classic games as well as add brand new levels specifically for help wanted and you're going to have your jump scares you're going to have the cameras and robust things and it's all going to be in vr so if you're somebody who's into five nights at freddy you should definitely check this one out this one is coming sometime spring of this year so not too far away potentially save the day so speaking of release dates we got a bunch of release dates for games that we already knew about so i'm just going to quickly list out all the stuff that we learned so we got sky world a real time strategy game from the makers of arizona sunshine that's actually already released on the 26th of march so that one popped out fast falcon age then the falconry game is coming on the 9th of april so that's very soon ghost giant will be with us on the 16th of april table of tales is also coming on the 16th of april jupiter and mares which is that dolphin game you may have seen that's coming on the 22nd of april everybody's got the ore now that one looks kind of cool that's coming on the 21st of may blood and truth which looks really good that got a new trailer by the way definitely check that out that's going to be coming on the 28th of may and pre-orders are available for that so if you want to grab that now you can go ahead and do that trover saves the universe which is the next justin roland game now that's coming the 31st of may so not too long after blood and truth then you've got mini mech mayhem that's coming june 18th the vacation simulator is also coming out that same day and gollum that's the game i feel like i've been hearing about that game for about a thousand years that one is coming spring 2019 so surprisingly close and especially seen as we didn't even see a trailer for that i think we just got that release days on the playstation blog come in spring 2019 so keep an eye out for that one luna is another game coming spring 2019 luna is going to be some kind of relax and puzzler type game i'm not too interested in this one if i have to be honest and that's it for release dates as you can see some of them are a bit vague not all of them are specific release dates some of them are like spring or whatever just windows but still spring is not long well kind of already in spring now every but just let me know in the comments which of those kind of bunch of games are you guys interested in grabbing i know i'm interested in grabbing blood and truth obviously that's the big one but i was surprised to find myself kind of gravitate and towards that everybody's got the or game that has online stuff like that i could i can definitely see myself snapping snapping snapping that one on day one so tell me shithead do you like the look of these games what do you think about these games you know what that that actually raises a good question why are so many people salty over all these psv or games that are coming let's take a closer look are you getting thirsty from all that salt lads so playstations stays of play which was their nintendo direct type thing which is where we got all the info i've been talking about this whole video was grace as far as i'm concerned but there is no denying there was a lot of winging going on afterwards especially from the flat gaming portion of the gaming demograph out there so to discuss this i've invited back fan favorite doctor reginald seagull to talk about this phenomenon of why people are getting so salty over so many supporting their own platform professor seagull you're very welcome can you please tell me professor seagull why'd these bitches be getting all salty well yeah we've been doing some studies on this back in harvard university and to be honest with you we've been finding a direct correlation between penis size and the level of saltiness that these people are exhibiting the more salt you see the less size of the penis in terms of length and girth you know is it really as simple as that professor seagull well we've got bear charts you know we've got the bear charts the bear charts don't lie that's what we do at harvard university so that's all we've got time for on this episode of the pumpkin patch show but before i go let me give a very special thanks to all my patreon supporters who are on the screen right now they're putting food on my table and they're helping this channel grow now right now we're working towards getting a sexy new microphone so big thanks to them and if you want to help out then you can check out the link that'll be in the description but if you don't want to do that which is perfectly understandable then you can simply help me out by doing the old-fashioned stuff the like and the subscribing and the commenting and sharing the video to places where people might want to be watching it if such a place exists i don't know and i will appreciate that a lot as well thank you very much for watching i will see you lads and ladies in the next one bye for now


So now we can start. So we were talking about the symmetric group. Oops. We had introduced the cycle cycles in the symmetric group. So this was just to have an element, say, A1 to AR, where this is just a set of elements of set of numbers from 1 to n, such that sigma of ai will be equal to ai plus 1. And if i is smaller than r, and it's equal to A1 for i equal to r. So that really means the permutation permutes these elements cyclically. And obviously sigma of n is equal to n of k is equal to k if k is not an element in the cycle. We had then seen that every element in Sn has a cycle decomposition. So we had this theorem. Every sigma in Sn is the product. So sigma is equal to sigma 1 times sigma s of disjoint cycles. So disjoint cycles just means that the supports are disjoint, so the sets of elements which are actually permuted by them. And we also had seen that this product is essentially unique. And it's called the cycle decomposition of sigma. And we want to see that if we want to now use this to describe the conjugacy classes of Sn. So the claim is that what we want to show is that there's a one-one correspondence between the conjugacy classes for elements in Sn. So a conjugacy class is just an equivalence class under the operation of being conjugated, so where a is equivalent to g, a, g to the minus 1, or this case in Sn. We want to have one-one correspondence between the conjugacy class in Sn and the partitions of the number n. So I have to remind you what the partition is. So the partition of the number n is a tuple, cn1 to cnr of positive integers. So that means that ni is bigger than 0, which are somehow ordered. So with n1 is bigger equal to n2, bigger equal and so on, bigger equal to nr, and the sum of the ni is equal to n. So for instance, the partitions of 4 are we have 4, 3, 1, 2, 2, 1, 1, and 1, 1, 1, 1. You can easily see that these are all, so in this case, there are five different partitions of 4. And now we want to associate to every conjugacy class of elements in Sn a partition, and want to see that this is a one-one correspondence. So let sigma in Sn be a permutation, and let sigma equal to sigma1 times sigmaR be its cycle decomposition. So the cycle decomposition really means that these are. That sigma is a product of these disjoint cycles, and that the union of the supports is the whole of the set 1n. So every integer from 1 to n occurs in one of these cycles. Some of these cycles might be one cycles, which then means they are the n-cells identity, but that's OK. So we can look at the, so let L of sigma i be the length of the cycle sigma i. So that means sigma i is a L of sigma i cycle. So for instance, this thing would be an R cycle. The length of this thing is R. So obviously, this composition is well-defined up to reordering them, because it's the sigma i commute. So by reordering, we can assume that the length of sigma1 is bigger equal to that of sigma2, and so on. We can just decide in which order we want to enumerate them. And then the tuple, say L of sigma1, L of sigma2, and so on. L of sigmaR is called the cycle type of sigma. Now, I claim that this thing is a partition of the number n. So clearly, these numbers L of sigma i are positive integers, because they are the length of the cycle of the cycle. So it's some positive number. So the L of sigma i are positive integers. And we have ordered them in such a way that they are descending in this way. And we have also clearly that L of sigma1 is bigger equal to L of sigmaR by our definition. So the only thing we have to see is that the sum of them is n. But we have that sigma is equal to sigma1 times sigmaR is a product of disjoint cycles. So every integer from 1 to n occurs only in one of them. And the support, the union of the supports of the sigma i is the set 1n. This is what we wanted for a cycle decomposition. So that means that if we take all the numbers, so every number from 1 to n occurs precisely once among all the numbers occurring in the cycles. So that means that the length of the cycle of a cycle is how many numbers occur in it. So every number from 1 to n occurs precisely in one cycle. So the sum of the length of the cycle is the total number of numbers that occur in all the cycles. So this number is n. So thus the sum i equals 1 to R L of sigma i is equal to n. So this is a partition of n. OK. And now we want to see. So we can look at maybe some example to make sure we understand what we're doing here. So if we, for instance, take again the element sigma to be the element in S6. So 1, 2, 3, 4, 5, 6. And then what is it here? 6, 2, 5, 3, 4, 1. So then we can make the cycle decomposition. So 1 goes to 6, and 6 goes to 1, and 2 goes to 2, and 3 goes to 5, 5 goes to 4, and 4 goes to 3. So this is the cycle decomposition. So the length of the cycle are 3, 2, and 1. So the cycle type is this partition 3, 2, 1 of the number 6. So now we want to see what the cycle type has to do with the conjugacy class. So the claim is that two partitions are conjugated to each other if and only if they have the same cycle type. Lemma, two permutations, sigma and tau in Sn, are conjugated if and only if sigma and tau have the same cycle type. OK. So we want to see that. So we have to see what does it mean to be. So we take our element sigma, element s, bs. So this is the cycle decomposition of sigma. So now we want to take an element conjugated to it and see that it has the same cycle type. So let, maybe I call this in pi. Let tau be an element in Sn. And so we want to conjugate sigma with tau to see what happens to the cycle type. So if we form tau, sigma, tau to the minus 1, still conjugated element, what does it do? So we want to find out what this is. So we have this cycle decomposition. So we write sigma i equal tau. So a i, a 1 i, until a, say, m of i, i, OK? So I think it has a double index, no? So the i-th cycle goes from a 1 to a mi, but it depends on this. And then we put, so for all i from 1 to r and j from 1 to m of i, we put bi, b, that was maybe, bj i to be tau of a g i. And we put, say, pi equal to b 1 i until a b m of i, i. So we make a new cycle by replacing the ai's, the aij's, by the bij's. And now what do we have? So if we take tau sigma, tau, so tau, say sigma tau to the minus 1, and we apply this to some element bij, what is it? First we apply tau to the minus 1 to this. The tau to the minus 1 of this is aij, because tau of ai of r, here it is, yeah. So it was the other way around, but anyway, ji. So the tau of aij of aji is bji. So the tau to the minus 1 of bji is aji. So this is tau of sigma of aij, aji. Now we know what sigma does to this, because sigma is the product of this disjoint cycle, sigma i. So applied to this element, it's just sigma i applied to this, which sends this one to the next one in the same cycle. So this is tau of, so there are basically two possibilities. This is tau of aj plus 1i, if i is smaller than mi. And it is the first one, so a1i, if j is equal to mi. That's how it is. So you just have the cycle. You turn it around, you go one further. This one to the next, and so on, the last one comes back. And this tau applied to this. But now tau of this is the corresponding thing with b. So this is bj plus 1, i, j is smaller than mi. And it's b1i, if j is equal to mi. So that means this thing permutes just the bij's, the bji's in the same way as here the aji's, aji's are promoted. So it follows that tau sigma tau to the minus 1 is equal to say pi 1 times pi r, where pi i is equal to precisely this thing. We're going to write it from the next line, where, as I already wrote here, pi i is equal to b1i until bm i. So we see that what happens is actually very simple. If you have a cycle and you conjugate it with a certain element, then what happens is just that you get, again, a cycle of the same length where the elements in the cycle are replaced by the images under the element by which you conjugate. So in particular, we see that these have the same lengths as before. And so clearly, we have that the cycle type pi equal to pi 1 to pi r is equal to that of sigma. So we see that if two permutations are conjugated, they have the same cycle type. And now for the converse, it's actually not very much. And basically, we trace the steps. So assume, we basically just have to see that all the steps can be reversed. So assume that sigma and pi have the same type of type. So we can write sigma equal to sigma 1 tau times sigma r, pi equal to pi 1 times pi r, where the length of the sigma i is equal to the length of the pi i for all i. So that means I can write, so I can write sigma i equal to, say, a 1 i until a, say, mi i. And pi i, I can write as b 1 i until b mi i for just some elements a 1 i. So for this, just some elements in one n, and the same here. So as these have the same number of elements, and the union of all these, when the union of the supports of them is up to the set 1 n, we can find the bijection from 1 n to itself by just sending each aji to the corresponding bji. So let tau from 1 n to 1 n. So I send aji, where in this composition, to the corresponding bji. We know that these are disjoint subsets of 1 n, and the union is the whole of 1 n. So this is a bijection of the set 1 n to itself. And pi, what we have proven here, we know precisely, so by the above, we know if we take tau, sigma tau to the minus 1, and this is equal to pi. Because we have precisely seen that this is, we are precisely back in this situation. And we know that if we conjugate in this way, we get that the cycle with the aji's is replaced by the corresponding cycle with the bji's. So it's anyway good to know that this conjugation has this very simple effect on the cycle. So as a corollary, we now can count the conjugacy classes of Sn. So the number of conjugacy classes of Sn is equal to the number of partitions of n. And basically, we have shown it proof. So we have seen that the map, which associates to a partition, so the map from the conjugacy classes in Sn to the partitions of the number n, which associates to a conjugacy class of an element sigma, the cycle type of sigma. So we have seen that, so what have we seen? We have seen that if that the cycle type of a partition depends only on the conjugacy class, so that means this map is well defined as a map from conjugacy classes to partitions. And we have seen that, in fact, two elements are conjugated to each other, so in the same conjugacy class, if and only if the partition is the same. So that means this map is well defined and injective. So we have to see it's subjective. And that is very simple. We just have, for every partition, we have to write down an element in the symmetric group, which has that cycle type. But that's kind of trivial. No, we just have to. So for instance, so let's say if p equal to n1 to nR is a partition of n, then we can take sigma. So we just have to have something where the lengths of the cycle are given by this. So it means the sum of d and i is equal to n. And we just take, so say, 1 until n1 as a first cycle. Then we take n1 plus 1 until n1 plus n2. So the length of this will be n2. And you can kind of see how it goes on. In the end, we have n1 plus nR minus 1 plus 1 until the sum of all d and i, which is equal to n. So this is a cycle decomposition of some permutation. Because these are just disjoint cycles. And we see that the length of the corresponding cycles is here n1, here n2, and so on. Finally, here it's nR. So this has cycle type. So we have this projection. So this was as much as I wanted to say about this cycle type and the conjugacy classes in SN. Now finally, to finish with the symmetric group, I want to talk about the sign of a permutation. So this is something that you might be even familiar with from linear algebra. Sometimes one of the definitions of the determinant is that you take the product over all ways how you can take one element out of every row and column. And you multiply by the sign of the permutation to sum them all up. So it already occurs there. Now I want to introduce it here. So there's one n invariant. So a number you can associate to a partition. Of the partition is its sign. And in fact, the way how one usually understands it is it is minus 1 to the m, where m is equal to the number of transpositions, which one uses in, so the partition is called sigma, used to write sigma. So if I write sigma as a product of transpositions, there will be many different ways how to do it. But the claim is that whenever you write it, the number of transpositions you need is either even or odd. And this does not depend on how you write it. And minus 1 to the number of transpositions you need is the sign of the partition. Now, so this is what it is for some reason. It's not now the easiest to work with for me at this moment. So I give another definition, and then I show it's equal to this one. Because obviously here the problem is that whether this definition is well-defined or not. You could expect that sometimes you need an even number, sometimes a not number. But this is not the case.  So let n be equal to 2 be some positive integer. And so for sigma in Sn, the sign of sigma is, so it's called, I denote by epsilon of sigma. And I write down some crazy formula. So this is the sum over all i, j, sigma of i. Minus sigma of j divided by i minus j. So these are all integers. So you can certainly do this. And the sum means it's the sum is over all pairs of integers i, j with i and j, just integers from 1 to n, and i bigger than j. So I take this sum. So if you look at it in the first moment, you look at it, you might even wonder whether this is a sign. This is supposed to be plus or minus 1. So in the first moment, you might even not be sure whether this is plus or minus 1. It looks like a rational number. But we'll see the moment is not the case. So now let's study it. So the first claim is that epsilon of sigma is equal to minus 1 to the m, where m is not precisely this, but something similar, where with m equal to the number of pairs. Let me write it equal to the number of pairs j with, so again, i, j pairs in 1n with i bigger than j and sigma of i smaller than sigma of j. So this is the number of elements where kind of the order of the elements is reversed by the permutation. Well, you're right. This would not make any, yeah, yeah, this would not make any. I think I certainly want the product there. What? What? OK, let me see. No? Yeah, yeah, if I sum, I get 0, yeah. So if the product is OK, this term will never be 0, because i is different from j. And this is a permutation, so they always are different. So this was just a misprint, which, however, I did not. OK, this is certainly the product. I think now in the proof, it also is everywhere the product. And then the second statement is that this epsilon, so the map which associates to permutation its sign, is actually a group homomorphism. So secondly, the map sigma, no epsilon, from SN, to the set consisting of the elements 1 and minus 1 together with the product. So this is a subgroup of the, whatever, the rational numbers with multiplication.  So this map is a group homomorphism, so where these are really the integers 1 and minus 1 with the multiplication. Well, it's obviously subjective, yeah, but I mean, that's if n is at least 2. But I mean, that's kind of trivial, we'll see that in a, but it just means if you have a, for instance, if you just have a trans, according to this story, if you just replace two things, if you place 1 and minus, if you have two elements, say, you always have the transposition 1, 2, and this transposition will have, according to the part 1, the sine minus 1. And obviously, the identity, according to the same story, has obviously sine plus 1. So it's a trivially subjective. So this is a group homomorphism. And third is the statement that I used as a definition. So if sigma equal to tau 1 times tau k is a decomposition as a product of transpositions with tau i transpositions, then it follows that epsilon of sigma is equal to minus 1 to the k. So which was what I first said would be the definition. OK, so let's prove it.  So let's see. Let's just write down, as before, product i bigger than 0, sigma of i minus sigma of j. So we are afterwards supposed to divide it by this. So we want to compare it with this. So let's look at what this is. Well, clearly I can write this as a product of all i bigger than j, where sigma i is also bigger than sigma of j of the same thing times the product where now sigma i is smaller than sigma of j, sigma i minus sigma j. That's not very difficult. But now, if you exchange the role, if you here take the product the other way around, you get each time a sine minus 1. So this is equal, if I want product i bigger than j, sigma i bigger than j, OK. So anyway, so I can replace this by sigma j minus sigma i. And I get to sine minus. So if I do this here, I get minus 1 to the number of elements here. So I should maybe say that m was the number of pairs. So I claim this is minus 1 to the m times the product over all sigma i bigger than sigma j of sigma i minus sigma j. Because what do we have? So this is the product over all. So what I mean is that if we get this, I think I'm confused. So if i is bigger, so here we have this. In the second product, we can exchange the role of i and j. If we exchange the role of i and j, then we get now that j is bigger than i is smaller than j. Sigma i becomes bigger than sigma of j. And now we have exchanged i and j, so we get to sine minus. And so we get minus 1. And we do this precisely m times, where m is the number of elements which occur here. m is the number of pairs ij, where i is bigger than j, and sigma of i is smaller than sigma of j. So these are precisely those which occur in the second product. They are precisely m factor. Is it clear or should I start again saying it?  Should I explain it again? Or is it clear? What? It's fine. I hope it's fine for everybody. Anyway, so we have this. So this will be this. And now, so now what do we have here? What? Yeah. OK, maybe I write a little bit more. I write back what I had before. Sigma of i minus sigma of j. So note that, so first I should notice that these are precisely m factors. Because m was precisely the number of pairs ij, such that i is bigger than j and sigma of i is smaller than sigma of j. These are precisely those which are here. And now, in this last product, we want to exchange i and j, so i with j. So what does it mean? It means now that first we get that now i is smaller than j and sigma of i is bigger than sigma of j. So just the last factor looks like this. And here we have sigma of i minus sigma of j, which precisely changes sign. So we still should write sigma of i minus sigma of j, which is the same as minus m. Because now this i, so we have exchanged the role of i and j. So now what was i before is now j and vice versa. So i is smaller than j, and we have this, OK? So we have precisely m factors. So this part is minus 1 to the m times product i smaller j sigma of i bigger sigma of j sigma of i minus sigma of j. But now if we put this together, here we have the product. So I've only dealt with the second factor. So this is the product of the i smaller than j sigma of i bigger than j. And here we have the same product with i smaller than j, but the sigma i's are the same. So the condition is now only on the sigma i. So if I now, this was a computation in between, but if we go on here, this is equal to product over all ij such that sigma of i is bigger than sigma of j sigma of i minus sigma of j. So this product means it's the product over all pairs ij such that sigma of i is bigger than sigma of j. Because there's no condition on whether i is bigger than j or smaller because we have used both. But now sigma is a permutation of the set 1m. So here, if we apply i by sigma of i, we have just reordered the factors. So this is the same. So reordering the same as the product over all i bigger than j, i minus j. Because I just replace sigma of i by i. If I do this for all i, I get the same factors only in a different order. And so this means if I take the quotient, except that, obviously, I made one mistake, because I had this factor minus 1 to the m here, which I dropped. And so the definition was that I'm supposed to take this product and divide it by the product of the i minus j. And what remains is just minus 1 to the m. Maybe you can go through it again also, but OK. So this is 1, then 2. So I want to see it's a group homomorphism. So it's somehow always the same trick, which takes some time getting used to. So maybe you can recognize it the second time. So we want to show it's a group homomorphism. So if I take a product of two permutations, then I should get also the product of these signs. So if I take sigma and tau elements in a symmetric group, then we have the product. So we want to compare epsilon of, which one do I want to draw, sigma tau. We want to compare this to epsilon of sigma times epsilon of tau. In fact, we want to show they're equal. So let's try to compute this. So according to this definition we have given, this is the product over all i bigger than j sigma of tau i minus sigma of tau of j divided by i minus j. There's nothing wrong with that. So this is just the definition. Now we can certainly multiply this as one often do in analysis, multiply this with one in a slightly complicated way. So I hope you remember the statement. So we just multiply this with one. So this is product i bigger than j sigma of tau of i minus sigma of tau of j divided by i minus j. So it was by i minus j instead. We multiply it by tau of i minus tau of j. And we multiply by tau of i minus tau of j divided by i minus j. If I want, I can also write product i bigger than j. So we have this. Now this is just, we've just multiplied. We have multiplied by this and we have divided by the same. So there's nothing happening. And now let's look at this factor. So I claim, first I write it down and then I give it the reason this is the same as a product over all tau of i bigger tau of j. So it means again, the product over all pairs ij of different elements in 1n such that tau of i is bigger than tau of j of sigma of tau of i minus sigma of tau of j divided by tau of i minus tau of j. Why is that? So if i is smaller than j and tau of i, if i is bigger than j and tau of i is bigger than tau of j, then nothing changes. If i is bigger than j and tau of i is smaller than tau of j, we can replace, we can exchange the order of i and j. So we place i by j. So then i is smaller than j and tau of i. So that means, so in this, to compare these two. So if i is bigger than j, tau of i and tau of i is bigger than tau of j, we just take the factor here, you'll get the same factor here. Don't change anything. You don't change the factor. And if i is bigger than j and tau of i is smaller than tau of j, we replace, so i by j and j by i. Then we have now that i is smaller than j, so then the new i is smaller than the new j. And the tau of i will now, again, be bigger than tau of j. But what is the effect of this? Replace i by j. So now the denominator changes sign because we exchange these two. And the numerator exchanges sign because we flip these two. So then both numerator and denominator change sign. So we get this whole product, we get this product from this product by exchanging i and j in all the factors where this holds. And when we do it, each time we exchange them, both the numerator and denominator change sign, so the whole thing doesn't change sign. And so they're equal. And then we have the same argument as we had to finish the argument here. If we take the product over all pairs with tau of i bigger than tau of j, tau is a bijection from the set 1n to itself. So we can apply this bijection. So it means that if we replace tau of i by i, we have just permuted the factors. So tau is a bijection. So for each value tau of i, for each tau of i, tau of j, we get the factor. And if we replace tau of i by i for all i, we get also effectively get the same factors because tau is just the permutation of 1n. So there's a bijection between the factors in the same way as we did it here. So it means this is equal to the set of all i bigger than j sigma of i minus sigma of j divided by i minus j. So let me say it once more again. I just say that these factors here are just the permutation of these factors. They are permuted by the permutation tau. And then obviously these two things together say that this product, so this together says that epsilon of sigma tau is equal to epsilon of sigma times epsilon of tau because here this is epsilon of tau. And this one we find is epsilon of sigma. So this is part 2. And then the third statement was this one. So if we write sigma as product of transpositions, then epsilon of sigma is minus 1 to the number of transpositions we use. And for this, we only have to see that for transposition, this number is minus 1. So for 3 to prove 3, it is enough to show that epsilon of tau is equal to minus 1 for any transposition. Because as this is a s by 2, this epsilon is a homomorphism. So if I take epsilon of sigma, this is epsilon of sigma 1 times epsilon of tau 1 times epsilon of tau k, each time we get minus 1. So we get minus 1 to the k. And so we just see it. So for instance, if tau is equal to 1, 2, so the transposition which exchanges 1 and 2, then we just see then obviously epsilon of tau is equal to minus 1. Because if you look at this thing, then all stay the same. So if i bigger than j, then also tau of i is bigger to tau of j for all i, except for 1 and 2. And 1 and 2 are exchanged. So you get 1 factor minus 1, and all the other factors are 1. So 2, 1 is the only pair of i, j with i bigger than j and tau of i smaller than tau of j. And so this gives us minus 1. And so now on the other hand, we know by the above that if you have a transposition, so now if tau is equal to i, j is a transposition, we know it's conjugated to this one. In fact, if sigma is a permutation in 1n with sigma of 1 is equal to i and sigma of 2 is equal to j, then we know, we have seen this, that if I take tau 1, 2, tau to the minus 1, then this is equal to i, j. So if you apply again our epsilon to it, this is a group homomorphism. So it follows that epsilon of say this permutation i, j. So epsilon tau, epsilon of 1, 2, epsilon of tau to the minus 1 is equal to epsilon of i, j. So this is some number plus or minus 1. This is minus 1. This minus 1 times epsilon of tau times epsilon of tau to the minus 1, which is epsilon of tau to the minus 1. So now we are just in the group of 1 minus 1 with multiplication. So this commutative, so we can cancel these two and we get just minus 1. So we see that for any transposition, we get minus 1 and this proves the result. So finally, one can use this to introduce the alternating group. So definition of permutation sigma in Sn is called even if its sign is 1. So it means it can be written by an even number of transpositions. So the set of even permutations is called the alternating group in N datas and denoted En. And note that this is a normal subgroup of Sn. So An is a normal subgroup because it's the kernel of a group homomorphism with An is equal to the kernel of the map epsilon from Sn to 1, set 1 minus 1 with the multiplication. The kernel are those which map tau, the neutral N. OK. So one of the things which, so this group occurs sometimes, one of the things it occurs in is also in Galois theory, I don't know whether we will mention that later. But the fact whether you can solve the polynomial equation by taking roots and so on, it depends on whether a certain group is solvable, which has something to do with having enough normal subgroups. And it turns out that the fact that you cannot solve all polynomials of degree 5 in terms of taking roots, N roots of something is connected to the fact that this group An is simple. That means it has no normal subgroups that we will see. That's actually not so is for N equal 5. So it's just a statement that, for instance, A5 is a simple group. I was thinking of making it an exercise but it's somewhat difficult and so maybe I don't. What? I still don't understand you. So what is not commutative? Well, I will have to think about the statement. So you say that, yeah, you could say like that, yes, if you want. OK, yeah, there are certainly, in principle, well, I don't know precisely what that means now. Because if you, for instance, take, the fact is, if you take S3, for instance, as a simplest example.  So they're also commutative. The fact is only that the S3 is actually to isomorphic to the dihedral group in three letters. So it is somehow, it comes from two cyclic groups but altogether it's not commutative. So I don't know how this answers your question but you can see it somehow. The way how the two come together is somehow, I mean, sometimes, so in this case, you see that the even permutations are actually commutative. And the quotient is also commutative but the whole thing isn't. OK, OK, so I took a bit longer than I expect. So now we will want to start to prepare for, in some sense, for the statement and proof of the seal of theorems. And we do this by studying operations of groups on subsets. So if we see operations on subsets. So if we have a, so see if G acts on a set X, on a set S, it will also act on the set of all subsets of S. We want to a little bit investigate this because we will use it when we try to prove the seal of theorems. And anyway it's interesting to know. So let's state this. So let G act on a set X and let U be a subset of S. Then for an element G in G, we can form GU to be the set of all G. Maybe I'll write with time because in operation G times U. So it's not U doesn't have to be in G. It's not automatically a coset. G times U, the set of all G times U, where U is an element in U. So we have, we can associate, we can make an element G in G act on a subset of S. But just taking this set we obtain by applying G to every element of U. So it's easy to see that the map G comma U, maps to G times U, defines an operation of G on the set of subsets of S, of G on the set of subsets. Because you know what you have to see. If you take the identity, it maps every U to itself. If you take the identity element in G, it maps every one to itself. And if you take the product of two elements, it's the same as if you apply first one element and then the other. It's kind of trivial. We can also see we can restrict this action to subsets of S with a fixed number of elements. So I claim if we just take the set of subsets of S, which have five elements, then this operation defines an operation on that set. And this is because as we have learned, the set, the multiplication, the operation is always by bijection. G times from S to S, which sends an element S to G times S, is a bijection. It follows from the fact that you have an operation that each element operates by bijection. And so if you have a subset with a given number of elements applying G to it, the image will have the same number of elements. And now we want to define the usual things that we have for operations. So we can look at the stabilizer, GU. So if U is a subset of S, no. The stabilizer, GU, is the usual thing. It's the set GU, which is the set of all G and G, which with respect to this operation act as the identity. But what does it mean? It means such that G times U is equal to U. That's the stabilizer. So it means that the set U is mapped to itself. As usual, this is a subgroup of U. And maybe there might be some good. Yeah, well, it couldn't have to be one of U. So I just, as a warning, the stabilizer, GU of U in this thing is not the set of all elements G and G, which fix all elements of U. So it is not. So GU is not the set of all G and G, such that GU is equal to U for all U and U, which would be the stabilizer of every element. It just stabilizes the set. So each element of the set is sent to some element of the set. But rather, the set of all G and G, such that GU times U, is an element of U for all U and U. Is that a problem? No. OK. Anyway, it's maybe obvious, but I just wanted to indicate. Well, OK, I make one trivial proposition. It's a more remark. So let G act on a set S, and let U in S be a subset. Then I'm asked myself, when the stabilizer of U is the whole of G, so then G is equal to GU, if and only if. U is a union of G orbits, OK? So we can see that's quite obviously, and that's basically obvious if you think of what it means. So G, by definition, G is equal to GU, if and only if.  That's essentially the definition, and for all G and G. So the set of all GUs, G times U, for all G and G is the orbit of U. So that means it's the same as the set of all that GU is contained in U for all U and U. And the statement that for each element, its orbit is contained in U is the same as saying that U. So for every element, its orbit is contained in U is the same as saying that U is the union of orbits, namely precisely, the G of U for elements U and U. OK, maybe not a good moment to start something new now. So maybe I will stop now.


What's up, everyone? My name is Alex. I'm one of the co-founders of MyInvestingClub.com, and I want to let you guys know about something special we're doing for our viewers on YouTube. So the most common question we get asked is, you know, how do I start day trading? So what me and my mentor about it is we create a free two-hour mentorship course for the brand new trader. It's going to be available at MyInvestingClub.co. The link is going to be right here. This is a free webinar that reveals our 12 secrets that every single brand new day trader should know before they start. I also want to let you guys know about something that's very unique to MIC. So if you have any questions about trading or you're curious about trading or you don't know if MIC is the right fit for you, now you can text our head mentor, Tash, whose number is going to be right here, and he'll answer all the questions that you have in less than 24 hours. Thank you and enjoy the video. What's up, everyone? It's Tom Diesel here on another weekly video for you guys.  Today, I'm going to do a little bit differently from the previous videos. I'm going to go over one of my trade that I took on Friday that was a losing trade and you guys probably have noticed that I post some red P&L in the main channel as well. So I just want to go over that quickly and also there are some, you know, a few of the important topics that I would like to kind of address, you know, on that as well. So let's get into it. Yeah, so my, the ticker that I lost on was INO. I kind of traded this in the pre-market because why did you do that? I'm just going to bring the chart, you know, up real quick here. I mean, the stock was pretty extended for the last few days and basically, you guys can see it went from, you know, it was like June 22nd from 14 to like, you know, almost like 35, you know, 32, 33 here. So you have one day, two, four, five. I mean, it's, you know, to me, it's pretty overextended and also the volume as you see coming in is like, you can easily tell that those are people longing the stock or like chasing the stock drive up the momentum, right? And so for the first red day setup, we want to see the overextension like this instead of, you know, just like one or two days is not even extended enough for me to kind of interested in that move. So I kind of waited, you know, for the stock to run up like that. And it's been one, two, three, four, five days, and it was pretty overextended to me since like 15 to like 30, so it's like 15 bucks there for the drop, right? So and so that's why I kind of anticipate the first red day move a little bit in the pre-market and knowing that Friday was the day. I mean, my thesis is, you know, was correct, but my executions are for, you know, are really far from, you know, from, from the best or like a far from, from being great. So I just want to go over with that with you guys and so that, you know, hopefully you can, you know, pick up some lesson from this or so about the first red day, you know, set up about the pre-market trading and also how to deal when you kind of miss your entries, right? Usually what you do after that or like what I did on Friday, like I was frustrated. I mean, technically I didn't chase the entries, but, you know, it's just like I felt so down that my thesis was so right, but I couldn't execute it, you know, it well and let it, you know, slip basically. So I'm just going to go over that in details with you guys and hopefully, you know, it's going to help. So yeah. So I posted this one in the main chat on Friday and regarding about pre-market trading. Okay. So what do you think, you know, what, like a lot of people, a lot of members are asking me, what do I think about it? I mean, pre-market trading, I mean, the concept here is pretty much the same. As long as you have a plan, as long as you have predefined risk, right, like you're going to do, you know, well, but keep in mind that in the pre-market, you can't, like you don't have a hard stops, right?  So, and because you cannot use a market order, so that's why. And I posted this one, as you can see, I was anticipating the move. Yeah. So 7am, you know, I'm using treasure. So it opens 7am. I saw it was popping here, you know, rejects and popping back here. This line again. So the moment it kind of rejects to this line, I shorted some here and risking over just this line. So it's like 33.8 and my average was here, like 33.58, something like that. I was basically risking like, you know, maybe 20 or 30 cents here. And right after that, stock wash, which is, you know, confirm my thesis, right? Wash down here, you know, and I saw these rejections through view app. And I said, all right, so the green to red move is on 31.25, right? So I was hoping for the stock to kind of, you know, wash down here more and then pop back and then continue to fade in and maybe a big wash at open. And so that was my thesis. And as soon as stock cracked this view app level, I was pretty confident that, okay, so my thesis is kind of work now. Maybe I mean, you know, it's still the first red day anticipation because it's not even, it didn't even went red yet, right?  And when it stopped kind of trying to bounce back here, I said to myself, okay, so I'm going to add back all the shares I cover here as the stocks, you know, continue to fade more down here, maybe crack this line. I probably, you know, this is, you know, pretty much the same as also equivalent to the green to red kind of line. And if it cracks, you know, under that, I'm going to add back more onto that pop. So that was my thesis basically. But since I was trading a premarket, I had, I need to have a risk in place, right? So whatever I do is, you know, I had a good entry here, a good average, okay, I'll cover some here, add it back here. My new average now is around like 33, 22, something like that. And it was like, you know, well above view app, so right. And to me, if the stock breaks this level, probably my thesis is no longer valid. So that's what I did here. I tried to add back and stock couldn't do what I think it would do, like fade down here more instead of pop like that. So I just ended up covering the rest and for break even basically. So that was the first trade of the day in the premarket. I was up, you know, like small here. I mean, that's, that's fine, right? I didn't lose any money, but I had a plan. I stuck to it. And you know, I was pretty happy and that was it. But the biggest one came right after this, right? As you can see, like this one, right? So after I got stopped out here, yeah, I just want to again emphasize on the premarket trading one more time. The premarket trading, okay, guys, if you knew, don't try it. Because you are not used to the volatility in the premarket.  And the stock could really pop like crazy. Let's say this one VXRT and especially not on the day one, right? That's why we don't recommend trading premarket because of that.  Stock is like a wick here a little bit, right? Crack through view app, pop back here and you're thinking, you know, you're shorting down here and expecting, you know, move down here. I mean, that's, you know, the thesis is okay, but you have to ask yourself, where would you like to stop, right? But you know, if you not fast enough with stopping, I mean, look at the slippage you could get, you know, from 8.2 all the way to 10. There's no hard stops in premarket trading and, you know, daily max loss. I don't think they will trigger in the premarket as well. So basically you pretty much, you know, done, right? And instead of waiting for the market to open and put a hard stop there. I mean, yes, you can put a hard stop here. So if it pops like that, it will stop you out, maybe with slippage somewhere around here, but that's fine. But still the hard stop is there to protect you. The max loss is there to protect you as well.  So you have to keep that in mind. If you still want to take that risk, I mean, I would prefer it for more seasonal or like more experienced trader to kind of do it instead of someone who's just, you know, been trading for a few months or even a year. Yeah. So that's it for this week, guys. I hope this video helped everyone, guys. Take care.