In the previous lecture, we looked at solving quadratic equations. Those are second-order polynomials. But what about cubic equations? Those are third-order polynomials. Let's take a look at how to solve those. Let's start thinking about cubic equations. Let's do our definition right there of a cubic equation. We're going to say that that is equal to AX cubed plus BX squared plus CX plus D equals zero. We know that A cannot be equal to zero because of that zero, that first term disappears and that's just a quadratic equation. So here we have a cubic equation. The highest power of our variable is three. Now we've already looked at some of these. Let's look at A cubed. But let's make that X cubed plus A cubed. Now we can indeed factor that. You might remember that's going to be X plus A multiplied by X squared. That's minus AX and then plus A squared. And we cannot just factor that very easily. We also had X cubed minus A cubed. And remember in both these instances, A is just going to be some constant.  And again, it's not easy just to factor that. Let's look at an example of one of these. Let's do X cubed. Let's do plus 27. And this is an equation. So we're going to set that equal to zero. Can we find solutions for X that satisfy this equation? Now we can rewrite this. Of course it's X cubed plus three cubed equals zero. And now it's just that what we have right up here, A is now this constant three. And so we can just plug in three everywhere where we see A. So we're going to have X plus three and we're going to have X squared minus three X and then plus nine because three squared is just equal to nine. Now this equals zero. Now we just have to think about what to do here. Now we've got two terms here, one and a second one. And we multiply those and that equals zero. That means we must have that either X plus three equals zero or we must have the fact that this polynomial X squared minus three X plus nine is equal to zero. So we will have that X is equal to negative three. I'm just subtracting three from both sides. Or I must have this equal to zero. And what we're going to do now, let's do some work on the side. This is AX squared plus BX plus C. So don't think about these A's and B's now. Let's just concentrate on this quadratic. My leading coefficient for X squared is going to be one. My B is going to equal negative three and my C is going to equal nine. I'm just doing AX squared plus BX plus C. Let's put it out there AX squared plus BX plus C equals zero. So don't think about and worry about what we had up there. It's a brand new problem that I'm dealing with as far as this quadratic's concerned. And I do remember that we have that X equals minus B plus or minus the square root of B squared minus four AC. Let's draw neat lines. That's going to be divided by two A. So let's substitute for A, B, and C. So we're going to say X equals negative B. Now B is already negative three. So negative negative three, that's just going to be positive three. Then plus or minus the square root of B squared is going to be negative three squared. So that's nine minus four times one times nine. So four times one times nine. And that of course is 36. And we're going to divide that by two times A and A is just one, so that divides by two. This leaves us with X equals three plus or minus the square root of, and now we have the square root of negative 27. And so we will absolutely have to involve complex numbers. Now let's look at what 27 is. Remember 27, that's just going to be three times three times three times negative one. Now this negative one, if we take the square root of negative one, that's just going to be I. And if we consider two of these, we can bring one out and then be left with a single square root of three. This would come from the fact that if we have the square root of three squared, well that's just going to equal three because if I take this very nice square root, I get three squared. So we know that we can say X equals three plus or minus. That's going to be three square root of three times I divided by two. And so now we have three solutions and we really want three solutions because this, the problem that we started with is a cubic equation. So we're going to have X equals negative three or we're going to have X equals, let's do three over two. Let's do the plus first plus three square root of three divided by two I or X is going to equal three over two minus three square root of three divided by two I. And we see our three solutions. Now in some cases, we're going to be very fortunate. Let's write out again AX cubed plus BX squared plus CX plus D equals zero. In some cases, we're going to be very lucky in as much as D equals zero. So let's look at an example there. Let's do a very simple one. Let's have X cubed, let's do minus eight X. Now in this instance, we also see that B is equal to zero. There's no X squared term, but definitely D is zero. We don't see any constant there. Now this is very easy to do because I can take out a single X as a common factor because do remember this is X times X times X minus eight X. In this term, I have a single X, I have a single X on this side. I can take that out as a common factor. So there's my X. What am I left with? Well, I'm left with X times X, which is this X squared and then I'm left with negative eight. Negative eight equals zero. So now either one of these two has to equal zero. It's either gonna be X equals zero or we have X squared minus eight is equal to zero. I can add eight to both sides and I get X squared equals eight and then X equals plus or minus the square root of eight. I'm just taking the square root of both sides. Now what can I do with eight? I can say eight equals two times two times two. Well, there's two twos there. I can bring that outside of the square root symbol. So that's gonna X equals plus or minus two and I'm left with a single square root of two. So my final solution, I get X equals zero or X equals two square root of two or negative the square root of two, as we can see there, underline that. So let's do another example. Let's have X cubed minus four X squared minus five X equals zero. Again, you can see the D term is not there. The constant term is not there, is equal to zero. Once again, I can take out a common factor, a single X, which means I'm left with X squared minus four X minus five is equal to zero. So either X is equal to zero or our polynomial here X squared minus four X minus five is equal to zero. Fortunately, we can look at this polynomial on the site, this quadratic I can do X and X. And if I look at five, well, I can think of this as negative five plus one because negative five times one, that's gonna give me the negative five. But I have negative five X plus one X leaves me with negative four X equals zero. And this one is very easy. I'm gonna have X equals zero or X equals five, looking at that term. And for this term, or X equals negative one. That was a simple one to do. Sometimes we're very fortunate that we can do grouping. Let's have a look at this example. Let's have X cubed plus X squared minus four X minus four equals zero. Now let's just do some association. Let's put X cubed plus X squared minus, let's do four X, let's do plus four. Why the positive there? Well, I've taken out negative one. If I distribute this negative, I'm gonna get negative four X, negative four. And I'm left with the original problem. Let's look at this first term. Let's take out X squared as a common factor. So I'm left with X plus one, X cubed, X squared. That looks fine. Let's take out four as a common factor here. And I'm left with X plus one. You can verify that, just do the distribution. You're gonna land up with the exact same problem. Now look at this. I have some term in common for both of these. So let's take out X plus one as a common factor. Yes, you can do that. And I'm left with X squared minus four equals zero. Again, with two terms equaling zero, that means either X plus one is equal to zero or X squared minus four is equal to zero. I can subtract one from both sides. I'm left with X equals negative one. And on this side, X squared equals, add four to both sides. X squared equals four. I can take the square root of both sides. That leaves me X equals plus or minus. The square root of four. And that means X equals plus or minus two. And so it's a cubic and I have my three solutions. X equals negative one or X equals two or X equals negative two. And one by one, you can substitute negative one. On the left hand side, you should get zero. Two, you can substitute. You should get zero. Negative two, you can substitute into X and you should get zero. So you can always verify your work. Now let's take the exact same problem.  And I'm going to do a different technique. I'm going to look at my constant term D. My D is equal to negative four and I'm gonna look at my constant term A. In this instance, it's one. Now I'm going to look at all the factors of these two numbers. By factors, I mean what integers will divide these numbers leaving me with no remainder. So let's start with D. Well, certainly positive and negative one will divide negative four without remainder. So we'll positive and negative two and so we'll positive negative four. My A term is very easy. This is gonna be positive negative one. You can divide either positive or negative one by one and you're not going to have a remainder. Now the next step in this division technique is to take each one of these terms and divide it by each one of the terms for A. Now this one's gonna be very easy because we only have positive negative one. So that will leave us with positive negative one, positive negative two, positive negative four, which means I can write six terms.  So it's just gonna take each of these values and we're gonna pop them in there. There's my positive one, negative one, positive two, negative two, positive four, negative four. Now I want to know, do any of these terms divide my original problem? So let's start with this first one. I'm thinking X cubed plus X squared minus four X minus four. Let's start with this very first one, X plus one. So I'm gonna say divide that by X plus one. I want to know, does this divide the numerator without any remainder? Well, I'll have to do some long division.  I'm gonna divide that by X plus one. So just some long division from school. Now we do that by looking only at the first term. How do I get from X to X cubed? Well, I multiply that by X squared. So X squared times X plus one. What would that be? X squared times X, well that's X cubed. X squared times one, that's positive. X squared, and now I'm going to do subtraction. X cubed minus X cubed, that's nothing. X squared minus X squared, well that's the zero X squared left there, now I bring the negative four X down. And I'm thinking, how do I get from X to X squared? Well, I multiply that by zero X. Zero X times X, that's zero X squared. Now zero X times one, well that's just positive zero X. And I do my subtraction once again. Zero X squared minus zero X squared, that's nothing.  I bring down my negative four. How do I get from X to negative four X? Well, that's just minus four. Negative four times X, that's negative four X. Negative four times one, that's negative four. And look at that, if I do the subtraction, there's nothing left, there's no remainder. So indeed, X plus one does divide X cubed plus X squared minus four X minus four. And I can write X plus one times X squared. I can drop that zero X minus four equals zero. If I multiply these two terms, I get back to there because I did the division and there was no remainder. And now look at this, it is what we had before. X equals negative one or this term, X squared minus four equals zero or X squared equals four or X equals plus or minus the square root of four or X equals two or X equals negative two. And you can see as before, I have the very same results. Let's do one more example. Let's do two X cubed, let's do plus three X squared minus 11 X minus six equals zero. Now my D term there is gonna be negative six, it's right there, my A term is going to be positive two. Let's think about all the integers that I can divide into negative six that leaves me no remainder. So let's look at the negative six. Obviously that's gonna be a plus or minus one because negative six divided by one, that leaves me with no remainder. Negative six divided by negative one leaves me with no remainder. It's very easy to do. Two will also divide, negative six plus or minus two, plus or minus three will divide negative six and plus or minus six will do the same. Let's look at the term, that's my A term there. Remember A X cubed and there's my D term there. This is gonna be plus or minus one and plus or minus two.  If I take plus or minus one and divided by plus or minus two, that's gonna leave me plus or minus A half. So I can move on to the next one, plus or minus two divided by plus or minus one, that leaves me with plus or minus two, plus or minus two divided by plus or minus two. remember there's four divisions going on here plus two divided by plus two plus two divided by negative two negative two divided by plus two and negative two divided by negative two all of them is going to leave me with plus or minus one so that's fine I can now move on to the three three and the one that's going to leave me plus or minus three three divided by the two that's going to leave me with plus or minus three over two and then six divided by the one that's going to leave me plus or minus six and six divided by the two, well that just leaves me with a plus or minus three. So I've actually got many, many terms here. There's going to be an x plus one or an x minus one. There's going to be x plus a half and x minus a half. There's going to be x plus two, x minus two. There's going to be an x plus three and x minus three. There's going to be x plus three over two, x minus three over two. There's going to be an x plus six. There's going to be an x minus six. One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve. You'll have to check twelve of these. It'll take a long, long time. Now do go one by one. I'm going to skip a few. I'm going to go straight for trying x minus two. But please, you can't just skip around. You'll have to do one by one. So let's do x minus two. We'll have two x cubed plus three x squared minus eleven x minus six. Let's see if x minus two indeed divides that expression. Let's put our long division. We've got our lines there. And so how do I get from x to two x cubed? Well, that'll be a two x squared because two x squared times x, that's two x cubed. Two x squared times two, that leaves me with negative four x squared. I do subtraction, two minus two, that's zero. Three minus negative four, so that's three plus four, that leaves me with seven x squared. I'm going to bring down the negative eleven x. How do I get from x to seven x squared? Well, I've got to multiply by seven x. Now that's it for this video. There's an equation that will solve for cubic equations. There's actually also one that will solve all of these that are polynomials raised to the power of four, fourth-order polynomials. Once we get to fifth-order polynomials, or higher though, there are no longer any such easy equations to solve them.


Johnny dollar dollar the investigator. That's right insurance investigator Well, it just happens I'm late for an appointment mr.. Mr.. What did you say your name is I didn't And you must come out here right away, mr.. Dollar protect me protect you Manchester, Connecticut My address is 13 421 13 421 Well, I told you I have an appointment to keep You'll come here as soon as you're through with it. Well, that depends mr.. Lynch what seems to be the trouble Well, isn't that something for the police to handle After all unless there's insurance involved Oh what company mr. Lynch Yeah, that's right. Well Okay, okay, who made this threat against your life and why Well, unless I know what this is all about and look you didn't answer my question mr.. Lynch Have you told the police about this do you think I'd call on you if I had but if your life has been threatened the thing All right then calling the police Find any legitimate reason why I should make an investigation And I tell you the people Mr. Lynch, okay, mr.. Go ahead and wait All the crack parts Wonder though Yeah, Johnny. Yeah, that's right At McCracken Johnny universal adjust. Oh say Pat Will you find out for me what company if any holds a policy on a Walter E Lynch over in Manchester? Sure sure Johnny But listen, I thought you were going to be here in my office at 10 o'clock sharp I'm sorry, but this whole signature on a deposition for that arson money you covered last I know I know I've got to have it right away. I see it is Pat. I'll be there in three minutes flat Okay, meantime look up that water Lynch for me. Would you okay? Just get over here, right the exciting adventures of the man with the Action-packed expensive car America's fabulous freelance insurance investigator. Yes, truly Johnny Dallock best-selling records. Here's a familiar tune about America's best-selling filter cigarette Winston Has filter blend up front Choice flavorful tobaccos Specially selected and specially processed for filter smoking. No wonder Winston tastes good like I Know act one of yours accounts submitted by special investigator Johnny dollar to Easton trust and insurance company home office Hartford, Connecticut Following is an account of expenses incurred during my investigation of the hapless ham matter Expense account item one a dollar twenty plus another dollar for the cab driver who got me from my apartment a universal adjustment bureau in record time I Barged in on Pat McCracken without being announced put my John Hancock on the deposition after which he witnessed my signature All right, there we are Now here you are miss. Delcey give these to mr. Peterson and tell him happy plane trip Yes, sir. Oh and see if Bartell has found any of that lynch information. I asked him about He's been a lynching mr. McCracken No, no judge. I just do as I say now go ahead, but I thought I heard you Yes, sir. Hey, what is the worry on this man lynched Johnny?  Yeah, who is threatening him? Oh, I don't know He was well, he was a bit evasive about it same as he was about naming his insurance company I know I never heard of him. What did you tell him? I told him to call the police then unless there was some insurance angle a requirement investigation, you know, right exactly, right? Although of course if Bartell does find something on him if he is a client of one of the companies That's still a police matter. Why call on me to play body. Yeah, you know Johnny If you would have your telephone number unlisted, you'd save yourself a lot of these nuisance calls. Oh, yes Here's a photo set of the lynch policy. Mr. Martell said you asked him for oh, then there is one Thank you, and I thought you were talking about some kind of a lynching some words. Mr. McCracken isn't that funny? Yes, yes, thank you. Thank you, Mr. Chelsea That'll be all Yes, sir. Oh, let's see Johnny Walter E. Lynch Policy issued 1954 Eastern trust and insurance face value Hmm. Yeah, I see 50,000 smackers a lot of money. Let's see the name of the beneficiary. Oh, well a nephew Fred Lynch New York City address I Wonder Johnny so do I how well Freddie and his uncle get along named him as beneficiary five or six years ago Things can change but there's one way to find out and with $50,000 to be paid out if anything does happen. Yes. Yes, go ahead Johnny go ahead and pay Mr. Walter Lynch a little visit at Company expense well now Well, oh sure. Why not? I'll notify eastern that we've assigned you to this but that you'll send your expense account directly to them Okay, okay There's only about 13 miles from my place to Manchester and I use my own car But I saw no reason for now using the old expense account so item two was 460 for a tank full of gas The address Mr. Lynch had given me turned out to be in a rather nice residential section The house was a small but attractive Cape Cod affairs surrounded by stinging the elm trees And then I saw them the shade of the alms at the front of the place a couple of police cars Oh, my name is dolly sergeant Johnny dollar insurance investigator. I came here to see Mr. Lynch Investigator, huh any credentials, but let me see him sure here. Okay. Yeah Hey, what's happening around here? Okay, dollar You boys act pretty fast. Don't you what do you mean? Is mr. Lynch in some kind of trouble? I'd call it worse than trouble dollar. He's dead He's what that's right murdered Well, I'd better get inside there and see what's what yeah, no sure. Maybe you better Inside the Lynch hall the police medical and my own eyes confirm what sergeant beakley. It told me outside Watering Lynch was dead. All right, and there was no question about his having been murdered As you as you can see mr. Dollar the the knife obviously a long very sharp one Plunged into his body here directly back of the heart Yeah, doctor. I see no doubt penetrated that organ death must have been virtually instantaneous Yes, any sign of the murder weapon? We haven't found any sign of it here in the house dollar Conroy is out back looking for it And for any footprints that might give us a clue my little rain. We had last night on a help in the footprints department only it doesn't However did this must have come right in the front door? Hi dollar hi Conroy sergeant. You want me to see what's holding up the boys in the lab? Yeah, I'll give them a call Right sure meantime dollar if you have any eyes. Oh, wait a minute Too late I guess What are any fingerprints on that telephone Conroy's probably covered them up probably covered up anyway, huh? Well, it wasn't the killer who called us about this was the old character lives next door a man named Halsey Right now he's back home in his rocking chair at the corner window muttering to himself trying to recover from the shock Maybe when he does recover he'll be able to give us some help How did he find out about this seems he came over here to talk to mr. Lynch about his nephew Fred Lynch Nobody answered his knock and the front door was wide open so he walked in here and found this so he picked up the phone and You know about Fred Lynch, I know he's been a fishery of his uncle's insurance Another good motor gone to waste. What do you mean? You don't know about a couple three years ago when Freddy's mother died left him a pile of money Only made the mistake of leaving it in charge of this one this stiff. We've got lying here. What happened Well, this old crook finagle that young kid out of every center of it so that by the time Fred was 25 Entitled to it. That was about six months ago My dollar Freddy ended up with nothing his own nephew. That surprised you dollar It's because you don't live here in Manchester. Don't know anything about the Lynch family What's left of them which is only Freddy now? What's that supposed to mean not one of the Lynch tribe was any good? Ask me the town will be glad to see this one out of the way Oh, I get it. Oh what they say Conroy somebody on the way over Harker will be over as soon as he clears up a hit and run over on the east side. Okay Well, I guess there's nothing more we Go see what it. Yes, sir You about finish doc Sergeant history this knife wound puzzles me puzzles me greatly. Yes. What do you mean, doc? Very unusual instrument very long thin with a two-edged blade. Yeah, both edges very sharp. That's so yeah Very unusual sergeant. You said a neighbor a man. Mr. Halsey found him here. Did somebody speak my name? Here's the man who was knocking on the door. Oh, Mr. Halsey come in. I don't know that I want a sergeant with that poor man in there dead. Mr. Halsey. This is Johnny dollar insurance investigator Yeah, insurance investigator. Yeah, I understand you came over here to talk with mr. Lynch about his nephew, mr. Halsey Freddy was that that worthless young whistler that why do you say that well because he Never did a day of honest work in his life the theater. That's all he cares about theater in New York City That bit of a liquid it spends all his time with sinful people Actors of course girls and the like. I take it. You don't have much use for people in show business That's what they are that's all there. Yeah Mr. Halsey if you saw him here this morning, what? He was here. No doubt come to beg some money from his uncle. Why didn't you tell me that mr. Halsey, of course his uncle cut him off from the money his mother left him wouldn't you have done the same mr. Halsey listen. Yes. Well, wouldn't you look here now? He is in that that that sinful city down there. Wait a minute. You say that Freddy was here this morning Yes, of course he was when that's why I finally made up my mind to come over here and talk to Walter about it What time was he here? From 9 30 to a few minutes of ten doc Have you been able to fix the time of death? Very close sergeant. Yes, sir by the body temperature. What time would you say doctor about? About 10 o'clock mr. Halsey. Are you sure it was a few minutes of ten when Freddy left here? Yeah, well, of course. I am how sure I was sitting by the window listening to my radio Like I always thought you didn't like anything about show business radio is different. Oh, thanks Freddy left his house just before the WDRC announcer gave the time as 10 a.m. And radio time is accurate time I see what you meant sergeant about a motive going to waste where Freddy was concerned What are you talking about dollar the doc says death was about 10 a I know so and mr. Halsey you're sure that Freddy didn't come back here after 10 What's the difference of course I am these eyes of mine may be pretty old, but they never miss anything Well, what's the difference Johnny Freddy's the one that killed him Lord knows he's the one logical No, no certain doc says approximately 10. So all right. It was a few minutes of before Freddy left here So Freddy killed him now look everything ties up. Yeah, and the kind of a knot that's gonna hang now Sergeant I wish it was that easy of course it is motive opportunity everything it all spells Freddy Lynch now Conroy put a tracer on him right sergeant. I'm afraid you're wrong. How can I be that is if mr. Halsey is sure of his time of course? I am on the radio Okay dollar why because Walter E Lynch was still alive after 10 o'clock What are you talking about? How do you know because he'd called me on the phone because I myself was talking to him What exactly eight minutes after 10? We'll continue with act two of yours truly Johnny dollar in exactly one minute Here's Hollywood star Mona Freeman who feels like acting with a miserable cold I relieve coldest rest the fast way with four-way cold tablets Yes tests of all the leading cold tablets proved four-way fastest acting Amazing four-way starts in minutes to relieve muscular pains and headache reduce fever calm upset stomach also Overcomes irregularity when a cold strikes do what I do take four-way cold tablets It's the fast way to relieve nasty cold distress and feel better quickly four-way only 29 cents Now here's a word about another fine product of Grove laboratories had dandruff for years now get rid of it in three minutes with Fitch dandruff remover shampoo three minutes with Fitch regularly is guaranteed to keep unsightly dandruff away forever Apply Fitch before wetting hair rub in one minute add water lather one minute then rinse one minute Every trace of dandruff goes down the drain three minutes with Fitch embarrassing dandruff gone Fitch can also leave hair up to 35 percent brighter get Fitch dandruff remover shampoo today And now act two of yours truly Johnny dollar and the hapless ham matter No question about it. I was the perfect alibi from Freddie Lynch Freddie had left his uncle's house a few minutes before 10. I had talked with his uncle at eight minutes after 10 Yeah, and when the police checked up they learned that Freddie had taken a bus to New York at 10 15 On foot and he was on foot. Oh man Halsey had seen that he had to leave the house before 10 to make that bus I Told you young man these eyes of mine may be old but they never miss a thing He was on foot all right But now wait wait a minute if someone else came in here to commit the murder after Freddie left Why hadn't old man Halsey seen that person? You're right dollar Nobody else came here. So it must have been Freddie killed his uncle. I know look sergeant only how could he Because if you talked to mr. Lynch at eight minutes after and Freddie was only his way to the bus station Wait, wait, then it's impossible Dollar this really bugs me. I listen the murder weapon very unusual. I know doc. You told me it must have been something I'd never seen before I wouldn't be too sure about that unless you feel the same way mr. Halsey does about show business Sinful iniquitous Yeah, maybe you're right on this one. What are you getting at dog? I'm getting out Was that an express that bus that Freddie took to New York? No, it was a local But I don't see what luck in my own car. I can beat him down there. Do you know where he lives, mr Also, I certainly do. All right, where at 12 31 wish 43rd Street. Good. I'll see you later Now all right sergeant. Maybe you can fix any tickets for speeding. I pick up along the way Somehow just lucky I guess I managed to reach the address of New York without getting pinched A $10 bill of the superintendent that's item three got me into the rooms of Freddie Lynch The walls were plastered with theatrical programs and play bills most of them from summer playhouses off Broadway shows In a closet. I found some costumes He'd used in various productions and one corner was a battered shield with a broken spear beside it Some heavy oriental belts and jewelry the long robe of an Arabian desert rider with a sort of turban to go with it Yeah, everything fitted in very nice.  I beg your pardon kind of careless, but it's all right. It's okay.  Jerry Allen the agent you know Agent yeah, where you've been Freddie when you didn't answer the letter in the telegram I sent you where you've been hot kid.  What else listen look hey kid get this I got a part for you great part a great one a good salary, too Of course, you got to let me represent you in the deal, but all I want is the usual 10% How about it I kid Jerry Allen you say yeah, sure, that's right Oh, we never been introduced, but I caught you in half a dozen things and you're good boy good real good Well, I like to think so sure sure you are I've been it's my business I got to keep adding to my stable of act is see what I mean, so if you want this part well, I I Don't think so. What what are you talking about? Well, I'm thinking of giving up the theater. Oh now wait a minute Wait a minute kid. You've fallen there to a fortune or something. Why do you say that? Hmm, of course if you don't want the part No, wait Perhaps I'd better do another player too. Good. You'll let me represent you. Well Freddie. Oh, why not? Well, if the part is good is the part good. It's great kid. It's great And if this thing doesn't hit Broadway the first time up hey listen listen, it's a new play written by John Stone You get that John Stone No, I'm afraid I don't recall. I'm sure you've heard of my man He's had more hits. Oh and listen Freddie listen listen to this part All right Okay, that's how the young man just about your age second act is 20 years later. Hey, you think you can do it Of course my good man good good in the third act well now that part may be a little tough You got to kind of make like you're about 60 Hey, there's no problem at all. I'm telling no problem at all only it's a shaky old man Like he's scared to death all the time. All right, listen All fear has crushed me by the throat the old power Willing oh hey, that's great. That's great. Come on. Keep it up Freddie boy.  That's great great great great now say my life is in danger I've been threatened mr. Dollar My very life isn't What?  I'll be coming into a lot of money His insurance and whatever money or property there is why don't you drop dead? I See But do you see this? So that's what you did it with yeah, yeah, that's right Arabian isn't it I stand long thin blade with a double edge From one of your costume players stand back. I say sure sure I Don't need to be close to you to use this Funny I didn't think I was that good a shot I'll have to make another deposition for the sake of another trial And I'm sure that hamming it up in court won't keep Freddie from playing out the rest of his life In front of a captive audience Expense account total including a flock of incidentals. Well, let's call it a hundred bucks yours truly Johnny doll The word from our star about the case you'll investigate on next week's program Thanks, Daniel. It's called the unholy to matter and involves a pair of characters who fit that description perfectly one of my killer And if you're at all curious about what to do when you've got the situation well in hand and then suddenly a gun is shoved into the middle You're back Well, join us. Won't you right here on your favorite station at the usual time on Sunday yours truly? Johnny doll In our cast were Virginia Greg, Chuck Stratton, Florence Dobkin, Sam Edwards, Herb Ellis, Ralph Moody, and Junius Matthews Be sure to join us next week same time and station for another exciting story of yours truly Johnny doll early speaking Next Ray Bradbury's science fiction classic zero hour as suspense follows on the CBS radio network


Look at this beautiful orange sunrise here in Milan in Italy. The sad thing is, is it's rising over a hellhole. There's slaughterhouse here. Huge slaughterhouse. They're all over the earth. A beautiful earth with a very dark underbelly. And this is 24-7? This runs 24 hours a day? No, no, no, no. How many a day? 10 hours? From Monday to Friday. 10 hours a day? 1500 to 2000 a day maybe. 5 days a week. 10,000 a week. 500,000 a year. So they murder a few here in this murder factory. So I'm guessing this translates into the only difference is your attitude. Bobby Calfs are killed in the dairy industry, first day of life or a few weeks later for Ville. Significant message. No matter what language you say it in. Hello, darling. Oh my god. It's okay. Oh, look at this one over here. I'm just laying down here in the middle of the truck. They are so terrified.  Come here. It's okay. You can actually tell they've all got different personalities.  Some are a bit more curious. Others are more sad. They've got these scratches on their body because they don't do well confined with each other.  They look really sad these ones, hey. And off they go. Inside there. To be electrical shocked in the head. Stuck in a throat. And just chopped up into products. Like how terrible are those poor animals. What their fate actually is in there. And people just consume them unknowingly, unconsciously consuming rural sentient beings with personalities. Terrified, scared, young animals. Just such a tragedy. Terrible. Abdullah, you were just saying how the back of that truck really smells. Smells like their feces. Pigs are very clean animals. They hate being in their own feces basically. If you have them at a sanctuary they build a nest out of hay and they will always go to the toilet outside of where they sleep. This is the animals feces on the side of the truck. They defecate because they're scared. But they're forced to live in their own waste in those trucks and on those farms and it's just horrible for them. Like smelling them their own waste and other animals waste and being so close together. So empty trucks coming out. They once had lives in there now they're gone. Their whole life, their whole existence taking someone's bacon sandwich. I can't think of a more, you know, higher injustice than that. Just so unfair. Wow, look how many. Hello. They've got holes punched out in their ears. See that hole? So hello, hello. It's okay. It's okay darling. I'm so sorry. These poor babies are going to be killed. It's so sad. They're so scared. They're so young and just innocent looking. They have no idea what they're about to face in there but it's just so tragic, man. And really unnecessary. They have a popular ham here called prosciutto and they keep the legs like at restaurants and they just cut slithers of the flesh off the leg. See and they serve the legs of the animals here. They slice the pieces off and eat them and they're unaware of where they come from. The slaughterhouse down the street. Crazy, right?   This truck carries their corpses to the restaurants where unknowing customers will die on the corpses of the victims. That was one of the first times in a long time that I've actually started tearing at my eyes. Like that angel, she had her ear. Her ear was covering her eye and she was sort of peeking outside of her eye. She was so shy and just depressed. They looked so sad. That was really heart-wrenching for me and I don't know, I felt like bawling my eyes out for them. Hey, so sad. That was so sad, hey. They're really sad, those animals. It never gets easier, does it? Like it's always sad. I actually think it gets harder to face them. I don't know, just every time the truck comes it's just like more and more deeper emotion to it. Thank you everyone for coming, it's really cold and it's important that we are at the front of slaughterhouses so that they know that we know, the public know that we know and that we are doing our best to speak up for the animals and share the photos as much as you can. That was quite a powerful vigil. Very sad animals, very sad experience. Pulled my heart strings a lot seeing those animals like that. Angel, little puppy. Pigs are my favourite animal but I shouldn't have favourites but I've always just loved pigs since I was a kid, aye.


Time blocking you have to time block your day when you time block your day You actually go into understanding how much you're worth an hour So when I would work in the dealership I'd get there in the morning by about 7 30 and by 10 30 I had done more in the day than three or four cells people combined because I would time block every Single minute and my time mattered and that's the reason why I was able to scale how much I'm worth an hour Because I actually was accountable for my time I didn't get stuck at the coffee machine for an hour having a conversation with somebody I knew I couldn't afford to do that This right here is probably one of the most important things that you could have as an automotive salesperson as anybody in life that's selling Confidence think about this What's next to confidence? Create new business. How can you create new business? Well, I create new business by posting a lot of people that know me I'm when I made over 700 grand a year selling cars. I posted on Craigslist non-stop I lived it Facebook Marketplace wasn't available then Erin. She said hey Andy, you know Craigslist don't work. I posted two cars for we sold both of them. I do Certainty think about this word with me. Okay Certainty if somebody was telling you something if they were telling you something and they didn't seem a hundred percent certain What they were saying, what would you do? Well, you would leave and this is what happens most salespeople they can get to 99% certainty But because they have 1% of Uncertainty the customer cleans to the 1% and they leave This is a time where you got to check yourself from the neck up and decide man. Hey, look, how's my believability skill? How's my confidence look? How's my believability? How's your conviction? Plan for Revenue. All right, you say Andy. What's that mean plan for revenue? That means how much money do you want to make? Every single day, right? Look, this is how your day is going your time blocking your day. You're going to create new business You have to plan which means how much money do you want to make each day? Let me ask you that question Do you want to make 500 a day?  Do the math Energy is important Momentum feeds the machine with you and the customer's relationship I am constantly building up momentum as I'm going through doing business with my customer.  I do need more energy. How do you get it? Where do you get it from? Well, it's real easy. I trained myself to have it and plus our customers deserve it And if you can give it to every customer most likely all of them will buy from you now Let's talk about some things real quick Word tracks are very important Plan to build all right. What does that mean? Well, here's the deal Everybody has to have time in their day to do what make money, which is make revenue Train okay, and also to build you guys are a business inside of a business I know a lot of people right now that have their own landing pages You don't have to have a landing page to be successful But it sure makes a big difference because people go to websites to get information But when they go down funnels what happens you capture the lead and leads create cells What happens when his words are wrong? You see because most salespeople that sell 30 plus cars a month The one thing that they know is this their words are everything and most sales people say 70 percent of the right stuff But 30% of the wrong stuff and every time you say something wrong it pulls the sell backwards again Look when people buy they make decisions one or two ways actually both one logically right logically like what you're saying Logically makes sense to me and then second with my gut. I decide as I hear you talking to me Hey, man, do I believe in this guy? Do I look up to this person? Do I think that what they're telling me right is right in Persuasion and influencing people are you taking the way you feel and pushing it into the customer? Think about it. I can take anybody in any state take the way that I feel walk up to them Put my arms around them and push the way I feel into them and they'll and they'll act that way You have to learn to make gross If learn to make gross guess what's gonna happen You're just gonna make many's your whole life now look maybe you are the rare exception and you work at a what? Unit volume store only that's like 3% of the population out there the other 97% guys We have to put up gross we have to make money. We're paid off of gross profit guys There's a million ways of skin and cat but you have to be great at closing negotiations Overcoming objections war tracks and not just closing but closing for all the money


Oh, yeah, and I want you guys to be able to see the view from my hotel room, because it's pretty, pretty awesome. In LA, 24 times, she's on the phone, but she's staying home. Good morning from here in New York. It is Sunday, and the Pride March is kicked off. AHS Float, which I'm going to be on, doesn't actually start in the March until 5, 6 p.m. And it's just barely noon now, and the march is starting, so this is going to be a long, long march. Some people have told me it's like 10 hours long. I never heard of anything like that in LA. It was like four or five hours long, and I thought that was so long, but this is crazy, and there's millions expected to be here. Yeah, I'm just going to eat something, see if I can get some cool footage, and see what's going on out there.  So as you can see, all these normal residential streets have just tons of floats sitting here, waiting until their step-off time. And it's just street after street after street after street like that. Actually, I'll post the map right now so you can see it. Just all the streets that have floats lined up, all waiting to go to be a part of this march. It's nothing like I've ever seen, or like I thought I would see. Coming here blows me away. Got a little ham and cheese going, and some sea salt chips. Surprise, I spoke to them because as I said, you know, I cannot believe that people are still stigmatized. Absolutely, yeah. I was very surprised. I had a barber in LA after he found out that I had HIV. He wouldn't no longer talk to me. He wouldn't see me anymore. And this is for real? Oh yeah. You know what? I don't get it. There's still a lot of work that needs to be done for HIV. I don't get it. They say you have no idea.  I was so surprised. True. Anyway, you look fantastic. Thank you. Appreciate it. And I wish you well. Thank you. You have a partner? I'm not available, though. I'm seeing someone. We're working on it. And he is not intimidated? No. You know what? Go strong.


Welcome to today's lecture where we will be discussing the Rayleigh-Bernard convection problem. We will just continue from where we left off in the last class and just to make sure that we have everything in place what I have done is I have written down the equation which describes the perturbation V star talks about the functional dependency of y of the perturbation and V star is going to be governed by this sixth order ordinary differential equation. Now this particular ordinary differential equation was obtained by eliminating the u component of velocity, the pressure and the temperature.  Before the Navier-Stokes equations were only second order to begin with because we eliminated the other variables the u component of the velocity, the x component of the velocity and the temperature, the second order equation became a sixth order equation. This equation is subject to these boundary conditions. The boundary conditions are the same at both the surfaces at the lower wall and the upper wall because both of them are solid walls. We have the velocity being 0, this comes from the fact that the walls are impermeable and therefore there is no vertical component of velocity. The fact that you have a no slip boundary condition which tells you that the x component of velocity is 0, we use that to transform the boundary condition to the vertical component of velocity v. So and for this we use the continuity equation and we got that the first derivative of velocity in the y direction is 0. Remember capital D is the operator d by dy and this boundary condition came from the temperature boundary condition. So you see that the boundary conditions which were there for the temperature, the boundary conditions which were there for the x component of velocity all of that have been converted to boundary conditions on the vertical component of velocity v. Now what we like to do is take a close look at this equation and I want you to see that this is a linear homogeneous equation with homogeneous boundary conditions. What does this mean? This means that v star equal to 0 is always a solution to this problem. It satisfies the boundary conditions and it satisfies this differential equation. Now so in some sense this is like an eigenvalue problem. You have A x equals lambda x where A is my differential operator now, lambda is something like this quantity and x is my velocity vector. What we are interested in doing is we are interested in finding out conditions for which this equation has a nonzero solution. You know that v star equal to 0 is always a solution to this problem. Why? Because this is a homogeneous system for but can this have a nonzero solution. What is the question or under what conditions can this have a nonzero solution? Clearly the only parameters which are at your disposal are the temperature of the wall, the gap between the 2 plates, the properties of the fluid, the thermal diffusivity, the kinematic viscosity, the density dependency on temperature and so for some combination of these parameters, it is likely that this system will have a nonzero solution. Now if this system would have a nonzero solution, then for that group, for that combination of parameters, remember we are on the point of marginal stability or neutral stability. Why is that? Because we have derived this equation by imposing the condition that sigma the growth rate in time for the perturbation is 0. Now if you recall what we did in the last class, this equation was derived by imposing the condition that the real part of the growth rate and since the growth rate turned out to be real and not complex, the growth rate was 0. So if the growth rate is 0, you are on the point of marginal stability or just that transition point, the critical point where it is going to change from stable to unstable. So the point of neutral stability and the fact that this for a particular combination, this is going to have a nonzero solution is going to correspond to the point where you have the transition of the steady state from a stable steady state to an unstable steady state, okay. What I want to do is I want to point out here that this particular group of parameters leads me to seek a dimensionless number.  So remember v tilde the perturbation was given as v star of y times e power sigma t times sin alpha z, alpha x, sorry. Now since we have the same variable on both sides, we are not particularly worried about making the velocity dimensionless because we would choose a characteristic scale for the velocity and that would occur both on the left hand side and the right hand side and that is going to cancel off, okay. So imagine you had a characteristic velocity and you made this dimensionless, it would not really affect. But what we have here is this term here which possibly has some units, okay. So remember this term d, the capital D is given by d by dy. So this has units of reciprocal length, okay. The alpha here also has units of reciprocal length. For example, the alpha occurs as a wave number, okay. The wave number is such that it is the reciprocal of wavelength, okay. So alpha is something like 2 pi divided by lambda. So the point I am trying to make here is the alpha actually has units of reciprocal of length and there is only one length scale in the system and that is the height or the gap between the 2 plates. So the length scale in the system is h. I am going to use this length scale and make my equations dimensionless, okay. And what does that mean? I am going to define alpha star as alpha multiplied by h. So this has units of reciprocal length and this is units of length, so this is dimensionless, okay. And I am going to write and if I want to make my equations dimensionless by scaling it with h, remember the domain of the problem now is going to go from 0 to 1. So now the domain of the problem extends from 0 to 1 instead of 0 to h, okay. And clearly what I am going to do is I am going to now write the alpha here in terms of my dimensionless wave number alpha star. I am going to write this as alpha star divided by capital H. The D squared is going to be the second derivative with respect to y and when I am making y dimensionless I am going to have an h coming out in the denominator, okay. So what I am going to do is I am going to define D star as d by dy star which is equal to 1 by h d by dy, sorry, is equal to h times d by dy, okay. This is a dimensionless derivative and this is what I get. So this is h times D. I am going to now, so remember this is the dimensionless derivative with respect to dimensionless coordinate and I am converting this to dimensional coordinate. I am going to substitute for D and alpha star in this equation here. So D I am going to write as D star divided by capital H, alpha I am going to write as alpha star divided by capital H, okay. And now I get D star-alpha star and this is all squared divided by H squared whole cube V star equals-beta g and alpha squared I like to write in terms of alpha again, alpha I like to write in terms of alpha star.  So all I have done is converted that equation to a dimensionless form and remember now alpha star is dimensionless, okay because I have scaled it with H and D star is also dimensionless. I would have gotten this if I had scaled my equations made in dimensionless in the beginning. And all I am going to do is recognize that this is h to the power 6. Take h to the power 6 on this side and remember I have an H squared and an H here. So I have an H cube. So I have H power 6. When I take it to this side I get H cube.  I have D squared-alpha star L square star squared-alpha star squared whole cube operating on V star equal to this. I like to group these terms together and I want you to realize that beta in general is negative and Th-T0 is in general negative because the upper plate temperature is lower than the lower plate temperature. So this is negative and this is negative. So the quantity inside my brackets is a positive quantity, okay. And those of you who have done a course in heat transfer will remember that this is extremely similar to this number called the Grashof number which you have come across. Only thing is in the context of this problem this number is called the Rayleigh number. So I am going to define this Rayleigh number beta times g times Th-T0 times alpha T times nu times H cube, okay. And this is a dimensionless number just like you have the Reynolds number, the Prandtl number and the Schmidt number. And the whole idea of making this thing dimensionless was to get a dimensionless number. So we could talk things in terms of this dimensionless number, the Rayleigh number.   All I have done is converted the boundary conditions also to the dimensionless form, okay. And this is straightforward because you just get a factor of capital H in the denominator and since that is not 0 this has to be 0. So now to go back to what I was saying, we are going to ask the question what is the Rayleigh number for which this has a nonzero solution. If I can find the Rayleigh number for which this has a nonzero solution then I will know that for that group of parameters equal to that number equal to that value I have the marginal stability or the transition from a stable steady state to an unstable steady state, okay. How do I go about solving this problem? Remember alpha star is a constant, okay. So what you really have is a sixth order equation with constant coefficients. Now you know how to solve this problem. So what you would normally do is you would seek a solution in the form of an exponential function.  There is a v star here and there is a v star there, okay. We seek v star as some function times some constant multiplied by an exponential function. You would substitute it in this equation and you would get a characteristic equation which would be a sixth order polynomial. You will get 6 roots which would give you 6 values for m and those would be your possible solutions to the homogeneous equation. Clearly the values of m would depend upon the Rayleigh number and the alpha star. Then what you would do is you would impose these boundary conditions and try to find a non-zero combination of C1 for which you will be able to satisfy the boundary conditions, okay. So we do not want a situation where the constants C1 are all identically 0 because then you would get v star as 0. Remember we want v star to be non-zero. So the strategy is to try and solve this equation by seeking solution of this kind, getting a sixth order polynomial and then imposing the boundary conditions. Since you have 6 boundary conditions and you have 6 constants C1, C2, C3 maybe I should just write this as C1 e power m1 y, C2 e power m2 y. You want to impose the boundary conditions and then you would find that you have 6 equations in 6 unknowns. The 6 unknowns are the arbitrary constants Ci. We want a non-zero Ci and what you would do is you would formulate the problem as if it is a matrix problem, as if it is a matrix multiplying a vector. The vector contains my Ci and look for a solution for which you get a non-zero solution. Look for a condition for which you get a non-zero solution and from linear algebra you know that this is given by the determinant of the matrix being 0, okay. So let me just write this down. We seek V star as Ci e power mi y, okay. Substitute in the sixth order ODE and get 6 roots for the mi. That is I will have m1 m2 up to m6, okay and you know that V star is going to be given by C1 e power m1 y plus C2 e power m2 y plus etc. up to C6 e power m6 y. What we do is we use the 6 boundary conditions and recast these as a matrix which multiplies C1 C2 up to C6 equal to 0. The right hand side is going to be 0 because remember my boundary conditions are all homogeneous. These equations are going to be linear in the C1, C2, C3, C4 because my boundary conditions were linear, okay. So I can actually write this as a matrix multiplied by a vector and remember the elements here are going to be functions of depend on Rayleigh number and alpha star. So you would get a nonzero solution if the determinant of this matrix is 0. So as you can see there are 2 unknowns. One is the wave number, dimensionless wave number and the Rayleigh number. So what you would do is you would actually fix alpha star at some value and find the Rayleigh number for which the determinant is 0, okay. That is the idea and you would do this for different alpha stars. You will do it for different wave numbers and you would get different values of Rayleigh number and you would make a plot of this curve. So for every wave number you would know what the Rayleigh number is for which you have a nonzero solution, okay. Now you would have to possibly write a small computer code to actually do this problem in MATLAB. But what I like to do now is refer you to this book by Subramanian Chandrasekhar, Hydrodynamic Hydromagnetic Instabilities and the primary reason for me to refer you to this book is for you to see that how about 100 years back when scientists did not have access to the kind of software tools that you have access to, they were still able to solve these problems. So reading this particular section on the Rayleigh-Binard problem in Chandrasekhar, you will see how we use this arguments like symmetry, okay and tries to answer this question and that I have just talked about, how to go about finding a nonzero solution, okay. What I want to do is I want to tell you that we can still get some insight about this problem and get a feel for how this curve, how the Rayleigh number depends on the wave number by solving a hypothetical problem, okay. And so at the beginning when I started I wanted to tell you that I would told you that we would possibly get an analytical solution. So I am not going to disappoint you, I am going to still go about getting an analytical solution but then it will not be for the problem that we were talking about where we had the fluid between 2 solid walls, 2 rigid walls. Imagine now a hypothetical situation where you have a layer of fluid situation where let us say you have a layer of liquid which is suspended in the atmosphere. This is exactly what we had last time, only thing is in the sense that I have a liquid film which is extending from 0 to capital H or in dimensionless from 0 to 1. Instead of having a solid wall I have a gas-liquid interface here, I have a gas-liquid interface here. Remember at a gas-liquid interface the boundary condition is that of the shear stress being 0, okay. So at the gas-liquid interface we have the shear stress vanishing. I am going to keep my life simple and I am going to restrict myself to the case where the interface is going to remain flat, the interface does not deform, okay. So now since the interface does not deform I am going to still have V star equal to 0, okay. So since the interface does not deform V star equals 0 but now the shear stress is 0, you do not have a no-slip boundary condition where I put u as 0 but you would now have du star by dy equals 0, okay. So this is the 0 shear stress condition and remember what I want to do is I want to convert this condition on u to the condition on velocity V star. So what we do is we use the equation of continuity du star by dx plus dv star by dy equal to 0, that is my equation of continuity. I can use this condition only if I want to differentiate this with respect to y. So if I differentiate this with respect to y, I would get d square u star by dx dy plus d square v star, you know and really should be a bit more careful. I should just leave this as u and v by dy square equal to 0. So this is my 0 shear stress condition on the actual velocity and also on the perturbation and this is my equation of continuity and all I am trying to tell you is this condition of the 0 shear stress this is going to be true for all x, du by dy is 0 for all x. So du by dy equals 0 for all x, therefore d square u by dx dy equals 0 and which means this condition implies that the second derivative of v with respect to y is 0. So this implies that d square v is 0. Remember for the rigid wall I got the first derivative equal to 0. Now I am getting the second derivative as being equal to 0 okay. So this hypothetical problem results in v star being 0, the second derivative being 0 and also the other boundary condition which is v star squared minus alpha star squared whole squared v star being 0. So this condition remember comes from my temperature equation. My temperature equation is such that my temperature is fixed. So my temperature perturbation is 0 at the lower wall, temperature perturbation is 0 at the upper wall and therefore that still remains the same and therefore this boundary condition remains the same. Only thing is the change from the solid rigid wall to a gas liquid interface has resulted in the change from the first derivative to the second derivative okay.   Now since v star and the second derivative are both 0, this equation simplifies to this equation and it further simplifies to the fourth derivative being 0 because this is 0 and this is 0 okay and so what I have is the fourth derivative equal to 0. Now therefore what we have now is a problem wherein my v star is a function which satisfies all my even derivatives being 0, the function the even derivative of the second derivative and the fourth derivative are 0 okay and remember these are going to be 0 at 0 and 1. So these boundary conditions are satisfied at 0, 1 okay. I mean since I made a dimensionless I have got to write this at 0, 1 not at 0, h okay. So these are the dimensionless version. Now can you think of a function which actually has this property. So let me give you a clue. This is a trigonometric function and the function which satisfies this is the sinusoidal function sin n pi y. Clearly a n equals sin n pi y satisfies the boundary condition v star is 0. The second derivative gives me the same thing, something multiplied by sin. The fourth derivative gives me this. So automatically this function satisfies the boundary conditions but we want a function which satisfies not only the boundary conditions but also we satisfies the differential equation. So I want to make sure that this satisfies the differential equation okay. So this satisfies all 6 boundary conditions. So I do not worry about the boundary conditions but in order to satisfy my differential equation I am going to substitute this functional form into my differential equation here. And remember this is actually sin n pi y star because I made a dimensionless okay. When I do this I am going to differentiate it 2 times. I am going to get n square pi square with the negative value, negative sin. So I get minus n square pi square minus alpha star square whole cube. Instead of v star I would get, I have already substituted this. So I am going to get a n sin n pi y equals minus Rayleigh number times alpha star squared times a n sin n pi y. All I have done is substituted this functional form in my 6th order differential equation, differentiating this 2 times gives me n square pi square and this is whole cube and this is what I get. Clearly we want a non-zero a n sin n pi y okay. And I can take out this negative outside. I will get negative of this thing whole cube okay. And so this basically gives me the condition Rayleigh number is of the form n square pi square plus alpha square whole cube divided by alpha star squared okay. So when I was talking about the problem with the 6 constants and the determinant and I told you that you get an expression for Rayleigh number as a function of alpha star. We were not able to do it because it involved possibly writing a computer program. But this hypothetical problem gives me explicitly what the dependency is of Rayleigh number on alpha star squared okay. And clearly it also depends upon n. What we will do now is make a plot of this particular function. I am going to plot Rayleigh number versus alpha square. You realize that as alpha star tends to 0, this function is going to go to infinity is going to be unbounded because of the denominator. And so for low values of alpha star, this is going to be negligible compared to this. This is low and that is going to make my Rayleigh number go to infinity. For very large values of alpha star, this is going to dominate over this. I can neglect this. I will have alpha star per 6 divided by alpha star squared. So it goes as alpha star per 4 and again it is going to go to infinity. You also realize that this is all going to be positive. So I have a curve which is going to be confined only to the first quadrant and which has a shape of this kind okay. Point I am trying to make here is that this curve is going to have a shape of this kind. And I am going to draw this for different values of n. I will have a curve like this for n equal to 1. I would have a similar curve for n equals 2. Clearly the curve for n equals 2 is going to be having a magnitude or larger for Rayleigh number for every value of alpha squared because of the way the n is appearing here. So the point I am trying to make here is that the n equals 1 curve is the one which is lowest okay. And what we want to do is we want you to realize that along this curve remember the growth rate is 0 because that was the condition which we had imposed on getting the solution to the equation when we were simplifying. So the growth rate of the curve, sigma, the growth rate of the disturbances is 0 along this curve okay. So let me try to tell you what we have done. I started solving a real problem of a fluid between 2 rigid walls and I said look I have to do it numerically and you have to do it numerically in the sense you have to solve this problem numerically and that is going to be one of the assignments. But I do not want to give up, I want to explain to you something about the physics of this problem and to allow me to do that I have gone to this hypothetical problem where my boundary condition now has changed from a no-slip boundary condition to a 0 shear stress boundary condition. This helps me get all even derivative 0, helps me guess a solution to my 6 order equation. The solution I have guessed is the sinusoidal curve okay and then I have substituted the sinusoidal curve and got into a situation where my boundary conditions are satisfied and I just want to make sure the differential equation is satisfied and that tells me how the Rayleigh number depends on alpha star squared. What we want to do is find out for n equals 1, find the point at which this minimum occurs and I am going to leave this as an exercise for you to calculate the value of alpha star at which you have the minimum and the corresponding value of the Rayleigh number. But I am going to tell you what the corresponding value of the Rayleigh number is and this value of the Rayleigh number is 657 approximately and this is remember for the hypothetical problem. So far we have not bothered about the alpha, the wave number of the disturbance. It would appear, we have not told you what this wave number is okay. We have just done the problem for a general arbitrary wave number and that is how this analysis has gone. Now how do I know or how does the system decide what the wave number is of the disturbance? That is the question and by looking at this picture we can get some insight. When you give a perturbation to a system, this perturbation is going to be composed of different wave numbers. It is going to be an arbitrary perturbation. You cannot in a real situation impose a perturbation of a fixed wave number in general. So you would give an arbitrary perturbation and this arbitrary perturbation can be decomposed using something like a Fourier series analysis into different modes. So those different modes corresponds to the sin alpha x. So when I have an arbitrary perturbation with the function of x, I would then say this has some component sin alpha 1x, some component sin alpha 2x. This is like decomposing a vector into a set of basis vector components okay. Now what we are asking is I want to find out how does each wave number behave? How does the disturbance in each wave number behave? Clearly I have told you along this curve sigma is 0 which means this curve is dividing the space into 2 regions. On one side of the curve sigma is going to be negative and on the other side sigma is going to be positive. Now we can use common sense to figure out where sigma is going to be negative. Clearly for low values of Rayleigh number, this corresponds to low values of the temperature gradient or the temperature difference. I expect the system to be stable. So if the system is going to be stable, this corresponds to sigma being negative and the region above this curve is going to correspond to sigma being positive. Now this curve therefore divides this into 2 portions where sigma is negative, 1 where sigma is greater than 0. Now if the Rayleigh number is sufficiently small, it turns out that the growth rate sigma is negative for all the wave numbers which means if I give an arbitrary disturbance, if I want to decompose it into different wave numbers into components of different wave numbers, each of them is going to have sigma negative. So in this region the system is stable. So for Rayleigh number less than 657 and all wave numbers have sigma negative. So system is stable. For Rayleigh number which is just greater than 657, so just slightly above this line, you see that if I give an arbitrary disturbance, I am going to resolve it into again different wave numbers. Wave numbers on this side are stable. But the wave numbers here are unstable. So you would therefore see in your real system the disturbance which grows and that is going to correspond to a disturbance which has a wave number closer to the value where this minimum is occurring. So the disturbance which grows corresponds to the alpha star of the minimum, the minimum value of alpha star. And this one remember is the one which is going to have the fastest growth rate because close to this sigma is 0. So if you have a Rayleigh number given by this value corresponding to this alpha star, this is farthest from this boundary. So the sigma value is going to be largest. As you come closer to this boundary, sigma is going to be 0. And so this disturbance with this wave number is going to be the one which is going to be farthest. So what I am saying is the one disturbance which is going to manifest itself in a real situation is going to correspond to the one with this alpha star. And this is the one which the system is going to decide by itself. So the non-linear interactions of the system tell you that the wave number which you are going to observe when it becomes unstable is going to be given by this alpha star. We will see more of this in the next class where I will discuss this in more detail. Thank you.


Okay. Okay. Well hello to everybody who has come back for the talk and hello to the CalRound audience. We are here at the 37th annual Bay Area Playwrights Festival, and we are talking to three of our playwrights, T.D. Mitchell. This is the screen, also known as the Hunter screen, and Philip Howes, on a topic called Translating the Untenable. And we have Amy Neeler, our executive director, and she's the director of moderating. And we're just going to, I'm going to let you take your way. Okay. Yeah. But come sit up close so we can hear each other, so I suggest. So the idea of this panel came about just actually choosing and reading the work for this year's festival. And, you know, making the hard choice really in a lot of ways. A lot of the theater we see today is, tends to be safe, especially in the mutual theater circle. And playwrights' foundation is a place where we allow playwrights to be, not allow them, but we celebrate playwrights who are writing work that isn't necessarily safe or tenable. But that used to be a theater, in a sense, to poetically reflect back some of the more difficult human interactions that will be basically weird about 247 on our email feeds. So we thought maybe we could get these three playwrights to talk a little bit about how they each approach the complexity of very, very difficult, possibly traumatic situations without, say, re-traumatizing the audience or the actors for that matter. And in fact, sort of investigating and interrogating traumatic human interactions in a way that gives it sort of the beauty of poetry, of theater, of theatrical poetry. So I just wanted to, you know, give each of these playwrights a talk a little bit about that. Isn't that, yeah, it takes a question for the audience. Oh, I'm sorry. No, please, please, please. He will start by introducing me to TV Mitchell. So your play talks about sexual trauma in the military sense to MS. The military sense of the recovery movement. Terminology itself actually experienced safer, years safer then. It's in terms of trying to translate it For me, it really starts with the research process. Because I'm very much a research-based writer. Directly portray any real person in my work. But I do interviews with a significant number of people who have gone through experiences that ultimately inform my characters and their stories. In terms of the question of how do you not wish to traumatize your audience with traumatizing the actors, you actually have to go further back with that. And how do I interview survivors without having that interweaving process to traumatize them? How do I create a safe place for them? And have them feel comfortable sharing these very interesting stories. It's a privilege to be given that trust, but it also creates for me a lot of pressure because I feel a great sense of responsibility to these people for entrusting me. So how do you use the medium of theater expressions, these ideas, or poetic ways? What would you say? Well, what I chose to do in my plays, I found a great myth that I thought posed an interesting question that's also posed in the temporary dramatic narrative of the play. So I used a myth that I just think we are as humans a storytelling culture. It is certainly how the Greeks taught morality through the earliest forms of theater. And ideally, theater is not me telling an audience how to feel. It's hopefully an exchange of energy and an exchange of questions. I'm not here to answer any questions for you. I'm hopefully here to present the same questions that I struggled with in the course of writing. So I think it's harder to create poetry in my play. That's interesting. So TD Mitchell wrote Young Things right here as a language. We're going to be seeing right after this. How about you? I need a hundred screen. Elizabeth was probably just watched, but this is good. So I do a lot of research, you know, and because I want to, and I'm going to ask the audience to go through something, and I actually asked the audience to go through something serious. I feel like I need to understand that world and its stories. You chose a structure that was quite fragmented and that is not only is it nonlinear, but it also takes place in several different places and among several different groups to leave this story. What did you choose that for? I chose it on purpose because I have this writer in me that loves to support story structure and wondering if could we tell the story by passing it through when the audience followed training by watching the film or do that. But I also think her play is an expert. I also think the fragmented quality of her work so reflects the fragmented emotional state in so many characters from there. I think it was not that linear. I don't think it would bring the writer true as much. I think that in their experiences, their emotional state is fragmented. Also, when I was writing it, I found that the most true thing when I found that silence is inability to be expressed in so many characters, you just can't get the words out. When I found that, I feel like there is also a kind of distancing that I am able to achieve as the viewer and the audience through the fragmentation and also by being forced to different story lines. You get very upset or triggered by somebody's behavior when they are doing what is happening, suddenly you are fooled or you are in the child's world. And then you are reflecting those two worlds against each other and it is kind of a distancing where you are not emotionally engaged or so deeply that you can't think. And I think that was very effective, a way to think about broken people like you, Mr. Howells. We haven't seen a bottomable yet this weekend. So how do you approach it? I don't do research, not initially. I typically will have women in on the play. Sometimes long enough to stand. And then it will sort of work very quickly at first. And then I'm like, wow, what is this? And then I'll do some research and rewrite the play and screw it up and then I'll rewrite the play and fix it and turn it back to you. You can't fix the play but you don't turn it back to you anymore. But you know, all along the way, sort of searching for what are the kernels of truth, what are the moments of grace, what are the things that are most pressing about the storytelling, about the story on Time to Tell. And I think that in this case, this play, I think it really converges as a tragedy, which, you know, TD brings up the Greeks.  I remember reading that during the Euripides' life, most of his life, that the war was at war. The publication was happening during his life and well after it. And that's what he wrote about. He could, we were writing, at least this play seemed to me about the present moment, which is that. And so, yes, I don't know if I would call it that. I mean, there's a lot of trauma involved, but to me the frame of it, but these two plays, too, I think one, two, they're very different structures and the stories couldn't be contained inside of the traditional narrative, which is why they emerge and how you see them during this festival. Yeah, and I think that's really exciting. I mean, the characters in your play, in particular the adults, are very desperate and they deal with the opposite ways, you know, which is, I love how the focus isn't necessarily on their desperation. The attempt to make meaning out of every single moment and make every moment count. And the attempts at normality among the traumatized, among the desperate are those moments that I think we agree that it's also very American because, you know, this idea of aspiration, the desire to achieve. Any impossibility in the American way. Yeah, yeah, or something like that. So, yeah, I mean, the characters, I think the adults in search for something and in search of something, there's a lot of country. There is a lot of country. And still, still binding. Maybe material things that we do have access to, in the case of the characters in my play, is language. And so then they use language very... The interactions with certain things, besides the individual, the important roots of the individual that was actually born, they would cut a dog in half, a dog in half, and they would march the soldiers through to recognize that they have done this thing and now it's over. It is now. Questions? Yes. T.D., I'm really interested in what made you take on this subject and why do you believe it's one of them, right? So for those of you who haven't seen it yet, don't spoil it. T.D., part of my... I haven't seen it either. But you know about it. It's about four generations of military women in one family and they all have been Marines. To back up and answer the first part of the question, what happened is an earlier play I wrote called Beyond the Seventeenth Parallel dealt with the Vietnam War Conflict and veterans in temporary time, about the time that we, just after me, invaded Iraq, and veterans in their families and their memories after this chunk of time. So I was doing a lot of veterans in their meetings here and some in Vietnam as well. And I kept encountering people who said, oh well my wife also served or my daughter was there and so I thought, oh gosh, I really want to interview these women and find out what this is like. And as I talked to these women, I realized that women needed their own play. I realized that the experience in the military family was in some ways different for women than it was for men and that they really required its own exploration. It wasn't going to fit into that other play. It needed its own space and its own time. I considered for a while which branches to look at. The two primary reasons I chose the Marines, one has to do with the history of the Marine Corps in our country in our various battles and the history of how the Marine Corps integrated and then decommissioned women during various conflicts through history. So we had the Marineettes in World War I and of course we have some previous to that history of women sneaking into the Marines. So I looked primarily for what historically did each branch offer in terms of the history specifically of women of service. So there was that and then there was also the fact that when I did one of the earliest interviews I did for the play was with a scientist who was heading up the first ever US based study of US service women and post-traumatic stress. And so we did an interview about Pat and she, you know, overall you've got 15 to 16% of the total forces for female but in the Marines only about somewhere between 6.5 somewhere around 6.5% are female and overwhelmingly the women the young women signing up for the Marines had already experienced a previous trauma and many of them were signing up with the idea that they would be toughened beyond all hurt or harm and instead they actually put themselves in the most vulnerable spot of the... Yes. Hi, I'm just wondering what was your visceral experience when you realized this should be a play? Your idea. How did you know, yes, I wanted to be a play about your topic? For me it's a non-comfortable thing. I mean, I call it the itch and it's this, it is this unsettled feeling it is this itchiness where I I usually try to abandon it two or three times Yeah, you're doing this. I don't want to do this. This is too big, I don't dare who am I to do this and it's something that I don't feel full of control of so actually it's a it's this level of irritability and discomfort where the only way to make that discomfort and chill out a little is to start writing this rule. Yeah. Yes. Wait, wait. What? It's a sequential. For me, I think it's also a compulsion of feeling like I don't get this out and see it on the page and I just I feel a little unbalanced like the world this needs to be out. It's a bloodline. Yeah. Just a part of these words and the sensations of how I'm taking in all the information, so to speak you know the beginning of the Iraq war and things that have asked me philosophical questions about these things that I just have to do and don't feel an outlet for me to explore these questions that I have. Yeah, same. I'm writing almost every day and so it's not always toward something oftentimes it's just all this stuff and then I'm like you know there's something there like sort of I'm I mean you can be surprised I think too as it starts and just being open to that I think is also really important being open to both the writing process like the ritual and routine of the writing and then the possibility that something may reveal itself to you. Yes. These particular plays that you're involved with how long roughly did it take you to actually come to this? So for me I've been working with these materials since 9-11. Oh wow, okay. But I really felt like I need to find the appropriate vehicle to express it and eventually it emerged that I have to express this little cycle of plays just everything I'm looking at will be contained in play in one play. And so it's been a while and then sitting with it and really steeping in the source material or the writing of the play and then leaving it alone it just takes my whole life. This one? Yeah. Sometimes it does have this inevitability like every single moment it's led up to me writing this play. I wrote the first draft of my play two years ago. I've periodically looked at it but there have been other things in between, sure. That's right. Yes. When we started out the clock it was about how you get what you have the research and all the stuff that you know is special and how you get across to the audiences. And the audience usually in the world brings stuff that's spliced with humor. You know the humor can do a lot of exposing really, really nasty things but in more plays you didn't use humor so I was trying to think what could you use that made me pay attention other than having a theater and I was thinking, I can't tell if you can tell me what it is that you think you put there that allows me to get sucked in. See to me there is humor. It's just very, very very neat. It's I love dark humor. It's also as a performer in this and I've done a number of these in many places ten hours with six people to do this kind of play. The last thing you find in like week three or four of rehearsal is the humor. There's a lot of it.  We just missed it. So it's actually written, it's actually in fact someone beautifully set up but we just blew it. A lot of times just not a lot of times. Some plays just are one play sheltered in place very, very funny play. It's meant as a kind of comic sort of play that it's still funny as a reading but this, as you say this play, I mean there's also just the difficulty in a play like this really all of these plays but I think in particular your play and December your play in a reading so these readings of these plays are really rehearsals. They're really giving the playwrights the opportunity to kind of throw this play up in a room and listen to the actors a lot of rewriting is happening a lot of cutting is happening new pages coming in and out and so the actors are actually every day encountering new material and so for the audience it's an interesting process because in some ways it's not really for you even though you're necessary to do this process it's like later when these plays are staged they will be for you and right now you're kind of for us you're our spies you're our unique you're for us in some ways you're doing a service to us by responding even just we learn so much so this is kind of exchange and so it's interesting that you ask that question because that's something for Elizabeth to really think about as she takes her next steps scene shifts scene shifts and that helps in being involved and visual stuff that right now in this time, present time they're doing it rather than talking about it that also draws me I think there were some parts like in the children's play it's hard to know what they're doing because they could be climbing or something and the play comes out of the moment for them and it is very happy it sounds like they're talking about it but it is actually happening and that was a hard thing when we have to pick and choose what stage direction really needs to be read in terms of describing the action it can very much break up the dialogue for them to see if we have the narrator jump in and say by the way this is happening over here in this other part of the stage and so it's very, it can be tricky to know what of those to leave in because the audience action absolutely needs to use their imagination and know what this looks like here versus what is more important to hear and it's intended to be low verbally, trying to buy things like that so that could be a bit of a issue but I do think you're right with what you're asking is what technology what it is that they do is I know all the research is what you do that helps me get it because I search by crazy trying to get it with you and with your plan what I quite helped me get it was when I went into paragraphs as soon as I read that and I wish I could put it ahead of time it's like I'm supposed to be here and it's supposed to be sort of destroyed and I'm supposed to have a lot of questions and this is the way it's supposed to be and just flowing with it I'm not good I don't flow so so this stage is going to flow it's like whoa I don't know how to do that but I did search and I think that's harder if you're a theater person than when we have a different field for theater then audiences just comes in because they want to be abused or entertained so how do you get those that is for sure and there's only so much of that I'm willing to do because if I did more it would be the claim that I wrote and so it's a difficult ratio of time during the retreat it was decided that she should have this translated into Polish because the Polish will love it so except I think I think there are a few of us that do you want to speak to anyone? No I was just going to say I mean I think that I mean I would offer that there's something about the experience of watching the play which is the reflection of what the artist is trying to do also so perhaps it's possible that some of the frustration you've got of what the artist wants you to feel and that's okay that's okay similarly I think in the culture there's something Poland you would think that we were in North California Poland has come up so many times you know sort of what we as Americans perhaps allow like my play like you know if you come and see you might be like am I supposed to laugh you know like is that I don't know if I can laugh at that but you can you know I think that we experience a kind of apprehension or don't allow ourselves to laugh there's a sense of singing permission and it's hard it can be hard as the playwright to ensure that some picture in the front of the play gives the audience that permission gives the audience that clue so I hear it you can laugh at that you can go yeah that other part and that's cool so it's tricky to know how to include that permission well that actually it almost bounces off that I think about this stuff too as it deals with really dark materials in fact how conscious you are in the process of I need something here to give the audience and the actors a break sort of whether it's through humor or through a switching perspective or whatever or like that just comes as part of you know do you find yourself thinking about that consciously in structure like oh we you know we need a breather for me the story comes out naturally I'm pretty much worthy I think the first thing that was the first thing that came out was the last bit that you saw I didn't give myself permission I was like yeah this is it and get ready I'll get here you have to live with this you know and I don't think about giving the audience I think for a humor that comes out is a dark sense of humor it's not for everybody but I've heard throughout you know people kind of like I would hear these little tidders and they would be like oh it's not really funny but I think we discover at least for me we discover it from the director and the actors and watching you guys you know I've circled around for years even doing things that actually you're part of the research and development you know that are helping us figure out how to modulate our plays and it's much less about what is said in the talk about afterward what's the most helpful thing sitting in a room with a community of people all having the same experience all you know having the exchange of energy with the words and the performers on stage and just feeling how the room shifts and feeling you know when I'm listening to the play but I'm also very aware of all of you and I'm aware of where people are shifting getting a little uncomfortable and whether that's good uncomfortable or bad uncomfortable and those things tell me more than anything I think it's so useful I think we have time for one one or two questions Yes, I'd like to ask a more general question at what point does a play become unsafe and does it need to stay safe for the rest of the play or become safe at the end has it lost its unsafeness I'm thinking specifically of Clive Wernth Park which has a very unsafe second back but has this nice Wernth Wernth Park What do we mean by safe exactly well we're talking about plays being safe or unsafe I suppose taking risk as theatre making it uncomfortable or edgy or not necessarily uncomfortable or edgy but taking risk specifically taking theatrical risk in telling in the story that it tells or how it tells it or what the characters do or what it's meant to to engender the audience Well the good news is that that I think one of the things that makes theatre art is the fact that it is an unsafe experience anything that happened up here anything to go wrong and I think some of us as audience are yeah we may be rooting for it all to go right but isn't it always also a magical moment when something goes wrong and you see how it's dealt with so I think it's that I think it's that lack of safety from the get that's part of the appeal in terms of when a play is safe or not safe I I don't know I really don't know and I think it's also going to be so different for the individual as well because I know there are certain parts of my play that will feel very unsafe for certain audience matters and other audience will be safe or unsafe in the realm of American theatre when it gets political and when it gets produced but that's a whole other sexy weekend and it's only something to discuss next year exactly we had a question it's not a process question you know you need a certain momentum to do writing actually so she's asking about how a writer can keep momentum going when researching but really very difficult issues or such as I just generally say a but you know about the depression I don't always but I've gone through periods where by nightmares where I just have to walk away from play in one case about two years so so I'm trying to get better at that very same I don't have the answer I just know that for me sometimes I don't know like I work this is a cycle of plays I'm in the middle of a second play and it's even darker and funny and no very good but yeah but I found myself being really beautiful like and I just realized I need to get this stuff away and I'll go for a walk or I'll play with my kids you know my kids are great for keeping it real like getting back like once for dinner totally refocus it's also been really helpful like I felt really alone and it's nice to come here and realize people are feeling really supported throughout the a lot of things you know that stuff other people's kids um that was a joke you know Ben Franklin used to use the smoke pipe and Mike Angelou wrote the hotel room in sort of solitude and then you know most writers I do exercise and then a lot of writers do meditation so I think there are a lot of things that sort of help stave off the depression and I might be I'm not joking because I'm a writer myself I'm thinking too also just in terms of the material like approaching the actual material looking for the thread of hope looking for the statement that you know to keep bringing you back to it a sense of responsibility for the future is going to tends to be what holds me back the thread of hope I don't really have in the play really dealing with stuff there is no thread of hope and that's how it feels and coming together in a room and being able to talk to people with no one the silences of the play the inability to express the feeling about the situations the material it's not like we aren't aware really what we do about it actually she'll have a final question I mean I would also share that I think you know these plays that have stayed previously are about the present moment and if the play is still then it's and you've written more than 30 pages of it or more than 50 pages I mean it's like it's hard to abandon it's hard to abandon I've done it many people I'm sure have done it but there's something about the urgence I mean if it's still urgent with you if it's still swirling around if it's if the questions are still haunting your dreams then it's you have to you have to get back in the sandbox with it I mean your play since 911 and the end of this material I mean I think I said this yesterday I write the hope that I have actually that's just what is the hope is that I hope that this play is unnecessary when I finish it it would be wonderful we didn't need this play that would give me that's the hope so far it hasn't materialized so I'm going to go through the cycle one last question I don't even want to ask that question I hope everybody can come and see all of the plays because they each tackle these difficult issues in completely different styles and we'll leave you feeling different ways and the question I was going to ask is a sort of overly simplistic question that is leading towards getting you to look for that what do you want to leave your audience with and I think writers are often not thinking about that and probably shouldn't think about what you want to leave your audience with if you want to just write your play that's something you want to say but sometimes you know sometimes very specifically with little computers trying to get people to get up and march out and change the world other forums are trying to get you to laugh at it so you can just deal with it sometimes it's I want us all to just be able to have a moment of admitting we all fill up what's about this thing and so I actually if you want to answer what you're in your board questions Howard Baker said that I want the audience to know what I'm trying to tell them I'm going to say mystery we are going to wrap this up because our play that will be read at 4pm needs to get in here right now