Так, я думаю, що я початкую після того, що я збережуся, як 5-10 міниці. Так що, вважно, я просто маю вибачити, що я говорив про це, якщо не легко, але, точно, на лише, на який я розумів, що це не випадаво. І потім, в піперівці, які ви можете розуміти, з цих референсів. Так, я говорив про нонлініопаршал-деференціал-еквейшан, які вважаються в форме 1. Мільйонів, литералі мільйонів реалістиків, чи селістиків. Реалістикаль-паршал-деференціал-еквейшан може бути в цій формі. Так, тому рівні дирівативи плюс лініопаршал-еквейшан, плюс нонлініариті, плюс нонлініариті, каже до вважній селістик. З кількою селістиків це ето-оф-т і це ето-омего-оф-т. Зараз є інішальна кондіція. Є інішальна кондіція. І я... Я рекомендував, що ця еквейсія вважає в цій цій формі. Я вважаються на нонлініопаршал-деференціал-еквейшан для функції еф-тк. Я дивився не за чим який співав, бо звідси є відео. Він пам'ятає, що у зв'язку аж я маю фіжи, що має вигадані функціону оператору Л. Луфіжи кулася до Л'Амда-J, Л'Амда-J фіжи, і Л'Амда-J є відео 1, і Л'Амда-J йшли до відео. Л'Амда-J, це завжди так. Звісно, мій звідси так, це пресайсно, що фіжа Грошов каже який має світ і який має контрол. Привідувайте, мій звідси є відео, Л'Амда-J, є відео, який має фіжи, це відео, який має фіжи, а фіжи, аніжи, і який має фіжи, і який має вигадані функціон, який має контроли, або і вигадані функціон, for me it is random force. Bj, Bj, what is my, what is my notation? It's a j, it's a j, it's a j, right? I can regard this, it's a j of t, I can regard it as a control, and then this is what what Professor Agracio spoke about, right? Because you see, you should clearly understand that that phi j, I can regard it as a vector field in the space a, a, a, j as a constant vector field, right? So this is, this is vector field phi j, right? So this is precisely what Professor Agracio called, called, called, called a fine control, called a fine control, right? Then, then I told you that first summation goes from 1 to m. m may be either finite or infinite. Then I said hm, hm, this is linear envelope of this, of this elements of basis, which are involved here, phi j, j from 1, j from 1 to m, from one term, yes? So the noise sits in the space hm, but I study, but I study the equation in the whole, in the whole space h, so and this is how they, how they interact, this is the equation. I make one assumption which is, which is innocent, no physicists will ever, at least for the level, you see, in 20 of 30 years, well, all this will be heavily developed. This assumption, that all it's a j's, they independent identically distributed, they people may, may, may relax it. But now for the moment this is absolutely perfect. So different, so different, different random processes which stands here for different, for different j's, they are independent, identically distributed. It means the following. I take one fixed process, eta 1 omega of t, it's a random process. And I make, and I make, it I make independent copy of it. And these are, and these are, and these are processes eta 2, eta 3, and then eta 4. They are four to describe properties of these processes. I can, I can describe only properties of the process eta 1, because all other processes, they are, they are, they are the same. They are the same. Important quantity for me is quantity b, which is summation of bj squared, which is finite. Which is finite quantity. So it means that, it means that I am, I am spoken about. Okay, well just, we will, we will see a bit later, what does it mean? What does it mean? So then apart from this, rather, apart from this rather, rather, rather innocent restriction. I impose another restriction, which is much less innocent. Much less innocent, but which is not, which is not physically absurd either. So physicists do not really object. But it would be better to, to remove it. So this is the following. I define segments jeta l, which is a segment from l minus 1, l minus 1 to l. Я розумію, що ристрикція у процесі jeta1 у цих сегментах, де l еквону до 1, 0, 1, 2, і так. Після часу я можу рігати ці процеси, як процеси на цих сегментах 1, а цих сегментах 1, якщо 0, 1. Після часу я думаю, що я рігати ці процеси, як випадати ті, від 0 до 1. Після часу я garantити, що ці процеси, як Vis lactaten у цих сегментах TáOL одягурно imaginable дозб devasterooba який був дуже добре, щоб відбувати цю сенсію, з чимось більше фізичним. Тож ристрикцію цих процесів на цих часах інтервіалів, які з'являють у професорі Грошові, вони тоді індепентно, ідентикально дистрибюватися. Спочатку, вона є великі експедиції у процесі також, які випадують ціфактори від фізичних цих процесів. Їто воно. Дякую. Їто воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. Воно. аж аж альжебра, який вирішив з виспецію міжі, з виспецію М. Я сказав, що якщо ми зберемо тривіло, якщо виспеці М є файній, або виспецію, якщо ми звичайно виспецію Тіday, якщо ми виспецію М, то всі виспеці М є асимофіг. В ц manageable example of polling space. So, from the point of view of measures theory, all spaces of measures are the same as spaces of measures on a network. This is deep series, highly retrieval result, Actually, i will not use it, but it is just good to know. happens to be lucky to know this also, what I told that, of course, seems locastic setting. Then we have somewhere here on the corner, we have probability space on which my random variable depends. Probability space is an omega fp. Omega is the space and f is the sigma algebra and p is the probability, if in any measurable map from the space omega to Zoo-ack-a-respondent Polish Space M, which is given Barille sigma algebra. It is measurable, it means that pre-image of every measurable space belongs to this sigma algebra. Then I told that extremely important is the notion of the law of random variable psi. The law of random variable xi, this is a measure on the space m, which is defined by the following relation. This is such a measure, so that the measure evaluated for any set q equals to probability that the random variable xi belongs to q. This gives me a relation that for every random variable corresponds to a measure. This mapping, this correspondence is surjective. This is a very well-known fact, so that if you have any probability measure on the space m, you can find a good random variable such that the law of this random variable is this measure. And due to what I said above, when you do this, you always take for omega sigma 01. Actually, for all of series probability, probability space, which is sigma 01, is sufficient. No other space are needed. So now, why distributions of random variables are so important? This is because of the following formula, which is... Let us take a bounded measurable function defined on a polish space m. We have a distribution of some random variable, which is a measure. I have a function, I have a measure. I can integrate function against this measure. How to do this? This is trivial but extremely important relation. I will give it to the name A, I will refer it from time to time. Relation is obvious, but in some context, it will be not so easy to see that what I'm talking about. So what is this relation? So if I take any function, measurable function on the space m, if I integrate it against a measure, which is the law of a random variable xi, so this is simply I have to do the following. I have to take composition f of xi, and I have to take expected value of this object. Since this is what physicists are concerned with, as I said, as soon as physicists consider any model which depends on random parameter, any equation which depends on random parameter, any function which depends on random parameter, then immediately they have this tendency that instead of considering function like energy, like moment, they immediately have tendency to study the expected values. But for us, this is rules. So what they look at, this is obtained as an integral against this measure. And just a tautological remark that this is nothing but integral over the space of omega integral of what? I take f of xi of omega of omega pd omega, pd omega, right? So this is why the distribution of random variables are so important. Another construction, which is very closely related, and which I will use always.  You can find it in many courses in functional analysis in the section spaces of measures. This is not probability theory yet. Probability theory, good definition, is that probability theory starts when we start to consider independent random variables, right? But this is functional analysis. One more important construction, which will be important for me, is the following. Let us take m and n be two Polish spaces. Let f be a measurable mapping from Polish space m to the space n, right? It means that the image of every barrel set here is the barrel set here. Let us take a measure mu here in the domain of definition. This is a measure in the space n, right? Then given a map f and given a measure mu here, I can construct an image, push forward of the measure mu under the map f. So push forward of the measure mu under the mapping f. This is a measure on the space n, and it is defined by the following relation. It is a measure such that being evaluated for any set q, this is simply measure mu of the preimage. Of the preimage, f minus 1, if f minus 1 in few. If we consider this with this definition, right? If we consider this definition with this definition, right? Then we will simply see, we will immediately see that if I have a random variable like this, that the law of random variable xi is nothing but push forward, push forward of measure p, which is this measure, under the mapping xi, under the mapping xi, right? If I have a measurable mapping, if I have a map that sigma algebras then sets move from one to left, we consider preimages. And measures move from left to right, right? So preimage of berylion set is berylion set here. For any measure here, I can construct image of this measure under the set f. It is very dangerous to violate this rule, because even if f is a smooth map from real line to real line to the real line, right? If you consider here perfect berylion set, for example, open interval, it is not true, not true. That image of open interval is measurable set in the right-hand side, okay? So measurable set under any map f, they move from right to left, measures move from left to right. If you violate this rule, we should be very, very careful. So this is essentially, this is mostly this interesting pathos mathematics, which was intensively developed in Russia, in France 100 years ago. And just notation, just notation. But notation is the following. Notation is the following, notation is the following. Look, if I have, let I have a measure mu in some Polish space, Polish space m, and let I have a bounded continuous function on the space m. This is a notation of the space of bounded continuous function, right? Then I will denote integral, then I can integrate function against measure. I can integrate function f against measure mu. This is notation. Sometimes I'll write a test like this, right? And this is just integral. This is f of u, f of u mu of du. Okay, so this is just, this is just notation. This is just notation. This is just notation, right? Now, now let us go back, let us go back to equation and let us, now let us slowly move in the process of study of long time behavior of solution for this equation. Keep in mind these objects, these objects, which I mentioned to you about. Look, what I'm going to do. I'm going as precisely as in the lecture of Prof. Greshyov. I'm going to cut time x, time x to sigma, to sigma of length 1, of length 1, right? And I'm trying, I will construct my solution step by step. First on this segment, second on this segment, thirdly on this segment, blah, blah, blah, blah, right? And I will try to understand, try to understand what happens. What happens, right? And let me recall you, let me recall you, right? That when I restrict the random process, which is the right-hand side of my equation. To, for example, to this segment, which is segment number two, still I make shift of variable. So I regard it as a function of t, where is, where is, where is from zero to one, where is from zero to one. Another symmetric notation. So you see, eta j, where j is lower index. This is component number j of the process eta. Maybe it is a bit misleading, but hopefully not. That eta 1, eta 1, this is the process eta restricted to the sigma, to the sigma j, j1. So process, process eta, when t, where is here, this is, this is eta 1. This is eta 1. And now watch. For every omega, for every omega, for every omega eta 1, eta 1, eta 1, eta 1 of omega, eta 1 of omega, this is a function of t where zero smaller than t, smaller than 1, right? There for it is very natural to consider to define space E. This might permanent notation. I will use it always. Not, not so many. So E is L2 space on the sigma 0, 1, where it is the space Hm, which is either finite, either finite or infinite notation, right? And then watch. For every omega, j1, j1 of omega is an element of the space, right? For every omega, j1 of omega is a curve from 0 to 1. They will be bounded. So this is, this is L2. It belongs to the space, to the space always. There for eta 1, eta 1, this is a random variable, which sends omega to the space, to the space E, to the space E, right? So this is random process when time varies from 0 to 1. This is the same, this is the same as to say that it's a mapping from omega to E, from omega to E, right? Okay, so this is extremely important, another extremely important notation. Since this is a random variable, I can consider a law of this random variable, right? And I will denote it L, and this is the object, which I will walk with always. So this is a measure, this is a measure in this space E, and E, let me, let me keep it here. E is L2 space on the sigma 01 valued in the space, valued in the space, in the space, in the space HM. HM, right? So these are objects. Now, good and interesting question. Good and interesting question. Now I will put, I will start to put, I will start to put restrictions. Restrictions on the, on the space, on the space, on the space, on this, on the measure and just on the distribution. I imposed already some, already some, some restrictions on my, on my process. Restrictions, restrictions where like this. Restrictions where like this. P1, that the different components are independent identically, identically distributed. Ah, restriction P2, P2, that if I take process, process eta1, I restrict to J1, I take process, same process, this is restricted to J2. They are also independent identically distributed. Restriction P3, P3, this is some non-degeneracy, non-degeneracy assumption, which will be, which will, which will appear, appear later. Appear later, so non-degeneracy, non-degeneracy, non-degeneracy. And now, and now one more, one more restriction. Now one more restriction, so look, so what is this, what is this one more, one more restriction? Did they, did they put it? Ah, eta J, maybe, okay. Maybe, maybe let me, let me put it. Ah, I think I told this, I told this already. This will be restriction, maybe, maybe P0. So as the title of my lecture, of my lecture set, I will consider equations where the noise, where the noise is, is bounded random process. That is, if I take, if I take noise eta omega, eta omega of t, then, then this, then the force, the force, which I have in the right-hand side, the norm, the norm of the forces, which I have in the right-hand side of my equation, they are bounded by some uniform constant c. This happens for all omega, for all omega, for all t. For all omega, for all t, right. Now watch. I denote d of eta1, measure the, the distribution, the, the distribution of the, of the process eta restricted to the sigma 01. This is a measure. This is a measure in the, in the L2 space of the sigma of the sigma 01, 01, right. Due to my assumption number zero, this is measure supported by some, by some set, by some set of bounded functions. By set of bounded functions, I need more. I need more, I need condition, condition number p4. That if I denote, if I denote by k, if I, if I denote by k support, support of this measure, support of this measure, because this is minimal closed set, such that L of this closed set equals to 1, right, support of this measure. Then my restriction is that this is a compact subset, compact subset of the space, of the space E. Of the space E, right. Of the space E. So, so and, and one more, one more notation. So you see eta1. This is the process eta restricted to the sigma NG1, and naturally etaR, etaR is the process eta restricted to the sigma, to the sigma GR. Not so many notations, I think it is. We can get, we can get, we can get used to them. Now some very important point. So look, process eta. I naturally restricted to the sigma 01, because on the sigma 1, 2 it will be the same, on the sigma 2, 3 also the same, right. I restricted this on this segment, and I consider this, consider this is a random variable in the space L2, in the space L2, right. But I told you many times, number of times, that in fact this is a bounded random variable. So for any omega, this is L2 curve, such that for every omega and every t, this norm of this L2 curve is bounded by some constant. So it means that in fact I'm talking about, I'm talking about subspace, subspace of bounded, but bounded curve lives in the space L infinity, right. And I insist that I want to consider the law of this force in the space L2. But it sits in the much smaller space L infinity, question. Why I do not want to define L as the law of the random force in the space L infinity? Guesses, why? Why I avoid space L infinity, despite I know that my force is bounded? Because it is not Polish. Space L infinity is not separable. If I try to use properties of the space of measures in the space L infinity, I will find myself in very serious trouble, in very, very serious trouble. Therefor, what we are forced to do, we consider the law of this process in the measure of the space L2, but we always remember that in the fact that it is supported by the space of bounded functions, this is the best solution. So because of the theory of probability, it is very dangerous to leave this comfortable realm of Polish spaces. As soon as we start to talk about spaces, we are not Polish, we are in danger. We are in danger. No, no, no. This is just abstract equation. For me, if you keep in mind complex Ginsburg-Landau, this is HM. No, this is not. Okay, look. So the force sits in the space HM, which is subspace on the space H. Yes, it equals to H. Equals to H. Yes? No, no, no. This is an additional... Right, right. This is not the end of the story. You're just absolutely right. There will be... I impose restrictions layer after layer. I could... I thought it is... But you're right. This will appear very soon. You will see. Now these are definitions, and now we start to construct our objects. Now we start to construct our objects. Now look. As I said, I will examine solutions for my equation. First on Sigma 01, next on Sigma 02, next on Sigma from 1 to 2, next on 2 to 3. So the basic tool to do this is very natural. This is the shift mapping. Mapping S. Mapping S. Mapping S is the following map. It takes our phase space H, where initial data sits. Right? It takes space E. It takes space E, L2, where it sits right-hand side being considered from 0 to 1. And this map sends this back in the space H, precisely, by the way, as in the case of Professor Grachov. So such that in initial condition you 0. And my force restricted to the Sigma 01. 01, right? So this... Oh, maybe to stress, let me do it like this. Right-hand side restricted to the Sigma 01. Deterministic. Deterministic. So Omega is fixed. Just I consider one equation with some one choice of the right-hand side. It is being sent to the value of my solution at t equal 1. So just this is the map S, where of course U of t is the solution. Where U of t for 0 smaller than t smaller than 1 solves equation 1. Solve equation 1. Solve equation 1. Okay, so given initial data U0, value of my solution at time 1 equals to the map S applied to U0 and restriction of my force to the first Sigma. U of 2 equals to the map S applied to U of 1, right? And with right-hand side, which is the restriction of my process to the Sigma number 2, etc. And just more general, we just recursively have that. When we found our solution at time r, and this is S. S means I am going to use this subject. Then to find solution at time and time r plus 1, I should apply map S at the value of solution at time r and to the right-hand side restricted to the Sigma number r. Where r is integer and just bigger than 0. Therefore, to study values of my solution at integer moments of time, I should simply iterate the map S, right? So to study long-time behavior of solutions for equation 1, this is absolutely the same as to study multiple iterations of the map S. Of course, after this, when I do this, I will learn, I will calculate solutions of my equation also in only integer levels of time. Then I have to understand what happens in between. But this is easy. You can find discussion in the papers which I quoted. This is not a big deal, so I will not specify that. I will not specify that. Okay, so this is map S. What is our restriction on what happens here? One restriction is just general. And I will not repeat it because it is absolutely innocent. This is Assumption 0. The map S is too smooth. Why this assumption is innocent? Because if this is a nonlinear partial differential equation and if you know that these equations will post such that for any solution L0 from H and for many right-hand side, Ita from the space E. I have only one solution that always this map is smooth. This is just a fact of life. This really goes together. When you start to learn nonlinear partial differential, you will see that this always holds. You have technique how to do it again and again. It can be found in the papers. It is in these two or three papers which I refer to. Now I have to impose serious restrictions on the map S and to impose them, I have to define, I have to examine and to discuss one more object. So look, since the map S is too smooth, since the map S is too smooth, here is S, I can consider differential of S which is a linear map at any point, linear map from H times E to Ita, but I also consider, I can also consider differential of Ita, of the map S, evaluated at some point E prime Ita prime, which is the map from E to H, which of course Ita prime belongs to H, U prime belongs to H, Ita prime belongs to E, Ita prime belongs to E, right? And this mapping will be important for me. This is as I learned many years ago from Prof. Grachev. This mapping also very important from the series of control and optimal control. Let us talk about this map. Let us talk about this map. Okay. Let us examine how this map acts. Let us take any psi, which is element in the space E, this is just L2K, right? And let us ask ourselves how to construct D Ita S of U prime Ita prime applied to this psi. We know that this is an element of the space H, but how to construct this element? Let us denote it V. How to find out V? Well, for this end, let us repeat the construction of differential of the mapping S on the second variable Ita. To do it, let us do the following. Let us consider Ita epsilon, which equals to Ita prime, this Ita prime, plus epsilon times psi. Psi, this is this guy. Psi, psi is this guy, right? This is the curve in the space E. This is the curve in the space E. By definition of the second differential, this guy V, the value of this differential is defined like this. This is derivative in epsilon, when epsilon equals to 0, applied to S, evaluated at U prime, U prime Ita epsilon. This is simply the definition of the second differential. How to construct this object? How to put hand on it? How to understand which kind of person it is? Okay, let us consider let us consider let us consider let us consider S, let us consider let us consider this guy. To consider this guy, I have to consider equation star, which is the following equation.  And with the same initial data.  Then if U epsilon of T is a solution for this equation, then V, which I have here, this is simply Ita epsilon equals to 0 of U epsilon of 1 of U epsilon of 1. This is this is V. This is V. V, not 0, this is V. Well, let us let us consider the whole curve. Let us consider value of this curve, not at T equals to 1, but the whole curve. Let us consider the curve value of T equals to D over D epsilon. Ita epsilon equals to 0 to this to this my solution.  Let us consider this curve. Let us find out equation for this curve. But this is easy. Let us simply differentiate the relation star in epsilon and to calculate this derivative when epsilon equals to 0. Equals to 0. Then look, let us do this. When I differentiate in epsilon the first term I have DT U epsilon tilde, right? Because derivative in epsilon at this guy at epsilon equals to 0 equals to U tilde. No, no, U tilde plus, plus effort L tilde is a bit more complicated. I have to differentiate in epsilon at epsilon equals to 0 this object. But to create all the basic analysis this differential of F evaluated at U of T because when epsilon equals to 0 U0 equals to U apply to U tilde. And this equals to Ita epsilon differentiated in epsilon. But this is just psi. Initial condition does not depend on epsilon. So U tilde of 0 equals equals to 0. Equals to 0, right? So we see that this method which we have here. This is the following one. This is the following one. We'll just maybe write it with right here.  Which solves equation to star. Right? Okay. So now we will we are very close to the same panorama of what happens here. So this is my equation. This is time one map along trajectories for this equation. Of course time one map in this setting depends on the right hand side restricted to the corresponding segment. I took out of blue but we will see very soon why it is needed. That linearization of this time map on the second variable is very important. Please note that this is rather complicated map because it's a linear map okay. This linear map but this linear map sends one Hilbert space to another. But these two Hilbert spaces they are very different. Because this is a Hilbert space that is roughly formed by function of T. Because if M is finite for example if M capital is 1 this is simply L2 space of function of T. But this is space of function of X. So this is clever mapping. Clever mapping which transforms function of T to the function of space variable. It cannot be easy. It is not easy. Can we generate some sort of optimistic businessman якщо можна, який ми говоримо. Ви знаєте, що вбачимо ростови ростови. Підберемо ростови ростови на моїй еквейшені. Ці ростови ростови були викими ростовами, а це викими ростовами.伸іш Most of history, англermakers американською яormalською А deserve би нікому Uherput two, because more interesting is result concern on VCA group of restrictions. I will discuss the proof for the case of stronger group of restrictions, but then I will show that it's rather easy to explain what has to be added to couple with this two to handle with this more complicated problem and they have VCA group of restrictions. Так, першу кондіціону є регулярність. Биван без прайм. Наймати цю кондіціону є регулярність. Регулярність. І тут є Биван прайм. Биван прайм. Так. Регулярність є також у босі «Строн» та «Вік». Є банох, чи хілбертспець. Є хілбертспець. Це може бути банох, але це немає. Хілбертспець. Хілбертспець «Ві», який випадав у моєму «Строн» із компакт і дець. І цей випадав є компакт і дець. Комакт і дець. Комакт, компакт і дець. І дець. Що я говорить? Ви пам'ятаєте, що у випадаві комплексу «Гінсьбро-Кландоу» «ХМ» – «Мій Х» – «Соболіспець ХМ». Коли ми випадаємо цей цей рем у комплексу «Гінсьбро-Кландоу», то «Ві» – «ХМ-1». Це типична. Це біць-смуза. Заткнутися біць-смуза. Заткнутися біць-смуза. Це компакт і дець. І зараз віка, зараз сильна реструкція. Заткнутися біць-смуза «ХМ-1» «Ві» – це біць-смуза. Я вже сказав, але випадаєте, що це більше. Є смуза. Бо я сказав, що це біць-смуза таке, але це біць-смуза. Але якщо ми говоримо про нонлініапараболику, то це біць-смуза. І я випадаю, що «С» – «ХМ-1» «Ві» – це аналітик. Це аналітик. Це біць-смуза. Не біць-смуза, це біць-смуза. Бо я сказав, що це біць-смуза. Але, звичайно, ця асамптація біць-смуза сильна. Але біць-смуза – це біць-смуза. У факті, за всі практичні ціпи, це біць-смуза і це біць-смуза. Трактикально – немає ціпи. Тому, якщо ви говорите про функціональні спосіби з якою реалістикою мабі, який був зроблено з реалістикою маніпуляцією. Якщо це – С-1-смуза, це біць-смуза, це біць-смуза. Кондіціон бі-2 це таке. Це випадання. Ність нічого струн-фома і віг-фома. Кондіціон бі-2 – це дисепатівиті. Дисепатівиті це це дуже добре. Це так. Я, дивіться, що каї я діносив вupport на межі ЛК. Це компакт в екрані і ристрикція що зиропіволон в компакті. Ще випадуєш? я можу втискувати рандум форсу. Це дуже важлива кондіція. Ви виймеш, тому що збільшово згадується мовою межі. З цим повідомом я можу втискувати, я можу, це був, що в тисточній кінці є так рівні, як я виш. Це означає, що я можу втискувати рандум форсу. Та технічно, це дуже важливе, так? І від цього, Проте з цього, я вважаю, що є гамма, різніше ніж один. Ніж негативного біта, також два холи. Спершу, вважніше, вважніше від цього, але я притворюю це так. Ніж С, Й та гамма times u, гамма times u plus біта. Гамма times u plus біта. Контракціон, який вважає від диспаціону або іншого. Видається дуже ристриктивно, але це завжди холи для вилпосту на лініопараболіку. Він холи для дименціональних негативних біта. Якщо ви можете виважати, що 3-дименціональних негативних біта є вилпостом, я працюваю, що це також так. Так. Так. Уважніше, уважніше, уважніше, уважніше, який вважає від дименціональних негативних біта для лініопараболіку. Але це треба бути хм, але ти прав. Трібні canonical, також директор. Зараз так, наприклад, есця і біта. Кажеться, якщо я вважію вару вважію вару, то вважене. Гамма. Так, як я уважаю就是 ці іважністи, All my solutions will just exponentially quickly converge to zero. So this is B2. The whole point is the condition B3. So this is innocent. This is just innocent, believe me. This is always true if you are talking about nonlinear parabolic equation. My similiano asked me what happens if it is nonlinear parabolic equation. This is violated. І це grandson. this is always true if you are talking about nonlinear parabolic equation. I am trying to prevent, and I'm trying to prevent, and I'm trying to prevent... B3. B3 is a whole point. B3 is a whole point. This is called non-degeneracy. Is called non-degeneracy. Але зараз не генеральна в сильній формі, а тут генеральна в вікафорі. Генеральна в сильній формі. Для ній у, для ній ета, для спорту, для спорту масер Л, якщо я вважаю, що ця мета, ця ді-эта, ця лінірізація мета, то ця лінірізація мета, ця лінірізація мета, цю обмеку визнати liberated. Це дуже оптимальний контрол. Це пресайсно, що оптимальний контрол втратить. Ви знаєте, тут є саптальний пункт, тому що випадки С. З'являється, що імеч у саптальному пісті С має бути на рівній спеціал. Тут треба денсити, в який є віка-спеціал, а в який є віка-спеціал. Why? Because if I replace here h by v, this is usually wrong. This is the density, which we can really prove for a realistic equation. So with this equation I can prove mixing relatively easy. The problem with this condition is that probably this equation holds, usually holds, if m equals to infinity, but not if m is fine. Not if m is fine. This condition is nice. This is right, absolutely yes. This is linearized controllability, which is the tool to study approximated controllability. The eta of SU eta of E is dense. Right? So second condition is more sophisticated, but it holds very often. It holds for a huge class of forces like this with finite m. For example, for two-dimensional Navier's talks we can take m equals 3. For complex Ginsburg-Landau we can take m equals to 5. The condition, which I will write down here is fulfilled. So look, for any eta of h, for any initial data, there exists subset of good forces, which is big of full measure. Subset of forces of good measure, such that for any u, and for any force from this KAU, the image is dense. D, the linearized mapping, S of u eta, its total image is dense in h. This condition looks not that nature, but this is what we have in real life. And this precisely what quite a number of years ago, maybe ten, Prof. Agrachev explained us how to do this. First it took for me some efforts just to convince him that this is a good question, which was of efforts just to find how to prove it, but then he did it. Okay, let me make it more compact, because I need space. For any u, for any eta, eta belongs in KU. KU, range of d eta of S u, of S u is dense. Now comes the last assumption. This is regularity, regularity of the components of the force. This is regularity, regularity of the force. It is the same, it is before, it is the same. It does not exist in weak and stronger forms, so I write it without separation. Name for this condition is decomposibility. Decomposibility, terrible. Decomposibility of eta1 and non-degeneracy. And non-degeneracy. Okay, so look. Let us consider the process eta1, this one. Let us consider the restriction of this process of the segment J1, because we remember that restriction on the segment J2, this is the same process, but independent copy. Let give to this guy name eta11, which is nature, right? So low index, this is Fourier component, upper index, this is which segment in time. So eta11, this is a random process. This is just a random process, which takes value in the space, in the real line. Then the process eta11, for every omega, it belongs to the space L2, L2, 0, 1. L2, 0, 1, right? L2, 0, 1. Then assumption is there exists a basis, there exists a basis in the space L2, 0, 1, in the space L2, 0, 1, 0, 1. And we call this basis E1, E1, E2, etc., etc., such that, such that I can such that the process eta11 is simply random series with respect to these basis. Random series with respect to these basis. So look what does it mean? It means that eta11 of omega of t, this is summation from 1 to infinity, total decomposition of the basis Ej and now quite similar, here comes Bj tilde, which measures how big is the coefficient of the composition number J and here comes xi j, xi j omega and here comes xi j omega. Here comes xi j omega, here comes xi j. Here I assume that all Bj tilde are non-zero, so we have real non-degeneracy, and here I assume that again is xi j's, xi j's, xi 1, xi 2, etc., а, again, independent, identically distributed, random variables, random variables from omega to real line, to real line, right? Such that, oops, not good. Such that, I will move, I will move here. Such that, such that the law, such that the law of every coefficient of every random variable xi j, it has a density with respect to the Lebesgue measure, it has a density with respect to the Lebesgue measure and this density is supported by a sigma minus 1, 1, supported by a sigma minus 1, 1, right? It means that these random variables xi j, they are bigger than minus 1 and smaller than 1, they are bounded by 1, bounded by 1 in modules. Rho of R is a Leipschitz function, is a Leipschitz function, right? And also Rho of 0 is positive. Rho of 0 is positive. So these are our restrictions. These are our restrictions. This, again, after this, now, with these restrictions, of course, you see, we have to check consistency of these sets of our restrictions. We have to check that this set of restrictions is consistent. Which restrictions on equations, as I said that quite a lot of nonlinear partial differential equations satisfy restrictions b1, b2, b3. What about restrictions before? Restrictions before it is always fulfilled if you take some non-trivial basis in the non-trivial basis Ej in the space of function, namely if you take hard basis. Excuse me? No, no, it's absolutely different. Ej is a function of t, of time, of time. So this is a bit tricky. This is a bit tricky, so my functions are, you see, watch the following. Let us consider space of function u of tx. This is what we really keep in mind, right? Let us consider l2 on 0,1 times td. Let us consider the space of my solution when time is from 0,1 and x is in z. What is the basis in this space? The basis in this space precisely is formed by all pairs of functions Ej of t, phi j of x, phi l of x. Right? These products, when 1 smaller than jl smaller than infinity gives me basis in the space of function of tx. I need both. I need two basis to construct a base in the space of functions of the, in the space of functions of two variables. Right? So this is, so these are, these are the conditions. These are the conditions. They will be imposed, imposed always. So now in words, in words, if this condition holds, then the equation one is mixing. But then first I want to talk, develop language of Markov chains just to define properly. Maybe, maybe just let me, let me state, let me state the theorem right now. And let me just, and why this, just to show you that this is something really important. I will, I will state, I will state a theorem maybe in not quite, in not completely rigorous way right now. And then I will develop rigorous language and I state, and I state the theorem, the theorem properly. Properly. Then theorem. Theorem. So yes, it is absolutely rigorous statement. It's correct. If either strong condition holds, be one before, or weak condition holds, or be one prime before prime, before prime holds, right? If this condition holds, then, then, then. There exist one and only one, there exist unique measure, mu star in the space of measures in H, in H, right? Such that, such that for any initial data U0 in the space H, for any initial data U0 in the space H, if I do the following. Now let us, let us consider this solution, this solution U of T satisfies. This solution, this solution U of T satisfies. Satisfies. Satisfies. Satisfies. Okay. I take my solution U of T. It's a random variable, right? Since it's a random variable, it has a law. It has a law. Law of random variable is a measure in the space in the space H, right? I have this very special measure in the space H, mu star, and this measure converges to this measure weekly, weekly in the space of measures. Precisely it means the following. Precisely it means the following. It is the same as to say. If I take bounded continuous function, if I take any F, which is bounded continuous functional in H, right? Then if I take F of U omega of T, if I take expected value of F of the omega of T, that this converges to the integral of function F against the measurement star, right? And I stress that it holds for every initial data U of 0. Another important, extremely important point here that the convergence is exponentially quick. The convergence is exponentially quick. It's exponentially fast. It's exponentially fast, right? This is what physicists know very well. Just one example. One example is that in physical community they believe that essentially they just agree, so practically everybody. That right language to describe water turbulence is the following. We take 3D Navier-Stokes equation and they believe that it's well-posed, right? We force this equation by random force. This, they would be happy with this random force, right? And then they believe that this system is mixing, that this property holds. And then when you open and if you take this pile of book on the theory of turbulence, the pile like this one maybe ten times like this, right? That all these books are about this measure. So all these books are only about, they call it limiting statistical equilibrium, right? But this is really great, right? So you see, our system forget initial data. Our system, if you talk about statistical properties of solution, then from this point of view, our system forgets initial data. Forgets initial data, right? So this is mixing, right? And you see, and this is crucial. So this is actually, I'm sure, I'm really sure that these conditions be one, they're very close, very close to necessary also. So you cannot, you cannot violate, seriously violate one of these conditions and think that they, I think that if you violate them more or less seriously, then the result may be wrong. Then the result may be wrong, right? And I repeat that the condition B3, this is essentially condition in terms of, in terms of optimal control. There is, this is why, this is why the school is joining these two lecture courses. But there are three ingredients in proof of this theorem. Optimal control, how this condition plays. Technique of Markov processes, which I will explain right now. And the component number three, this is the method of quadratic convergence. The method of quadratic convergence, this is also very, also known, often called, often called as KAM theory. And this I will clearly explain you. Because KAM theory, if it abbreviates for, if you understand, for example, it's in the version of Kolmogorov, it is extremely hard to explain it on the lecture. But here everything happens in such and such a form that it is very possible. So my lectures will be about this. Now I have to explain who are here Markov processes. I will develop the language of Markov processes. And I will explain the doubling approach to proof mixing. Because I did not know if you really have sources, but in his lecture Andrei Grachov explained how to prove mixing and finite dimensional evoking tools of optimal control. My proof is different, is different. I will tell you about different proof of this result, which originated on the approach of Dioblin. Dioblin, Dioblin. If you do not know this name, I suggest to look to Wikipedia. Because it's a person of remarkable and tragic life and absolutely brilliant mathematician. It was a really remarkable French expert in probability, Paul Levy. I think he was the only student of Paul Levy. Maybe Levy has two or three, but no more. Just look in Wikipedia, you will see why. Okay, so now I will start. It is not that a big deal. No, maybe just what is left from this lecture is not enough. I'm starting to tell you about Markov's processes and just Markov's chain. Strangely enough, if you take... No, no, no. It's another story. No, no, no. This, as I told, this is what this tons of books on probability are about. Okay, this object exists. No doubt, they even don't talk. But then they start to how energy is distributed. And they write one volume and another, 10, 50 volumes. So this gives you an object and then you start to study the object. And then you start to write books. Since no proofs are there, so you can write 50, 100, 200 books. Okay, so now... Okay, so now you see. So now we are talking... Now look, so where is this S? I have erased this, which is not a good idea. Okay, so anyhow. So we have this system S. We have the system S, which is iterated iteration, which are made by iterations of the map S. So system S is obtained by the following rule. To calculate U of k, I have to apply the map S to U of k minus 1 and to eta k. This is the fourth restricted to the segment number k. And this is for k bigger or equal than 1. And then U of 0 is given. U of 0 is given. You see, now who is this U of 0, who is given? This U of 0 is who? It is very important. Very important that this given initial data, U of 0 in h, is either a constant, is either a constant, is a constant, a constant, or is a random variable, or is a random variable in the space h, independent from the forces. Independent, independent, independent from from eta 1, eta 2, etc. And this is how it should be. This is how it must be. This is the law. This is the law. If you try to consider initial data, which are not independent from the forces, you arrive at absolutely different problems. This is jungles. But this setting is good and we can slowly move forward in this setting. Right. Now look. Now first trivial observation, which is very important to what follows. You see, let us consider the law of the random variable U k. The law of the random variable U k. Look. This is the law. This is the law of the of s applied to U k minus 1, which is a random object. And to random force eta. And to random force eta. Right. And to random force eta. Ah. Ah. Okay, but now let me... No, no, I have to start with something different. So look at this relation. Look at this relation. From this relation we see that U k.  Is a function... Is a function of... Of what? Is a function of initial data of the force eta 1 up to the force... Up to the force eta k. Right. Up to the force eta k. Right. Function, some complicated function. This is why... This is why it's a big job to start this. But it's a function of U 0 eta 1 eta k. But so it is independent. It is independent. Thus it is independent. It is independent from from eta k plus 1, eta k plus 2, eta k plus 2, etc. Right. You see. So this is... This is where from Markovness very soon will come. Now look. Now this is... Now it's precisely like this. Eta k. Right? Eta k. But now look. It means that this is... That this is the law. It means that this is the push forward under the mapping S of the law of the pair of the pair U k minus 1 omega omega eta k. Eta k. Right? By... From what is written here, from what is written here, the random variable U k minus 1 is independent from the random variable eta k. But this is one of things which everybody should remember. Since the random variable eta k minus 1 is independent from the random variable eta k, then the law of the pair of this random variable is the direct product of the two measures. This measure is the direct product of the two measures. But the measure number E k I gave a name to this measure. This is the measure L. This is the measure L. Right? So this is... This is what I said. So this is... In the theory of probability, this is the common wisdom. So theory of probability... So functional analysis ends. And the theory of probability starts when we start to talk about independent random variable. You see, so this is... This is the vector. This is the vector in the space H... Ой, excuse me, U. U. U, U, U, U. U, U, U of k. This is the pair which belongs in the space H times E. Times E, right? So this is a measure в кIVE, в кive, у кive,damnym applicants of the law, in the direct product of the spaces. Because of independence it is the product of the measure here times times measure here, this is construction how to arrange a direct product of the measure. This is generally and just rather easy. Now let us go here. So this is a mapping S, this is a mapping S applied to the law to the law of the applied to the law of my solution My solution when time k-1 times times times measure l. Look this is explicit formula. It is explicit formula. But it is not easy explicit, it is not easy profic from it, because it is extremely complicated. Because okay this is saying, so we iteratively construct, we see that we can iteratively find distributions of the random variable eta1, eta2, eta3, etc. iteratively using this formula. Using this formula. Therefore, therefore, the sequence of the laws, the sequence of the measures, the sequence of the measures, which is the law of the solution at time equal 1, the law of the solution at time equal 2. The sequence of the measures in the space H. Depends only on. Depends only on. Only on. It depends only on initial data U0. Depends only on the measure L. Depends on the mapping S. What is very important here? There's something very important. Something very important. If the system S is defined like this, and if we assume by assumption that eta1, eta2, eta3, they are independent, identically distributed, random variables in the space of C of E, right? Then, in fact, our solution, the distribution of our solutions in integer values of time. They do not depend on specific choice of these random forces, but only on their law. On their law, right? This is very convenient because at some stages, at some stages of our construction, it will be convenient for me to tell, let us now consider, let us now consider a bit different, a bit different forces eta k. Still that they are independent, identically distributed, and their laws are L. So, for example, OK, right. You see, so this formula, which we have here for every physicist, so what is written here? That the law of solution at time k plus 1 depends only on the law of solution. The law of solution, what is written here? The law of solution at time k depends only on the law of solution at time k minus 1, and on the law, and on the law of the force, which we used to stir our system, right? For every physicist, it means that this, that what we have here, u k omega, 0, 1, 2, і так, is the Markov process with discrete time. The name of Markov process with discrete time is the Markov chain. So, physically, this precisely means that u k, that just u k is the Markov chain. Unfortunately, if you take any book, any text book of Markov chains or Markov processes, you will find the much more complicated definition, much more complicated definition. Because this is a very special. It's an easy case of Markov of Markov chains, Markov processes. So, what I will do next time, I will introduce what is needed from the theory of Markov chains and Markov processes just to follow what follows. Because really, indeed, so this situation, and this is not, by the way, because this book on Markov systems, they are good. Simple, they are they are written to to treat general case. General case is really much more complicated. But this is very easy, very nice and very clean, very clean case of Markov processes. We will discuss them next time, but now I will do something just to finish just to finish our lecture. I will do something simple. I will do the following. So, you see, as I explained, we are free to replace this sequence of random forces to sequence of random forces which are like, which we like. So, then there is one choice of random forces which are like the best. Which are like the best. This is, let me call this canonical model. Canonical model. So, canonical model, now I will show you how to define this sequence of independent identically distributed process of these forces. It's okay, which are really more convenient, which is the most convenient to to work with. Easy. Look, we have our probability space omega, omega, omega, omega fp. Let us take let us take independent copies of this probability space. Take independent copies independent copies of it. Independent copies of omega. That is, consider probability space omega1, omega2, omega3 which are simply independent copies. Omega1 with sigma algebra f1 probability p, omega2 with f2p2 but they are the same. And let us do the following. Let us assume that our force eta k eta k, eta k, which I have here. It is eta k, eta k, eta k, which depends on the variable omega k from the random space, from the random space omega k. Then clearly they are clearly they are independent. Right? Because simply because they are defined on just different probability spaces. And let us simply and of course let us choose this is sort of a copy of the same random variable. Copy of the same random variable, right? So eta k is independent copy, that is what I said. Independent eta k omega k, omega k to omega k, omega k to omega k to r. Independent copy independent copy independent copy of of eta 1. This sounds trivial but this model helps a lot to make everything cleaner, cleaner, more more more elementary and easier. Just one remark, so to say you may say that formally this canonical model does not fit to what I told you. Because I told you that we have probability space omega. We have random variable defined on omega. But now instead of this I suggest you countable many probability spaces and my random variable is defined in this way. Solution is very easy. Let us consider real probability space omega nu. Which is direct product of all this. Omega 1, omega 2 blah blah blah. And with sigma algebra which are direct product blah blah blah. Then simply this fourth number k depends only on the component k of this big probability space. Right? This will be very convenient to write explicitly the constructions which I want to write. But I think I will finish on this. I will finish here because next topic is called preliminaries on Markov chains. And after this preliminaries I will state the theorem in a way which is the most suitable and I will explain how to prove it. So this is the end of the story for today.


The fun is, Hattin' in Maggisites, the black viking is back and ready to react, and where did I get this shirt? I got it from Walmart.  Now, one of you sent me screw attacks, Deathstroke vs. Deadpool, so you know this shit's gonna be funny. Now, I'm only gonna watch the fighting part, cause the video's like 15 minutes long or whatever, so I love screw attack.  You know what I mean?  Who do I got? See, who would I put my money on? I put my money on Deathstroke because Deathstroke is, he's gonna be, he's gonna take the battle more seriously. I think I've even read before he's the only one that's been able to, he's one of the only people that's beaten Batman in hand-to-hand combat before. Yeah, and I remember watching a versus battle where Batman beat Deadpool, so I think that Deathstroke is smarter than Deadpool, too.  As a matter of fact, I'm gonna start calling myself Deathstroke when girls come over and she's gonna be, why do you call yourself Deathstroke? Oh good. Alright, I'm done. Okay, okay, let's watch this. Alright, the combatants are set. Let's end this debate once and for all. It's time for a- Who makes this chop worth ten bucks more than me? Come on. I'm gonna change this. Why did you beat me? Oh, I think most people would ping me more as a Michelangelo. You know, that's definitely- Exactly. Gotta be by you, old people. Snipe, this is great. That's what I was gonna say, it looked like the leg. Yeah. Whoops. Hang on, give me a sec.  Deathstroke's healing factor was perfect for repairing damage, Deathpool's trumped his by being capable of replacing entire organs at a much faster rate. Sometimes the original isn't always the best. Deadpool is just a cut above the rest. The winner is, Deadpool Shaxx. Check out the new expansion, The Dark Below, with all new loot, level increase, new story missions and quests, a new co-op strat, six player reign, and new competitive multiplayer arenas. I'm getting- I can't wait until a Deadpool movie comes out.   I always knew he was perfect for that. Alright y'all, post your comments below, let me know what you think. Hey, I was wrong, it was close, you know what I mean? That's one of those give or take, like it couldn't- couldn't went any way, but hey, I was wrong, so. I opened up to it like a man. One million subscribers. Boo!


Myself, Mr. Akshay Kumar Suvade, Assistant Professor, Department of Mechanical Engineering. Today, we are going to study strain energy and impact loading. During outcome, at the end of the session, students will be able to estimate strain energy stored due to impact load. So, figure shows one vertical bar which is fixed at the upper end and there is a collar at the lower end of the bar. Let P is being the load dropped over the collar of a vertical bar from a height of h as shown in the figure. Now, at this moment, we will take a pause and think about what is the impact load and what are the various examples of impact load.   For example, hammer blow or an accident take place in which the impact load is there in case of accident. So, consider a body subjected with an impact load and due to that, there will be extension in the bar by the amount x due to the impact load P as shown in the figure. Sigma stress developed in the body due to impact load. E, Young's modulus of elasticity of the material of the body. A, cross sectional area of the body. P, impact load. X, deformation of extension of the body. L, length of the body that is vertical bar. V, volume of the body that is vertical bar which is equal to A into L. U, strain energy stored in the body that is in the vertical bar. So, within the elastic limit strain energy stored in the vertical bar is equal to work done by the load in deforming the vertical bar. So, in this case, the load P is falling through height h on the collar and due to that impact, the bar will get deformed by the amount x which is shown in the figure. And therefore, strain energy stored in the vertical bar is nothing but load multiply by displacement. And therefore, load is P and displacement that load falls through height h and due to that extension of the bar take place by amount x and therefore, displacement will be h plus x. And hence, strain energy stored in the vertical bar U is equal to P multiply by inside the bracket h plus x. As we know that strain energy stored in the body U will be provided by the following expression as mentioned here. So, U is equal to sigma square upon 2 E into V, where volume is equal to A into L and strain energy stored in the vertical bar that is nothing but P into h plus x. So, putting the value of strain energy in the above equation, you will get P multiply by inside the bracket h plus x is equal to sigma square upon 2 E multiply by A into L. So, as per the Hooke's law, stress is equal to Young's modulus into strain, where capital letter E is Young's modulus of elasticity of the material and hence, sigma is equal to Young's modulus into strain. So, in this case, strain is nothing but ratio of extension due to the impact load which is given by x divided by original length L. And hence, x is equal to sigma multiplied by L upon E. Now, let us use this value of extension or deformation x in the above equation and we will have P multiplied by inside the bracket h plus sigma into L divided by E is equal to one half sigma square upon E multiplied by A into L. So, left side of the equation multiplied by P to the bracket, you will get P into h plus P into sigma into L divided by E which is equal to sigma square upon 2 E into A L. Divide this equation by A L, you will get sigma square upon 2 E minus P into sigma divided by A E is equal to P h divided by A L. And now, multiply this equation by 2 E. So, multiplying by 2 E, you will get sigma square minus 2 into P into sigma divided by A is equal to 2 into P into h multiplied by E divided by A L.  Now, if you see the left side of the equation, this is the equation in terms of sigma and P upon A.  So, this left side of the equation can be reduced to sigma minus P by A bracket square is equal to P square upon A square plus 2 P h E divided by A into L. So, taking the square root, you will get P sigma minus P by A is equal to square root P square upon A square plus 2 P h E divided by A L.  And taking the term P by A outside the square root and rearranging the equation, you will get sigma is equal to P upon A bracket 1 plus under root 1 plus 2 A E h divided by P into L, where P is load dropper or impact. So, when the extension x is very small as compared to height h, then work done will be P into h. So, as we know work done is equal to strain energy stored. So, sigma square upon 2 E into A L is equal to P into h. And therefore, sigma square is equal to 2 times E into P into h divided by A into L. And therefore, sigma is equal to under root 2 into E into P into h divided by A L. And hence the strain energy stored is equal to sigma square upon 2 E into A L. So, by determining the stress induced due to the impact load, you can calculate the strain energy stored in the body due to the impact load. The material is referred from the book of Strength of Materials by Dr. Avkay Bansal and S S Bhavikatti. Thank you.


Okay, I think it's about time to get started and let me welcome you. Thanks for joining us today. I'm Cliff Lynch. I'm the director of the Coalition for Networked Information and you have joined us for one of the project briefing sessions that's part of the final week of the Spring 2020 CNI virtual meeting. We'll hear three speakers today and I'll introduce the topic in just a moment and when they're done we will take some questions. I would just note that there is a Q&A tool at the bottom of your screen and you can use that to enter questions at any point. We'll respond to them all at the end but certainly there's no need to wait till the end to put your questions in. I also would note there is a chat window over on the side and we will be putting out a couple of URLs during the presentation that may be helpful for some of you. Finally I'll just say that Diane Goldenberg Hart from CNI will be moderating the Q&A when we get to the end of the presentation. Now just a quick word about the presentation which I'm going to be very interested to listen to and I think you are too. One of the great sort of disagreements among CNI member institutions and we've explored some aspects of this in executive roundtables in recent years has been what's the right role of the cloud?  Do you do a balance between local and cloud? What's best situated where? What are the pros and cons? This has been a really lively thematic discussion. We're going to hear here I think from one of the institutions that has been more aggressive than many in taking a cloud-first strategy and I think that we all can learn from their experiences here so I'm really grateful for really grateful to Gene Phillips, Fivensio Calvo, and Lewis Brooks for sharing that experience and that thinking with us and I think this is going to be a very timely presentation as well because I think that in the aftermath of what's happened over the past few months and the pandemic, we are going to see some institutions reassessing how aggressively they want to be moving services into the cloud and shifting off some of the operational facilities risk to vendors. So with that it just remains for me to thank our presenters and thank you for joining us and say over to you Gene. All right thank you Cliff. So I am Gene Phillips and we are here to share our experience and moving to AWS three years in the cloud and so I want to introduce next slide. So I'm the associate dean for technology and digital scholarship at Florida State University Libraries. Lewis Brooks is the head of systems and many other things. Fivensio Calvo is the director of software development. Next. So in the beginning, in the beginning is when I came to FSU Libraries in 2013. Okay our computer room does not look like that but I think you get the idea. We had an environment where we have a computer room in the libraries and we had we started using virtual machines for our critical servers that the campus supports. This is a very expensive solution for us especially as we needed more and more storage. This solution also had some problems with reliability and a lack of a failover option. Our computer room itself was not optimal. We had AC problems. We had power problems. It really it was an office that was turned into a computer room and we had a growing number of services we wanted to offer. We had just recently added programmers to staff and we were able to add sys admins so we had some capabilities in staffing and now we wanted to offer more services. One of the biggest areas that we saw that we needed to expand into is digital libraries and preservation. Next. So that was our atmosphere that we were working in. This is our traditional setup of servers. Our two critical services were our easy proxy server for off-campus access to e-resources. No surprise there. Even more important today. And our web service and we see the web server as the front door to our virtual library services as many and in addition many other things. And next. Okay so we what we did we started looking at the cloud and I have to say that our first sub bullet point storage cost is probably the thing that inspired us the most. We decided that there were many advantages to the clouds but this two cents a gig versus four dollars a gig a month for the campus IT we just you couldn't beat that. The campus IT could not compete. They also didn't have very flexible pricing. They had very it was four dollars a gig no matter what. We looked at it and decided it would cut our costs of 50 percent for the campus hosting service. In addition we'd have more reliability. We wouldn't be working with hardware or networking problems. We would be able to have our servers and our data in multiple data centers with no extra charge. It would have doubled our cost to do it in in the campus IT virtual machine area. We'd have better monitoring of the services in the cloud and of course we wouldn't be dealing with power or climate issues. We also had never really been able to consider disaster recovery or automatic failover really for continuity operations because of the cost and it wasn't offered by our campus and so now we now the cloud had these abilities. We were able to spin up servers pretty darn quickly. Our servers are not housed in Tallahassee which every once in a while there is a local disaster such as a hurricane as we start the hurricane season watch right now and it was inexpensive to mirror servers. Those are the things that we saw as the advantages of moving to the cloud. Next. Oh there's my dogs. Not my dog. The one on the left is Lewis's and the one on the right is mine but that's just a little break and a place that tells me to hand this over to Lewis to talk about how we moved into the cloud. Thanks Gene. After a lot of discussion internally some testing going back and forth between our developers and our sysadmins and management we decided to dip our toe into the cloud. You can go ahead and move to the next slide. Very basic what I'd call a 0.1 architecture cloud based it's recreating a virtual uh recreating a physical server in a virtual environment where you have within one system you have your cpu your file store at your ram networking software everything is in a server. This is really easy to set up. We did a lot of testing with their free tier of service that AWS and many other cloud vendors provide and we we using this model we moved up all of our servers critical mission servers up into the the cloud and in fact our easy proxy server still uses this architecture even though we have transitioned all of our other servers to uh more of a cloud-centric setup. So next slide. One thing I do want to talk about real quick is cloud security because this is they use a what's called a shared security model in the cloud. AWS itself is compliant with a bunch of different programs FERPA and HIPAA being the two that we deal the most with for student and for medical data since we have a medical school on campus. They guarantee that the the layer that's below the applications which includes the the compute the storage database networking virtual hardware all of that stuff is is secured they're responsible for that whereas the customer that's you is responsible for the applications that you run your identity and access management any kind of server side encryption excuse me. So it's a partnership but there is built-in security into the cloud so you no longer have to worry about the lower layers of what you're building on you can just focus on your applications and your data. Next slide. So this is our 1.0 architecture. I call it 1.0 because as opposed to basic moving a physical server in the cloud we did start taking advantage of their relational database service which is a database service AWS offers that emulates a lot of different popular databases. MySQL, Oracle, MariaDB and allows you to to connect to those with your existing scripts that are set up for the for a specific databases but it obfuscates the entire database back in so you don't have to worry about it or do anything with it but we had you know our primary web server and then we had in another availability zone we had a secondary web server we also had a primary search server and then a secondary search server these were all backed up and connected with scripts mirrored nightly. We tried to do it with easy proxy if you've ever worked with easy proxy it's not the most flexible software I'll give it its dues and the fact that it's rock solid for what it does but it's pretty dated and we had a lot of trouble getting it to adapt to the virtual environment so like I said we're still running a very basic architecture with easy proxy next so once we get your servers up into the cloud the next thing that you tend to move into is moving into data in the cloud which is what we did we started up backing up servers those over the cloud made sense and then our physical servers so we'd have them all backed up in the location off campus then a few years ago there was the ransomware encryption scams that were going around so we made a decision sorry my dog's backing in the background we're having to publish them file backups we moved 14 terabytes in a week up to the cloud to protect us from from this encryption scam and we've had it up there ever since it's been slowly growing the reason why we have 14 terabytes is because as you know librarians never throw anything away so then we just moved to staff desktops we just started moving into preservation I say that with capital P because that's library and preservation not IT preservation which basically means it needs to be there forever and we're looking at research data now we also spun up a bunch of new services because it's easy to do new servers such as we became a DPLA hub we set up a Wikipedia wiki server which has been hugely successful with our internal communications and sharing knowledge within the library and our request tracker or RT ticketing system which was originally developed to be an IT ticket system but it has been expanded organization-wide it's now used by special special collections human resources facilities a bunch of different groups next so this is where things started to get complicated because it's so easy to set up servers um back in 2017 you know we thought we had everything set up we had our primaries and our backups but they were all in one excessive they were all in north virginia and north of virginia went off the grid for AWS it was on all the newspapers so then we decided well we should have a separate disaster recovery site and what we did was we mirrored everything we had in north virginia and Oregon which got very complicated because we had scripts not only backing up from the primaries to the mirrors but also from the production site to the disaster recovery site and you'll see that I say this is a hot site because all the servers were up and running all the time we had three years commitments for everything it's very bulletproof again we still didn't have the capability of actually failing over automatically from one system to the other but it I will say that we had two hurricanes come through in one year and while the campus's web servers all went down ours all stayed up so I was pretty happy with that next slide and then things really got crazy because we had not only our primary and our backup we also had servers set up in California and Ohio and Canada which were used for testing or for experimenting by our system men's and our developers we had a lot of new systems as I mentioned but we'd also gained a lot of knowledge this was well two years into production three years into working with AWS because we spent about a year spinning up getting people ready to work in the cloud and we had a lot of new knowledge and a lot of new experience we better understood how things were starting to work in the cloud and we decided that we really needed to simplify and work on a new plan and I'll turn it over to Benzie now who will talk about our 2.0 architecture okay thank you Lewis um yes so as Lewis uh you know said we had been working in the cloud now for a few years and um you know we realized that we had you know we had a few gains we were doing better as far as money was concerned we were providing better services but it was certainly getting unwieldy and we realized that there was much that we could do to improve right and so you know the three-year anniversary was coming up and and this was important because I don't know if you caught this in our previous slide but the way we were doing our budgeting it was the three-year cycle so we were buying our servers for three years and as the new budget cycle came up we kind of thought well you know we wanted to reassess what we were doing and think if could we do this better right so we wanted to optimize for the cloud and so what does that mean right well when you want to optimize for the cloud I think you have to like basically shift your thinking into a whole new way right and and that happens gradually I think I think Lewis had mentioned how we had already started to use RDS and all these different things but I think what happens initially is that you're still thinking in the old way of you know we have a server here and we want to move it over there and you still think you're still thinking in the old you know hardware way of thinking at least that was the case for us but you know with with the we with the three-year budget cycle coming up and and also something that happened was that we had a denial of service attack that brought down the the web server for a few hours we realized that there was still fried a little bit more fragile than we wanted it to be and there was things that we needed to think about that so what does it mean to to the cloud is not just servers I think the cloud is really about the platform right and it's about using AWS and what it all the tools that it gives you so what what we had to do is start thinking about the servers and start thinking about services and this is what happens when you start moving to the cloud that you start hearing all this new words all this new acronyms you start hearing about ec2 and s3 and kms and all of these things right and and it's it's it's a lot to take and and it's understandable that we initially would want to sort of lower complexity and sort of move everything the way we had it before but but as you learn more about it you start thinking well okay you know at first we had we were using ec2 to move our virtual servers we were using s3 and ebs to create disk space we were using we started using the relational database service for the for the databases but all of this is very much one to one it's all like this is what we're doing before and this is what we're doing now and it's all one to one and so what you start realizing is that there are tools that can help you improve and optimize your systems if you think in a different way right so instead of using something like key management service which is what you initially use to give people ssh access like your developers in your sys admins to your system you might start using something like systems manager which is a more advanced way of providing this type of access you might start using things like the global accelerator so that rather than assigning a specific IP addresses to your servers you can provide one IP address to a service in general and then and then redirect traffic from that IP address to any host any number of servers that can live anywhere right so you start sort of like decentralizing your your systems and make it more resilient and make it also more able to be flexible so you you might also start using things like application load balancers and auto scaling and that sort of thing and that also gives you also ways to do things like infrastructure as code right so you start thinking of new strategies that's what you really need to think about is what is a way to think about cloud-centric services that can help you provide services that use only what you need right and that's really the key here when you think about cloud formation about cloud-centric strategies is how can we stop buying servers that are only that are going to be bought only for peak times right so like the old way of doing things whenever we the new budget cycle will come up we would say okay because the website gets at most hits during any period of time would be a thousand hits you know per minute then we are going to buy this that's just an example but we would buy a server that can handle that type of traffic even though that would only happen for maybe 30 minutes a day in the peak time traffic right so for the rest of the time that server was sitting there spinning doing nothing what you really want to be able to do is do things like build smaller servers and then use something like the use something like sorry i'm going to bring it up to the next like to the to use auto scaling groups which lets you build smaller servers and then you can spin up new ones whenever traffic becomes necessary right so so there's there's all these different types of optimizations that you can start to do the biggest one going back to this this side was being able to build our infrastructure as code right so before we would do something where we would build servers as individual sort of like i want to call them beautiful flowers right like you would build you would start up a server you would configure it manually and now that thing lives in a way that cannot be rolled back or cannot be destroyed because it has all the history inside of it sort of living as a configuration the way we do things now is that we create cloud formation templates that allows us to reuse components so that as we bring up new applications we can if we search for an example would be an Apache web server if you need one Apache web server you don't need to reinvent the wheel every single time we can use our cloud formation templates to roll a new one and then as changes happen to that server we can we can keep that as part of the code so that if you ever need to rebuild it again we can destroy our entire infrastructure and rebuild it in in another availability so we wanted to or simply rebuild if we wanted to use another type of software like if we instead of using Apache we wanted to use IIS or another type of web server there's ways for us to now be able to be more flexible about the type of applications we use so that that this is what we're thinking now and this is where most of our services are going we are you we're making it much more easier to manage our services because rather than that each service being an individual thing that lives separately we make it so that we reuse the systems and make it much more flexible and efficient for us to manage so this is where our new what our new applications look like you can see that we're we're thinking about using things like lambda functions whenever you we have things that let's say you have a specific process that you want to run and it only runs for a few minutes let's say you know once a year for example our ATDs we receive those from ProQuest only three times a year and we need some some processes that need to be run at a specific time during those years rather than spin up a whole server there are ways for us to now run applications on a server so this way so that we don't have to pay out for servers and think about configuration at lower layers we just simply write code and run it in in this cloud services so so that that has made us a lot more a lot easier to manage cheaper better and with that I am going to hand it over to Gene for lessons learned and I know we're running a little late here I'm going to try to rush through these things our lessons learned at administrative level were how difficult it is to explain what cloud services is to people who are really thinking about technology as hardware stuff they can see and so first we had to start with library management and talk about things that we really didn't know that much about but talk them into it they were willing to take that risk get campus approval they campus had not had a relationship with a with a cloud vendor at that time they just signed a relationship with AWS last year and they have a relationship with Azure also how to go through purchasing when you can't get a fixed price quote from multiple vendors to do some price comparison and everything is sort of a la carte that's also very very challenging and then and I say misconceptions about the cloud but I really mean concerns about the cloud privacy and cost are the two things we hear the most about we had to rethink how people work together to sediments and programmers it's a very different way of working and as Favizio says when you say that the services is the code that's kind of interesting and we had to rethink that we had to make sure we had funds for training because as we said it's very it's essential to being able to kind of shift the way you look at things next I think I we've talked about the strength a lot I we do have I we do consider the last two the the services are now under our control and we can respond to emergencies faster than we have in the past and our emphasis is on the service and not the OS or the facilities but there are definitely weaknesses and I think this is some of the things that the cliff was talking about in our introduction we're a vendor lock-in you know we've been talking you know most of the things we say about AWS you can get from other cloud servers services now but every one of them is done differently and so we are locked into this vendor at this moment there is cost variability the slide that Lewis showed can show how things can get out of hand and luckily you know by the end of the month that you've done something that's going to up your bill very quickly we're highly customized to AWS and it's very complicated there's a suite of services that you can use and how we mix and match those and use those is probably different in each institution it's not right for all services we were running virtual desktop servers that are on prem right now because that's the best place for them so we can't use those for the cloud and there's a steep learning curve in doing this next and so we have but we have a lot of future dreams we're going to take over our digital asset manager it's consortial manager right it's consortial service right now and move that to the cloud because this is an area of growth for our libraries we are creating a data repository specific to one subject for learning disabilities data we're going to reimagine our whole born digital library workflow we're looking at open publishing virtual research workstations and managing the data the lifecycle of data and research we know we need to develop an exit strategy and that is the heart one of the hardest things and there's always more there's always more things we want to do next okay so we're the star there in florida if you can see the little red star and that's a hurricane coming at us we're not that concerned because our critical services are in the cloud sorry but we're not in that cloud that is a hurricane so are there any questions that we can address i'm sorry about the time not at all thanks thank you for making that distinction gene and wow that's really concerning um i'm glad you got your stuff in the cloud thank you for this wonderful presentation we really appreciate it and it really is very thought-provoking um i see that we already have a question in the q&a box so let me just go straight to that and that comes from mary be mary beth snap mary beth says very informative we're along the same journey i'm curious how many team members do you have so we have currently really three developers and two sys admins is that right guys and then these two managers that are and that's on working on a huge variety of services wow fascinating okay thank you did anybody else want to weigh in on that thank you um oh sorry uh yeah so we one thing that we've really had to do though with with the staff that we have is the type of people who work in this environment you can't it's really hard to hire them and honestly they probably couldn't afford them as a library anyways we've had to develop our own and what we've done is we've taken developers and giving giving them some sys admin skills and then we've taken our sys admins and giving them developer skills uh some are stronger in areas some are stronger in other areas but uh it was the way that we were able to get people to get this done uh in an effective way and it's been very successful for us i'll say yeah i think that's a very good point is that uh i think what we have you what you have to do uh when you deal with budgets like ours is that we hire potential rather than experience we we see people who we who we think we can train and that can learn and then we hire them and sort of bring them up uh so that they can work with us and this new technologies and hope they stay with us thank you thanks uh mary beth for that and she comments spot on on your assessment on staffing so thank you for the question thank you for those responses um from sarvani he asks uh for research lab virtualization what apps have you considered and thank you for the presentation so we we've looked at a number of things but uh uh louis do you want to talk about the whole campus has been looking at how to manage data and uh we've been keeping an eye on a project that's going on with emery and others um to create a um a whole management suite in aws i don't actually know what the what the apps are that they're thinking about louis you um so yeah we uh we've been taking a look at this for about two years uh it kind of started off with us being pulled into a big data research initiative to build some infrastructure for that on campus and then we started taking a look at how we could support being libraries where we're more interested in supporting all kinds of research data um and we um um they're putting data into the cloud uh i know what emery is doing with um their project is mainly a HIPAA compliant way to store uh and work with data in the cloud where they set up an s3 storage bucket and a virtual server that the person can log into remotely and run queries on for different softwares such as r we're taking a look at how we can do that as well to try to to see do it up on a much smaller scale we're just starting to spin that up though okay thank you um i see we we have a few questions in chat uh that um that louis has um responded to so let me just read those out to be sure everyone knows what those are um let's see how frequently do you back up the servers and data what about the network infrastructure and costs to support these transfers can you please talk more about that the reply is we back up nightly there's no cost to upload or move data within an availability zone there's a small charge to move data between availability zones such as Virginia to Ohio we feel it's worth it and it goes on to say the big cost with data movement is out of the cloud but there are ways to minimize that as well um and also let's see Martha Anderson asked are there any other platforms or programs you're considering now for your exit strategy strategy from aws the reply from louis is while we're not planning to leave aws we are looking at what it would take there are other options the other options would be microsoft azure or google iCloud um and then we have a question from cliff cliff asks are are you part of an institutional aws contract or are you on your own contract so we um we were the first uh official contracted at fsu with aws um the institution just started last fall with a contract that they're writing from a texas university texas contract we haven't joined that yet and we're not we just we've had some conversations with them they're just starting off on aws the very beginning of their journey they're trying to do disaster recovery there um and um they're really just getting started so we're not we're on our own right now interesting um i will say that we're looking at it um mainly uh for simplification uh we've had and for getting the bills paid on time we've had some issues with purchasing paying stuff on time uh and getting things taken care of since this is a new way of doing business uh one of the things that we do that are i personally like about the uh the campus um or the agreement at the campus level is being able to pass through costs so if we have a researcher with a grant who wants to set up a um something like ld base or they want some research storage and a website uh we can pass that cost down directly through to their billing without us having to handle the money or anything it's all taken care of at the campus level great thank you we have a question another question from Mary Beth snap have you experienced resistance or hesitation to having master digital objects i.e preservation capital p in the cloud versus locally we're really getting started on this process right now we did as as louis said we took our highest um level of preservation need um and we moved those over our electronic theses and dissertations and we moved those into glacier and uh we're now getting archived matica set up which is a tool to to manage your preservation um of your objects and and we're trying to understand what tools we're going to run we're setting that up in aws right now and we're trying to figure out all the the dependencies because um so and it's ours it's our people managing the digital library and the digitization that are interested in doing this so i don't think we have resistance because we haven't figured out a reason not to do it we will be doing checksums etc aws has tools to do those uh native tools and then um we're trying to figure out the best fit for all of us great thank you so much thanks for that question and thank you for those answers uh a lot of great questions here today um really appreciate everyone being here and um to our panelists as well we're still around to please uh stay with us if you would like to chat with jean favenzio or louis if you'd like to make a comment or ask question uh all you need to do is raise your hand and i will unmute you take care everyone bye


I want to take some time to talk about Dr. Dean Waldman. Now, you may not know who this individual is, but he is a medical doctor, he's a professor, and he is also a published author who has written extensively about the U.S. healthcare system.  He must be an expert. And he certainly thinks that he's an expert and he thought that it was really important that he share his expertise with all of us. So he penned an op-ed for The Hill and it'll kind of tell you the extent of his knowledge. I'm just gonna read you the title. You ready? Quote, prepare for buyer's remorse when Joe Biden and Kamala Harris nationalize healthcare. A so-called expert thinks that Joe Biden is going to nationalize healthcare in the United States. I'm not kidding when I say there's a greater likelihood that Trump will nationalize healthcare before he leaves office. To think that Joe Biden is going to nationalize healthcare, that's not just laughable, that's delusional. Like you're in the same delusional territory as the MAGA chuds who think that Donald Trump actually won the selection. Like that's a joke, it's laughable. Joe Biden is literally the candidate that the health insurance industry bet on during the Democratic Party primaries to save them from the popularity of Medicare for All. So I mean, this really goes to show you how delusional right-wingers are and we don't have to be more charitable here. We don't even have to go beyond the headline because I think it speaks for itself. Nonetheless, we're gonna be kind and let's just hear him out. He writes, nationalization of healthcare will involve some combination of the Biden healthcare plan with Bernie Sanders single payer Medicare for All, Vice President-elect Kamala Harris strongly supports Medicare for All as detailed in HR 1384, a February 2019 bill that was never brought to a vote in Congress. While there are some technical differences between the two plans, there are numerous common elements. The overarching theme is the federal choice replacing individual or patient choice. The very first word of the Biden plan sets a tone for Americans dependence on Washington. The federal government quote gives, despite Biden's campaign promise to the contrary, private health insurance policies will cease to be available, I wish. Section 202 of Medicare for All prohibits private insurance by law. The Biden plan would create a public option which is a taxpayer supported government insurance company. By low balling prices, Washington will drive private insurance companies out of the market. Either way, Americans will have only one insurance option what the federal government decides for us. Now the article goes on a little bit longer but he basically just talks about why he believes nationalization is bad. But he kind of has this thinking that he has already established, it as a matter of fact, that Joe Biden is going to nationalize healthcare in the United States, which is why he goes on to explain why that's bad. But that's all the evidence that he presented that Joe Biden is going to nationalize healthcare. That's it. He supports a public option. First of all, we don't even know if Joe Biden is really committed to a public option. Second of all, if Joe Biden actually does go through with a public option and we get a public option, there's no guaranteed that that's going to be a robust public option that actually offers buyers a lot. And if he does offer us a public option, that isn't nationalization. That's an attempt to save our capitalist system as it is. Really what a public option would do in actuality is it would drive the sick, disadvantaged people onto the public option. And a public option would be underfunded and overburdened because private insurance companies would market cheaper plans that are less comprehensive to people who don't need healthcare, people who are younger and healthier. And as a result, the so-called public option will ultimately fail unless it is constantly updated and tweaked. But we know in our late stage capitalist society, Republicans will do everything in their power to strip away benefits from it and it won't last. So the fact that he thinks that a public option will basically act as a gateway to nationalization is a joke. It's a joke, even in the best case scenario for a public option, if it was the best, most comprehensive public option imaginable, immediately when Republicans take power, they're going to chip away at it. So it's a joke.  Kamala Harris' plan is not real single pair. It's a fake Medicare for All bill that was created to placate members of the left who she pissed off when she moved away from Medicare for All. But he says that the reason why Bernie and Kamala's plans are the same basically is because the overarching theme is the federal choice replacing individual or patient choice. What does that mean exactly? Choice for what? What choice is being taken away from us because we don't have choice currently? We have a limited array of options if we're lucky enough to even get health insurance and we don't get to choose who our doctors are. That's restricted to whoever is within our network and offered to us. Furthermore, we don't even get the choice of choosing everything that's covered unless we want to pay an arm and a leg. So the choice that he's talking about is really your choice of private insurance but actually what we want in terms of choice in healthcare is to choose our doctor, choose which hospitals and doctors offices we can go to. If everything is comprehensive and covered, that's when our choices as consumers are maximized but this dunce, he doesn't get it. He's supposed to be an expert but the things that he's saying, these are like basic things that he gets wrong.   Is that so? Why don't you talk to anyone in the UK? Just pick anyone off the streets and ask them if they have buyers remorse about their national healthcare system. If you try to take away their NHS system, they would riot. So it's not going to be the case that Joe Biden gives us a national healthcare system. I wish that were true. I mean, he would be awesome. I would love Joe Biden if that were the case. But he's just a neoliberal, like any other Republican when it comes to healthcare. I mean, he might not want to take away healthcare as much as Republicans, but he's not for offering comprehensive healthcare. But if you ask someone in the UK or you talk to someone in Canada about their single payer system, there's no buyers remorse there. It's incredibly popular. And this guy is stupid, but he's got to know that that's why he wants to stop it from happening because if it did happen, there would be no buyers remorse. Once you give people healthcare, taking it away from them would be extremely unpopular, which is why there's always backlash whenever we talk about or we hear about cuts to Medicare being talked about. So I don't know how a so-called expert could be this stupid when it comes to basic facts about Joe Biden's position on healthcare and our healthcare system overall, but it's entertaining at least, I guess.  Get nervous, man.


The U.S. military said that it had seized a boatload of advanced conventional weapons sent from Iran to the Houthi rebels in Yemen on January 11, 2024, while conducting a flag verification. U.S. Senkham Navy forces conducted a nighttime seizure of a Dao conducting illegal transport of advanced lethal aid from Iran to resupply Houthi forces in Yemen as part of the Houthi's ongoing campaign of attacks against international merchant shipping, the U.S. Central Command said on X, the seized items. The propulsion, guidance, and warheads for Houthi medium-range ballistic missiles and anti-ship cruise missiles, as well as air defense-associated components, Senkham said initial analysis indicates these same weapons have been employed by the Houthis to threaten and attack innocent mariners on international merchant ships transiting in the Red Sea, it said. This is the first seizure of lethal, Iranian-supplied advanced conventional weapons to the Houthis since the beginning of Houthi attacks against merchant ships in November 2023, the statement noted.