| 00:00 | Mhm. Okay. Thank you. . How things work. So today |
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| 00:10 | a few remarks about that's fine. then Yes, well there were a |
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| 00:21 | of things. I didn't dude, structure and then I will continue to |
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| 00:30 | about. That's the subject of the . Mhm. And in particular a |
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| 00:38 | bit about what's known as backward ever , something known as Horny as |
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| 00:43 | when it's from Chapter one of the and I only have a couple of |
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| 00:51 | slides in the slide deck that is . So I'm not going to comment |
|
| 00:55 | all of the slides. But I the one thing here that was |
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| 01:02 | I just pointed out by last but there's a number of yeah, |
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| 01:09 | a way reserved values in my that's we should avoid trying to use |
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| 01:16 | variable menus. That's pretty much what like. I tend to say |
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| 01:22 | And one of the things that was the side that I as a constant |
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| 01:25 | kind of hard coded and available with precision in that place. And I |
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| 01:32 | that's pretty much it in terms of on this slide, that's like, |
|
| 01:37 | know, you guys talked about most these um The only thing that I |
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| 01:41 | that did not comment on was there's a function after the for a race |
|
| 01:50 | are kind of convenient construction. So a bunch of but building functions and |
|
| 01:54 | don't talk about them all in the here that as to the convenience of |
|
| 02:00 | maps have been manipulating matrices then I this one perhaps it's not totally intuitive |
|
| 02:13 | functions operators works on the race. so that has these two. |
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| 02:20 | I mean, all the philologist They're raised with each other and their |
|
| 02:26 | call one morning. So for this trip, for example, with you |
|
| 02:31 | two x 2 matrices, it operates wise. And then just the condition |
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| 02:38 | each pair's elements going down col so the any this that means that the |
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| 02:45 | that somewhere in the column, there some elements at least of a that |
|
| 02:52 | less than the corresponding. So for particular cases that honestly, this number |
|
| 02:59 | not smaller than the corresponding number. , But this number that 20 |
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| 03:06 | So that's why the condition is true the first column, but it's not |
|
| 03:11 | the second column, because in that it becomes there is comparison results in |
|
| 03:17 | being vigorously. On the other they all means that this condition needs |
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| 03:24 | hold all corresponding pairs. The best obviously falls for the first column because |
|
| 03:32 | . In this case, the first first column of A and B. |
|
| 03:37 | the element is the biggest of us because of not just something to be |
|
| 03:45 | of how that works. And then have another example here. That's um |
|
| 03:54 | me, um It's not totally that's for example. So there's a |
|
| 04:00 | simple logic. And uh where did different things depending on what the value |
|
| 04:08 | access compared to a couple of So the question here is What happens |
|
| 04:16 | these four cases. And in So that's and I want to ask |
|
| 04:22 | so for the first case A Here, what do you think? |
|
| 04:29 | would be 500? Okay. That sense to me. But that's not |
|
| 04:39 | McMahon thinks what. So we can another one and we can try to |
|
| 04:52 | and then I'll tell you how much about this. So fine. |
|
| 04:57 | I'll give it to you About 30 them, Right? That's what you |
|
| 05:07 | , I would also believe that that's what's going to happen. So what |
|
| 05:14 | matter eventually? Do? There's one and then the next time I want |
|
| 05:22 | show you. Yes. So it Each of their conditions, right? |
|
| 05:30 | compares whether X is greater than zero not. And it also tests whether |
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| 05:37 | is smaller than 10 or not. if either one of them is |
|
| 05:42 | that's an order, then it proceeds that cost. So in this case |
|
| 05:49 | -1, It's not smaller than but it's smaller than 10. So |
|
| 05:55 | that case because it's an or then it proceeds to do Why for |
|
| 06:02 | seconds. And you can go through same logic happens with all the other |
|
| 06:07 | and that's fine. All of this up during the first statement after the |
|
| 06:13 | . Of course. So that's something , To me, one needs to |
|
| 06:19 | very aware of because our normal sense my normal stance would be the same |
|
| 06:23 | yours that they would fail the first and move on for the x equals |
|
| 06:29 | to the vast but that's not for . So when you have the objects |
|
| 06:35 | to get what I think improves, one may have intended is that they |
|
| 06:42 | to force, it says that both and right sides of the street and |
|
| 06:46 | comparison as to the tool together they from the loss. So to me |
|
| 06:54 | kind of a things are easier. can results in books because you think |
|
| 07:02 | it as an air conditioned utilities, that's not the way for me. |
|
| 07:10 | hmm. So if you do that him then it comes out and that's |
|
| 07:17 | think pretty much the comments of heads this place. Something here. And |
|
| 07:23 | there's somewhat references here when we find said there's many more slides that covers |
|
| 07:31 | text of the exercises that three. you're just going through last time as |
|
| 07:40 | , the ones who do not. , mm hmm. Just all |
|
| 08:03 | Yes. So the exercises of the , I would say it's most difficult |
|
| 08:11 | some getting used to using um, I call the reputation for indexing or |
|
| 08:20 | location. Uh, and the functions manipulated it raises. That's why can |
|
| 08:35 | go ahead, Right. Um, a few things that were left last |
|
| 08:42 | and this might be useful for you and some of the later assignments |
|
| 08:47 | So let's see. Um So first here is um using the symbolic |
|
| 08:56 | so you'll need to install that in matter of you can simply do |
|
| 08:59 | Um So in math lab you can get a symbolic representation of different expressions |
|
| 09:07 | you want to look at them in we're not just not just solve them |
|
| 09:12 | by putting numeric values in them for . You can simply use the symbolic |
|
| 09:17 | . And the first thing I'll show how you can solve uh linear equations |
|
| 09:22 | quadratic equations or even higher order Um So that's the first thing you |
|
| 09:28 | to do is define a symbolic So here in this case i they |
|
| 09:33 | X as a symbolically able and then the function solved, you can actually |
|
| 09:41 | a linear or quadratic equations in this I'm solving here for X. So |
|
| 09:46 | X -5 equals two. That's a equation. And remember you need to |
|
| 09:52 | double equals for the for getting the answer. Um So if you learn |
|
| 09:58 | it's quite straightforward, you get uh solution of x S seven in that |
|
| 10:04 | the solution of a quadratic equation at square was two, X plus one |
|
| 10:09 | zero and you solve it for Um You don't really need to give |
|
| 10:15 | . Specifically practical by default X is first variable that meth lab looks |
|
| 10:20 | Um But still you can you can that. Um So yeah, the |
|
| 10:25 | for a practical radic equation is the of the quadratic equation. So in |
|
| 10:30 | case, why do contains minus one minus one, which are the two |
|
| 10:35 | um of this particular equation? If are no roots, then you also |
|
| 10:39 | lab also gives you the imaginary uh and the imaginary roots for that equation |
|
| 10:45 | well. So it doesn't give you additive. There are no uh no |
|
| 10:50 | . Now there are two different ways can define ah sort of functions or |
|
| 10:58 | . So one is if you are already aware of, is called a |
|
| 11:03 | handle. And this is basically very when you just want to evaluate an |
|
| 11:09 | for some numeric values. So it's a shorthand version of defining a small |
|
| 11:14 | function. And in this case, I've done is I've defined F1 as |
|
| 11:19 | function handle and this is the syntax you need to provide it. So |
|
| 11:24 | the rate uh and then inside parenthesis need to provide what the variables are |
|
| 11:30 | that, in that functions in this it's just X, that's the |
|
| 11:35 | So what, what I've done here I've defined F one as a function |
|
| 11:40 | to a function that defines whose definition basically X square and if you want |
|
| 11:47 | evaluate that function just simply call that handled just like any other function and |
|
| 11:53 | here I've done F12 and that gives the answer for that And see that |
|
| 12:01 | is the class of F1 here is called function handle and that will be |
|
| 12:07 | from what I what I'll show you is symbolic variables and symbolic functions. |
|
| 12:13 | yeah, you can directly evaluate the handle by providing a numeric value and |
|
| 12:18 | output of whatever you get by calling function. Again, just calling class |
|
| 12:24 | on F12 tells you that it's a type of variable there. Uh Similarly |
|
| 12:32 | can have a function that has multiple in that. So here F two |
|
| 12:36 | a function of X and Y where X squared plus Y. Is the |
|
| 12:40 | of that function. And again the thing, it's a function handle. |
|
| 12:45 | then you can evaluate it by calling like any other function as F |
|
| 12:49 | comma three. And then again the is a double bag of variable. |
|
| 12:57 | , so that's function handle it. the symbolic functions are a little bit |
|
| 13:01 | and they are useful when you want um see sort of an expression type |
|
| 13:07 | um um view of that function so say, I'll show you what that |
|
| 13:15 | . Um So here we have defined symbol, T. You don't necessarily |
|
| 13:19 | to have to use X or you define any different any other symbol. |
|
| 13:25 | you have different T. Which is symbolic valuable. And then F. |
|
| 13:30 | . Is a symbolic function of So F is a symbolic function of |
|
| 13:36 | whose definition is two times sine Please mm hmm. And if I and |
|
| 13:44 | print it out right. Okay. just done it again. Yeah. |
|
| 13:59 | . So now if I just simply F. D. Then you get |
|
| 14:04 | same symbolic representation of that of that . So it doesn't get evaluated for |
|
| 14:09 | yet because obviously we haven't passed any to it. Um before I show |
|
| 14:16 | how to evaluate it, we can can also perform differentiation. So by |
|
| 14:20 | just calling the function on a on symbolic function gives you a symbolic function |
|
| 14:26 | it has returned. But whatever you is a derivative of that function. |
|
| 14:30 | if you perform differentiation of to sign d square. The differentiation of that |
|
| 14:36 | a 4D costly square obviously. But this what what you get is already |
|
| 14:44 | is still a symbolic function. So can't go and directly evaluated yet. |
|
| 14:52 | the first step in evaluating it is , I like to go by these |
|
| 14:56 | . You can just technically call the final result in just one statement |
|
| 15:03 | Let's say I define a variable equals . And now if I call this |
|
| 15:09 | that we got uh and substitute the because there was only one variable. |
|
| 15:15 | can just provide a s the input that derivative. So what model of |
|
| 15:21 | is it substitutes the valley of tea that derivative with the value of the |
|
| 15:29 | . So here it will substitute through the value of lee. And if |
|
| 15:34 | substitute to what you get is eight fast forward. So again, still |
|
| 15:40 | only a symbolic representation of what whatever derivative was. It's still not uh |
|
| 15:46 | hasn't been evaluated. And the type uh this substitution Um substitutions result by |
|
| 15:57 | . It's now it's a symbol. it's not a symbolic function. It's |
|
| 16:01 | a symbol. And now you can it once you have a symbol. |
|
| 16:08 | just evaluate it simply call the data name that you want to evaluate it |
|
| 16:13 | and pass the uh the substitute of to it. And that will finally |
|
| 16:19 | you um the american answer for that that expression. There are quite a |
|
| 16:27 | steps involved, but it's helpful if just want to look at it in |
|
| 16:31 | symbolic representation. If you have a , you want to see what it's |
|
| 16:35 | looks like. And you can also derivatives of higher order. So if |
|
| 16:41 | don't specify anything that's just the first derivative, uh you can just provide |
|
| 16:47 | comma And then the order of the to get uh let's say a second |
|
| 16:51 | derivative in this case. If you too. Mm hmm. Ah |
|
| 17:00 | If you have a function right uh they've written more number of variables in |
|
| 17:09 | . You can also derive performed delegation it with respect to a particular variable |
|
| 17:16 | there is one difference between the previous and this one is that there I |
|
| 17:20 | defined my expression as a symbolic function . I am directly calling the |
|
| 17:27 | The difference uh function that perform differentiation on the expression. So what I |
|
| 17:34 | as a, as a result, directly a symbolic expression, not a |
|
| 17:40 | function. It's a little bit very little if you spend a few |
|
| 17:45 | on it that it will make Uh So yeah here the function or |
|
| 17:51 | expression I should say that I wanted perform differentiation for was sine X times |
|
| 17:57 | square but I wanted to perform differentiation respect to T. And so you |
|
| 18:03 | provide uh commodity as the second parameter differentiation function. And differentiation would be |
|
| 18:11 | with respect to T. And X be considered a constant in that |
|
| 18:16 | So what you get here is the of that expression with respect to |
|
| 18:22 | And if you check the class of you got as a result is a |
|
| 18:27 | in which you can directly evaluate um when you have multiple variables inside your |
|
| 18:34 | there is a slightly different way to ah that the symbolic representation for your |
|
| 18:42 | . So for that you can use function subs stands for substitute um uh |
|
| 18:48 | expression name a list of the variable order. Um It can mean whatever |
|
| 18:56 | want but and then the values of of those variables but the values of |
|
| 19:01 | variables should be same as this. order of those values should be same |
|
| 19:06 | the order of the parameters there and you call double on that whole |
|
| 19:13 | And that gives you the numeric result that whole expression. So there are |
|
| 19:21 | different ways. You can first do function followed by expression and then evaluating |
|
| 19:26 | or just directly evaluate a symbolic Yes. So using different can perform |
|
| 19:34 | . Similar things you can perform. For integration it's just the difference is |
|
| 19:42 | the function is called? It's not stand for typecasting for interior as other |
|
| 19:48 | language starts for stands for integration. it's pretty much the same thing you |
|
| 19:52 | perform integration of a symbolic expression that you the integration of the two. |
|
| 19:59 | . Uh integration is X square, get a symbol in return and then |
|
| 20:03 | can evaluate it for whatever value you . So here I valued it for |
|
| 20:11 | Yeah I didn't evaluate this one. this is actually an indefinite integral |
|
| 20:17 | Did did not provide any limit for integration. You can also perform uh |
|
| 20:23 | integral integral of the same function for . You just need to provide the |
|
| 20:29 | and upper bounds for that integration. here what I've done is perform integration |
|
| 20:33 | uh two times X in the range comma two and that gives you that |
|
| 20:41 | not four, the numerical result for that expression. So these these things |
|
| 20:51 | will need it because at some point get to differentiation and integration in the |
|
| 20:56 | lecture. So the processes will show will obviously have some um error |
|
| 21:02 | And then you can compare uh the that you get from those processes that |
|
| 21:08 | direct functions and then compare and see the error looks like. So I |
|
| 21:16 | a look, it might be a bit confusing in the in the beginning |
|
| 21:19 | spend some time on it at all clear. And just one last thing |
|
| 21:24 | you want to time things in in labs, it's a very simple |
|
| 21:32 | You just need to wrap the whole between pick and dog that basically times |
|
| 21:40 | section that's that's between them. And think we'll ask due to time and |
|
| 21:45 | what the performance of different quotes looks during in the assignments. Any questions |
|
| 21:53 | that? Yeah. Yeah yeah This one, as I said, |
|
| 22:12 | a function handle. So it doesn't any symbolic representation. It just it's |
|
| 22:17 | like a shorthand function like you would Or function in any other programming language |
|
| 22:21 | you provide parameters and it gives you results back the one on line |
|
| 22:29 | It's a symbolic function. So you see it as a symbolic expression when |
|
| 22:33 | evaluate it in Medlab and then you provide some numeric value and then evaluate |
|
| 22:38 | on that value to get final numeric for that. So I just said |
|
| 22:43 | symbolic expressions are generally helpful when you to see a symbolic expression for your |
|
| 22:48 | function. Like you want to check the derivative or what was the integral |
|
| 22:53 | a particular function or an equation. you any other questions? That's that's |
|
| 23:04 | I think on the exercise. Yes, variable. Um It just |
|
| 23:20 | of registers variable name as a symbolic in math labs. Whatever, registry |
|
| 23:26 | memory. So mad lab knows that a symbolic variable. Okay. Um |
|
| 23:31 | if you if you go in directly define a function like this F. |
|
| 23:36 | . Equals to 70 square metal will that it doesn't know what he |
|
| 23:40 | You need to tell it that it's symbolic valuable. So online line. |
|
| 23:52 | it was like the second parameter is very black. Oh yeah. As |
|
| 23:57 | said so my club has sort of don't remember what it's called but think |
|
| 24:04 | it as an environment variable list or kind of thing in that it has |
|
| 24:10 | default names for the variables that it consider first when evaluating a symbolic |
|
| 24:16 | So let's say if I if I give that to my ex it will |
|
| 24:21 | in its environment variable list it automatically X as the first video. But |
|
| 24:27 | it should look for while evaluating a expression if you want. If your |
|
| 24:32 | is defined let's say using t. you need to go and specifically tell |
|
| 24:37 | that they solve it for tea not . So it it already already knows |
|
| 24:49 | it needs to look for text That's that's defined in its environment. |
|
| 24:54 | analyst. I don't know what it's called. Yes, yes. It |
|
| 25:04 | look for X in your in your . Yeah, I think it's called |
|
| 25:10 | called semberas as who I am. So I think this is what it's |
|
| 25:17 | . Look for look for this keyword on uh google and look look for |
|
| 25:22 | it means in Medlab. I think a list that defines what names it |
|
| 25:28 | for. All right. Uh If guys want to do some exercises, |
|
| 25:36 | you guys get a chance to do Maybe 5, 6 and seven that |
|
| 25:42 | didn't get to do last time? do that. Okay. Um |
|
| 25:47 | maybe try to do these To exercise eight and 9. So it is |
|
| 25:54 | just trying to solve for quadratic So you can do that by using |
|
| 26:00 | function for her function. Salt. We're doing it as a life. |
|
| 26:06 | you guys want to try now or . Okay. American. Right. |
|
| 26:17 | the function solved to define these quadratic and find out what their roots |
|
| 26:25 | And once you're done uh see if can get the derivative of these two |
|
| 26:32 | and exercise mine. That should get started with some of the syntax ah |
|
| 26:44 | . Um so on 9 17 you the variable Y um but I noticed |
|
| 26:50 | would be like a sims ahead of . Yes. Uh I think that |
|
| 27:03 | don't think you need to define Ah I might be wrong. You |
|
| 27:08 | , these symbolic expressions are okay, , no, this this is actually |
|
| 27:14 | function handles. So in function handle don't specifically need to define the stems |
|
| 27:20 | variables. Those are needed only for symbolic functions and symbolic expressions. |
|
| 27:28 | say that again. I did not them yet, but you know, |
|
| 27:33 | take a look here. Mm See if you can get the syntax |
|
| 27:50 | right for self function and the Remember for derivative you need to define |
|
| 28:01 | symbolic variables and then the symbolic maybe I can put them side by |
|
| 28:13 | for reference. All that will be giving you the answers. But the |
|
| 28:19 | time you guys are doing it Mhm, mm hmm. Of |
|
| 28:54 | Oh wait, you guys can't see right now. Nobody told me I |
|
| 29:01 | pay attention back there. I was , I don't know. I thought |
|
| 29:08 | was on my screen, but now you guys should be able to |
|
| 29:13 | it. I didn't realize it wasn't up there. Mhm, mm |
|
| 29:24 | Nice. Maybe you can raid five minutes. Yeah. What? |
|
| 30:50 | Okay. Thank you. Mm Whatever. Mhm. All right. |
|
| 31:37 | huh. All right, baby. . For available day mrs Wallace. |
|
| 32:08 | , it's it's just a simple Yeah. If yeah, if you |
|
| 32:15 | you create a symbolic expression or a function from some radial, then you |
|
| 32:20 | to define as a symbolic variable. simple. Uh well, presently, |
|
| 32:28 | this employment function. Absolutely. You have to ask the parable. What |
|
| 32:35 | you just have to? Right, . Oh, Mhm, mm |
|
| 32:55 | Yeah, I think I should have clear with that. Um What the |
|
| 33:00 | of those symbolic variables and mad levels you it only comes into play when |
|
| 33:06 | have multiple variables in your equation. say a function is defined using T |
|
| 33:13 | X. Let's say then by default will always perform differentiation or integration for |
|
| 33:19 | . Because that's the one that's uh in its president's list for the for |
|
| 33:24 | variables. If you want to perform and integration, 40, then you |
|
| 33:29 | to specify it that you need to it for tea. Yeah. Um |
|
| 33:36 | think I said that it will give . I don't think it will give |
|
| 33:40 | will perform differentiation or integration for X and everything else will be considered as |
|
| 33:46 | constant. Mhm. Yeah. Thank you. All right. I |
|
| 34:14 | we can to the solutions. Um . For Exercise eight. I think |
|
| 34:21 | was quite straightforward. Uh Let's say . I specified X. Um and |
|
| 34:32 | to be able to solve for the for the Children automatically solve for |
|
| 34:38 | Uh So yeah just remember for solving you need to provide uh double equals |
|
| 34:46 | in between uh the left hand side right hand side of the creation, |
|
| 34:52 | all you need to take care And then as you can see the |
|
| 34:56 | equation had imaginary roots so they provide with um imaginary looks at the |
|
| 35:03 | Okay exercise mine you have to perform for law one over X. Power |
|
| 35:13 | . Yeah. So yeah I just didn't define a function using uh using |
|
| 35:19 | one over export, I just provided as an expression to the dysfunction and |
|
| 35:25 | explicitly mentioned experts introduced me to do . Yeah. So if one obviously |
|
| 35:34 | you a symbolic expression, that's the of one over X. Powerful, |
|
| 35:39 | -4 over X. And then you substitute uh that symbolic expression, read |
|
| 35:48 | value you will ask for so far for X equals two. You can |
|
| 35:51 | the subs function uh to get the But I forgot one step here and |
|
| 35:57 | you got from subs is again a expression. So this -2 although it |
|
| 36:01 | like a number but it's actually a radio voluntary check its class. So |
|
| 36:06 | actually evaluate the whole thing, you to add double or whatever data type |
|
| 36:11 | want and then it will give you think everything goes well. Yeah, |
|
| 36:17 | it's it's a double precision number Ah thing that the 2nd 1 ah here |
|
| 36:26 | asked you to Performed the differentiation of explicitly. Part four with respect to |
|
| 36:32 | and here that's the way you can it. Provided the expression and the |
|
| 36:36 | and the parameter for origin it should for substitute function to substitute the values |
|
| 36:43 | that symbolic expression for the other two . X 20. And then if |
|
| 36:47 | add double uh in front it's solves for those values and give you a |
|
| 36:53 | precision result. Any questions on Okay. It's unfortunate. Oh |
|
| 37:14 | Mhm. Yeah, yeah. Any additional questions from that time? |
|
| 37:40 | hmm. Good. Okay. A number of the exercises here talking |
|
| 38:04 | , that's what's on the web and encourage you. Yeah. Um on |
|
| 38:15 | own let us know. Um so want to switch everything from the |
|
| 38:25 | So the first thing I've thought about what's known as a very funny their |
|
| 38:32 | and give a couple of examples and simply the purpose is to try to |
|
| 38:37 | out um but it's known that backward that given the results that they have |
|
| 38:44 | to get some sense for what range input sizes could result in that particular |
|
| 38:55 | . That's the backwash that tried to you know what potential inputs could give |
|
| 39:00 | stuff. Mm hmm. And so is A couple of simple examples uh |
|
| 39:09 | think three or four of them just going through this notion of what |
|
| 39:16 | And um, so the first thing simply that feeling and get sort of |
|
| 39:22 | errors when numbers are not exactly represent ble in the computer. So, |
|
| 39:30 | know, there's a limited position with certain number of or depending on the |
|
| 39:35 | type that you have and you're the projection can be represented exactly by |
|
| 39:43 | of the time it's not. And it comes to floating point and that's |
|
| 39:51 | in front of this apprentices investment calling notion that most of the time in |
|
| 39:58 | course we didn't want members. So that one they have the relative error |
|
| 40:07 | representing the variable Z. And there's magic about that. Just five minutes |
|
| 40:14 | the store and forward zone that depending in this case the actual collective error |
|
| 40:22 | depend not only on the value, then also on the browning road that |
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| 40:27 | talked about. That's something I checked the standard stuff. The principle, |
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| 40:33 | yeah, approximated in the representation and your particular in that case. So |
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| 40:39 | can be exact. And the bounding the era of the maximum error in |
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| 40:47 | case is depending upon some of the you have going for you the next |
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| 40:55 | , but they don't have what And that's what the machine epsilon comes |
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| 41:00 | play. That was mentioned before. there is nothing difficult. So one |
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| 41:06 | is the first thing in best representing number in the computer. And the |
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| 41:11 | thing is what happens when you do of a couple of machine numbers like |
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| 41:18 | and Y on the second floor that outcome of that meeting of the and |
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| 41:26 | machines of the um potential translation around happened depending upon the scheme you're |
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| 41:36 | So for instance the application, the of the industry gaps those and so |
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| 41:46 | you can't represent all indigenous to translate rise. So that's just an example |
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| 41:51 | happened operations and then we get another and so yes, I guess an |
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| 42:02 | that did. So in this case first thing ah just looking at what |
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| 42:10 | when you do operations on the So assuming that whatever X, Y |
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| 42:15 | Z in this case or what they be. So there is no other |
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| 42:21 | representing the variables. Let's start what are. And things may errors may |
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| 42:27 | introduced because of the operation to do and that the outcome of the operation |
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| 42:32 | be represented from this case. In what to do first on X and |
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| 42:37 | . And then they're a modification of outcome of that physique. So this |
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| 42:44 | kind of the first step then that outcome of the addition results in the |
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| 42:50 | that are not necessarily exactly represent So that's what trends are being the |
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| 42:55 | of the true value of plus some and it's a pretty standard and you |
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| 43:07 | , their possession and single position to . And that's actually so it is |
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| 43:19 | by the emotional next. Mm You know, putting around the next |
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| 43:24 | is you take the outcome of which is what now is in the |
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| 43:30 | of the true value and you multiply the true value and then again, |
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| 43:35 | may not be representative of the machine that the product is them and the |
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| 43:41 | secondary. So, um, and can basically plug all these things and |
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| 43:49 | messy thing and it should be Then you get what the outcome of |
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| 43:55 | drone number but a couple of rooms actually multiplication also in the area. |
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| 44:01 | this was for the results of the of potentially insulting the modification has |
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| 44:08 | another potential area. But since these machines absolute tends to get kind |
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| 44:15 | small, the product is very So that small. It's roughly approximately |
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| 44:21 | potential. There's some of the errors the tube operations. They had the |
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| 44:27 | . So it's just but this sign just trying to come into that. |
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| 44:33 | is a chance that there will accumulate more operations to do every operation |
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| 44:40 | And of course I'm a if you're and they kind of cancel each |
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| 44:45 | But if you want to put the on it, I have the worst |
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| 44:49 | for each operation and then another. uh huh to include that the starting |
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| 45:04 | , Z and X. And Y actually be some numbers that come from |
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| 45:08 | and it's not already in the Not exactly. So then you have |
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| 45:13 | approximation areas and representative checks required. gotten the disease somewhere in here. |
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| 45:19 | and then you do the operations on outcome of and speaking X into the |
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| 45:26 | as well as why. Okay. was just some so that he gets |
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| 45:31 | then and um that is present. plug it all in. Um you |
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| 45:37 | a similar to before but you have done really. But the negative effects |
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| 45:49 | from design opponents. Okay. Sorry they haven't seen this implied but there |
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| 46:03 | a multiplication that they had before adding complexity of effects and life and our |
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| 46:10 | exact numbers. So then so if then start to look at the relative |
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| 46:18 | in the some representation here. The is that you do have the fact |
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| 46:30 | developing era ends up being the one that's three that came from the previous |
|
| 46:36 | and than the relative error. Uh the product of these two guys bargain |
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| 46:46 | to some other X plus one. that's kind of where things are bad |
|
| 46:55 | it looks like this is, you , a safe operations and additional |
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| 46:59 | But uh sort of why it may may be very close to the negative |
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| 47:04 | accident or that's a plus but it be negative numbers of this number can |
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| 47:09 | very small. So that means when divide by this number relative american factor |
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| 47:15 | large because X plus one will be close to zero. So that's the |
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| 47:21 | thing to be aware that things can up when you are relatively speaking, |
|
| 47:27 | get more of the cancelation of some the operations. And I think the |
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| 47:35 | fight is just a simple in America wow for the sensor when it's again |
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| 47:45 | talked about it and the loss of in the first lecture. So it's |
|
| 47:51 | , numbers almost canceled each other out relative areas can definitely large. And |
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| 48:00 | pretty much all I wanted to say . Yeah. Back when their analysis |
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| 48:06 | just to make it where there is seem to do pretty much in every |
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| 48:14 | operations. In addition to the earth my D. When the first representing |
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| 48:21 | expressions. So the next thing was talk about this Hornet's ruler, Horner |
|
| 48:32 | . That is a commonly used and commonly used in terms of dealing with |
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| 48:37 | normals that we talked about. Um maybe I think a couple of lectures |
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| 48:44 | their own. But polynomial approximations are common in many. Not bad went |
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| 48:53 | the university like that. You make recordings of something and they get the |
|
| 48:59 | of stuff and sometimes it's convenient to collection of numbers by some kind of |
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| 49:07 | function. That means that gives you way of estimating or Yes. What |
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| 49:20 | function or the observation might be if had looked at some other X values |
|
| 49:25 | figure out what value would be they the truck. So if you have |
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| 49:29 | kind of a function you can for to say, hey, why would |
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| 49:33 | be something like this? Whether that's or not, that's a different |
|
| 49:41 | Whether the polynomial function actually represents whatever is that visits or something behind it |
|
| 49:48 | coming to different things. But it's it's often used as they don't use |
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| 49:54 | substance degree to represent the table of or for that matter I think the |
|
| 50:01 | that I have on the next So and one of those computer graphic |
|
| 50:06 | . It's amazing to use. This some kind of follow normal expression to |
|
| 50:12 | at the surface there on some particular . Not for the whole surface testament |
|
| 50:18 | the surface and down describe it for follow normal function and some of this |
|
| 50:24 | just you can do it in this or another example of our common, |
|
| 50:30 | sensitive, commonly used in a number senses. If you have some very |
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| 50:37 | function like this expression here that it's so easy to evaluate. Um then |
|
| 50:44 | if one does some camera plug in few values in this direction perhaps and |
|
| 50:51 | to be recursive into those values and might be described so much easter nature |
|
| 50:57 | they were. Yeah. Well the right. The data is one of |
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| 51:06 | that we talked about better pull normals well as other polynomial. That's a |
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| 51:16 | point. So this is just simply working polynomial is a common thing. |
|
| 51:23 | this whole thing is um something that's useful and trying to evaluate the manipulate |
|
| 51:32 | office and then we'll talk about different of point. All this. So |
|
| 51:39 | is just mm hmm straightforward point in and any degree polynomial um with |
|
| 51:47 | And they're pretty much standard way in case. And the um that's the |
|
| 51:53 | politicians. That was the very first esperanto You know, collective value of |
|
| 52:07 | A one and 9 if they want X amount of bias They want with |
|
| 52:12 | and zero and keep calm down doing operations as their friends. That's The |
|
| 52:22 | one operation or or well extra then you have to multiply unless the |
|
| 52:31 | For yourself. And Okay. And questions there will be. So if |
|
| 52:38 | want to do it as this thing written Uh you would have I guess |
|
| 52:43 | -1 for him. Whatever. Um end additions. Right. But if |
|
| 52:54 | evaluate the soul, here's one multiply this would impact the fact three |
|
| 52:59 | Right? Because you have to square . That is one and then you |
|
| 53:03 | to find the results. A So that's another one. So each |
|
| 53:07 | of these terms then would have more more modifications. So if that's |
|
| 53:16 | to be quite, um, a number of operations. So he's in |
|
| 53:22 | better way of doing it. For spending decline one of them to |
|
| 53:31 | a better way of evaluating yes. , in some ways right. |
|
| 53:56 | when you're done say x square Next is X. You if he somehow |
|
| 54:05 | kept her X squared value, then only need one more to the |
|
| 54:10 | But unfortunately the computed expect so variables. But another way of doing |
|
| 54:20 | is this super hornish corner schema that and that's implicitly yeah. Says you |
|
| 54:34 | hmm. Like many powers of accepting . So I'm looking at this expression |
|
| 54:42 | everything and the nesting level historically in most seriousness. one month to find |
|
| 54:48 | ad when you go to the next by one as basically one month supply |
|
| 54:53 | one at all the time until you the scenario and and acquisitions, you |
|
| 55:06 | ? So yes, computers are You mean all the characters but you |
|
| 55:11 | what? It doesn't take much to instruction today. Well, um, |
|
| 55:23 | does affect performance. Right? So have to be basically and squared versus |
|
| 55:27 | end operations that counts in the It's large in terms of fire that |
|
| 55:35 | . And um, these days, society LTD and their computer designed |
|
| 55:43 | How are you can make it So it also means extra operations also |
|
| 55:49 | unnecessary participation and then so trying to conscientious about they're not doing more obvious |
|
| 56:02 | it takes about their expressions. It's performance and their power bill for now |
|
| 56:12 | terms of the performance. I would California or something. You want to |
|
| 56:18 | that for most computations and typical computers . The thing that limits performances and |
|
| 56:26 | the operation counts to the member So But never let us being prudent |
|
| 56:34 | highly right two expressions. There's still . And this thank you. A |
|
| 56:44 | of examples just taking more completely expression just writing it down. That's the |
|
| 56:50 | question. Yeah. Obviously it also you know this simple code that is |
|
| 57:01 | simple. It's just a long line american inefficient. So any questions on |
|
| 57:18 | . Okay, so now something that no expectation. Again, manipulating polynomial |
|
| 57:29 | that would be used much later as . Many times over. And that's |
|
| 57:34 | kind of factoring a polynomial zor finding of polynomial. You can use the |
|
| 57:43 | . And I want to came out the schema for actually during this factory |
|
| 57:51 | expression however normal. So in this are factor X miners are and then |
|
| 58:01 | get the new polling all about you some other kind of expression that turns |
|
| 58:08 | to be a predictive value for for nobel path, the value of our |
|
| 58:15 | . And we'll tell you how those of things work. So and then |
|
| 58:19 | looks funny the thing is it's our to be in groups this polynomial then |
|
| 58:28 | is the remainder, which is the . In fact because it's so that's |
|
| 58:34 | is kind of a procedure you can also use some finding rules to |
|
| 58:42 | Okay. So how do you have and settled down in the polynomial |
|
| 58:48 | Is that the polynomial motor one order than your written for the normal by |
|
| 58:54 | . So some place in Q. the senate basically X. To the |
|
| 59:01 | and but at the highest degree of is and -1. Kind of reduce |
|
| 59:07 | paranormal every time you try to. . Yeah. So that's what I |
|
| 59:18 | . So here's kind of how everyone doing it. Princeton and papers already |
|
| 59:22 | to steam up. They were kind write down all the politicians of the |
|
| 59:26 | polynomial starting the coefficient for the highest power of X. And then there |
|
| 59:34 | torch is now the highest part in Cupola Noma is one over the |
|
| 59:39 | So it kind of goes in the for. Okay, so the power |
|
| 59:42 | N -1. And you keep breaking down of course you don't know to |
|
| 59:48 | us. And the other one's supposed figure out what they are. But |
|
| 59:52 | turns out that the coefficient for the order um X. And the two |
|
| 59:59 | . It's the same as in the polynomial. And then you're fine. |
|
| 60:08 | The other coefficients in the cube of lower of the polynomial by multiplying the |
|
| 60:18 | in the queue polynomial with our And take the monster previously computed. So |
|
| 60:24 | one, you know? So now know but the computer this product that |
|
| 60:28 | can find the and -2 and then go successfully, don't They're always and |
|
| 60:35 | just to find all the politicians for coupon. But it's very simple |
|
| 60:41 | This is under the manger term and it's zero that that happens to be |
|
| 60:46 | exception. Um So this is against schema. It's a service dog. |
|
| 60:54 | then so then give you 50. that's the way of it. So |
|
| 61:03 | fact that impala normally would be the of distension. So you know, |
|
| 61:15 | Britain or in the comments on that I guess a concrete example. Um |
|
| 61:20 | will start with the coefficients. So b now written starting with the highest |
|
| 61:27 | . So first so it's a one minus 47 minus five minus two. |
|
| 61:33 | We want to top it off x three. Um Some of the three |
|
| 61:39 | they are so Copy that this coefficient , multiplied by three And add them |
|
| 61:46 | to -1. Take Finance one comes -3. Had adopted four down |
|
| 61:53 | No until they got 19. So are now the coefficients in the polynomial |
|
| 61:59 | degree three. Um So hopefully we'll This enormous factor of X -3. |
|
| 62:10 | coefficient was one. Next Professionals four. Mayhem. So I want |
|
| 62:20 | graduate T. Queen then what we to we might see is zero. |
|
| 62:28 | this is called So I think is out what I just said whether coefficients |
|
| 62:45 | and if one doesn't believe it um can plug in three. And this |
|
| 62:55 | directly obviously. So three to the of four is 81 ah 32 and |
|
| 63:03 | three is 27. Right? That's . That's 108 And seven times times |
|
| 63:12 | . And he has pocketed. it's interesting this one double check themselves |
|
| 63:22 | as the proceedings Francis straightforward. So , let's see what else is |
|
| 63:37 | No, I guess the same Right. So in this case to |
|
| 63:43 | now is that so same coefficients. you want to follow the things and |
|
| 63:49 | turns out now they're reminded of So that turns to is a room |
|
| 63:57 | the question. So that means ah its support actually constitute. Then |
|
| 64:06 | a question about the zero and the of zero. So yes, once |
|
| 64:18 | . So um Right, so that like that then let's see what |
|
| 64:28 | Um Yeah. Um So one x cannot tell you this and we'll talk |
|
| 64:38 | about that in a later election and but these foreigners rule is unuseful and |
|
| 64:48 | are falling over. So if we the polynomial deflated the fact that our |
|
| 64:56 | are and now they want to take derivative Um It can be the chain |
|
| 65:01 | , right. But the derivative of phenomenal then is ah 1st derivative of |
|
| 65:08 | first Factor X -R. Which is times two. And and then expands |
|
| 65:14 | transit derivative of the second. And is a constant respect to exit the |
|
| 65:21 | of that zero. So now we go a lot and then continue If |
|
| 65:28 | want to do it. Or someone evaluate this may be uh the reputation |
|
| 65:34 | about your arm being that evaluates to . And R. And to do |
|
| 65:39 | you can go through and factor And then the remainder is what we |
|
| 65:45 | to do in terms of. So can keep doing this to find higher |
|
| 65:51 | derivatives that particular values are being exploited using mm hmm. And in this |
|
| 66:02 | simple paranormal. So this can also done population publicly using some other countries |
|
| 66:10 | than the point in the marriage. I tell you later remember talking about |
|
| 66:18 | later on, differentiation is in America tricky and it's very unforgiven versus integration |
|
| 66:27 | smooth himself. So whenever I want do things, symbolic things for the |
|
| 66:33 | is proposed. Mhm. Um That's . Yes, showing do the |
|
| 66:43 | Yeah. The first to get the and then yeah. Ah the next |
|
| 66:53 | for one more around all factoring off are in order to ah I'll get |
|
| 67:04 | water. Let's see where we are example here. Um Oh then the |
|
| 67:14 | of design before and the steps uh front of Q. And A. |
|
| 67:23 | have this one and then the next was basically to do ah the organization |
|
| 67:32 | the derivative by doing one more step long time I got this from the |
|
| 67:41 | and then come on, can you it? Who doesn't believe? It's |
|
| 67:48 | about things incorrectly And consider the first in my life. What is |
|
| 68:01 | It's simply cold again. That's every you basically Get one More Life. |
|
| 68:08 | it's just 1014. Uh huh. this one it's been much more. |
|
| 68:18 | have for today actually. So next we'll talk about taylor series that was |
|
| 68:24 | here as one of the type of series. It's very good. It's |
|
| 68:30 | familiar with it and it cannot make you do get it into being very |
|
| 68:37 | and so on top of your mind that will be used extensively and building |
|
| 68:44 | properties. Province of approximation states. often falls back to using tendency this |
|
| 68:52 | try to listen to. I got approximation. So it's kind of, |
|
| 68:59 | is used to all the chapters in book. Yeah the best thing. |
|
| 69:05 | that's why I've been familiar with it be comfortable that taylor series is important |
|
| 69:13 | then this is called an elimination of it was quickly. Hopefully everybody remember |
|
| 69:20 | song central system of equations using and elimination of steam was invented by, |
|
| 69:30 | know, recovered. Thank you. , how many do you remember that |
|
| 69:43 | ? Some systems of the Persians. . But so we'll talk a little |
|
| 69:52 | about athlete control errors in elimination so for the next lecture and stay from |
|
| 70:07 | on for complex. Besides for lectures years old, don't. There will |
|
| 70:19 | more it actually. Yes. yes. Yes. Song as |
|
| 70:32 | Yeah. That's doable of our convenience students and may be uncomfortable for whatever |
|
| 70:42 | that will provide some access recordings and . But I will be here for |
|
| 70:48 | . So you would prefer. Yes. And we use the |
|
| 71:00 | I guess we need to think about assignment. I don't remember the first |
|
| 71:09 | somewhere but first assignment tonight. It's a comfortable, a little confused that |
|
| 71:31 | . I'm sure I get my Mm hmm. I lost my |
|
| 71:48 | That's good to remember. Okay, you find my history and maybe I'll |
|
| 71:58 | to you? Um, uh, . Oh, that's the curse. |
|
| 72:11 | Spinoza. Yeah. Well, that's story |
|
