Coffee, it makes all this seem less daunting. Oops. Coffee, it makes all this seem less daunting. Hi, hello, greetings, and welcome to episode 21 of Reverse Engineering Disassembling, Reassembling, whatever. I'm on ROK calculator. So on this side, this lever, I think, actually needs to go on the other side of this. So I'm going to fix that later. And I still haven't quite figured out what this wire is for. I did find, I did remember in looking at the videos that it sort of hooks over this thing so that apparently this lever can't go any further. I don't know why, because on the KA, which is the automatic version that I have, this wire doesn't exist. There is actually a part that sits on here that goes down this way. But I still don't know the purpose of this wire, and hopefully as we put this thing together, we'll figure it out. In the meantime, we get to put together the other side of the frame. So this is, I think, going back to video seven, I believe. So we've got all of these parts, and all of these parts need to go on here. So let's get to it. Got my oil. And I think, well, the easiest thing we could do is put this stand off together. So here we have a stand off plus the little washer that fits on it, and that goes right in here. So I'm just going to screw that on. That's pretty easy. I'm going to tighten it. Okay, that was pretty easy. We also have, let's see, we have this pair. This is a screw. In, let's see, in bag nine, we actually had two of these smaller screws and one of these bigger screws. So we're going to use the bigger screw and this funny looking nut. And that goes in the fourth hole down. So the screw goes in like that. And on the other side, the nut faces this way. And so we're just going to put that in and maybe tighten that up a little bit. Okay. So that's that part. Then we have this part, which is this complicated screw bushing, axle, shaft sort of thing. It's got a large washer that goes onto one of the shoulders like that. Then it goes into this hole right over here, just like that. And then on the other side, we put the lock washer, which I insist is a useless thing, and the nut. So we tighten that up. And I ran all of these parts through the degreaser to get all the oil off.  This can be tightened more. Oh, it's turning. Of course it's turning. So I'm just going to hold on to the nut using the wrench. And then I'm just going to tighten it up using the screwdriver. Perfect. All right. So there we go. All right. What's next? All right. What's next? Okay. So what's next is we have this complicated mechanism, which is the repeat, non-repeat, clear section. So this key stem is held in place by a little wire and a washer, so it doesn't come out. We also have these two pieces. So we have this piece and we have this piece and they do look different. So the one with the rectangular hole goes in the middle. So I'm just going to do that. I'm going to put, let me see if I can, there we go. Okay. And it goes in the slot just like that. So that's the one. Okay. I'm going to hold it in place with my thumb and then the other one. And you can see that all of these sort of shoulders here face the same way, same deal with this, except of course it's like this because the little rectangular hole here has to go in here. Okay. So now I'm just holding on to all of these things with my hand. Also we have this thing, which does go in here, but it has to go through all three of these holes. Then we can put them on here on the frame. So the pin that's right here, that goes in this hole right here. And then on this part we have two screw holes, one here and one here. And they're going to screw in here and here. I'm sorry, here and here. And you can sort of see a little outline of the part over here. So I'm going to carefully just place this with the pin. Before I do that, let me oil that hole because that is going to be a moving part. Okay. So now I'm going to carefully take this and put it in. There's the pin. Now I'm just going to align these. Here let me raise the camera a little bit so that we can get a better view, hopefully. All right. And then we've got these two small screws and they screw this plate in. I'm just going to put the screw in a little way. And I'm going to put the other screw in. And I can screw them both all the way down. Oops. Magnetized. That's one. Okay. So that's that. So the next part is this part and this screw with shoulder. And they go on along with this screw. So the screw, so the part goes on the screw's shoulder. And then we just screw that in right there. And this is the lever which allows you to remove the handle for the machine. Just like that. And there's a spring that goes here. But first, let me go ahead and tighten this up so that it is not so. Actually, before I tighten it up all the way, I'm going to put a drop of oil in there. So that way, that way it will move nicely. Now, I can go ahead and screw it down. And of course, even if I tighten the screw, the lever itself is not going to be tight because the shoulder is ever so slightly wider than the thickness of the sheet metal. Here is the spring. So I'm just going to go ahead and put the spring on. Well, I really did a number on the end of this. It's quite bent. So what I'm going to do is try to bend it back. Yeah, that's good. Just hook ends. All right. So now let me hook one end around the one post. There, please. There's one end. The other end goes around here. They're called hook ends for a reason because they're supposed to hook around. It's actually quite a difficult job to get them to actually hook. There's probably a technique to this that I don't know. All right, so there we go. That works pretty well. And the very last set of pieces are we have a gear, a washer, a very large washer, and the clip that holds the gear in place. So now what we need to do is flip this over again. And the gear is going to go onto this bushing. So first thing I'm going to do, of course, is oil that bushing really well. So I'm just going to put some oil on it. And I'm going to take the gear and put some oil on the inside, coat it nicely. Okay. So this gear also has this moving part here, which is between this press fit thing. So I'm just going to put some oil also in there just so that it can move a little better. Put some oil in there as well. Just basically any place where it interfaces with a stationary part. So there we go. All right, so this piece goes on like this. Okay, so we just put it on there. There we go. I think that's correct. Rotate it a bit to get all the oil distributed. Good. So the next thing that goes on is the washer. And the washer just goes on like that. Then we need to put the clip on, which will hold the gear in place. So what I'm going to do is clip it around one side. There it goes. All right. Well, that was pretty easy. A little too easy. I think that was a little too easy. So what I'm going to do is I'm going to actually bend this. I can just do this by hand, I guess. So just bend this a little in. Make sure that the two ends of the wire are aligned. Then I think I can put it back. Yeah, that seems to be pretty good. Yeah, that doesn't seem like it's coming out. So all right. So that was all that was in bag nine. So let's proceed to the next stage. So now we are moving into video six, which took this apart. This other section. So we now need to take all of these parts here and put them all back together. So the first thing that I note is that we take this cross member with the pins on this side, and not like this, but actually like this. This is actually the smooth side. So the smooth side goes down and it goes. So you can see that there are these two pins here and two screws over here. And they go into these four holes here. Now you notice that if I flip this around, this is quite clever. These two pins, these pins are actually on the opposite side. So it wouldn't actually go in this way anyway. So that's quite nice. The other thing I note is that this pin is pushed in quite far and it shouldn't be. So I may want to fix that. Let's see if I can actually do that. Let's see. So to fix that, I am just going to take, to attempt to fix that, I'm going to take a block. And let's see, the pin is over here. So I'm going to set this down just like that. So that now I might be able to bang on to that with something. I'm not really sure what because this doesn't really want to stay in place. So, there we go. So I'm just going to take one of these large things and where's my hammer? I don't seem to have my hammer so I'll just use this. And let's see if that actually helps to push the pin out. I'm going to have to rotate this. Not really. Doesn't really seem to be coming out. There we go. That actually does work. So let me just knock it a little more. Okay. That's pretty good. That's good enough. The pin is out. Okay. So anyway, as I was saying, oh, I did want to, I did want to move this. Let's see where did it go? I think it goes up above here. So let me do that now. Let's go ahead and loosen the set screw so that I can just push this out. I think that's how it goes. Like that. So now I'm going to tighten the set screw. Again, I don't know quite where this thing is supposed to go. So I'm just going to tighten it up so that the part doesn't actually move. Boy, it's hard to do this with one hand. There. Okay. So, okay. So, it's also an interesting mechanism because if I pull on this, this thing can go here. And then if I pull this back, isn't that neat? And this thing, again, I don't know why it would go around there, but I'll put it there just for now. Okay. So we've got these. So what did I say? It went like this. And let's see. Oops. Okay. So I am the dumb one because I'm not working on that part. I'm working on this part. You were probably screaming at your screen. Okay. So we have, this just fits nicely right in there. Okay. And now in our bag of parts, we had several different size screws. The most important thing is we had four screws of this size, two screws of this size, which I know happened to go on this, and one screw of this size, which means that we're going to use these four screws. In looking at this, I noticed something really funny. And that is, you see these slots? Look at this. This slot was actually cut into the shaft while this slot is still in the head. None of the other screws are like that. So there you go. That's how they made slotted screws. They just sort of cut a slot in it. So I'm going to take this and I'm just going to take one of these screws. And how does the screw go? Okay. This doesn't seem right. Must have gotten this backwards. Yep. I did get it backwards. So the pins go in the holes. Great. And the screws go in there. Nice. I'm wondering if I should oil the screws. It couldn't hurt. And it might prevent rust. It's just a thought. So, yeah, I'm just going to put a little bit of oil on the threads of the screws. It's probably a good idea to do this for all the screws. We'll start doing that from now on for all the threads. Okay. So now let me change my bit out to give me a bigger bit. Okay. And then I'll tighten it until I get the other screw in. Okay. We also had this rod, which goes in here, just like this.  There are two washers. There are many washers in the bag. So there is a big washer, a smaller washer, a smaller washer, and something that looks like a washer but actually is a bush because it's got multiple levels on it. So that's not the one. We're going to use the smallest washer. So the smallest washer goes right on there. And I promised that I was going to oil threads. So there we go. And then we take this standoff. And in the bag we have actually two standoffs. We're going to use the smaller of the two. So there we go. And I'll just tighten that up a little bit. Just a bit. Okay. So all right. The next cross member is this cross member. And you can see that it is also asymmetrical. We've got a wider space here and a small space here. The pins face the other pins here. And the small space goes over here. So I am just going to put it right in this set of three holes. Just like that. And then we use one of these two thinner screws. And I will put some oil on the threads. And then put that in there. Okay. So those are the cross members. Then the next thing that we do is we put this gear on. The gear has associated with it one of the two washers. In fact, it is the bigger washer because it needs to fit on top of here. And this one doesn't. And the clip. So also if you'll recall from last time we have these timing marks. So we've got one here and one here. So we've also got a timing mark here. Right there is a timing mark. And I had to actually scratch a little bit in order to make it visible. So the question is which timing mark is the one that goes up to here? Well, I'm going to try one. The timing marks don't quite line up. And the reason they don't quite line up is that this timing mark points right at a tooth. And this timing mark also points right at a tooth. So instead I'm going to try the other timing mark which actually points right at the valley between two teeth. So I think that's the right orientation. So I'm going to oil as usual. Let's just put some oil on there. Put some oil in here. I'm going to put this on. Make sure that the timing marks line up because that's really important. So the next thing that happens is I put the washer on and then I put the clip on. So this clip is smaller and it just fits right on like that. There we go. I'm just going to rotate these just to get the oil distributed. That's pretty good. Now it seems pretty good. One thing that I might want to do is put some oil on the teeth. Because even though they don't actually rotate they do mesh over here and they do rub against each other in order for one gear to provide force to the other gear. So I may end up not now but later oiling the teeth. All right so the next part is we join the two halves. So what we're going to do is we're going to turn this part around like that. And we've got this part and they need to join up. So we can see that the two pins here and the screw hole are of course going to line up with these over here. We've got this end is going to line up with that hole right over there. We've got the four holes over here which are going to line up over here quite nicely. And then we've got this end here which is going to line up with this hole. So let's go ahead and just put that together. So one part goes in. Then this part goes in. It's nice that all four of the parts don't need to get lined up right away. It seems like one after the other will go in. So that's quite nice. So I'm just aligning this last part. So they all pretty much have to go in. So something seems to be just a little stuck.  I think it's this middle rod that gets stuck. So what I'm going to do is I'm going to put some oil in the hole and oil on the part. And I may as well put some oil on all the other things as well. So that perhaps they'll go in more smoothly. Alright, let's go ahead and try it again. So here we go. Put this in, put that in, jiggle it, put the other thing in. Keep jiggling it until it all goes in. I think they're kind of sort of aligned. So what I'm going to do is I'm going to now screw this one in. So that takes the thin screw and oil that screw. And then by screwing it, I'm basically going to pull the parts together. I'm just going to screw this in. That does seem to work pretty well. I'm not going to tighten it because I just want to make sure that all the other parts are in alignment. Actually, all the other parts are in alignment with the exception of this central cross member. So I'm going to use the screws for that one. Put some oil and put a screw in there. And that way, the parts will be aligned. And when I screw them in, it will pull the parts together. Move this guy out of the way. There we go. So now I'm just going to screw these. Okay, now I can tighten them. So I'm going to tighten this a little bit. Tighten this a little bit. Tighten this other far guy a little more. I'll tighten it all the way. And I'll tighten these guys all the way. There we have it. We have a frame. Let me see if I can raise this up a bit so that you can see the entire frame is now put together. Okay, very nice. So let's see. Clearly, we have this larger standoff, which goes on the end here. Or, yeah, it does. It does. It goes on. It's kind of funny, because if you look at this, you can see that there is the hex end, and then it's rounded over here. While on here, the hex end actually goes on the inside. Kind of a funny way of doing things, but that is the way it is. So I hope that's the way it is. I hope the small one goes on this end. Should be easy enough to correct if that's wrong. So I'm just going to tighten this one up. There we go. It's all tightened and together. So what have we got left? We've got a bunch of parts left in the bag. So let me just drag this over. So we've got what appears to be a washer, a screw, a washer looking thing that really isn't. We've got a lever, and we've got another sort of part. So we have this lever, and we have this screw, and we have this bearing. And we also have this part. So it appears from the video that this part goes with these tabs face down, and they just sort of hang out over here for now. I'm not quite sure why that is, but they just sort of hang out here. The lever goes with the spring down and with this round part over on the left side. We have the bushing. Oops, I just dropped it. That's not good. So we have this washer with its shoulder, and the small end of the shoulder goes in there just like that. So what I'm going to do is just apply some oil on there just like that. And then this fits in right over there. So I'm going to take the screw, oil the threads, stick the screw through just like that, and then place it right there. And you can see that it goes over this thing, which is why I guess I had to put it in first. Something like that I guess. So now we go ahead and screw that in, get it aligned on its shoulder. I think that's how it goes, just like that. So the real question is what is this doing? Why is it all loosened and things? And where does this washer go? And this washer actually went here. So I'm going to unscrew this. So this washer goes right on here. There we go. Put this back. Is there a washer on the other side? Yes, there was. All right. I'm not sure why. Oh, yeah. I guess that makes sense that there would be a washer because whenever you're, I suppose, screwing something against a surface, you want a washer so that you're not grinding into the surface. Okay. And then so that's all the bags. And then the last step is that there's the spring that goes on this post right over here. So I'm just going to get my pick and I'm going to hook that over somehow, somehow. I guess the key is to find the loose end, which is this side. And first put the non, I guess the spring side on first and then pull on the end. This is kind of tough because it requires holding one of the ends and pulling on the other end. It also doesn't help that I've got oil all over my hands so things are slipping a little bit. Well, I'll figure out how that's done somehow. Maybe I just need to open it up a little bit. Sometimes that helps. There. It fits quite nicely. All right. So that's that part of the machine. So there was a tiny bit of a screw up and that is when I put the screws on for this cross member. In fact, this loose part, that was supposed to be screwed on as well. So that's pretty easy to fix. I'll just remove the screws from the cross member. Pull this apart a little bit. Let's see. So these go right in here. Okay. So this screws on right there. So I'm just going to take the screw and screw it in there. It didn't actually hurt that I put the screws on first because they did help to bring the two frames together. So that's all right. So now I can tighten those and tighten those. There. And you can see how those two screws went. This goes on like that. And this is like that, I guess, or possibly like this somehow. And then we'll figure that out. We'll figure out where all these little levers go. Next time on Calculator Reverse Engineer Disassemble Reassemble Clean Restore. I have to come up with some sort of a nice title for all of these videos because it's not just this calculator. It's the other calculators as well. So until next time for Video Series on Calculators, I'm Rob. Take care.


Hello, I'm David Wong and welcome to my channel. I'm starting a new series on teaching you how to play piano by ear. Today I'm going to show you my process of what I do when I hear a song that I never heard of and then start to play it on the piano without any music. So I'm gonna start with this new popular song that just came out called Promise by BTS. And I only heard this about two times. So I'm gonna show you what I do when I hear the song and then start to decipher it and convert it into piano. So first, what I do is I listen to the song to figure out the basics of what the structure of the song is and what key it is in and what time signature it is in. So let's just listen to it. So first, the simple part is what key is it in? What key is it in, okay? How do you know what key is it in? Well, we've got to figure out what chords are playing and what notes are being played. So the first note, we're just going to, if you don't know any piano, you can just randomly hit something on the piano and see if that hits the pitch. So the first pitch is, so you're singing it out. So try to find where on the piano that is, okay? So it's higher than that, right? So I gotta go higher. That's getting closer, but not quite enough. So I'm gonna go to the next one. So yeah, that actually sounds like the note. So I think that is the note that we start on. So we go, it goes with one note and goes higher by one note. So let me play that again and see if that's correct. Good, so it sounds like a match. So now we figured out that that is the note to start in. So the first two notes are, which are A, B. So does that sound like it? How about B major? But no, that doesn't sound like it. How about this one? E minor. Maybe. How about E major? That doesn't sound like it. That sounds like it. So you got G major, maybe, E minor, B minor. Those sound kind of close to it. How about C major seven? No, that doesn't sound like it. How about B seven? How about B minor seven? Duh-duh. Nope, not really. E minor seven. Duh-duh. That sounds close. How about E major seven? Duh-duh. That doesn't sound like it. So there's four chords that are possible with this first chord and the first note in this song. So let's see which one's the closest. Let's listen to it again. So let's play that again. It doesn't actually sound very close to it, so I'm gonna cross that out. So these are gonna be G major or E minor. So an easy way to figure out which one it is, is you wanna ask, does it sound happy or does it sound sad? First chord. Does it sound happier or does it sound sad? Okay, does it sound happy? Or does it sound sad? So most people would say that it sounds happier than sad, so I'm guessing that this is G major, okay? And if you actually get the music, it's gonna be G major seven, okay? Because if you listen to the guitar, they have this on top. Listen to it again. Duh-duh, right? So that's what's being played with the guitar. And now we know that G is the chord. So actually what this first chord is, is G major seven, okay? So we play a guitar and then the melody is. So you add them together, it's gonna be. So that sounds like what's happening in the first bar there. Okay, so we know now that the first chord is G major seven. So what I would do is I would go and write out a basic chart for bars like this. And then in the first bar, I'm gonna write G major seven, okay? And then I'm gonna write the notes, what are the notes? A, B, okay? I'm not gonna worry about the rhythm because I memorized the rhythm in my head. So A, B are the notes. Now we have the first chord. Let's figure out the rest of the melody. So that first part, it goes. Du-du-du, du-du-du, du-du-du. Okay, so we're gonna play it out. It's gonna be, oops. So if you don't know how to figure out, just think about what scale we're in. Because we're playing G major seventh and the scale of G major seventh have these notes. It has G, A, F sharp, G, okay? So those are the notes that you can use when we're playing the melody. So we can write that somewhere. So if it's G major seven, G major seven, the notes that we can use are basically G, A, B, C, D, and then E, and then F sharp and G. Okay, so I'm gonna use those notes and then figure out what notes are being played when the melody comes. It goes du-du-du, du-du-du-du, du-du-du. So I know it first two, and then du is lower. So I'm gonna go lower than the next one. No, that's not low enough. That's not low enough. Okay, that's low enough. And then it goes from here. It goes du-du-du-du, du-du-du. So it sounds like just the regular scale. Du-du-du-du, du-du-du, du-du-du. That's still A, B. So I'm just gonna go because that's still A, B. And it goes back to A, right? So that's how I figured it out. I wanna know what the relative position of the note is, whether it's going up or down, what the interval is, because if it's just a short interval, then it's probably close together. If it's a longer, bigger interval, that means it's farther apart. So you can just test out and see which note. Is it this one? Is it this one? Is it this one? Is it this one? Right, find the interval between the notes to find the proper one. Du-du-du-du, du-du-du, du-du-du, du-du-du. In this case, it's du-du-du-du, so I know I'm going to a D. Okay, so now that I know what the melody is. Du-du-du-du, du-du-du, du-du-du. Okay, that's the melody, so I'm gonna write it in. Now, if you're good with writing it on a staff and with all the notes on different lines, then that's good, but for me, it's easier for me to write like this. I don't need a staff paper, and I don't need to write the rhythms because it gets complicated to write. Yeah, it's complicated to write rhythms, especially for pop songs and the phrasing is very difficult. So, I'll just memorize the rhythm on my head and then I'll just write what the notes are and then I'll remember what to play when I see that. All right, so now we just figure out the first part. Let's figure out what the chords and the melody is for the second part. Okay, so you know the first one? It's du-du-du-du, du-du-du, du-du-du. And it goes du-du-du-du. You listen to the guitar, that's what's happening. It goes du-du-du-du, du-du-du. So, since I know it's going down, it goes du-du-du, and then it goes du-du-du. Then, my intuition tells me that it's actually going down also in the bass. It goes du-du-du. And that's actually exactly what it is. It just went from G major seven, du-du-du, sharp minor seven, du-du-du. And then what's the next one? Let's... So it goes down again. So actually that is a F major seven. So let's play the bass. That's the bass, and then we add the melody. So see, I'm figuring it out as it goes. So let's do it one more time. Get used to playing the chords along with the rhythm, along with the melody. So you have the phrasing correct, and you have the timing of the notes correct. Once you practice that and get good at that, the next step is to add the chords together with the bass. So what you can do is play the chords like this. So G major seven, and then F minor seven, F major seven, E minor seven, and once again, G major seven, minor seven, and then F major seven, E minor seven. So what you can do is just play the chord in the left hand while you play the melody. So we can do this. That's the basic. Now you have the chords with the melody, but for some reason it doesn't really sound that good because the chords are really modeled up in the bottom. There's too many notes on the bottom. So what we wanna do is spread out the chord. Instead of playing in clustered like that, we wanna spread it out. And usually what I like to do is play only the root, the fifth, and the third. Okay, or if you can't reach that far, just the root, the fifth, and then the root again. So if you convert it to that, it will just be like this. And then E minor seven. Okay, so you spread it out. It won't sound so modeled up in the bottom. So let's try doing that with the melody. So as I'm changing the rhythm in the bass too. So it's not just a, it's boring, right? So I'm matching the rhythm with the melody. So it'll sound something like this. Next step after that, you got some rhythm happening in the bass. It's kinda like a bass line. And then you have the melody. Now what you wanna do is fill in the melody with some chords. So now you wanna play the chords on the right hand and play the bass on the left hand. So let's practice first by just playing the bass. That's one note, so keep it simple so you can focus on the right hand, okay? So let's play the chords on the right hand. It's gonna be like something like this. G major seven. F sharp minor seven. F major seven. And then E minor seven. E minor 11. Get back to E. G major seven. F minor seven. F major seven. E minor seven. E minor 11. So that's what you're gonna play in the right hand. Left hand, you just play the single note for now. So you go one, two, three, four. Some rhythm to it. You just do the chord and then now you can add the bass line. You play some chords and now you got some bass line. Now you want to do is add the melody together with the chords and with the bass line. So it should sound something like this. Sometimes I'm playing just the chord to fill in the gaps. Sometimes I'm playing just the melody and sometimes I'm playing the melody with the chord itself. So that just takes practice and you can play by playing everything separately, right? So just start beginning by just playing the melody. So now see where you can add in the chord. And the best place to add in the chord is on the first beat of the bar. So you go on the chord there and then add in one here. So you have one chord per bar, chord there because too many notes. So I go to the next one. There's a chord there. So this one is actually an E minor 11 chord. E minor is 11, that's right. Okay, so if you practice more, then you can add more interesting rhythms and you can add more interesting lines and then add more expression in your phrasing. For example, you can go, okay. So there's phrasing, there's some notes are short, some notes are longer. There's crescendos and basically getting louder, getting softer. You want to sound like you're singing it, but with the piano, okay. That's what you want to do. The process for the rest of the song, eventually you're going to get the whole song and we'll get to that in the next video. So thanks for watching. You'll be interested in learning more about how to play piano by ear without any music and how to play any song and press your friends and have fun and to play all the music that is popular, that you want to play on the piano and subscribe, like and comment and share and ask any questions that you want on this video in the comments sections and we'll see you in the next lesson.


is June 1, and it's about 5 o'clock in the afternoon, and I'm about to start reading for the Olympic Games read-a-thon. As a reminder, I am in Apollo's cabin. I am a daughter of Apollo, and so I will be following those challenges. If you don't know what I'm talking about, I already released my TBR, and so you can go ahead and check out that, and I'll have all the details, including a link in the description to the original video for this entire read-a-thon. So you can go ahead and look at that video and then check that out. But first, I'm gonna get started with the first couple books that I will be reading. So for the beginning of this read-a-thon, the first prompt, as I said in my TBR, is follow the prompt music to my ears, listen to an audiobook, and as I said for that I will be listening to The Winner's Curse, and this is by Maria Rutotsky. I'm probably gonna say that wrong many times, and I apologize if I am saying this wrong, which I probably am. So I'm going to start this as an audio, but the thing is that while I do really enjoy listening to audiobooks, I do need some physical book to get reading, and because of that I'm also gonna get started on the second prompt of this, which is read a book that makes you happy, and for that I chose The Invisible Library by Genevieve Cogman. And so I'm gonna start reading this probably tonight. I'm gonna go ahead and get started with both of these books, this one as an audio and this one as physical to just really get a jump start on this month-long read-a-thon. So I will update you guys when I have finished one of these. Hey guys, so it is June 14, and it's about 5 30 o'clock at night, and I can finally say that I finished a book. I don't know why I do this, but I do this for every read-a-thon. I will start right away, and I'll work very slowly, and it just takes me a while. So like we're two weeks into the read-a-thon, and I just finished my first book. I'm hoping to pick up the reading in the last half of the month, but for now I am making progress. So I finished listening to The Winter's Curse by Marie Rutotsky. I hope I said that right, and this was for the very first prompt music to my ears, listen to an audiobook. So I listened to this as an audiobook. So this book is about an empire where a group of people conquered another group of people and have turned them into their slaves, and it follows the general's daughter, and she kind of goes to the market one day, and she goes to the slave market, and she sees the slave, and she for some strange reason just has to buy him. So she buys him, and she takes him to her home to be her slave, and they kind of form a connection by playing a game called Byton's Thing, but they've only played it like two or three times. Then really that's all the summary I can give without spoiling a lot of it, just because at any point after that, and I feel like I'm starting to spoil the book, but there isn't much else that's done in the beginning, the thing about this world is that her people, it kind of reminded me a lot of Sparta, because it doesn't matter if you're male or female, by the age of 20, she has to either enlist in the army or get married.  So that's what the book is about. And I gotta say, I had heard some kind of reviews about it, like people thought it could have been better, but it wasn't. And so I didn't go into it with super high hopes. And I really have to agree with a lot of people. I rated it two out of five stars, but I really, it's a higher two, like I really struggled with between two and three. Just because I felt like this book had no meat to it and had no substance, it was very surface level, simple plot, you know, it just, I wish it had gone in depth more with the characters, with the plot, with the world building, you know, it just seemed very simple straightforward. And so just kind of lacked that emotional tie because there was no depth to any of the characters or anything like that. I wish, like I said, I wish the characters were more fleshed out, but I also wish their relationships were a little more fleshed out because we have the main romance, which just kind of went from like zero fifty a hundred, you know, like there wasn't much there, but also the main character and her best friend Jess. I loved her best friend, but she did not get a lot of time in the book and they really didn't get a lot between the two of them, and I felt like Jess, the friend, was a very nice, good character that could have been fleshed out more, but then Kestrel was a bad friend, and there was like no reason for it, and so I was like, I was super bummed about that. The ending of the book wasn't actually that bad, like there's this whole event that happened right before the end, and I felt like that was pretty unnecessary, but then like the very end of it was like, okay, that's interesting. That's a little different. That keeps me intrigued, so I might read the second one, but again I'm not gonna promise anything because this is a read-a-thon, and I don't want to be adding extra books on top of the nine that I'm gonna do, but like it really didn't seem super interesting until that very ending point, and I feel another thing was that everyone was telling her she's gonna be something, or she is something. She's an advocate for the slaves, or you know, she is the general's daughter and she's gonna provide information, but then she just wasn't anything. I felt like she wasn't an advocate for the slaves, but she also wasn't helping her people at all either, so I just felt like there was so much like, oh she's caught in the middle of everything, she's gonna make a difference, and really she wasn't doing anything, she was just sitting there petting herself, so that was kind of sad, but yeah. I've been making progress on the Invisible Library, which was the prompt for the second challenge of read a book that I think will make me happy. I'm slowly working through that one, and I think it's because I think it's gonna make me happy, so I want it to last, but again that's not how readathons work, like I should be reading it and getting it done, so I can move on to the next book because I'm really excited for that, but that's just not quite how I'm doing it, and it's kind of a bummer, but I did start my other book because I'm going to listen to it as an audiobook. It's Angel Mage by Garth Nix, and so I've started listening to that, and that is for the prompt of read a book that starts with A, and I am enjoying that, so I think I'll finish that one pretty quickly, and then I can hopefully pick up the pace on the Invisible Library so that I can continue on to the rest of the books in this challenge, and especially so because I want to do the Hogsmeade challenge, or readathon, by Book Roast, because like I, as you know, I have done my owls, and I'm going to do my noots, but there is there is a Hogsmeade challenge coming up here, and like I think it's from like June 22 to June 28, and so with that we can, well I'll go more detail of that, but that is adding more books onto my TBR than what I was expecting for this month, but I'm gonna try and correlate it so this book goes towards the Hogsmeade challenge and the Olympic Games challenge, so that's really what I'm trying to do, but it's it's a lot, so yeah I don't have too much for this update. My thoughts were pretty clear and simple just like the book was. It's June 21st, and it's a little after 10 o'clock, and I am still reading the Invisible Library. I think my thing with this book is that I'm expecting to like it so much, and I am enjoying it so much. I'm almost halfway that I don't want it to end, so I'm reading it really really slowly, like I've got like two audiobooks almost done before I finish this one physical book after 21 days, so it's going super slowly, but I'm really hoping to pick it up in the next day or so, so I can start doing what I always do in these read-a-thons and reading like a book a day and just barely making it. So honestly, I don't have much of an update. I'm really enjoying this. I'm almost done with Angel Mage by Garth Nix, which is the book to cover the prompt of a book that starts with A, and I'm listening to that as an audiobook, so that's the only way I'm getting these down apparently. So yeah, that's really all the update I have. Angel Mage is good, like I really like the concept in the world-building, but it's not always the easiest to understand, like you really have to be super invested in the story to pick up all the details and everything that happens, and I haven't always been that, so it's been a little confusing for me because of that. Yeah, hopefully I'll definitely start picking up the pace soon, slash maybe tonight, maybe tomorrow. But yeah, I am gonna start the Hogsmeans read-a-thon. That'll probably be a separate video just because my computer has been really picky with its amount of content that I've been uploading to it, so that'll probably be a separate video, but it's gonna tie into this one quite a bit. So I'm gonna finally sit down and do some reading and hopefully not have it be where I have to read a book a day, because some of these books are really long, so we'll see. I'm gonna say goodbye for now and I'll update you when I finished a book or possibly more. It is Wednesday the 24th and I'm sitting here on my couch. I just got a bunch of reading done. I actually finished a book and I got up to get my next book and then this flufferdoodle right here decided to stay on my spot, but now I'm back. I finally finished The Invisible Library by Genevieve Cogman, and this fulfilled my challenge of reading a book that I thought would make me happy and it did. I think it didn't make me as happy as I was hoping, but I still liked it in the end and I think that was because I chose this book because it was about books and the library and going out on missions to get books for the library and I love those kinds of books. I love that kind of reading, so I thought that it would kind of fulfill that joy, but I found the majority middle part of it to be more about the murder rather than like finding the book, you know? So like it was good and it was fulfilling, but it wasn't quite what I was hoping for, but it was still really good. I rated it 4 out of 5 stars. I liked the characters. I liked how the characters really stayed true to themselves except for Kai. He was kind of back and forth a little bit and I was kind of confused on his character, but I dealt with that and it was okay. Yeah, so this book was about this woman named Irene who works for the library, the Invisible Library, and basically she goes out and retrieves books from different alternate worlds to help keep balance between chaos and order within all these different alternate worlds. It's a very intricate world building in terms of how it's all set up, but the world that they were in for like most of the book, I just had a really hard time picturing because it was like how it described what the women had to wear, but then like it had advanced technology, but it still seemed very Victorian and it was just kind of confusing and I had a hard time like getting one image of London that they were supposed to be in kind of set throughout the book, but I still enjoyed it. The plot was interesting. Yeah, it took me a very long time to read it and I think it's because it did dip in the middle for me a bit and I was really hoping that that wasn't the case. I was like no if I just if I go through it slowly or I don't pressure myself too much then I'll get it done but I'll like it more and it'll be at a better pace for me and really it just took me forever to read it. So right now it is just about 10 o'clock and it's almost dark outside. Like it wasn't dark outside like 10-15 minutes ago like the sun is kind of setting so I'm gonna start the tiger curse. This is the one to fulfill the prompt of my prediction to be a five star and as I said before I don't know if it's necessarily gonna be a five star but I do think it is going to be the highest rated book, the book I will most like of the ones that I chose for the TBR for this month. So I will start that but if it does get darker out and I'm still up I might switch to kill the queen because this is what I'm reading for the prompt to honor Artemis and that was to read when it's dark outside. We'll see how late I end up staying and how much of each I read. I have finished, I finished three books so I finished three of the five prompts so these are my last two books for the month if I'm just doing the basics for Apollo that I need to do but I would really like to do some of the extra activities so I'm hoping to get these two books finished before the end of the month so I can have some time to try and fulfill some of the other activities so that's my update for now and I will catch up with you guys once I am I have a better game plan or I'm farther into a book and I have something to say. Also I totally forgot to like mention that I also finished Angel Mage by Garth Nix today. I finished that audiobook way earlier today and I was like yes I got one book done today maybe I can do another and then yeah I did another so that was also a good one I believe I gave that three out of five stars no I gave it four out of five stars because I really enjoyed the world building the ending seemed kind of rushed and it was very quick and it seems like such a simple six that it took like one slightly creative mind to come up with when like this person's causing all this destruction and like fear in the hearts of all these people and it's like all you needed to do is this but other than that I really loved the characters each of the four main characters had such strong distinct personalities from each other and yet they work together so well some of the transitions was in the book were a little foggy so I kind of had a hard time like following along with those and kind of keeping up plus I wish there was a little more description of like the world itself like I had a harder time picturing places in my head but otherwise I thought it was like a really enjoyable book I just love the concept so much so I finished listening to Angel Mage which was fulfilling the prompt of reading a book that starts with a so I have finished the winners curse I finished the second book the invisible or then I finished Angel Mage which accomplished the third prompt that was also accomplished by format of audiobook but then I went back and I finished the second prompt and that was the invisible library so I've completed the first three prompts now so there we go all right now I'm gonna go back to reading I feel better though it's 2 30 in the morning of Sunday June 28 I have not gone to bed because I finished Kill the Queen by Jennifer Estep I read the first chapter two nights ago and then last night I stayed up until three o'clock reading the first half of the book and then tonight I finished up or I finished the book and it's 2 30 I started reading at like 11 so there I've gotten four out of my five required readings for Team Apollo done why do I do this to myself why do I torture myself like this I don't know but I'm a good maybe a third almost a third of the way through Tiger's Curse by Cullen Hoke so I will definitely get this done tomorrow even if I have to stay up late like this again I will get it done so that I will at least complete all the challenges for my cabin and all of the requirements for the readathon I will be able I will be able to do some of the extras I don't know if I'm gonna be able to do them all because there are only a couple days left in this readathon so I don't know if I can read four and a half bucks one of them will be an audiobook one of them is 72 pages and is mostly pictures so I feel pretty good I'll get those done but then the other two are both like 400 plus pages like it's the toll and before she ignites which are both pretty big books so I would not count on that but I again I can't start those until I finish the Tiger Curse so there's my late night update for you guys I will talk more about what I thought about Kill the Queen either in the morning or just flat out at the end of the video when I kind of do a generic overall this is what I thought of this book and this book and this book because I want to go to bed now so goodnight hey everyone it is July 2 and that means that the Olympic Games readathon is officially over and it has been since the last day of June and I am now going to film this wrap-up so yeah I finished more books than the last update you got I just never really had time to film or I was too tired because I would either finish them late at night or I would finish them and then I'd move on to something else in my life so I'm just gonna go ahead and do a full kind of wrap-up more the overview on some of the other books that I read that I haven't really filmed about so let's get started with the first book that I read and finished was the Winner's Curse and I listened to this from the audiobook prompt and I'm actually almost done listening to the sequel not because I like it because I feel bad about starting series and not finishing them so I did finish the Winner's Curse and then the second book that I finished was actually Angel Mage and this one I also listened to as an audiobook it's by Garth Nix but I read this one to fulfill the prompt of read a book that starts with a so then I finished the book for the second prompt the Invisible Library by Genevieve Cogman I read this for the prompt of read a book that will make you happy I enjoyed this book I didn't enjoy it quite as much as I was hoping I just feel like there was a lot more mystery of the murder rather than like really pursuing the book or getting to know the library or the magic behind all of it so I was a little disappointed but still made me very happy because it was it was still a good book like it was still a good book it just took me forever to get into it I think that was because I was afraid that either wasn't going to be as good as I was hyping it up to myself to be or because I didn't want it to end because it's a book about books and those make me happy so that's why I chose it for the prompt then the fourth book for the fourth prompt I read was Tiger's Curse by Colleen Hoke I'm very sorry for bad pronunciation I read that because for the prompt of read a book you think will be the highest rated that well a five star rating or the highest rating that you give I didn't think it would be a five star read I but I read it because I thought it was going to be the highest book that I rated off of the entire TBR and actually I rated this three out of five stars which I was kind of bummed about because for the first half of the book it was great there was a very few issues about it like I felt the main character Kelsey she gets a temporary job working at a circus where she meets this tiger and she kind of grows attached to the tiger because it's a white tiger and tigers are cool I can't blame her then this man approaches her and he is like I want to take you to India and you're gonna help me break the curse of this tiger and she agrees to go along and so it's their adventure of trying to break the curse and it is a series so it was not completed at the end of this book but I felt like Kelsey turned 18 at the beginning of the book but I still feel like she acted like a 14 year old like she acted well below what her age was which you don't always see in books like I always feel like characters act above their age because I have to but for her I just felt like she acted more childish than young adult-ish I guess and then there was this one line that got me a little bit like at the beginning of the book and that's so the tiger's curses this man was cursed to be a tiger but he has 24 minutes of the day with Kelsey to be human but then throughout the second half of the book is when it started to decline for me personally because I felt like people were causing drama for the sake of drama not because of actual things and they started to make relationships and actions more complicated than they needed to be just for the sake of being complicated and a lot of that also stemmed from a lack of communication which if you guys know I hate it when conflict is stemmed purely for lack of communication so when a lot of the conflict that doesn't come from the main plot of the book is from a lack of communication or doing dramatic things just to be dramatic and keep writing it started to go down for me and it started to become less interesting and more annoying to read like as the book went on I started to dislike Kelsey as like again for the first half maybe two-thirds of the book I thought she was a great character you know good protagonist yeah I can get behind her and I can get behind her and read but then she just started to decline and it really just kind of made me be like oh no I don't know if I want to keep reading about Kelsey so that that's why I rated it three out of five stars because I thought like the first half of it was great but then the second half it just started kind of going downhill for me personally moving up so then the last challenge that I fulfilled I did manage to film a little bit of previously and it was the last of the five required challenges for my cabin and that was to read a book honoring Artemis and I chose to do Artemis's prompt of reading a book only at night or when it's dark outside and so for that I read Kill the Queen by Jennifer Estep and I enjoyed it I had a few like a major problems with it but overall I thought it was just fine good read I would be intrigued to continue reading the trilogy there are two more books after it that I know of that I would I would be okay with reading if I had the time to devote to that which I I could but we'll see and then so I finished all of the five challenges that I had to but then I had four prompts to choose from for some bonus challenges for this read-a-thon and I was able to get two out of the four of those done I was hoping to do all four but in the back of my head I knew I wasn't gonna get all four so I was hoping to do at least three I only did two one was Archery and that was read a book under the 200 pages and for that I read the Tea Dragon Society and I loved it so much that I made my mom read it immediately after and then I made my dad read it immediately after it's just a cute story it's by Katie O'Neill it's just such a cute story about you know these dragons and you like harvest leaves off their horns to make tea but like you gotta care for them and it was just adorable that's all I can say and the artwork was beautiful and amazing too and then the last book that I finished I literally just sat there and listened to the audiobook all the way through until like 3 in the morning it was just sitting in my bed with the lights off listening to the audiobook being like I want to finish I want to get this done and that was the Queen of Nothing and this was for the climbing wall I was to read the next book in a series so this is the third book in the trilogy by Holly Black the air folk trilogy and I as soon as this came out I had read both of the previous books and I was so excited to read this and then I just never bought it so I never read it no like a day or so ago like the last night of the read-a-thon I just sat there and listened to this audiobook and I gotta say it was a good ending to the trilogy I liked the ending it was a very satisfying ending I feel like it was just kind of a of kind of a lighter book you know not quite super complicated plot or not as much trickery as in the previous books but I mean it's also just a shorter book and it was kind of the ending so I really didn't mind it I I kind of want to read all three books again one right after another because there was so much time and space between the second one and the third one I really did enjoy it and I I would highly recommend the trilogy because all three books in the trilogy were amazing and especially like the last one like for it being short and kind of a lighter read I thought it was great I'm debating on making a video just reviewing each of the three books in the trilogy and then the trilogy overall so if you're interested in seeing that more specific to the air folk trilogy for Holly Black then leave a comment down below saying yes I would love to watch like a more in-depth review of the trilogy as a whole or of each individual book because that's something I've been more interested in doing lately and there were two books like I said that I did not get to and one was before she ignites I did start reading this I'm literally a chapter into it so I barely started reading it so we'll see and then the other one which was one I really didn't expect to read but it would have been nice if I could have but I wasn't going to was a toll by Neil Schusterman because this thing is thick it is big it is huge and for a read-a-thon and especially with how little time I left myself after I finished the first five books I knew it wasn't gonna happen but I'm just so excited to finish this trilogy so this would be honestly another trilogy that I've been sitting here like okay I want to film a video of the trilogy as a whole in each independent book just like doing a more in-depth review of because I really enjoyed this trilogy and I really want to talk about it more so I'm bummed that I didn't get to it but hopefully those will be two books I get to soon but as June is over my fantasy reading is over that is the end of this video please comment click subscribe like the video and do all that jazz if you want to keep following me during my year-long journey of exploring different genres and reading different genres or if you're as excited for the reading rush coming up as I am because I am so excited for that and with that I will leave you with a happy reading


The LEA programme is a programme for women leaders and male allies and the purpose is to help our women leaders develop more confidence and credibility as leaders within the business and also to open our eyes to our male leaders to see some of the challenges that our female leaders experience. LEA is a programme that takes a look at the importance of women in leadership and takes an intentional focus on how to bring movement along on this journey. It shows that the organization is willing to invest in its leaders, its future leaders. It's willing to train them. It's willing to provide them with networking opportunities. It's willing to provide them with opportunities to be empowered, to learn about themselves and to grow. The other thing that has been really great for me is the external coaching. Having someone outside the company who doesn't know me or what I do or what we do beyond just what they've been told to be able to sort of look at me through different lens and say, why would you think that way or to challenge why I would have a certain perception or a certain assumption about something and to challenge me to try to do something different which has been great. Men have a spirit of influence where they can affect change and really influence the ways in which we all work together. And so this is not just a women's issue. This is an everybody issue. And men should also be very proactive about engaging women in leadership because at least a better outcome is for all of us. Let me look back and reflect on my life and even this past weekend that like here's the things that I've done, here's things I could learn from it. And so I think just having the ability to focus on these issues and hear somebody's perspectives and viewpoints and give some of these tools has been an incredible experience. It's the fact that I work in a company where the leaders are investing in women to make sure that women get the opportunity to be seen, to be heard, to be developed. It's really important that we're our own best advocates and our own best allies, advocating for yourself, your value, your work and your goals. It's going to be very important to advancing into leadership as a woman. It doesn't stop here. It's got to keep going, got to take that self-reflection, the learning from the program and think about what do we do with that now? How do we take that next step and what's the opportunity for our next series of women leaders that we've got coming through as well?


Hey folks, welcome again. So in this session, we are going to take up one theorem related to certs and understand very critical and important theorem says Square root of a rational number. So when you find out a square root of a rational number It would either be a purely rational number or it would be purely irrationals You will never get square root of a rational number M food M that is where M is what I'm saying M is a rational number M is a Rational Number you will never get root M as a plus root B Okay, we'll never be equal to a plus root B where a is rational and This is irrational root B irrational you will never get Yeah, yeah, the square root of M as a plus root B square root of M will be let us say square root of N Or square root of M will either not be possible that is square root of M will be simply square root of M Or if M is a perfect square, then you will get n is equal to n such that n square is equal to M Okay, so if you see Either you will not have the square root only or if you have you will have as a Rational number so hence root M will never be equal to let us say a plus root B form It will never happen Right, it will never happen. So how to prove this? If possible, let us say So we'll use the process of let's say method of contradiction Contradiction so I am saying let us say root M can be expressed as a plus root B. So you are getting a Let us say this is possible root M is equal to a plus root B where a belongs to Rational number set that is a is a rational number and root B is a is an irrational number root B is an irrational Number, let us say it is possible for for for the time being so squaring both sides squaring Squaring both sides Both sides. What will you get? You will get M root M square is equal to a plus root B whole square that is M is equal to a Plus root B square plus to a root B That means M is equal to a Sorry a square. It will be a square plus B plus to a root B That means what does it mean? M minus a square minus B is Equal to 2 a root B and hence root B, which is an irrational number can be equated to M minus a square minus B upon 2 a Now if you closely look at this equation here, this is a this is a irrational number irrational Irrational number and it is on the right hand side.  So how can an irrational number be equal to a rational number? So hence there is a contradiction contradiction you You arrive at a conclusion that an irrational is equal to rational and that is happening because You assumed this to be true. This was you you assumed that root M can be expressed as Root M can be expressed as a plus root B. So hence clearly root M cannot be expressed as a plus root B. So hence hence root M will either be Root of M will be root M itself. That means it is not Possible to find the square root or if M is a perfect square M is a perfect square That means root of M is equal to n such that n square is equal to m where n is a Rational Number right example root of three not possible. So root three is an purely irrational purely Irrational root of four is two.  So irrational Root of six is irrational Root of seven also is irrational Root of eight can be written as two root two, which is again irrational and Root of nine if you see is three, which is purely rational So hence either it will be purely rational or it will be purely Irrational so square root of a rational number will be either a purely rational or irrational irrational It cannot be a mix of rational and irrational


We're beginning with expanding maps on the circle. Let's begin with something. I thought that this was something of us mathematicians. We care too much about classification. But in fact, this is a thing of scientists. We all care about classification. We classify insects. We classify even mental illness. Is that, can you read this? Yes? OK, so in general, we classify things. And we aim at classifying things. And in order to classify things, we need a criterion. So in math, you will know. In algebra, for instance, you have isomorphic. If you have a differentiable structure, you have diffeomorphisms. So the criterion focuses on what is important for us. Well, in dynamics, we have a criterion. We focus on the dynamics. So we will say that two objects belong to the same class if they are what is called conjugate. I think that you all know what a conjugate is.  Let me just remind you what it is. So we have two dynamical systems, f and g. And we will say that f and g are conjugate if there exists a homeomorphism such that it makes this diagram commute. So h composed with g equals f composed with h. So this produces classes of dynamics. So we aim at classifying expanding maps on the circle. This is what we are going to do. We are going to classify expanding maps on the circle. But sometimes we don't have all the information. We cannot tell whether two dynamical systems are conjugate, but we can say that some dynamics is contained into another dynamics. In that case, we're going to speak about this is conjugacy. Sometimes we will have this, the same picture. But instead of h being a homeomorphism, it will be only a continuous surjective map. In that case, we are going to say that g is semi-contrugate. Also over here should have set y to x. That makes the diagram commute. It is never clear to me what does this mean. Who is the one who has more information? The one who is up has more information. So we are going to say that f is a factor of g. I presented both definitions. I will tell you why. Here, as Corina has said, important objects of study are the orbits. So if there is a conjugacy, the conjugacy will take orbits into orbits, periodic point into periodic point. Orbits into dense orbits. And the other way around. But in this case, you will take periodic orbits into periodic orbits, dense orbits into dense orbits, but not necessarily the other way around. You could have a dense orbit that does not lift into a dense orbit. Can you think of an example? I will put lots of exercises. You just pick a couple of them. But I will put lots of exercises. So one exercise. You can do it with very simple dynamics. Not necessarily the circle. We can think of even simpler dynamics. So think of periodic point p for f such that it's not periodic. So it's a pre-image is not a single periodic point. And does not contain a periodic point. Can you think of a periodic point p for f such that pre-image does not contain also? Can you think of a dense orbit for f such that its pre-image does not contain a dense orbit for g? You can do lots of questions. Even at this point, many of you are familiar to these notions. And there's some question I don't know the answer. Perhaps it's known. I don't know. You can think of the semi-contrugacy as an order relation. Because you have that, well, it is transitive. If you have this is a factor of this, this is a factor of this, then this is a factor of this. But and of course, every map is a factor of itself via the identity. But what about the symmetry? Is it true that if f is a factor of g and g is a factor of f, is it true that f and g are conjugate? Are conjugate? It's a question. Anyways, we have this conjugacy and semi-contrugacy are will be useful to classify maps. And so let's begin with Corina has begun with the description of maps on the circle and on the interval. So forgive me if you are already very familiar with this, but I want to introduce the lift of a map in the circle. You remember that the circle, as Corina has said, you can see the circle as this quotient between the rears and the integers. And you have this projection. And this projection can be what I put it like this. But you can look it as this, or x mod 1. This can be seen as a semi-contrugacy between a map that commutes with this and the map in the circle. For instance, Corina has presented three maps. One was let's put f1. The notation will be the same. But you can define this in the circle or in the rears. And let's call it f tilde when it's in the rears and f when it's down. And then you will have that this commutes. This is a factor of this. It's a non-interesting factor. But let's call it like this. And in the same way, she has defined the expanding map, which was 2x mod 1. And then, sorry, 2x mod 1. And then you can lift it to the rears like this. And then you will have this 2. And the same with the rotation. The issue is that you might be already familiar with this result, but let me state it anyways. When you have a continuous map on the circle, you can always lift it to the rears as a continuous map and in a unique way. So if you have a continuous map on the circle, then there's a unique map in the rears, which is continuous and unique modular integer translations. I will tell you what it is in a minute. But such that they are semi-contrugated. Such that f is a factor of this continuous map. Small f is a factor of big f. And f is unique up to integer translation. I will tell you in a minute what this is. And this is called a lift of f. What does it mean that f is unique up to an integer translation? Let's keep the notation here. So that means that if we have two lifts of the same map, then their difference will be an integer number. Will be a constant integer number. This allows us to define the degree of a map. Roughly speaking, the degree of a map is the number of branches. And the map has, when you draw it, as a map of the interval. But we can define it. If we have a lift, the degree of the map will be that integer. I don't know how to, here. This is an integer. Can you tell me why this is an integer? You know why? x plus 1 minus f of x belongs to the integers. Do you know why? Yes, both of them are lifts. Because if you have this as a lift, then immediately, this will be a lift, OK? You can check it yourselves. f of x plus 1 will be also a lift. Will be a continuous map. It will commute with this. We'll make this diagram commute. And so since the lifts are unique up to an integer translation, this will also belong to the integers, OK? And this is called the degree of f. And roughly speaking, it's the number of branches that small f has when you draw it in the interval. So for instance, what is the degree take? And this is independent of the lift. If you take another lift, it will have the same number. And if f is a homeomorphism, then the degree will be 1 in modulus. So what is the degree of this map? Can you tell me? You can see it as a map of s1. You can define this as s1, as a map of s1. 0, isn't it? Because it's 0 in 0 and 0 in 1. So the degree of this is 0. And what's the degree of this one? Can you see why? Well, you have a lift of fx2x mod 1. Well, a natural lift is just 2x, OK? This is a natural lift. And so if you calculate this, this will give you 2. And what's the degree of this map? Well, you have it. If it's a homeomorphism, it's in modulus, it will be 1. But this will be 1 because you take out this mod 1 and you will get a lift, OK? So the number of, and as Corina has drawn, the degree is the number of branches, OK? This has two branches, has degree 2. This looks like if it had two branches, but it has only one because the branches are horizontal in some sense, OK? So this has only one branch. And this has zero branches. This is hard to do. It makes more sense when it is expanding. So the number of branches is the degree, OK? So we will classify expanding maps according to their degree. We will see that all expanding maps of the same degree are all conjugate. This is what we are going to see. This I'm not going to, if you're interested, it will be there. I will not enter into that. So let's go into linear expanding maps. These are what are the rigid models. The linear expanding maps are the rigid models. So first of all, what's an expanding map? What's an expanding map? We are going to say that this is an expanding map. If it is continuous, plus C1, and the derivative is greater than 1 for all x. And since s is compact, we will have, in fact, that there will exist some constant doing this, OK? You will have a constant and uniform bound, because it is C1. OK, so I will go first into two examples of linear expanding maps. You are very familiar with the two of them, but one was suggested to me by Stefano and I find it a great idea. So the first linear expanding map we're going to see, we have seen it already, is this one. 2x mod 1, the one with two branches. But probably it is a good idea what Stefano suggested, because you're more familiar with that. It's this one, each 10 of x, which is 10x mod 1, which is a degree of this, OK? It has 10 branches. Well, it will be hard for me to draw them. No, that's OK. That's OK. Sorry. 1, 2, 3, 4, 5. OK, so here I will not draw them all, but you will have branches like this. There will be 10 branches like this. It will expand by 10. OK, so let me show you an example of how this works. This is the dynamics of a point. Have you ever seen this? Have you ever seen the dynamics of a point with the green line y equal x, like this? Yes, OK. So you have this, the line y equal x, and the two branches here. And then you will have the dynamics of a point. We'll go here, here, here, and so on. OK, so in order to classify these dynamics, we will do what is called symbolic dynamics. In the case of 2, and I will do first in the case of 2x mod 1, but then I will do it in the case of 10x mod 1, because you will find that you are very much more familiar with this than with this. How do we do this? Oh, sorry. I will do this later, but let me first tell you how we are going to do symbolic dynamics. If the point is in the first half, we will assign them 0. If the point is in the second half, we will assign them 1. Then you iterate once. And if it is still in the 0 part, we will assign them 0. And if it has changed, we will put 0, 1. And then the trajectory of the point will give us a sequence of 0's and 1's. But this is not hard. However, you will be more familiar with this in the 10 expression. Let's get symbols for this sequence. So this symbol, this sequence, so in the first, we have to put them inside one of these 10 boxes. So it will be not here, not here, here between 3 and 4. So the first symbol will be 3. This is a symbol 0, symbol 1, symbol 2, symbol 3, 4, 5, 6, 7, 8, 9. Then we apply the map, then we will have pi. And then we do it mode 1. So the second symbol will be which one? 1. The second symbol will be 1. We multiply by 10. We do it mode 1. And the third symbol, 4. And then, in fact, the symbols will be just its expansion. So it will be very simple for us. The symbolic sequence of this orbit will be just its decimal expansion. And in the case of this, it will be its binary expansion. So we will go get back into this later. Let me first just see a relationship between the periodic points and the degree. So we know what periodic points are. And one thing we want to know is the number of periodic points some linear expanding map has. And in fact, we are going to calculate the number of periodic points each expanding map has, not necessarily linear, whatever expanding map has. This will come as an easy consequence once we have classified all expanding maps. But we can do it in advance. We can calculate the number of periodic points each expanding map has only knowing their degree. So let's call this the number of fixed points of fn pn of f. We are going to calculate this pn of f in terms of the degree. The linear map for linear expanding map 2x mode 1, the number of fixed points of e2 to the n is 2 to the n minus 1. The minus 1 comes because we are identifying this and this, the 0 and the 1. So we lose 1. As a consequence, it's not as a consequence. They are what Corina told equidistributed. So since they are also equidistributed, they are going to be dense. So I leave this as an exercise. This calculating exactly for the linear expanding map is not hard. You can calculate it. This can be a hint, but if you find any other way to calculate them, I will be happy with it. So let's do it with a general map. Well, first with other linear expanding maps. But then we are going to do with any expanding map. So what happens when we have like this 10x mode 1? What happens in this case? Can you tell the number of periodic points in terms of the degree n to the n minus 1? OK, let's see when this will absolute value. In fact, no. I'm sorry. It depends on the absolute value of m on the sign of m. And the periodic points of em are dense. In fact, this will happen with any expanding map, not only for the linear expanding maps, but this will be a consequence of our classification. So we are going to show that these are dense. And we are going to show that this is the number. And in fact, it's not going to be hard. But the main part of the proof, you are going to make it, because it's an exercise. Let me tell you how we are going to do it. Well, I have already told you what an expanding map is. It's there, and we already know what the degree is. This is the main part of the proof. If we have two maps, f and g, on the circle, the degree of the composition is the product of the degrees. The degree of the composition is the product of the degrees. And this is going to be an exercise. In particular, the degree of the g composed with f is equal to the degree of f composed with g. And also, we have this for the iterations. Degree of f to the n is the degree of f to the n. And this, which is the key part of the proof of this formula, is going to be an exercise. OK, so let's prove why we have this. I don't know if I will have enough time, so I will give you some hints. So if it is an expanding map, the degree is greater than 1. Can you tell me why this is? Not necessarily linear. For any expanding map, the degree is greater than 1. One hint is that if f is a lift for an expanding map, we also have that f prime of x is greater than 1, because the derivative is something local. And the lift is locally the same as the small f. Pardon? Yes. No, the degree is general. The degree is global. Yes, we can calculate it at any point. It will be constant. So the degree will be x at any point. Yes? No, it's strictly greater than 1. Yes? Yes. And it will be continuous. So it will be all the time about 1 or all the time below minus 1. And there you go. Then you will have that this. Either this is greater than 1 or less than minus 1. So the degree will be in absolute value greater than 1. And so why is this? Now remember what we have said, that the degree of fn, we have this. We have just said this. So let me. This we have already done it. But let's see that we only need to prove this. Why is that? This is the number of fixed points. So this is, in fact, p1 of fn. If we prove this, we will have that p1 of fn will be, if this is true, we will have this. This implies this. But this is equal to this. OK? So we only need to prove this for p1. We only need to prove this for p1. So let me see if in this 10 minutes I can do this. So is it clear why is this? If we prove this for any f, in particular, we will have this. But this is equal to this. So just to prove it for fixed points is enough. So we have to count the number of fixed points in terms of the degree. We begin by taking a lift of f, call it big f. Let's consider g. We need to count in some sense the zeros of fx minus x. Or what is the same, the number of times this cuts z. OK? Between 0 and 1. And so we have to count the number of zeros or integers of gz, gx. OK? And but this equals the degree of f minus 1. You know why?  OK? So this is the degree of f. This goes with this. And this is minus 1. OK? So this implies that there exists at least amount of points such that g belongs to z. Can you see this? What happens if the first one and the last one are the same? You still count it. You will not have that because the degree is greater than 1. But you will not have it because the degree is greater than 1. So you will have the map is expanding. So yes, all this. But this you are counting twice. That's why you subtract 1. You are counting this twice. You have to subtract 1 because you are counting the end points twice. When you have this, well, you will have this gx plus 1 and gx. And you count all the times this cuts. How many integers are there in between? And then you will have degree of f minus 1. Because this is this. It's because of continuity of g. It's at least, at least, in fact. Because you will have, by the Bolsonaro, you will have at least degree of f minus 1 points that belong to z. At least. But on the other hand, we have that gx is greater than 0 in absolute value. OK? Because gx is fx minus x. And this f prime is greater than 1 in absolute value. So this will be different from 0. So it will be either increasing or decreasing. So we have that this cuts at least degree of f minus 1 points in absolute value. But if it is because of Bolsonaro, but if it is strictly increasing or decreasing, it will cut each of them exactly once. And so the number will be this, the number of fixed points of f, which is the same as the number of zeros of this, will be exactly degree of f minus 1. The number of times these cuts and integer coordinate will be exactly the number of zeros of g, which is the number of fixed points of f. And this is exactly proofs this. So we have, we counted, this is not for linear, it's not for linear expanding map. This is for any expanding map. The only thing that we have used is the derivative of f. We have used that the derivative of f is in absolute value greater than 1. So all expanding maps have degree f minus 1 fixed points. And the number of periodic points are degree of f to the n minus 1 in absolute value. So we could count the number of periodic points just by using the definition of degree. We will later see that, in fact, we have another path of calculating this because it is very simple to calculate the periodic points of a linear expanding map. And then we will see that all of them are conjugates. So they will have the same number of periodic points. But this is a direct proof of this. OK, so to finish, I have three minutes. But I will define, you already know this. So let me just, it's just a remark. We will say that f is topologically mixing. If for any two open sets, we have this, we have this. And in particular, there's another exercise. If f is a homeomorphism on the circle, then f is not topologically mixing, is it? The definition is up here. It's for the future, OK? So from one end on, you already know this definition, don't you? And you have, OK, rotations are not topologically mixing. Isometries are not topologically mixing in the circle. But let's put it some more difficulty. Homeomorphisms on the circle are not topologically mixing. You can get a flavor with the rotations. And so expanding maps on the circle are topologically mixing. This is not hard. I have it in the following slide, but I will end it here because I want you to do it as an exercise. Expanding map on the circle, are topologically mixing. OK, and then I finish here. Thank you.


Olga box clips. We are moderate buy side. So if you're taking trades right now or now buy side only For the best place By side finger over the buy. Let's see where we're at high time frame We've got space right here. Okay. We got space up inside. However, this could just continue down We might be right there. You can kind of aim and see so we don't have a good gauge on that That's not hyper extended off of the golden ropes, but if we do go up We've got the space. Is that an ignition bar? No, not an ignition bar, right? We gotta wait for the third bar There's one there's two but this one would have to be solid top to bottom. So not an ignition bar either We don't have anything off the 89. Let's look at the next one. What do we have? We've got the first set of alpha omega right here in front of the ropes So we could be turning back down. We've got lots of space down below there have to be careful of that But we want to do that between 940 and 950. We still got three minutes till then Prior to that, we've got a little bit of some low speed here Let me add that one additional indicator item that is hopefully not calling us issues here But there we go. We've got a screen refresh now that is hyper active and should help us on our entries Okay, waiting on an enigma. We are now sell side. We have shifted. You see the shift already shifted over So sell side finger over the cell. Do I have my tiny ATM? Yes I got my tiny ATM right now to start the morning Press the like button, share, subscribe and comment below with your thoughts and what you'd learn


Because she finished up being someone else's version of you and that then becomes, trust me, the biggest problem your life will ever have is not being who you were intended to be. Because once you get that problem everything from then is out of whack. And I spent a lot of my life battling to say no, no, no to people who believed their version of my life was better than my version of my life.