16:07:52 Okay, Here we go.
16:07:54 So I want to start by reviewing
16:08:00 the theorems one to seven.
16:08:05 Although we only need number seven today.
16:08:08 So, but just for completeness. So, this is from the paper called post processes, and evolution algebra.
16:08:21 So.
16:08:25 So first of all, the first three theorems are just repeated from the earlier paper by Robert and brown.
16:08:33 And so, we're not going to use them today but so theorem one. So, P. This is the vector that tells you the pulse at each vertex have a diagram.
16:08:47 At time t. This is the initial pulse at each vertex. And as the Jason see matrix of diagrams, you may not remember all of these things and that's okay, because when the time comes, you will.
16:09:00 you can review them.
16:09:02 So then we're interested in this matrix A, and it's eigenvalues, and that determines whether the, the DI graph, or the, the pulse process, which is the boss process is stable or unstable I will review those definitions down.
16:09:22 So there's five and six.
16:09:25 Were characterizations of when the diagram is pulse, stable or values table.
16:09:34 and the condition is expressed in terms of eigenvalues of this matrix say, Okay, so, and then there in five
16:09:50 just says tells you exactly when an evolution algebra has a unit and identity element or a unit.
16:10:02 And so, but what we need today is this theorem seven so damn six. We don't need to. Don't even state it.
16:10:10 So, so the stem seven is.
16:10:16 Okay, I did say these are character. Yeah, I did.
16:10:19 So, this theorem is tells you what the multiplicative spectrum is and an evolution algebra.
16:10:50 And, namely the multiple multiplicative spectrum which I'm not going to remind you the definition right now but
16:10:40 we might do course, anyway.
16:10:44 So, we have, we have an evolution algebra, we have a structure matrix.
16:10:50 So of course here, a denotes and algebra whereas before a was a matrix so I don't think that should be confusing.
16:11:01 So, well, this multiplicative spectrum.
16:11:08 Whatever. Whatever it is, and we will need to recall what it is, at some point, Probably not today.
16:11:18 It's it's the same as the eigenvalue the ordinary eigenvalues of this matrix. Now this matrix is a product of two matrices, and the second one is a diagonal matrix so it's going to be very easy to multiply them.
16:11:37 I could never do it in my head, any case.
16:11:41 So, we'll refer to this in a moment.
16:11:47 And by the way, this is encased lambda is not zero it's lambda zero Well it might or might not be in here.
16:11:57 And I'm not going to worry about that right now. So, this is a one page file.
16:12:05 And so there are no yes one page plan.
16:12:09 So, okay, I may have to refer back to this but not necessarily.
16:12:16 So now.
16:12:21 So actually,
16:12:21 before I made that previous file which was just a couple hours ago.
16:12:27 I was going to recall theorem theorem five
16:12:35 by just pasting a previous file, and so you can ignore these first two pages there and five and six are stated here, this is the first time stated them so that's what I want to do is now look at section four of the main paper.
16:12:53 And so let me pull that went out.
16:12:55 So that would be this one.
16:13:00 This This is the main objective of this class and I don't know how far will get.
16:13:06 But you'll recall that section two was about pulse processes and stability. Okay.
16:13:17 So, and there's where you find their own long term, two, and three, and four.
16:13:22 And then, then section three would be on evolution algebra.
16:13:28 Excuse me, there's no yes stability.
16:13:33 evolution analogy was. And
16:13:36 so here we have a theorem five there I'm six about home muffins.
16:13:43 I'm ignoring today, as well as the other ones and theorem seven This is the main tool for today. I forgot to mention that a was.
16:13:54 There was a basis, natural basis the eye and then a coordinate sound fine that's what the office are.
16:14:01 I failed to mention that but I'm sure it was understood. So actually, that brings us up to what I want to do today. And that would be this section. So, there's only a couple pages.
16:14:17 And so, let me go back to my notes here.
16:14:22 And so, so here, here's, here's the idea.
16:14:26 Number last time I said, this is just a translation of language from graphs to algebra.
16:14:35 And so, this is, this is
16:14:40 so so there's not much going on here, in the sense that there's nothing to prove. Once you make the identification.
16:14:49 In fact, diagram and evolution algebra is our in one to one correspondence, so
16:14:58 let's say you have an evolution algebra, a with a natural basis be
16:15:06 a structure matrix, which is denoted by the symbol hair m sub Boa or wij.
16:15:17 And what that means is ej squared is the some WIJEI, which gives you the columns of the, so.
16:15:31 So here's what the matrix looks like of course wij looks like this.
16:15:37 So, now, the definition is the definition so the graph, we want to associate a graph, a diagram.
16:15:51 With this algebra.
16:15:53 And so, we're going to take the basis.
16:15:56 So it depends on the basis, the basis, will be the vertices.
16:16:14 And the adjacent see matrix will be the transpose of the, of the structure matrix. So, it's basically the same thing, except for the transpose.
16:16:18 And to be honest with you, I don't know why you need to transpose here.
16:16:22 it doesn't affect the eigenvalues.
16:16:28 And so, but.
16:16:32 And I think so it's it's not a crucial point, we can work, work around it.
16:16:38 Even not having that so.
16:16:41 So if I take the transpose of this matrix, I can write it this way, the first row becomes first column and so forth.
16:16:50 So here's an example.
16:16:53 So, we have if you start with an algebra.
16:16:58 And then you just labeled the vertices, with with the names of the basis factors.
16:17:05 And then you draw arrows between them.
16:17:09 And if you're going from A to E one, then the weight, there is defined as w two one
16:17:23 where who one is is from the structure matrix.
16:17:31 So it's pretty obvious and you wonder how anything important can come from this but we shall see.
16:17:43 All right. So of course, any anytime there's no arrow between like a one, or let's say a three to one there's no arrow so w three one would be zero.
16:17:58 Okay, that's a convention.
16:18:01 Now, conversely if you start with a way to die graph.
16:18:08 And the vertices, we indicated maybe by x one objects in there just symbols and we can think of that as a basis.
16:18:20 And so this graph will have an edge agency matrix, remember what that was, in fact, it's exactly these numbers here.
16:18:34 So, again, why we need to transpose is not clear to me. I'll admit it, and that's okay we'll figure it out if it's important.
16:18:44 It's, it's an easy thing to figure out if shot when the time comes. So, if you have a graph. Then you you take its vertices, just label, just put it as vertices act as a basis.
16:19:01 So you're really just talking about and dimensional vector space here, except that the multiplication.
16:19:14 Let's see.
16:19:16 So.
16:19:19 So the thing is the Jason see matrix, you just take the transpose of it and use that as your structure matrix.
16:19:31 Again, I apologize this transfers I don't know why you need it.
16:19:35 Nevermind, as good. So, this is. This example will explain everything if it needs to be. So let's say we start with the grab list follow these arrows here.
16:19:46 Okay, So let's take a graph that looks like this so CF and D constitute.
16:19:55 Some, some quantities which which can have a value.
16:20:01 Like, energy or fuel economy and so forth, things like that.
16:20:07 And then in any context. Okay, so now we have some arrows here, which are determined by some real life situation, maybe.
16:20:18 And so that, that's a graph. and so we just construct its head a Jason see matrix.
16:20:28 And so we see that 3131
16:20:35 entry is point 05, so there it is down there.
16:20:42 And, and so forth. The two. So what I've done is I've labeled the indices 123.
16:20:49 Instead of CDs, which is in the right alphabetical order but doesn't matter.
16:20:55 Okay.
16:20:57 So, okay, so clearly, you can have a bunch of zeros here, because there's, there's no arrows between some of these vertices.
16:21:08 And so now if we just take the transpose this,
16:21:14 which I just wrote wrote in this way.
16:21:17 But with the numbers, it's, it's equal to this and. And what this means is this first column is a square of EC
16:21:31 E sub c is the basis factor corresponding vertex. See, and so it's zero times etc so he is the first day basis factory the second and so forth.
16:21:44 So, this gives you the coordinates of EC squared. This gives you the point okay.
16:21:53 So, nothing, nothing very deep going on there and remains to be seen how useful this is.
16:22:02 So here, here's the first bit of information on that the notes important are going to be important.
16:22:13 So if you have an evolution algebra.
16:22:17 And you have a basis, which is fixed.
16:22:20 Now just add up the basis elements, with no coefficients and call Daddy, and that's given a name. It's called the evolution element.
16:22:32 Okay, relative to the basis of course, and.
16:22:38 So let's go back to theorem seven.
16:22:44 You see if you have an element Fe, which is written as a linear combination of the basis.
16:22:52 Then the multiplicative spectrum.
16:22:58 Whatever that is, there is the eigenvalue of this now, if A is equal to the just the sum of the eyes and all the alphas are equal to one, and this is the identity matrix.
16:23:12 And so, the multiplicative spectrum is simply the eigenvalues of the structure matrix. So that sounds pretty good.
16:23:23 That sounds so that's that's what the so this is a corollary, if we take all those houses equal one, then
16:23:34 multiply the spectrum. Now, a is equal to E evolution element to find here.
16:23:41 And that's equal to the eigenvalues of the structure matrix and that's the same as the eigenvalues of its transpose, which would be the, Jason see matrix.
16:23:55 And sometimes you'll have to add the number zero, And I'm not going to worry about that exact. It's claimed here that if a is a trivial evolutionary or that a say.
16:24:20 If the structure matrix is diagonal nonzero entry entries.
16:24:22 Then you don't you don't have zero but I don't remember why that's true and I don't care.
16:24:28 So now, once that once you have this correspondence graphs diagrams of the kind.
16:24:37 We thought about our in one to one correspondence with evolution algebra is provided.
16:24:48 You know you you fix a base. I mean, with respect to some basis.
16:24:56 Well, and the only difference being a transpose between me Jason see matrix and structure matrix.
16:25:07 So, we have a concept of stability for graph
16:25:23 No, it's important.
16:25:25 And so you just transfer that over to to the, to the algebra so nothing going on here so I have my doubts about whether this can really be useful but
16:25:38 people have told me that it should be. And that's the reason we're doing it.
16:25:44 So, Now, I'm gonna I'm gonna just not going to go over this.
16:25:52 There are made in this paper.
16:25:54 Simply.
16:25:57 Now, take them three and four. Remember,
16:26:01 we're the characterizations of stability.
16:26:08 All stability and the characterization of values to build a data respectively.
16:26:11 And once we have this corollary that we just stated corollary.
16:26:21 Then, this is just a translation of them three and four.
16:26:26 In the context of the corresponding algebra.
16:26:30 So I'm not going to go over it and I'll even the corollaries, and then there's another example.
16:26:37 But I don't think I want to go over today.
16:26:42 Maybe some, some day one.
16:26:47 So for this example is pretty simple one.
16:26:51 You have three vertices here, which we indicated, like they were basis, elements, which we can do.
16:26:59 And so, the structure. Well, if this is a graph, then I'm going to scroll down here, then.
16:27:14 So, the thing is W one two should be over here. I guess that's why it takes to transpose.
16:27:21 But anyway, the strength. So, this is the structure matrix.
16:27:30 And,
16:27:34 well, actually start with, with the graph with the lesson is called Jason to matrix. And then to get the algebra that corresponds to this need to just take the transpose.
16:27:50 And then we know that the spectrum of either one of these spectrum meaning eigenvalues, is just going to be zero, because this is a trivial matrix, you compute the eigenvalues and yet lambda cube equals zero.
16:28:08 So, now, then that means multiple that of spectrum which is of the evolution element.
16:28:20 It says, only have 01 point.
16:28:22 So, it remains to be seen.
16:28:27 If this is going to be important for anything so.
16:28:50 Okay, not, not includes, this is what I was, I was going to do this last Wednesday, but I mean feel comfortable for various reasons and I'm still not very comfortable, but that's the way things go.
16:28:49 All right.
16:28:52 So, let me just go to this paper again. And
16:29:00 you can see here the evolution element is the sum of the basis elements.
16:29:06 And then here you have the definition of stability of the algebra, which is just transferred from the stability of its corresponding diagram.
16:29:20 and the this there may just
16:29:25 repeat repeats those two earlier
16:29:29 on characterizing stability.
16:29:33 Okay, so
16:29:37 now.
16:29:40 Actually, let me just take a quick look at the rest of this paper I think there's maybe two more section. Now this one involve some algebra.
16:29:51 More algebra,
16:29:54 which doesn't seem very interesting but there is a point to it so that'll be the next thing we do, turn on and off will do it this quarter.
16:30:06 But of course, our quarter.
16:30:11 Most of us, since we're all graduating. So, then, then there's some, some terminology. There's some discussion, some concrete problems involving energy.
16:30:29 Now, this. So, and then there's more algebra in here.
16:30:34 So, Alvarez is fun, can get a little bit messy.
16:30:40 And then now we're back the graphs here and you have certain terminology.
16:30:45 So there's a lot of material here still to do.
16:30:49 And there is a point to all this and it's called the reduction process.
16:30:58 But I don't know if we'll have time to get to that.
16:31:02 So, let me leave it for the moment, and just stop sharing.
16:31:08 Okay, so now I'm about projects.
16:31:15 So, I want to say the following.
16:31:20 So, I would say,
16:31:29 well, so I've already so remember, I want to have a one on one, zoom, with each of you.
16:31:40 And then after that,
16:31:45 you know, we'll make a presentation together.
16:31:53 So, and so far actually I've made appointments with four of you already.
16:32:02 So, one zone namely, One is on tomorrow is Monday what is Tuesday one is Wednesday.
16:32:11 So the rest of you, you can plan on probably it's good idea to keep it to one a day to.
16:32:20 To save time for me.
16:32:26 I've got to prepare the paper as well as you.
16:32:31 So,
16:32:31 to be precise. So, Wednesday is June 2 So, June 3 to do the middle of the following week which is finals week.
16:32:44 I would like the remaining.
16:32:49 One on One Zoom's to take place. So, just think about when you'd like to do it.
16:32:55 And, of course, you can do it anytime.
16:33:02 We should have some kind of deadline of june june 9 middle of finals week.
16:33:07 If you're in the middle of finals. Well, you can always delay this if you need to.
16:33:13 And then
16:33:17 from the middle of finals week to the middle of the next week
16:33:23 would be the, the window for the presentations and as I said before, we'll have a one on one and I'll sort of advise you about what what I expect.
16:33:43 And then, and we can have more than one, one on One zoom if it's if it's needed, that's appropriate.
16:34:00 But then, then, when we're. When you're ready, we'll we'll schedule your presentation, which I'm thinking about 30 minutes.
16:34:14 To me, and I can ask you questions, because it is like an exam but.
16:34:23 And
16:34:26 I will let everyone know when it's taking place in case they want to listen.
16:34:32 But you'll be muted. No, no, actually, if they want to ask a question it's okay but they can do in the chat. Okay.
16:34:44 So, all right.
16:34:47 I'm done.
16:34:49 So, anybody.
16:34:53 We should anybody wish to raise an issue of some kind.
16:34:59 I don't mean it that way. I mean, anybody have a question about this or is.
16:35:06 So from now on.
16:35:08 So let's see, it's Saturday so next Wednesday. Yeah, we should meet at the real time.
16:35:16 Let's see this.
16:35:19 That's, that's.
16:35:20 I'm confused now next.
16:35:23 Yeah, June, Wednesday June 2 it's week 10 so you're still, you're not doing finals yet, but you will be after that.
16:35:36 Okay. Anything I sir, so well we made on that Saturday.
16:35:47 Yes. But, you know, the meetings will be recorded if you can't make it for some reason that's the beginning of finals.
16:35:58 So I understand if if you can't make it the meeting will be recorded and we may not, I don't, not sure what we'll do.
16:36:06 But there will, there will be a meeting unless I cancel it like I have in the past.
16:36:13 After all, I'm the boss can do it.
16:36:19 All right. Alright.
16:36:22 Professor.
16:36:23 Yes.
16:36:24 So I also question, a whole lot usually how long will that presentation be.
16:36:31 I'm sorry. Say that again.
16:36:34 I didn't chat.
16:36:39 Hmm.
16:36:41 I didn't even chat.
16:36:47 There's a chat.
16:36:50 Oh, how long will that presentation will be okay I forgot about the chat.
16:36:56 Yeah, I'm thinking, 30 minutes.
16:37:01 If you can do it in 15.
16:37:05 That would be okay too but I'm not sure that's reasonable.
16:37:10 I suspect you you know with with me maybe interrupting and might take a little bit longer maybe 45 minutes.
16:37:19 But, you know, I'm going to talk to each one of you beforehand and focus on
16:37:30 one one concept.
16:37:32 And so, Yeah. So I think, but but
16:37:39 it can't be too short.
16:37:41 I mean sometimes if you go to meetings and so forth, there's like 10 minute talks and stuff, which are just little hard lines of things.
16:37:53 So, I expect
16:37:58 you know some, some analysis of of what you studied.
16:38:06 But it's just representing what's in the paper and just showing me on you understood something.
16:38:18 Okay.
16:38:25 Now,
16:38:25 Mr. Chu Did you have a project.
16:38:34 Oh, Well, you can you can let me know.
16:38:37 But, yeah, yeah you can let me know.
16:38:40 So, actually, I would like to ask
16:38:47 is there. Is there anyone enrolled for two units.
16:38:53 I am,
16:38:59 who is talking.
16:39:02 It's Bob.
16:39:04 Bob.
16:39:07 Okay.
16:39:10 Let me so.
16:39:17 So Bob Bob has two units.
16:39:20 Yes, and
16:39:25 jam true two has two units also.
16:39:31 I'm, I'm stuttering tongue
16:39:36 situ thing.
16:39:39 Ok, ok,
16:39:44 ok so
16:39:48 I think I'll drop this.
16:39:51 It's not important to two units or four units.
16:39:57 I originally I said the two units would not require a project.
16:40:04 But, in fact, the the project is really the main there aren't wasn't that much homework.
16:40:10 There was some, and that would be fine.
16:40:17 So, if you want to do a project, whether you're taking two units or four units okay with me.
16:40:25 Okay.
16:40:31 Are we done.
16:40:37 If there's no objection.
16:40:39 Okay, so let's keep in touch with email and