So questions from the homework and I recognize that some of you are going oh, I didn't do the home I I'll take questions still on Tuesday as well. Any of these you want me to go over? question three, okay Quiet, please Question three, and I think I think I think I sort of like question three I think I sort of like there in fact, you know what I do like question three I do like question three. I do like question three It says a 3.2 kilogram mass moving at 4 meters per second east hits a wall The wall exerts a force of 250 newtons to the west for point zero seven seconds Find the final speed and the direction Okay, we have a collision and we have a force. I think I'm going to be solving this using impulse I think I'm going to be solving question number three by going That's the equation you have on your formula sheet. Is that okay? Um Did they tell you the force? Okay, you have to say it louder and be a bit quieter the point. Sorry. What? Direction direct because now direction is going to matter. I see east and west now Uber paranoid. Sorry. What? Okay, I'm going to draw a little compass over here And I think I'm going to let east be positive and west be negative So I'm going to disagree with your 200 few newtons and I'm going to say technically what is it 350 or 250? I just can't okay. Hang on stop moving Stop talking say it loudly every time you said that first syllable someone's trouble 250 thank you now you keep moving and did they give me the amount of time involved point Point zero seven Seconds and what's changing anything and since momentum is mass times velocity. This is going to be mass the final minus mass The initial so far so good Did they tell me the mass 3.2 kilograms Excellent so I'm going to write this as negative 250 point zero seven equals 3.2 v final minus 3.2 and they did tell me v initial did they not Direction direction direction positive four negative four which one How do you know? Okay, we decided to be consistent here with this so positive four Otherwise, I put a negative four in there and out of a minus minus which would really change the answer quite a bit Okay Let's see what this side is negative 250 times point zero seven I get negative 17.5 equals 3.2 v final minus 3.2 times four 12.8 plus 12.8 to both sides. Yes I'll get negative 17.5 plus 12.8 and you'll notice I got nervous enough about this one that I didn't do this Algebraically I went to numbers because I was a little worried that with all the negatives and positives that I might do something sloppy I figure I'll be more careful with negatives and positives with numbers was by thinking I Get this Negative four point seven equals three point two The final now what to net And you're gonna get a negative. What's the negative velocity telling you direction is You know what it hit the wall and bounce off the wall its final velocity must have been To the wet whatever the four and the force that applied and that hit it I'm assuming it's a wall or head on collision was enough to not only stop it, but to set it backwards negative That okay, that's a great lead into today's lesson more questions Want you to put on your thinking caps a little bit here because what I'm gonna do is take you from To one dimension everything in a nice straight line Where the worst genetic that we have to do is let one direction be negative in one direction be positive and Two dimensions or we're gonna be doing lots of diagrams tip to tail tip to tail tip to tail And that's really the first question you start to ask yourself in this unit you say if I'm using momentum Can I is everything in a nice straight line? I'll let to the right be positive into the left be negative or to the left bean that positive into the right Be negative whichever is more convenient If I'm not in a nice straight line Raise your level of awareness and we'll talk about how we'll handle it first of all Example one a 30 gram bullet moving at 250 meters a second strikes a 50 kilogram Wood block which is at rest and sticks into it If the bullet and the block move together after the collision on a frictionless surface surface a Right a momentum equation B find the speed of the block after the impact and See just for giggles. Let's find the heat energy created during the collision. Okay, we can do all of that First of all, is there a collision? momentum Secondly Are all my motions in a nice straight line or do you see weird angles mentioned in this question? So straight line means I'm not going to be as paranoid I'm probably gonna let to the right be positive into the left be negative Every once in a while when I'm doing a question I'll notice it's way nicer to let to the left be positive into the right be negative But that's rarely usually my default is going this way Give me an elbow for me, would you? Yeah, it can wait. No, he was reading a history book Not the it wasn't the phone He was reading. He was reading very small type font with no spaces Which meant it couldn't be this because it didn't have any pictures and I would think that if Joel's reading it would have picture Oh, okay. Anyway, all right Joel you ready What does part a say to me my friend? I always start out by saying my momentum equation is gonna be this the sum of all the initial momentum Equals the sum of all the final momentum. That's where I start That's my hey, this is one size fits all and from here. I'm gonna break it down a bit further If a bullet and a piece of wood, so I'm gonna use B for bullet and W for wood or you could use mass 1 mass 2 whatever Before the collision, what's moving the bullet the wood or both? Bullet it's gonna be the momentum of the bullet initial Wham collision Equals is the collision After the collision, what's moving the bullet the wood or both? Both are they stuck together or are they separate? So if they're stuck together, I can write them as one momentum if they weren't I'd write them as separate It's gonna be the momentum of Both final Because I'm in a nice straight line I can go straight to doing algebra I can use dividing and subtracting things to both sides and adding that I can use my normal math eight I don't need to use vector math This is gonna be mass of the bullet the initial equals mass of the bullet plus mass of the wood The final okay now a said momentum equation the coal you could argue It's this you could argue. It's this you could argue. It's that I'm just gonna put the a there I'm gonna say there's my starting point B wants me to find the speed of the block, okay? B which of these is the speed of the block and the Bullet I assume Because they're stuck together which of these is the speed of the block and the bullet afterwards Vf are these in a nice straight line? Then I can just divide to get the VF by itself don't need to go all vectory yet Back the final is gonna be the mass of the bullet the initial All over the mass of the bullet plus the mass of the wood the final Is going to be Thirty grams is point zero three V is two hundred and fifty All over point zero three plus five Point zero three times two hundred and fifty all over Point zero three plus five You get one point four nine Yep One point four nine Meters per second what does C want us to find Jacob the heat what? You said energy now. I'm going to use conservation of energy not work because they didn't talk about work now I'm going to say for part C Kinetic energy initial plus potential energy initial equals kinetic energy final plus Potential energy final And plus heat Connor any of these zero Yeah fact, you know what I wouldn't have minded if you would jump straight to saying all of my kinetic At the beginning is going to be some kinetic afterwards and whatever is missing must have gone into heat In fact, this is almost the same physics of when I spammed those two metal spheres together Except there the final kinetic was zero and all of it went into heat. This is going to be a Half m v initial squared. What's starting at rest? Is everything at rest before the collision? What's moving before the collision? What isn't the bullet moving before the collision? Yes But Nicole you've raised a good point Here's what we're gonna have to do before the collision Nicole. What was moving before the collision only the So I better make a note. This mass is only the bullet Okay equals a half After the collision what was moving ah? Mass of the block sorry the bullet and mass of the wood the final squared Plus heat This is the first time that our masses aren't gonna cancel because we have different masses before the collision Only one mass was moving after the collision two masses were moving, but they both have kinetic energy Now can you all look up? This is the most common mistake. I saw in yesterday's test to get the heat by itself How would I move this over? What's happening in front of the heat right there? What sign is that a plus sign? How would I move this over then? Yeah, I had about 10 kids divide okay, I Had a whole bunch of students divide which told me a couple of things first of all told me they probably hadn't done much of the homework because he probably would have found out that that didn't work and Hey, that's sloppy throwing away math Not gonna divide the time thing the heat. How could I divide this over? I have to subtract it over in fact I'm gonna get this The heat is gonna be a half mass of the bullet the initial squared minus a half mass of the bullet Plus mass of the wood v final squared, and I'm running out of room here Moving up a little bit. I would love to show the numbers, but I just don't have room here I'm gonna go to my calculator I guess point five times mass of the bullet was point zero three times initial velocity point zero three Mr. Duke times initial velocity two hundred fifty square two hundred and fifty squared minus point five bracket mass of the bullet Plus mass of the block times the final squared you get nine hundred and thirty two joules of heat. Yeah He equals nine hundred and thirty two Joules Emily's looking a little bit clamped Didn't work on your calculator. Oh, you're really hungry. Oh man Did you have breakfast? example two the diagram below shows two pop cans of unequal mass initially at rest and Explosive charge is detonated between them boom and They fly apart. What can you tell me about their velocities after the explosion? Yeah How about let's call it small can and big cans that you're really having a tough time working your tongue around the words so Go for it. What we got. I agree small can't first of all I can I think we can even be more specific than that. Let's see What I think the small can and go twice as fast the big can and go half as fast in Opposite what? You know why opposite directions? First of all, is there an explosion? What I said to you was I use momentum when there's a collision or an explosion in the shorthand I said when there's a jump explosion collision, I think momentum. What's my momentum right now? Nicole, what's my momentum right now? So what's my final momentum have to be? Has to be okay, so I'm going to go like this Zero equals the sum of all the final momentum And this is a nice also segue from Jeanette's question earlier today because since we have them traveling in opposite directions But in straight lines all that one direction be positive in one direction be negative.  How big is the small mass? What did we call it? And what do we call the big mass? Oh?  They do and in fact you get this final answer the negative small velocity is twice as big as The big velocity. What's the negative telling me in the opposite direction the small one is moving? Twice as fast as the big one, so we've actually done an algebraic proof of what you intuitively saw I don't know if I look for something quite this detailed and again You could always just make up reasonable numbers and crunch the numbers and say oh, it's twice as big That's that's fine, too, but there's a lovely algebraic Which one is bigger? This is going twice as fast as that right in other words if this is going 10 You just per second how fast is this going five to make this equal I better put a two there so I can get 10 equals 10 Don't feel bad everybody does that twice as fast. Why isn't the two on the twice as fast side? because The two needs to be on the twice as slow side to make the equation balance So that's all been a nice straight line some terminology here and now we're gonna go two dimensions important terminology There are two main types of collisions There are two main types of collisions. The first type of collision is An inelastic or completely inelastic collision The second type of collision is an elastic collision Unfortunately, first of all the words sound an awful lot alike when I say inelastic and an elastic you have to listen very closely I'll we try and pause there's any elastic collisions and then there are elastic collisions What's the difference between the two any elastic is when the masses stick together for example? if two cars t-bone each other and the wreckage sticks together and they move off together or Nicole like the block and the bullet that we just did those stuck together. That would be an inelastic collision Elastic collisions are when the masses bounce off of each other curling body checking and hockey Playing pool and hitting a billiard ball with another billiard ball. Oh any elastic Tackling ideally you're hanging on and sticking together. Yes so We have any elastic Then we have partly elastic and perfectly elastic. You know how I remember the difference What's an elastic band? What's that you all know when it tell me what when I say the word elastic band. What's an elastic band? Matt you said the magic word what? Here's what I remember elastic rubber bounce.  I'm thinking elastic elastic band rubber band a bounce They didn't stick together. They bounced off each other Okay, now in all collisions in all collisions Momentum is conserved in other words the sum of all of them into minstrel equals some of all the final in a perfectly elastic collision in a Perfectly elastic collision you have something else that's conserved in a perfectly elastic collision Kinetic energy is conserved now What's a perfectly elastic collision well for a perfectly elastic collision to occur You need the following both objects must have the same mass They must rebound it apart at a 90 degree angle So if the question says it's perfectly elastic you can assume the angle they fell apart at was 90 degrees I'm not gonna do a huge song and dance about this You are gonna see in the review a couple of times once or twice There'll be a question and you'll go mr. Do it They didn't tell me the second mass and they didn't tell me what angle they flew apart at and I'll say does it say perfectly elastic? And you'll say yes, I'll say well then they don't need to tell you the second mass and They don't need to tell you the angle example 3 a 60 kilogram hockey player traveling 20 meters towards the north Collides with a 50 kilogram hockey player traveling one meter per second towards the west the two become entangled together That means this is any elastic because they stuck together Can you see that this is no longer in a nice straight line Kara? See the directions Okay We're gonna start out the same But we're gonna be way more paranoid and they want to find to be final with what velocity to move up after the collision Eric do you see the word north mentioned in this question? I'm gonna draw compass rose over here. Is that okay Trevor? Okay Trevor is there a collision? I'm gonna start out by going the sum of all of the momentum before the collision has to equal the sum of All of the momentum after the collision so that hasn't changed Before the collision what was moving mass one mass two or both? Sorry Both where they stuck together before the collision Okay, so I'm gonna say this before the collision. I have mass one initial. That'll be the first hockey player Andrew Plus mass two initial that'll be the second hockey player equals Wham collision after the collision Who's moving mass one mass two or both? Both stuck together or separate Now right now some of you are saying mr. Dewick This is exactly almost exactly the same equation as you've used on the straight linear ones so far Now though we are not we are not we are not going to say momentum is mass times velocity We will eventually Now we're going to dull We're going to dull We're gonna draw a picture What direction is the first hockey player traveling? plus What direction is the second hockey player traveling?  Problem I don't know how big the arrows are. Oh, yes, I do momentum is what times what? So the mass of the first hockey player is 60 Times the velocity is two. You know what? This arrow is 120 long Is that okay connor? And what about the second guy well his mass was 50 His velocity was one This arrow is 50 long How long is the first arrow? How long is second arrow? So if I want this to look roughly to scale this arrow should be a little more than twice as long as this guy Okay So ready? 120 50 See the difference by the way. This was the same but now we're actually adding pictures What's my resultant? What am I how am I going to draw it first of all it's always from The tail of the first one to the tip of the second one. You know what? They're going to move off together in that direction and i'm going to call that on my diagrams a Momentum of both final Now this is a lovely right angle triangle, which means I can use so katoa and pythagoras So let's do that. How can I find out how big this is? Well momentum of both Squared is going to be 50 squared plus 120 where it's pythagoras And ze when you did that and then you took the square root What did you tell me the momentum of both is equal to? Okay Sorry Don't care yet because they didn't ask me to find the momentum final What did they ask me to find ze? Final velocity that's momentum Momentum is what times what? Ah I'm going to say this the mass of both The final equals 130 Emily, how can I find out the final velocity? okay The final is going to be 130 divided by 60 plus 50 What's the final velocity that they move off together? at Connor would you get oh never mind? I thought you were typing sorry You could be if you wanted to john would you get I can't hear you 1.18 meters per second And we're still wrong technically we haven't found the velocity You know what we found the speed Velocity has two things magnitude and what? so the final Vector equals 1.18 meters per second at There's my direction Ze which trig function can I use to find this angle? tan theta Equals remember all the spokes opposite over adjacent How do I find an angle? Yeah shift tan Making sure I'm in degrees. I am And I get 23 22 points. I'll go 23 degrees 23 degrees what of what? Jacob west of north By the way, do you see how as soon as you go non-linear not in a straight line? That's a lot more work But it opens up some very cool questions that we can solve because I'm going to tell you right now Most car crashes aren't in a nice straight line. You get t-bone. You get broadsided. You don't hit head on Turn the page Let's suppose in example four Oh, by the way, I love example four. I love example four except I'm going to put a little star next to this. This is advanced I'm not going to cross it out But we may not do it today finding the final amount of kinetic energy in the initial amount of kinetic energy And then subtracting there is one like this in your review. In fact, you know what? I am going to temporarily cross this out Let's do that But on your test in this unit the written section is going to be a collision at an angle Can you see Brett just by glancing at the diagram that we're not in a straight line? Is there a right angle here? This is going to be sin law cosine law stuff Is there a collision? What do they want me to find? Okay, you said there was a collision. I'm going to start out exactly the same by going The sum of all the initial momentum equals the sum Of all the final momentum Mitsu, that's my fallback stand by get my thoughts organized my questions no longer blank But now I'm raising my level of paranoia of being careful You know what I'm saying? The angle is going to be careful here. Before the collision Nicole, what's moving mass one mass two or both? Which one? No, how how which which of your answers that you said was correct mass one. I agree. Why how do you know stationary? Okay? yes so mass one initial equals After the collision, what's moving mass one mass two or both? Nicole are they stuck together or are they separate? This is an elastic collision not perfectly elastic because the masses aren't the same and it's not a 90 degree angle Which would make everything way easier anyways. It's an elastic collision. So I'm going to write this Nicole momentum of the first one final plus Momentum of the second one final Underneath this I'm going to dull The first momentum looks an awful lot. It seems like that equals Looks like the final momentum of this small mass looking at the picture Looks like that looks like the final momentum of mass two looking at the picture Looks kind of like that. I can fill in some of these because momentum is what times what breed? 16 Yeah right Uh, oh this is mass. Oh, I can do this one two times six point seven Which is I think 13.4 13.4 Can I fill in this one? Nope. So I'm just going to label that as mass two final Okay How will I draw this vector triangle? Well, how will I add these two together? And here's the nice thing When I add these two together, they're supposed to give me a horizontal line because that's what that is So here's how I would draw this watch I would go, okay I got a 13.4 I know to stop drawing this right here Not this far And not shallower the reason I know is Apparently when I add momentum one and momentum two final, it's supposed to add to exactly horizontal 16 This plus this has to give me that Oh, and they gave me an angle did they not Somewhere in my diagram originally see it see it see it Apparently this angle right here is 10 degrees. Oh 10 degrees So katoha No No right angle Sign law well, I would use the sign law if I knew both of those or both of those or both of those Do I have a pair anywhere? This is cosine line in fact It's going to be cosine law almost every time and you're going to get good at the cosine law What is the cosine law you ask? What is the cosine law you ask? Yes Cosine law says this how much room do I have lots of room still good C squared equals a squared plus b squared minus two a b Cosine big c Where mitzvah There's c right there what I'm trying to find Momentum of object two final squared equals What's a Nice thing is as long as you identify c correctly as being across from the angle It doesn't matter what you pick is a and b. It's going to be those two So i'm going to go a is I hack 16 squared b is 13.4 squared minus two times 16 times 13.4 Cosine what's big c 10 degrees The nice thing is if you've listened to my advice about getting a good calculator You should be able to go do this in one fell swoop 16 squared plus 13.4 squared Minus two times 16 times 13.4 cosine of 10 And I get 13.27 which is wrong What what what what bret? Yeah, I found momentum squared The final momentum of object two is going to be The square root of that 3.6 4 3 4 And i'll carry a few extra sig figs Because they didn't ask me for the final momentum Brett you're looking puzzled am I wrong Then we'll check uh, are you have a graphing calculator? Okay, make sure you're in degrees because the graphing calculator is by default going to radians if you've ever reset them Or replace the batteries or anything like that. I think i'm right in fact. I'm sure i'm right And then square root yes Brandon momentum is what times what our vendor momentum is what times what? So this is the momentum and I want to find the velocity. I think i'm going to go like this then Velocity final is going to be 3.6 4 3 4 divided by mass 2 that's going to give me the final velocity of the second object Actually, it's going to give me the final speed Get through some more tree What do you get for speed of object 2? What was the mass of object 2? I've scrolled down it was oh 10. Oh, I can do that in my head then 0.364 Meters per second is not moving very fast Which it shouldn't because it's a big heavy mass and it only got grazed by a much smaller mass at I'd like to find That angle I'd like to find don't write this down because i'm really going to gum up this diagram That angle Where does that angle appear in my triangle? Can you see the zed? I think if I want to find this angle What I really need to find is that angle And then go What of what for that? Does that make sense? Connor are you good? Sure because you're not looking good You are Okay, so i'm going to go like this That's theta That's theta How can I find theta soca toe? No sign law Wait a minute. We couldn't use the sign law because we didn't have a pair We do now have a pair because we just figured this out Which is why I carried a few extra decimals on my calculator The sign law says this The sign of our mystery angle divided by Okay, mr.  I'm going to call this theta. Sorry. I forgot to turn up. What's going on here Change colors, mr. Do it. Let's try this again Hmm problem Let's do this mr. Do it. Ah, there we go. I don't know what was going on there So i'm going to call that theta And that theta This is the angle that I really want but I can only find it by finding that one It's going to be the sign law because I'm going to have this the sign of my mystery angle divided by what's across from it 13.4 equals The sign of 10 degrees divided by what's across from it, which I didn't know at first but now emily I know it's 3.6 4 3 4 make sure you use the momentum and not the velocity When I cross multiply I'll get sine theta equals 13.4 Sign of 10 degrees divided by 3.6 4 3 4 By the way, this is almost as much work as you'll do for any one question all year 13.4 sine of 10 divided by 3.6 4 3 4 and oh Theta is point 60. Oh, no, no, no. I didn't find theta. What have I found? Sign theta, how do I find theta? Second function sign of my current answer Ah 39.7 or 38 degrees. Sorry for or 40 degrees I'll go 39.7 39.7 degrees What of what? What is that? What of what? Mitchell north of east Seven marks ready one mark Two marks three marks Four marks Five marks Probably actually a half mark and a half mark Probably you also one mark for that six marks And I find another way to give you part marks in there. Can you skip steps? Yeah Would I not unless I got really good? Yep Okay You know what your written question is going to have probably three collisions Probably one on a nice right angle, which takes um, dare I say it way less time It's going to be two Where they are not in a nice right angle Now I'm going to also though say although the math is tricky if you're careful It's reasonably plug-and-chug. There's just lots of room to make sloppy mistakes, too We're going to come back to example five We're going to come back to example six example seven A 10 kilogram curling stone is sliding along the ice. No friction When it hits a stationary 18 kilogram bucket of sand after the collision The curling stone's velocity is three meters per second east and the bucket has a velocity of 2.3 meter second at 43 degrees south of east What's the initial velocity? So here instead of saying be final it's saying. Hey, how fast was the stone coming in at? And this one is trickier also because no picture We're going to have to really think ready Is there a collision? Are we in a nice straight line emily? In other words, can I just let to the right be positive to the left be negative and just do my usual cross multiply divide and subtract junk How can you sorry easily at almost a glance tell that we're not in a straight line? south of east east north Angles So you ready? Are we in a nice straight line?  The momentum of the stone initial Wham there's a collision After the collision what was moving mass one mass two or both? stuck together or separate in elastic momentum of the stone final Plus momentum of the bucket final Why because emily this is not in a nice straight line Did they give me the initial direction of the of the stone as it was coming in? I'm going to hold off on that drawing that I'll come back to it. I am going to put A little equal sign here Did they give me the final Direction of the stone? Oh, yeah due east And velocity is not conserved momentum is conserved mass Velocity due east with a momentum of 30 plus Did they give me a direction for the bucket? Yep What 43 degrees south of east? There's east 43 degrees south of east How big is the momentum of the kit? What's the buckets mass? 18 what's the buckets velocity? Oh, okay 18 times 2.3 41.4 I'm still not quite sure what this one looks like but I can tell you what this looks like how would I add these together brie draw them Okay As a picture this is going to look like this 30 41.3. Oh What's the resultant here look like? That's momentum of the first What did I call it? I use the letter s didn't I? Momentum of the stone initial Which if I go back here means it must have looked like this Caitlyn is this nice right angle so katoa Oh And in fact, I got a real problem right now. I don't even have an angle. Oh hang on. Hang on. Hang on. Hang on I got this angle Yeah Which is This angle How big is this angle? Caitlyn what you're right Doesn't help me at all How big is that angle? What are those two add to? Okay, so how big is this angle inside the triangle here 137 And I can tell I'm gonna run out of the room so bear with me while I do this No, how will I find momentum of stone initial? Well cosine law Momentum of stone initial squared equals b squared plus c squared minus 2 b c Cos I'll hang on mr.  Nicole.  Oh, no, no, no, no, no, that's a square root square root square root, right? Oh, and that's not its final velocity. Anyway, that's its final moment initial momentum 66.468 41.4 or 41.3. I thought it was 41.3, wasn't it? Oh, I put a four there and a three there Oh, that's silly of me This was uh This was 18 times 2.3. Yes I think it's 41.4 Kara candy for you If you see me do something dumb like that try and point it out sooner so that all the people at home Aren't freaking out when they're looking at this lesson online Which is going to change that as well and that as well Put a four there mr. Dewick Let's go try that again Four four So you get that and then square root How about 66.563? Thank you Kara. Okay, that's the that's the momentum initial. How do I find the velocity initial? If I know momentum, how can I find well first I can find the speed divide by the mass so So I'm running out of room velocity of the stone initial Is going to be this number divided by the mass of the stone, which was 10 Initial velocity is going to be 6.5 Sorry 6.66 Get the homework off someone at Sean what angle this angle right here? Which is going to be what of what by the way South to east yeah, it was coming in south of east and then I guess it got deflected straight sideways And continued this way so let's see Sign of theta I should have left more room clearly over 41.4 equals The sign of the angle that I do know Over the side that I just figured out which was 66.563 When I cross multiply I'll get 41.4 times the sign of 137 divided by 66.563 And I take the inverse sign of that sorry. I'd love to be able to fit this on I think you get 25 degrees south of east We're going to temporarily pause here. I'm going to get a couple of questions to try and then we'll pick up with this on tuesday What questions can you try right now? Uh strangely enough, there's no homework attached to this is there. Ha that's a little silly Okay revising that I have a take home quiz for you not on today's stuff But on the last few days worth of stuff


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


In the last lecture, we were looking at direct methods for solving sparse linear systems. Sparse linear systems are ones in which a matrix is filled with lot of zeros, very few non-zero elements and towards the end of the lecture, we looked at the Thomas algorithm. The Thomas algorithm is for tri-diagonal systems. Before that, we also looked at block-diagonal systems. Block-diagonal systems, you can exploit the structure to come up with a computation method which is significantly, you know, computationally efficient as compared to the conventional Gaussian elimination. So, now, to continue with this, we looked at Thomas algorithm yesterday and I said that the number of multiplications and divisions is linearly proportional to n, which is significant reduction from n cube by 3, particularly for large dimensional matrices. If you have n to be 1000, n cube by 3 would be a very large number as compared to 5n minus 8. Just compare, just compare the number of multiplications and divisions that you have to do for a modest n of 1000. So, just continuing on this, I am going to talk today about Thomas algorithm for block triangular matrices or block-tri-diagonal matrices. So, for sparse systems, we looked at tri-diagonal systems and now, we look at this spatial class which is, so this is a block, this is a matrix. So, I am trying to solve an equation A x where A is given by this matrix. I am going to split again x into sub vectors. So, I have this vector x 1, x 2, up to x n. So, this is a block tri-diagonal system. Earlier case, we had scalars here. In earlier case, beta B 1, B C 1, A 2, B 2, C 2, all these were scalars and then what we had derived earlier was Thomas algorithm, when the diagonal matrices are scalar. If you look carefully, this matrix which you get here is not really tri-diagonal because each one of them could be a dense matrix. So, if each one of them is a dense matrix, the raw matrix is not tri-diagonal. It may have multiple diagonals. But we are able to identify this B 1, C 1, A 2, B 2, C 2 and so on such that this is a block tri-diagonal matrix. What I mean by block tri-diagonal matrix? The so called elements of this matrix are themselves matrices.  Is everyone with me on this? Is this clear? See these are sub-matrices within this huge matrix. These are sub-matrices. So, this could be a 3 cross 3, this could be a 3 cross 3, this would be then let us say this is 4 cross 4, then this would be 4 cross 3 and so on. We will have to have appropriate dimensions here. They need not be all of the same dimension. They can be of different dimensions. So, now I want to exploit the spatial structure this has a lot of zeros or zero matrices or these are zero matrices. That is why I am writing them in square brackets. So, this is a huge matrix whose elements are sub-matrices. These are sub-vectors. X superscript 1 and X superscript 2 are sub-vectors of appropriate dimension. If this is 3 cross 3, this X 1 will be 3 cross 1 and so on. If this matrix is 4 cross 4, this X 2 will be 4 cross 1 and so on. So, this matrix has appropriate dimensions. So, these are sub-matrices and these are sub-vectors. The right hand side also has been divided into sub-vectors and now I am just going to apply the Thomas algorithm that we came up with. The first step, the step 1 is called block triangularization. What do you do in Gaussian elimination? What do we do in Gaussian elimination? In Gaussian elimination, we make the diagonal below the main diagonal zero. Here we are not going to make the diagonal zero. We are going to make the block diagonal below main block diagonal zero. So, we are going to retain matrices along the main diagonal. I am going to retain matrices along the main diagonal. I am going to make everything that is below that is zero. So, this block triangularization would be first matrix is gamma 1 is B 1 inverse C 1. And gamma k is if you look at the Thomas algorithm which I developed for the scalar case, you would notice that what I am doing here is just a matrix analog of what I have done for the scalar case. It is not different. So, earlier I had written scalar gamma. I am writing here capital gamma. You have to be careful now because you have matrices. You have to talk about pre multiplication, post multiplication very very carefully. You cannot interchange the order and whatever was a division earlier would come out to be a matrix inverse here and so on. Then you have this beta 1 matrix vector will be B 1 inverse D 1. So, you have to worry about these elements when we do the transformation to block triangularization and beta k will be B k. Yes, we will say B k. So, my first step is block triangularization. So, in the block triangularization step what I am going to do is if I go back here I am going to replace this by I 0 I. So, all these will be replaced by identity matrices of appropriate dimension. Everything below the main diagonal will be null matrices and these upper one will be replaced by gamma 1, gamma 1, this will be gamma 2 and this will be gamma n. These vectors will be replaced by beta 1, beta 2, beta n. So, I have done block triangularization. This is block diagonal identity. This is gamma 1, gamma 2 which we are calculating here. Now, notice one thing even though you are doing matrix inversions here, even though you are doing matrix inversions here these are small dimension matrices. Each one of them is 3 cross 3, 4 cross 4 whatever even if it is 10 cross 10, these are small dimension matrices as compared to the big matrix. So, the individual inverses which are involved here are small dimension inverses. So, these you could compute by some standard method Gaussian Gauss Jordan method or Gaussian elimination or Gauss Jordan method basically. So, this you can do quick computations because the number of computations involved is relatively small as compared to the big matrix. If I were to apply Gaussian elimination to the big matrix the computations will be much more than doing these small inverses. So, that is why the next step is of course, backwards sweep. So, in backwards sweep we start from this end. Look at this i times x n is equal to beta n. So, this is my first. So, this itself is the solution for x n component. Using x n component in the second equation I can recover x n minus 1, x n minus 2 I go back and then I recover I recover the entire state vector. So, this algorithm it exploits large number of zeros. So, actually if you if you if you go back and look at this matrices this matrix in its original form right now I have written it in a block triangular or block tridiagonal matrix form. In original form this will be a banded matrix. So, there are few diagonals which are non-zero rest all above and below are zero. We are able to express those diagonals in terms of these matrices in terms of these matrices sub matrices and that is why we are able to come up with a computationally efficient solution for solving this particular problem. There are many other such forms and as I told you that if you go to Matlab toolbox you will find there is a Matlab toolbox for sparse systems and there are many more forms which one another simple form is to solve are triangular matrices. Triangular matrices are either lower triangular matrices or upper triangular matrices. Now where do you get upper triangular lower triangular matrices? In Gaussian elimination you get lower triangular upper triangular matrices right. In Gaussian elimination you will get an upper triangular matrix and then you do backward sweep right which is. So, if you have a lower triangular matrix you can do computations very fast and then likewise if you have a block lower triangular matrix you can come up with a algorithm which is again very fast and exploit the structure not worry about the zeros which are there and then try to come up with the solution. Just look at the notes I am not going to do it on the board. We have to move on to something else there is one more concept I want to introduce here. Now block triangular or lower triangular upper triangular these are spatial structures and likewise you can go on exploiting these structures to come up with efficient Gaussian elimination algorithms or efficient modifications of Gaussian elimination algorithms which are suited for a specific structure and you can do fast computations that is the idea. So, as I said my motivation behind talking about sparse matrices was to sensitize you that there exists something called sparse matrix computations. Now what is the origin of these problems is problem discretization that is discretization of boundary value problem discretization of partial differential equations on finite element or finite difference all these methods orthogonal collocations on finite elements all these will give rise to certain nice structures which has lot of zeros and then you can exploit that to come up with solutions which are efficient. This is particularly useful when you have iterative procedures. In iterative procedures you may have to solve AX equal to B kind of equations Newton-Raphson you are actually solving multiple times AX equal to B. You never actually when you actually do Newton-Raphson step you never in a large scale problem you never do A inverse it is fine to do it for a beginner problem which has three variables or four variables but when you have large number of equations and when you are doing Newton-Raphson you actually solve A delta x equal to minus f at x k solve a linear problem and then substitute delta x and get a new x. So, you have to solve multiple times linear algebraic equations and that is where exploiting the structure. Now if the if your solution scheme has a gives rise to a specific structure to A matrix that will happen in every iteration it is not going to be different in only the numbers will change iteration to iteration the structure of the matrix will remain same. So, if you are calling a specific subroutine sparse subroutine within your Newton-Raphson that subroutine will remain same it is not going to change. So, if I am doing the problem this TRAM problem in TRAM problem suppose I decide to do say orthogonal collocations on finite elements. So, three finite elements every time I will get a matrix in the iterations whose numbers might change but the structure will be same. Then the wherever zeros appear zeros will appear every time that is because of your structure of discretization or if you decide to do it using finite difference within the iterations that matrix is always going to be tridiagonal it is not going to change. So, those features will remain same those features are not going to change just remember that. So, you can actually make use of those features and come up with there is one more one more trick for reducing the computations this is if you have a large system now I am not talking about not necessarily talking about sparse matrices the thing what I am going to talk about need not be a sparse matrix but it is a trick to make computations fast. So, solving a matrix by a matrix partitioning I am going to explain the basic concept it can be extended to more complex partitioning I want to solve a x equal to b of course where a is a large matrix I do not want to invert the large matrix or I do not want to use Gaussian elimination on the entire big matrix. One possibility is that I transform this equation into two sub equations. So, this a matrix is written as a partition a 11 a 12 a 21 a 22. So, this big matrix a I am going to partition into four sub matrices. Well if some of them let us say this one is a almost a null matrix or this is a null matrix great you know you have a simplified solution x 1 x 2 and b 1 b 2. So, I am going to partition this equation how will I proceed now the way I am going to proceed is well I will say that we have two equations a 11 x 1 plus a 12 x 2 is equal to b 1 can you try and solve this can you make an attempt how will I proceed now let us say a 11 is invertible. Then can you eliminate and write can I eliminate x 1 using the first equation how do you solve suppose I decide to write x 1 as a 11 inverse b 1 minus a 12 x 2 what next how do I solve the next part I just take the second equation a 21 x 1 plus a 22 x 2 is equal to b 2 and then I substitute this x 1 here. So, it is a 21 a 11 inverse b 1 minus a 12 x 2 plus b 1 minus a 12 x 2 plus b 1 minus a 12 x 2 plus b 1 minus a 12 x 2 plus a 22 x 2 is equal to b 2 and now I can rearrange I can put all the terms of x together I can put all the terms of x together. So, I will get an equation which is a 21 a 11 inverse a 12 minus plus a 22 x 2 is equal to b 2 minus what is the advantage of doing this see n cube you just remember that Gaussian elimination would require computations to the proportion of n cube by 3 suppose this is 1000 cross 1000 matrix I decide to divide it into 500 cross 500 500 cross 500. So, this will be the first problem first problem I have to do a Gaussian for a 500 cross 500 matrix and next time. So, I will get any a inverse a 11 inverse once I will store it I will use it in the next step again I have to do a Gaussian elimination of 500 cross 500 to Gaussian elimination of 500 cross 500 requires less computations than 1000 cross 1000 to 500 cross 500 is less than 1000 cross 1000. So, likewise I could I could I have just done a simple partitioning here I could think of a 11 a 12 a 13 a 21 a 22 a 23 I can make multiple partitions actually matrix theory is very very interesting and it has history of almost more than 100 years or 100 150 years Kelly was one of the founders of matrix theory you probably know Kelly Hamilton theorem Kelly and who was the second mathematician I forget his name I will try to remember and tell you they were friends and incidentally Kelly's friend. So, he did not get he though he was a mathematician that time in England if you have to be graduate you had to take a oath to be a devote Christian and he was a Jew and he refused to take that oath. So, he was not given degree in mathematics so he took to law and during the recess during the you know during the you know cold recess between between two sessions these two guys used to work on matrix theory that is what travels us now. So, but they came up with voluminous matrix theory which is now well they are known as twins of I will tell you the name tomorrow. So, this is lot lot to matrix theory and a lot of lot of things have been developed. The Bible of matrix theory would be there is a book called Gantt Matcher this book was published in I think 1945 we have this book in the library it is like it is two times Perry's hand book it is huge and this is in 1945. So, you can imagine what must be matrix theory by now I do not think you can have it in this. So, what actually we look at as a matrix theory is just tip of the iceberg and linear equation solving is something that you that is just bread and butter of numerical computing. So, you need it everywhere it is you cannot leave without ok. So, this is about sparse matrices or matrix partitioning and you have all kinds of tricks to make your competitions faster. Now, I am going to move from but this this method which I talked about matrix partitioning was still belonging to the direct methods I had not gone to I have not gone to iterative methods yet. Now, I am going to move on to iterative methods. So, iterative methods the idea is that you start with the guess solution ok. So, I want to solve again a x equal to b and then many times it is difficult to come up with a general number of number of multiplications and divisions for iterative methods, but in general the number of multiplications and divisions for iterative methods can be much smaller than the Gaussian elimination based methods. So, iterative schemes and next few lectures I am just going to spend on how to solve this problem iteratively. So, what is iterative schemes I start with a guess solution I want to solve this I start with some guess solution. So, let us say x naught is my guess solution ok and somehow I form a sequence. So, I want to form a sequence such that x k plus 1 is generated from x k ok. I start with x 0. So, x 0 will give me x 1 x 1 will give me x 2 x 2 will give me x 3 and so on ok. A very very simple crude way of doing this is you know I will say I will rearrange this equation as i plus a minus i into x equal to b. So, I will just write this as x equal to i minus a x plus b I have just rewritten the same equation a x equal to b. I have written as ok and then I can form an iteration scheme from this as x k plus 1 is equal to i minus a x k plus b. So, I start with a guess solution x naught I start with a guess solution x naught I will get x 1 I put back x 1 I get x 2 I get a sequence of vectors ok get a sequence of vectors and then well I need to talk about convergence. So, I will have to I will have to say when to terminate this. So, I am going to terminate this when x k plus 1 minus x k this becomes this ratio becomes very very small this is less than or equal to epsilon. So, epsilon is typically very small number 10 to the power minus 10 10 to the power minus 8 or something. So, I am going to go on doing this iterations well I will I will write a more generic form of the iterations I am going to put a a generic iteration here x k plus 1 is equal to i minus m a where m is a matrix where m is a matrix which I have to choose such that such that this sequence will converge to the solution ok sequence will converge to the solution that is what I want ok. What will happen if m is exactly equal to a inverse if m is exactly equal to a inverse then you will get the solution right because it is a inverse b ok here m is called as approximate inverse of a how to do this iterations how to construct the iterations we will see now, but basic idea is this that I am starting with a guess I construct a new guess and I go on iterating till it converges to a solution ok. Another way of looking at the convergence is through a residue. So, I look at this residue vector which is I look at this residue vector b minus when you when you form when you arrive at the solution what should be this difference it should be exactly equal to 0, but well in numerical computations we do not look for 0s we look at a small number. So, another way another criteria for conversions could be this r k r vector I it is good to normalize this with b is less than or equal to some let us say this is epsilon 1 and this is epsilon 2. So, either I look for this criteria to be satisfied or I look for this criteria to be satisfied for terminating my iterations and I am going to iterate till I reach a solution. This is the philosophy of iterative schemes now next few lectures will see how to form these iterations under what conditions you are guaranteed to converge to a solution. See the problem you might face is that well I am trying to solve a large system of equations ok and I have to guess a solution. So, tough problem because if you have to guess a vector which is 1000 cross 1 or 10000 cross 1 how do you guess well that is where you have to use your knowledge from physics engineering chemical engineering, but fortunately here even if you give a wrong guess what we will see is that if you take care of correctly choosing this m matrix ok you are guaranteed conversions to the true solution ok.  First we are going to look at these methods algorithmic part of it after we are done with algorithmic part we will move to analysis under what conditions these methods converge to the solution ok. Can you do some tweaks tricks to make the solution converge to the solution. So, those things we will look at ok. So, let us start developing this methods one by one I am going to talk about three methods which are very prominently used one is called a Jacobi method some of you might have done this in your undergraduate curriculum. Now, these iterative schemes are very often used while solving partial differential equations because you just want to go very quickly to the solution close to the solution ok. Well you might wonder by this iterative methods am I going to the true solution or am I going close to the true solution well you are getting an approximate solution no doubt, but even when you do Gauss elimination then too you are getting an approximate solution because you are doing truncation there are all kinds of errors even when you want to solve a problem a x equal to b ok. You normally end up solving a tilde is equal to a tilde x is equal to b tilde ok that is because for example, if you have a element pi coming in your matrix you cannot represent pi exactly. So, you when you do multiplications ok the computer has a finite precision. So, you truncate overflow errors. So, even when you solve using Gauss elimination you cannot construct the true solution there also there are approximations here also there are approximations. So, nothing to feel bad about approximate solution ok. So, let us look at this Jacobi method which is the simplest one. Let us say this is my Gauss solution k th Gauss solution ok where k starts of course from 0. So, this is my k th Gauss solution now starting from this Gauss. So, this is my Gauss solution starting from this Gauss I want to create a new Gauss ok. Now, where I am going to do it is I am going to look at each equation in this set of equations line by line ok. I have how many how many equations I have n equations in n unknowns right. I have n equations in n unknowns I am going to look at each equation line by line let us look at the first equation. So, a 11 x 1 k a 12 x 2 k up to a 1 n x n k is equal to b 1. This is my first equation everyone with me on this this is my first equation. Well actually since it is a guess this is not exactly equal but for the time being just understand how the method is developed. I have this guess solution I want to construct a new guess what I am going to do is I am going to rearrange this equation and say that x 1 k plus 1 is equal to b 1 minus I will put this a 11 here and this b 1 minus a 12 x 2 k a 1 n x n k is this fine ok. So, now my new guess is going to be x 1 k plus 1 is 1 by a 11 into b 1 minus ok. See I am starting from the previous guess I am starting from the previous guess I am constructing the new guess for x 1 is this clear what I have done just look at one equation ok. Well of course my assumption is that a 11 is not 0 a 11 is not 0 how will I use the second equation I can do the same trick ok. So, my second equation if I just skip in between steps I can write my second equation x 2 k plus 1 is equal to 1 by a 22 into b 2 minus a 21 x 1 k minus a 23 x 3 k minus a 2 n x n k use the second line. Take x 2 on the left hand side use the second row in the equation second equation take x 2 on the left hand side and you will get this equation I have just done the same thing which is here ok. If you are confused with I can take this inside plus helps better to compare with the previous expression then there is a close brace here close brace here ok. This is the first element this is the third element second element has been taken on the left hand side is everyone with me on this is this clear ok likewise I can go likewise I can go and my in general I can write that x i k plus 1 is 1 by a i i b i minus well equation 59 if you are the notes just correct them this should be not b 2 should be b i it is b i minus a i 1 ok. So, I am written the expression in general for the i th i th row I have written the expression for the i th row in i th row what is the implicit assumption here the implicit assumption is that all the diagonal elements are non-zero. So, if it is not like that you have to do a rearrangement of your equation such that all the diagonal elements are non-zero that is a critical thing here you cannot implement this algorithm unless that is done. But you can see here it is very very simple it is very very simple to do this calculations I am just generating from the old guess I am generating a new guess ok yeah that is another modification. So, she has rightly guessed the next modification which is called as Gauss Seidel method ok Smita right. So, Smita like yeah yeah but this is the next the obvious thing to do next is that if you are generating a new guess here in the next expression when you go here in x 2 why continue with the old guess right. See I am generating here in this step I am going to start from one ok. So, I am first going to generate x 1 k plus 1 having generated x 1 k plus 1 here I need not use old guess I could replace this by k plus 1 ok I could just replace this by k plus 1 likewise when I go to x 3 I will use x 1 x 2 new and x 4 to x n old and so on ok. So, in general here i th expression I will use k x 1 k plus 1 up to i minus 1 and i plus 1 onwards I will use the old values ok i plus i minus 1 I will use the old values this is obvious modification this is called as Gauss Seidel method and what you can show is that convergence of this method is much better than the Jacobi method ok this method will converge much faster than Jacobi method and this is very very very simple to implement. In fact, even in terms of computer storage this is easier to do because here in the earlier case you have to maintain two separate vectors one is the old guess other is the new guess here you can just keep using the new value which is created those of you who know programming well will appreciate that you do not have to maintain two vectors you can just have one vector and just write these equations old the new value will be automatically used in the next equation ok. So, this is implementation wise this is a much much efficient this is also convergence properties are better we will see why convergence properties are better a little later. So, this is one modification that we do to come up with iteration scheme now there is one more modification called as over relaxation method and I will talk about it in my next class or relaxation method. So, we say that well from Jacobi to Gauss Seidel you are actually you end up making the convergence faster. So, why not even further accelerate by putting some acceleration parameter ok. So, this acceleration parameter business will it is actually called relaxation parameter this relaxation parameter business will see later, but these iterative methods tend to work much faster than the conventional methods ok. So, for large scale systems many times these iterative methods are preferred as against and the solutions that you get are pretty much close ok. If you do a MATLAB experiment in which you solve a problem using iterative methods and probably you can also check how many number of multiplications and divisions are required. So, you will get an idea which particular method is faster. So, the next class onwards will start on how to analyze this convergence how do you make sure that convergence will occur is there are there any tricks to make enhance the convergence. So, all those things will see from our next class. So, it would help if you bring these nodes because there are too many summations i, j, k business and instead of writing it I want to explain it more than spending time on writing on the board. So, just get copy of the nodes and then we can do it. So, the so other thing which I am going to do is this is what I have written is the algorithmic part of it ok. I am going to rearrange these equations in the matrix form because once I write these equations in the matrix form it is easier for me to analyze this equation the convergence behavior. It is difficult to analyze in this raw form. This raw form is useful for implementation you can write algorithm in this way ok, but analysis when will it converge and all will require rearranging these calculations into some nice matrix forms and we will look at the matrix properties and say when will convergence occur ok. So, that is what we will do next.


Hello everyone, this is Alice Gao. This video is the beginning of a brand new unit where I'm going to talk about probabilistic reasoning and inference using Bayesian networks. After doing quite a bit of review on probability theory, we're finally ready to talk about Bayesian networks. Let me give you a few examples of the kind of problems where we might want to use Bayesian networks as a model. You might not have seen a lot of examples of Bayesian networks in your daily life, but hopefully once you see these examples, you will realize that they are pretty natural kind of models for certain scenarios. This first example is sort of genetics, modeling the inheritance of handedness, whether someone is left-handed or right-handed. We have notes here, modeling the handedness in the parents. And given the handedness in the parents, those genes will decide on the genes in the child. But whether these genes will be expressed and how they're expressed in the individual might be different. So we have additional nodes, modeling sort of how these genes are expressed in that individual. So we have how the genes are expressed in the mother and the father, and also how the gene is expressed in the child. So we have notes for the genes, and then we have notes for the expressions. And we have edges representing the relationships. This is an example of a car diagnostic network. You will see that a lot of the Bayesian network examples are about diagnostic or diagnosis of problems in certain scenarios. So in this scenario, we may have problems with our car in a particular part of the car, and that may be caused by many different reasons. So the battery might cause the lights to have different problems, or it may cause the engine to have problems, and that may in turn cause the engine to be able to start or not. Obviously I'm not an expert on cars, but maybe some of you know better than me to understand everything in this network. This is a network on diagnosing the problems in nuclear power plant. Someone much more serious problem than when we have trouble with our car. So this network has some interesting structure. We have three tiers in this network. The top tier represents the root causes, the underlying causes of some problems. And then the middle tier represents, well, given the underlying causes, what kind of events we might have because of these underlying causes. And finally, the third layer, well, the bottom layer says that if we have certain events, then we will realize these events are happening because of the reports or outputs from some sensors. So a lot of real-world scenarios are often like this. There are a lot of root causes, the top things that we never get to really observe ourselves. And then there are events happening and often these middle events, we also don't observe them directly. Instead, we only observe the things in the bottom layer, which are outputs from sensors or output from some alarms, indicating that there might be something happening underneath. This is a much simpler scenario where we have a fire alarm situation for a particular in the building, and then it's saying that an actual fire can cause either smoke or alarm, and then it could cause people leaving the building, and that could cause people to report to authorities that something might be happening. Another possible cause for the alarm could be somebody tempering with the alarm. Our home scenario is quite similar to this, where we have an alarm and then we have two different causes for the alarm. It could be caused by burglary or it could be caused by earthquake. But we do not directly observe the alarm. Instead, we only observe calls from Dr. Watson or Mrs. Gibbon, notifying Mr. Holmes, maybe the alarm is going off and then maybe there's a burglary happening. Another prominent application of Bayesian Network is a medical diagnosis scenario. If you remember when I explained the base rule, I talked about the fact that often we know that certain diseases will cause some symptoms, but we never directly observe whether somebody has a disease. We only really observe the symptoms and we want to infer whether they have a particular disease. A lot of medical diagnosis scenarios are like that, where we have underlying causes, where we have some disease, and these diseases could again have some underlying causes. Like, it depends on your genes, it depends on your age, weight and things like that. It could cause you to have or does not have diabetes and then diabetes could cause additional symptoms, like so. And as doctors, we often only observe the results of diagnostic tests and also symptoms and we're using all of these to try to infer whether somebody has a disease. In this scenario, it's interesting that we also observe things from the top layer. We could observe things about patient information, their age, their weight, their parents' health information, so on and so forth. We sort of observe things in the top tier and in the bottom tier, and we're trying to infer things in the middle. That's everything for this video. I hope this video gave you a sense of what kind of applications there are for patient networks. Thank you for watching. I will see you in the next video. Bye for now.


I am Senior Master Sergeant Demetrius Booth and today I'm going to talk to you about our social pillar of CAD. And so now more than ever they need you. You fill out that line of faith. But how you connect with them is just as important because situational leadership is needed. So get back into practice. If they're old school write them a letter, send a message in a bottle, jump into some gaming, FaceTime them, whatever it takes you to reach out and stay connected as we are physically distant. Because you are someone's lighthouse.


Hello everybody, E here. Welcome back to another Stephen King review. If you're confused and wondering, hey, didn't he already review all these books? No, I have not. I have done Thursday Theorist episodes, but I have not done the reviews for the majority of his work. So today we are talking about The Eyes of the Dragon by Stephen King, of course. So this is a fantasy novel, standalone fantasy novel, even though it has a flag and it has some other connections to the Dark Tower universe, which we won't be talking about today because this isn't a Thursday Theorist video. If you don't know, my Thursday Theorist video is where I go through every single one of King's books and tie them back into the Dark Tower or to the Stephen King universe in general, but mostly all of them tie into the Dark Tower in some way, shape, or form. This this story, I do not remember a whole lot about it. In fact, I had this talk to my friend Josh about certain things. I do not have any want or need to ever read this again. I probably will if I go back and do another full read through of Stephen King's catalog from the earliest to the newest, which I do about every five to seven years or so. I'll go through his entire catalog. In 2015, I did it and I did every single one of his books, even his new releases that year, in a year. I blogged about it. I'm not sure if it's still up or not, but if you want to go to wordpress.edwardlauren.com, you might be able to find it. I called it a King for a year and then I did each decade as a... Anyways, but I did all the reviews every decade talking about how he releases certain books. There's always a thousand page novel every ten years, every seven years, roughly about that. There's always a short story collection and a novella collection. I found it interesting. If I can find it online, I will link you guys down there in the doobly-doo. But this book, I don't have too much to say about it, which is why I'm talking about other things, but there's two things that I remember specifically about this book. One of them is the escape, how he uses the napkins. And then another thing is what I'm going to focus on here in this review because King says in the forward, or at the beginning of the book, that he wrote this one for his daughter Naomi and he didn't want to talk down to her. So he wanted to write an adult fantasy novel for basically a young adult before the term was even coined young adult.  So it kind of feels like he's writing down. But in the very first chapter, Stephen King goes full Dean Coons and talks about someone squirting some seed, which was very odd, especially right up front at the beginning of the book. I believe it's the king squirting his seed into his wife. Back in the day, seed used to be a normal nomenclature for semen. And it's really weird going back and reading these older books and seeing it just everywhere, just seed just spattered all over the place. And if it's with Dean Coons, it's usually within the velvety interior of someone's vagina. Anyways, so I didn't care too much for this one. But then again, I don't care too much for fantasy period. I like his Dark Tower series, except for two books. And we'll get to those when I review those. I'm planning on rereading them all in order before I actually get to those reviews. So if you're wondering where the Gunslinger review is at, where the drawing of the three review is at, they're coming. Okay, it won't be until the end of this. And then I'll do the Thursday theorist for those books. But yeah, this one feels like a middle grade, more than a YA book. It truly feels like King is writing down. And had he not pointed that out at the beginning of the book, I might not have noticed. I don't know. I can't unread something that I've already read. This I'm trying so, so hard to try and remember literally anything else about this book. My buddy Josh said he liked all the parts with seeing through the actual eyes of the dragon, all that stuff. I vaguely, vaguely remember that. So what I'm gonna do now is I'm gonna read the description to you guys and see if I can think, see if I can, if it strikes any memories that pop up in my head. So a tale of archetypal heroes and sweeping adventures of dragons and princes and evil wizards as only Stephen King can tell it. This is probably the least Stephen King book I've ever read. So that's whatever. The passage through the castle is dim, sensed by few and walked by only one. Flag knows the way well. In 400 years he has walked it many times in many guises. But now the passage serves its true purpose. Through the spyhole it conceals the court magician observes King Rowland, not Rowland Deschain. It's not the same Rowland. Old, weak, yet still a king. Rowland's time is nearly over though and young Prince Peter, tall and handsome, the measure of a king in all ways stands to inherit the realm. I do remember that Flag is a wizard and you know he's trying to usurp the throne and so on and so forth. Yet a tiny mouse is enough to bring him down. I don't remember the mouse at all. That's a very green mile. I wonder if I wonder if I could tie that back to the dark tower that way. Hmm. I don't know. Let's talk about it down there in the doobly-doo. A mouse that changes upon a grain of dragon sand behind Peter Shelves. Chances upon a grain of dragon sand behind Peter Shelves and dies crying tears of fire and belching gray smoke. A mouse that dies as King Rowland does. Flag saw it all and smiled. For now Prince Thomas, a young boy easily swayed to Flag's own purposes, would rule the kingdom. But Thomas has a secret that has turned his days into nightmares and his nights into prayed for oblivion. The last bastion of hope lies at the top of the needle. The royal prison where Peter clans a daring escape using napkins. I remember that much. Writing with the wit and power that have enthralled millions, Stephen King has taken the classic fairy tale form and transformed it into a masterpiece of fiction that will captivate readers of all ages. I disagree. Everyone I talk to that actually likes this book. They like it because they read it as you know a younger person as a kid. So I would I guess my question today is also some gorgeous artwork in here. I guess my question today is did you read this as an adult and enjoy it? I would love to hear from you. Yeah, this story is for my great friend Ben Straub, which I believe is Peter Straub's son and for my daughter Naomi King. Like I said, he wrote it for her. Let's see here. I'm trying to find the exact line where he where he spreads his seed but I'm not I'm not good at skinning. Anyways, but it was just a very odd thing in the very first chapter for a middle grade, why whatever book, especially back then for him to be talking about you know, copulation. I thought that was odd and then the rest of the book is written in a very fairy tale manner. It's it's it feels dumbed down for Stephen King, which again is confusing because he said that he didn't want to write dumb down for Naomi and I'm a I'm a firm believer in not writing down to kids but especially young adult books seem to go but but then again, you know, nowadays we have the TLD TLDR crowd, the Too Long Didn't Read crowd. So maybe it's better that the language is more simplistic in YA and middle grade. I don't know. Let me know your thoughts down there in the doobly-doo. Also, let me know whether or not you loved, hated, felt meh about Eyes of the Eyes of the Dragon by Stephen King. Let me know all that down there in the doobly-doo, but give me detail as to why you felt that way so that we can have a discussion. But until next time, I have been E, you have been U, this has been another Stephen King review. I'll talk to you guys later. Bye bye.


When you know how you see the world, but you know who you are, the world just feels different. The world just opens up. You have so much more agency because you go, no, not for me, no, not for me.  The prefrontal cortex doesn't fully form until your late twenties. It does, like that's just biologically. So we're still sort of figuring out ourselves and the world. And we don't have all the hardware to do it. The heart well isn't fully developed to make that happen. So it's when you're in your thirties and your forties, not to be demographically focused, but that's when biologically, we're more inclined to say, no, I'm who I am. No, I'm good. No, thank you. Right.  And you prioritize those things because we find ourselves having better outcomes when there is congruence in that way. And who you're looking for as part of that tribe. I think when you're younger, there's a lot more impulsivity, right? Because your emotions are really running the show with your prefrontal cortex not fully there developed. So you'll swing from, you know, hipster to goth to Johnny's probably been through all of them along those rock and roll years. You know, you will have this impulsivity trying to find that tribe and that culture. But as you get older, you start to really know what exactly you're looking for in that tribe, what you want to surround yourself with and what clearly doesn't feel right and you want to run from or move from, relocate from. So many people in our audience find themselves sometimes in that situation, either through career or through choice that, you know, this place just doesn't feel right. I need to relocate. And then they'll ask us, well, how do I find my new tribe? How do I really identify and find this new tribe? And we'll often say, well, look at the cultural production, right? So look outward. What what movies do you like? What activities do you like? Look at the artifacts. What are those people wearing? How are they symbolizing that they're part of that? And then invite them to hang out with you, to be part of your tribe. And that's how you build the tribe. But tribes don't just open themselves up to you. You don't just stand on the corner. Tribe walks by and says, hey, you're wearing the right artifacts. Come on in and join our tribe. It takes effort. It takes action on your part. But you have to be paying attention and identifying these parts of culture that feel right to you. So when you are surrounded by those people, you can be the truest expression of yourself and it feels natural. It feels easy. That's right. And language becomes such a perfect way to do that. You know, when you hear people talk and you go, man, I've always said the exact same thing. Where are you from? Who are you? What do you do? Like you hear people evangelize the same point of view you have. And you go, man, these are my people. You know, I mean, this is kind of what dating is like, too. You know, we go on dates and we ask, what do you listen to? Where do you go? What do you like? Where do you stand on this? Then you go, no, this isn't going to work out. No, no, no, no, no. I mean, and it's the same thing when it comes to, you know, when it comes to platonic relationships, like we're trying to find connection. We're trying to find congruence. You know, we often talk about social networking platforms being like an echo chamber, but like our real life networks are echo chambers, too.  Like if you can agree on anything, everything is a fight. You go, ugh, I can't with this guy, you know? And the same thing goes with our people. We try to find connection. You go, you love Frank Ocean. I love Frank Ocean. What do you think about the Soldier Ultra? Oh, my God, I love Channel Orange. And like we find ourselves connecting. And you go, these are proxies for us to say you're one of us. Right.  Unless we are wearing the artifacts that signal tribalism. Right. So if you're in, I don't know, like if, if, you know, AJ, you're from Michigan, you're from Michigan, went to Michigan. If you're walking through an airport and you see someone with the block, you may go, go blue. Now go go blue. You go, look at us together. Right. Those those branded products become marks of identity. Right. If you're in South Central Los Angeles and you're wearing red, you may be like, you're a blood too. I'm a blood too, you know, whatever the affiliation may be. We find connection. And I think it's it's we that that's what we're wired to do. We are social animals by nature, as Aristotle says. Right. We do everything we can just to crash into each other. Evolutionary anthropologists were argued. The reason we were able to evolve was our ability to socialize, to cooperate. Everything about us is about connecting. And we use every shortcut that we can to surmise are we together? Are you one of us? And we use these outward expressions, cultural production, artifacts, behaviors and language as proxies to basically say, you believe what I believe. So therefore I'm safe for the brain. It just means safety and survival. I'm safe around you. You I'm not I'm not in harm's way around you because we are one in the same. I wouldn't do it to you. Because I know you wouldn't do it to me. And that's really what we're after from a very evolutionarily focused viewpoint.


Hello, my name is Sayuri, and I am from Mexico City. I am an English teacher, and I teach pre-first. I just completed the TEFL certification online. This was a course that really helped me structure my classes. It really gives me a guide now to prepare my classes. And also, because it is an online class, I think it was really good to understand. The units were really easy to understand. The videos were easy to guide you around the new themes in the certification. Also, there was this part in Unit 18 that I felt it needed more information. For example, it was about the phrasal verbs. It really didn't explain much about when to use them, how to use them. I was expecting to see it in the video, but nothing really came as an example. Also, it came in the exam, and it was hard to actually be able to respond because I had no examples on it. But other than that, I found it quite interesting. I found it quite helpful for my future classes. Also, I want to introduce some of my students here who want to say hello. Everybody! These students are going to be the ones who are going to benefit from my certification. I hope they like the classes. Certification gave me lots of ideas that I can apply now and that I am applying right now. Thank you!


В тази лесене ще го говорим ото как да се конвертим хексидесималната номера в хексидесималната номера. Т. е. към са, че имаме... не знаю хексидесималната номера 1, 2, а б и това, That we want to convert it to the integer number will use function int and will put inside that function our caxodetimal number that we want to convert to decimal number. Let's use this number and let's try to convert it to decimal number. и ще го добраем нестъпно. Почему? Потому, когато върху това функционата е аргумен, ще го добраем нестъпно. Но се добраем нестъпно, както е 1, 2, а и б. Това, че е, че нестъпно. It's hexadecimal number, but we don't give a clue to our function that this is a representation of hexadecimal value. If you want for any number to give a notice that it is hexadecimal number, we need to start that number with number 0 and letter X and then to put that hexadecimal number. And as you can see, Python's idle is smart enough to convert it automatically to integer, but we'll use int function and we'll provide that exact same hexadecimal number as an argument and we'll get that same number. Now let's open calculator to check if our conversion is fine. So let's say that we have hexadecimal number 1, 2, a, b and let's convert it to decimal number and we'll get exactly the same number by this conversion with using function int and providing as an argument hexadecimal value and you should be careful if you want to provide hexadecimal value. You need to put before that number number 0 and X as any representation of hexadecimal value. That's all for this video. Thanks for watching. Happy coding.