 the motivation about open cup bodies anymore, but still I need to introduce some of the features. I'm going to use this complex graph, the Euro open cup bodies, we need some extra data, so we give them, we fix a big line model of LLX, and eventually Y and become a single point. Of course we can't take a very general flag of this form, but we need some extra assumptions. But here I will not try to be very precise, so it surprises me if it's such a general flag. And another almost equivalent way of doing so is to fix valuations on the function view. Speaking, if you have a flag here, then we can take a general original function here, and first argument of the valuation as the order of managing along the first component of the flag. And then you continue with the second component, and eventually you end up with a vector with one component. That's how you get a general valuation in this form. And in this form I will fix either a flag or a valuation that works with the corresponding open cup bodies. To use the notion of open cup body section, I will not continue here, but I will be trying to explain a little bit about it. The construction is not really relevant to the construction of the partial open cup bodies as well. But before we use the statistics, then why open cup bodies are important? Because that is the line of those for the study of the valuations. So first the main theorem is due to Rata's field and it's doesn't in their original nature. There are even partial irreverence of the numerical class of L. This is the theorem of each other. But that should probably be what depends on the flag. Yeah, it depends on the flag, so that's why it depends on the next view here. What is it? It depends on the flag, so that's why it depends on the next view here. What is the valuation used by the flag? Oh, okay, so why do you fix the valuation of the flag? Yeah, what I fix the valuation is that it depends only on the numerical class. That's what I mean by this theorem, which says that the collection is new here. We have a class of a line model and one should expect all of the numerical information for the line model can be expressed in terms of these open cup bodies. And in the statistics, for example, we have the volume of the line model, essentially the volume of the open cup body. Once the valuation is added to all of the valuations, they need a more complicated study for more than a very, very hard case. It might happen that you even need all of the valuations instead of only one. But if we're moving on to the partial open cup bodies, let me remind you of the construction of the open cup body I divide them into some simple steps. The first step to construct a open cup body is that from the start with the line model L and you pass to the ring of sections. And it's important to remember that this object is a ring, it will be very relevant to the construction. As long as you have such a ring, the next step is to make use of the valuation as an extra input. To start with, let me call this ring R of F. To start with the ring RL and the valuation of mu. Then we construct the semi-group. The semi-group, the first component K and the second component mu of S, where K is an integer and S is L to the K. For any section, you can take the valuation. It becomes a vector in Z to the N. And you put an extra data tag here on the first variable and you end up with something in Z N plus one. And as a consequence of the fact that the object RL is a ring, you know that gamma is always a semi-group. And all we do is just, we take a closed form X call and we call it com. And this will be a form in RL plus one. And we take an intersection with RL. This is by definition, our open-call value. Here you set something like this is our R to the N and you will have some next one here. And for the destruction and the semi-group gamma is something like you take up all of these points, some special points from the data. And then the open-call value, the construction in this directory of open-call values, you take the closed form X call, something like this, and you take an intersection with the vector N, RN here. And the part you get, this part is the open-call value you end up with. To generalize the partial open-call value, we follow a similar, we follow a similar step as I always do in partial open-call values for several times, but I only use what it is. And then you try to do it now, but is there, yeah. What would you like to do? You can set the micrometer, this is the micrometer. This is? Yes. The similarity of the class, I want to actually determine the similarity of the magic H. So, delta middle of LH. That is the partial open-call value I'm going to position it. Oh, you're going to find it. I haven't found it. If we count the total number of bodies, if H has the minimum number of bodies, what is the maximum number of bodies? In the end of case, when H is found, then it is the same as the euro from the body. I guess you see very similar. Yeah, in the next case, let's see what happens in the case when there is an extra input of the single magic, the stack Y, through the second ring of L. And here, there is a single magic, so I have to put it in a really good somewhere. And it turns out that the most next way to do it is to put it in the form of a particular new sheet. So what does this mean? This means to take all of the homomorphic sections that are in all of the homomorphic sections, such that it's integral. It's final. This is the meaning of the notation here. And this is the number of rings in the center, so you can't expect it to be a ring in general. We still want to carry out the second step. We want to pass from the ring or from this object to some semi-group. We can still take the same definition we passed from this object to still some gamma. The same way we're at the section of L, A, but with a marker on each ring here. Since this is not a ring, this is not a semi-group in general, we want to carry out the step three to show you what would happen. In the worst case, let me draw something like this. This is not a semi-group. Why is it not? Why not just from Cauchy Schwarz and the following? Why is it not a ring? It is not defined to sections. So for example, you have two sections, X and S prime, which are the X, the integral, which are going to each of these, and S prime also start to, such that this is finite, and we take another section so that this is finite, and then we carry out a little bit. That's not the condition, though. You said K. Yeah, it's K. So S is a section of L-13. Yeah, so I'm not going to tell you that even if it's K-21, it's not what you think it is. Oh, we don't want K-1. Why don't we? At the same K. So we get S to the power four or on the other side? Yeah, that's all I need. I'm going to go to the other side to see if it's in power. They're going to the other direction. Okay, I'm just going to take a picture of what was happening in general. So you take this white point, you put points in the top, and then you can take a white star as well. And then you end up with something like this, and you take a intersection with a bigger plane here. You end up with something super big. But the red, you know, are essentially the same projects. So these points, they are actually not in the, so if you don't look at the first level, then you expect something like a single point So, this video is a disaster. If that was not the same group, you would not be able to talk about it with characterized some useful information in it. But the paper is that I find some magic that can get around these problems. Now I'm going to talk, I'm going to tell you about the magic. Did you find a book yet? I haven't defined the aconcopathy yet. After doing the magic, sort of magic on the gridded subspaces of RL. There's also this RL. Yeah, RL is a virtual. It's a false-action ring. It is respect to the pseudo-magic, respect to the topology defined by the pseudo-magic. It is not the opposite of magic to each other. And then the flow, that might be some magic by continuity method, which is approximate boundary point by some extra rings, and then you end up with, and it turns out that it is indeed the case. Just to give you a hint about the pseudo-magic and write it right out into the page. So just a question, so a gridded linear subspace is not the ring? No. Now in general I've just been subspaced. I've just been that subspace of the form L-t where L-t is the sub-sector space of RL activity. So they have nothing to do with it? No. It has nothing to do with the ring structure. And you write it in this way. And then the pseudo-magic is defined by the ring too. And I actually mean it here. I can imagine the dimension of V i plus dimension W i dimension of V i at W i. So far away the two gridded linear subspaces are. And it measures the six-pology called far-raded R. It is long and tedious exercise. You linear drill to show that this is indeed a pseudo-magic. And that would not try to do it here. Then there is the E-satnatic. Some of you, that's a pretty good idea for me. The partial-couple-pology test, it is initially defined on the set ring. I mean, there are rings that are in gridded rings, gridded sub-grams of the full-section ring, of course. Especially here in this body here. Who likes bodies? This is our original definition of the third time linear. And it turns out that it can be extended continuously sub-grams of R at all. As for the other part, it has to be a gridded sub-linear subspaces. But here I'm not setting to be very precise about it. On the set of complex bodies. And there is a very digital one, which is the so-called holstop podge. And it is even a mass-couple-g. So there is a holstop magic. It can be very precise about it. But just use your imagination to think about what it can be. This turns out to be true, that the holstop body map can be extended from the set of rings to the plural of the rings. This is in no sense true. It takes me a long time to figure this out. And, but now you see that we have the two magic together. We know that the RLH, although it's not a ring, it lies in the plural of rings. And we know how to define the holstop bodies of the plural of the ring, in the plural of the rings. And we have the definition of the holstop bodies. So finally, we come to the most important definition here. The partial holstop body in the third column of the permissioned line model is defined as the holstop body of RLH. So this is our definition. It seems to me that the hardest part is the birth theory. Yeah. That says that it is holstop. Yeah, indeed. In the case of magnetic similarities, this is not very hard. And then in the case of, in the general case, you need to use the micropersonation to approve that. Preventive version of the proof. And you're right. There are many difficulties. Does this mean you don't use the relation to like an air-confinition in the version of the model by any machine? Well, in the general case, if I have a similar method, then it can happen that the RLH version of the matter-conditioned distributed. So it doesn't have much to do with this equation. But in the case of magnetic similarities, I, in the case of magnetic similarities, I do make use of the RLH version of the micropersonation to prove that it is holstop real. And I begin 10 minutes late, right? So I still have a few minutes to go. 30 minutes? Okay. Okay. That's something about partial proof. All of these depends. I mean, there's prejudice into it like this. And on this side, we have a similar result. Third time, new, L, H. This dependence on H is only singularity time. What do I mean by that? Suppose that we have two different similar methods, patient H prime. We say that there are I equivalent, if all of their larger RLHs are equal. This is by definition the, this is the definition of the I equivalence. And it turns out that this reflects the property of the I singularity time of H. The main theorem about Ocunco bodies is a theorem of L. This says that third time, new, L. What do you mean by this? I mean, for you, the second problem is the line model. Yeah, this is the line model, yeah. And I'm only talking about the dependence on the method, please share. Yeah, and the second method is the theorem of J. The collection of Ocunco bodies, Eternals, is numerical class. And in our study, the first funding result is the funding. The collection, Eternals, is a very similar type thing. So you see that in the partial Ocunco bodies study, we have read a lot of the theorems in the Euro-Ocunco bodies study. And we see that these objects can be very useful in detecting the similarities or the isimilarity type of the method H. And this is why these partial Ocunco bodies are useful for my own study. I'm going to say the... I'm going to say my thing a little bit. So the extra theorem I wrote is that here you need some possibility to stop the model. I didn't imagine that. Yeah, I just tried to be as simple as possible. Okay, not very precise. The definition of the partial Ocunco bodies are very abstract, and we might not know how to compute them in reality. But it turns out that there is one very concrete body of computing them. Fixing the nation L is U as before. And the theorem says, third time U, the partial Ocunco body can be computed in a very simple way from the body on the right-hand side. This is the so-called big divisor. It is simply a super-classics on all of the variation models of text so that they are compatible and they're push-forward. And there is a kind of a way of constructing a big divisor out of the singularity of H. So roughly speaking, the component of the big divisor on the manifold X itself is simply the, by definition, if we take the pseudo-decomposition of ADC of H, there is an observable part and the remaining part. And the big divisor of Cunco de la H is, by definition, L minus D. And all of the variation models, you can imagine some similar instructions. You just take out all of the divisor parts and take them away from the line model H and it turns out that this is a well-defined B divisor. And then what is the definition of the B divisor? What we're talking about is a B divisor. So you guys use a B divisor or a B divisor class? It's a B divisor class. That's what I mean by a B divisor voice. Yeah. The general problem is the divisor solution can have extremely many components, the observable components. So it doesn't really make sense to talk about B divisors in the honest sense. So I'm always talking about the B divisor in the numerical sense. It's defined by a section to the line model. Yeah, in that case, from divisor D, in that case, we know that there's a new L minus B plus some vector. Yeah, so I guess these B divisors are where we have approximation. Yeah, that's right. So we need a similar approximation to a very new sense of the B divisor, because we generally define their personal code. So think about this B divisor when it is 9. And to prove that it is 9, we need a similar approximation procedures. An intersection of all of these components, the definition of the first one here. The definition of the first one, I hope it still relates to the components of the B divisor. And in some sense, the usual think about this, they are more or less computable, at least in the case of surfaces, you know how to compute them, in the general case, they are well-studded. So in that case, I can really say that these things are computable. And the last term is just a vector, a fixed vector, you are in, and they can be computed in multiple ways. And so in some sense, the partial combolius, although it is defined in an abstract way, more concretely, it can be computed in terms of the euro or computable bodies, and up to a translation, of course. So the second term does affect the volume, for example. Or this is the second term, does it affect the volume? No, it doesn't affect volume at all, just a translation. Okay, and the trigger which should be, where each one can compute the partial combolius in the case of universe-only curve, with some level structure or it could have to rate it, and it seems like it is possible to compute the partial combolius with this method. In that case, we have a very good understanding of the B divisors. So to come to the last subject, the last problem is that we have a vision. I know it's a vector called Osmium, and wherever we begin, there are several differences. Several different regions of the technical model, so I'm not using a smooth calculation, but using multiple of them, using one of them. And the vector is the volume of the curve, but at least the volume of them are all of the variations, the volume of the partial combolius. At least mine, we have many of them here, that is, we talk about the euro combolius, instead of the partial combolius. So in the special case, it affects something like this. What about this, the inside one? Yeah, I haven't defined them yet, but if you, yeah, for you, it's easy to define them. So you take the eye projection first, and then it is the boundary coordinates. And then after taking the eye projection. You say that again? You say that again? It's a little more slow, right? Yeah, we're not made of this projection, unless you tell us. Yeah, of course. So, let me start to state in the most simple case, where H-I, there are actually H-I models, in that case, the least volume is just defined by, a not very clear message, but with some impact value. You cannot define them in terms of the number of the dimensions based on sections. I don't think so. Can you find it as a term in some part of the new, I don't really think so. It seems to be something already, not linear. In this case, the volume, I mean, already means the normal intersection for that. So, in turn, it's not linear either. Well, at least it's also linear. Yeah? What do you mean? I mean, I was just guessing you take, like, bundle T1, L1, KC, and LN, and then the metrics. Take that volume. Ah, okay, I get it. I think it's getting that out. I think it's the sounds of that thing. Uh, I don't really think so. If that happens to be true, then we don't even need a soup. Take that, there's a hierarchy text, right? Oh, sorry, I didn't hear it. If we've got this hierarchy text, this is all still there. Yeah, it's not something that happens. That's all I mean here. It stays here. Yeah, if you don't know about the volume here, it doesn't matter. If you can solve this connector, it's already a very good result, I think. Here, this is the mobile intersection theory. Mobile intersection for that. And on the right-hand side, is the euro, it's the volume. You take the soup over valuations. Yeah, it is. I have some concrete examples in the case of torque surfaces. In my case, I already used the torque surface, right? You want to take the flat as the flat, the zero times zero in the diagonal. If you can give a piece of volume here, it's a six piece model than the volume here. The volume could take just the integration with the means of the singularity of the classical simulation. Yeah, it's a projection. So that is not... No, it's not volume here. So it's already over there. Yeah, sorry, I didn't hear it. Okay, yeah, I can hear that. Sorry. Okay, okay. Oh, okay. Oh, I'm sorry. I didn't hear it. Sorry, I couldn't hear it. Shh! Shh! Did we get any points? And then we have an e-comedy, but the entire time it becomes more... I don't even have an e-comedy. They explained it all. If I had a counter example, I would never know about it. We have a draw theorem, so as long as you know all of these classical models, you shouldn't know anything about everything, about the different class of the light models. So in principle, there should be a sort of formula expressing the mixed volume in terms of these optical bodies. I just don't know if this is the practical version of the formula. Okay, I think I will stop here. Thank you.