What's up everybody Jason here for Jasper's Case Breaks.com. We're gonna wait three 2009 draft spots in this 2020 Hennini select hobby one box break Random teams number three and again guys the way this works is first. We randomize the customer names and teams everybody gets a team of MLB Break open the box distribute the hits out sure we're no better in commons will ship out of here Then at the end of the break will take all 30 customer names Randomize them only and the top three at the end of the randomizer will win spots in this break Right here, which we're pretty close now guys Three full spots left one more filler after this and that's it I have a chance to get a get a trout so Actually guys, I wanted those car show these so these base Mike trout bowman's first Graded PSA 10 or 9-5 is there almost like $25,000 now So even at that price if we were to hit one in a pack We'd be pretty strong So it's pretty wild. So let's do this break first guys Roll it select and it's a three and a six nine times. Good luck one two three four five six seven eight and nine Matthew Donokobi All right nine times here one two three four five six seven eight Twins down to the Astros All right, so quick little one box break with no trades, but I'll let you look at your teams though for a quick minute See who you guys got In the meantime, I'm gonna try on the camera and start ripping the select box really quick All right, so let's quickly go through this box Joe del silver here He has money grand all and Abraham Dotto to 99 little patch auto Paul Goldschmidt little jersey up and We got a Vladimir girl junior red to one of your nine Cavendizio and we got a gold Robin Bologna For the Padres six out of ten and we got a tricolor Anthony Rendon your Don Alvarez And We got a Javier Baez blue little color match to one 49 Luis Robert ready Freeman John means And Louis Thorpe Jersey relic Mike Trout See something nice coming up Juan Soto and wow, this is a cool one coming up Arsidis Aquino It is a white Number 250 Premier level 19 out of 50 Cincinnati Reds with that one. It's Christopher I Sweet sweet so now let's switch scenes Give away those couple spots in the draft 2009 All right So there you go dice roller here's the list of customer names again Good luck Let's roll it It's a four to four eight times top three get in one Two three four five six Seven and eighth and final time Eight times Top three Brandon Vassilis and Vassilis Wow, I think Vassilis claimant came top two with his only two spots Crazy, there you go Congratulations to the winners up there one more time to what I think we need to do it to unlock it So let's get it going guys appreciate it


Can everyone hear me? OK, great. So welcome back for the second establishment of this lecture series. So let me recall briefly where we were before we started and tell you where we're going. So last time, we sketched the characteristic P version of this sort of motivational result of bot yanger and maw, where we showed that if you look at a netherian ring in positive characteristic, an analog of Koons' theorem is that it's regular if and only if it admits a faithfully flat map to a perfect ring. And so the goal now is to give a generalization of that or a twist on that in mixed characteristic, if you will, to show that if R is netherian and P is in the j, it's in radical. So either P is 0 or maybe you're in mixed characteristic, then R is regular again if and only if there exists a faithfully flat map to what's known as a perfectoid ring.  And so in some sense, the point of this talk is to do that. But I want you to keep in mind, like I ended the last talk with, the easiest cases. So if your characteristic P, a perfectoid ring is the same thing, it's just a perfect ring. And then the interesting ones for us in some sense will be the sort of orthogonal direction. Let's say A is P torsion free, then the easiest case to say here is that A is perfectoid if and only if it is P complete and separated. And P has a P-th root up to a unit inside of the ring. And when I look at the Frobenius map, after getting rid of that P-th root of P, what I get is an isomorphism. So the Frobenius map on A mod P is surjective and the kernel is generated by pi. Is everyone happy with where we were? And so here's just a couple examples to keep in mind in both of these fall into this last framework down here. Just to give you a flavor with what these rings look like. So here you could look at, take the P-addicts, adjoin infinite P-th roots of P, and then P complete. Or for instance, you could take a nice mixed characteristic regular complete local ring. So the P-addicts series adjoin say x to up to x d. So d is the dimension of this ring because P is also a parameter in here. Now adjoin the infinite P-th roots of P and all of the x i's.  And both of those will be P-torsion free rings. So you can check what happens when you go mod P and compute the kernel Frobenius to check the perfectoid using this easy criterion down below. So but to give you the full generalization or the full statement here, a perfect means that the Frobenius map is an isomorphism. Isomorphism. So the traditional thing is to say that a ring with a surjective Frobenius map is semi-perfect. Any other questions? So let's get to the definition then if we can. And what I need are the Witt vectors. So to start here, let's say that R is any ring. So the Witt vectors, so the construction is maybe not so enlightening when you first see it. But I still think of it as the best way to start off. So is this out of all countable tuples of elements of the ring? Let's do. So if you will, just a countable product of things in there. I didn't write product for a reason, namely the addition and multiplication on these tuples is not going to be the usual addition and multiplication. So the twist is that addition and multiplication are determined by universal polynomials in the entries of the sequences so that the following maps are ring maps. I'll write these as capitals. So start off. For each fixed i, look at x0 to the p to the i, then drop the exponents by 1, put x1 to the p to the i minus 1, and add a p out front and make a pattern like that and write those out. And what you want is that such expressions give you a ring map. So these are called the ghost maps, so for all i. So knowing this, so if you believe that such a thing works, you can work out what the universal expressions have to be. So let's do just a couple of them. And I've written the x's here and tried to write capital x's because I want to think about this as just these xi's variables. And I want to force this to be a universal expression that forces addition and multiplication to be satisfied. So for instance, if I write down a tuple of the x's and a big sequence of the y's, I know I have to get some other tuple, say, with the s's. Well, if I start off here with i equals 0, that map is just the map which sends x0 to x0. And I want that to be a ring map. So the easiest thing to say is that I better have the map applied to each of these x or plus y0 equals the map applied to this guy. And similarly, I have to do the same thing, say, for i equals 1. So with i equals 1, I'll get an expression that looks like this. And the exercise is to go through and recursively solve these formulas to get out what the formulas for the s's have to be that force this to be an additive map. So for example here, so we already know that s0 is x0 plus y0. So I can plug that in over here. And now I have some formula that only involves s1 and polynomial in the entries of the x's and the y's. Now, if this is going to work, for any ring, in particular, it had better work in the case when my ring is, say, p torsion free. So when you solve for this and throw everything onto the other side, you'll find that you have some polynomial which has a p in front of all the coefficients. So let's just do that. I get p s1 here is p times this guy over here. And the whole point is, if it's going to work for a p torsion free ring, then I notice this expression has p's in all the entries after I do the expansion. I do that formally. And I just divide by the p's to get a formula that works for s1. Is everyone OK, or have an idea of how I got this formula over here? Similarly, I could do the same thing with multiplication and recursively solve, again, the first level. I know projection onto x0, projection into y0 has to equal projection over here, has to equal t0. And on the other hand, I have some product expression, which again, when I put in x0, y0 over here, that gets rid of the t0 and I can solve for t1. Just spit out an expression, which will tell me how to write down a polynomial for t. So again, the point of this is just to say that the traditional approach to exhibiting the vectors is just to say that, well, I have these vectors and I'm forcing each of the maps that look like this to be additive multiplicative. So after checking that this gives me a ring, I also have the first map that I forced, gives me a ring homomorphism from WR back to R, and just I see directly that this thing is surjective, because I can put any sequence on the left that I like, and these two things together are pretty easily seen to be functorial in R. So everything just involves these universal polynomials with coefficients in Z. And I would say the same thing, well, maybe I won't write it. I could also talk about the truncated vectors, where I only pick out certain numbers, so for instance, W2, where I only use the first two entries inside of this thing. That will give me a similar functor, W2, which is also functorial in the ring that I write down. So in particular, using the functoriality, if you know that the characteristic of R is p, well, so I have a ring map from the R given by R goes to R to the p, that lifts to a ring map upstairs, which on the entries just gives me the p powers. They just apply the ring map down below that commutes with the usual forbenius downstairs. In particular, note in the same vein that if forbenius is an isomorphism, IER is perfect, then the lift of forbenius up here also gives me an isomorphism for the same reason. So the inverse map will also give me the inverse. So with this, so the real magic happens is that when I input a perfect ring, what I get is really quite nice. WR is the unique p-complete and separated p-torsion free ring with a surjection down to R so that the kernel is generated by p. So it really has a strong uniqueness property here. So if you like, you can prove the uniqueness without going through the construction, just assuming that such a thing exists, in fact. And that will be implicit in much of my presentation later. But if you believe such a thing exists, you could also imagine trying to come up with formulas that define it. And of course, let me tell you a couple of other properties about this. So before we move on, the weight vectors come with a shift map, generally referred to as V, for Vershibang, German word for shift. So which just sends a tuple, sticks a 0 in front, and passes everything off to the right. Now, this is just an additive map, not a ring map, not a multiplicative map. But it has the nice property that when I do this shift, what I get gives me, in either order, f of v or v of f is multiplication by p on the weight vectors. So again, if you believe here that the kernel here was generated by p, obviously, the image of v over here is something so that if I project onto the first coordinate, I get 0. And this is somehow saying that that is related to how the formulas behave with respect to Frobenius. So the other thing I sort of need is the Teichmeler map. And I'm going to give a, we've done a little work already related to this in the first lecture, although you may have missed it. So you can define the Teichmeler map in the following way. So I'll denote it with just a bracket. Well, since R is perfect, it's isomorphic to R flat from the first lecture. OK? Now, our definition of R flat for an arbitrary ring was go mod p and then take the inverse limit over Frobenius. So this is just definitional. This is because R is perfect. All right? And we constructed the flat in the first part of the lectures. This musical notation came together with a sharp map. Whenever I had a ring that was, say, p complete, all right, got rid of the flat. OK? So the Teichmeler map here is exactly this string, right? So where I have a map from R back up to omega R. OK? And again, the key point of this thing was that this was multiplicative. OK? But you can explicitly write this down in coordinates as well. The image of little R gives me the tuple, which is just R in the first entry. OK? And in fact, you can show more generally, right? So that the image of sharp is always the set of, as we talked about last time, the set of elements that have compatible p power roots. So this precisely gives me the elements of WR that have compatible systems of p power roots. OK? Is everyone happy with this expression? OK? So let's just note a couple of things. So first off, well, if I stick R in the first entry and then I get rid of all the other entries, I get back R. I started off West. So in particular, pi is injective. As we said, the map here is multiplicative. More generally, you can show that it's easy to multiply times any sequence in the following way. So what I get gives me, right? So point being that while multiplication in general is complicated inside of the wet ring, if I give you a representative in terms of the dichmular map, it's easy to multiply times the tuple. OK? And in general, you can use it to show many other things. In particular, I give you an arbitrary tuple, say R0, R1, and so on and so forth, right? Well, you can realize this as an infinite sum where, in fact, you break this up into putting each one of the R's in the components and then shift, well, putting it in degree 0 and then shifting up the degree R. So this tuple can be realized as stick Rn in degree 0, then shift it up to degree n and sum all those up. And of course, I also know, since my ring is perfect, that I can always take roots inside of the dichmular representatives, since my ring R was perfect. So the Fn, or my ring with this dichmular map is multiplicative. It's another way of saying the same thing. And now, v to the n and f to the n, I know composed to give me a multiplication by p. So starting off with the tuple, R0, R1, yada, yada, yada, I can represent it over here, in fact, as a power series in p by looking at, well, I take the p to the nth root of the nth entry over there and then take the corresponding dichmular representative for that guy. So this gives you an alternate way to go back and forth or to take a tuple and represent it as a power series in terms of p, using that the ring is p complete. Is everyone happy with this formula here? But maybe in some sense, the right thing to say is this gives a unique way to represent things over here. So this is a separate set of coordinates in some sense on the width vectors. So these are called the dichmular coordinates. And in many ways, their manipulation algebraically is much easier to play with than the original addition and multiplication formulas. The other nice thing about the dichmular representatives is that it tells you a little bit about why you have uniqueness over here. So again, my representation of the dichmular map only involved the map sharp. And the map sharp only required from the first lecture some statement about the thing being p complete. So you can define an analog of the dichmular map for any ring satisfying these properties over here, leading to similarly unique expressions which tell you exactly how to do the isomorphism with WR. Okay, okay, great. Almost there, right? So one note in one definition. So note an element of WR written in these coordinates here is a unit if and only if the first entry here is a unit. So, well, again, easy to see. So in terms of sort of the functoriality of the representations. Okay, so with this, I make the following definition. So an element of the Witt ring here is distinguished if two things. So first R is complete with respect to the first entry D0 and the second entry D1 is a unit. Here's the representation in terms of the original coordinates on the Witt ring, right? Equivalently, right, so I could also write, right? So I can pick off the first entry as the dichmular representative of D0 in the expansion and everything left over, right, has a P in it, okay? And you can check that requiring D1 as a unit is the same thing as just requiring that this guy over here after I pick off all the other components of the unit. Okay, so here I've told you what it means to be distinguished in either of the two ways of writing this down. All right, so there's the first part of my definition is what a distinguished element is and the second part is perfectoid if it has the form WR mod D for some perfect ring R and the thing I kill over here is the distinguished element. So something of this form, right? Is everyone happy with this? So let me just make a couple of remarks. So this is the full definition. If you have a perfectoid ring over here, right, then in fact the ring R is uniquely determined, right? So let's just think about what happens when I look at A mod P, right? So if I kill P, what I'm left with is, well, WR, I've killed P D0, right? So of course I could have killed P first so this is mod D0, right? And if you go back to our construction of the flat map, all right? So what does this mean? Well, A flat then is R mod D0 flat, right? And in the first lecture, we argued again using some construction using the sharp map that whenever I take something which is perfect and I kill something principal, what I get back is the perfection of the first guy with respect to this guy. So this here gives me back R. So this uses that R is D0 complete, all right? So again, this was embedded in the first lecture. All right, so if I have such an expression, right, without seeing what the R is, I can tell you what the A, or without seeing what the R is, given A, I can tell you what R had to be, all right? And more than that, the presentation, right? So now we know that R had to be A flat, so what? This uses that my perfect ring R was D0 complete, right? I guess there's no brackets over there, I'm not. After I go mod P, there's no more brackets, right? So the presentation here also in the same way doesn't actually involve any choices, right? So I can tell you what that presentation has to be. It's determined by what's known as Fontaine's theta map, right? So what does that look like? Community square that looks something like this, right? So as we've said, everything in the wit ring has the form, has some type-mular expansion, right? So write it out as a power series in P, right? And what I'm gonna do is map that over to, right? So take the P to the nth roots of those representatives, again the ring's perfect, so I can take those at will, right? Times P to the n, right? And this gives me a map fitting inside of this square, which, all right, so here's my theta, which determines what this presentation is, right? So there's, the point of here of remarks one and two is to say that, well, the definition of perfectoid is that there is some ring R, right? So that it's WR mod a distinguished element. What the R is is completely determined by A and what the map, giving you that, is completely determined by the setup as well, and it's what's known as this Fontaine's theta map, okay? Let me think about that for a second. So I, this guy is, right? So, and I think this one is two, right, at the bottom, I think so, okay? Right, so I don't know that that's true for the other, well, yeah, right, so here is one definition of perfectoid, let me give you another equivalent definition, right? Which is maybe at least useful to note, in part because this is not how you'll see or recognize these things in the wild, right? So A is perfectoid, if four things, one, P is P complete and separated, the Frobenius map on a mod P is surjective, three, the kernel of Fontaine's theta map is principal, there exists a pi and a unit in A, so again, P is pi to the P times U, okay? So very similar to this other characterization in the easiest cases of what a perfectoid ring is, but slightly generalized, okay? All right, so, and I should say again that going through the equivalences of all these definitions is not so easy in the notion of what is perfectoid and whether it's relative or whatever has changed over the years, so maybe the equivalence of this definition with the others is due to Batmohr and Schulze. Okay, so, great. So again, the point of the theorem is, or my argument is supposed to say that you're supposed to replace in your mind the word perfect in characteristic P with perfectoid in either characteristic P or mixed characteristic zero P, all right? So you'd expect that perfectoid rings satisfy some of the analogous properties, all right? So here, maybe the first one, which I won't prove is not quite so straightforward to show as in the characteristic P case is that at least if you're perfectoid, then again you're reduced, just like you were for a perfect ring in characteristic P. Two, while it's not necessarily completely clear from our definition, okay? Maybe you'll see this in a second. If you're in characteristic P, then we would like to get again that perfecting perfectoid or the same thing, all right? So how do I see that down here? Well, I could take P as my distinguished element down here and I know that if I have a perfect ring, then W of R mod P gives me back R, the thing I started before and P is distinguished. This is the characterization I said before, right? So without writing it out for a third time here, so we have the characterization for P torsion free rings as well.  If you have a perfectoid ring, there exists a short exact sequence or a push out diagram, however you wanna phrase this thing, all right?  And one can show that, all right, here I've killed P or I've taken, killed a radical of P in particular, this is characteristic P, right? And the claim is that all of the rings, oh, I forgot the A. So A sits inside of two other rings mapping to a third, okay? These both of these rings are all three, these rings are perfectoid. Two of them are perfect in characteristic P and the third one is P torsion free perfectoid, all right? So in other words, A is the gluing of these two rings along the third, all right? So but in principle, this means that any perfectoid ring comes together by matching up cases two and three, all right? So in fact, to prove that the thing's reduced, you do it for torsion free and perfect perfectoids, all right? And then deduce it from such a sequence. To end maybe today's lecture, I wanna finish with my sketch of the proof of the Bat-Yinger Moth Theorem. Where here? So it's that guy, tensor that guy over this guy. So A bar radical P A times A bar, all right? Okay, so let's give some sketch here, all right? So without loss of generality, you can assume if I'm trying to show everything that the ring is regular local and maybe for simplicity, although this is pretty easy reduction, I'll also assume that the residue field is algebraically closed, all right? I'm trying to show that you're regular if and only if you admit a faithfully flat map to a perfectoid ring. So first start off by mapping R to R hat, all right? And there are of course three cases. Either it's a power series ring over K where K is characteristic P. So if this thing is regular local here or it's a power series ring over a whip ring, all right? So this is the unremifed case. It has the form, well, I have to add an extra variable and kill a relation, F is in the square of the remaining variables and let's say P does not divide F. Okay, so the ramified case. And in each of the three cases, the first direction of the theorem, right? So this is R regular implies you have a faithfully flat map for a perfectoid. You just exhibit it explicitly. So in particular, all right? So here we're in characteristic P, I can just take the perfection. Here you use my second example I used in the easiest cases from before, all right? So just to join all the P th roots of P and all the variables and then P complete, all right? So the only real interesting one to do is this one, all right, so the third case, you do something similar. I have all these variables, X one up to X D again, join all the infinite P th roots of all the variables, but not the P, right? Complete with respect to P and then kill the same relation, right? And of course you have to check that such a thing is perfectoid, right? So, but it's pretty easy to see in this case that this thing is P torsion free, so you have the easy check to see that this thing is perfectoid if you just play around with the elements, all right? So I'll leave that as an exercise. The other direction, right? So, we're gonna mimic the earlier argument we did when R is perfect and you'll find that you need to use the following sort of key lemma if A is perfectoid, right? And I have some ideal, which is the radical of the ideal generalized P, F two all the way up to F D, right? Again, when A was perfect, remember this was the ideal of all of the P power roots of all of those elements, right? Then the lemma says that the flat dimension is bounded and by the number of elements you had to take. So literally the same argument we gave goes through all you need to know is this finite flat dimension part to conclude that the ring A was regular, right? And the proof, I'll leave for you to figure out, right? But again, the point here is that, well, the radical of P, P has a piece root and in fact has a compatible system of, I can assume it has a compatible system of P power roots, right? So already has this form. So as we saw before, this is flat as an A module, right? So if I look at A bar, probably equal to one, right? But I know this thing has a two step resolution where both of the things are flat. So the flat dimension here is at most one. On the other hand, A bar is perfect so that implies that A mod I, which is, well, A bar mod I bar, right? So when I kill this radical ideal down here, I end up with A bar modulo. Again, something that looks like this, right? So once I know that A bar is perfect, right? So now I have a better expression for this thing and what we saw before gave us the flat dimension over A bar of A mod I or A bar mod I bar, however you wanna think about it was at most D minus one. So now if you use the standard, if you will, spectral sequence argument for how to compute or for torrent base change, right? So that'll give you, combining these two things, the flat dimension of A mod I over A, right? Is bounded above by D, okay? Great. So that's my sketch of the proof of this theorem, right? And we'll start there next time and move on to finally getting to big Kamakali things, the title of lecture series in the third talk.


For this DC motor example, we are simply verifying that we have the conditions that are needed. First is that you have g add fg and add f square g to be linearly independent which turned out to be a little bit complicated. I will really ask you to try to verify this numerically. And then the second condition that we needed to verify was that the distribution formed out of g and add fg is in fact, involutive. And that is pretty straight forward turned out to be 0 0 0. So, yeah you have involutivity. Now, we have the, we have by Frobenius theorem that this is in fact, completely integrable right. So, therefore, we have some output beta for which you have d beta multiplied by this terms of the distribution to be 0 ok. This is supposed to give you a partial differential equation. So, I am going to expand this. What is g?  So, this is 0 ok. So, how do you find beta? Just write out the partial differential equations ok. Let us see what am I going to get? The first one is going to give me 1 over ls del beta del x 1 is 0 correct right. This means what? What does this mean when you are trying to solve? Absolutely, implies beta equal to in a very very bad notation ok. Yeah, you should not write this like I mean in a paper and all in exam paper it is fine not in an article yeah. But anyway I am just saying beta is independent of x 1 ok. What do I get from the second and third now? So, you see that del b del x 1 is already 0 right. So, when I expand this one I do not have to worry about this term right. So, I am going to simply write what minus k l r x 3 del beta del x 2 plus k j x 2 del beta del x 3 equal to 0 ok. So, this is whatever the k goes away right. So, I have some condition right. How do I go forward now? You can guess yeah. What do you think? What do you think I should have beta I will make this bigger if you do not see it well enough? What do you think you can get beta from here? See there is it is very symmetric right. There is x 3 partial with respect to x 2, x 2 partial with respect to x 3 ok. So, to me it seems like there should be you know x 2 square plus x 3 square something like x 2 square plus x 3 square yeah. Because if I have x 2 square it becomes twice x 2 this gives me x 2 x 3. If I have x 3 square I get twice x 3 I get x 2 x 3 again. Now, this will cancel out modulo some constants right yeah make sense yeah that is how I am guessing yeah instead of trick that is all I am not doing anything magical here yeah. But you can solve anyway you want I guess yeah I do not know again what is an obvious way to solve otherwise yeah typically how you do solve PDEs is you assume to be you know some that the it is the two variables appear independently and things like there are some methods of solving PDEs ok some only in some cases you can analytically solve them yeah. But here I will guess so what is this by the way what is the scaling on x 2 square and x 3 square you think can somebody tell me L r x 2 square plus j x 3 square perfect this will work why because it will give me you can put a half if you want but nobody gets this will give me the first term will give me L r x 2 twice L r x 2 so L r L r goes away. So, this becomes x 2 x 3 this will give me j x 3 2 j x 3 so j j goes away so x 2 x 3 again so that becomes 0 that is a fair choice ok. So, again what did we end up doing what is this beta what was this beta what is all this work for output this is the output with respect to which the system is feedback linearizable. Now go back to what we had done we had taken the system fine the friction was missing no problem I mean there is this friction term that is missing from here that I think has 0 impact on anything yeah it is this friction term is missing in the third dynamics not here no problem. But the output was pre specified to some x 3 alright now what we are saying is that we are not going to pre specify the output we are going to try we are not given an output we are going to try to find the best output under which the I get complete feedback linearizability ok and that is this guy L r x 2 square plus j x 3 square ok and now that we have you know this output why do we try ok try I am going to say h x is L r x 2 square plus j x 3 square ok what about h dot let me put a half if you do not mind yeah I will put a half just what about h dot this is L r x 2 x 2 dot plus j x 3 x 3 dot. Notice that x 2 dot and x 3 dot do not contain the control ok right. So, this is going to give me some big mess I am quite sure yeah I am not going to write it sorry I am not going to write it yeah but I am going to compute h double dot yeah this is L r x 2 dot squared plus L r x 2 x 2 double dot plus j x 3 dot squared plus j x 3 x 3 double dot yeah yeah just product rule used yeah I do not want to write this mess. So, I am avoiding writing it yeah now what do I know I know that x 2 dot square and x 3 dot square are not bringing the control right because x 2 dot is this x 3 dot is this does not have the control on it. But when I take the double derivative of x 2 and in fact the double derivative of x 3 I have to take derivatives of these guys right and here I have x 1 dot appearing here I will have x 1 dot appearing ok and that will give me the control ok. So, what did I end up finding that control appears ok. So, what is it relative degree is what where I do this correct by the way should the control be appearing here on the third derivative no this is fine sorry I got all my coordinates yeah I got all my coordinates right should I have wait wait wait I should be very careful did I get all my coordinates what would be my coordinates in this case they will be h h dot and h double no control should not appear here either no no no no no no no no no no no no no no will the control appear here this computation is so painful that I am not very keen on doing it what is the control term can anybody compute it here and here can somebody tell me what is the control term forget every other term what will be this term what will the control term coming from this L r x 2 and in x 2 double dot I have this guy right I will have so I am going to write it as minus k L s L r x 1 dot x 3 yeah that is the term that is containing the control is going to contain the control. Similarly, if I look at this guy this is J x 3 x 3 double dot will be k L s over J x 1 dot x 2 okay yeah this x 1 dot is going to bring the control I am just writing the terms in the control my feeling is these two terms are cancelling aren't they these two terms cancel out isn't it yes minus k L s x 1 dot x 2 x 3 k L s x 1 dot x 2 x 3 these terms cancel out right okay that is that is the right thing to happen okay. Now not going to write it but h third derivative has control okay okay it look like this second derivative contains the control but those terms will cancel each other out. So control does not appear it can't because I need three coordinates right I started with the three dimensional system. So my coordinates are now these three h, h dot and h double dot yeah unfortunately as you can see very very ugly looking yeah nothing much you can do yeah doesn't look very nice but nothing much you can do if you folks are enter about it we can try the other example also do you want to try the other example or do you want to do it yourself why don't you do it yourself then we will make it an exercise yeah I mean we will try actually we don't know so that was the other system which was what was the other system we were looking at we had this other example right this guy even worse no no no we are not trying this you can see that the G is not even a constant no no maybe something else yeah so I mean see fortunately or unfortunately you can do you know analytically you can work this out for very specific examples only yeah but you can also see that you are you know you are sort of coming up with some rather unusual transformations okay which you would not never have guessed otherwise without looking at this structural details you will never have guessed that this sort of an output would give you complete feedback right we were working we did work with some other you know y equal to x3 and some something arbitrary right yeah you would never have guessed that you would get feedback but yeah with this very very crazy looking coordinates yeah you will get a linear system in the end yeah and this is not like I said this is not linearization or approximation of any kind this is actually the system is linear in these coordinates yeah it is it is almost like I think a lot of people say that if you if you if you squint your eye carefully enough every nonlinear system is linear yeah so if you look at it very carefully every nonlinear system is linear it is like you know that is basically the idea here yeah so a lot of them do have this condition satisfied as you can see it is not easy to verify yeah some of these I would still recommend that you verify numerically yeah but for systems where you can do this analytically you have a pretty powerful result alright a lot of times it is not a scale issue or anything it is not like it is like n becomes large there is a computational issue it is not a computational issue it is just that for example if you have a thousand of the same kind of dynamical systems and you can sort of by some suitable transformation you can make each of these dynamical system appear linear then you have now thousand linear systems that you are working with yeah so it is not going to affect you in terms of scale yeah if you have some you know robot dynamic like a mobile robot dynamics which is which you can feedback linearize yeah then you can say a lot of things right even if you have thousand such mobile robots you can still control them with easier control loss seemingly easier loss yeah of course this nonlinear transformation will sort of play big role but the good thing is you have it is just a transformation now it is just going from one function forward and one function backward yeah everything can be transformed like that so yeah I mean there is a little bit of a pinch of salt associated with it and this is also I believe yeah there is a little bit more on okay I mean there is just a little bit more on zero dynamics and so on if you notice in the last slide of this what I have done is I have looked at the rigid body dynamics yeah and I mean although I have written it as an exercise it is almost like I have done a little bit of the work myself you can or probably the entire bit of work actually at least a little bit of work I have done myself so please take a look at this exercise yeah there this is the rigid body dynamics again yeah in terms of some parameterization you already seen the rigid body dynamics it is part of the current assignment also and this is in some parameterization do not worry about what this row is but what I have asked you to do is to see if it is you know full state feedback linearizable and find the change of coordinates yeah which is this lambda yeah so that is what I have sort of asked you to do and I have actually checked a little bit of the you know involutivity and things like that I don't know if I actually found the coordinates doesn't look like it as of yet so yeah you can probably give that a shot yeah basically the idea is to find an output for which you have feedback full state feedback linearization yeah actually I think I have this this I would encourage you to look at this although it is like a solved exercise yeah but I would encourage you to look at this that basically it is I have not used the integrability condition here because it life becomes very hard with that condition I have simply guessed that y equal to rho if I take my output as rho itself that is a that is going to make give me complete feedback linearization because the second derivative will contain the control yeah so I have sort of guessed at it again instead of actually going by the integrability condition that you have yeah but this also works so a lot of there is a lot of literature in spacecraft community where they use you know this this instead of using the rotation and the angular velocity as variables they take the rotation and its second derivative as sorry as its first derivative as the variables yeah because in those variables you can claim some kind of linear system yeah though this is little bit more complicated than that yeah anyway maybe we will give you some simulations based on this all right


What's going on and welcome to QuickPix. My name is Tom Vecchio. I'm here to bring you three NFL player props for week six of the season. Let's jump right in starting off with Justin Jefferson, any time touchdown sitting at minus 110. The Vikings have a 24.25 implied team total there on the road to take on the Dolphins. The Vikings are three point road favorites. Surprisingly, Justin Jefferson hasn't caught a touchdown since week one. He did have a rushing touchdown a few weeks ago against the Saints, but hasn't caught a receiving touchdown since week one. This is a player that's playing a 97% of snaps running around on 97% of the drop backs of the 29.6 percent target share and a 41.3% air yard market share for the Vikings. All those lead the team and with back to back weeks of 13 targets, he still hasn't found the end zone via the receiving game. The matchup against the Dolphins is a good one. They've allowed 867 receiving yards to wide receivers. That's the ninth most in the league. They've allowed five passing touchdowns to wide receivers. That is the fifth most in the league. So all this shapes up to be a good matchup for Justin Jefferson to finally find the end zone via the passing game. Next up, Robert Tunion over 23 and a half receiving yards at the tight end for the Green Bay Packers. The Packers are at home. They are seven and a half point home favorites with a 26.25 implied team total against the Jets. This season, the Jets will have 270 receiving yards of tight ends. That is the 13th most in the league. We look to Robert Tunion on the Packers offense overall. And he comes in with a 12.1% mark, target mark share, which is rather modest overall. But Alan Lazard leads the team and he's only up at 19%. The Packers are really spreading the ball around a lot this year. There's no one dominant target. And realistically, Robert Tunion is one of the favorite targets for Aaron Rodgers. All add all this up and Tunion comes in projected for 29 and a half receiving yards hitting the over on his 23 and a half prop. Finally, Saquon Barkley over 29 and a half receiving yards. The Giants are six point home underdogs going up against the Ravens, which should put them in a positive passing game script. They should be trailing past the ball to keep up. The Giants have just dealt with so many injuries among their wide receivers. Sterling Shepard is out for the year. Canary Stoney has been ruled out for this week. Kenny Galladay has been ruled out for this week. Wandao Robinson is listed as questionable. We look to Saquon Barkley. He comes in with an 18.3% target share, which is the highest on the Giants, you know, ruling out Sterling Shepard. He's the highest on the team. And all these players are ruled out. We've won them listed as questionable. He's already running around on 63.5% of the drop backs, Daniel Jones drop back. So this really should put him in a spot with all of these players being out and them being in a positive passing game script to see plenty of action in the receiving game. We look to the Ravens and they're allowing the fifth most receiving yards to running backs. All of this projects Barkley for 32.6 receiving yards hitting the over on his prop. All right. So that does it for this week. I'll be back next week with three more player props.


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So guys, back in 2008 we had a similar, well it was a lot battered in this as far as just the recession and jobs were slim and everybody was concerned about losing their jobs and it was not good and even for developers we're always in demand so we always get the top tier salaries we know people need us but when it gets bad like this you had recession you had depression people want to cut fat and they don't really think clearly so a lot of us get cut prematurely even though you know from a impact perspective we still can help that company and just just unfortunate side of being on in a depression or in a recession that being said guys the junior developers get hit the hardest because you have no experience you have really no value and people is not ready to take a risk on you during these tough times so you're wondering hey Ry how can I position myself so I can get a job and still be successful and have a salary that's competitive so I can take my career to the next level. I got three things I want to kind of discover or discuss with you guys today hopefully it'll give you more well it's going to give you better perspective and it's going to put you in the best position so that you can still be or you're still gonna get to that high level even during a recession the last one guys I still was able to increase my income throughout that recession I just I was strategic about it and I was able to really kind of communicate my value to the company so that not only they keep me on but obviously give me a raise under the right circumstances and I kind of go over that with you guys in this video so number one guys let's start with what jobs we'll do with junior python developers first of all guys when you're in python you're working on a lot of AI a lot of data science a lot of new data science project unfortunately during these times all the sexy projects get cut guys unfortunately this is the nature of the beast and a lot of you guys has been working on AI project data science projects for months and you're gonna quickly find out they're just gonna cut it all or setting and not that is your work or what you've been doing it's just that companies getting survival mode they only want to focus on the stuff that's making them money and a lot of these new sexy projects is not going to be what they're needing this being said guys on your resume you have to have a wide range of projects that you do in order to communicate that hey even during tough times I have skills that's aligning with what you need at the time if you have project python projects that only dealing with AI only the sexy new stuff and not necessarily getting to the core business projects processes and automation really saving that company money you are going to be doomed I tell you that you will be doomed so let's make sure we diversify our projects to make sure that you can do a lot of automation a lot of reporting a lot of business process making that more efficient guys this that's the kind of stuff that even during a recession people need companies need because their labor force has shrunk but they still want to produce the same level of products and services technology can bridge that gap without that labor if you can communicate you can do that guess who's going to continue to have a job guess who's going to get a raise as they start really putting these projects into play this is the type of stuff that's going to put you in that top mastery level when it comes to software development and that advanced stage guys so you have to be able to have those low level skills and be able to communicate that number two like I was telling you before focus on the core business problems guys a lot of the sexy stuff that we've been working on so far is not going to be in the core business projects because quite frankly this is not the sexy stuff guys this is the routine repetitive stuff that they do day to day that puts money in their pockets so as much as you can do to improve this process so that they can make more money it's the less likelihood you will get laid off and you even if you're going in for a job if you can communicate this is going to put you in a better position to be able to get that job guys even as a junior developer you got the projects to showcase during your interview process you can go in and get a job even over senior level developers because you have this specific niche that's going to allow you to get that job get those projects done for the employer you're speaking that language they like you you have no risk associated to you because you know quite frankly you know even though they're going to be paying you less than a senior developer yeah let's get me on the top end of the junior developer salary which is probably right there at a senior level salary a lower end of senior level salary that's going to give you flexibility in the future to jump up to the higher end senior level of salary at the beginning fun fact guys you don't necessarily if you don't necessarily have to come in just getting a higher title let's get the top end of the lower the lower salary or lower title so that you can have flexibility to jump to the max on the second so kind of keep that in mind number three guys generalist resumes will not stand out during a recession it's just not you're going to be doing guys if you're coming in with these general junior python developer resumes it's not going to work it's not going to work yeah you'll probably end up getting an interview they'll just bring you in so that they can meet their quarter of interviews but are they going to take you serious probably not because they're going to see your resume you're going to have the basic frameworks basic stuff on there next and specific to their company and what they do it's just that they they look at you as a project yourself that they're going to have to teach you everything and then there's a possibility you may leave versus on the other hand if you're specific to them their industry and what they do the risk has dramatically lower guys they know coming you know they think you coming in you have an expectation at a company what you're going to be doing all this good stuff you've done it in the past it makes them feel at ease they're willing to give you more money because they know that you're going to be a better fit than all those other 99 general resumes that's on their desk and they're going to offer you the job first even as a junior python developer guys this is stuff that you have to put in perspective put yourself in the shoes of a hiring manager and just imagine if the hire manager interviewed or just imagine a higher manager hired you as a junior developer you come in you quit within three weeks because either number one you your expectation does a job is not good or you just couldn't make the cut three weeks guys that looks bad on a hiring manager because it seems like they didn't do their due diligence they're not going to take a risk on you even if it's the low the cost is low from a salary perspective but you're not looking at the training in all the time that's invested in actually recruiting somebody getting to me and interviewing all the HR stuff lots of work guys people want to hire a employer that's going to be around for a while so they don't have to do this as often you know everybody have to do it regardless of how good of a company are people leave they get better jobs it's going to happen over time but I guess the way I see it is anybody who stays around probably two plus years I think that's a good hire you know you're starting to get then three plus five ten that's really good hire if they provide value too you get people come in they leave less than a year that might not been a good return on your time let's say a rockstar developer which again you know you're going to know that pretty quickly and rockstar developers you're not going to bring them in at lower salaries anyways um but my point being is you have to reduce that risk for them and showcase the ability skills which you what you have done in the past what you're willing to do and it just makes things that much better for you guys so that being said guys covered a lot of stuff here at the end of the day I want to make sure you leverage your python development skills so that you can get a job even in a recession if you haven't already go ahead and sign up for my seven step guide and also I got a lot of premium courses that's going to help you guys really take your skills to the next level and put you in a position to win we want to really get you guys started doing that if not you're going to be at the house with no job and wondering why hey I've done all this work now I still can't get a job we do not want that guy so if you agree with me comment below if you disagree with me comment below let's start a conversation guys like subscribe to the content go ahead and check out my seven step guide right here click on that and sign up and I'll see you guys in the next video peace


Thank you for checking out this no spoilers movie review for the movie best worst movie Yes, that is the title of it best worst movie. It's a 2009 release It is currently streaming on shutter when I'm recording this Hopefully it's still on there and it's a documentary film And I would say that if you're watching this trying to figure out if you want to watch or not Please know I I really would recommend that you see the film that this is covering first it's a documentary about Not so much about the the making of the film troll 2 but kind of about what has happened since troll 2 came out It's kind of about The the fandom that has kind of cropped up after the fact because like the title says it's been considered the best worst movie out there which actually I Don't know. I might contend that the movie the room by Tommy Wiseau is the best worst Best worst movie out there, but it troll 2 probably has to be like the best worst horror movie I would think I mean it is crazy if you haven't seen it Definitely go see that and then check out best worst movie because it is a very good companion piece It gives you some really good insight into What are some of these people like who were in the movie who created the movie who worked on the movie? Yeah, and and the documentary is actually done by the guy who plays the main character child He was a child actor in this and his career didn't go anywhere because of this terrible film So he kind of covers that a little bit and talks about his involvement with it But then he also you know goes around to a bunch of the cast members and it's just kind of cool to see their personalities their feelings about the movie and the And especially like the director and his wife who is the writer of the film who didn't even speak English Okay, I'm not gonna spoil too much more I mean there was a language barrier that really kept directing from happening is what it sounds like Which is part of the reason you get such a bad product So there's a lot more of that type of stuff kind of talked about in the documentary Which is very interesting because then you can kind of understand why it's so bad It's not it's not just more bashing and just looking at the fandom that's cropped up since it's kind of looking at why is it so bad You know some ideas of well, this was going on and this was going on So it's cool for that reason So if you're especially if you're a fan of how bad troll 2 is best worth movie is a must It is a must documentary for you and I feel like it is well put together So I'm gonna go through some of my notes. Hopefully this doesn't go too long I actually took a lot of notes for some reason really enjoyed watching this a lot of fun It's kind of a funny introduction They introduce What's his name George Hardy? So he's kind of the focal point of the documentary and he played the father in The movie troll 2 and he's got some famously bad acting in that film some really Killer lines and you'll see what that is if you if you know troll 2 then you know what a bunch of the lines are I'm sure and if you can't remember them when you watch the documentary they go over that so that's another good thing It kind of reminds you of some things if it's been a while since you've seen it And so the I love the fact that they chose George Hardy as their focus for this documentary Because he's a very interesting guy. He seems like a crazy nice guy like super nice He's very quirky as well He's a very like bubbly energetic dude, but he's also very scattered at the same time And he's just a really interesting dude to be honest I think if they just did an entire documentary just on this guy his life and how he lives and how he is as a person That would be a great documentary and you know, you get a little bit of that in the film So he kind of ends up being a little bit of a narrator for it Because he's followed so much in it and I like that. It was really great. Oh a firmware reference troll 2 came out in 1990 so it has been some time since it was out um Okay, I wrote down that Mike Michael Paul Stevenson who's the guy who did the documentary who was in troll 2 as the kid I Wrote down that it's really cool to see someone who is actually in the film going back and really Revisiting it in the form of doing a documentary. I've seen documentaries about films before but it's not usually Done by people who are involved with it or were in it and I feel like done this way It seems a lot more personal and it seems a lot more like cool and insider-y if that makes sense It's fun to see what people are up to now Whether the film like ruined their careers or they're still working or they have somewhat of a career or a convention a Horror convention career based off of this film, which that kind of happens with some of them And it's especially interesting for for people who admit that they kind of did a terrible job Like it's cool to see like who's accepting of how bad they were in the film and who is Denying that they were bad in the film and and that's just kind of fun to see and also just like I said What they're doing now with their lives that kind of stuff is always interesting It delves a lot into the fandoms That arise from it being a terrible movie Kind of hits how like it just didn't do well when it first came out and with a lot of really bad movies It started gaining some cult status over time and then people were like oh my god You have to see how bad this film is then that person takes it and says hey now you you need to see how bad this is and it just like grows and grows and That's a really cool aspect that they show a lot of and people just being like excited about troll too and it's this funny thing that I realized watching it is that You you kind of like ironically like it at first because you're like oh well This is just so bad, but it's so entertaining because it's so bad I can laugh at it But at some point when you continually watch it and you continually say you love it It kind of goes all the way around it becomes like full circle and then you just end up feeling like you Legitimately like the film and you could kind of see that emotion on display Within the documentary and it was cool So yeah Another thing do do do do let me see It's crazy to think that people can amass a following from being in a in something this terrible It really is but you know, I'm I'm someone who's who's guilty of it I love the room by Tommy Wiseau so much if I had the chance to meet Tommy Wiseau I'd be all about it and that movie is terrible. It's just so bad, but I Don't know like it's it's something about the passion too. That's another thing And they kind of touch about touch on that in the documentary The documentary drives point drives home the point that entertainment is entertainment. It kind of doesn't matter What is good and what is bad anyway? It's kind of a question. That's put out And I wrote down Michael Bay makes off awful movies, but people see those he just has money behind him Yeah, and that's kind of one of the things that kind of touch on is just like what is entertainment It's kind of like an individual thing So some people may legitimately be more entertained by movies that just aren't good Some people may just be sucked in with the enthusiasm of a film whether it's good or bad In cases like I was saying like Michael Bay I feel like he makes terrible movies in my opinion, but people go see it And I think it's a combination of the practical effects all or the the CGI always looks good because he's got tons of money They're all these explosions and he usually gets really fun properties to work with like transformers even though I think he just like Murders those things terribly, but you know, there's always gonna be a fan base for that So it's like what is entertainment for you in particular? these people people involved in the movie are real characters themselves I wrote down and It makes the documentary kind of on par with troll too in that sense because you can see The essence of troll too in some of these people's psyches some of these people's personalities Manorisms all that obviously you see it in their faces because they were on there, but it's cool So the writer I found this really interesting the writer of the film had said that the movie is a vampire film But they are but since they're vegetarian, but they're vegetarians like the vampires are actually vegetarians in it In troll too and she did this because at the time she had a bunch of friends who had become vegetarian And it was really annoying her so she was all pissed off So she wanted to write this script basically like a vilifying it like Vegetarians, it's I don't know. It's it's weird, but you got to hear for yourself It's crazy how the director Claudio Fragasso Still thinks it's good.  I mean he's done more films He's he's doing Big Shark, which is a shark movie. I think Unless that's kind of a red herring no pun intended, but So I wrote down oh no the woman who played the mother needs some help like legitimately. There's some cringe-worthy stuff in here where you're like someone is Having some problems mentally and Someone needs to help out and that's kind of like a cringy thing in this documentary But it's interesting and I mean it's real life, you know, what are you gonna do? It works very well to make George the focal point. Yes and he kind of like It follows him to the other people kind of it seems that they kind of used him as a gate as like the key to get to the Other people to have them involved with the documentary and you kind of get that sense And then the last thing that I had written down here and this was amazing to me So there's a part where George Hardy goes to a horror convention and he's you know They're to sign autographs for troll too and I guess he's not actually a horror film fan because he's a reaction to all the horror All the horror icons there and also the people attending it who are all dressed up and everything is Funny but also kind of like oh Because he seems like a cool guy But then he's like very judgy about these horror fans and I was watching it and he's just like all these people And he goes like on and on and on about so it's like funny it for a little bit And then it was just kind of kind of becoming like extremely judgy and a little offensive and then I was just like These are my people like these people he's talking about these are my people people. I love the most I can't tell you how Comfortable and happy I feel when I'm at horror conventions because everyone's there for the same love of horror And I don't think I've ever had any or interactions at a single horror convention I've been to that has been negative with with other patrons there Like everything's just so nice to each other because you're all there for the same thing So that kind of like rub me the wrong way where I was just like these are wonderful people man Like why do you be so judgmental?  I really think that this is a really good Well really not really really good, but it's a very solid documentary. It's a lot of fun Especially if you've seen troll too and I mean you can watch it if you haven't seen troll too and you'll get enough of an idea I think actually if you watch it and you haven't seen troll too It's really gonna make you want to see troll too so you can experience how terrible it is But I would recommend seeing troll too first and then seeing this I think pairing them together is very nice They work well So that said my five-star rating with half stars in play I'm going to go ahead and give this a four-star rating. It is good I recommend it Yeah, it's it's good stuff. Anyway, I'd like to do some more Documentary reviews. This is the first one I've done actually and I really like documentaries, especially horror documentaries I'll tell you right now one of my favorite ones that I can just recommend to you is the It's called you're so cool Brewster And it's a documentary that has to do with the behind the scenes of the filming of the original Fright Night film It is like over two hours, but it doesn't feel like that because it is so well put together It is jam-packed with awesome information and just really cool stuff cool interviews. It is phenomenal It I think it's the best horror documentary I've seen. It's really amazingly good. Ooh Actually that or nightmares in red white and blue. That's another really good one that one has to do with just covering the horror genre in general and then sub-genres and It goes through each decade of horror and what these sociopolitical influences were and why these horror films are so popular Tying into society and people's fears and anxieties and stuff like that at the time Super awesome. That's more of like an intellectual level thing So kind of very two different things, but I would highly recommend them You're so cool Brewster and nightmares in red white and blue amazing. So anyway, thank you guys for checking this out Please hit the subscribe button That's the least you can do for me if you like anything that I do if you don't like anything I do why are you watching it? But I'm not complaining because keep watching it. That's great. Put some comments down there. Have you seen this?  What are your thoughts? We can you know get a little bit nerdy talk about some stuff and then give me a thumbs up if you want to Don't have to but whatever Anyway, thank you everyone for checking this out and until next time take. Oh, no I just totally screwed. I was gonna say take it easy. No, I'd never say that keep it brutal


If you're looking for something to surprise you to kind of throw you off guard, this is definitely a movie that will do it. How's up guys, it's Josh here today I want to do a little review on malice from 1993. Now, if you're looking for a mystery, if you're looking for something to surprise you to kind of throw you off guard, this is definitely a movie that will do it. I would be pretty impressed if you were able to within the first 10 or 15 minutes, tell me everything that's going to happen in the movie because it definitely does twist and flip around quite a bit. So let's jump into the pros ahead with this movie. The pros is that it has multiple twists to it, which I really like. This movie has like two kind of mysteries to it. The main story, the main mystery doesn't really reveal itself as a mystery or as a problem. So about halfway through the film, I think the film starts off with somebody getting murdered about halfway through this movie. Even two thirds, you find the answer to the original killer. It's a very interesting little side story because it's almost like a mystery within a mystery. There's kind of like a much more interesting mystery going on with the three main characters. The acting was overall good. Like I don't have a problem with anybody's acting. And I got to say I was absolutely blown away by Nicole Kidman in this movie because Nicole Kidman can oftentimes play a very quiet, very simple kind of girl and she plays that very well. But in this movie, she has to go to some extremes with acting and it's amazing that she could pull it off. She looks like she's like 22 in the movie. And she has like by far the best intense acting scenes that I've seen from anybody in this movie. I don't know if I could show you too many of those scenes because you know, it's in a heated moment. You might find out some of the secrets, but wow, wow, was I just blown away by her performance. And then another little thing I liked and maybe this is just me. It had a lot of little cameos like I think Gwyneth Paltrow's in here. She looks super young. I feel like this is probably her first movie or one of her first movies. You have Sandy Cohen from the OC. I think his name is Peter Gallagher. Always loved seeing this guy. Haven't really seen him in a long time. And it's interesting. He plays a lawyer in this movie and he plays a lawyer in the OC. So I like that. And then also you have Jigsaw in this movie from Saw. So I liked all the little cameos, really the main takeaways. It's got a really good acting, especially Nicole Kidman. It's got multiple twists that, you know, give you multiple things to guess on, multiple things to be like, oh, I had no idea was going this way. And it did just what I came here for. I appreciate it for that. My mixed aspects is I'm a little bit mixed on the murder mystery in the beginning because it doesn't have much to do with the main story. It seems like it should take president, but it's just kind of something that goes on in the background and just kind of gets resolved in a very anticlimactic fashion. So although it is nice to have someone end up being this killer, you get to guess who it is. And then you have a whole another mystery afterwards. I just kind of didn't see how it fit in the movie. You know, it's just kind of like an extra thing thrown in. That's why I put it in mix because I don't hate it, but I don't really a hundred percent know if it was the best move they could have done. All right. So let's jump into my cons now with this movie. And it is very, very slow in the beginning. You don't really know what's really happening at one point. There is some turmoil between the three main characters, Alec Baldwin, Nicole Kidman and Bill Pull. Things kind of get more intense from that situation. But until that happens, it's kind of very simple, very slow. It ends up kicking off. So I appreciate it. The last half is great. Some of the twists here are really far fetched. I'm not going to get into any specifics, but there's one person who permanently does something to their body, just to get some sort of monetary gain or to make their BS strategy slash plan believable. And it just doesn't seem smart or like, why would anybody ruin part of their body to pull this off? It just is a little far fetched. The whole thing was a tiny bit far fetched. But at the same time, everything matched up in the end.  I don't mind Bill Pull as like a guy working in like an office, but besides that, like, he doesn't do very much extra for me. I don't want to root for him. Nothing against Bill Pull. He made his way. He made money. He's successful. He's not a bad actor, but I can't watch him as like the main role, main character. I watched one called like lost highway where he was the main character. And then I've watched this one. And it's like, I just can't get on board with this guy. He's like, he has a personality, but he doesn't. The fact that Bill Pull was the, one of the leads here was very difficult for me. It's also very dated. I would say even more dated than something like Die Hard, which came before it's, you look at speed that came out a year after speed. Looks like it came out 10 to 15 years after this movie. Then the last thing, and again, this is just me, something about the really old Alec Baldwin movies where he's kind of like the star or the lead. I love him as an actor. Love him, but there's just a vibe to his movies tour. I just can't get into it fully. I don't know why, but then you fast forward to say 2017 or whenever they did the Mission Impossible movies, he gives one of the best features in Mission Impossible history. Love it. He's great in the Mission Impossible movies. Something about him in the older movies. I just can't get into. So if I were to rate this movie, guys, I probably rate it between a 7.5 to an 8. Maybe I'll just stick with a solid 7.5 by Trier Pass. I think this is a try if you're somebody who likes mystery movies and doesn't mind a more dated movie with a little bit of a slow beginning. It really does have something special for you if you do like mysteries. So anyways, guys, have you seen this movie? Have you not seen this movie? Is there any unknown mysteries that I haven't reviewed or that you like? I'd be very interested to see down below from the road of 50,000 subscribers. And I couldn't do it without any of your guys help. You guys are the best having a great day out here. Hopefully having a great day at home. See you all in the next video. Peace.


I've seen enough of this advice in my comment section, as well as online in general, that I have to address why the amount of vitamin D from foods is not significant and shouldn't really be mentioned in any sort of context where you're implying it's an acceptable soul source of this vitamin in someone's diet and lifestyle. There are a lot of raw primal dieters, Western-day price people, carnivore, who want to avoid supplements and keep things natural when in reality they're using that as an excuse to not put as much effort into their health and understand things from both the scientific and the experimental perspective. This is kind of similar to those raw vegans that think their diet is so pure. It's not too far off from that mentality. The main reason is that the amount of vitamin D specific to any food, even the foods highest in vitamin D is not comparable to the sun, it's not even close. We're talking hundreds of IU in foods versus tens of thousands for minutes in the sun. One hour in the sun will get you more vitamin D than a month of eating foods high in vitamin D. Even so, any food that does have a large amount of vitamin D will have a correspondingly high amount of vitamin A which negates it. If you're deficient in vitamin D, you have to balance your fat-soluble vitamin stores, which means you need to ingest or get vitamin D from the sun without getting the other fat-soluble vitamins. Yeah, I've seen some pretty ridiculous things like people saying, oh, well, IU 10 raw eggs a day and I get my vitamin D from those or I ate one tablespoon of pork lard and I got 5,000 IU. This is completely ridiculous guys. These are all lies. I think there's like a hidden agenda to some of these diets that don't actually want people to be healthy. So here I made a list of the top vitamin D foods just for comparison and one thing worth mentioning is vitamin D2 versus vitamin D3 which is the form of vitamin D that occurs in plants versus the form that occurs in animal foods and the former is not really available, not really active, but again, this whole thing to me is not really relevant because it's not important, you know, you're supposed to be out in the sun. So cod liver oil, trout, salmon, milk sardines, after the fish it kind of falls off really hard and that's because the diet of the wild fish, the algae that's like on the top of the ocean in the sun is naturally very, very high in vitamin D and in comparison to the rest of this stuff, conventional land animals, factory farms aren't really out in the sun and even if they were, their diet isn't really adequate to metabolize the vitamin D well and have healthy meat. So you know, if your chickens were out on pasture, if the cows were out in the sun, if the pork, would it be higher in vitamin D? Yeah. Would the meat be a lot healthier? That's more important. Much higher in minerals. You know, we're not worried about vitamin D from animal foods. We're getting it from the sun.  If that was the truth, someone would have tested their meat already and used it as a marketing gimmick. Oh, this is the highest vitamin D egg. Someone would have done it already guys. Maybe I should try to do it. Now what's pretty hypocritical is these people will be on the carnivore diet, on the western price diet, raw prime or whatever it is, and they'll still go with the RDA for vitamin D, which is like 600 IU. It's pathetically low. How can you flip flop that much? You're going completely against the government eating 15 eggs a day, but then you're going with the government's vitamin D recommendation. Come on, let's think a little bit here. Now in regards to how much vitamin D we actually need, I did a video about a year ago now titled, How much vitamin D do we actually get from the sun? Definitely check that out. Over how a person can get tens of thousands to hundreds of thousands of units of vitamin D for just a few hours of sun exposure. Now each of those foods, as I said earlier, will have a corresponding amount of vitamin A. More vitamin D in the food means more vitamin A in the food. And the reason that vitamin A negates vitamin D is because vitamin A is used in healing tissue. It needs to be used up. More relevantly, your skin, when it's damaged from the sun.  And a few years of doing that will result in significant fat-soluble vitamin imbalances in the body. We spoke about magnesium being needed over the past two weeks. That's very important as well. It's not just about the vitamins. So animal food increase in the diet, you also want to increase hunting activity. How would you have obtained these animal foods? Which means prolonged full body sun exposure. Being out in a bathing suit for five, six, seven, eight, nine hours a day in the sun. And highly physically active. When you're walking 10 hours a day, when you're working out, when you're moving, when you're doing things. That's using up vitamin A and fat-soluble vitamins. So if you just sit on a couch all day and eat trout and cauliflower oil and salmon all day, your body is basically just going to be storing vitamins, waiting to be used in these physical activities. And you're never going to end up using them. And you're probably going to develop severe imbalances, which can lead to organ and liver issues. So you also have to keep in mind that vitamin D supplements bypass the body's natural regulation. So initially, if you're very deficient in vitamin D and you're not tanning in the sun, yeah, taking a supplement can help kickstart that. And you definitely want to take it with the synergistic ones, vitamin K2 and magnesium as well. But after you get to a moderate level of vitamin D, you want to absolutely stop supplementing. Because when you're in the sun and your skin's darkening, your body's able to regulate that vitamin D that comes in and store whatever it needs. But if you're just dousing your stomach with concentrated vitamin D supplements, it can get very bad and your levels can get very high.  The main difference between minerals and B vitamins versus fat soluble vitamins is that the former are much easier to remove from the body. When you're ingesting fat soluble vitamins, the body tends to want to hold on to them because they're sparse outside of our natural modern dietary context. So we do have to be mindful of that. So first, you definitely want to stop all supplements. And then you want to get blood work to check your levels. Because if you get blood work now and then next year and the year after, and your vitamin D levels are still really, really high, despite not taking vitamin D or getting that much sun, that's an indicator that your body's stores are very, very high. And you need to use them up. Magnesium vitamin K2 to balance. Definitely needed several hundred milligrams of magnesium, same with a few milligrams of K2.  That's a big reason you guys see me eating this new diet and make it fun to me for eating bread. Well, there's a reason behind it. There's no real way to see how much vitamins are stored in your body without getting a biopsy of your tissue. So it's easier to just even out the diet, make sure you're not getting any of this stuff, and then taper in supplements and see how you respond so you can isolate individual nutrients.  And hopefully I can continue to provide you guys with information as well as foods that will make you happy and healthy. As always, thank you guys for joining me today. If you could please drop a like on the video, leave a comment down below, subscribe so that YouTube can unsubscribe you next week, and be sure to check that notification bell so they don't notify you of my videos. Thanks again, guys. And I think that's it. So I will see you for the next video. Bye.