| 00:42 | Yes. Does it work now? hmm. Oh okay. It's |
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| 00:57 | I think the transcript should be on right. Yes. Oh okay. |
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| 01:25 | I don't know continue to talk about today. Mm hmm particular now. |
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| 01:38 | . 33 spines. Right. And kind of a more general former spines |
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| 01:54 | . Alright, so last time again to return quickly. That's that's |
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| 02:01 | There's put these concepts of both notes notes and notes for the thing interpretation |
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| 02:09 | terms of the independent variable and not the independent variable points where you can |
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| 02:16 | smooth recommendations for the joining criminal. yeah. Traditionalism. And we kind |
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| 02:27 | went through 1st and 2nd degree lines the first of these plans was fooling |
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| 02:34 | safari by the interpolation conditions. Or not true for The second degree. |
|
| 02:41 | not true for high response at 12 you will see today. No |
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| 02:48 | So the approach that was taken was figure out how to get there conditions |
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| 02:55 | need to be satisfied that function in determined all the they're pulling all the |
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| 03:02 | that are needed. And then the step was to you can start with |
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| 03:10 | highest derivative which one acquires some form growth this condition, highest order derivative |
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| 03:20 | ST that should be continuous. So there's wrote an equation for each of |
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| 03:32 | intervals or the highest of alternatives. then I came today. Second degree |
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| 03:41 | integrated it and then we find the of the conditions and got a set |
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| 03:46 | equations. And then we work for to find coalitions and those questions. |
|
| 03:54 | this schema for and then 3rd degree has gone to response completely the same |
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| 04:04 | since the 3rd° finds that means Each polynomial has a degree of about |
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| 04:13 | That means for them coming up on specified continuity or sort of the for |
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| 04:20 | function itself for the spine and spoke that the derivative and cannot on the |
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| 04:26 | and very sort of but not this . But we can also of course |
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| 04:31 | second derivative to be the same with right side of the not. So |
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| 04:36 | kind of and a way of making spine even smoother and there's some more |
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| 04:45 | nice, more nice to behave then quarters explained. And right. |
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| 05:00 | Alright I think that's so we'll do . That's the current population. And |
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| 05:09 | through a procedure actually that determined 33 . And there's particularly one that will |
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| 05:16 | with the smallest unnatural treaty explains. I'm gonna inflation a bunch of intervals |
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| 05:23 | for each interval now from no degree for this construction here. Even in |
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| 05:34 | thoughts were used. Mm hmm. you have nots and notes to the |
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| 05:40 | . So there's just one simple independence . X values that I used both |
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| 05:49 | the population and for specifying those And so the small test conditions now |
|
| 05:58 | it's too big spines mentioned, that the function itself. But basically was |
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| 06:03 | a but also the 1st and 2nd um should be the same on by |
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| 06:11 | side of it now. No, was yes. Yeah. The three |
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| 06:19 | , right. One for each one the case. That's the function |
|
| 06:22 | 1st and 2nd derivatives. So that's conditions of each knot and eternal um |
|
| 06:30 | of knots. And so all of condition points And is one of them |
|
| 06:37 | we have the total and the last ah in population. So we need |
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| 06:48 | bit more. Yes. Yes. better for the normal some degree |
|
| 06:53 | That means that four coefficients A. C. D. And we need |
|
| 06:57 | department create intervals so we need foreign in order to but things that were |
|
| 07:02 | defined and what they have here we'll you in that forum finest eight plus |
|
| 07:09 | nurse finals too. So we're listening conditions in order to have in africa |
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| 07:19 | able to generate enough conditions or equations victim computer all the conditions. So |
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| 07:25 | this case that's covered 3° of freedom after we had have applied both in |
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| 07:32 | conditions and smoke conditions. But we a second degree polynomial was one. |
|
| 07:39 | in that case we use it to I think for us the derivative. |
|
| 07:43 | one of the one point. So way of doing it is not specifying |
|
| 07:52 | conditions and they can use that to Values for the 2nd derivative. Both |
|
| 08:03 | of zero. No one does have . The spine is known as a |
|
| 08:08 | to explain. So that's what we're to use to do some more particularly |
|
| 08:17 | for your future. But there are ones too fortunately periodic spines. And |
|
| 08:24 | means best products you have conditions for left and right. And whatever the |
|
| 08:33 | is for me is that it's a values are the same. Maybe derivatives |
|
| 08:39 | well. And there's also something known clients, clients and who expressed they |
|
| 08:46 | . Um Right. Um you just a part of the derivative and the |
|
| 08:56 | things. Uh The end points in case for the same. So that |
|
| 09:01 | enough conditions for other than natural too before. But we will run well |
|
| 09:09 | . It was fast. And then it. Once again, there's I |
|
| 09:14 | last time one of the stories of . You can use something the other |
|
| 09:20 | for this. Fun. Then the interpolation was hoping. Alright, so |
|
| 09:28 | spines. Um So mhm. Now get to the machinery and let's remember |
|
| 09:41 | same machine approach as for secondary based . We start with looking at the |
|
| 09:50 | order derivative for which you are smooth conditions. And then to explain certain |
|
| 09:56 | polynomial that mr derivative you can have the second derivative. And that is |
|
| 10:04 | second derivative is the linear function and the derivative of the third degree polynomial |
|
| 10:13 | and was left with the first order . So this is what I have |
|
| 10:21 | . So now like in the Case the 2nd degree polynomial. This old |
|
| 10:28 | that this category derivative in this case linear polynomial. And it has the |
|
| 10:39 | values which we don't know. Obviously gave us to us. We have |
|
| 10:44 | have conditions to determine so. And is the one way of writing down |
|
| 10:54 | equation for the second. Mm It's easy to verify. And if you'd |
|
| 11:01 | to tell them this way that plug the eye this time disappears, ci |
|
| 11:10 | nothing but H. I. So president you plug in X equals |
|
| 11:14 | I. That necessarily secondary initiative is host assumption. And uh probably very |
|
| 11:24 | so. That's what I think Similarly Anti High Class one and this term |
|
| 11:32 | and this is the first one. the right hand point is the response |
|
| 11:39 | the construction is correct in terms of second derivative, Assuming these values have |
|
| 11:46 | airports. And I just talked to about that. So and this comment |
|
| 11:56 | as you will discover in the next slides, there's a reason for writing |
|
| 12:01 | down in this particular form rather than straightforward Newton. And so for me |
|
| 12:08 | was there's prime Minister. So um we did second degree polynomial, then |
|
| 12:20 | integrate the expression for in this case secondly grateful in Omaha. And then |
|
| 12:27 | now we have to integrate it twice to any question for the spine or |
|
| 12:35 | paranormal for each one of the So now it's look forward to seeing |
|
| 12:44 | this corresponds to this part integrated But then there's three more terms that |
|
| 12:49 | in. Yeah. Okay. I the first derivative D disappear. Someone |
|
| 12:56 | gets up with the C. And another derivative. But yeah, but |
|
| 13:00 | forms are it's just the additional terms comes in from the integration. So |
|
| 13:07 | in fact they have both disease the . And D. That needs to |
|
| 13:13 | determined in order for the paranormal. . So for comedians, in terms |
|
| 13:26 | manipulating expressions to get to form the , that would be easy to then |
|
| 13:35 | what disease eventually that's what they And that was like in the second |
|
| 13:42 | for a normal and then uh equation and using constraints in order to get |
|
| 13:51 | set of equations that allowed us to the derivative about this. Uh |
|
| 14:00 | So now this is again rewrite of equation. Ah That is now having |
|
| 14:11 | forum that you have some of the . Yeah. And this equation also |
|
| 14:19 | up now and this form. So easier development. So you have a |
|
| 14:25 | between this. See when the names I. And C. And the |
|
| 14:31 | . So you sure about this. can see there's basically uh X. |
|
| 14:38 | X. Now has the best correspondent taxi and the index. Mm |
|
| 14:48 | Well, so the point is how try to ah But this is kind |
|
| 14:54 | a form that is easy to get fire. And mark he survived the |
|
| 15:01 | for me ah that they used the conditions. So we inspiration conditions and |
|
| 15:14 | the polynomial put in it. I double At the less than .3 equals |
|
| 15:22 | should be um The function value of has an endpoint And at the right |
|
| 15:31 | of all I. And the I am a wife. That's what |
|
| 15:40 | that are best going to take this forward as I said um let's begin |
|
| 15:48 | . I. So the left stand and the left hand side here because |
|
| 15:52 | I this time disappears and this time . This one and this one and |
|
| 16:00 | . I. Plus one minus I. The X. Value we're |
|
| 16:05 | with. It's basically a char. a different choice of distance between the |
|
| 16:14 | . So this becomes best is er the six times H. R. |
|
| 16:21 | . The one on the doctors have disappears. And then we have this |
|
| 16:28 | I can say. So that means have left hand side. That is |
|
| 16:34 | . We don't have any C. . S. Because that disappeared. |
|
| 16:37 | we have now in this very way saying that why I yeah equals And |
|
| 16:46 | Z. I divided by six times square. So they can solve the |
|
| 16:53 | come down. Yeah it's still if have advanced. So this is good |
|
| 17:01 | you can see it in that. this year this is H. |
|
| 17:05 | So the division. We had my on the left hand side to develop |
|
| 17:10 | thing that is a child. And there was this other term here that |
|
| 17:15 | the idea but it was six times and square. But at this |
|
| 17:18 | so this is the original and similarly got see now by looking at the |
|
| 17:26 | time for So now we have an for the polynomial for the interval |
|
| 17:39 | The only houses disease, the derivatives they postulated in the beginning as they |
|
| 17:46 | known. The rest of it is the coordinates are known but obviously we |
|
| 17:52 | to separate them between. There are so now we have some questions, |
|
| 18:03 | only contains disease at some point. , so any questions so far wrap |
|
| 18:15 | up and just a little bit. best that we have and then we |
|
| 18:18 | to do all the inquisition. So have started off with doing the continuity |
|
| 18:27 | the derivatives. Would you explain that knots? So that guy's a conditions |
|
| 18:35 | on this one. So that was answer and money zone conditions. And |
|
| 18:41 | using population conditions for CDS and DVDS that was to conditional left and right |
|
| 18:49 | each interval. So that's two. So how do you stop three and |
|
| 18:54 | is one condition. So that means Commission and Plus one Conditions. And |
|
| 19:03 | what didn't you not used so So we did not use were um |
|
| 19:20 | they did not use what the continent the First Amendment continuity and in preparation |
|
| 19:30 | Mhm. Find themselves they use continuity the secondary relative. But we didn't |
|
| 19:37 | what the first degree for the first then give us but the knots for |
|
| 19:45 | in front of the not So that's that there's help beforehand. But Michael |
|
| 19:51 | , so then they don't additional conditions imposed from the natural sciences were too |
|
| 19:59 | but all of that then we are for it. So what they need |
|
| 20:05 | do is to go and the for part, the rest of it you |
|
| 20:10 | used. And in some ways they to of course the degrees of freedom |
|
| 20:16 | used. But the actual achievements. , so now I was going to |
|
| 20:25 | at name continuity condition order first derivative space ah at the men you may |
|
| 20:36 | T. I the left and the . Ah well, in all March |
|
| 20:42 | understanding. So then that same question they have at the bottom of the |
|
| 20:52 | . The point of contact, is anything longer, respectively? Maybe it's |
|
| 20:59 | that cycle that's missing and then they the fights are the best we have |
|
| 21:06 | term comes from here and then we this term here and then we have |
|
| 21:12 | rest of it here and we'll take derivative evolved and that some of the |
|
| 21:16 | that relate to X that survives in innovation. That's right. So so |
|
| 21:30 | know these ability to find it right . So this is kind of the |
|
| 21:39 | and I'm going to go products um in T. I. But this |
|
| 21:46 | correspondent for this And then we need do another one Change in the Index |
|
| 21:52 | -1. Also ci and then put meat say that and make sure that |
|
| 22:00 | expressions are the same. So plugging T. I. Into this question |
|
| 22:09 | , then this fund disappears and get squared here. So you can see |
|
| 22:16 | a few times a child, a . Um These are they are best |
|
| 22:23 | . A guy that's fun is showing here. The thing was the I |
|
| 22:29 | have one here and one here and . And what is this 1? |
|
| 22:37 | then you have the rest that is for this part. And this part |
|
| 22:43 | that can be consolidated in this case B. I. As you can |
|
| 22:49 | at this time this time and has so now they have to find a |
|
| 22:56 | equation for the derivative of these. over endpoint and then we need to |
|
| 23:08 | the same for as yourself take That's polynomial. That's the same. |
|
| 23:16 | it's I'm not sure about that. you just switch out our Prime -1 |
|
| 23:22 | it gets here and then plug Yeah. So in this case it's |
|
| 23:31 | plug in. First to change the are into I minus one. So |
|
| 23:36 | becomes T. D. I. then you for excellent. Since this |
|
| 23:41 | disappears and this one marble counts. the I minus t. Oh minus |
|
| 23:53 | . So anyway so then you yeah expression satisfy the straightforward and now the |
|
| 24:02 | of the doctors these this impression of expression should be the same. So |
|
| 24:08 | we go to the best to get questions for the cities they had at |
|
| 24:15 | bottom of the previous slide just putting equals and rearrange terms. So now |
|
| 24:21 | just get this equation that is followed consequence of the Small Movements Foundation on |
|
| 24:31 | 1st so now we can see this in fact again coming back. It |
|
| 24:40 | so that explains this now is a of equations. This holds for of |
|
| 24:47 | the internal, not for this and is one of these questions um and |
|
| 24:56 | had a choice from the effective response upon these two. So the same |
|
| 25:02 | is one here and then they students fact they had them plus one conditions |
|
| 25:08 | mps one unknown disease. So now kind of done. We just need |
|
| 25:15 | put it and to do that and protection a little bit. So one |
|
| 25:23 | is done. Thanks. The question the diagonal thermal was on diagonal and |
|
| 25:31 | diagonal matrix called a huge and all right time. Sorry thank you. |
|
| 25:37 | then I get like this one simple . Um that is not sufficient to |
|
| 25:44 | all the facts. You know, two of them are already given some |
|
| 25:49 | degrees of freedom used to specify So it's only 10 -1 that actually |
|
| 25:57 | that's handed down the processes the fully . Oh okay. My question that |
|
| 26:07 | want to talk to this particular Mhm Right. So find a little |
|
| 26:19 | first thing the notes and notes and is not to respond. So that's |
|
| 26:26 | two of the three unknowns. I know what's coming from you and here |
|
| 26:32 | reforms the procedures. You uh is from really no position of the, |
|
| 26:48 | we're not special. There's basically related the distance between. It may not |
|
| 26:54 | the same and then the B follows the values that is used for the |
|
| 27:04 | so the right time so I can computed from words given at the table |
|
| 27:08 | so can be used. So this and this system of the question, |
|
| 27:15 | basically only the was he used a that no, no, now again |
|
| 27:22 | the next slide machine that we have nodes are nuts from the tree and |
|
| 27:30 | we have the right values interesting and you need to figure out what the |
|
| 27:37 | of all life evolved from the one and then it's just getting to be |
|
| 27:43 | right hand side values um um yeah of course that's it to go back |
|
| 27:53 | look up the definition the beast governments is successive you know why values divided |
|
| 28:01 | their separation those corresponding nos or not this case they are going to say |
|
| 28:10 | we have a bunch of these and we'll plug it in and get these |
|
| 28:14 | eventually that this technical system in this . Um I guess two of the |
|
| 28:22 | derivatives are in the Cayman there is one of the computer industry so there's |
|
| 28:30 | little bit all of the simplistic example that shows the machine there wants to |
|
| 28:37 | set up to try the angle system how to find the right hand side |
|
| 28:44 | I still wanted that. I had proficiency in the tide animal systems and |
|
| 28:52 | we can write down once we have disease that we have this sparking to |
|
| 28:59 | expression for it polynomial for each one the sub intervals. This case is |
|
| 29:05 | the two combined is 10.0-1 so invested these two polynomial on the conductor 27 |
|
| 29:16 | . So that's kind of the way to the find those spots unless it's |
|
| 29:26 | inside the equations. And so this kind of the general form obviously for |
|
| 29:36 | particular conditions what you have and this kind of an equation. You don't |
|
| 29:41 | it needs to be with me the object. Strip those things off Just |
|
| 29:47 | system for the online as one Yes. Ah So then if you're |
|
| 30:01 | you got some elimination. let's try out of the system in the elimination |
|
| 30:08 | . The only thing you need to the diagonal values for the matrix. |
|
| 30:13 | this is just a simple equation that kind of the factory ization part in |
|
| 30:19 | of doing the interview or meditation. then you also in the form of |
|
| 30:24 | needs to work on the right What a very simple question. I |
|
| 30:29 | to have that disease to the And again, that's simple, wow |
|
| 30:38 | So well that's true practice just putting for the number of a specific |
|
| 30:51 | The interval length. These are our somewhere and the Yes there are millions |
|
| 31:01 | millions. Very straightforward and disease and . Mhm. Right. Any questions |
|
| 31:14 | not? But where common companies machine same first sort out. Mm |
|
| 31:27 | Find the conditions after me and make you have enough equations to able to |
|
| 31:34 | walter. There are unknowns and the is always the highest order derivative of |
|
| 31:41 | parts the and applied conditions there was words second or third degree or any |
|
| 31:50 | degree spun. That you start with highest total derivative that is as the |
|
| 31:56 | linear function. Yeah. And that serious. The conditions for every Not |
|
| 32:06 | have one other conditions applied. And we then use whatever this force to |
|
| 32:13 | Paula. Normally instinct it's when I second degree polynomial that was there so |
|
| 32:20 | . Then you only needed to integrate wants to get something that is quite |
|
| 32:25 | in this case is stupid. So in your functions. And it's first |
|
| 32:33 | the polynomial choice. And then you all the interpolation conditions and disposed of |
|
| 32:42 | for. Mm hmm The polynomial and the derivative for which you have conditions |
|
| 32:51 | the highest fund because you already have the construction and then there's degrees of |
|
| 32:58 | left. And you use stuff too the polynomial and some other properties such |
|
| 33:02 | they want. But it's not so from a normal for us. 619 |
|
| 33:13 | . So one thing and one wonders the things to behave nicely is based |
|
| 33:17 | whether they use since you have this formula fear not updates to use um |
|
| 33:28 | wanted to kill you. There will sort of problematic if it turns out |
|
| 33:33 | it's used. They came very close zero because and during the divided on |
|
| 33:40 | . So there is of interest on kind of normal what happens to the |
|
| 33:48 | ah basically we're gonna get away from . This is kind of what this |
|
| 33:55 | . Oh, calculation says in terms that they use are basically guaranteed. |
|
| 34:05 | The bounded away from zero services. was the update formally use or a |
|
| 34:14 | of things in there to ensure that are always running away from the inter |
|
| 34:20 | . So they have a lot of there. As long as you have |
|
| 34:25 | . So it kind of works for . Doesn't have to worry in terms |
|
| 34:29 | things going back in terms of there a nomination for solving the system. |
|
| 34:41 | right. So um now. Mm . Yeah. Collective. And this |
|
| 34:54 | just a little bit example. So colleges are on the same thing and |
|
| 35:01 | the index than their space, what ? The system looks like this, |
|
| 35:07 | like this. And that's all they that the dominant and it's very |
|
| 35:13 | Um it's huge. Go back and up the next year about there was |
|
| 35:22 | denominational direct solvers when we talked about system. You have a system of |
|
| 35:30 | form than they use emerge to the . So they don't really have to |
|
| 35:40 | to do that. Yes, cultivation update this calculation. You can terminate |
|
| 35:54 | a modest number of ice. So if you have a very large |
|
| 35:57 | I'm not, if they want something detailed then you can still get away |
|
| 36:04 | only executing this statement and the statement have to carry through all of |
|
| 36:10 | But in terms of the characterization and can terminate after this step. So |
|
| 36:24 | that my questions and this the next will do is best for saying, |
|
| 36:32 | , maybe this form is not physical for doing in my rations are doing |
|
| 36:38 | more like ah this corresponds to the function or one thing that is one |
|
| 36:48 | the evaluation of their starting points. the next one here is just starting |
|
| 36:56 | hmm. Then figure out the correspondence thinking get values for a. B |
|
| 37:02 | C and D. From this form the expression for the for each of |
|
| 37:08 | intervals. So, so this service kind of the form of the taylor |
|
| 37:18 | expansion. So no, that S M T I. The rest of |
|
| 37:25 | people's ai So that's kind of we know that from the population |
|
| 37:31 | What is um then we can also the government and what they also |
|
| 37:43 | So basically this gives it the coefficients this formulation as a function of the |
|
| 37:50 | celebrators. Oh, really? So finding Hayley, as I |
|
| 38:00 | it's given by the interpolation conditions what rest of the boys. So that |
|
| 38:06 | fine. No. And this is . I. Which is the second |
|
| 38:13 | second derivative of this plug in tr are gone in the second derivative From |
|
| 38:21 | disappearance because this time zero. So clear that the second ceo and that |
|
| 38:27 | the second derivative ah of Divided by because this is two times to the |
|
| 38:36 | . So senor, Yes, that's . That the property that they're going |
|
| 38:41 | . So now the second derivative was . That the But the machinery to |
|
| 38:48 | . So this is kind of a they can directly go get A and |
|
| 38:59 | . And the so and this equation only appears here and it's only appears |
|
| 39:18 | I understand multiplied by D. And so that this I can see |
|
| 39:25 | where activity has kind of voice. if you are still identifying terms But |
|
| 39:33 | fact coefficients from X three and this from this the gap is this |
|
| 39:41 | This part someone you can invest get directed for comparing the coefficients for the |
|
| 39:49 | of the term in these two So then the only thing left, |
|
| 39:55 | you're on is how did it can in this question from this form? |
|
| 40:01 | polynomial expression. Yeah. So so this case we have to look at |
|
| 40:09 | first derivative that is being equals. then we have this form. So |
|
| 40:18 | with this we can since we are disease and their homes. So everything |
|
| 40:22 | this question is home from that. you can also go, so it's |
|
| 40:29 | too complicated from this form. So form with us from what of the |
|
| 40:37 | . So then we can write it me office. Okay. Just there's |
|
| 40:48 | examples of uh huh. Yeah, up the traditions and this one is |
|
| 41:00 | of the evaluation first determine which finally which polynomial in the spinal you're going |
|
| 41:08 | use. So you need to figure this interval explains in and this is |
|
| 41:14 | this kind of look here does make a lot which docs will be used |
|
| 41:20 | what correspondentsassociation being used to do the evaluation. So this point guests can |
|
| 41:27 | from, this is written from um from the highest. Okay. |
|
| 41:36 | Most not. And then those left of these conditions too. And they |
|
| 41:40 | found the interval. Mm hmm. ? Yes. So is this a |
|
| 41:50 | recursive? Is that why is that there's like or is that um in |
|
| 41:58 | case this is an assignment. So, so I guess that this |
|
| 42:18 | right. No, never question comin there's such self concrete example I'm |
|
| 42:31 | I think we have this before another cooker the polynomial approximately in sign next |
|
| 42:42 | explain that to explain to be The approximate next for uh huh. |
|
| 42:50 | our world different intervals of four equally apart for some intervals. And then |
|
| 43:03 | . Yes, here it is. evaluating the difference between the sine function |
|
| 43:19 | uh 10 points and so the endpoints so take a different set of points |
|
| 43:27 | comparing design function and find hotels The mark of the not in the |
|
| 43:39 | . And where is this kind of ? Go through the interval and look |
|
| 43:46 | a bunch of points in this Yes. The difference between the spine |
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| 43:52 | is a critically important topics and then keep clear the maximum. So we'll |
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| 43:58 | to figure out where they are absolute is the largest walking through from left |
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| 44:04 | right and these evaluation points. The by ah this council, it is |
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| 44:15 | to be used for generating the coefficients the spines. And then they have |
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| 44:23 | sufficient and music. Yeah That is times as many points when you speak |
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| 44:34 | mm mm mm mm mm mm mm mm mm. Mhm. So so |
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| 44:44 | was actually used to that is generating and then you compare the polynomial that |
|
| 44:53 | sources favorable function values. And a of points other than the ones used |
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| 45:01 | college. So for more questions on . So so in this case um |
|
| 45:18 | is an example of the other Whether they have just introduced the character |
|
| 45:22 | fine for this except for values. I think the next five is showing |
|
| 45:28 | the outcome was using after to respond this collection point. But it doesn't |
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| 45:35 | too done. It's not very this to be 74 and that's not so |
|
| 45:44 | , wow. This example there's more but it was a little bit |
|
| 45:54 | So I'm yourself, this second example it's sometimes it may be beneficial even |
|
| 46:05 | it finds the best. You, the population problem as kind of dealing |
|
| 46:14 | true separate segments of the internet in case choose to look adorable twins, |
|
| 46:23 | for. And the one spy this now that's buying approximation for the last |
|
| 46:35 | . And the reason is that may be always looking at the table but |
|
| 46:39 | kind of and look at this So in this case against kind of |
|
| 46:43 | nice approximation but he sets up points another offer, he sets of |
|
| 46:51 | But if you have tried to do um spying across all points if they |
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| 46:58 | not today. That's nice. Of we have this point thousands continued. |
|
| 47:06 | so I guess that I should point even those spines gives so much better |
|
| 47:12 | and our behavior now that's the simple . It's still not necessarily the case |
|
| 47:20 | you just want to I want to for better off. Uh huh. |
|
| 47:31 | the substance of what to do with . So let's see what's meant. |
|
| 47:47 | , just summarizing all things dar oh government or not. And there's a |
|
| 47:57 | of more things and then we talked little bit about this point. So |
|
| 48:04 | next thing here is basters asbestos and can do kind of parametric approximations if |
|
| 48:11 | want to first. So the parliamentary here is the best of the one |
|
| 48:21 | relating really things or is not to next variable. Another spying that relates |
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| 48:35 | asked to provide values that jewel And third or put the X five points |
|
| 48:41 | the plane. He is under way from the press Between the X five |
|
| 48:51 | . So this is kind of It's used in this particular case up |
|
| 48:57 | this case district functions. What's perfect relationship between that so that the data |
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| 49:09 | a parameter two. Move along The one for you. This particular |
|
| 49:20 | Remember the trip the you can see so X squared. Then it's gonna |
|
| 49:32 | four of it, tangent square banging signed on the cold side. So |
|
| 49:39 | we're gonna launch forward here and there have it. Thus another 1. |
|
| 49:43 | times sine squared of the coastline square is 1 plus um signed up the |
|
| 49:54 | manipulated, it gets the best in bottom here you have signed plus coastline |
|
| 49:59 | squared plus cosine squared, which is of the one and then on top |
|
| 50:03 | you have mm hmm the one half a sign of the coastline and then |
|
| 50:13 | that's that's a multiplied colored transparent so best of what to get on top |
|
| 50:19 | do sign call sign, which is science escorting. So it's important to |
|
| 50:25 | . So this is just an unaware parameter rising this particular expression. And |
|
| 50:38 | in this case the vividness and use t as for these points here, |
|
| 50:47 | values And there's 14 values than investigating amateurs and that was still, it's |
|
| 50:57 | . That is versus just phenomenal. saying just this one world in which |
|
| 51:06 | sometimes it's useful. See that parameters how they're moving on. And then |
|
| 51:16 | was this other things about the smoothness that is again, underlying this whole |
|
| 51:23 | exercises in spines instead of a collection polynomial instead of a single point |
|
| 51:31 | Mhm. And then we talked about , we talked about corona and that |
|
| 51:39 | for so um they have the arguments looking at derivatives in this case we |
|
| 51:48 | a function like this. And then stick to everything that supports comes from |
|
| 51:55 | . But my point is really This part in terms of understanding the |
|
| 52:02 | conditions. So in this case the has the property that the interval um |
|
| 52:09 | is the second derivative square is always in some sense the area or under |
|
| 52:17 | second derivative of the function itself. spine is satisfying this unless variation. |
|
| 52:27 | dr caribbean function has worked most the . Not necessarily that stuff just as |
|
| 52:34 | . So very many then they certainly to the function itself. So I |
|
| 52:45 | that was it in terms of all state former relative notion or Our 1st |
|
| 52:58 | and 3rd. This appliance that we've about so far is that's the thinking |
|
| 53:05 | the common on each other. Uh of standard form. But you |
|
| 53:12 | A plus Bx plus C. The combination of mono. He also powers |
|
| 53:21 | was underlying this whole exercise 1st, and 3rd. And then we had |
|
| 53:25 | machine to figure out the minerals as . The spines almost every difference. |
|
| 53:39 | for holidays, january spines that the selection. So yeah, is that |
|
| 53:50 | are not the switch and different things these points okay from predator colonials. |
|
| 54:00 | the basis or the degree of spine have seen so far. That's to |
|
| 54:06 | last. So coming back to some and they went back content view the |
|
| 54:17 | , the powers of X as the functions from being approximations. And that's |
|
| 54:22 | they did in the 1st, 2nd 3rd degree. Both the polynomial approximation |
|
| 54:27 | final. Well, what he pointed when we did talk about polynomial that |
|
| 54:37 | were no, no, not necessarily behaved and they're very briefly commented on |
|
| 54:46 | championship polynomial that again, whichever degree want. But they have, it's |
|
| 54:55 | equal oscillation properties And yeah, That's they want being developed between and to |
|
| 55:05 | smartest one and 1 and they never of grew out of bounds. So |
|
| 55:12 | have this phenomenon has said they're always . So that sounds very nice to |
|
| 55:21 | and whereas polynomial so they get more more extreme behavior behind this coral. |
|
| 55:31 | that's why many times state championship on on them. Its development in the |
|
| 55:36 | in terms of, I compute the in the polynomial now. Please. |
|
| 55:48 | plants were going to talk about them ah, very nice people would |
|
| 55:55 | And that's fine. Quite popular as alternative interpretation. So and that was |
|
| 56:12 | talked about the naming of these points and one is based on the kind |
|
| 56:20 | look like belle cars which you will them. So I said come himself |
|
| 56:30 | . So racist sets. But that's for on all meals as well as |
|
| 56:36 | well. Um, but they call romantic and otherwise they kind of bell |
|
| 56:43 | check. So I'm going to start zero degrees spots of this one 0°, |
|
| 56:58 | constant first degree. It's just X. Is in there zero degree |
|
| 57:05 | X. That's going zero cost, . So, so here is the |
|
| 57:17 | of zero degrees. Bye. So it is, it's basically one |
|
| 57:30 | one interval zero everywhere else. And also one more particular thing to pay |
|
| 57:40 | to is that is defined on it open and half closed, whichever you |
|
| 57:47 | to call it intervals. So the hand point is voted at the interval |
|
| 57:57 | the The spine is one but the endpoint is not. So that's what's |
|
| 58:05 | half open. So it's defined building last stand points to be won on |
|
| 58:11 | way, Bring the right 10.5, including the right at the right end |
|
| 58:19 | . This is important for us to through how to use this, these |
|
| 58:24 | and multi construction. So a lot being in the subscription construct, you |
|
| 58:34 | think response standing kind of extent. as I said first, it's not |
|
| 58:45 | . Um, now when we talked first of responding so far and I |
|
| 58:50 | the sponge, then the spine itself continuous, but that's not true. |
|
| 58:54 | the is fine. The president is constant self. If she forced it |
|
| 59:00 | be continued one value across the board or area of interest start continues not |
|
| 59:11 | it says here, it is continuous the right. So if there are |
|
| 59:17 | and go left uh in the limit the zero that in the limit to |
|
| 59:24 | . And that is the fact that bass pond was about to be |
|
| 59:27 | So distance is continuous. And also this point, if you approach one |
|
| 59:33 | the right then since it's one all way it will become one and just |
|
| 59:39 | finds defined for the Wanda, there continuous from west but not okay. |
|
| 59:47 | of that someone is just one of just stopped too. And then there's |
|
| 59:55 | notion of support for this kind of form and that system defined with support |
|
| 60:04 | region or the section of it X or accept of independent variables for which |
|
| 60:11 | function is not safe. So that the support for this zero degrees. |
|
| 60:21 | ? Yes. Half open. Because defined in the one at the left |
|
| 60:25 | point in all the way to direct the right. So we'll use that |
|
| 60:33 | proving some properties, all of these stuff. So obviously the first one |
|
| 60:43 | pretty trivial. The way they were find that it's always not negative. |
|
| 60:50 | it's also true. So again, indicated the left standpoint of the ancient |
|
| 60:56 | for which ah mm hmm. Everything fine is down there while 20%. |
|
| 61:09 | this is an infinite sum. This about the final four because for any |
|
| 61:17 | select. Um is there anything It's only 90. Where in the |
|
| 61:28 | exercise for us very well. It's . So it's only one of |
|
| 61:35 | zero. That's the final accounts for nations started trading. So this is |
|
| 61:40 | like this. one says no. the infinite someone's contract. So |
|
| 61:54 | So this is then you know I you fire cause that's where construct I |
|
| 62:00 | he's blind as best function. That the spice constant and mr than the |
|
| 62:08 | from the right. It's simpler to to bother proper zero spine 0 one |
|
| 62:19 | what the value of the constant is that pinto. And then you can |
|
| 62:25 | it up because you just plug in X. Value and only one of |
|
| 62:29 | foundation terms survives Because all the other are zero. So in this case |
|
| 62:35 | that the swiss constant function. Where it is the constant values for anything |
|
| 62:47 | as well defined again are continuous Right? In on their properties. |
|
| 62:52 | a change. Oh Alright. So I want to watch Honey. You |
|
| 63:02 | 20 different spots. And that's the construction and now get someone tested |
|
| 63:10 | We're sorry about that. This looks I think they're not safe. Next |
|
| 63:18 | that So there's someone constructs to take spine are too um same one as |
|
| 63:28 | or less like order particularly less. hmm. This fun function they want |
|
| 63:36 | use two of them. This one best way is starting uh T. |
|
| 63:48 | . At all. That point is . I. Um And the other |
|
| 63:55 | mills is shifted by one. So starts up the point or the |
|
| 64:02 | Well, it's a watch. A day likes axes. The other point |
|
| 64:09 | urgency here is yeah, this the is scale. So this factor and |
|
| 64:23 | is explain this one as well. see in the reports of construction is |
|
| 64:30 | best to define um non zero on interval. And the one that has |
|
| 64:39 | by one is just replacing ci or with I plus one. So this |
|
| 64:45 | I plus one. Where did you him you get this impression exactly to |
|
| 64:52 | the sign of it. So And and look at these impressions of |
|
| 64:58 | it means. The one thing between two these spikes is such that when |
|
| 65:06 | move from T. I. Towards right hand point. So X. |
|
| 65:10 | continuing always faces factor zero. And the other end points the factory is |
|
| 65:17 | then also I found this case or to the scale that this is the |
|
| 65:24 | Viniar waiting between the two endpoints of ah display this other fellow works in |
|
| 65:33 | opposite direction. So this that But the left hand points ah Which |
|
| 65:41 | the G. R. Plus That's where it starts. Then This |
|
| 65:47 | is one at its right 10 The X equals 10 point years of |
|
| 65:54 | at zero. So the white thing this one g stuff has to move |
|
| 66:00 | exports. And this one goes So that's the problem. It's not |
|
| 66:12 | we're going to do it for the degree polynomial because to generate the first |
|
| 66:19 | response, sorry To depend on the on stuff that has talked about the |
|
| 66:24 | constant. Thank you. So that , actually I Then it's just defined |
|
| 66:35 | one and then the bottom rent So that was K. one for |
|
| 66:40 | he wants to get for this um idea. So what does this look |
|
| 66:48 | ? So there were in the 70s fine. So now the one so |
|
| 66:57 | know, But this one is zero this interval and this one is zero |
|
| 67:07 | they can evolve from ah from the like plastic. So if they are |
|
| 67:14 | to the right of the right hand to the left of the left hand |
|
| 67:19 | , it's definitely zero. Mhm. this interval this one the one that |
|
| 67:33 | left point that um Is going from to 1. This one that's the |
|
| 67:42 | of stuff. So so that's the that something like this. So so |
|
| 67:51 | is coming from the B. Ah zero degrees spy. And this |
|
| 68:00 | from them, the IRS correspondent zero term goes from 1-0. So It's |
|
| 68:12 | constant. So it's one is defined be one at this point. So |
|
| 68:18 | is continuous. And as well the . But at the right hand point |
|
| 68:27 | almost starting here is zero outside. this one gets this so have its |
|
| 68:36 | And sometimes problem that is the first use but so that's good too. |
|
| 68:42 | Thanks for being here, wow, . And the first The support is |
|
| 68:51 | zero and this open interval for any on this. Uh huh. So |
|
| 69:14 | the zero degree point These points. have the property that the sum of |
|
| 69:20 | all was one anywhere for any But something is always one because in |
|
| 69:26 | place was only one of them that was not there and that one now |
|
| 69:35 | any given an X value, ah 1st responders are over that so on |
|
| 69:46 | everyone is not totally obvious that this also holds true for firstly responsible all |
|
| 69:55 | one everywhere nes But like it's from , one thing is to look at |
|
| 70:02 | here that for any interval that you since The underlying 0° find it's one |
|
| 70:12 | so the great which one rights rates is the same as the other one |
|
| 70:18 | are So in that case the sun always one of the Yeah, it's |
|
| 70:26 | it doesn't matter what the length of interval between the knots. It's almost |
|
| 70:31 | . So funny people the slope So that's something. So that's Uh |
|
| 70:43 | . So that's something important, that song. So then it's asbestos |
|
| 70:57 | no part. So that for example to get some in tradition going in |
|
| 71:02 | of first three at this part. of this don't have functions. The |
|
| 71:11 | thing is that it is okay. now you're looking at in general for |
|
| 71:16 | higher degree spine. Some of faith his spine. Yeah, it is |
|
| 71:24 | fact that so that the support like function of the area for the set |
|
| 71:32 | X values for which it is zero . Right? They started with the |
|
| 71:40 | that was on the non-0 for one . And as we show from the |
|
| 71:46 | function, The first degree spine was for two in intervals of on |
|
| 71:52 | Already done on the next one. in general as the degree of the |
|
| 71:58 | increases the set of X values for is um non zero. Obviously talking |
|
| 72:11 | the fact that it's incredible the states . But So first place non zero |
|
| 72:17 | the widening intervals of zero. As as you're outside start painting. |
|
| 72:25 | again, for the first degree spine that was non-0 Between I and Ii |
|
| 72:31 | two and outside. I think the is the signals of texas not and |
|
| 72:39 | in turn off That it is All right. So, and this |
|
| 72:48 | I'm going to talk about that by And the production assumption look at 10 |
|
| 72:54 | one and then figured out what happens the next K. So then what |
|
| 73:01 | have is for the combined as well to the assumption with induction. Then |
|
| 73:09 | this is the welcome. I thought stay and then the next find that |
|
| 73:15 | used to construct the chase. It's . Is there screening which jumped by |
|
| 73:23 | . So that is outside. It clear. And then obviously done by |
|
| 73:33 | . If X is outside both which is a statement here, then |
|
| 73:38 | they left the first one that I the one at five plus one. |
|
| 73:42 | zero has been straightforward zero and that's inside the interval. It's down |
|
| 73:51 | They consist either this one or that is zero. And I can see |
|
| 73:56 | forgiven the construction here. That it at the left hand point. You |
|
| 74:02 | this Factor for this Formula zero. in fact so and strengthen this to |
|
| 74:10 | outside the open interval Picks up four . 40 Where the 0° spine. |
|
| 74:21 | ? That is not true. That's everything else. It is a |
|
| 74:26 | Sorry, nonsense. So this shows began that these 0° span the harbor |
|
| 74:35 | well defined region In which were there non zero. And then there's hair |
|
| 74:42 | else stolen. That's not a good . I wanted to be use them |
|
| 74:46 | a basis function. And figure out weighting of different what these points turns |
|
| 74:57 | because you only have to consider your number because for any energy given X |
|
| 75:04 | , There's a finite number of these that are around zero. So |
|
| 75:08 | there was only one zero. That fine. There were two In this |
|
| 75:16 | . Right? That there is two . One Planning for the next five |
|
| 75:22 | You have two spines are around Well, this was coming on to |
|
| 75:27 | used population. All right. This is pretty much to go back |
|
| 75:39 | the same formula Basically greater than And the open interval Between the two |
|
| 75:50 | because of computation factory here Translates to also. And then that's important. |
|
| 76:07 | . So here is a special or intuition Find one of these fights. |
|
| 76:18 | we have the first one. The one was the help function. But |
|
| 76:23 | as you can see the heart of that's fine sir. It's both of |
|
| 76:27 | accounts and then we can also senior and then give an exile. You |
|
| 76:37 | right. You can imagine I guess There is more social degree three |
|
| 76:52 | So if you have then then there's obviously the figure. But then there's |
|
| 76:56 | third degree spine that starts here. will also be non-0 here. So |
|
| 77:02 | really that's so hard think of the of these points that are involved for |
|
| 77:10 | the next father is in response to degree of the spot. It was |
|
| 77:16 | one for 0°2 for one. the number of spines. Um, |
|
| 77:22 | you can use for interpolation and then the next value there is there's |
|
| 77:45 | Okay, so this is now getting the point that the sentiments unfortunately a |
|
| 77:54 | bit that's the equations quickly. So if I want to use this these |
|
| 78:06 | are sending queen kei to do an of the function than comments here. |
|
| 78:16 | . It's not going to be that large number of nanomaterials but it depends |
|
| 78:24 | the degree of this plan is coming the need for human expansion and what |
|
| 78:31 | is which ones depending on what So the best thing. There are |
|
| 78:37 | um a number of them to the of our exes that reaches the |
|
| 78:43 | Value that needs to be working Bring out the approximation. So there |
|
| 78:52 | kind of service looks nicely on the . There was the definition of |
|
| 78:58 | the spine of the barricades. So want them an approximation linear combination of |
|
| 79:04 | finds in the free trade and manipulate expression of it fine. That's when |
|
| 79:13 | second term here where we have I one. Since it's an infinite |
|
| 79:19 | You can play your armor than investing . Again, change the I. |
|
| 79:24 | here to be the same as in first. Ah So we're replacing The |
|
| 79:30 | part here. I passed one with the investing becomes this becomes the |
|
| 79:40 | Will they survive. And this becomes . O. I. Plus one |
|
| 79:45 | I. Plus J. Sorry. then we have to adjust the come |
|
| 79:51 | This family replaced eight last 1 So now the investor has an |
|
| 80:00 | Um So the function here, see happens to be also function events. |
|
| 80:09 | then you have now a lower degree . So now instead of starting doing |
|
| 80:16 | with respect to amazing. I guess . Find my $10 to have also |
|
| 80:23 | to claim them into another expression And order our response. And then you |
|
| 80:30 | repeat that a bunch of times. I congratulate then you'll get to something |
|
| 80:35 | They only have to worry about one the 0°. finds that coverage the interval |
|
| 80:43 | it's it's But the one thing on difference, yeah, these spines are |
|
| 80:51 | in these proficient. And here was formula for which generates these core efficiency |
|
| 81:00 | it's only the highest order one where started some of them back to this |
|
| 81:05 | . So we start here. Those the ones that are not depending on |
|
| 81:10 | expert As the index was down to . Then there or in this case |
|
| 81:19 | experiment here, the wise divine. these are dependent on the X. |
|
| 81:27 | this is for so then they get formula here that our dependence on X |
|
| 81:31 | clear. But it's a starting That is the highest thing that state |
|
| 81:38 | are not. So then when they dispersion formula to compute the coefficients, |
|
| 81:45 | of them waiting together. The you the scheme like this to compute the |
|
| 81:56 | . Singapore technology. And unless my is a massage stop. No, |
|
| 82:04 | God has been much less side. . So, so, but it's |
|
| 82:11 | of messy, but it's a recursive that for the popular indian proficient that |
|
| 82:19 | the different Despina implicitly from the limiting . And use the fact that the |
|
| 82:26 | response. That's the one where it . Oh, Sylvester, that's But |
|
| 82:36 | the next and they get the one for the Dr. one correspondent |
|
| 82:41 | what that means? It's okay. . I examine exception appliance under the |
|
| 82:53 | . They don't, yeah, this be on the phone. Uh Most |
|
| 82:58 | . Yes. So everything addressed through next week. Have you as legal |
|
| 83:09 | put underneath term whether it will be or not, since I haven't constructed |
|
| 83:12 | . But most likely it will be of this point. But I don't |
|
| 83:17 | I will put the most complex piece it. |
|
