Yeah, welcome back to Think Tech. I'm Jay Fidel. We have Community Matters today. And we're going to talk with Bob Toyofuku, who is the founder and CEO and chief honcho. Can I say that? And Pacific Law Institute. And we're talking today about, you know, we'd like to get to know what it's doing. We'd like to get to know some of the programs that you have done, will do, and how they affect the members of the bar and the members of the public. Welcome to the show, Bob. Thank you very much, Jay. My pleasure.   In fact, in the 70s, the 80s, and the 90s, you were the major source of continuing legal education for the whole bar association.  But can you talk about, you know, where you are now, where you're going with Pacific Law Institute? Well, you know, Jay, in the last five years or so, or maybe a little more, I have been so busy at the legislature in government affairs and lobbying that I haven't had much of a chance to design and put on different continuing legal education seminars, except for the trainings I do every year for the court annex arbitration program.  We had more attendees at that seminar and the two replays than we have ever had doing it in person. We ended up, Jay, with 445 people, attorneys, arbitrators logging in to watch the program. So that was very satisfying for me. But... It was a very good program, Bob. It was really excellent. It was engaging, it was interesting, and it taught the lawyers and arbitrators who were watching, you know, how things had changed and how they could catch up on the various dynamics that were going on about COVID. You know, that's the first virtual type seminar I've ever done. And I have always thought about doing webinar type seminars, videotaping the speakers and then playing it on a website somewhere. And never did get around to it. And hooking up with you and Think Tech really energized me about doing more programs because it really takes away a lot of the administrative, I mean, the functional things that you need to do. Rent the place at the hotel, set that up, send out the brochures. Everything is changed with technology. So, you know, in the past year, I've done hundreds of continuing legal education seminars and I have done different types of seminars as well, although using primarily lawyers or legislators. So, since we did this arbitration training, I got a, well, can I use the word energized? And so I am going to do a continuing legal education seminar in December with you on legal ethics. And the Supreme Court rules require that all attorneys that are licensed and active get three continuing legal education credits, CLE credits every year. And one of those credits every three years must be in ethics or professional responsibility. So I decided to at least put on a course in December in case certain attorneys do not have that one credit of legal ethics. So that's gonna be with Think Tech on December 11th. And then moving forward, Jay, you know that I am planning to do a course for the public really, not just for lawyers, on how to access and lobby at the 2021 legislative session. The Capitol is still closed. It's unclear at this moment how they are going to conduct the session. Is the timetable gonna be changed? How are we going to testify? Is it all gonna be virtual? And how is that gonna be handled? Is there gonna be a time limit? If we do want to meet personally with any legislator, can we do that? You know, make an appointment ahead of time. Do we have to be cleared before we go to the Capitol? There's so many questions. So probably in early January that's scheduled with you, hopefully somewhere around January 8th or so. And I have three legislators, two from the House, one from the Senate. And I'm waiting for the Ways and Means Chair to call me to see if he's available to participate. And I have the Speaker of the House and the President of the Senate to make some introductory remarks. Well, you know, just like the arbitrator program that you did a few weeks ago, you know, that has to wrap around, I'm sure it will wrap around the possibility of a virtual session and therefore a virtual testimony and therefore virtual lobbying. I can easily see a situation where you call up a legislator and say I'm a lobbyist. I would like to talk to you about a given bill. Let's make a Zoom meeting and we'll have our discussion. And, you know, I don't think people have done that up to this point, but that does present itself. It certainly has swept through the arbitration proceedings. And now I expect it may sweep through the legislator proceedings just the same way. Oh, yes. And you know, one of the things that I had mentioned to certain legislators several years ago, let's see, seven years ago is that because the neighbor island individuals who want to testify, you know, they have to fly into Honolulu and that's costly, it takes time, they have to wait around. And when I was in Alaska seven years ago, I visited the state capital in Juneau and they had a system, not Zoom and not video, but audio. So somebody from Fairbanks, which is, you know, like, I don't know how many hours away, you know, either by car or plane, they could call in at a certain time. And so I talked to certain legislators about thinking about setting something like that up for the neighbor islands. And not only neighbor islands. Somebody from Waianae who can't get in very easily, they're not in central Honolulu can call in. So anyway, long story short, now with the pandemic and the capital being closed, there is no choice but to do it either audio or video by Zoom or some other platform. So that's gonna be, I think, the standard. Yeah, and, you know, just as it happens in the legislature, there's also an issue about all these courts and boards and commissions and committees and what have you all over state government, you know, which call for participation, presence, testimony. And I remember I served on a couple of boards that were in a big island and the whole troop would get on a plane in the morning from Oahu, fly over to the big island and then fly back with a huge waste of money. And they did not have either A, the technology, you know, or bring people together by electronic means, nor was the statute ready for them on sunshine. It wasn't clear that how to do that on sunshine. It may require further legislation to clear this up in terms of commissions and boards and committees and the like that otherwise are required to comply with sunshine. But I suggest to you, Bob, it isn't only the inner workings of the legislature that, you know, are in a transition over COVID. It's the whole state government and city government that's gonna have to, that will probably benefit by using Zoom. Well, absolutely. And Jay, after we did the arbitration seminar, people, attorneys talked to me and said, you know, it's gonna change legal practice as well because instead of flying to say Los Angeles to take a deposition, they're gonna do it by Zoom.  And I think it will be just as effective. The other area that I think more and more organizations in a health area looking at is telemedicine, telehealth. We have looked at that for years here because of the neighbor islands. And I think now it will expand. You know, you have to have certain protocols in place but then I think it will definitely start to take off. Yeah, and these things will, they will continue after COVID is reduced.  It's limited, but you can have telemedicine. And I agree with you as time goes forward and you comply with the protocols and they develop systems and best practices. We're gonna do more and more medicine, more and more personal medicine, seeing doctors and the like electronically. Oh yeah, I think so. I think so, and you know, all the records and everything now are all done electronically. Yeah. So, you know, it's just a matter of time before you perfect certain things and move forward using technology. So, you know, I think this is the right time. This is a perfect inflection point for Pacific Law Institute. You know, it's very important that lawyers and all licensed professionals stay current on their topics. You know, I think for a long time it wasn't required and now it's required and that's a good thing. And you wanna know when you go see a lawyer that he's, you know, at least or she has had some exposure to the latest developments and, you know, can wrap around, you know, changes in the law. And the only way to do that is the training. That's why CLE is so important, not only to the lawyers, but to the public. See, I was involved in continuing legal education, CLE, since 1977. I went up to the law school to teach and I kind of left my private practice of law and I went up there to teach a first year course and more specifically to set up the Hawaii Institute for Continuing Legal Education. And so I ran and operated that Institute for, let's see, about five years and developed all these different courses, developed a little, what I call manuals and so let's assume I did a seminar on collection law. I was there. You were, that's right, I was gonna say and you were a panelist. And so we created a little handout which had forms in it and little explanations and attorneys loved it because I knew from private practice that a lot of attorneys knew how to file a complaint or submit an answer. But after they got a judgment, they were a little fuzzy on how to collect it. And remember we spent time on garnishment, attachment, execution and we had forms and that was a big hit. I remember that first seminar. That was really, really great. But- It was bread and butter is what it was. It's all bread and butter, real estate transactions and a divorce law, how to draft the will and trust. You know, all those basic things that a solo practitioner especially for a small firm needed to know. So that's what I concentrated on. And so Hickel, Hawaii Institute for CLE really became self-supporting. And then after five years, I left to, I still love education but I left to work for then Congressman C. Seftel and for just a little over two years. And then after that, I started lobbying just my chance and in 1984. So I started lobbying for the trial lawyers and then I formed Pacific Law Institute at the same time so I could continue doing seminars because I really enjoyed the education part of, you know.   You're gonna get it, you know, down there and you're gonna tell them what in torts to talk about and what points to cover, what issues to, you know, address. And this is very valuable because the end of the day, it's a much better program that way.  I talked to the panelists, I choose, get their input. But I have, you know, I always consider myself a jack of all trades, master of none in the law. So I have all these experts, you know, telling me certain things. And then I formulate the program, create the program. And then I do a script, you know, by time and I'm the moderator. So I followed the script. They can follow it, but it's meant as a guideline, not a real tightly controlled type of presentation.  One of the first seminars I did was on divorce law and when I was at the law school at Hickle. And I had some of the best divorce attorneys on the panel. And I said, we need to do a rehearsal. I actually did a rehearsal with one of the law students videotaping so they could see how they perform.   You know what that is anyway. And the panelist, the expert says, no, we don't need to do that. I said, well, believe me, it would be good to spend a little time on that because we don't know who's gonna be watching live. And you know, it's important. That's a basic step one. Well, the panelists couldn't agree on all of the law. And that was so funny. So the law student, you know, I knew, I said, turn the camera off because they're arguing up there as to what the law was. And I said to myself, you know, that this is a very valuable lesson to, you know, I felt like at least I was on the right track anyway. So it was kind of fun. But you've been through the wars on so many programs, Bob. I mean, everybody, and it was, a lot of it was in Waikiki hotels for a long time and other venues, but now, you know, it's different now.  It's easier. You know, moving forward, and Jay, you and I have, well, I've talked to you a little bit about it. I'd like to do some other courses, not only for attorneys, but for the public, you know, as a public service. And I think that, you know, I have enough background in certain areas that I could moderate it and I create the program anyway. So in March of next year, I think about doing a legislative update at the midpoint, you know, to inform the public of what's out there, what's on the table, what could pass, what will not pass, you know, what is the state of the financial condition of the state? And I'll get the best legislators to be on the panel. And then at the end of the session in May, maybe I'll do a wrap up. It doesn't have to be long, could be an hour or hour and a half at the most, you know, and I use legislators and they're the ones that talk, I'll lead them on. So those are some of the things that I think about. And one of the things I've always been interested in Jay is climate change for a long time. And, you know, I dealt with Ford Pujigami who was the chief of staff at one point for Governor Ege and was the director of the department of transportation and just a very knowledgeable person and very a doer. He gets it. So he and I talked about climate change a little bit because of the roads. I asked him, what's going to happen to the roads? And, you know, we had just an open discussion and, you know, the question came up whether the roads have to be moved at one point. You know, I'm talking about the North Shore, especially. Yeah. And, you know, that's a serious issue because of the cost involved. And some people feel even Kalana Ono only highway going to Hawaii Kai, depending upon which year we're talking about 40 years from now, the water may become right up to the road. Yeah. You know, who knows, right? Yeah, but I mean, the public may not be fully apprised. The public may not realize this will affect them and their property. And so a discussion, a public discussion over this issue is very important. And it is ultimately a matter of law. And I think that if the public is aware, then they will ask the legislature and the administration, hey, you've got to do something about this, you know? And because it's happening, it just is happening. And, but that's one area that, you know, I kind of like to do a hour half program on getting experts and you can get experts from the mainland. Sure. They don't have to stand up either. No. Well, you're connected to all these judges and law school deans and professors all over the country, Bob. I mean, these years have connected you everywhere and strikes me that if you want to do a program with national experts on issues that are either local where we can learn from other places or national issues that we should learn from other places, we can do that. You can do that because of your, you know, your familiarity with not only the issues, but the people. Yeah, and, you know, Jay, I'm still a commissioner on the Uniform Law Commission. I'm a life member now. And there's a wealth of talent in that conference, in that Uniform Law Commission. Attorneys, academics, judges, you know, are from different states and one person I always wanted to bring here is Robert Stein, who was Dean of the University of Minnesota School of Law and also was the executive director of the American Bar Association for many, many years. And it's just such a talented, smart, brilliant guy. And he has the course on the great cases that were decided by the Supreme Court going way back to the present time. And I've listened to his course because he did a couple of hours at our annual conference and it was just so fascinating. So I never could get him out here, excuse me, I never could get him out here, Jay, because he's so busy, you know, he goes to Europe, he goes here, he lectures all over the place. He's familiar with several United States Supreme Court justices and anyway. So it's really terrific. And a long, long time ago, I dealt with Professor Larry Tribe from Harvard. I remember.  He's an incredible lawyer. Oh, just an amazing guy. And so like these are some of the people that we now can hopefully get because they don't have to necessarily fly here although they would like to, you know, to do a program. Well, you know, a thinking of John Drew Binsky, who was on our three digit program a few years ago, that is lawyers who had bar numbers of less than 1,000, you know, the older generation. And one of them was John Drew Binsky, who has passed since. And his thing was, gee whiz, if the Supreme Court of the state or the United States Supreme Court, you know, makes a decision, we ought to have a public discussion of that. Is it good? Is it bad? How did they reason it? What's the result? What's the effect? Very valuable discussions. And although we get some of that in the press, we never get drilled down in the press. And so unless you read professional journals. And so, you know, one of the things that I'm hoping you will do is have panels of these, you know, high level lawyers like Larry Tribe or any number of law school deans and professors around the country, who can speak to the national issues, can speak to cases that are decided in, you know, in appellate courts on the mainland and educated our people here and elsewhere exactly how the law is doing these days. I think I call that what, C-P-L-E, continuing public legal education. Jay, that would be a very interesting series. You don't have to do it every month. You can do it once a quarter even, four times a year, because as the cases come up by the Hawaii State Supreme Court, or even the Intermediate Court of Appeals, you could pick one case or two and I have a discussion on them and the policy impact and why the court decided it that way. It'd be very interesting for lawyers. And that actually, lawyers would, I think, log on, especially if we gave them credit, C-L-E credit for that. What about non-lawyers? If you have a C-L-E program, continuing legal education ostensibly for lawyers, and I'm not a lawyer, maybe an aspiring lawyer, either a law student or a student of the law, sort of thing, can I come to her? Oh yeah, you can log on. The only time that there's any restriction because of the nature of the program, the Court of Next Arbitration Program trainings that we did are only for arbitrators, not for the general legal population. Well, let's assume I did a course on wills and trusts, right? Doesn't have to be for lawyers. It could be for the general public. Say take auto insurance. A lot of people don't understand auto insurance, the different parts, what you should get, what you should definitely get and why it's important. And that would be a public course, but if I do it for free, right? And an attorney wants to log on, that's fine too, because a lot of attorneys may not know either. Yeah, well, we live in a complex society. It's more complex all the time. And given the changes in our national government and the changes in Congress, the changes in the state legislature, we're gonna see more dynamic all the time. And the average person really has to follow that, because at the end of the day, it means dollars and cents and quality of life and all that. We all have to be interested in what government is doing and what the law is doing and how it's changing. So I think there's a huge amount of material out there for you, Bob. And I think a lot of people are motivated to know about it. Yeah. Well, Jay, I think we're probably close to the end. I wanna thank you for having me on to just expound a little bit on Pacific Law Institute. And I'm looking forward to doing the seminars that are coming up and many more seminars and public forums, so to speak, in 2021. Well, Bob, we're delighted that you're working with us. We're delighted to have you on our shows and cover your seminars and educational programs. On the other hand, I've been a student of your programs since the 70s, so nothing has changed. You have a way of looking at things, you have a way of teaching, you have a special skill. And I think it's very valuable to the community. And I just wanna compliment you on your interest in educating people and your skill in doing that. Not only lawyers, but judges, legislators, the whole community, and you are the center man. Thank you, Bob. Bob Toyafuku, the founder, president, and the leader of Pacific Law Institute. Thank you so much for coming on. I hope to see you soon. I know I'll see you soon. Yes, thanks a lot, Jay. Aloha.


So, I'm going to talk about off-gala extension. I will present this structure. I will also talk about the minimal structure off-gala structure on separable field extension. Then I will introduce normal basis for this field extension and then also application in finite field arithmetic. So, we start with off-gala extension, some background concerning off-algebra, co-commutative of algebra. The important fact is that here I have two structure, a structure of algebra and a structure of co-algebra. So, here I have the multiplication and here the multiplication. Here I have the unit and on the other side I have the co-mortification and the co-unit. I have also an empty pot here and I will say that the off-algebra is, the H here is an off-algebra if this diagram is commutative, which means that which is expressed here. And I also want the off-algebra to be co-commutative, which means that the application of the switch map with the co-mortification is right. We have a simple example. We have the example of off-algebra. Here it is an algebra, as we know. We just want to describe the co-algebra structure. We have the commutification here. Since this algebra here is generated by the element in J. So we just say what is the behavior of our G. G is sent to G times G. And the co-unit is sent every element in G to 1. And it is, the anti-pod sends an element to its inverse. So it's easy to see that we have the diagram which commutes. And since we have this definition for the co-mortification, it is easy to see that we have this variant. The co-mortification is a variant by the switch map. And we have an interesting, which is also basic example of off-algebra. We have the cube root of 2. And we consider the field obtained over Q. We have this linear map and the morphism of this field. They find here. And this is an information which will be easier to clear later. In fact, it is a sort of way to say that Q is the field or a variant of this thing. But we know that this is not Galois over Q. And it is a co-commutative of Galois, because it is, we have, sorry, we have from this under morphism, C and S, we construct H, which is a co-commutative of algebra, because it is a quotient ring, since it is an algebra. And the co-mortification, we just want to define it on the generator. On C, it is C times tensor C minus 3 times S tensor S. And on S, it is like this. The co-unit sends C to 1 and S to 0, the antipode stabilized at C, and sends S to its 2 minus S. So here, the commutativity of the diagram is given by this relation here. In fact, we make the quotient because we want it to satisfy the commutativity of the diagram, the diagram, sorry. And here we have this variant of the co-mortification because remember that we define C in this. And if we switch in C, it's OK. Because if we are C and C, here it's OK. But here we have C, we will get 0 here. But remember that the co-unit sends S to 0, so it's free work. We can talk about of Galois extension. So if I have a finite field extension, L over K, and a group of co-automorphism L, we have a linear representation here. And this linear representation can be extended to a morphism of algebra, which is a morphism of the two states. And we know that the dimension of this group algebra is the order of G. And we have this equivalence. The extension is Galois if and only if. This is an isomorphism. Clearly, we can identify the element as the multiplication by them. So it is included in this. This also includes it. We have this inclusion, but the dimension is the same. So it is an isomorphism. And from this, we generalize by saying that we will have a structure of up Galois here in this field extension. If we have a co-commutative of algebra H, such that the dimension as a K vector space is the same as the dimension of the degree of this field. And we have an up action like this, such that we have an isomorphism here. And it is also important to see that this, in fact, since H will act on L, we have the fact that L is H module, which will be interesting later. So this is what we call an up Galois extension. Sorry, for when you took G in the beginning, it was not like the whole Galois group, right? No. Some group of the Galois group. This one, sorry. You are talking about this G? Yeah. No, it is a group, maybe a short group if you want. But it is just a group of automorphism. This is to do the setting. I just want an extension and a group of automorphism. And what is important is here. Here is here, because this extension is Galois if and only if I say it. In the beginning, it is not the Galois group. So you imagine that I introduce the group algebra and the structure of this because this gives us an example. Here, we want an up Galois structure on this extension. This can be defined by this group algebra because everything is OK, as we said. This is a very easy example. Everything is OK. And here, we have also that H2 is restored that it is an up algebra. And we know the dimension. It is generated by c1, cns, dimension 3. So here, we have this which is a vector square dimension 3. And all of these go in here. We have the isomorphism. And it is co-commutative. Everything is OK. But we can see that, in fact, of Galois, extensions generalize Galois extension. Now, on separable, this is what we want to the topic which interests us. So what do we do? We have a finite separable field extension. So we can take the normal closure. And we know g is the Galois group of the top field, the normal closure on the base field. And we have g prime, which is the top, the normal closure to the intermediate L. And so it is easy to see that. g operates faithfully on this set of coset, coset mode g prime for this map. And the kernel, in fact, the kernel is included in g prime. And since we know that, we know the fundamental theorem of Nara theory, which gives us the subgroup from the subgroup we have a subfield, from the subfield we have a subgroup, this cannot be non-trivial. It is, of course, it is necessarily trivial. So the action is the full. And what is important for the following is the regular subgroup of the permutation group here, this permutation group here, the regular subgroup of this permutation group. And I recall the definition. We say that subgroup H is a regular subgroup. If the stabilizer of any element is trivial, and the permeability, here is H. It is not N. It is H. The permeability of H is the index of g prime. So we can see that any normal complement N of g prime in g is a regular subgroup. Because the stabilizer is in this intersection. But since it is a normal complement, this is trivial, which means that this is trivial. And we saw that here. And as the order of N is the index of g. This is an example. Now, so in fact, of Galois extension theory, start with Sweetler and another guy, I forgot the name. But for the case of separable extension, the work start by Kreitzer and Paragis. In fact, since when we have a separable extension, we can take the normal closure. So they prove that the of Galois structure on this extension can be studied only from the of Galois constructed from the normal closure. This is what we see here. Because the result says that we have an of Galois structure if we have a normal subgroup N of the permutation group, which is normalized by g. And the of Galois structure is given by this of Galois, co-commutative of Galois algebra. And you see, in fact, they use distance theory. We want of algebra for L over k. But defining structure on L, what we do? We go to the normal closure. We take the subgroup, which is the group algebra. And we look for the fix of algebra. This is the structure. Now, it is important when in of Galois theory, they need to count the number of subalgebra of a given of algebra. It is important for them. And here we have a result for that. Here, I give you the way they are counting. In fact, when we have a Galois extension, we have only one Galois group, which is normal. But when we have a field extension, you can have on this field extension many of Galois structure. You can have many of Galois structure. And here it is a way to count what we do, since things are encoded with the subgroup, the isomorphism class of subgroup in the way I just described. We are going to just first do a classification of the isomorphism class. And then we are going to count for each isomorphism class the structure coming from them. And then we have all the numbers, the whole number of Galois structure. This is the result. Here, what we want to do, we have a field, we have a group. The sub-of-algebra of this of algebra are the one coming from the subgroup of the group. There is it by Prespo-Rio and Vela. And then there is a projection between the subgroup of this end and the sub-of-algebra of the fix of algebra. And this is the way, this is yes. For the inebrious side, what is the half-algebra of type N? Of type N means that this half-type N means that, in fact, the half-Galois structure is generated by this of algebra. Because the group algebra is generated by N. That's when an half-Galois structure is generated by this of algebra. We say that it is of type N. And we know that we have a Galois correspondence when we have a Galois extension between the subgroup and the subfield. And it is a bijection. But here it is not a bijection. We are just an injection, not a bijection. So when we have a field extension, which is h of Galois, which means we have a h-option, we say from this half-algebra, we have this field fixed by h prime. And we have a correspondence between the sub-of-algebra to the subfield. And this correspondence has the same properties, but it is not bijective. And now, what we did with greater is to consider the minimal structure, the minimal structure idols for which there is only two sub-of-algebra, the top one and the base field. This is what we call the minimal of Galois structure. And we have this result. So as usual, we have the field extension, the normal closure, the Galois group to this field. And the minimal of Galois structure, remember that we are in the case where the off-Galois structure comes from, off-algebra, defined in the top. From the normal closure. So the n is very important. The n gives us the structure. And now, the minimal one will be the one for which the n has no proper non-trivial subgroup, normalized by g. And we take this particular case because it works. We want n to have no proper non-trivial if we take h to be a characteristically simple group. It works. This is what we describe here. And if we take also n characteristics that we have for h, we replace by the group of automorphism, we have at least two minimal of Galois structure. And here we have only one. And this only one comes from the previous result from Bayard. Because Bayard gives us these properties. We know how to count. And here, if we count, we can identify only one in the case. In the case, remember, in the case, not here. We do not have the case. I will show you the point. Yes, in this case, yes. Bayard continues as well by saying that when n is a Bernstein number, which means that n is prime with phi. And phi here is the other phi function. This means that this work for n prime, we have only one. And this only one is minimal. In fact, it uses the degree prime. And here we have an example to show that from the claim for group. We know what this means, automorphism group. And we see that we can write the Galois group as this semi-direct product. And the conditions are OK as what we just described to get a minimal of Galois structure. We know that for Galois extension, every separable Galois extension we degree to here. Every separable extension of degree to is Galois. This is true for the degree until 4 in case of Galois. Every field extension of degree at most 4 is of Galois. Because we have the case where n is prime. And you have the other case. But in the other case, the normal closure as a Galois group, which can be written as a semi-direct product. So we can conclude. But that is not true for 5. We have some degree 5 extension which are not of Galois. This separable field extension of degree less than 4 is Galois. Sorry? Is it true this way? What do you say? The separable field extension of degree less than 4? We? I think you can find like. It's hot Galois. I'll talk about oxythritol here. Can you understand what I'm doing? OK. Now I see choose a second part of the talk. And I want to talk about normal basis. What is a normal basis? It is a basis generated by an element we have. Sorry, we have a field extension, Galois field extension, which is Galois here, Galois field extension. And what we call a normal basis is a basis of L over K, which is generated by an element of L and the action of the Galois group. That is what we call a normal basis. This means that, in fact, our L is a module of rank 1 over the group algebra, sorry. And we know that we have this theorem, which is here you have the story of the theorem. This theorem says that such an element always exists. It was conjectured by Eisenstein, proved by Hansel. And this is for the story of the theorem. And it's also true for infinite extension. Now, some construction. I decided to show you some construction. But these construction are not mine, so by just to know that people are working on this. So we have an extension of local field, which is parable and total reunified. This is different. This is a fractional ideal of the ring or integer of this top field. And I do not see. Yes, I put the name. Biot and Ender construct, in fact, they give a characterization of elements who can generalize a normal basis by using the group of ramifications and the group of ramifications. And the number, the number, the lower D max here is a lower ramification number. But there are also from the group of ramifications, we have the lower ramification number and the upper ramification number. And this is how we characterize the man who can generalize, who can generate a normal basis. Also, yes, here, now we have the case where the extension, our extension for the literature, is of Galois, the structure is given by an H. And in fact, L is a module of rank 1 over H. So we have also a normal basis in this case. And here, Biot describes how characterize the element who can generalize a normal basis using the variation of the different. And now we have the case of normal integral basis. In fact, we have, again, an extension of locality. What is the definition of normal in the non-Galois setting? In the non-Galois, for example. Yes, this means that, in fact, we found an element in L such that the action of H in this element will give us a basis. That's definition of normal. Yeah. Here we have, in fact, a crater in 2022 describe or characterize how we can construct a normal integral basis. This means the normal basis of fractional ideal. Because the difference is the fractional. It is a variable. And this, it describes how to compute out a explicitly construct normal basis of this fractional ideal. Our construction, in fact, what we do here is to consider one-dimensional algebraic group. One-dimensional algebraic group over a perfect field with positive characteristics. We have t, a rational point on the algebraic group of order n. We have the quotient isogenic. If the fiber above this a is irreducible, then the field obtained from k by adjoining the fiber is a cyclic extension of degree n. And this fiber above a is a divisor which is invariant by translation. In fact, this translation is the action of a Galois group of an element, an element in this fiber. And in fact, since we have a group here, which is an algebraic group, which is a variety, we have some equation. So using the equation of the variety, we can construct a function which will generate this linear space, meaning the function which dominates minus d. We work divisor, we dominate minus d. And this space is n-dimensional vector space. And if the fiber above a does not meet the n torsion, then the evaluation of this basis at any b in the fiber will give us a normal basis of n over k. So I want to show you the argument of the proof. So the extension is Galois because we assume that the fiber above a is reducible, which means that the Galois group stabilize this. This means that this extension is Galois. And remember that our field is perfect, which means that any extension is separable. So it is Galois. It is normal because this fiber is assumed to be reducible. And it is normal because the base field is a perfect field. Now, using these things here, we prove that the Galois group is obedient because the Galois group is defined by a joint element which is in the n torsion. And when we do it for each element, we can switch by composition. We have that it is Galois. And we show also that the exponent sorry divide n. But in fact, what we do here is a kind of commercial theory for one dimensional algebra group. We are doing commercial theory. That's what we do. So what we have here, we have a pairing from this product to this. That this pairing is non-degenerate. That means that we have this isomorphism here. But this group is a cycle of order n, which is the standard algebra group of order n. And cycle, because here it is cycle. Now, we take a one dimensional algebra group. Why? Because we know the classification. We know that over perfect field, we have only three for a fine algebra group, the additive, the multiplicative, and the lucastoride. And in projective, we have just elliptic curves. So we do the work for all of this. That's the result I show you. What do we do? We know the fiber above A. It is this divisor. It is invariant over this, because the support of the divisor is stabilized by this map. And using the equation, as I said before, we can construct the function. But what is the way? For example, if I assume that the algebra group is an elliptic curve, we have this equation. From the equation, we have this function, which is normal, natural, small x and small y. We have the divisor, which shows us that we have this function will have a pole at the unit element. But we have also that x is a pair function, but y is not. So we know that we see that we have the function as divisor. I do put it here. The divisor of this function is the polar divisor is the unit element minus t. And if we consider the action of the algebra group or the action of this translation, we have a system of n function. In fact, this n function are linear independent, because if we assume that we have this 0 linear combination, this means that the function, this function here, we know it's divisor. But it is a divisor of a function on an elliptic curve, which means that the sum of the points in the support, which gives us 0. But if we sum, we have that nB, it is the unit point, which means that B is in the n torsion. Remember that here, we assume that the fiber does not meet the n torsion. So this gives us the fact that it is a basis. And we show we have shown that it is a normal basis. So now I want to show you the properties coming from a normal basis to do finite field arithmetic. But why are normal basis useful? Because if you have an arbitrary basis and we want to do the arithmetic here, we have to consider addition and multiplication addition is easy because it is constant otherwise. But multiplication can be more difficult. But in fact, what is difficult here is that we have to consider the product of the element in the basis. We have to consider this. And to do that, we have to introduce some scalar and see that we have some bi-linear form, CK, who help us to compute the product. Compute the product means that we compute the coordinate of this product in the base from the coordinate of the two components. But the fact is that when the base is a normal basis, in fact, all this bi-linear form comes only from 1, which is C0. And that is why it is important to use normal basis to do arithmetic in a finite field extension. So now I want to share with you some things about multiplication, finite field extension. First, I consider the general setting, which is I have a field, I have a two vector space, finite dimensional vector space, and I have a k-bi-linear map here. And what I want to do is to compute the product. At the end, it will be the product. But here, I start from general setting, just a bi-linear map. And from this model, bi-linear map, I have what I call a pure bi-linear map. It is a map in this field, in this way, to help for which we can compute the image of V1, V2 by using some linear form in this way. And when the phi 1 and phi 2 are the same, I say it is a pure symmetric k-bi-linear map. The k-bi-linear map, the space of k-bi-linear map at this dimension, which means that I do not define. Yes, I define here the algebraic complexity. In fact, I want to compute this. To compute this, I want to use pure bi-linear map. And what I call the complexity is the expression and the composition of this map into a sum of this map. If I have this decomposition, then it's really easy for me to compute. So it's very important to know how to express. And I call the complexity of this map the smallest integer such that this map is rkT. Time, it can be written as a sum of rkT pure map. Now, the symmetry. What I did for the general bi-linear map, I can do the same for the symmetry. But for a symmetric, we have an interpretation. So the algebraic symmetry complexity, it is a decomposition. But if we have this decomposition, this is equivalent to have to map a dash up and dash down, who will go from the vector space to this power of the base field and the power to the base field to this? Such that to compute the product, we just need to apply this map. And here we have some bound coming from what we know concerning pure bi-linear map and algebraic bi-linear map. Now, remember that we are interested in multiplication in finite field extension. We know that every finite field extension is a Galois extension. And the Galois group generated by the Frobenius automorphism. So what we are doing now is to consider a group. We have the group algebra. And we are inside the convolutional product who can allow us to define the component-wise product. And we are here when this will be the complexity. Remember, complexity is the number of pure map in the decomposition. And the diamond sigma is this map here, component-wise, for each part of the vector. And we see that a bi-linear map, where V is a KC, a KC module, T will be said to be C equivalence if we have this relation here. This means that when we place the same element of the group by the first component and the second, we can give a dissolution. And now, equivalent complexity, we define what is pure equivalent, pure C equivalent K bi-linear map. It is a bi-linear map given by the, remember that V is a KC module. So it is also a K vector space. But what we call a pure C equivalent is the case when we have W a vector in W. And we have 5, 1, and 5, 2, which are linear form, but as a KC linear map. We have pure symmetric also. And if we have here the equivalent complexity, in the same way as before, it is a decomposition, the number of elements is a decomposition. We have also the symmetric equivalent complexity. It is a number element in the decomposition, pure equivalent K-linear map. And for the case where L is, for the case when we have a finite extension, we denote by this the linear C equivalent complexity. It is the new QC of N. N is a degree. Now we have some bound. In fact, we have our group C. We have K. And we take L and M to finitely generate a KC module, such that N is free. Since N is free, we can write M as this direct sum with this vector space, W. And we have a KC form, KC linear form, 5, is defined by a family of K form. Why? Because when we apply 5 to N, we have 2, 2, 2, to get this. And in this way, we have also that for this family, in fact, all this family is described by just one element. I remember that E is the unit element in the group. In this way, we have an isomorphism of vector space between the space of KC form and the space of K form. This means that every KC form is described only by one K form. Now for the C equivalent binary element, we have this map here. It is given by a family of map here. But as before, we can generate all this by one. And in fact, we have an isomorphism between the C equivalent binary element and the binary element. And the result is that since this is an isomorphism, and we have this means that to get the complexity of this, of the map T here, in fact, this complexity comes from this, or is bounded by the complexity of this. This is very important for the following. I recall that a C equivalent K binary element here, its complexity is bounded by the one coming from the K binary element associated to. And here, I just to convince you that in fact, symmetric equivalent complexity controls the multiplication. If I know the symmetric complexity and I know a basis of L over K, which means a normal basis, in fact, I can compute, remember that the complexity, giving the complexity means that we have dashed down and dashed this to map, dashed down and dashed up, who help us to compute the product. And since we have the base of the departure space and the variable space, we can use metrics and we can compute the product. The reason we just publish is the following. We have K, a finite field, and prime to the characteristic. We have this Galois-Cover, unit 5, between curve, as we integrate over the fifth K. We assume that the Galois group C of E cyclic, the Galois group E cyclic of order n, we have, you can remind what we did before. We have a point here, a rational point so that the fiber is reducible. And if you have R, an effective user on the, R is an effective user, it is on X. R is an effective divisor, sorry. No, R is an effective divisor on Y, such that R and the action of the Galois group on R are different for every element in the Galois group except the unit. And we have D, a divisor on the curve X. We write R, capital R for this system. It is the divisor generated from R by the action of the Galois group. And this is the inverse image of the divisor D. If we assume that we have this condition and this condition here, then the complexity of multiplication in the degree, because you know that we have the degree and extension of K. We have all the conditions to get this extension and the complexity of this extension, multiplication in this extension, is less than or equal to sigma, where sigma is, the multiplication is this small ring. So some idea for the proof. So we have, remember we are working with curve. With curve, we have the Riemann-Rohr divisor. And in fact, from the definition of the Riemann-Rohr divisor, we can see that this map is well defined because this function is divisor in the minute minus E. But what is E? E is a divisor and here is a local equation of the divisor. So this map is well defined. Here also it is well defined for the same reason. Here we put 2 because we have E to E squared. And this map, EB, the first one, E is subjective. Why is it subjective? Because the dimension of its image is equal to the dimension of L. And this comes from the requirement, the hypothesis. We assume that the degree of D is greater than this. And using Riemann-Rohr's theorem, this means that we have this. But since if we have this using Riemann-Rohr's theorem, this will give us what we want, which says that the dimension of the corner is equal to this. Now we can evaluate at R also. And we have these two maps which are well defined. And this map here is injective for the same reason. We assume in the theorem that we have this condition. But this condition means that we have this. This is some interpretation of the definition of R, D, capital R and E. But when you have a degree less than 0, which means that the linear space associated is trillium, which is just 0, but this is the corner of this map. Since the corner of 0, it means that it is injective. And now what do we do? What we want to do is to compute this product. We have the first fact is that since the order of C is prime to the characteristic of the field, this group, the group algebra, is a semi-simple ring. This means that any sub-module is a direct factor. This means that, in fact, here the corner of E b is a direct factor here. And remember that we saw that E b is sub-jective, which means that we have, in fact, a right inverse. Using the same kind of argument, we have that E r squared has also a left inverse. And when we have these two inverses, we can compute what we want. Remember that we have sigma, the complexity of multiplication here. We have two elements. We have to compute the product. What do we do? We apply, since we can go from here to this, we apply E b star times E b. We are here. Then we apply E squared. We come to this. And that E r squared is a left inverse. And we come back here. And we apply E b squared. We go here. So, in fact, we do nothing. But doing this, we come to this ring here. And in this ring, remember that we have this fact. Yes. That if I have a module, a kc module, which is a direct sum in this way. In fact, the complexity of multiplication in L will be given by the complexity of multiplication in w. And this is what we do. Because what we have is that r is this sum, which means that k r here is a direct sum of small k r. And this means that, in fact, multiplication in here has a vector space r. Sorry. So what we have, yes. This is it. So multiplication according to this comes from multiplication according to this. But when multiplication here bound, multiplication here. Because we have this sum. And it is the same thing we have here. Because capital R is obtained by the action of a group of small r. This means that the group algebra here, capital K of small r, gives us, in fact, capital R is a direct sum of the group algebra generated by small r. And when we have this in mind, we are done. Because when we come here, we want to do the multiplication. But here, we have the two elements here in this ring. And do multiplication in this ring. The complexity is bounded by multiplication is K of small r. We can do it using our hypothesis, which we know this sigma. And we are done. And we come back. What we do is to come back here and then apply eb. This means that, in fact, when we have two elements, we apply these two. And then we apply this. We apply this map here, the dash up and dash down. And then we come back to L to compute the product. But in fact, we can see that we have two maps, two linear maps, one to L, two kc to the sigma. And another map will come, will go from kc to the sigma to L. And these are linear maps. And these two linear maps help us to compute the product. This means that, in fact, the complexity of multiplication in L is less than this sigma. So the question is please. It was tonight. If you take, like, a philic session of finite pieces, say, fq n over fq. So we have sine q in group. So the Garouk group is sine x. In this case, how many of Garouk have in this case? Is it fair? Is it formula for that? Yes. In fact, we have to count. What you are saying is that it depends. If the, what I show here is that if the degree is prime, we have only one. See if the degree is prime. But if the degree is not prime, it can be Galois that have many of Garouk's structure. So for every sub extension, there is one, because it's targeting groups. It is a cyclic group. That's why I say, when the degree is prime, this means that we are a simple group. So no sub, no kind of thing. Normal sub. No normal subgroup, which means that no normal sub, no Galois sub extension. And yeah. I mean, there is an analog of finite fields in the global world, in that cyclotomic fields. Do I understand how to do similar things, multiplications, lessons in cyclotomic fields efficiently, because we know how to do them in finite fields? The question is from the? Is there a passage from finite fields to cyclotomic fields? Do I understand how things, how, I mean, are there similar questions and answers for cyclotomic fields? In fact, I do not see the work in this way. But you are talking about arithmetic in cyclotomic. I cannot answer to this question. Thank you. OK. There are no more questions. Let's thank the speaker again.


the wrong button and, oops. So anyways, like I was saying a second ago, I was on something and I saw this lady, she had a Christmas tree and her name is Metal, or Medley, M-E-T-T-L-E, Modern Design, and she does a Christmas tree. I guess maybe she only goes live on Facebook? I don't know, I didn't do that much homework, but let me just show you one of her trees that I was like, OMG, I wanna do that. But I thought I would give proper shout outs. It's hard to tell there, but it's all fire glass and resin and canvas. And so you guys should go check her out. I know she's on Facebook, beyond that, no idea. So we're gonna try to do like a me version of that. Hello, Christina, how are things out West? Stephanie, JJ, Sweet Pea, how are you doing? I don't know what happened, operator error, which is me, I have errors a lot. So I went ahead before this live feed and made these little clay balls, and then I swirled them around in colored paint. I just did pink and gold, boop, they're still drying a little bit, but they'll be dry fully in a second. And I wonder if it would do better on a canvas rather than a cradle board, cause it'd be probably better for it to dip in the middle so that all your resin will collect instead of like spreading out, but we'll see. I also went ahead and put some something on, there we go. I put some adhesive from Artistic Painting Studio on my this board so that it could go ahead and be drying. Jen Gerhams, how have you been? So many fam in there. How's Missouri? Melly, is it warm? Is it cold? That's not warm, it's Missouri. That was silly. Okay, so since I went ahead and did some sizing on the back of this, you could use a bottle with a lady or a deco sticky pen, but I just used this. And then this is some foil from Artistic Painting Studio as well. It's best to let it set up for a little while. I have learned and then use one of these scrubby brushes. Thank you, TG, for getting me a proper brush cause I was using something way too soft. This is just gonna kind of be a background. I noticed that metal modern designs, I don't even know if I'm saying that right. If you ever see this video, ma'am, your work is beautiful and I'm sorry if I am mispronouncing your name. So yeah, there wasn't anything in the background but I wanted to have something in the background because I'm painting over pieces that I've just had for a while. Hope this looks okay, cause I didn't let it, that'll work. That'll be an okay background, yeah? Sweepy, you have eight degree temperature tomorrow, that's bananas. I know it's been a while, Ginger, how have you been? So it's cute, it's sparkly. I wanted it to have like a brushed look. I wanted it to be kind of spastic. Apparently I nailed it. So next thing is to, I'm just going to let my, I went ahead and used that mixer that we were sent today to mix my resin. I need to remember to rinse off my mixing paddles cause I know it's gonna, I'm gonna forget. 1,000% I'm gonna forget. So yeah, hello Cara, did I say that right? Welcome, glad you caught a live. Okay, so I didn't really watch any of my inspiration person's full videos. I just saw a screen grab of her showing off. Some of her beautiful pieces. And I want to see if I can make my version of her awesome work. I would love to, I'm going to watch some of her videos. They look really awesome. I just haven't had time cause Christmas also, my birthday and Thanksgiving. But I just got in three cases from just resin and two from color passion. So this was gonna be a longer video, but I need to get those bad boys in the system so that I can get orders shipped out. I do have a sale going on right now for one day only. The code is hurry for 10% off because I'm, well, I'm, we are going out of town for Christmas. They're going to my parents house and we, I'm sure Tudor is gonna go be with his family. So I won't have anybody shipping. So today is last day to get an order in if you want it shipped before we head out. Anyways, so here's our this meow. I'm gonna use clear glass. I wonder if this glass that I'm using is too big. I'm using the one inch fire glass that we have on our website. That's crazy, JD. That's way too cold. So the lady that did my inspiration stuff, she does long slender canvases. I would argue that that's the best thing to use because then you don't have to worry about filling all this negative space. Just a thought. If I had more room or more patients or more time, more everything, I would kind of vary how the parts of the tree like are. But since I'm strapped on everything, I'm gonna just do a typical just triangle. That has reasons. I don't know what I'm gonna use for a tree trunk though but I'll worry about that later. So I use my handy dandy mixer that was sent to us to review. I only let it spin once so you can still see some of that berry floss in there from where it didn't quite combine all the way. I didn't get a chance to scrape the sides when it wasn't mixing either. So I'll just do that. A low of eight, that's just absurd. I don't wanna see those temperatures unless I'm actively snowboarding. It's getting me a more popular name while I think it's beautiful. You have a high of three today and that's minus 15 tonight, oh no. Do you wear actual like North Pole clothes, Alaskan wilderness outfits? I would have to. Like it's like 30, no. It's 40 something here when I take Bowie out in the mornings and I have my snowboarding outfit on. It's like 60 right now. Yeah, it's 60 right now. But, but it's supposed to be like 12 tomorrow. All right, I think we have it. The rest of the way combined. I should probably go ahead and put out my little these. I then just rolled it around in a cup with, this was the Montana just MTN liquid gold. Actually, I'm not gonna put those on yet. Okay, I'm going to be very not aggressive with applying this resin because I don't want it to bloob out too much. Bloob out is a technical term. I just want just enough to make my glass sticky so that whatever I put on top of the glass sticks to it. And then the glass sticks to my board itself. Not going crazy. I'm just literally drizzling it on. I'm more aggressive with like drizzling caramel sauce or hot fudge than what's happening right now. I think that's good. Also, I think I made too much resin. I may have to do a larger one for the second. Okay, so now I'm at a precipice moment. I brought some of my green glitter glass. Should I sprinkle some of that on or just do gold? What do you guys think? There are some sparkle. Definitely gonna add something sparkly for sure. I think I may use the green because, what's up Robin? You can't really see the glass anymore because resin makes glass clear. Oh my goodness, this reminds me of those like little Debbie Christmas tree cakes. My sister was a full addict of those growing up. So usually I tell people that you cannot add anything on top of glitter glass. That only goes for the clear. So I'm gonna add a little bit more resin to this.  I'll have to get my little vacuum in a minute. So it's looking pretty different right now. Maybe I shouldn't have added metals, modern designs so that I don't embarrass her by messing up this. This inspired by her piece. But maybe it'll turn out okay. So all this green that's just here will dust off after the resin sets. Glitter glass is awesome. It's one of my favorite things ever. You can get it from me or directly from Myer Imports. Just use code ATD2022 for a discount from Myer on their website. On my website you can use ATD all the time or hurry. That's a code for just today. I thought these gold balls would be brighter. Should have used a brighter gold. Oh well, live and learn. I think I'm gonna do my second one on a black background long. Well, a 16 by 20 canvas. That way I have more room to do design stuff. I need a trunk for some like tea tiny boxes or something. Maybe I'll just do, maybe I'll do like brown glitter down here. Let's just put some resin. It's a geotree. I didn't think about that. And this down here for a trunk, that is a wide trunk. We'll just make them a little presence. Once the resin is cured I'll dust it away. So this one's a fairly much a hot mess. Ugh, got a little bit busy in the background. So let's try this again. Put that crazy person over there. Jeff with the remix. This is an old canvas that we are going to make into a Christmas tree. I think I'm gonna do, what's up Julie? How have you been? I think I wanna do this finer, like aqua sea color glass on this one for some reason. I don't know if it's gonna be a good idea, but we're gonna find out together. Whoa, blue piece in there. Cool, cool, cool. Is it better now? Hi Susan, how work's going good? Maybe I wanna do that. Maybe I'll do the, let me get this. I've always liked the look of those brushed Christmas trees, you know what I'm saying? It just looks like a sketch of a Christmas tree. You guys know what I'm talking about? Am I a crazy person?  You don't have to answer that. Let's just refine my squiggles. Maybe refine the squiggles, but also maybe not room for a star. This is why I don't put resin down before I add my glitter glass because you wouldn't be able to just kinda scoop it around, you have to commit. So this doesn't really make sense in this area, but I'm cool with it. I still like the squiggle of it. All right, I was not such a fan of. So you guys traveling for Christmas or staying home and people are traveling to you or neither? You could be neither. I wish I had some bows next year. Yeah, I like the negative space look it gives too. Kind of a backwards person, am I? That I'm using the tea tiny glass on this one and then the huge mugist glass on the last one. Special I am. Think I can like that. I'm sure I can like that. So now let's put some resin down. Just like before I'm being very specific with where it goes because I wanna keep the black background matte and it will never be matte if I put resin on it. So, trying to be real specific. This, I guess it's fine glass is mirrored on one side so it should give a little extra sparkle but I think I'm gonna add a little bit of our clear glitter glass to it as well because glitter glass is always an option, TM. You can see where I didn't have that much glass down. It's getting dark in those areas so I'll probably like pinch and add some more glass and those little bits. Well, let's wait and see. Maybe the dark areas will look nice. We'll just see. So I think we need to add some to those areas. That's an easy fix. We'll just put a little bit right on top there. Of course, that means that when I add the glitter glass it's not gonna stick right there so we're gonna have to add more, a little bit more resin in those areas and all that overflow glass will just dust off once the resin has cured. Still not sure what I'm gonna do for a trunk or base so if you have ideas, spit them out. So I did these pink ones, see if they look okay. I didn't do enough. I have spares. So I'm gonna take all of my little spare bobbles, put them in this cup, a little bit more paint in it, swish it around, put them on my tree. I'm just using Creotex fluorescent airbrush paint because it was the brightest pink I saw when I was running around trying to get everything set up. So I need something. This looks good. Just coating them. They're still kind of wet from being just fresh clay. So I can just poke them and put them where I want them. I wish I had, you know, the easiest way to do this would be to just take like Mardi Gras beads and cut each individual little doodad bead pearl. You know what I mean? And just color those, but I didn't have any so, oops. Oops. So we're working with what we got and there's nothing wrong with that. I've always loved aqua and like a neon pink together. One of my favorite is color combinations. I think all of my favorite color combinations are some form of a color in the aqua teal family and the pink red family. I feel like I need more little bobbles. Is that even what they're called? Christmas ornaments, bobbles? Words are hard. I think I might steal the gold ones off of the other one and put them on here. Well, I have these ones still. 1000% would have been easier to just cut up. Modeling paste, genius lady. Betty, pure genius. I happen to have some modeling paste. Have you seen those chefs that use like tweezers to add garnish to their, like the final dish? Those like Michelin star chefs. That's what I feel like is happening right now. Although this is in no way Michelin anything. I also feel like I wanna splatter paint in and around it. Maybe I'll do that with gold. Haven't decided. Y'all let me know. Maybe I can do it with Clara. I hope you're feeling well. As do I hope that for you, vamp. I don't have a star that's big. Okay, so what if I just draw a star on there? There's a really big, let me ask Jeff to do it actually. He's busy. Well, you are busy. One more practice. That'll be good. Okay, that looks fun and sketchy, right? Make it a little bit more sketchy. Okay, let me get in the modeling paste. Modeling paste, check. Need modeling paste applicator. I'm gonna use this. Stuff's like spackle. I bet I wouldn't be able to do it like I do like the modeling paste flowers. Maybe it would look more normal if I did this with, I don't know, it's too thin to do it like I do modeling paste flowers. So let's just get a dollop and commit.  That's too big even. Well, it's no going back now. So let's kinda like bark-ish. We'll add some presence to it as I figure out how to make presence. Just swirl it and make it look like bark. Cool, bark, check. When it's dry, I'll just paint it brown. This modeling paste does not take that long to dry, so. So now, feels like you guys are like forever away. I'm just gonna put a little bit of resin on my bubbles and I'm debating on flicking some gold. You guys have any thoughts, concerns about that? And of course, I'm gonna go get some clear glitter glass. I'll have a couple of dots outside of what I wanted, but that's fine. Splatter gold, I think you're correct. I wonder if I can do it with this guy. I always feel like a magical, like fairy person when I do this. Oh! Until my whole magic wand comes out. Still looks fun. Glitter glass, it is. I still love glitter glass. Okay, so if you guys don't know, this is our clear glitter glass. Sorry about that. Definitely just the bump chins in my bed. And this is the 10 gram jar from our website. Now I'm gonna pinch it, I'm gonna just go ham and then whatever doesn't stick, we'll just scoop up for later. I feel like it's missing something out here, but maybe it just needs more splatter. Vibbity, bobbity, boop. Wonder if I could just put some brown on it now or the trunk. Where's the these? Maybe I'll just brush on some like, if the end is two S's, I'm just gonna use some copper and just do a light dusting. That way it keeps the shape of the wood that we put in and it'll stick.  There's still something off about that. I think it needs, I need a lid. No idea, I'll put it right there. I need like more of the blue glass right here, I think. It's a little bit too much trunk in the trunk. Anyways, I'm good with that. So yeah, it would have been better to color the modeling paste first, but what are you gonna do? So, I feel like my star needs something, but you know, what would have been really cool is if I added, I like poked a hole in this and then added some lights, just like one there, one there and like one at the bottom, just boop, boop, boop. Maybe I'll do that after it sets up because I don't want the resin to run through it anyways. So, hope you guys enjoyed this video. A little bit more crafty than I usually go, but I was very inspired by metal modern design. I hope I'm saying that right. Yeah, I'll add lights after it dries. White splatter, I can see white, hold on.  This will give me smaller dots and then I went crazy with it and there are no longer small dots, but that's fine. It went from flurries to almost a blizzard, but just almost though, not like full blizzard. So, there we go. Just like the star, that'll have to happen in post-production.  So, that'll make all this look cleaner in between and then we will add probably three rows of lights and a star with lights. Anyways, if you guys give this a try, make sure you tag us and check out my inspiration for today's video, metal modern designs. I hope I'm saying that right or else I'm gonna feel really bad. I hope you guys have an amazing Christmas. I'll try to go live as often as I can from my mama's house and yeah, you guys have happy holidays, Merry Christmas and we'll see you guys as soon as I get back. We're leaving tomorrow and coming back. Not sure when, after Christmas and then, but I'm gonna try to go live from my mom's because we're trying to DIY some stuff at her place. I may have to redo her bathroom counters, but we'll see. Anyways, you guys have an amazing day and we'll see you guys next time. Till then, be kind to one another because you never know what someone's going through. So you don't have to, we'll see you guys next time. Bye. Here's my Bobo. He said bye too. He just woke up from a nap.


So we're here right in the streets we're trying to figure out how many hardcore net fans know some trivia questions Okay, let's do it. All right. Can I ask you a couple of questions about the Nets? All right, it's a quiz show and you win nothing. Where are you from Chris? I'm from Walden, New Jersey Okay, where are you from? No one cares. All right. No one cares. He's wearing Nick's stuff Who is the Brooklyn Nets all-time leading scorer? That should be another one. Who is the Nets all-time leading scorer? You know this has to be on television, right? It has to be Jason Kidd, I would assume. Yeah, Jason Kidd is amazing as this guy. No. Who is the Nets all-time leading scorer?  Yes. Okay. All the players that have ever played for the Nets are your choices. Oh, man What if I give you a hint? His name is kind of in the name of this city Brooke Lopez Yes, look at you My boy Richard Jefferson. No, I am not the leading scorer though Second all-time and playoff scorer. No one cares. All right. We're here joined with Brian Scalabrini He's a long-term Nets fan that who is the all-time leading scorer for the Nets. I Should know this you should I'll say his name is in the name of the team His name is Brooke Lopez Easy I know it kind of freaks me out too is a little disappointed, but yes, he is our lead. I'm kidding, Brooke We love you. Do you know the Nets all-time leading scorer who that is? It's Brooke Lopez You're the first person to get that right Okay, see this is a man that watches Nets on yes, all right name the five boroughs Manhattan Bronx Queens Brooklyn We're done. We're done to win a three-point contest Joe Harris Joe Harris. Yeah, who on the Nets team won the three-point contest My man Joe Harris. Yes, beep jerky Joe. That's our guy named three Brooklyn landmarks the Brooklyn Bridge workplace center that's a smart one and That showed you how much I know about Brooklyn named three Brooklyn landmarks Brooklyn Bridge the arch One of those who is the former net that is the oldest player in the NBA? Former net and he is now the oldest player He's not he's now the old is the oldest player in the NBA currently my boy, Vc Give me a hint his name is Vince Carter. How's that for a head? Okay, which former net is the oldest player in the NBA Vince Carter what former net is currently the oldest player in the league? Oh, Vince Yes, Vince Vince. You are the oldest player in the league make no mistake about that All right, we found a random fan sitting courtside and we decided to ask some some some Brooklyn trivia Can you name the five boroughs of New York City? No, please Manhattan Long Island Okay, that's perfect that that's actually correct. Do you know do you know what? Net player one the three-point contest Joe Harris this man knows some basketball, okay? All right, do you know who is the all-time leading scorer of the Brooklyn Nets? The Brooklyn Nets are out just of the net of the Nets His name is in the title his Congratulations, I don't know how you came up with that. So hey, what is your name? Mark well, thank you. What's your name Richard Richard? I'm sorry. You're Jefferson. I work for the Nets Yes, you were the guy when they traded Jason kid his numbers went all the way down. All right. Well, thank you for the Brooklyn Nets We love you guys


Categories of control theory are defined as open-loop and closed-loop control. In an open-loop system, the output is not measured or monitored for comparison with the system set point. Open-loop systems are commonly referred to as non-feedback systems. Closed-loop systems, on the other hand, are designed to automatically maintain the desired output by comparing it with the actual condition. Feedback devices are used to control the outcome and are referred to as closed-loop. A closed-loop motion control system is comprised of the following elements. The operator interface allows the user to command signals to the controller. Examples of an interface are industrial computers or an operator interface terminal. The motion controller takes the user input from the operator interface and creates a motion profile for the actuator to follow. A simple motion profile consists of position, velocity, and torque. Position parameters require a start and a finish position to move a load. The velocity profile resembles a trapezoidal shape. In this example, the acceleration, velocity, and deceleration is constant. In the torque profile, acceleration is positive, constant, or negative to meet the target velocity defined by the motion profile. Amplifiers, also referred to as drives, take the motion profile from the controller and generate the current required to drive the motor or other actuator type. The actuator is defined as a motor, cylinder, or other electromechanical device used to create motion. Linear actuators used in industrial applications convert rotary motion to linear motion by turning the actuator nut to move the screw in a straight line. The feedback device, usually an encoder or position-sensing device, recognizes the motor position and reports the result back to the controller. As the encoder rotates, position and speed information are relayed to the controller and compared to the user input.