We're good Okay, let me get organized here real quick. I do want to mention a couple of things about Some of the things I saw in the proofs that I would like you to modify Okay All I'm gonna say here is Okay, just a couple of things overall You guys did pretty well okay So I do expect over the course of the semester that you're gonna start to progress a little bit in your proof writing skills Here's one thing I noticed and this this is just this comes with time Let me Preface by saying that when I if I write something on your paper if I underline something and say meaningless or if I underline something And say nonsense.  I'm telling you this because you need to know it because I want you to get better Okay, I want to be very clear about that. I'm not being me Well, maybe you think I am I'm not trying to be mean Okay, it's just because I'm just trying to help you and that's why I spent a lot of time on the grading a lot of you'll see I have a written a lot of comments on your stuff I do that because I want you to get better this course for math majors a lot of you are math minors But for math majors, this is also another very fundamental course You need to get the proof writing down some of you are gonna be high school teachers And maybe a few of you are thinking God, this class sucks. I don't you know, I don't like writing proofs I just have to teach a high school You and you have trust me on this Part of being a good teacher Part of what goes into being a good teacher is having a good thorough understanding of mathematics It's absolutely true. Do you have a student that says why can't I divide by zero? Why can't why can't I just because you can't? That's not an and that's not a very good answer, you know or saying, you know, why does why does the vertical line not have any slope? Why is that true? Again, when you have a better understanding of why things work the way they do you can answer these things and you can you become a Better teacher. There's no question about that. So I'm trying to improve your your skills I'm trying to fill excuse me facilitate understanding of this stuff because it is important. It really is first thing I want to say and I noticed this on a lot of a Lot of your your papers. Here's one thing that I want you to try to do right Complete sentences Okay, this is something that a lot of you aren't used to in mathematics and I understand that And you several of you have seen on your paper that I've written this comment You're writing a proof. You're writing an argument, okay? It's not just you know, it's not the same thing is taking the derivative of tangent of e to the x We just write it down. You're trying to convince me of that. You know why something is true Okay, so what I expect you to do is Imagine that I have no idea why it's true imagine. I don't understand math very well Okay, and try to convince me that that is true Okay, don't assume that I know what how it works. I do but don't assume that That's not the point. Okay, you have to show me that you understand it and part of being clear is Having good exposition is writing things out completely. I posted solutions online, which I will if I can in a second I'll show you kind of what I'm looking for This I've seen I've seen a lot. This drives professors nuts Okay, and I'm gonna try to be very clear about this Don't in general don't use The conclude this is gonna look really stupid, but I'm just I'm just gonna write it this way because I want to Don't use the conclusion blah Equals blah to justify no, no, I'm not saying it's not true But I would you're making an argument and I'm gonna show you I have a point here. I want to make to justify Just because something's true doesn't mean it belongs in an argument right the cow is an animal that doesn't mean you should write in your proof right No, you guys you know that just because something's true doesn't mean that it belongs that it's relevant I've been writing proofs for a long time. I honest to God. I know how to do this stuff and I'm saying it for a reason, okay? Let me give you an example a lot of you were doing this You want to prove something so you just in your proof you just write it as if it's true And then you conclude for example k plus one equals k plus one and then you think that that justifies what you assumed and it may But it's not it's not that's not the best way to write the proof.  We can certainly do that right one squared equals minus one squared So one equals one that's true. Therefore the original assumption is true. No, it's not Okay, what this this is relevant to this course if you think well, but I didn't square both sides Well, how do you know that what you did is legal? Okay, just so the point is circumvent the whole issue by not doing it Okay, that's what I mean by streamlining your proofs and writing them in a Logically beautiful way.  Don't do that Okay, as this shows just because you end up with blah equals blah that does not justify your initial assumption counter example right here Okay, you guys with me on this Sorry, I'm getting excited again. I'm gonna start I want to start throwing stuff. I'm gonna knock my mic off and start breaking things Okay But overall You guys did pretty pretty well I was actually pretty happy with with your performance on this Okay, and so again take the the the know the meaningless the nonsense these comments just take them as face value Don't read some subtext text to these comments like you suck. You're terrible. I'm not saying that Okay, I'm really not that's not true No, I That's not to be inferred from the comments is what I mean Okay, so let me let me go ahead and move move on past this here Ah Geez, I don't know if I'm gonna have time to do this Solutions to the graded problems the proofs are all posted online on the course web page I've really strongly encouraged you to go online and look at this and you see how I'm writing the proofs Complete sentences. I'm not assuming what I want to prove in the beginning I only assume say the inductive hypothesis and I conclude That the statement's true friend plus one these kinds of things look at the proofs you if you read the proofs Which I encourage all of you to do you will look at and you'll say wow that's really nice That's really elegant. There's you know, you Really will be able to get better at writing these and you will start to approach this if you work hard And this will definitely help you with with your other more theoretical math classes for sure There's no question about it. So really I encourage you to look look this up online But again, just to be clear you guys overall you guys did a very good job. I was pretty happy okay, so Let's see. So yeah the on the syllabus you guys are all smart and on the syllabus I have the the website just go to the website the homework so when the solutions have been posted The homework, you know homework one homework two homework three etc. Those statements will be hyperlinked So you can whenever you can click on a homework. It's to the solutions. Okay? all right so Let's go ahead and move on So here's what we're going to do as I said last time today. We're going to finish up the division algorithm and then Yeah, so I spent way too much time on that but We're going to finish up the proof of the division algorithm, and then we're just going to do a few problems. Okay? So this is just a continuation of section 2.2 Okay, so again just to recap you don't if you were here, of course you don't need to write this down So if you recall this set s, I'm not going to redo everything but just to remember what we did a minus xb where sorry x is an integer and A minus xb is bigger than or equal to zero That's our set s What did we show so we proved Let me try to streamline this a little bit we found a Least element right I Think I called it R. Didn't I can somebody confirm that? Okay, we found a least element R in s by proving that it was not empty and then we use the well-ordering property, right? So a couple of things I think I finished off with this last time that we know about our first is that Since R is in s R is equal to a minus Qb right or some Inager K right. I think I use the same notation on Tuesday Right did I write this something like this? I think I use Q and so Shuffling this around A is equal to Qb plus R, right? Okay, so everybody see that Just add Qb to both sides. That's pretty easy Okay, and the second thing is by definition that zero is less than or equal to R Why is that because R is in s and by definition everything in s is bigger than or equal to zero Okay, they just follows from the definition of s that zero is less than or equal to R. Oops Okay, so Remember what it is that we want to prove Well, there's several things you want to prove one is that Okay, if you look back and for the sake of time, I'm not going to write it down again look back to the statement of the division algorithm Okay, what was it? We're trying to divide What a by b and The statement is that there exists unique integers Q and R satisfying a equals Qb plus R Which we have and the second is that zero is less than or equal to R and R is less than b That's the second part. This should be in your notes if you were here. So we have part. We have most of it We'd still need to show that R is less than b and then at least we have the existence part of the theorem You guys with me on this Okay, at least a couple of you are nodding.  Okay, so Let's and this is a useful technique in writing proofs. Let's suppose That that's not true This is called a proof by contradiction, which you should have been exposed to then well far as not less than b What do we know and then R has to be what? Greater than or equal to b, right? Okay Thus and just try to follow this. It's I don't think this is too bad We put a star here a minus Q plus 1 times b And we've got a lot of letters floating around here, but I like said, I don't think this is too bad. This is equal to a minus Qb Minus b I'll pause for just one second you guys by that Distributing right a minus Qb minus b Okay well What is this equal to? This is Okay, I'm gonna be a little sloppy here So you forgive me for this but just to make sure everybody's with me on this What's a minus Qb equal to we have another expression for that if you look up above This is all right. You see that right up there a minus Qb is R. So this is R minus b okay, and Somebody tell me why this is true R minus b is less than R This goes back to an assumption that we made that's not on that was actually back on Tuesday But remember we're dividing by b. What was the assumption on b is that b is bigger than zero? That was the assumption that's in your notes. It's there b is bigger than zero So if we're subtracting something positive from our we have to end up with something that is smaller than R right So this is true since B is positive Okay, so We can conclude that so a minus Q plus one times b Let's see What else do I want to say here? Okay, so let me just put this together first. So this is R minus b Which is less than R? Okay, that's the first thing I want to say Also a minus Q plus one times b equals R minus b which is bigger than or equal to zero Why is this? since R is bigger than or equal to b. Okay, let me call this one Let me call this two First thing we showed was that a minus Q plus one times b is less than R, right? And the second thing I want to I want to be clear about this a minus Q plus one times b We just of course we just figured out out that that's R minus b This is bigger than or equal to zero because we're assuming that R is bigger than or equal to b Okay, so R minus b just subtract b from both sides R minus b is bigger than or equal to zero Okay So what what went wrong here? There's something that bad that has occurred here that will allow us to conclude that our assumption was not correct And that R actually does have to be less than b But then Okay, look at look at this and think about it everyone once you guys to think about this you buy that this is an s a minus Q plus one times b is an s Why is how do we know it's an s? Look at the definition of s up here s is the set of all The set of all integers of the form a minus xb where x is an integer and a minus xb is bigger than or equal to zero It certainly has that form right? This is a minus an integer times b, but that's not enough to conclude that it's an s We also needed to be bigger than or equal to zero Right, but we have that from from this right here You see that I'm not going to write all that down, but you can just see it from what I have on on the screen Now there's a contradiction here Okay This does contra this information we have at the bottom of the screen does contradict something Can anybody tell me what the contradiction is when we know it's an s? There's also something else we know about it It's less than R Okay, listen what I'm saying R was chosen to be the least member of s Here now we have a member of s that's smaller than our contradiction because ours the smallest Okay, therefore It can't be the case that ours bigger than or equal to be R has to be less than b So let me write that down you guys mostly have this now Can I move on to another screen? You guys okay, you good so this and you can just keep writing this in your notes this contradicts that Yeah You said Hang on one second, let me just finish writing this I'll answer that okay, so we've this this I'll answer that in a second this so now we've proven the existence part of the theorem right we've we got our least element R We know that a is equal to qb plus R and we just proved that zero is less than equal to R Which is less than b that's exactly the existence part of the statement of the division algorithm now We have to prove that that these q and R's are unique So the answer your question the So you are you just sort of asking where did the q plus one come from or? Okay, well here's the here's the idea. Let me let me go back to this for a second. Okay, I mean just this just roughly I mean the the q plus one well, why do we have the q plus one? Well because it It makes the proof work. I mean that's a very unsatisfying answer, but that's really the case The idea though is that if our so think of R as a remainder here's the rough idea Okay, suppose that R is bigger than or equal to be B is in your head what you were dividing by okay? Then the point is then the quotient can actually be increased by one So if you can imagine right dividing through if you divide 17 by 5 and you say it's 2 Well, they have a remainder of 7 but then really you can you can extract 5 out of that 7 and increase the quotient by 1 That's the idea of where this comes from Okay Does that make sense? Well, no the point is though. We wanted to choose We don't know I mean the idea is just knowing that R is bigger than or equal to B Then we know that Well, the quotient can be increased by at least one for sure But we don't know what we don't know anything other than R is bigger than or equal to B So that's really about all we can say and that's all we need to say because it gives us a contradiction Okay Okay, so All right, are we okay with this first part any other questions? No, okay This thing is all checked up. All right. I don't know what's going on here second part Seems to be okay now. All right and This is the last part of the proof uniqueness Okay, so the sake of time I'm not gonna go back and write everything down again But the point is and I will write part of this down the point is the quotient and the remainder are unique And what is that? What does that really mean? Here's what it means and I'll write it down I just want to say it first it means that if You take a and you divide it by B and you get that a equals QB plus R and R is between 0 and B And if you also divide it and you get that a is equal to say Q prime B plus R prime And our prime is between 0 and B then the those two quotients Q and Q prime and the two remainders are the same They both match up Okay Yes, yes, okay, so I'm just gonna write this down Now suppose that we have one a equals QB plus R and Zero is less than or equal to R, which is less than B and Let's assume the same thing is true for some other integers Q prime and R prime. So let me call this one prime a is equal to Q prime B plus R prime and 2 prime will be That zero is less than or equal to R prime, which is less than B Okay, so here Just to be clear of course Q R Q prime and R prime are all integers Okay Now maybe it's a little clear what I mean by uniqueness now Basically what we're doing is we're assuming that we've divided a by B and we've gotten one quotient and one remainder on one hand But then on the other hand we've divided a by B and we've got another quotient and another remainder We're going to show that the quotients and the remainders have to be the same in other words Q and Q prime are equal and R and R prime are equal Okay, that's the uniqueness part. So there are a couple ways you can do this Okay, we're actually almost done and this is the last theoretical thing I'm going to do and then we're going to Just talk about some homework This is not this part's really not that bad By one and one prime we get this shouldn't be too much of a stretch QB plus R Equals Q prime B Plus R prime right Okay, I would encourage you to try to try to follow along I know it's very tempting to just kind of daydream and just write this down and not think about it But we'll help you to actually think about this as we do it You see where this is coming from right a is equal to QB plus R But a is also equal to Q prime B plus R prime so therefore since of course a is equal to a QB plus R is equal to Q prime B plus R prime right You guys see what I'm getting this they're both equal to a so they have to be the same right thus our prime minus R is equal to B times Q minus Q prime Okay, think about this for a second. This is this is not anything too complicated Okay, our prime minus R. We're just going to subtract the R off and then we're just going to take the Q prime B And we're going to subtract that off to the other side We're going to factor out a B on the left to get Q minus Q prime Okay, just basic algebra here Okay We're going to use this fact here in a bit hence Okay, let me put a A couple of stars around this The absolute value of R prime minus R is Equal to B times the absolute value of Q minus Q prime Christina you have a that's your name right? Yeah, okay. You have a question Not the way I have it because I've taken the R over here and I'm taking the Q prime B over here So this will be this so I'm taking this over and then this over so this will be QB minus Q prime B So when the B gets pulled out, it's Q minus Q prime Okay, so we can certainly take the absolute value of both sides Right, so now the question is well, how much do you have to prove and how much do you do not well? This is this well, I mean You don't Things going back to the foundations of the real numbers. You don't actually have to prove these things so the point is the absolute value is Multiplicative in the sense that well certainly this is true The absolute value of R prime minus R is certainly equal to the absolute value of B times Q minus Q prime That's certainly true. They're the same so the absolute values of course have to be the same Why can I do this? Well because B is assumed to be positive That's why The absolute value B is B because B remember don't forget that hypothesis B is positive okay, and The absolute value of X Y is the absolute value of X times the absolute value Y you don't have to prove that So let me just put this parenthetically since B is positive. That's why we can we don't need the absolute value signs around the B because it's positive Okay Now we're gonna look at a couple of inequalities and we will be done very soon so from See I keep saying that and I just keep going and going and going you keep thinking all it'll be done So I can listen So okay, well we'll be done kind of soon from to Okay, let's see. I'm going to go through this kind of slowly Minus B is less than minus R. Okay, do you guys buy that? From this second inequality up here Subtract R from both sides and then subtract B from both sides. You can certainly do that and You see what I you see what I did okay? Subtract subtract R. You get zero is less than B minus R now subtract B. You get minus B is less than minus R Okay, do you buy this Minus R is less than or equal to zero right? Okay, because zero is less than or equal to R just add minus out of both sides and minus R is less than or equal to zero okay, okay, and now from Two prime. We're not gonna monkey with this. We're just gonna list it The way it was I think anyway, let's see Zero is less than or equal to R prime Which is less than B. Okay, so and we'll see if I can squeeze all this in here And I'm going slowly here so that I don't lose anyone. So what do we have we've got? minus B is less than minus R Which is less than or equal to? Okay, this I should say something about minus R is less than or equal to R prime minus R because R prime is bigger than or equal to zero. I want you guys to look at this inequality right here And I this is really sloppy. Sorry about that. Let me try to fix this. Okay Minus B is less than minus R. Well, that just comes from this Why is minus R less than or this is horrible? Sorry This might yeah, okay, that's even worse. Okay, let me erase it Getting no, I don't want to do that. Okay. I'm not gonna worry about it's fine Okay, I could go back to the white, but why is this true? Well our prime is bigger than or equal to zero Okay, so what's the difference between these two sides of this inequality? This is just minus R is less than or equal to R prime minus R Well, this has to this can only get bigger if we add something non-negative to it. That's the idea Okay, make sense Okay I'm running out of room here. Okay. This is less than Yeah, well Sloppy and small is gonna be really bad Okay Why is this true? Okay, this isn't this is this Last while we have one more This inequality right here Well, we already know that our prime is less than B from this So what happens if we just subtract R from both sides of this inequality? We get our prime minus R is less than B minus R, which is exactly that Okay Okay, and So I'm gonna squeeze this on here Let's see. This is less than or equal to B This is terrible. I'm sorry since R is bigger than or equal to zero Right, so the same this is kind of the same idea We're subtracting something that's bigger than or equal to zero if we're subtracting something bigger than or equal to zero That's only gonna make it smaller than what it was Right, that's where that comes from Okay, so I'm gonna write this on the next slide. So what have we actually done now? Yeah, this okay does everybody have this down I'm gonna write this on the next slide. Here's the here's the idea This is the main point. I want to extract from this minus B is Less than R prime minus R, which is less than B The point is that our prime minus R is squished between minus B and B That's the main point to take away from this. Can I go to the next one? Okay, so minus B is Less than R prime minus R Which is less than B.  I think some of you have taken analysis. No, no about this R prime minus R. It's just it's just think of it in your mind. It's just one number Don't get confused with the R prime in the R. You have a number that's strictly between minus B and B There's something you can say about the absolute value of it Okay, anybody tell me what the next step should be then What can we say about the absolute value of? What can we say about the absolute value of R prime minus R? It's right. It's less than B. Okay I'm not gonna mess up the Lecture with this, but if this seems a little strange you just think about just think about Suppose you've got a number that you know is strictly between minus five and five Okay, well if it's If it's positive the absolute value is equal to itself and it's certainly less than five But because it's between minus five and five if it's if it's negative Maybe it's minus four, but the absolute value is four. That's still less than five Maybe it's minus four point nine the absolute values four point nine It's still less than five because it's squeezing me these between these two numbers the absolute value has to be less than This guy on the right. Okay, so now we literally are almost done. Okay, so recall that Okay, I think I denoted this with two asterisks the absolute value of R prime minus R It's equal to B times the absolute value of Q minus Q prime so what does that tell us? I'm gonna use so again.  Well, yes, we can because what we can do and this is really the proper procedure Which I'm not gonna write it right out, but we can multiply both sides by 1 over B because B is positive The inverse exists and it's and it's positive. So we have no problem with this Well, yeah, but the problem is if you didn't know then you wouldn't know if you had to switch the sign or not Because you wouldn't know if it was positive or negative But I mean I think what you're asking is suppose you knew it was negative then you would flip the sign Yes, that's that's right Okay, are we okay with this? okay, so All right, I will not say so again thus Dots whatever You guys believe this zero is less than or equal to The absolute value of Q minus Q prime Which is less than one right Would I get to zero from well absolute value can't be negative That's where the zero is popping in from right by definition. The absolute value is not negative You guys buy this this is an integer for sure right certainly not negative integer, but it's I mean it did certainly an integer The absolute value of an integer is an integer What's our conclusion Look at this previous inequality Okay, so I'll write this out. Yeah, so what is the absolute value of Q minus Q prime have to be? Has to be zero right There's no other possibility for it says because it's an integer and if it wasn't zero there would be an integer trapped strictly between zero and one which That's not possible right What do you know about the absolute value of a number if suppose you know the absolute value of a number of zero What can you say about that number? Itself has to be zero right and Voila Q equals Q prime that was one of the two things we had to prove right the quotients are the same Now my question is looking and actually you can do this just looking at what I have on the screen right now How can we there's another part we have to prove right? We have to prove that the remainders are the same too How do we know that? Yes, yeah, I mean that's mostly you have the right idea. Yeah, I mean the point is that this is zero You said a little more than you than you had to but this is zero so therefore B times it doesn't matter what B is doesn't matter that it's positive this thing is certainly zero because This guy is zero Therefore the absolute value of our prime minus R is zero and for the same reason our prime equals R Okay, so does that make sense? Okay Yes, I am good.  Okay That's the end of the proof. That's uniqueness Okay, yes, so that took a while Well, I ask you to write this proof on the exam. No Do what I expect you to state the division algorithm precisely on the test. Yes The definition what the division algorithm says that means a theorem really, but yes, you I may ask you that No, as long as it I mean as long as it's yeah I mean you don't have to use the same letters necessarily, but but it should be you should be saying the same thing Okay, can I move on here? Okay Not insulting me. I don't have to start bringing rocks to class start throwing in with people. I've died been known to do that No, I'm just kidding. I don't bring rocks. I'll bring vodka to class. I've been doing but I won't share it I just drink it myself and then I just get really silly Okay So Let's see what else we want to do. Okay. Well, that's that is it for this section. So like I said I want to Spend the remaining 25 minutes or so on a problem. So I feel since this is this is due on Tuesday I definitely want to do a problem using this division algorithm Luckily the idea behind most of the problems you're going to have in this section is this is it's the same idea for almost every problem So I'm going to do one in fact I'm going to do one of your handling problems for you Okay, and then that that will give you the rough idea of how you proceed with the rest of the stuff. Okay, so Let's this this is still 1.2. Okay You definitely want to pay attention to this because you're going to be handing this in Okay, you're going to have a lot of cases here. So this is going to be a little bit tedious the square of Any integer is either of the form 3k or 3k plus 1 This is exactly how the book has worded the problem. They don't tell you what K is. It's understood that K is an integer, okay? So let's see why this is true, and you might think well the division algorithm I learned this when I was 7 How could you possibly use it to do anything interesting? Well, you'll be you might be surprised I don't think this is that interesting of a problem, but the point is here's here's what the problem is saying If you square an integer and you divide by 3 the remainder is either 0 or 1 it can't be 2 That's that's really what this is saying.  I'm going to say it first and I'll write it down The division algorithm says that we can take a we can certainly divide by 3 3 is positive So it fits the hypothesis of the intermediate. Sorry of the division algorithm, and so we know that 3 that Any integer a is equal to 3 q plus r where q's an integer and ours between 0 and 2 Right okay zero is less than or equal to r which is less than 3 right. It's always less than what you're dividing by And then when you square you're going to see that a remainder of 2 is not possible And then I'll write that that down for you, but what you should be thinking about using the division algorithm Okay, so how you would prove this would be to do something like this you would just say okay let a be an integer by The division algorithm. We know that a is equal to 3 q plus r for Some integer q some integer r Satisfying 0 is less than or equal to r which is less than 3 that's exactly what the division algorithm says right Dividing by 3 quotient and remainder Okay, so we need to prove something about a squared though right the square of any integer has this form So a was just an arbitrary integer We're going to square it and we're going to show that in fact when we when we do the algebra What we get is three times an integer or three times an integer plus one and then you'll be done Yes, so we've got a equals 3 q plus r. We're going to square both sides, and this is just a pretty simple algebra 9 q squared plus 6 q r Plus r squared All right you guys by that Well, there are three possibilities And you're going to have to do these cases separately R is an integer satisfying that 0 is less than or equal to r which is less than 3 So what are the possibilities for our are 0 1 or 2 You're going to consider all three of these cases, and you're going to show that no matter what a Squared is going to have to have one of those two forms And that's it then you're done Well the two that the two could okay. No the two r could be two, but the point is when you square You you end up being able to sort of shuffle the algebra around so that you just have three times an integer plus one Remember the r is a division is what you get as a remainder when you divide a by three So are certainly could be two right so you divide five by three right the remainder is two But when you square everything Yes, when you square everything the two falls away. It's not possible. Yes, that yes, that's exactly right. Yes okay So case one and these I you know I'm sorry to say these these are a little tedious but If you see kind of what the mechanics are these Several of these problems aren't that hard then if r is zero then a Squared equals you guys by that a squared sorry a squared equals nine q squared Okay, because a squared it when we right was nine q squared plus six qr plus r squared when r is zero Those other two terms are going to go to zero.  You're trying to show you're trying to show that a Sqa on this case in particular. We're trying to show that a squared is three times an integer. We haven't explicitly shown that yet But how do we do that? We just factor out of three then we have three times an integer I expect you to write that in your proof, okay? so This is equal to three times three q squared and setting K equal to three q squared We get a squared equals three k And that's Satisfactory because we want to prove that it's either 3k or 3k plus one. We've got that as 3k. So we've got what we want it Okay, case two R equals one then Maybe you can see how this is going to go a squared equals nine Q squared plus six q plus one you guys by that Okay, because the r is one so we're just replacing We're putting in one for r in that equation Now what do we want to do so now? Yeah, so we Right exactly so if we pull out a three Then we have three q squared plus Plus two q Yeah, so when you pull out the three we've got three q squared plus two q left over right and then we saw the plus one on the Very end. Okay. I'm really sorry for being sloppy. I Because of time here. I know I should be right If we call this k right this is equal to 3k The three q squared plus two q. So if we call that k then we've got queues that we've got um, sorry I keep okay this I have to be careful about okay my cues and the a's are running together here No, you don't you don't have to write that. I mean that that is true Sure, right, but you don't have to you don't have to even worry about trying to prove that that comes back to the Basic foundations of set theory and it's way beyond the scope of what we're going to be doing in here Yes, yes, that's right Okay, did you guys yeah? No, I mean every case we're sort of erasing the variables and starting over again I mean if you want you could say we'll let k be three q squared and then the next case We'll let k prime be three q squared plus two q and the next case will let k double prime be Yes, yes k is just going to be we know that ace we want a squared to be three in this case We want it to be three times an integer plus one and by writing it this way We know that since this is an integer because q is an integer we have it in the form three times an integer plus one So we're just going to let k be that thing Well, but remember what it is that you're trying to prove for case one you're trying to prove that a Squared is equal to 3k or 3k plus one So you have to remember what you're trying to get to I mean so you have this I mean yes, sir You could you could take the square root of both sides But it doesn't get you where you want to go No, I mean no, no, we know that we know that a squared is 9q squared Okay, we know that because we're assuming that are zero that we already have the division algorithm So we already know that a squared is this in case one is r is equal to zero So therefore we do get this okay, but the point is you're you're going in a different direction than where you need to end up You have to always remember where you're trying to end up and that is to show that a squared is three times something or three times Something plus one so it's very tempting when you see squares to sort of just go back to You know high school mode and just take square roots and all that stuff But that doesn't mean that it's relevant to the problem you need to get to where we want to go And this is how you how you do it, okay? Or that yes Okay, do we have this we have this down Okay, case three. Oops. I gotta go back to this thing Okay Let's go. Oh, it's on white. It's on white. Yep. Yep. Sorry No, I refuse to do that on principle Let's see, okay, do we have class I'm done. I'm not gonna be here. I mean I have like 17 dates lined up There's no way I'm coming to class No way Let's see. Okay, so just kidding. I'm there's class this class Yeah, you think of course he doesn't have 17 days. He's a math professor.  But yeah, okay, not like that. But um, okay The next case the last case is r equals two, right? Then Okay, I try to get through this quickly and maybe I can talk about one of the problems from the other section then a squared equals nine q squared Plus Okay, you've got this in your notes If you go back to your notes and you replace the r with two you guys buy this Okay, so a squared is in this case when ours to a squared is nine q squared plus 12 q plus 4 Okay So How do we get now? We want it to be of the form three times something plus one, right? Or well, that's that's what we're gonna you might think well Maybe it's three times something because that's another possible option. It's not I'm just telling you three times something plus one Is where we want to go here. How do we do that? Yes, exactly rewrite the four is three plus one then you can pull out of three and then you've got your plus One on the end. Yep. Yep, that's right. Okay Hopefully you guys believe that Okay, good good Yeah, I'd rather not I know I would do it of course, but if you know if you if you got it then it's fine. Okay Okay, so there it is. There's the proof All right, so we're running out of time, but I do and I think this will well Be useful to you this is I'm gonna talk about one of your homework problems They're gonna hand in on on Tuesday from the other section. I'm gonna say something really quickly about this Okay, so this goes back to what was it 1.2? I think right 1.2 Number 4c. I'm not gonna have time to say a whole lot about this, but I do want to mention this prove that n choose n equals n choose r plus 1 if Sorry, this is getting sloppy Can you guys read this? n choose n oh Wait a minute. Yes. Sorry. Sorry about that. Okay The ink was bleeding on my on my paper Let me erase this. Thank you for that Okay, and choose r is equal to n choose r plus 1 if and only if n is odd and R Equals one half times n minus 1. Okay make this a big huge period. There we go. Okay r plus 1 Okay, here's what I want to say. I'm not gonna have much time to do a whole lot here But if and only if statements those of you that took to scream maybe are aware of this There are two statements you have to prove and if and only ifs, okay It's a it's a sort of a double implication if and only if means the first thing implies the second and the second implies the first That's what if and only if means so when you're writing this it's sort of a two You have a sort of two sub problems here The first thing you have to do is prove that n choose r is to do this assume N choose r equals n choose r plus 1 and then prove the other side of this that n is odd and r is one half times n minus 1 That's the first part of the proof The second part of the proof is assume n is odd and r equals one half times n minus 1 prove that those two binomial coefficients are the same So there's really a part a and a part b here that you need to do So what I will say I'm not going to have time to go into this But how do I do how would you do the first direction well assuming that n choose r equals n choose r plus 1? We want to prove we want to prove this What you do is you say okay? Well, we have n choose r equals n choose r plus 1 Write out the definition right n factorial times r factorial times n minus r factorial equals blah blah blah what your goal is is to get down to that Okay, what does that mean think about it? We've got a bunch of factorials to begin with this has no factorials So you should use the inductive definition of the factorial to try to cancel things out And if you do it right you will end up with that That makes sense. I had to say it quickly, but how do you know that n has to be odd? How do you know that? This is the last thing I'm going to keep you here until someone Yes You're going you're going way off in an area that you don't need to not that what you're going to say isn't true It's simpler. It's much simpler than that What's that? But yeah, okay, so when you so that's one thing you could do yeah, and that's well Yeah, that's that's one perfectly reasonable way to do it Hang on one second hang on there There's other things you can say to but the point is once you've got to this point Okay, solve for n you're going to get n is equal to 2r plus 1 so by definition n is odd That's it. All you have to do is solve for it Okay, another way of thinking of it is r for this day even have meaning r has to be an integer And if n was even then you'd take an odd number divided by 2 and that's not an integer Okay, so either way, but that's the idea so I Think I'll just we have a few minutes by thinking when to stop here I'm just not going to be able to get very far if I if I don't so That is that is not true I do I do not maybe one maybe two but that's about the extent of it Okay Yeah, sure


Want to stand up and say hi? Stand up. Hi, I'm Dara. Today, we're going to talk about Sean Fair and the alternative stem cell treatment he got for his damaged right knee. Sean Fair always identified himself as an athlete. Growing up, he played soccer, lacrosse, football, and in college, he was even on the tennis team. When he was 28, he was an aspen shredding powder on the slopes. I'm sorry. I'm from Arizona. I've never skied in my life. Basically, Sean was just skiing an aspen. He hit a jump, he caught some air, and then he mistimed his landing. The shooting pain hurt his right knee so bad that he ended up having to ski down the rest of the slope on his left ski. Sean's disabling knee injury left him unable to walk and certainly incapable of living the kind of life he had before. Remember when daddy had a bad knee and I had a lying bed? Remember? No, it's not anymore. My book was on this one. My book was on that one. How old was he when that one? He was about, just a year? Yeah. I remember the only time he saw daddy was when he came in and he fed you a bottle. Yeah. He was pretty young then. It was pretty sad though that Sean couldn't see him really at all.  Sean was diagnosed with OCD. Not obsessive compulsive disorder but rather osteocondritis dissequence. Sean had really damaged the articular cartilage in his right knee. Researchers aren't completely sure what causes OCD. But the condition often develops in teenagers and then later kind of presents itself in adulthood. Kind of like when you mistime a landing after you're shredding powder. Oh, sorry. Dr. Andreas Gamal, Sean's orthopedic surgeon at Brigham and Women's Hospital basically told me that the extent of Sean's injury was pretty bad. If you will, imagine it as, you know, a pothole in his cartilage. And actually the pothole was the size of a quarter. The size of a quarter Sean's doctor decided to try a different approach. One that he hopes will fix the problem permanently. Stem cell transplant. Okay, okay, okay. Before I move on, let's talk about stem cells. Stem cells can be thought of as undecided cells that eventually develop into more specific cells like heart cells or a liver cell. But that's based on the environmental factors that are placed on them. In Sean's case Dr. Gamal used stem cells from the umbilical cord. Although umbilical cord stem cells aren't able to become every cell in the body, the benefit of using those stem cells is that it doesn't require the destruction of an embryo. So, Dr. Gamal was able to transplant umbilical cord stem cells into the cartilage hole in Sean's right knee. Cartilage is in an environment where there is not much blood supply from the body and also has variable pressures. So, the stem cells that are in Sean's right knee differentiated into cartilage-like cells called chondrocytes. Basically, chondrocytes are cells that produce and maintain articular cartilage. As of now, Sean is thankful for his procedure. He's able to walk now. He's able to go up and down the stairs, you know, the small stuff. His right knee has healed, but it's not 100% just yet. My other knee, unfortunately, because of all the issues this one has, also needs reconstruction. We're waiting for this one to be 100% before we can tap into this one. So, yeah, my knee is definitely feel it. And even among all the physical injuries and the debilitation, Sean still says the majority of the challenge is mental, not physical. I try not to talk about it with my guys that I used to play tennis with, who are still athletic here.  I don't want to hear about you playing tennis. I know you are. It's enough to kind of get my mind wrapped around me here. I don't want to see you where I think I should be kind of a deal. So, I try to keep my mind focused and keep that outside of the blinders. Although Sean hasn't yet returned to his original athleticism, he is able to comfortably kneel down and play with his four-month-old daughter and his two-year-old son, Austin. Thank you. Okay, let's go help Mommy. I'm going to talk to them, okay? Get in your car. Austin, come here. Over here.


Exotons, quasi-particles of tightly bound electron-hole pairs, dominate the optical response of atomically thin transition metal decalcogenides. Here, the authors review strong light matter coupling in two-dimensional semiconductors, arising from confined exotons interacting with trapped photons, or localized plasmans. This article was authored by Christian Schneider, Mikhail M. Glasov, Tobias Korn, and others.


What's going on, everyone? Sam here with latent sports cards. We're in one box of 2023 Topps Star Wars Return of the Jedi, Chrome Sapphire Edition hobby for Sean L. So good luck here, Sean. Let's see what we got, buddy. You're on the bench. Darth Vader, nice. Castered by the Ewoks. And where's Princess Leia? Is Stormtrooper Attack, The Rescue, A Newfound Friend, and Lead Singer, Psy Snoodles, Toward the Sarlacc Pit, Portrait of Wicket, Royal Treatment, and Rethink the Plan. What we got? Is that an orange, I believe? R2-D2 meets Wicket. That is an orange! The Confrontation, 24 of 25. That's sick, dude. It's got Vader and Emperor Palpatine on it. Oh, is that Emperor Shocking Luke? Yeah, it is. That's pretty sweet. That's awesome. Congrats, Sean. Nice one there, buddy. We'll take that. The Confrontation, Force Creatures, and Luke Skywalker Jedi Warrior. Luke witnessed the true power of the dark. True. Very true. Han Solo's alive, sitting with royalty. It's quiet C-3PO, and surrounded by Ewoks. He's fluent in the Japanese. There's Snagged by the Ewoks. The plan begins. Ambushed by the Empire, and Headquarters... is it Frigate or Frigate? Frigate. That's like the new thing on TikTok and stuff, if someone's just talking about a lot. It's just a long video on top, and everyone's like, Bro's just yapping. Bro's fluent in the Japanese. It's so funny. Is Horror from the Pit, New Challenge, The Deadly Cannon, and R2-D2 Last Box, or Last Pack of the Box. There's a green in here. Escorted to the Ewok Village, Swing to Safety. That is a green toward the Sarlacc Pit, 42 of 60. Nice one there. Sean, congrats, buddy, and a checklist. That is going to do it for your break there. Sean, thanks again for the support, buddy. We'll get him right out in a minute.


Recently, a joint investigation by the Pike County Sheriff's Office and the Pikeville Police Department resulted in more than a dozen indictments for drug trafficking. Eleven were arrested in a recent drug roundup, but there are three still on the run. They are 19-year-old Harley Tubbs of Pikeville, 58-year-old Jimmy Hopkins of Maribone, and 27-year-old Latasha Thacker of Mouthcard. But overall, Sheriff Roddy Scott says he's happy with the results of the roundup. We had a good day. We probably took in about $3,000 in cash, gun, two vehicles. People need to know if they're out and they're selling drugs in their vehicle. When we catch them, we're going to take their vehicle. This is our second roundup within the last three or four months, and this is something that will be continually going on, and three or four months from now, we'll have another one. Sheriff Scott says putting a solid case together against someone for trafficking in drugs takes more time than many people realize. A lot of people don't understand that, hey, we can sit here talking about my neighbor selling drugs. It's not that simple. If you don't get someone to go in and buy all of them, audio, video, have the evidence, we're wasting our time and their time. In the court's time, whatever it takes to catch someone selling drugs, that's what we're doing. Anyone who knows the whereabouts of Harley Tubbs, Jimmy Hopkins or Latasha Thacker is urged to contact the Sheriff's Office or the Pikeville Police Department.


Welcome friends, in our previous lecture we have learnt about continuous casting process. Now today let us learn about centrifugal casting process. Now the question is what is this centrifugal casting process and when it was started. So this was developed by A.G. Eckhardt in England during 1809. Now what is its principle, it utilizes the centrifugal forces caused by rotation to distribute the molten metal to the mould cavities. As the mould is rotating, molten metal will be distributed into the mould cavities. So because of the centrifugal force, the molten metal will be sticking to the mould that is how we get the required geometry of the casting. So this is the simple principle of the centrifugal casting process. Now these are the types of centrifugal casting process. One is the true centrifugal casting process. In this again we have two classifications, one is the horizontal true casting centrifugal casting and the other one is the vertical true centrifugal casting. And the second category is the semi centrifugal casting process and the third one is the centrifuging process. So first we will see the true centrifugal casting process and also the horizontal true centrifugal casting. Now what is this true centrifugal casting process? What is the principle? The mould is rotated about a horizontal or a vertical axis and the molten metal is held against the walls of the mould by centrifugal force until it solidifies. The mould is rotated about a horizontal or a vertical axis. As it is rotating molten metal is held against the walls of the mould. Now true centrifugal casting is used for the production of cylindrical hollow parts. So this is the main application of the true centrifugal casting process. Now here we can see the principle, yes here the molten metal is being poured and this is a metallic mould, a cylindrical metallic mould we can see here. And it is supported using rollers, here there are the top rollers and here there are bottom rollers. As the top rollers and the bottom rollers are rotating even this metallic mould will be rotating. Now it is that time the molten metal will be poured into the mould and during pouring the what is a mould will be rotating at a predetermined speed. And this rotation continues till the molten metal inside this what is a metallic mould solidifies. After solidification the rotation of the mould stops then we get the solidified cylindrical hollow casting. So this is the motor which drives the metallic mould. And here we can see the mould details of this setup and here we can see this is the ladle and this is the pouring basin. And here we can see this is the metallic what is a metallic mould this is the cylindrical metallic mould. And here we can see these are the bottom rollers on which the metallic mould is supported and these are the top rollers. Now this is the metallic mould and yes we can see as the molten metal is being poured the metallic mould will be rotating. And this rotation continues till the solidification of the molten metal will be over after that yes we can stop this rotation of the metallic mould then what happens we get a finished casting like this this is a hollow what is a cylindrical casting. So this is an example of the centrifugal casting process and here we can see the animation. So this is the metallic mould the cylindrical mould and here the molten metal is entering as it is entering the molten what is a metallic mould will be rotating and the molten metal will be sticking to the walls of the mould and the rotation continues till the molten metal solidifies then what is a metallic mould will be what is a detached and we get the solidified casting. So long parts are produced using horizontal centrifugal casting process and here this is the what is a true photograph of a true centrifugal casting process here we can see this is the metallic mould this is the metallic mould and this is the what is a ladle through which molten metal is coming to the cylindrical metallic mould. And here we can see these are the bottom rollers and here is one roller and here is another roller on the other side also there will be two rollers and at the top also sometimes there will be rollers and as these rollers are rotating the metallic mould will be rotating it is that time the molten metal will be entering into the metallic mould cylindrical metallic mould. So far we have completed horizontal true centrifugal casting now let us see the vertical true centrifugal casting process. Now why we should go for the vertical true centrifugal casting process why not the horizontal true centrifugal casting process this is used for heavy parts whose diameter is more important than the length at such times we use the vertical true centrifugal casting process. The principle is the same there will be a cylindrical mould will be there and it will be rotating but here the diameter is more important than the length and here we can see this is the rotating what is a mould and the molten metal is being pulled as the molten metal is entering into the mould the mould will be rotating and this rotation continues till the molten metal solidifies inside the mould what is a metallic mould. And here we can see the animation yes this is the metallic mould vertical and it is rotating about a vertical axis and the molten metal is coming like this because of the centrifugal force and it will be going to the sides of the mould it will be going towards the walls of the mould and this rotation continues till the molten metal solidifies. Now true centrifugal casting is carried out as follows first step is we have to apply the syrimic slurry to the mould wall because it is possible that the molten metal may stick to the metallic mould. So that should not happen to prevent this we apply a syrimic slurry to the mould wall and it should be dried and also it should be baked. Next step is rotation of the mould at a predetermined speed and it can start from 300 rpm to and it can go up to 3000 rpm then once this what is a mould what is a keeps on rotating at this predetermined speed we have to pour the molten metal into the mould and most surprising thing is here no gating system is required. We have seen the design of the gating system for a sand mould and also what say we have seen the die casting here no gating system is required. The mould is stopped after the casting has solidified till the time the mould has to rotate. Then the next step is extraction of the casting from the mould and finally the next step final step is the finishing means what is that removal of impurities at the centre using machining grinding or sandblasting. Now in the molten metal there will be some impurities will be there. So these impurities will be subjected to lesser what say centrifugal force as the mould is rotating the particles having lesser dense are subjected to lesser centrifugal force so that is how they are collected at the centre. So these are to be removed using machining, grinding or the sandblasting. Now what are the advantages of the true centrifugal casting process? Flexibility in casting composition of course there is an exemption right. So maybe that exemption is for 0.4 to 0.85 carbon due to segregation. The next advantage is the wide range of available product metallurgical characteristics. We can get different types of the hollow what say cast components and also we can get what say different metallurgical features and can form very large parts. The next advantage is central core is not required for making a hole or pipe and we know that in the green sand casting if we want a hollow casting or if we want a hole inside a casting we used to place a core. So this core has to be made and that should be carefully kept inside the mould then we have to pour the molten metal. That is not enough after solidification we have to separate the moulding box, we have to what say break the mould then we have to separately remove the core. So all this headache is not there in the case of the true centrifugal casting process. We get the hollow cast components, we get the holes inside the cast components but no core is required that is the advantage. Next one, employments of gates and risers is not required, no gating system is there we have seen that is why that is how 100 percent casting yield in many cases. Now the question is what is this casting yield? Let us see this is the definition for the casting yield. Casting yield means it is the ratio of weight of the casting divided by weight of the poured metal hole multiplied by 100. Remember that weight of the casting is not equal to the weight of the poured metal. If the weight of the casting is 100 cases we always pour more than 100 cases may be 125 cases or 130 cases. So weight of the poured metal is always greater than the weight of the casting. So this is the casting yield, casting yield means weight of the casting divided by weight of the poured metal multiplied by 100. So this should be maximum if the industry has to get benefit. So in the case of the green sand casting the casting yield will be say between 70 to 80 percent but it is never 100 percent. But here we get close to 100 percent because there is no gating system as the molten metal is consumed for the gating system there will be what is this casting yield will be coming down. Next one relatively very light impurities move towards the center which can be removed easily. Now the question is how to what say remove the impurities if it is the green sand casting the impurities will be inside the casting. But here the impurities are subjected to lesser centrifugal force because of that they will be segregated to the center. Now after solidification yes they can be identified and they can be removed by machining or by brushing or chipping. So impurities removal is easier in the case of the centrifugal casting process. Next one fettling costs are reduced fettling means unwanted what say elements if there are any such elements we need to remove them like cutting up the gating system riser. So these are the fettling what say process. So these are reduced first of all there is no gating system. So fettling costs are reduced and little scrap is generated cost of production is less. Now these are the disadvantages or the drawbacks of the true centrifugal casting process. First one is suitable only for axial symmetrical components only for axial symmetrical and for making hollow components not for all the components. Second disadvantage is more segregation of alloy component during pouring under the forces of rotation. Now this is another drawback of the centrifugal casting process if the what say generally we use the alloys. Now in the alloys there will be different elements will be there. If the density of different alloying elements or what say almost close there is no problem. It is possible that the density of one alloying element will be very high and the density of another alloying element will be very less. At such times what happens the element having highest density will be subjected to larger what say centrifugal force and the element having the lesser what say density will be subjected to lesser centrifugal force. Then what will happen the element having the lesser density it will be moving towards the axis or the center of the casting and the elements having the higher density will be subjected to higher centrifugal force and they will be moving away from the center of the casting. That is how there will be segregation of the alloying elements in the centrifugal casting process. So, this must be handled very carefully. Next one secondary machining is often required for inner diameter yes inside the inner surface of the casting all the impurities will be collected because they are subjected to lesser centrifugal force. So, these are to be removed sometimes the machining is required. Next one centrifugal castings require very high investments that set up is very costly and expensive. Next one skilled labours are to be employed for this process right because the mould is to be rotated at a predetermined speed and it all depends upon what say a temperature for a particular temperature or for particular alloy it has to be what say rotated at different what say speeds. So, this all what say comes by experience. So, to operate this set up experience and skilled workers are required that is how the process will be costly or it becomes what say expensive. Next disadvantage is the inaccurate internal diameter and finally, it the another disadvantage is the long lead time may happen. So, these are the drawbacks are the disadvantages of the true centrifugal casting process. Now, what are the applications of true centrifugal casting process? Manufacture of pipes, bushings, nozzles, bearings, inner liners for IC engines, bimetal, steel, bronze, bearings and so on. Now, what are the materials cast alloy steel, carbon steel, cast iron, stainless steel, aluminum, copper, nickel. Now, let us see the part size diameter varies from 0.25 meters to 3 meters. You can see the diameter can be 3 meters means very large and length up to 15 meters and weight up to 5 tons means small castings can be made and very big castings can be made. Now, these are the components produced by true centrifugal casting process. You can see here these are the hollow what say tubes or the hollow components cast components are produced by true centrifugal casting process. We will see some more yes these are the tubes or the components produced by true centrifugal casting process. First one again these are the components produced by true centrifugal casting process. Again these are also the components produced by true centrifugal casting process. Again we can see here these are also the components produced by true centrifugal casting process. Now, bimetalic pipes using centrifugal casting process. Bimetalic pipes can be produced by centrifugal casting process and some bimetalic combinations are as follows. we can see here. So, these are the outer layer what say alloys and these are the inner layer. For outer layer it you can see 5 percent chromium steel and for inner layer it is stainless steel. For outer layer stainless steel and for inner layer mild steel, for outer layer mild steel and inner layer copper. For outer layer mild steel and for inner layer aluminum, for outer layer it is copper and for inner layer it is GCI and for aluminum outer layer and the inner layer is GCI, outer layer is stainless steel and the inner layer is GCI, outer layer is mild steel and inner layer is nickel hard steel. Now second metal is poured inside the mould after some time and this time lapse after solidification for the first metal and before pouring of the second metal is calculated by D is equal to K root of T, where D is the thickness solidified and K is the solidification constant and T is the time. So far we have seen true centrifugal casting process and the two subclassifications within that. Now, let us see the semi centrifugal casting process. Now what is this semi centrifugal casting process? Semi centrifugal casting manufacture is a variation of the true centrifugal casting process. Yes here also the centrifugal force will be applied on the mould and also on the molten metal, but there is a variation. What is that variation? What is that difference? The main difference is that in semi centrifugal casting process the mould is filled completely with molten metal which is supplied to the casting through a central sprue. In the case of the true centrifugal casting process, yes no doubt the what say there is a cylindrical mould and it is rotated, but finally the cast component will be hollow, there will be a hole inside, but here in the case of the semi centrifugal casting process we would not get the hollow components that is the main difference. You see again let us see in the main difference is that in the case of the semi centrifugal casting the mould is filled completely with the molten metal. The forces generated by the rotation of the mould ensure the distribution of the molten material to all regions of the casting because of the forces generated by the centrifugal force the molten metal will be what say uniformly distributed into the mould. And here we can see yes this is the revolving table will be there and this semi centrifugal casting process in this process the mould will be rotating about a vertical axis always and here there is a revolving table is there and here we can see this is the moulding box. Moulding box means yes there will be drag, drag means lower moulding box and there will be a cope there will be means the upper moulding box and here there is the pouring basin pouring basin and this is the mould cavity and as the mould is rotating about a vertical axis the molten metal is being poured into the mould. Then what happens as the mould is rotating molten metal is entering the molten metal will be subjected to the centrifugal force. So, it will be going away from the centre if there are any minute details tiny details on the sides of the casting yes molten metal will be definitely entering into those tiny features. So, that is the advantage of the semi centrifugal casting process and here we can see so this is the pouring basin and this is the cope and this is the drag and here we can see a component produced by the semi centrifugal casting. So, this is the wheel with spokes and here we can see this is the pouring basin and here internally we can see there are rips are there narrow rips are there. If we cast this component using ordinary what is a sand casting process no doubt the outer part we may get, but the internal rips it may not be possible for us sometimes there will be some discontinuities will be there, but these discontinuities can be overcome using the semi centrifugal casting process because of the centrifugal force that is falling on the molten metal. So, even this what is a place where there are there may be some discontinuities there also molten metal will be successfully flowing and this is the semi centrifugal casting setup. Advantages of semi centrifugal casting process in this number of modes can be stacked together one mode over the other can be fed by a common central sprue in order to produce more than one casting at a time means one mode can be placed over another mode and all these can be connected by a central sprue common sprue and the molten metal can be supplied to all these modes at a time means at a time we can produce more than one casting. Second advantage is similar to true centrifugal casting and hence it ensures purity. Poor structure forms at the center of the casting so this can be machined if there is any what is a impurity again that will be collected at the center and that can be machined off. These are the applications of the semi centrifugal casting process wheels you can see this kind of wheel. So, this is manufactured by semi centrifugal casting process pull is you can see here these are the pull is. So, these pull is are manufactured by semi centrifugal casting process. So far we have seen the true centrifugal casting process and the subclassifications within that then we have seen semi centrifugal casting process. Now, let us see the third classification of the centrifugal casting process that is the centrifuging process. What is the centrifuging process and what is its principle? In centrifuging casting manufacture modes employed to produce the desired castings are arranged around a central sprue. Here several castings at a time are manufactured within a using a single mode. If it is the true centrifugal casting process or the semi centrifugal casting process only one casting in a one mode, but here within a single mode there will be more castings. So, that is the special feature of the centrifuging process. These modes contain all the necessary geometry for the cast parts as well as the gating system. Runners travel from the central sprue to the mode entrances. This provides a means of increasing the filling pressure within each mode and allows for reproduction of intricate details. Centrifugal casting is often used in conjunction with investment casting process.  Sometimes castings with internal not internal tiny features intricate ships in which it is very difficult for the molten metal to enter or in to penetrate. At such times we use the centrifuging process. Then what happens as the mold is rotating because of the centrifugal force that is falling on the molten metal. The molten metal will be passing through the tiny details. It is used it is also used with molds made up of silicone rubber with sufficiently higher temperature resistance for repeated use in casting without mold deterioration. So, this process is known as the spin casting means same thing, but here the molds are made up of silicone rubber. In such a case we call it as the spin casting and cast materials for spin casting include zinc base alloys, lead base alloys, tin base alloys, aluminum and plastics. Now here we can see the what is a setup. We can see here there is a again this also what is a carried out about a vertical axis similar to the true what is a semi centrifugal casting process, but what is the difference between the semi centrifugal casting process and the centrifuging in the case of the semi centrifugal casting process. There will be a one mold will be there and it will be rotating about a vertical axis, but only one component is manufactured using one mold, but here there is only one mold, but there will be more components will be there. For example, this is one component, this is another component and this is another component, this is another component means all along the circumference there will be several what is a what is a mold cavities will be there each cavity meant for one casting. So, all these are cast at a time all these are joined. So, after the solidification we have to remove we have to cut and we get so many castings at a time and here we can see more details you can see here this is the central sprue and here this is the one casting this is one casting this is one casting this is one casting likewise there are several castings are connected to this central sprue. Now when we pour the molten metal as the molten metal is entering into this central sprue it will be rotating the mold will be rotating and because of that there will be centrifugal force sub applying on the molten metal and because of that because of the centrifugal force falling on the molten metal they will be passing through the tiny what is a cavities. What are the advantages of centrifuging process? Ensures better quality castings when generated on large scale will be economical just now I have told you in the case of the centrifuging process using a single mold several castings can be made with one sprue several castings can be made at a time. So, that become that is economical castings when generated on large scale will be economical. These castings generate high yield because for several castings there is only one central sprue. So, the what is a molten metal consumed for the gating system is very less that is how casting yield will be higher. The castings obtained by this method cannot be satisfactorily obtained by other casting methods. The what is a filling up of the mold by the molten metal is excellent because of the centrifugal force falling on the molten metal. Next one cleaning and fettling cost of the castings can be reduced denser castings having physical properties similar to those of the forged products. Next one rejection of percentage is very low because most of the castings are sound castings no discontinuities. Next finally easy way to achieve directional solidification. This directional solidification we have already learnt so I do not want to repeat again. Now let us see the centrifugal force centrifugal force acting on rotating body is C f is equal to M e square by R where M is equal to mass in kilograms V is the peripheral speed meters per second R is the radius in meters. Now gravitational force G f is equal to M g where g is the acceleration due to gravity meters per second square g factor is equal to C f divided by G f that is equal to M e square divided by R into M g that is equal to V square by R g. Solving further we get N is equal to 42.3 root of g factor divided by T. So this is the what is a formula for the what is a speed that is to be what is a operated. Now expected requirements of a centrifugal casting machine what are the what is a requirements? The machine must be able to accelerate the mould to a predetermined speed, maintain smooth spinning and this decelerate to a stop in a reasonable time frame. There must be a way to heat and coat the mould before pouring the molten metal. Otherwise what happens the molten metal may stick to the mould. So to prevent this we have to give a ceramic slurry coating this must be dried and this must be heated and also it must be baked. So thus that much provision must be there within the machine. There must be a means to pour the molten metal safely into the rotating mould at a controlled rate position and orientation. There must be a means of adding inoculants or fluxes for some special applications. The role of the inoculants and the fluxes we have already seen. So even this centrifugal casting machine should have means of adding these inoculants and fluxes. Once the molten metal is poured a proper solidification and cooling rate must be established in the mould to obtain a desired casting microstructure. Next one there must be a means of extracting the solidified casting quickly from the mould at elevated temperatures without deforming the casting. Common mould materials first the most common mould material is the metallic permanent moulds. So these are widely used. Next one refractory lined metal moulds. Next one sand lined metallic moulds and finally graphite moulds. The metallic permanent moulds are most widely used because of their reusability. Great casting geometry and high productivity. So these are the common mould materials among these metallic what is a moulds are widely used. Now let us see the defects in the centrifugal casting process. So these there are three types of major defects in the centrifugal casting process. One is the segregation banding, second one is the reining, third one is the vibration defects. First let us see the segregation banding. Bands are annular segregated zones of low melting constituents such as eutectic phases, oxide or sulphide inclusions. So these are the segregation banding. It occurs in true centrifugal casting generally where the casting wall thickness exceeds 50 to 75 millimeters. Banding is more prevalent in alloys with a wide solidification range. Minor adjustments to the casting operation variables such as rotational speed, pouring rate and metal mould temperatures will usually reduce or eliminate this defect. Now let us see the second defect that is the reining. Now what is this reining? Reining is a phenomenon that occurs in horizontal centrifugal casting process. If the mould is rotated at too low speed or if the metal is poured into the mould too fast the metal actually rains or falls from the top of the mould to the bottom. Just like rain the molten metal will be falling down. Why this is happening? Too low speed of the mould or too high speed of the pouring of the molten metal. So these are the main reasons for the reining of the molten metal from top to the bottom. So proper process control can eliminate the reining. So that is why sometime back we have seen that to operate this process skilled labours are required. Only those skilled people know at what rotation there will not be any reining. So if the mould rotation is too low there will be reining or if the rate of entry of the molten metal is too high that time also there will be reining. So these can be controlled by experience. That is why experienced and skilled workers are required to operate this machine. Finally let us see the vibration defect. What is this vibration defect? Vibration can cause a laminated casting and it can be reduced to a minimum by proper mounting, careful balancing of the moulds and frequent inspection of rollers, bearings and other equipments. So using what is a these parameters we can minimize the vibration defects. Now recent trends in the centrifugal casting process. Combustion synthesis in centrifugal casting it is used to fabricate inter metallic components metal and the other one is the ceramic layered components. So to make to fabricate inter metallic components this centrifugal casting process is used in which there will be what is a combustion based synthesis will be taking place. And here we can see this process in details using a diagram. So this is the combustion synthesis in the centrifugal casting process. So here we can see this is the reactor and this is the sample container and this is the igniter which what is a initiates the combustion. Now this is the reactant mixture. Now what will happen when this reactant mixture because of this igniter what is a will be reacting and there will be combustion inside the chamber and because of that as we have seen previously we can fabricate inter metallic components means where one layer is made up of metal and the other layer is made up of ceramic layered components. So this is the recent trend means using the combustion synthesis in the centrifugal casting process. And the next one is the using a yes in the previous case we were using the combustion and instead of combustion why not to electrical heating. So that is the next what is a development in this process and here we can see yes this is the green compact and this is the mould part and third one is the alumina container this is the alumina container and the fourth is the heating coil this is the heating coil means this is heated using electricity. Fifth one is the thermocouple and sixth one is the inertial container this is the inertial container seventh one is the outer steel container eighth one is the electric source this is the electric source and the ninth one is the graphite brush this is the graphite brush and the tenth one is the slip ring this is the slip ring. Now the principle is same we want to manufacture what is a bimetallic parts or the case place where there will be components where there will be metal ceramic layered components will be there. Now using the electricity what and there is a coil heating coil this will be heated up and here there is a green compact is there and because of that green compact yes this will be heated up and we get the required component required component means one layer will be metal and the other layer will be ceramic layered component. Now with this we are coming to the end of this lecture we have seen the principle of the centrifugal casting process and we have seen what is a branch classification of the centrifugal casting process. One is the true centrifugal casting process the other one is the second one is the semi centrifugal casting process and the third one is the centrifuging process. In the case of the true centrifugal casting process there will be either horizontal mould or a vertical mold will be there and these will be rotating at a predetermined speed and the molten metal enters into these molds, but in this case we get a hollow component or a cast component with a hole inside. So that is the specific feature of the true centrifugal casting process and we have seen the semi centrifugal casting process. What is this? This is a variation of the true centrifugal casting process and the mold always rotates about a vertical axis, but what is the difference between the true centrifugal casting process and semi centrifugal casting process? In the case of the true centrifugal casting process, we always get a hollow cast component or a cast component with a hole inside whereas in the case of the semi centrifugal casting process we get a, there won't be any hollow what is a cavity will be there. We always get a, the means the molten metal is always completely filled with the what say by the mold and the third one we have seen that is the centrifuging process. In the case of the, in both these two cases semi centrifugal casting process and the centrifuging process the mold will be rotating at a about a vertical axis. Then what is the difference between semi centrifugal casting process and the centrifuging process? In the case of the semi centrifuging casting process, yes there will be a mold will be there and it will be rotating about a vertical axis, but only one cast component can be produced using a single mold, but in the case of the centrifuging process using a single mold several castings can be made and again in the case of the semi centrifugal casting process the what say cast component will have a symmetrical geometry whereas in the case of the centrifuging process the cast components may not have a symmetrical geometry. So that is the difference between the semi centrifugal casting process and the centrifuging process. So with this we are completing this lecture and we will meet in the next lecture. Thank you.


Hi, I'm Thierry Carréz. I'm from France. I'm the director of engineering for the OpenStack Foundation and also the chair of the OpenStack Technical Committee. Well, thanks for joining us today. I was hoping you could tell us a little bit more about the Big Tent. We've been hearing a lot about it. What is the Big Tent? So the Big Tent is an idea that we should have within the OpenStack community. We have a lot of different projects, a lot of different initiatives that are created within our community. The idea of the Big Tent is to consider those people being an official OpenStack project, rather than having a segregation between a blessed project on one side and the rest of the community project. These are all OpenStack projects as long as they're created by the OpenStack community using the OpenStack ways and that's what the Big Tent is about. So within the Big Tent there are core services projects. Can you tell us a little bit more about what they are and what it means to be a core services project? There are two different initiatives. From an upstream perspective we just produce OpenStack projects. The Big Tent is everything the upstream OpenStack community produces. On the other side we have the board of directors trying to come up with a base interoperability set. So what does it mean to be an OpenStack cloud? What does it mean to be powered by OpenStack? What does it mean to be an OpenStack power storage? And to do that there is an effort called DevCore which goes into defining what is the set of those core projects that need to be in every OpenStack powered cloud or every OpenStack storage powered cloud etc. So the difference between the two sets is the core services is a very smaller subset of projects and it's usually when a project has reached a level of maturity and deployment that you can basically assume that any OpenStack cloud would include those services then it becomes a core service. So this is new for the Liberty Cycle Release, this Big Tent and previously it was the Integrated Release. So could you talk a little bit about what precipitated the change from the Integrated Release to now a Big Tent style release? Sure. So the Integrated Release back in Ice House or Juno, back in Juno we were in a situation where we couldn't grow the Integrated Release anymore and we could not really make it smaller either. There was this set of projects, it was really difficult for other projects to reach the level where they could be integrated in that set of projects, the requirements were pretty high. At the same time we didn't have a great focus on critical pieces either. So the Integrated Release did not serve any purpose anymore, it was excluding other projects, promising projects, new technologies. So it prevented OpenStack from being an integration engine and we wanted to be able to integrate new technologies as they come and as long as you have this artificial barrier between projects that are blessed, part of the Integrated Release and things that are outside in the community, we could not really solve that. You could not get to the level of maturity and deployment if you were not part of the Integrated Release and the Integrated Release required maturity and deployment so you could not get there. We were basically stuck by the Juno Release and there were a number of blog posts by various members of the community on how do we solve this, how do we solve the TC being the ultimate judge of all good things in OpenStack and let multiple flowers bloom in our community. And that's where the idea of the Big Ten comes from and we switched the question from judging if a project is OpenStack to judging the community producing it. Is it part of our community? Is it us? Is it solving the OpenStack mission? Is it using the OpenStack ways? And as long as it participates in solving the OpenStack mission using the OpenStack way, then it's an OpenStack project for all means. Well, thanks for telling us more about the Big Ten and I have one more question which is, can you speculate about any changes that we might see in the Mitaka Release cycle? It's a hard one. It's a second day. There were a lot of interesting discussions yesterday at the cross-project workshop track in the design summit and questions that we never really asked ourselves before. Like, should we as a project never ever drop an API again? And it's a hard call for a developer community to make because that means you will have to maintain APIs forever. But that means you care about your users enough that you don't want to put them in a situation where they have to evolve their applications from one API version to another. And so it's a sign of maturity that we are having those discussions and the result of that discussion was that we would probably push for never duplicating an API again in the future life of OpenStack project. That's a major change. We also discussed the introduction of a common distributed block manager for OpenStack projects. So we currently have OpenStack projects are backed by databases and queue services but we don't have a distributed lock system and we emulate it using queues and databases and that's a failure. And so we discussed introducing a common distributed lock manager that all OpenStack projects could take advantage of to better synchronize data and scale better. So that's the two things that I saw emerge yesterday and I need to run to see more. Thank you so much for joining us today. No problem. Thanks.


Item No. SCP-4429 Object Class Euclid Special Catant Procedures Foundation agents embedded in William Shatner's PR firm are to ensure he remains isolated from any reference to pulp song common people. Free interviews are to be conducted with interviewers and hosts to determine whether or not they intend to reference common people during his appearance. Anyone who engages him in a conversation about the song is to be contained and amnesticized. Description SCP-4429 is the anomalous persistent belief that musician and actor William Shatner recorded a cover version of the song Common People, originally by Brent Pott Band-Pulp. While no such recording exists, affected individuals have consistent memories of this cover. The song's lyrics deal with an affluent female college student who glamorizes poverty and wants to be like common people. In the non-anomalous pulp original, these lyrics are delivered in a disaffected ironic manner, but all those under SCP-4429's influence describe Shatner's cover as aggressive, biting, and far more acerbic in its mockery of the student. SCP-4429 affects upwards of several million people. Those affected allege that this cover appears on Shatner's 2004 album Has Been, which in fact exclusively contains Shatner's own original spoken word pieces. Among those affected is musician Ben Folds, who produced Has Been and claims to have produced the Common People cover as well. It does not, however, include William Shatner himself, who is the only known individual associated with this album to be completely resistant to SCP-4429's effects. SCP-4429 has a secondary related effect that manifests when an effective subject attempts to converse with William Shatner about the supposed song. Those affected will become verbally hostile towards Shatner, escalating towards threats of physical harm. It is unclear if this would eventually lead to actual physical harm, as Shatner has consistently been unwilling to remain in proximity of affected subjects once they start threatening him. While the Foundation became aware of SCP-4429 due to Shatner's failed attempts at prosecuting several such individuals, he has nonetheless declined all requests to test out the limits of his effects, even in controlled environments. When interviewed, affected individuals express no remorse. Addendum-4429-A Transcript of Radio Interview, November 2, 2004 The following is a transcript of an interview on the Music Radio Program Q between William Shatner, Ben Folds, and host Gian Gomeschi. This is the first known occurrence of SCP-4429's secondary phenomenon. Gomeschi Moving on, we need to talk about that incredible cover of Common People, which absolutely blew me away the first time I played it and really has only gotten better and better on each listen. Folds Yeah, that one was really fun to put together, building with Joe Jackson singing and then the children's chorus at the end. But it does all come down to the delivery by Will, and that makes it Shatner. I'm sorry, I… Gomeschi Sorry to cut you off, but I just want to make this point and then get your feeling on it, and we can discuss the song in that context as well. It really feels, in many ways, like an extension of the lovely monologues you did on Star Trek, where you absolutely lay into some hapless alien who doesn't really understand humanity. Did that play into it at all? Shatner I really don't understand what we're talking about here. Gomeschi Has anyone ever told you that making music involves actual singing? You're not supposed to just say all the words into the mic, you know. Folds I needed to tell him that, but he really wouldn't listen. Shatner Fellas, I could take a joke as well as anyone, but I… don't really care for these radio prank things, I'll be honest. Gomeschi Oh God, is this because I mentioned your acting career?  Folds He is a bit of a prima donna when it comes to stuff like this. Yeah Shatner What? I'm not willing to talk about my work on Trek, I'm very proud of it, it's just that…  Shatner Aren't… Aren't you Canadian too? What is this?  Gomeschi Tell me, Captain, have you ever seen what your intestines look like? Would you like to?  Shatner I… Why…    Gomeschi repeatedly smashes his face into the table in front of him, resulting in multiple cuts, contusions and broken facial bones. Gomeschi Duh, look at me, I'm William Buck and Shatner, and I'm the biggest dipshit in the world. Folds That's you, Shatner. That's what you look like, that's what you will look like after we're done with you. Shatner Jesus… Christ… Shatner rapidly exits the interview room, Folds and Gomeschi make no effort to follow him.  Addendum 4429B Data recovered from William Shatner's personal archives, R.E. Ronald Coleman Data was noted that in multiple instances of SCP-4429 induced hostile behavior towards William Shatner, the phrase, Fat Canadian Fuck, was used even in cases where it did not fully make sense as an insult. As in the example in Addendum 4429A The Foundation explored Shatner's extensive collection of personal fan correspondence and discovered this phrase multiple times in the letters of Ronald Coleman, a 37-year-old unemployed male from Seacoccus, New Jersey. Coleman had written hundreds of letters to Shatner, largely consisting of interpretive insults and mockery, several of which liberally quoted from common people. In these letters, Coleman accused Shatner of falling out of touch with his fans and deliberately ignoring them. Shatner was not aware of Coleman and had read none of these letters as his personal assistant had filtered them from his inbox. The last letter Coleman wrote was posted shortly before the release of Has been, and consisted solely of a two-word profanity. In sharp contrast to the majority of his writings, Coleman's first dozen letters to Shatner were uniformly positive, praising him for his acting talent and musical prowess. As far as the Foundation has been able to determine, the tipping point for Coleman came after Shatner left the Seacoccus fan meetup early, before Coleman was able to get him to sign an autograph. Coleman disappeared shortly after his final letter, and his current whereabouts are unknown. Foundation agents searched his house and recovered a copy of Has been, which contained a cover of the song Common People, apparently by William Shatner, that fits all known descriptions of SCP-4429. This copy has been extensively tested for cognitohazards and is completely non-anomalous. It is the only known recording of this song to actually exist.