doi
stringlengths 18
24
| title
stringlengths 3
182
| url
stringlengths 28
29
| video_url
stringlengths 116
217
| license
stringclasses 25
values | subject
stringlengths 5
70
⌀ | genre
stringclasses 9
values | release_year
stringlengths 4
4
| author
stringlengths 2
232
⌀ | contributors
stringlengths 4
275
⌀ | abstract
stringlengths 31
6.18k
| transcript
stringlengths 305
159k
| transcript_segments
sequence | keyframes
sequence | language
stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
10.5446/59320 (DOI) | O-minimal flows on nilmanifolds | https://av.tib.eu/media/59320 | https://tib.flowcenter.de/mfc/medialink/3/de6e7c1fc3e3c35e08b92cfe10be5e1533a69729f48727661e9482dfb99060ece7/201810150900-Peterzil_lrv.mp4 | CC Attribution - NonCommercial - NoDerivatives 4.0 International:
You are free to use, copy, distribute and transmit the work or content in unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor. | Mathematics | Workshop/Interactive Format
Lecture | 2018 | Peterzil, Kobi | null | Let G be a real algebraic unipotent group and let Lambda be a lattice in G, with p:G->G/Lambda the quotient map. Given a definable subset X of G, in some o-minimal expansion of the reals, we describe the closure of p(X) in G/Lambda in terms definable families of cosets of real algebraic subgroups of G of positive dimension. The family is extracted from X independently of Lambda. | So, okay, what I'll talk about is a subject that, again, I have been working on for the past two years and slowly improving various results. And I spoke about basically the same results in Oxford, so I apologize to people who have been there. And let me start right away by stating a problem. I hope people in the back can see the board. So this is the problem in its most generality. Let G in GL and R be real algebraic. Let gamma in G be a lattice. Now, I will not need it, but this is a discrete group. Eventually, we'll specialize to a simpler case, but this is a discrete group such that the hard measure which is induced on G mod gamma is finite. And let pi from G to G mod gamma be the usual map, which is pi of G equals G gamma. Now let's fix for the rest of the talk, R bar, to be O minimal expansion, expansion of the real field. And we take X in G to be R definable. Can you see in the back? So wrap down here, not so much. Let's do narrow boards. We have X in G R definable. Now we are looking at the image of X inside G mod gamma and of X inside G mod gamma. So gamma is not definable. No, no, no. Very much not definable. It's a discrete group. And the question is what is the topological closure of pi X in G mod gamma? OK, as we will see in a second, this is two general. But I should say that this is already, in some cases like this, we know this even from, well, recent work in model theory. For example, we know that you can look at this and onto an abelian variety. And if you replace the question, you start, let's say, with an algebraic variety inside CN. And instead of asking about the topological closure, you ask about the Zariski closure, then this is what's called the X Lindemann-Weierstrass theorem, because we know that if the variety was irreducible, the image is a coset of an abelian sub-variety of A. And then about a year ago, maybe two years ago, Umer and Yafayev looked at this problem and they asked the question, what can we say about the topological closure of the image of an abelian variety? And here and here, of course, the lattice is like R to the 2n in general. Z to the 2n. Z to the 2n. OK. And first of all, this is two general. The problem, as I stated here, in the sense that we cannot give good answers in general. And the example usually, you take G to be S of L2R. You take gamma to be S L2Z. And already here, we have very simple, definable subsets of G whose image is the closure of the image is something like a fractal, like a counter-set. And for example, if you take D to be the diagonal group and you choose G properly and you look at the definable set being just the coset of DG, then you can find G such that the closure of pi x is very complicated, this fractal. So x is definable. It's definable. It's a very simple set, but the closure of the image is very complicated. So what about D itself? No, the image of D, I think that this will be a lattice in D. So the image of D probably will just be a circle, the closure. I would think that this is closed, I think. OK. Now, of course, the problem, this type of problem, a problem is coming from egodic theory. So this is a dynamical system problem. And the classical theorem of Ratner, 1 from 1994, and I'm forming it, OK, I will say something later in the closure version of the theorem. The theorem is about, is formulated in terms of measure. Also I'll come back to this, but I'm forming it in the topological language. If H in G is a unipotent group, then the closure of the image of all orbits is nice. And then there exists another algebraic group, there exists F in G, algebraic group, such that the closure of pi, so then, sorry, then for every little g, then for every little g in G, there exists F, algebraic group, such that the closure of pi H G is just pi G. So the closure is itself the image of another orbit. So the closure is an F, an F flow, I can say. So the closure of pi of the orbit is itself an orbit, an orbit under F inside g mod gamma. Is there any unipotent group? Maybe they can do, it's enough that it's generated by unipotent groups, but for us this is what we do. F depends on H and on G, but the dependence on G is up to conjugate, not more than that. We tend to be looking at cosets, yes, yes indeed, indeed, we'll see why. But this is the starting point of what we need. Now what I want is to start working inside G and not in the image, and well let me start with that. So if we take G to be just the identity, then we will get that the closure of pi H is just the closure, it's just, sorry, pi of F. And if we pull it back into G, which will be more convenient for us, then instead of working in pi, in the quotient we'll work in G, what it says is that the closure of H gamma is exactly F gamma. Instead of talking of the closure of pi H, I can talk about the closure of H gamma and then take pi, it's the same thing. So now this is the closure I should say inside G. So we're in a situation where the closure of H gamma is a nice, it's a group times gamma. Now notice that if G was a billion, like in Cn or in Rn, this is very easy, why? Because H gamma is a group itself. Right? It's in a billion group, H gamma is a group, so its closure is a lig group. And then you just take the connected component of the identity and this will be your F in this case. But of course once we move out of the a billion case, this is not a group anymore and so there is no reason why you should be able to describe this as a group times gamma. In this case, all connect, okay, so you're right. But in the case of a unipotent group, which we will have eventually, all groups, all connected groups are real algebraic. Okay, now I want to give a name for this, so I will not go F actually, it can be seen. One way to read F is the smallest subgroup, real algebraic group containing H such that gamma is a lattice in F. What do we mean by that? It just means that again when you look at the measure restricted to F mod gamma, you get a bounded measure. But I will not go into that, I want to give a name for this F. It's uniquely determined and I want to write F from now on as H gamma. So given H, we find an F containing it, which is nice in this sense. The closure of H gamma is F gamma, so this is what I want to take out of Rottman's theorem is the existence of H like this and a name for it. Containing H. Containing H, yes. That H is unipotent. Being here is for unipotent H. Contains F, contains H, yes, contained. Because once you work with the cosets, it's not true unnecessarily. Anymore you have to bring conjugates into. Yes, yes, yes. Okay, now I'm going to simplify much the situation. We're going to leave the general Lie group situation and assume from now on that G is a unipotent group itself. So in particular, all the real algebraic subgroups of G will be unipotent. So from now on, and this is the setting of our theorem, G itself is unipotent, which for us will mean that G is a real algebraic subgroup of the upper triangular matrices with one under diagonal. Up to conjugation, this is true, which for us will mean that G is a real algebraic subgroup of the upper triangular matrices. So this is just 1, 1, 1 here, 0, and real algebraic. Actually all connected subgroups are real algebraic. Another way to characterize a group like this, if you don't want to work with matrix group, these are connected, simply connected, and unipotent groups. And now, maybe I'll put it here because I want to leave the theorem on the left, it is a fact that for unipotent groups, lattices are what we think of lattices in the Abelian case. So fact, a discrete gamma in G for G unipotent. In the lattice, which means the measure is bounded if and only if G mod gamma is compact. So from now on, we may as well assume that G mod gamma is compact whenever I say lattice. And now I will state the theorem that we'll be proving, we're not proving, talking about its proof. Can I have some water? Is this water? No. Is this some water maybe? Sorry, I should have prepared in advance. I think my mouse is slowly, slowly disappearing. Yeah, thanks very much. Great, much better. Okay, so let's say I want to fit everything into this board, so let's say that R bar is again O minimal and we have X in G definable. And to make it easier, let's make it closed already, because anyway we're going to take closure in a second. So let's just assume that X is closed because we're interested in closure anyway. And I don't have a lattice yet. Oh, great. Thanks very much. I don't have a lattice yet, but already we can extract information from X. Okay, then there exists a number R in N. There exists finitely many real algebraic subgroups of G, positive dimension. There exists C1, CR definable sets closed. Might as well take them closed. Closed. So such that all of this before gamma was chosen. For every lattice gamma in G, we can now describe the closure of, and now I will describe it inside G and I will say something about the projection, but it's more convenient for me here to describe it, the closure inside G, describing the closure of X times G. Okay, I can write it here. The closure of X gamma is the following finite union. First of all, you take X, of course you need X. And to X, you add the following finitely many sets. I goes from one to R, of CI HI gamma, so now I'm using the notation from there, so every HI, depending on gamma, I take the smallest gamma rational group containing HI and you need to add. I'm actually using the graph that's given me. This does not make sense otherwise. And I guess we have to multiply by gamma. So in some sense, what we are doing is reducing the closure problem on arbitrary definable sets to groups. As we will see, these will be exactly groups that sit on X at infinity, somehow that's stabilizing some sense, are affiliated to X at infinity. If I wanted to present it in the projection, then we will take the projection here and then we'll just get the projection of this. Okay, so the closure of the projection is just pi of this. Notice one thing, which is, I think, not obvious, that even though we have this gamma on the way, this is a definable set. It's families, you have finitely many groups and you have finitely many families of these cosets of these groups, so this is itself R definable. Of course, the closure is not R definable because we have gamma, we cannot avoid gamma, but it's an R definable set times gamma. And moreover, it's important, we can say the following, for each i from 1 to R, the dimension of C i is less than the dimension of X. Here, I multiply everything by gamma. Ah, ah, ah, this is not necessary. Thank you, this is not. It's good, someone counting parentheses. It's important. The second thing is, I'm not sure if we'll get to it, but if you take the maximal, so some of these have no h i which is contained in some have contained h i above them, but for h i's which are maximal with respect to inclusion, the C i's are actually bounded and then this set is actually closed set already. So some of these sets are not closed, only they become closed when we take the union like we're adding boundary components, but the maximal ones are closed, so C i is bounded and so it's easy to see. When you take a compact set times an algebraic group, you get still closer. Okay, let me say some remarks about this. Maybe we'll get more than remarks, maybe not. In the collection? Yes, in the collection, yeah. Why do you actually take h i and that they are real algebraic and h i gamma is also real algebraic, so why don't you take h i gamma to start with? I could, but then it would be a weaker theorem because now the same h i work for all gammas. You just have to take the gamma closure of the h i. So you could have formulated the result for each gamma separately, but there's something strong about that because you have like an a priori family that we will see we extract from x and we just have to apply the Ratner results to each of the groups to get the closure. Okay let me make some comments. Let's take the case when x is a curve. Let's look what this theorem says when the dimension of x is 1. So x is just a curve. We'll see an example, I'll put an example in a second. In this case, by one, the c i's are finite. So then we really, all we are doing is taking the curve and adding to it finitely many cosets and the closure then is obtained by adding to the curve finitely many cosets and that's it, you don't need more. Then because the dimension of c i is 0 is finite and then the closure of x gamma is just x union some finitely many maybe not the same r from k g i h i gamma. So you only need to add finitely many cosets and all of this. So let's do the example that if you've heard Sergei and I talk about it, this is the example I usually give, so let's look at this so we can embed this in g. This is not unimportant but it's algebraically isomorphic to an important group. Now let's take x, the hyperbola, all the x y, x y is 1 and to simplify let's just look at the first quadrant. Let's take gamma to be just z square. So we have this square and it's not hard to see that when you take z square closure you basically moving along these directions and what you will be adding is these two groups. So this is like h1, there is no point here, it's really a group and this is h2 and the closure let me write it additively because we are now inside r square is exactly x union h1 union h2 well plus z square. Here's however, okay I will stick with it, we'll do maybe another example later. I should say that of course what helps here is misleading that h1 and h1 gamma are the same because h1 is already z square group, the intersection of z square with h1 is a lattice, the intersection of h2 with z square is a lattice so in this case there was no need to take h1 z square or h2 z square. If I change the lattice and make it an irrational lattice then h1 and h2 both will become the whole of r2 so if I replace this by an irrational lattice then the closure of x plus the lattice will be everything, will be the whole of r2. So again notice that this operation is missing here because this is the same as h1 if you want z square and this is the same both of them are rational with respect to the lattice z square. Okay second remark. Actually turns out there's a theorem like this exists in the literature of robotic theory of well more or less there's a theorem sorry of Shah from 1994 actually maybe Ratner is not 1994 maybe slightly earlier 1992 which is more general than what I would say now but for us it will be enough. He did not even need to say that G is unipotent. Assume that you have p of x a polynomial map in some variable xd a real polynomial map and you take x to be exactly the image of the polynomial map. R2G yes thank you so it's polynomial map in in it's not in R sorry it's several variables so I should say pij and maybe write pij a map from Rd things into G. Assume that the image of the polynomial land inside G. It's good someone is counting for it. No you're right you're right you're right. Okay then there's a very strong version of this result because what you have to do is the following then let gh be the smallest set of a real algebraic subgroup subgroup H of G containing x. So just take the intersection of all cosets of real algebraic groups which contain x then for every lattice for okay so let me just say whenever I write gamma I mean a lattice in G the closure of x gamma is exactly the closure of gh gamma which is just gh gamma gamma. So actually when you take the image of a polynomial map as your definable set which is obviously definable then all you need is one coset and it captures the whole closure. And if I have time we'll come back and see how we can deduce this result from our work. But I should say that its theorem did not even assume unipotent but then the notion of a polynomial map you have to be slightly more careful about. Sorry yeah all my gammas now as I said it all my gammas will be lattice. Third remark. This was two I guess. This will be three and I'll say it in words maybe and then I'll say something more precise. As I said the theorem from a Goddick theory both of Scha and of Ratner are not formulated so much in terms of closure actually in Scha you have to extract the closure but it's formulated in terms of a Goddick theory and in terms of convergence of measure or what is called equidistribution results. And he says here that he makes sense of what it means for X to be equidistributed and then he says that X is actually distributed inside gh gamma. I'll just put I don't want to define equidistribution but still I want to talk about it. I know I know we don't like that. It turns out that how shall I say that that for definable sets in O'Brien's structure I'll give examples even without defining closure I'm sorry that I'm being vague and equidistributions are not the same. Differ of course you don't know what it is so what do you care if it differ. But let me give an example again without defining. Yes yes yes I didn't say equidistribution in the closure actually equidistribution in the closure this is what I should say. But let me give an example and whatever let's look at R2 square with gamma being Z square and X being the curve of flan T ln T greater than 0. So this curve of course is a geometrically a very simple curve is definable in Rx and it follows from what we are doing that the closure of X plus Z square is R square but X is not equidistributed in whatever language whatever it means without going into it but X not equidistributed in R square. Again I'll put it and actually it was interesting I mean without even knowing we were in a meeting in Oxford and Alex will he spent the first part of his talk talking about equidistributions and immediately pointed out that this is not equidistributed so these are not complicated statements what we prove is well at this point again it's a funny to say that theorem when I didn't define it but let me just say without a theorem its observation in polynomially bounded case at least for Rn these are the same so as long as the set or the curve is defined in a polynomially bounded setting then closure if the closure of X is R square then X is equidistributed so I will just say very vaguely but same for polynomially bounded so these are all minimum structures in which function is eventually bounded by a polynomial. Okay so I want to spend the last I guess 15 minutes to say some things about the proof and so I will leave the result and at least maybe give one idea that helps to describe the closure. Okay so let me put some other theory so far there is not much let's take R to be elementary extension of R but to me elementary extension with respect to everything so if you want a big ultra product with respect to full structure not only the minimum structure I just need it in order to talk about the lattice in elementary extension and in order to because I will move from R to elementary extension so let me say put a notation for X in Rn let X sharp denote the realization of X in the big structure and now as usual we have the valuation of all the alpha in R such that is bounded by some n there is n in n and we let mu be the ring of the ideal of infinitesimal so the maximum ideal alpha in R such that for every alpha sorry for every n. But actually I am interested in O and U on G on the group G not so much in R and what helps us we could have managed without it but what helps us is that when notice that when G is in U, T and R then actually G is closed in Rn square M and R right because the diagonal is one we cannot approach the determinant is one we cannot approach elements with determinant zero and then let me write OG just to be O intersection G and mu G to be mu plus I, I is the identity matrix intersection with G. Yeah yeah O to the n square thank you O to the n square and mu to the n square thanks. And now we have the standard part map going from OG into G so G is the real points I remind you I am calling G sharp the extension so we have a standard part my going from here to here and the kernel is exactly mu of G and in fact O of G is well it is a group mu of G is normally in O of G and O of G is the semi direct product of mu G and G and the R points this is the R points. So above it should be G sharp. Here you mean? Yeah. I, I, yeah this should be G sharp this is your answer this should be G sharp I prefer the notation to leave it so G but you are right this should be G sharp thanks. Yeah yeah. And just to leave this to put this notation I am for why any set inside G sharp I will denote in abuse of notation standard part of Y standard part is a partial map but I will write standard part of map standard part of Y to mean the standard part of Y intersection OG. Okay so I should not write this because standard part is not a full map but I will write this and simple observation is so this is just simple facts that we do often when we teach this stuff that for any X in G one way to get the closure of X is to go to elementary extension and take the standard part. This is very easy to see this is the case. So now all of this was to go back to this problem we are trying to understand the closure of X gamma so now we are back to the problem so assume now again X assume X in G is R definable and the closure of X gamma can be written as the standard part of X gamma sharp but this is the same as the standard part of X sharp times gamma sharp and what we want to do and it comes out to have really very nice geometric meaning is to now we obtain the closure as an image of a map and what we want is to define to divide the domain of the map according to the complete types on X so I will write it like this over all complete types in the O minimal language now in the O minimal language of PR times gamma sharp. I am going to take the standard part of X sharp gamma sharp type by type and for each type we will try to understand what this is and notice if we understand for each type what is the standard part then we will get what we want so the problem the heart of the problem is to understand what is the standard part of one type times gamma sharp so what is of P R times gamma sharp and I will do a very very simple example first of all which of course we will not get anything but assume that P is the type of an element which is bounded for alpha inside O of G so it is infinitesimally close to some element of G this is very easy to see then we all what we get here is like the monad neighborhood of the standard part of alpha or not even full monad neighborhood we will get part of the monad neighborhood but we will not get anything more than that when we take the standard part of this PR gamma sharp is exactly the standard part of alpha which is element in the closure of X but we said X is close so it is element of X times gamma so this is obvious because when we take the closure of X gamma we in particular get so this is part of X gamma so here of course this is exactly the contribution of here this part so the bounded types the types which leave inside the standard part will come here will contribute X itself which is of course we have to have X so the interesting part is the unbounded part the unbounded types types which leave at infinity and here we introduce the following notion we will call it nearest coset to a type definition for any alpha in G sharp so in elementary extension and for G in the real world and H in the real world real algebraic we say that G H is near alpha if up to infinity decimal on the left you get there so if alpha is lies in mu G G H or we could put mu G here if mu G alpha intersect G H for example if we take the curve XX square X say 1 over X and we take non-standard element here alpha then of course the X axis is near this point no times the H star yes thank you yes it could be at infinity thank you yes G sharp and the result we prove first that the intersection ok I'll say it like this so one theorem is that indeed there is a nearest for any alpha in G sharp there exists a smallest coset G H near alpha notice that the whole group is near alpha ok so every element has some coset near alpha near it the only issue is what when can you get less than the whole group ok so there is one coset G H which is contained in all other cosets which are near alpha and we can denote it let's call it G alpha H alpha of course G alpha is not unique but H alpha is unique every representative can be chosen and it's easy to see that if alpha is equivalent over R in the O minimal language actually even the semi algebraically semi algebraic language then G beta H beta equals to G alpha H alpha so actually G alpha H alpha is the property of the type of alpha so we will denote it from now on for the next two minutes by G P H P where P is the type of alpha over R so it's really the nearest coset to the type I should say for example this is not true in SL2R if you allow any groups any algebraic group so you can have two cosets of algebraic groups which are near an element but the intersection is empty but not yet in a second in a second so the theorem that we prove and I will finish with that that for every complete type or O minimal type Px in P and G the standard part of PR gamma sharp is exactly well I do it in two steps the closure of GP H P gamma which is the same as GP H P gamma gamma so at infinity what matters is the nearest coset and the nearest coset it what is what determines the closure of PR gamma and as a result just going back to where we left it's part of the work is the same as the closure of X gamma is just the union an infinite union at this point gamma P realizing P type on X so R the realization of P in R P realizing X P type on X let me write it and let me understand V dash P V dash X P implies X okay I'll stop so at the end we have to go of course to here and right now it's a union which is not clear how you handle it it's a union over types so there is more work to do and more models here we actually to get this statement. | {
"id": [
0,
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44,
45,
46,
47,
48,
49,
50,
51,
52,
53,
54,
55,
56,
57,
58,
59,
60,
61,
62,
63,
64,
65,
66,
67,
68,
69,
70,
71,
72,
73,
74,
75,
76,
77,
78,
79,
80,
81,
82,
83,
84,
85,
86,
87,
88,
89,
90,
91,
92,
93,
94,
95,
96,
97,
98,
99,
100,
101,
102,
103,
104,
105,
106,
107,
108,
109,
110,
111,
112,
113,
114,
115,
116,
117,
118,
119,
120,
121,
122,
123,
124,
125,
126,
127,
128,
129,
130,
131,
132,
133,
134,
135,
136,
137,
138,
139,
140,
141,
142,
143,
144,
145,
146,
147,
148,
149,
150,
151,
152,
153,
154,
155,
156,
157,
158,
159,
160,
161,
162,
163,
164,
165,
166,
167,
168,
169,
170,
171,
172,
173,
174,
175,
176,
177,
178,
179,
180,
181,
182,
183,
184,
185,
186,
187,
188,
189,
190,
191,
192,
193,
194,
195,
196,
197,
198,
199,
200,
201,
202,
203,
204,
205,
206,
207,
208,
209,
210,
211,
212,
213,
214,
215,
216,
217,
218,
219,
220,
221,
222,
223,
224,
225,
226,
227,
228,
229,
230,
231,
232,
233,
234,
235,
236,
237,
238,
239,
240,
241,
242,
243,
244,
245,
246,
247,
248,
249,
250,
251,
252,
253,
254,
255,
256,
257,
258,
259,
260,
261,
262,
263,
264,
265,
266,
267,
268,
269,
270,
271,
272,
273,
274,
275,
276,
277,
278,
279,
280,
281,
282,
283,
284,
285,
286,
287,
288,
289,
290,
291,
292,
293,
294,
295,
296,
297,
298,
299,
300,
301,
302,
303,
304,
305,
306,
307,
308,
309,
310,
311,
312,
313,
314,
315,
316,
317,
318,
319,
320,
321,
322,
323,
324,
325,
326,
327,
328,
329,
330,
331,
332,
333,
334,
335,
336,
337,
338,
339,
340,
341,
342,
343,
344,
345,
346,
347,
348,
349,
350,
351,
352,
353,
354,
355,
356,
357,
358,
359,
360,
361,
362,
363,
364,
365,
366,
367,
368,
369,
370,
371,
372,
373,
374,
375,
376,
377,
378,
379,
380,
381,
382,
383,
384,
385,
386,
387,
388,
389,
390,
391,
392
],
"seek": [
0,
0,
0,
0,
0,
2640,
2640,
2640,
5460,
5460,
5460,
5460,
7464,
9384,
9384,
9384,
9384,
12080,
12080,
12080,
14092,
14092,
14092,
14092,
14092,
16220,
16220,
18896,
18896,
18896,
18896,
18896,
21436,
21436,
21436,
21436,
21436,
21436,
24264,
24264,
24264,
24264,
24264,
24264,
24264,
27228,
27228,
27228,
28916,
28916,
31488,
31488,
31488,
31488,
31488,
33744,
33744,
33744,
33744,
33744,
33744,
36224,
36224,
36224,
38512,
38512,
41396,
41396,
43976,
43976,
46320,
46320,
46320,
48784,
48784,
48784,
50804,
50804,
50804,
50804,
53284,
53284,
56248,
56248,
56248,
59028,
59028,
59028,
59028,
59028,
59028,
59028,
61756,
61756,
61756,
61756,
61756,
64496,
64496,
64496,
64496,
64496,
66900,
66900,
66900,
66900,
66900,
66900,
69792,
69792,
69792,
69792,
72520,
72520,
72520,
72520,
72520,
72520,
75512,
75512,
75512,
75512,
75512,
75512,
78172,
78172,
78172,
78172,
80992,
80992,
80992,
83452,
83452,
83452,
85640,
85640,
88200,
88200,
88200,
88200,
88200,
88200,
88200,
88200,
90884,
90884,
90884,
92868,
92868,
92868,
92868,
92868,
95740,
95740,
95740,
95740,
98408,
98408,
98408,
98408,
100720,
100720,
103488,
103488,
103488,
103488,
103488,
106268,
106268,
106268,
106268,
108896,
108896,
108896,
108896,
108896,
111672,
111672,
111672,
111672,
111672,
114668,
114668,
114668,
114668,
117168,
117168,
117168,
117168,
119936,
119936,
119936,
119936,
122568,
122568,
122568,
125184,
125184,
125184,
125184,
125184,
128088,
128088,
128088,
128088,
128088,
130696,
130696,
130696,
130696,
130696,
133260,
133260,
133260,
135964,
135964,
135964,
135964,
135964,
135964,
138936,
138936,
138936,
141100,
141100,
141100,
143724,
143724,
143724,
143724,
146260,
146260,
146260,
149176,
149176,
151128,
151128,
153880,
153880,
153880,
153880,
156400,
156400,
156400,
156400,
158912,
158912,
161688,
161688,
161688,
164552,
164552,
164552,
166428,
168036,
168036,
170264,
170264,
172412,
172412,
174548,
174548,
177440,
177440,
177440,
180388,
180388,
180388,
182932,
182932,
185884,
185884,
185884,
185884,
188304,
188304,
188304,
190792,
190792,
193404,
193404,
193404,
196156,
196156,
198644,
198644,
201212,
201212,
203780,
203780,
203780,
203780,
206540,
206540,
206540,
208464,
208464,
211024,
212372,
212372,
214360,
214360,
214360,
217168,
217168,
219336,
220636,
222484,
222484,
225236,
225236,
225236,
227976,
227976,
229816,
229816,
232228,
232228,
232228,
234776,
234776,
234776,
234776,
234776,
234776,
234776,
237324,
237324,
237324,
239940,
239940,
239940,
242100,
243664,
245188,
245188,
245188,
248088,
249840,
251568,
251568,
251568,
254276,
254276,
254276,
257212,
257212,
257212,
258964,
260512,
260512,
263364,
263364,
263364,
263364,
266360,
266360,
266360,
268840,
268840,
270444,
270444,
273168,
275036,
275036,
277984,
277984,
280464,
280464,
283312,
284280,
286816,
286816,
286816,
289008,
289008,
291144,
291144,
291144,
293840,
293840,
293840,
296216,
297520,
299780,
301440,
301440,
303736,
304660,
306996,
308084,
308084,
310520
],
"start": [
0,
8.279999732971191,
16,
20.84000015258789,
22.31999969482422,
26.399999618530273,
31.040000915527344,
38.439998626708984,
54.599998474121094,
58.880001068115234,
63.63999938964844,
68.19999694824219,
74.63999938964844,
93.83999633789062,
105,
114.44000244140625,
119.23999786376953,
120.80000305175781,
125.68000030517578,
130.39999389648438,
140.9199981689453,
155,
157.52000427246094,
158.52000427246094,
160.52000427246094,
162.1999969482422,
177.60000610351562,
188.9600067138672,
195.8800048828125,
198.0399932861328,
208.16000366210938,
212.47999572753906,
214.36000061035156,
220.55999755859375,
225.47999572753906,
230.9199981689453,
236.75999450683594,
241.63999938964844,
242.63999938964844,
247.63999938964844,
250.63999938964844,
253.63999938964844,
255.39999389648438,
258.760009765625,
264.760009765625,
272.2799987792969,
277.4800109863281,
285.44000244140625,
289.1600036621094,
303.760009765625,
314.8800048828125,
324.2799987792969,
325.2799987792969,
326.2799987792969,
333.0400085449219,
337.44000244140625,
347.32000732421875,
349.32000732421875,
354.44000244140625,
355.44000244140625,
359.4800109863281,
362.239990234375,
379.6000061035156,
381.7200012207031,
385.1199951171875,
392.4800109863281,
413.9599914550781,
428.239990234375,
439.760009765625,
457.760009765625,
463.20001220703125,
473.7799987792969,
482.3599853515625,
487.8399963378906,
495,
496,
508.0400085449219,
520.1199951171875,
523.8800048828125,
531.8400268554688,
532.8400268554688,
551.5599975585938,
562.47998046875,
568.4000244140625,
586.5599975585938,
590.280029296875,
592.6799926757812,
598.5599975585938,
603.3599853515625,
609.6799926757812,
612.4000244140625,
613.4000244140625,
617.5599975585938,
622.1599731445312,
623.7999877929688,
630.239990234375,
633.8400268554688,
644.9600219726562,
650.1199951171875,
653.0399780273438,
658.7999877929688,
665.8800048828125,
669,
675.1199951171875,
676.5999755859375,
681.47998046875,
686.760009765625,
691.5599975585938,
697.9199829101562,
713.760009765625,
715.4400024414062,
719.3599853515625,
725.2000122070312,
731.0800170898438,
735.6799926757812,
739.52001953125,
742.239990234375,
750.3200073242188,
755.1199951171875,
760.0399780273438,
761.7999877929688,
766.3599853515625,
775,
780.5599975585938,
781.719970703125,
790.0800170898438,
793.2000122070312,
801.52001953125,
809.9199829101562,
813.9600219726562,
825.0800170898438,
834.52001953125,
842.1199951171875,
843.5999755859375,
856.4000244140625,
874.0399780273438,
882,
887.3599853515625,
888.3599853515625,
891.3599853515625,
893.3599853515625,
894.3599853515625,
897.0399780273438,
902.8800048828125,
908.8400268554688,
915.8400268554688,
921.8400268554688,
928.6799926757812,
939.47998046875,
944.7999877929688,
946.239990234375,
955.3599853515625,
957.4000244140625,
959.5599975585938,
960.5599975585938,
968.8800048828125,
984.0800170898438,
993.719970703125,
1001.8800048828125,
1005.0800170898438,
1007.2000122070312,
1017.6799926757812,
1034.8800048828125,
1039.1600341796875,
1045.3199462890625,
1049.6400146484375,
1057.52001953125,
1062.6800537109375,
1067.199951171875,
1080.0400390625,
1087.9599609375,
1088.9599609375,
1092.760009765625,
1097.47998046875,
1107.43994140625,
1112.9599609375,
1116.719970703125,
1121.6800537109375,
1133.4000244140625,
1138.800048828125,
1141.0799560546875,
1146.6800537109375,
1156.760009765625,
1161.5999755859375,
1167.9200439453125,
1171.6800537109375,
1177.760009765625,
1181.5999755859375,
1195.9599609375,
1199.3599853515625,
1209.3599853515625,
1218.3599853515625,
1222.199951171875,
1225.6800537109375,
1234.9599609375,
1241.280029296875,
1251.8399658203125,
1262.6800537109375,
1265.52001953125,
1269.43994140625,
1278.8800048828125,
1280.8800048828125,
1290.8800048828125,
1299.9599609375,
1301.9599609375,
1303.9599609375,
1306.9599609375,
1314.6800537109375,
1319.1600341796875,
1324.8399658203125,
1328,
1332.5999755859375,
1336.3599853515625,
1351.0400390625,
1359.6400146484375,
1364.1199951171875,
1368.4000244140625,
1370.4000244140625,
1376,
1382.6800537109375,
1389.3599853515625,
1396.0799560546875,
1398.800048828125,
1411,
1421.6800537109375,
1431.4000244140625,
1437.239990234375,
1442.719970703125,
1447.52001953125,
1456.800048828125,
1462.5999755859375,
1469.3199462890625,
1483.239990234375,
1491.760009765625,
1501.239990234375,
1511.280029296875,
1528.47998046875,
1538.800048828125,
1546.3599853515625,
1553.1600341796875,
1559.199951171875,
1564,
1571.0799560546875,
1576.800048828125,
1582,
1589.1199951171875,
1596.8800048828125,
1616.8800048828125,
1624.9599609375,
1631.4000244140625,
1645.52001953125,
1657.52001953125,
1660.5999755859375,
1664.280029296875,
1680.3599853515625,
1696.4000244140625,
1702.6400146484375,
1711.800048828125,
1724.1199951171875,
1732.0799560546875,
1745.47998046875,
1762.1199951171875,
1774.4000244140625,
1784.52001953125,
1797.6400146484375,
1803.8800048828125,
1813.1600341796875,
1824.199951171875,
1829.3199462890625,
1837.4000244140625,
1858.8399658203125,
1867.800048828125,
1873.1199951171875,
1878.239990234375,
1883.0400390625,
1890.47998046875,
1902.0799560546875,
1907.9200439453125,
1921.0799560546875,
1934.0400390625,
1945.1199951171875,
1951.6800537109375,
1961.56005859375,
1968.1199951171875,
1986.43994140625,
2002,
2012.1199951171875,
2019.8800048828125,
2037.800048828125,
2044.3199462890625,
2049.199951171875,
2061.239990234375,
2065.39990234375,
2070.919921875,
2076.760009765625,
2084.639892578125,
2095.39990234375,
2110.239990234375,
2123.719970703125,
2139.60009765625,
2143.60009765625,
2155.800048828125,
2161.280029296875,
2171.679931640625,
2181.52001953125,
2193.360107421875,
2206.360107421875,
2224.840087890625,
2236.9599609375,
2252.360107421875,
2260.320068359375,
2265.47998046875,
2279.760009765625,
2285.8798828125,
2298.159912109375,
2313.080078125,
2322.280029296875,
2329.47998046875,
2335.52001953125,
2347.760009765625,
2353.8798828125,
2356.679931640625,
2360.719970703125,
2361.719970703125,
2362.719970703125,
2368.080078125,
2373.239990234375,
2382.719970703125,
2392.360107421875,
2399.39990234375,
2405.080078125,
2406.080078125,
2421,
2436.639892578125,
2451.8798828125,
2467.199951171875,
2472.719970703125,
2480.8798828125,
2498.39990234375,
2515.679931640625,
2530.159912109375,
2535.39990234375,
2542.760009765625,
2551.199951171875,
2568.840087890625,
2572.1201171875,
2577.800048828125,
2583.56005859375,
2589.639892578125,
2605.1201171875,
2623.080078125,
2633.639892578125,
2647.0400390625,
2653.39990234375,
2658.760009765625,
2663.60009765625,
2674.199951171875,
2683.43994140625,
2688.39990234375,
2697.360107421875,
2704.43994140625,
2721.9599609375,
2731.679931640625,
2750.360107421875,
2765.47998046875,
2779.840087890625,
2796.639892578125,
2804.639892578125,
2816.239990234375,
2833.1201171875,
2842.800048828125,
2868.159912109375,
2873.840087890625,
2882.239990234375,
2890.080078125,
2902.43994140625,
2911.43994140625,
2923.679931640625,
2928.919921875,
2938.39990234375,
2948.320068359375,
2954.1201171875,
2962.159912109375,
2975.199951171875,
2997.800048828125,
3014.39990234375,
3029,
3037.360107421875,
3046.60009765625,
3069.9599609375,
3080.840087890625,
3100.840087890625,
3105.199951171875
],
"end": [
8.279999732971191,
16,
20.84000015258789,
22.31999969482422,
26.399999618530273,
31.040000915527344,
38.439998626708984,
54.599998474121094,
58.880001068115234,
63.63999938964844,
68.19999694824219,
74.63999938964844,
93.83999633789062,
105,
114.44000244140625,
119.23999786376953,
120.80000305175781,
125.68000030517578,
130.39999389648438,
140.9199981689453,
155,
157.52000427246094,
158.52000427246094,
160.52000427246094,
162.1999969482422,
177.60000610351562,
188.9600067138672,
195.8800048828125,
198.0399932861328,
208.16000366210938,
212.47999572753906,
214.36000061035156,
220.55999755859375,
225.47999572753906,
230.9199981689453,
236.75999450683594,
241.63999938964844,
242.63999938964844,
247.63999938964844,
250.63999938964844,
253.63999938964844,
255.39999389648438,
258.760009765625,
264.760009765625,
272.2799987792969,
277.4800109863281,
285.44000244140625,
289.1600036621094,
303.760009765625,
314.8800048828125,
324.2799987792969,
325.2799987792969,
326.2799987792969,
333.0400085449219,
337.44000244140625,
347.32000732421875,
349.32000732421875,
354.44000244140625,
355.44000244140625,
359.4800109863281,
362.239990234375,
379.6000061035156,
381.7200012207031,
385.1199951171875,
392.4800109863281,
413.9599914550781,
428.239990234375,
439.760009765625,
457.760009765625,
463.20001220703125,
473.7799987792969,
482.3599853515625,
487.8399963378906,
495,
496,
508.0400085449219,
520.1199951171875,
523.8800048828125,
531.8400268554688,
532.8400268554688,
551.5599975585938,
562.47998046875,
568.4000244140625,
586.5599975585938,
590.280029296875,
592.6799926757812,
598.5599975585938,
603.3599853515625,
609.6799926757812,
612.4000244140625,
613.4000244140625,
617.5599975585938,
622.1599731445312,
623.7999877929688,
630.239990234375,
633.8400268554688,
644.9600219726562,
650.1199951171875,
653.0399780273438,
658.7999877929688,
665.8800048828125,
669,
675.1199951171875,
676.5999755859375,
681.47998046875,
686.760009765625,
691.5599975585938,
697.9199829101562,
713.760009765625,
715.4400024414062,
719.3599853515625,
725.2000122070312,
731.0800170898438,
735.6799926757812,
739.52001953125,
742.239990234375,
750.3200073242188,
755.1199951171875,
760.0399780273438,
761.7999877929688,
766.3599853515625,
775,
780.5599975585938,
781.719970703125,
790.0800170898438,
793.2000122070312,
801.52001953125,
809.9199829101562,
813.9600219726562,
825.0800170898438,
834.52001953125,
842.1199951171875,
843.5999755859375,
856.4000244140625,
874.0399780273438,
882,
887.3599853515625,
888.3599853515625,
891.3599853515625,
893.3599853515625,
894.3599853515625,
897.0399780273438,
902.8800048828125,
908.8400268554688,
915.8400268554688,
921.8400268554688,
928.6799926757812,
939.47998046875,
944.7999877929688,
946.239990234375,
955.3599853515625,
957.4000244140625,
959.5599975585938,
960.5599975585938,
968.8800048828125,
984.0800170898438,
993.719970703125,
1001.8800048828125,
1005.0800170898438,
1007.2000122070312,
1017.6799926757812,
1034.8800048828125,
1039.1600341796875,
1045.3199462890625,
1049.6400146484375,
1057.52001953125,
1062.6800537109375,
1067.199951171875,
1080.0400390625,
1087.9599609375,
1088.9599609375,
1092.760009765625,
1097.47998046875,
1107.43994140625,
1112.9599609375,
1116.719970703125,
1121.6800537109375,
1133.4000244140625,
1138.800048828125,
1141.0799560546875,
1146.6800537109375,
1156.760009765625,
1161.5999755859375,
1167.9200439453125,
1171.6800537109375,
1177.760009765625,
1181.5999755859375,
1195.9599609375,
1199.3599853515625,
1209.3599853515625,
1218.3599853515625,
1222.199951171875,
1225.6800537109375,
1234.9599609375,
1241.280029296875,
1251.8399658203125,
1262.6800537109375,
1265.52001953125,
1269.43994140625,
1278.8800048828125,
1280.8800048828125,
1290.8800048828125,
1299.9599609375,
1301.9599609375,
1303.9599609375,
1306.9599609375,
1314.6800537109375,
1319.1600341796875,
1324.8399658203125,
1328,
1332.5999755859375,
1336.3599853515625,
1351.0400390625,
1359.6400146484375,
1364.1199951171875,
1368.4000244140625,
1370.4000244140625,
1376,
1382.6800537109375,
1389.3599853515625,
1396.0799560546875,
1398.800048828125,
1411,
1421.6800537109375,
1431.4000244140625,
1437.239990234375,
1442.719970703125,
1447.52001953125,
1456.800048828125,
1462.5999755859375,
1469.3199462890625,
1483.239990234375,
1491.760009765625,
1501.239990234375,
1511.280029296875,
1528.47998046875,
1538.800048828125,
1546.3599853515625,
1553.1600341796875,
1559.199951171875,
1564,
1571.0799560546875,
1576.800048828125,
1582,
1589.1199951171875,
1596.8800048828125,
1616.8800048828125,
1624.9599609375,
1631.4000244140625,
1645.52001953125,
1657.52001953125,
1660.5999755859375,
1664.280029296875,
1680.3599853515625,
1696.4000244140625,
1702.6400146484375,
1711.800048828125,
1724.1199951171875,
1732.0799560546875,
1745.47998046875,
1762.1199951171875,
1774.4000244140625,
1784.52001953125,
1797.6400146484375,
1803.8800048828125,
1813.1600341796875,
1824.199951171875,
1829.3199462890625,
1837.4000244140625,
1858.8399658203125,
1867.800048828125,
1873.1199951171875,
1878.239990234375,
1883.0400390625,
1890.47998046875,
1902.0799560546875,
1907.9200439453125,
1921.0799560546875,
1934.0400390625,
1945.1199951171875,
1951.6800537109375,
1961.56005859375,
1968.1199951171875,
1986.43994140625,
2002,
2012.1199951171875,
2019.8800048828125,
2037.800048828125,
2044.3199462890625,
2049.199951171875,
2061.239990234375,
2065.39990234375,
2070.919921875,
2076.760009765625,
2084.639892578125,
2095.39990234375,
2110.239990234375,
2123.719970703125,
2139.60009765625,
2143.60009765625,
2155.800048828125,
2161.280029296875,
2171.679931640625,
2181.52001953125,
2193.360107421875,
2206.360107421875,
2224.840087890625,
2236.9599609375,
2252.360107421875,
2260.320068359375,
2265.47998046875,
2279.760009765625,
2285.8798828125,
2298.159912109375,
2313.080078125,
2322.280029296875,
2329.47998046875,
2335.52001953125,
2347.760009765625,
2353.8798828125,
2356.679931640625,
2360.719970703125,
2361.719970703125,
2362.719970703125,
2368.080078125,
2373.239990234375,
2382.719970703125,
2392.360107421875,
2399.39990234375,
2405.080078125,
2406.080078125,
2421,
2436.639892578125,
2451.8798828125,
2467.199951171875,
2472.719970703125,
2480.8798828125,
2498.39990234375,
2515.679931640625,
2530.159912109375,
2535.39990234375,
2542.760009765625,
2551.199951171875,
2568.840087890625,
2572.1201171875,
2577.800048828125,
2583.56005859375,
2589.639892578125,
2605.1201171875,
2623.080078125,
2633.639892578125,
2647.0400390625,
2653.39990234375,
2658.760009765625,
2663.60009765625,
2674.199951171875,
2683.43994140625,
2688.39990234375,
2697.360107421875,
2704.43994140625,
2721.9599609375,
2731.679931640625,
2750.360107421875,
2765.47998046875,
2779.840087890625,
2796.639892578125,
2804.639892578125,
2816.239990234375,
2833.1201171875,
2842.800048828125,
2868.159912109375,
2873.840087890625,
2882.239990234375,
2890.080078125,
2902.43994140625,
2911.43994140625,
2923.679931640625,
2928.919921875,
2938.39990234375,
2948.320068359375,
2954.1201171875,
2962.159912109375,
2975.199951171875,
2997.800048828125,
3014.39990234375,
3029,
3037.360107421875,
3046.60009765625,
3069.9599609375,
3080.840087890625,
3100.840087890625,
3105.199951171875,
3119.639892578125
],
"text": [
" So, okay, what I'll talk about is a subject that, again, I have been working on for the",
" past two years and slowly improving various results.",
" And I spoke about basically the same results in Oxford, so I apologize to people who have",
" been there.",
" And let me start right away by stating a problem.",
" I hope people in the back can see the board.",
" So this is the problem in its most generality.",
" Let G in GL and R be real algebraic.",
" Let gamma in G be a lattice.",
" Now, I will not need it, but this is a discrete group.",
" Eventually, we'll specialize to a simpler case, but this is a discrete group such that the",
" hard measure which is induced on G mod gamma is finite.",
" And let pi from G to G mod gamma be the usual map, which is pi of G equals G gamma.",
" Now let's fix for the rest of the talk, R bar, to be O minimal expansion, expansion",
" of the real field.",
" And we take X in G to be R definable.",
" Can you see in the back?",
" So wrap down here, not so much.",
" Let's do narrow boards.",
" We have X in G R definable.",
" Now we are looking at the image of X inside G mod gamma and of X inside G mod gamma.",
" So gamma is not definable.",
" No, no, no.",
" Very much not definable.",
" It's a discrete group.",
" And the question is what is the topological closure of pi X in G mod gamma?",
" OK, as we will see in a second, this is two general.",
" But I should say that this is already, in some cases like this, we know this even from,",
" well, recent work in model theory.",
" For example, we know that you can look at this and onto an abelian variety.",
" And if you replace the question, you start, let's say, with an algebraic variety inside",
" CN.",
" And instead of asking about the topological closure, you ask about the Zariski closure,",
" then this is what's called the X Lindemann-Weierstrass theorem, because we know that if the variety",
" was irreducible, the image is a coset of an abelian sub-variety of A.",
" And then about a year ago, maybe two years ago, Umer and Yafayev looked at this problem",
" and they asked the question, what can we say about the topological closure of the image",
" of an abelian variety?",
" And here and here, of course, the lattice is like R to the 2n in general.",
" Z to the 2n.",
" Z to the 2n.",
" OK.",
" And first of all, this is two general.",
" The problem, as I stated here, in the sense that we cannot give good answers in general.",
" And the example usually, you take G to be S of L2R.",
" You take gamma to be S L2Z.",
" And already here, we have very simple, definable subsets of G whose image is the closure of",
" the image is something like a fractal, like a counter-set.",
" And for example, if you take D to be the diagonal group and you choose G properly and you look",
" at the definable set being just the coset of DG, then you can find G such that the closure",
" of pi x is very complicated, this fractal.",
" So x is definable.",
" It's definable.",
" It's a very simple set, but the closure of the image is very complicated.",
" So what about D itself?",
" No, the image of D, I think that this will be a lattice in D. So the image of D probably",
" will just be a circle, the closure.",
" I would think that this is closed, I think.",
" OK.",
" Now, of course, the problem, this type of problem, a problem is coming from egodic theory.",
" So this is a dynamical system problem.",
" And the classical theorem of Ratner, 1 from 1994, and I'm forming it, OK, I will say something",
" later in the closure version of the theorem.",
" The theorem is about, is formulated in terms of measure.",
" Also I'll come back to this, but I'm forming it in the topological language.",
" If H in G is a unipotent group, then the closure of the image of all orbits is nice.",
" And then there exists another algebraic group, there exists F in G, algebraic group, such",
" that the closure of pi, so then, sorry, then for every little g, then for every little",
" g in G, there exists F, algebraic group, such that the closure of pi H G is just pi G.",
" So the closure is itself the image of another orbit.",
" So the closure is an F, an F flow, I can say.",
" So the closure of pi of the orbit is itself an orbit, an orbit under F inside g mod gamma.",
" Is there any unipotent group?",
" Maybe they can do, it's enough that it's generated by unipotent groups, but for us this is what",
" we do.",
" F depends on H and on G, but the dependence on G is up to conjugate, not more than that.",
" We tend to be looking at cosets, yes, yes indeed, indeed, we'll see why.",
" But this is the starting point of what we need.",
" Now what I want is to start working inside G and not in the image, and well let me start",
" with that.",
" So if we take G to be just the identity, then we will get that the closure of pi H is just",
" the closure, it's just, sorry, pi of F. And if we pull it back into G, which will be more",
" convenient for us, then instead of working in pi, in the quotient we'll work in G, what",
" it says is that the closure of H gamma is exactly F gamma.",
" Instead of talking of the closure of pi H, I can talk about the closure of H gamma and",
" then take pi, it's the same thing.",
" So now this is the closure I should say inside G. So we're in a situation where the closure",
" of H gamma is a nice, it's a group times gamma.",
" Now notice that if G was a billion, like in Cn or in Rn, this is very easy, why?",
" Because H gamma is a group itself.",
" Right?",
" It's in a billion group, H gamma is a group, so its closure is a lig group.",
" And then you just take the connected component of the identity and this will be your F in",
" this case.",
" But of course once we move out of the a billion case, this is not a group anymore and so there",
" is no reason why you should be able to describe this as a group times gamma.",
" In this case, all connect, okay, so you're right.",
" But in the case of a unipotent group, which we will have eventually, all groups, all connected",
" groups are real algebraic.",
" Okay, now I want to give a name for this, so I will not go F actually, it can be seen.",
" One way to read F is the smallest subgroup, real algebraic group containing H such that",
" gamma is a lattice in F. What do we mean by that?",
" It just means that again when you look at the measure restricted to F mod gamma, you",
" get a bounded measure.",
" But I will not go into that, I want to give a name for this F. It's uniquely determined",
" and I want to write F from now on as H gamma.",
" So given H, we find an F containing it, which is nice in this sense.",
" The closure of H gamma is F gamma, so this is what I want to take out of Rottman's theorem",
" is the existence of H like this and a name for it.",
" Containing H.",
" Containing H, yes.",
" That H is unipotent.",
" Being here is for unipotent H.",
" Contains F, contains H, yes, contained.",
" Because once you work with the cosets, it's not true unnecessarily.",
" Anymore you have to bring conjugates into.",
" Yes, yes, yes.",
" Okay, now I'm going to simplify much the situation.",
" We're going to leave the general Lie group situation and assume from now on that G is",
" a unipotent group itself.",
" So in particular, all the real algebraic subgroups of G will be unipotent.",
" So from now on, and this is the setting of our theorem, G itself is unipotent, which",
" for us will mean that G is a real algebraic subgroup of the upper triangular matrices with",
" one under diagonal.",
" Up to conjugation, this is true, which for us will mean that G is a real algebraic subgroup",
" of the upper triangular matrices.",
" So this is just 1, 1, 1 here, 0, and real algebraic.",
" Actually all connected subgroups are real algebraic.",
" Another way to characterize a group like this, if you don't want to work with matrix group,",
" these are connected, simply connected, and unipotent groups.",
" And now, maybe I'll put it here because I want to leave the theorem on the left, it",
" is a fact that for unipotent groups, lattices are what we think of lattices in the Abelian",
" case.",
" So fact, a discrete gamma in G for G unipotent.",
" In the lattice, which means the measure is bounded if and only if G mod gamma is compact.",
" So from now on, we may as well assume that G mod gamma is compact whenever I say lattice.",
" And now I will state the theorem that we'll be proving, we're not proving, talking about",
" its proof.",
" Can I have some water?",
" Is this water?",
" No.",
" Is this some water maybe?",
" Sorry, I should have prepared in advance.",
" I think my mouse is slowly, slowly disappearing.",
" Yeah, thanks very much.",
" Great, much better.",
" Okay, so let's say I want to fit everything into this board, so let's say that R bar is",
" again O minimal and we have X in G definable.",
" And to make it easier, let's make it closed already, because anyway we're going to take",
" closure in a second.",
" So let's just assume that X is closed because we're interested in closure anyway.",
" And I don't have a lattice yet.",
" Oh, great.",
" Thanks very much.",
" I don't have a lattice yet, but already we can extract information from X. Okay, then",
" there exists a number R in N. There exists finitely many real algebraic subgroups of",
" G, positive dimension.",
" There exists C1, CR definable sets closed.",
" Might as well take them closed.",
" Closed.",
" So such that all of this before gamma was chosen.",
" For every lattice gamma in G, we can now describe the closure of, and now I will describe it",
" inside G and I will say something about the projection, but it's more convenient for me",
" here to describe it, the closure inside G, describing the closure of X times G. Okay,",
" I can write it here.",
" The closure of X gamma is the following finite union.",
" First of all, you take X, of course you need X.",
" And to X, you add the following finitely many sets.",
" I goes from one to R, of CI HI gamma, so now I'm using the notation from there, so every",
" HI, depending on gamma, I take the smallest gamma rational group containing HI and you",
" need to add.",
" I'm actually using the graph that's given me.",
" This does not make sense otherwise.",
" And I guess we have to multiply by gamma.",
" So in some sense, what we are doing is reducing the closure problem on arbitrary definable",
" sets to groups.",
" As we will see, these will be exactly groups that sit on X at infinity, somehow that's",
" stabilizing some sense, are affiliated to X at infinity.",
" If I wanted to present it in the projection, then we will take the projection here and",
" then we'll just get the projection of this.",
" Okay, so the closure of the projection is just pi of this.",
" Notice one thing, which is, I think, not obvious, that even though we have this gamma on the",
" way, this is a definable set.",
" It's families, you have finitely many groups and you have finitely many families of these",
" cosets of these groups, so this is itself R definable.",
" Of course, the closure is not R definable because we have gamma, we cannot avoid gamma,",
" but it's an R definable set times gamma.",
" And moreover, it's important, we can say the following, for each i from 1 to R, the dimension",
" of C i is less than the dimension of X.",
" Here, I multiply everything by gamma.",
" Ah, ah, ah, this is not necessary.",
" Thank you, this is not.",
" It's good, someone counting parentheses.",
" It's important.",
" The second thing is, I'm not sure if we'll get to it, but if you take the maximal, so",
" some of these have no h i which is contained in some have contained h i above them, but",
" for h i's which are maximal with respect to inclusion, the C i's are actually bounded",
" and then this set is actually closed set already.",
" So some of these sets are not closed, only they become closed when we take the union",
" like we're adding boundary components, but the maximal ones are closed, so C i is bounded",
" and so it's easy to see.",
" When you take a compact set times an algebraic group, you get still closer.",
" Okay, let me say some remarks about this.",
" Maybe we'll get more than remarks, maybe not.",
" In the collection?",
" Yes, in the collection, yeah.",
" Why do you actually take h i and that they are real algebraic and h i gamma is also real",
" algebraic, so why don't you take h i gamma to start with?",
" I could, but then it would be a weaker theorem because now the same h i work for all gammas.",
" You just have to take the gamma closure of the h i.",
" So you could have formulated the result for each gamma separately, but there's something",
" strong about that because you have like an a priori family that we will see we extract",
" from x and we just have to apply the Ratner results to each of the groups to get the closure.",
" Okay let me make some comments.",
" Let's take the case when x is a curve.",
" Let's look what this theorem says when the dimension of x is 1.",
" So x is just a curve.",
" We'll see an example, I'll put an example in a second.",
" In this case, by one, the c i's are finite.",
" So then we really, all we are doing is taking the curve and adding to it finitely many",
" cosets and the closure then is obtained by adding to the curve finitely many cosets",
" and that's it, you don't need more.",
" Then because the dimension of c i is 0 is finite and then the closure of x gamma is",
" just x union some finitely many maybe not the same r from k g i h i gamma.",
" So you only need to add finitely many cosets and all of this.",
" So let's do the example that if you've heard Sergei and I talk about it, this is the example",
" I usually give, so let's look at this so we can embed this in g.",
" This is not unimportant but it's algebraically isomorphic to an important group.",
" Now let's take x, the hyperbola, all the x y, x y is 1 and to simplify let's just look",
" at the first quadrant.",
" Let's take gamma to be just z square.",
" So we have this square and it's not hard to see that when you take z square closure you",
" basically moving along these directions and what you will be adding is these two groups.",
" So this is like h1, there is no point here, it's really a group and this is h2 and the",
" closure let me write it additively because we are now inside r square is exactly x union",
" h1 union h2 well plus z square.",
" Here's however, okay I will stick with it, we'll do maybe another example later.",
" I should say that of course what helps here is misleading that h1 and h1 gamma are the",
" same because h1 is already z square group, the intersection of z square with h1 is a",
" lattice, the intersection of h2 with z square is a lattice so in this case there was no",
" need to take h1 z square or h2 z square.",
" If I change the lattice and make it an irrational lattice then h1 and h2 both will become the",
" whole of r2 so if I replace this by an irrational lattice then the closure of x plus the lattice",
" will be everything, will be the whole of r2.",
" So again notice that this operation is missing here because this is the same as h1 if you",
" want z square and this is the same both of them are rational with respect to the lattice",
" z square.",
" Okay second remark.",
" Actually turns out there's a theorem like this exists in the literature of robotic",
" theory of well more or less there's a theorem sorry of Shah from 1994 actually maybe Ratner",
" is not 1994 maybe slightly earlier 1992 which is more general than what I would say now but",
" for us it will be enough.",
" He did not even need to say that G is unipotent.",
" Assume that you have p of x a polynomial map in some variable xd a real polynomial map",
" and you take x to be exactly the image of the polynomial map.",
" R2G yes thank you so it's polynomial map in in it's not in R sorry it's several variables",
" so I should say pij and maybe write pij a map from Rd things into G. Assume that the image",
" of the polynomial land inside G. It's good someone is counting for it.",
" No you're right you're right you're right. Okay then there's a very strong version of",
" this result because what you have to do is the following then let gh be the smallest",
" set of a real algebraic subgroup subgroup H of G containing x. So just take the intersection",
" of all cosets of real algebraic groups which contain x then for every lattice for okay",
" so let me just say whenever I write gamma I mean a lattice in G the closure of x gamma",
" is exactly the closure of gh gamma which is just gh gamma gamma.",
" So actually when you take the image of a polynomial map as your definable set which",
" is obviously definable then all you need is one coset and it captures the whole closure.",
" And if I have time we'll come back and see how we can deduce this result from our work.",
" But I should say that its theorem did not even assume unipotent but then the notion",
" of a polynomial map you have to be slightly more careful about. Sorry yeah all my gammas",
" now as I said it all my gammas will be lattice. Third remark. This was two I guess. This will",
" be three and I'll say it in words maybe and then I'll say something more precise. As I",
" said the theorem from a Goddick theory both of Scha and of Ratner are not formulated so",
" much in terms of closure actually in Scha you have to extract the closure but it's formulated",
" in terms of a Goddick theory and in terms of convergence of measure or what is called",
" equidistribution results. And he says here that he makes sense of what it means for",
" X to be equidistributed and then he says that X is actually distributed inside gh gamma.",
" I'll just put I don't want to define equidistribution but still I want to talk about it. I know I",
" know we don't like that. It turns out that how shall I say that that for definable sets",
" in O'Brien's structure I'll give examples even without defining closure I'm sorry that",
" I'm being vague and equidistributions are not the same. Differ of course you don't know",
" what it is so what do you care if it differ. But let me give an example again without defining.",
" Yes yes yes I didn't say equidistribution in the closure actually equidistribution in",
" the closure this is what I should say. But let me give an example and whatever let's",
" look at R2 square with gamma being Z square and X being the curve of flan T ln T greater",
" than 0. So this curve of course is a geometrically a very simple curve is definable in Rx and",
" it follows from what we are doing that the closure of X plus Z square is R square but",
" X is not equidistributed in whatever language whatever it means without going into it but",
" X not equidistributed in R square. Again I'll put it and actually it was interesting I",
" mean without even knowing we were in a meeting in Oxford and Alex will he spent the first",
" part of his talk talking about equidistributions and immediately pointed out that this is not",
" equidistributed so these are not complicated statements what we prove is well at this point",
" again it's a funny to say that theorem when I didn't define it but let me just say without",
" a theorem its observation in polynomially bounded case at least for Rn these are the",
" same so as long as the set or the curve is defined in a polynomially bounded setting",
" then closure if the closure of X is R square then X is equidistributed so I will just say",
" very vaguely but same for polynomially bounded so these are all minimum structures in which",
" function is eventually bounded by a polynomial.",
" Okay so I want to spend the last I guess 15 minutes to say some things about the proof",
" and so I will leave the result and at least maybe give one idea that helps to describe",
" the closure.",
" Okay so let me put some other theory so far there is not much let's take R to be elementary",
" extension of R but to me elementary extension with respect to everything so if you want",
" a big ultra product with respect to full structure not only the minimum structure I just need",
" it in order to talk about the lattice in elementary extension and in order to because I will move",
" from R to elementary extension so let me say put a notation for X in Rn let X sharp denote",
" the realization of X in the big structure and now as usual we have the valuation of",
" all the alpha in R such that is bounded by some n there is n in n and we let mu be the",
" ring of the ideal of infinitesimal so the maximum ideal alpha in R such that for every",
" alpha sorry for every n.",
" But actually I am interested in O and U on G on the group G not so much in R and what",
" helps us we could have managed without it but what helps us is that when notice that",
" when G is in U, T and R then actually G is closed in Rn square M and R right because",
" the diagonal is one we cannot approach the determinant is one we cannot approach elements",
" with determinant zero and then let me write OG just to be O intersection G and mu G to",
" be mu plus I, I is the identity matrix intersection with G. Yeah yeah O to the n square thank you",
" O to the n square and mu to the n square thanks.",
" And now we have the standard part map going from OG into G so G is the real points I",
" remind you I am calling G sharp the extension so we have a standard part my going from here",
" to here and the kernel is exactly mu of G and in fact O of G is well it is a group mu",
" of G is normally in O of G and O of G is the semi direct product of mu G and G and the",
" R points this is the R points.",
" So above it should be G sharp.",
" Here you mean?",
" Yeah.",
" I, I, yeah this should be G sharp this is your answer this should be G sharp I prefer",
" the notation to leave it so G but you are right this should be G sharp thanks.",
" Yeah yeah.",
" And just to leave this to put this notation I am for why any set inside G sharp I will",
" denote in abuse of notation standard part of Y standard part is a partial map but I will",
" write standard part of map standard part of Y to mean the standard part of Y intersection",
" OG.",
" Okay so I should not write this because standard part is not a full map but I will write this",
" and simple observation is so this is just simple facts that we do often when we teach",
" this stuff that for any X in G one way to get the closure of X is to go to elementary",
" extension and take the standard part.",
" This is very easy to see this is the case.",
" So now all of this was to go back to this problem we are trying to understand the closure",
" of X gamma so now we are back to the problem so assume now again X assume X in G is R definable",
" and the closure of X gamma can be written as the standard part of X gamma sharp but",
" this is the same as the standard part of X sharp times gamma sharp",
" and what we want to do and it comes out to have really very nice geometric meaning is",
" to now we obtain the closure as an image of a map and what we want is to define to divide",
" the domain of the map according to the complete types on X so I will write it like this over",
" all complete types in the O minimal language now in the O minimal language of PR times",
" gamma sharp.",
" I am going to take the standard part of X sharp gamma sharp type by type and for each",
" type we will try to understand what this is and notice if we understand for each type",
" what is the standard part then we will get what we want so the problem the heart of the",
" problem is to understand what is the standard part of one type times gamma sharp so what",
" is of P R times gamma sharp and I will do a very very simple example first of all which",
" of course we will not get anything but assume that P is the type of an element which is",
" bounded for alpha inside O of G so it is infinitesimally close to some element of G",
" this is very easy to see then we all what we get here is like the monad neighborhood",
" of the standard part of alpha or not even full monad neighborhood we will get part of",
" the monad neighborhood but we will not get anything more than that when we take the standard",
" part of this PR gamma sharp is exactly the standard part of alpha which is element in",
" the closure of X but we said X is close so it is element of X times gamma so this is",
" obvious because when we take the closure of X gamma we in particular get so this is part",
" of X gamma so here of course this is exactly the contribution of here this part so the",
" bounded types the types which leave inside the standard part will come here will contribute",
" X itself which is of course we have to have X so the interesting part is the unbounded",
" part the unbounded types types which leave at infinity and here we introduce the following",
" notion we will call it nearest coset to a type definition for any alpha in G sharp so",
" in elementary extension and for G in the real world and H in the real world real algebraic",
" we say that G H is near alpha if up to infinity decimal on the left you get there so if alpha",
" is lies in mu G G H or we could put mu G here if mu G alpha intersect G H for example if",
" we take the curve XX square X say 1 over X and we take non-standard element here alpha",
" then of course the X axis is near this point no times the H star yes thank you yes it could",
" be at infinity thank you yes G sharp and the result we prove first",
" that the intersection ok I'll say it like this so one theorem is that indeed there is",
" a nearest for any alpha in G sharp there exists a smallest coset G H near alpha notice that",
" the whole group is near alpha ok so every element has some coset near alpha near it",
" the only issue is what when can you get less than the whole group ok so there is one coset",
" G H which is contained in all other cosets which are near alpha and we can denote it",
" let's call it G alpha H alpha of course G alpha is not unique but H alpha is unique",
" every representative can be chosen and it's easy to see that if alpha is equivalent over",
" R in the O minimal language actually even the semi algebraically semi algebraic language",
" then G beta H beta equals to G alpha H alpha so actually G alpha H alpha is the property",
" of the type of alpha so we will denote it from now on for the next two minutes by G",
" P H P where P is the type of alpha over R so it's really the nearest coset to the type",
" I should say for example this is not true in SL2R if you allow any groups any algebraic",
" group so you can have two cosets of algebraic groups which are near an element but the intersection",
" is empty but not yet in a second in a second so the theorem that we prove and I will finish",
" with that that for every complete type or O minimal type Px in P and G the standard part",
" of PR gamma sharp is exactly well I do it in two steps the closure of GP H P gamma which",
" is the same as GP H P gamma gamma so at infinity what matters is the nearest coset and the",
" nearest coset it what is what determines the closure of PR gamma and as a result just going",
" back to where we left it's part of the work is the same as the closure of X gamma is just",
" the union an infinite union at this point gamma P realizing P type on X so R the realization",
" of P in R P realizing X P type on X let me write it and let me understand V dash P V",
" dash X P implies X okay I'll stop so at the end we have to go of course to here and right",
" now it's a union which is not clear how you handle it it's a union over types so there",
" is more work to do and more models here we actually to get this statement."
],
"tokens": [
[
407,
11,
1392,
11,
437,
286,
603,
751,
466,
307,
257,
3983,
300,
11,
797,
11,
286,
362,
668,
1364,
322,
337,
264
],
[
1791,
732,
924,
293,
5692,
11470,
3683,
3542,
13
],
[
400,
286,
7179,
466,
1936,
264,
912,
3542,
294,
24786,
11,
370,
286,
12328,
281,
561,
567,
362
],
[
668,
456,
13
],
[
400,
718,
385,
722,
558,
1314,
538,
26688,
257,
1154,
13
],
[
286,
1454,
561,
294,
264,
646,
393,
536,
264,
3150,
13
],
[
407,
341,
307,
264,
1154,
294,
1080,
881,
1337,
1860,
13
],
[
961,
460,
294,
16225,
293,
497,
312,
957,
21989,
299,
13
],
[
961,
15546,
294,
460,
312,
257,
34011,
13
],
[
823,
11,
286,
486,
406,
643,
309,
11,
457,
341,
307,
257,
27706,
1594,
13
],
[
17586,
11,
321,
603,
37938,
281,
257,
18587,
1389,
11,
457,
341,
307,
257,
27706,
1594,
1270,
300,
264
],
[
1152,
3481,
597,
307,
33991,
322,
460,
1072,
15546,
307,
19362,
13
],
[
400,
718,
3895,
490,
460,
281,
460,
1072,
15546,
312,
264,
7713,
4471,
11,
597,
307,
3895,
295,
460,
6915,
460,
15546,
13
],
[
823,
718,
311,
3191,
337,
264,
1472,
295,
264,
751,
11,
497,
2159,
11,
281,
312,
422,
13206,
11260,
11,
11260
],
[
295,
264,
957,
2519,
13
],
[
400,
321,
747,
1783,
294,
460,
281,
312,
497,
1561,
712,
13
],
[
1664,
291,
536,
294,
264,
646,
30
],
[
407,
7019,
760,
510,
11,
406,
370,
709,
13
],
[
961,
311,
360,
9432,
13293,
13
],
[
492,
362,
1783,
294,
460,
497,
1561,
712,
13
],
[
823,
321,
366,
1237,
412,
264,
3256,
295,
1783,
1854,
460,
1072,
15546,
293,
295,
1783,
1854,
460,
1072,
15546,
13
],
[
407,
15546,
307,
406,
1561,
712,
13
],
[
883,
11,
572,
11,
572,
13
],
[
4372,
709,
406,
1561,
712,
13
],
[
467,
311,
257,
27706,
1594,
13
],
[
400,
264,
1168,
307,
437,
307,
264,
1192,
4383,
24653,
295,
3895,
1783,
294,
460,
1072,
15546,
30
],
[
2264,
11,
382,
321,
486,
536,
294,
257,
1150,
11,
341,
307,
732,
2674,
13
],
[
583,
286,
820,
584,
300,
341,
307,
1217,
11,
294,
512,
3331,
411,
341,
11,
321,
458,
341,
754,
490,
11
],
[
731,
11,
5162,
589,
294,
2316,
5261,
13
],
[
1171,
1365,
11,
321,
458,
300,
291,
393,
574,
412,
341,
293,
3911,
364,
410,
338,
952,
5673,
13
],
[
400,
498,
291,
7406,
264,
1168,
11,
291,
722,
11,
718,
311,
584,
11,
365,
364,
21989,
299,
5673,
1854
],
[
14589,
13
],
[
400,
2602,
295,
3365,
466,
264,
1192,
4383,
24653,
11,
291,
1029,
466,
264,
1176,
27489,
2984,
24653,
11
],
[
550,
341,
307,
437,
311,
1219,
264,
1783,
16828,
443,
969,
12,
4360,
811,
9733,
640,
20904,
11,
570,
321,
458,
300,
498,
264,
5673
],
[
390,
16014,
769,
32128,
11,
264,
3256,
307,
257,
3792,
302,
295,
364,
410,
338,
952,
1422,
12,
8517,
4014,
295,
316,
13
],
[
400,
550,
466,
257,
1064,
2057,
11,
1310,
732,
924,
2057,
11,
624,
936,
293,
398,
2792,
320,
13379,
2956,
412,
341,
1154
],
[
293,
436,
2351,
264,
1168,
11,
437,
393,
321,
584,
466,
264,
1192,
4383,
24653,
295,
264,
3256
],
[
295,
364,
410,
338,
952,
5673,
30
],
[
400,
510,
293,
510,
11,
295,
1164,
11,
264,
34011,
307,
411,
497,
281,
264,
568,
77,
294,
2674,
13
],
[
1176,
281,
264,
568,
77,
13
],
[
1176,
281,
264,
568,
77,
13
],
[
2264,
13
],
[
400,
700,
295,
439,
11,
341,
307,
732,
2674,
13
],
[
440,
1154,
11,
382,
286,
11323,
510,
11,
294,
264,
2020,
300,
321,
2644,
976,
665,
6338,
294,
2674,
13
],
[
400,
264,
1365,
2673,
11,
291,
747,
460,
281,
312,
318,
295,
441,
17,
49,
13
],
[
509,
747,
15546,
281,
312,
318,
441,
17,
57,
13
],
[
400,
1217,
510,
11,
321,
362,
588,
2199,
11,
1561,
712,
2090,
1385,
295,
460,
6104,
3256,
307,
264,
24653,
295
],
[
264,
3256,
307,
746,
411,
257,
17948,
304,
11,
411,
257,
5682,
12,
3854,
13
],
[
400,
337,
1365,
11,
498,
291,
747,
413,
281,
312,
264,
21539,
1594,
293,
291,
2826,
460,
6108,
293,
291,
574
],
[
412,
264,
1561,
712,
992,
885,
445,
264,
3792,
302,
295,
413,
38,
11,
550,
291,
393,
915,
460,
1270,
300,
264,
24653
],
[
295,
3895,
2031,
307,
588,
6179,
11,
341,
17948,
304,
13
],
[
407,
2031,
307,
1561,
712,
13
],
[
467,
311,
1561,
712,
13
],
[
467,
311,
257,
588,
2199,
992,
11,
457,
264,
24653,
295,
264,
3256,
307,
588,
6179,
13
],
[
407,
437,
466,
413,
2564,
30
],
[
883,
11,
264,
3256,
295,
413,
11,
286,
519,
300,
341,
486,
312,
257,
34011,
294,
413,
13,
407,
264,
3256,
295,
413,
1391
],
[
486,
445,
312,
257,
6329,
11,
264,
24653,
13
],
[
286,
576,
519,
300,
341,
307,
5395,
11,
286,
519,
13
],
[
2264,
13
],
[
823,
11,
295,
1164,
11,
264,
1154,
11,
341,
2010,
295,
1154,
11,
257,
1154,
307,
1348,
490,
308,
21787,
299,
5261,
13
],
[
407,
341,
307,
257,
5999,
804,
1185,
1154,
13
],
[
400,
264,
13735,
20904,
295,
24269,
1193,
11,
502,
490,
22736,
11,
293,
286,
478,
15745,
309,
11,
2264,
11,
286,
486,
584,
746
],
[
1780,
294,
264,
24653,
3037,
295,
264,
20904,
13
],
[
440,
20904,
307,
466,
11,
307,
48936,
294,
2115,
295,
3481,
13
],
[
2743,
286,
603,
808,
646,
281,
341,
11,
457,
286,
478,
15745,
309,
294,
264,
1192,
4383,
2856,
13
],
[
759,
389,
294,
460,
307,
257,
517,
647,
310,
317,
1594,
11,
550,
264,
24653,
295,
264,
3256,
295,
439,
43522,
307,
1481,
13
],
[
400,
550,
456,
8198,
1071,
21989,
299,
1594,
11,
456,
8198,
479,
294,
460,
11,
21989,
299,
1594,
11,
1270
],
[
300,
264,
24653,
295,
3895,
11,
370,
550,
11,
2597,
11,
550,
337,
633,
707,
290,
11,
550,
337,
633,
707
],
[
290,
294,
460,
11,
456,
8198,
479,
11,
21989,
299,
1594,
11,
1270,
300,
264,
24653,
295,
3895,
389,
460,
307,
445,
3895,
460,
13
],
[
407,
264,
24653,
307,
2564,
264,
3256,
295,
1071,
13991,
13
],
[
407,
264,
24653,
307,
364,
479,
11,
364,
479,
3095,
11,
286,
393,
584,
13
],
[
407,
264,
24653,
295,
3895,
295,
264,
13991,
307,
2564,
364,
13991,
11,
364,
13991,
833,
479,
1854,
290,
1072,
15546,
13
],
[
1119,
456,
604,
517,
647,
310,
317,
1594,
30
],
[
2704,
436,
393,
360,
11,
309,
311,
1547,
300,
309,
311,
10833,
538,
517,
647,
310,
317,
3935,
11,
457,
337,
505,
341,
307,
437
],
[
321,
360,
13
],
[
479,
5946,
322,
389,
293,
322,
460,
11,
457,
264,
31704,
322,
460,
307,
493,
281,
45064,
11,
406,
544,
813,
300,
13
],
[
492,
3928,
281,
312,
1237,
412,
3792,
1385,
11,
2086,
11,
2086,
6451,
11,
6451,
11,
321,
603,
536,
983,
13
],
[
583,
341,
307,
264,
2891,
935,
295,
437,
321,
643,
13
],
[
823,
437,
286,
528,
307,
281,
722,
1364,
1854,
460,
293,
406,
294,
264,
3256,
11,
293,
731,
718,
385,
722
],
[
365,
300,
13
],
[
407,
498,
321,
747,
460,
281,
312,
445,
264,
6575,
11,
550,
321,
486,
483,
300,
264,
24653,
295,
3895,
389,
307,
445
],
[
264,
24653,
11,
309,
311,
445,
11,
2597,
11,
3895,
295,
479,
13,
400,
498,
321,
2235,
309,
646,
666,
460,
11,
597,
486,
312,
544
],
[
10851,
337,
505,
11,
550,
2602,
295,
1364,
294,
3895,
11,
294,
264,
9641,
1196,
321,
603,
589,
294,
460,
11,
437
],
[
309,
1619,
307,
300,
264,
24653,
295,
389,
15546,
307,
2293,
479,
15546,
13
],
[
7156,
295,
1417,
295,
264,
24653,
295,
3895,
389,
11,
286,
393,
751,
466,
264,
24653,
295,
389,
15546,
293
],
[
550,
747,
3895,
11,
309,
311,
264,
912,
551,
13
],
[
407,
586,
341,
307,
264,
24653,
286,
820,
584,
1854,
460,
13,
407,
321,
434,
294,
257,
2590,
689,
264,
24653
],
[
295,
389,
15546,
307,
257,
1481,
11,
309,
311,
257,
1594,
1413,
15546,
13
],
[
823,
3449,
300,
498,
460,
390,
257,
5218,
11,
411,
294,
383,
77,
420,
294,
497,
77,
11,
341,
307,
588,
1858,
11,
983,
30
],
[
1436,
389,
15546,
307,
257,
1594,
2564,
13
],
[
1779,
30
],
[
467,
311,
294,
257,
5218,
1594,
11,
389,
15546,
307,
257,
1594,
11,
370,
1080,
24653,
307,
257,
11742,
1594,
13
],
[
400,
550,
291,
445,
747,
264,
4582,
6542,
295,
264,
6575,
293,
341,
486,
312,
428,
479,
294
],
[
341,
1389,
13
],
[
583,
295,
1164,
1564,
321,
1286,
484,
295,
264,
257,
5218,
1389,
11,
341,
307,
406,
257,
1594,
3602,
293,
370,
456
],
[
307,
572,
1778,
983,
291,
820,
312,
1075,
281,
6786,
341,
382,
257,
1594,
1413,
15546,
13
],
[
682,
341,
1389,
11,
439,
1745,
11,
1392,
11,
370,
291,
434,
558,
13
],
[
583,
294,
264,
1389,
295,
257,
517,
647,
310,
317,
1594,
11,
597,
321,
486,
362,
4728,
11,
439,
3935,
11,
439,
4582
],
[
3935,
366,
957,
21989,
299,
13
],
[
1033,
11,
586,
286,
528,
281,
976,
257,
1315,
337,
341,
11,
370,
286,
486,
406,
352,
479,
767,
11,
309,
393,
312,
1612,
13
],
[
1485,
636,
281,
1401,
479,
307,
264,
16998,
1422,
17377,
11,
957,
21989,
299,
1594,
19273,
389,
1270,
300
],
[
15546,
307,
257,
34011,
294,
479,
13,
708,
360,
321,
914,
538,
300,
30
],
[
467,
445,
1355,
300,
797,
562,
291,
574,
412,
264,
3481,
20608,
281,
479,
1072,
15546,
11,
291
],
[
483,
257,
37498,
3481,
13
],
[
583,
286,
486,
406,
352,
666,
300,
11,
286,
528,
281,
976,
257,
1315,
337,
341,
479,
13,
467,
311,
31474,
9540
],
[
293,
286,
528,
281,
2464,
479,
490,
586,
322,
382,
389,
15546,
13
],
[
407,
2212,
389,
11,
321,
915,
364,
479,
19273,
309,
11,
597,
307,
1481,
294,
341,
2020,
13
],
[
440,
24653,
295,
389,
15546,
307,
479,
15546,
11,
370,
341,
307,
437,
286,
528,
281,
747,
484,
295,
497,
1521,
1601,
311,
20904
],
[
307,
264,
9123,
295,
389,
411,
341,
293,
257,
1315,
337,
309,
13
],
[
4839,
3686,
389,
13
],
[
4839,
3686,
389,
11,
2086,
13
],
[
663,
389,
307,
517,
647,
310,
317,
13
],
[
8891,
510,
307,
337,
517,
647,
310,
317,
389,
13
],
[
4839,
2315,
479,
11,
8306,
389,
11,
2086,
11,
16212,
13
],
[
1436,
1564,
291,
589,
365,
264,
3792,
1385,
11,
309,
311,
406,
2074,
16799,
3289,
13
],
[
2639,
3138,
291,
362,
281,
1565,
29456,
1024,
666,
13
],
[
1079,
11,
2086,
11,
2086,
13
],
[
1033,
11,
586,
286,
478,
516,
281,
20460,
709,
264,
2590,
13
],
[
492,
434,
516,
281,
1856,
264,
2674,
11197,
1594,
2590,
293,
6552,
490,
586,
322,
300,
460,
307
],
[
257,
517,
647,
310,
317,
1594,
2564,
13
],
[
407,
294,
1729,
11,
439,
264,
957,
21989,
299,
1422,
17377,
82,
295,
460,
486,
312,
517,
647,
310,
317,
13
],
[
407,
490,
586,
322,
11,
293,
341,
307,
264,
3287,
295,
527,
20904,
11,
460,
2564,
307,
517,
647,
310,
317,
11,
597
],
[
337,
505,
486,
914,
300,
460,
307,
257,
957,
21989,
299,
1422,
17377,
295,
264,
6597,
38190,
32284,
365
],
[
472,
833,
21539,
13
],
[
5858,
281,
29456,
399,
11,
341,
307,
2074,
11,
597,
337,
505,
486,
914,
300,
460,
307,
257,
957,
21989,
299,
1422,
17377
],
[
295,
264,
6597,
38190,
32284,
13
],
[
407,
341,
307,
445,
502,
11,
502,
11,
502,
510,
11,
1958,
11,
293,
957,
21989,
299,
13
],
[
5135,
439,
4582,
1422,
17377,
82,
366,
957,
21989,
299,
13
],
[
3996,
636,
281,
38463,
257,
1594,
411,
341,
11,
498,
291,
500,
380,
528,
281,
589,
365,
8141,
1594,
11
],
[
613,
366,
4582,
11,
2935,
4582,
11,
293,
517,
647,
310,
317,
3935,
13
],
[
400,
586,
11,
1310,
286,
603,
829,
309,
510,
570,
286,
528,
281,
1856,
264,
20904,
322,
264,
1411,
11,
309
],
[
307,
257,
1186,
300,
337,
517,
647,
310,
317,
3935,
11,
29025,
1473,
366,
437,
321,
519,
295,
29025,
1473,
294,
264,
2847,
338,
952
],
[
1389,
13
],
[
407,
1186,
11,
257,
27706,
15546,
294,
460,
337,
460,
517,
647,
310,
317,
13
],
[
682,
264,
34011,
11,
597,
1355,
264,
3481,
307,
37498,
498,
293,
787,
498,
460,
1072,
15546,
307,
14679,
13
],
[
407,
490,
586,
322,
11,
321,
815,
382,
731,
6552,
300,
460,
1072,
15546,
307,
14679,
5699,
286,
584,
34011,
13
],
[
400,
586,
286,
486,
1785,
264,
20904,
300,
321,
603,
312,
27221,
11,
321,
434,
406,
27221,
11,
1417,
466
],
[
1080,
8177,
13
],
[
1664,
286,
362,
512,
1281,
30
],
[
1119,
341,
1281,
30
],
[
883,
13
],
[
1119,
341,
512,
1281,
1310,
30
],
[
4919,
11,
286,
820,
362,
4927,
294,
7295,
13
],
[
286,
519,
452,
9719,
307,
5692,
11,
5692,
34900,
13
],
[
865,
11,
3231,
588,
709,
13
],
[
3769,
11,
709,
1101,
13
],
[
1033,
11,
370,
718,
311,
584,
286,
528,
281,
3318,
1203,
666,
341,
3150,
11,
370,
718,
311,
584,
300,
497,
2159,
307
],
[
797,
422,
13206,
293,
321,
362,
1783,
294,
460,
1561,
712,
13
],
[
400,
281,
652,
309,
3571,
11,
718,
311,
652,
309,
5395,
1217,
11,
570,
4033,
321,
434,
516,
281,
747
],
[
24653,
294,
257,
1150,
13
],
[
407,
718,
311,
445,
6552,
300,
1783,
307,
5395,
570,
321,
434,
3102,
294,
24653,
4033,
13
],
[
400,
286,
500,
380,
362,
257,
34011,
1939,
13
],
[
876,
11,
869,
13
],
[
2561,
588,
709,
13
],
[
286,
500,
380,
362,
257,
34011,
1939,
11,
457,
1217,
321,
393,
8947,
1589,
490,
1783,
13,
1033,
11,
550
],
[
456,
8198,
257,
1230,
497,
294,
426,
13,
821,
8198,
962,
1959,
867,
957,
21989,
299,
1422,
17377,
82,
295
],
[
460,
11,
3353,
10139,
13
],
[
821,
8198,
383,
16,
11,
14123,
1561,
712,
6352,
5395,
13
],
[
23964,
382,
731,
747,
552,
5395,
13
],
[
2033,
1744,
13
],
[
407,
1270,
300,
439,
295,
341,
949,
15546,
390,
8614,
13
],
[
1171,
633,
34011,
15546,
294,
460,
11,
321,
393,
586,
6786,
264,
24653,
295,
11,
293,
586,
286,
486,
6786,
309
],
[
1854,
460,
293,
286,
486,
584,
746,
466,
264,
22743,
11,
457,
309,
311,
544,
10851,
337,
385
],
[
510,
281,
6786,
309,
11,
264,
24653,
1854,
460,
11,
16141,
264,
24653,
295,
1783,
1413,
460,
13,
1033,
11
],
[
286,
393,
2464,
309,
510,
13
],
[
440,
24653,
295,
1783,
15546,
307,
264,
3480,
19362,
11671,
13
],
[
2386,
295,
439,
11,
291,
747,
1783,
11,
295,
1164,
291,
643,
1783,
13
],
[
400,
281,
1783,
11,
291,
909,
264,
3480,
962,
1959,
867,
6352,
13
],
[
286,
1709,
490,
472,
281,
497,
11,
295,
37777,
44376,
15546,
11,
370,
586,
286,
478,
1228,
264,
24657,
490,
456,
11,
370,
633
],
[
44376,
11,
5413,
322,
15546,
11,
286,
747,
264,
16998,
15546,
15090,
1594,
19273,
44376,
293,
291
],
[
643,
281,
909,
13
],
[
286,
478,
767,
1228,
264,
4295,
300,
311,
2212,
385,
13
],
[
639,
775,
406,
652,
2020,
5911,
13
],
[
400,
286,
2041,
321,
362,
281,
12972,
538,
15546,
13
],
[
407,
294,
512,
2020,
11,
437,
321,
366,
884,
307,
12245,
264,
24653,
1154,
322,
23211,
1561,
712
],
[
6352,
281,
3935,
13
],
[
1018,
321,
486,
536,
11,
613,
486,
312,
2293,
3935,
300,
1394,
322,
1783,
412,
13202,
11,
6063,
300,
311
],
[
11652,
3319,
512,
2020,
11,
366,
42174,
281,
1783,
412,
13202,
13
],
[
759,
286,
1415,
281,
1974,
309,
294,
264,
22743,
11,
550,
321,
486,
747,
264,
22743,
510,
293
],
[
550,
321,
603,
445,
483,
264,
22743,
295,
341,
13
],
[
1033,
11,
370,
264,
24653,
295,
264,
22743,
307,
445,
3895,
295,
341,
13
],
[
13428,
472,
551,
11,
597,
307,
11,
286,
519,
11,
406,
6322,
11,
300,
754,
1673,
321,
362,
341,
15546,
322,
264
],
[
636,
11,
341,
307,
257,
1561,
712,
992,
13
],
[
467,
311,
4466,
11,
291,
362,
962,
1959,
867,
3935,
293,
291,
362,
962,
1959,
867,
4466,
295,
613
],
[
3792,
1385,
295,
613,
3935,
11,
370,
341,
307,
2564,
497,
1561,
712,
13
],
[
2720,
1164,
11,
264,
24653,
307,
406,
497,
1561,
712,
570,
321,
362,
15546,
11,
321,
2644,
5042,
15546,
11
],
[
457,
309,
311,
364,
497,
1561,
712,
992,
1413,
15546,
13
],
[
400,
544,
3570,
11,
309,
311,
1021,
11,
321,
393,
584,
264,
3480,
11,
337,
1184,
741,
490,
502,
281,
497,
11,
264,
10139
],
[
295,
383,
741,
307,
1570,
813,
264,
10139,
295,
1783,
13
],
[
1692,
11,
286,
12972,
1203,
538,
15546,
13
],
[
2438,
11,
3716,
11,
3716,
11,
341,
307,
406,
4818,
13
],
[
1044,
291,
11,
341,
307,
406,
13
],
[
467,
311,
665,
11,
1580,
13251,
34153,
13
],
[
467,
311,
1021,
13
],
[
440,
1150,
551,
307,
11,
286,
478,
406,
988,
498,
321,
603,
483,
281,
309,
11,
457,
498,
291,
747,
264,
49336,
11,
370
],
[
512,
295,
613,
362,
572,
276,
741,
597,
307,
16212,
294,
512,
362,
16212,
276,
741,
3673,
552,
11,
457
],
[
337,
276,
741,
311,
597,
366,
49336,
365,
3104,
281,
15874,
11,
264,
383,
741,
311,
366,
767,
37498
],
[
293,
550,
341,
992,
307,
767,
5395,
992,
1217,
13
],
[
407,
512,
295,
613,
6352,
366,
406,
5395,
11,
787,
436,
1813,
5395,
562,
321,
747,
264,
11671
],
[
411,
321,
434,
5127,
12866,
6677,
11,
457,
264,
49336,
2306,
366,
5395,
11,
370,
383,
741,
307,
37498
],
[
293,
370,
309,
311,
1858,
281,
536,
13
],
[
1133,
291,
747,
257,
14679,
992,
1413,
364,
21989,
299,
1594,
11,
291,
483,
920,
4966,
13
],
[
1033,
11,
718,
385,
584,
512,
19151,
466,
341,
13
],
[
2704,
321,
603,
483,
544,
813,
19151,
11,
1310,
406,
13
],
[
682,
264,
5765,
30
],
[
1079,
11,
294,
264,
5765,
11,
1338,
13
],
[
1545,
360,
291,
767,
747,
276,
741,
293,
300,
436,
366,
957,
21989,
299,
293,
276,
741,
15546,
307,
611,
957
],
[
21989,
299,
11,
370,
983,
500,
380,
291,
747,
276,
741,
15546,
281,
722,
365,
30
],
[
286,
727,
11,
457,
550,
309,
576,
312,
257,
24286,
20904,
570,
586,
264,
912,
276,
741,
589,
337,
439,
8019,
3799,
13
],
[
509,
445,
362,
281,
747,
264,
15546,
24653,
295,
264,
276,
741,
13
],
[
407,
291,
727,
362,
48936,
264,
1874,
337,
1184,
15546,
14759,
11,
457,
456,
311,
746
],
[
2068,
466,
300,
570,
291,
362,
411,
364,
257,
4059,
72,
1605,
300,
321,
486,
536,
321,
8947
],
[
490,
2031,
293,
321,
445,
362,
281,
3079,
264,
24269,
1193,
3542,
281,
1184,
295,
264,
3935,
281,
483,
264,
24653,
13
],
[
1033,
718,
385,
652,
512,
3053,
13
],
[
961,
311,
747,
264,
1389,
562,
2031,
307,
257,
7605,
13
],
[
961,
311,
574,
437,
341,
20904,
1619,
562,
264,
10139,
295,
2031,
307,
502,
13
],
[
407,
2031,
307,
445,
257,
7605,
13
],
[
492,
603,
536,
364,
1365,
11,
286,
603,
829,
364,
1365,
294,
257,
1150,
13
],
[
682,
341,
1389,
11,
538,
472,
11,
264,
269,
741,
311,
366,
19362,
13
],
[
407,
550,
321,
534,
11,
439,
321,
366,
884,
307,
1940,
264,
7605,
293,
5127,
281,
309,
962,
1959,
867
],
[
3792,
1385,
293,
264,
24653,
550,
307,
14879,
538,
5127,
281,
264,
7605,
962,
1959,
867,
3792,
1385
],
[
293,
300,
311,
309,
11,
291,
500,
380,
643,
544,
13
],
[
1396,
570,
264,
10139,
295,
269,
741,
307,
1958,
307,
19362,
293,
550,
264,
24653,
295,
2031,
15546,
307
],
[
445,
2031,
11671,
512,
962,
1959,
867,
1310,
406,
264,
912,
367,
490,
350,
290,
741,
276,
741,
15546,
13
],
[
407,
291,
787,
643,
281,
909,
962,
1959,
867,
3792,
1385,
293,
439,
295,
341,
13
],
[
407,
718,
311,
360,
264,
1365,
300,
498,
291,
600,
2198,
18885,
72,
293,
286,
751,
466,
309,
11,
341,
307,
264,
1365
],
[
286,
2673,
976,
11,
370,
718,
311,
574,
412,
341,
370,
321,
393,
12240,
341,
294,
290,
13
],
[
639,
307,
406,
517,
41654,
457,
309,
311,
21989,
984,
307,
32702,
299,
281,
364,
1021,
1594,
13
],
[
823,
718,
311,
747,
2031,
11,
264,
9848,
65,
4711,
11,
439,
264,
2031,
288,
11,
2031,
288,
307,
502,
293,
281,
20460,
718,
311,
445,
574
],
[
412,
264,
700,
46856,
13
],
[
961,
311,
747,
15546,
281,
312,
445,
710,
3732,
13
],
[
407,
321,
362,
341,
3732,
293,
309,
311,
406,
1152,
281,
536,
300,
562,
291,
747,
710,
3732,
24653,
291
],
[
1936,
2684,
2051,
613,
11095,
293,
437,
291,
486,
312,
5127,
307,
613,
732,
3935,
13
],
[
407,
341,
307,
411,
276,
16,
11,
456,
307,
572,
935,
510,
11,
309,
311,
534,
257,
1594,
293,
341,
307,
276,
17,
293,
264
],
[
24653,
718,
385,
2464,
309,
45558,
356,
570,
321,
366,
586,
1854,
367,
3732,
307,
2293,
2031,
11671
],
[
276,
16,
11671,
276,
17,
731,
1804,
710,
3732,
13
],
[
1692,
311,
4461,
11,
1392,
286,
486,
2897,
365,
309,
11,
321,
603,
360,
1310,
1071,
1365,
1780,
13
],
[
286,
820,
584,
300,
295,
1164,
437,
3665,
510,
307,
36429,
300,
276,
16,
293,
276,
16,
15546,
366,
264
],
[
912,
570,
276,
16,
307,
1217,
710,
3732,
1594,
11,
264,
15236,
295,
710,
3732,
365,
276,
16,
307,
257
],
[
34011,
11,
264,
15236,
295,
276,
17,
365,
710,
3732,
307,
257,
34011,
370,
294,
341,
1389,
456,
390,
572
],
[
643,
281,
747,
276,
16,
710,
3732,
420,
276,
17,
710,
3732,
13
],
[
759,
286,
1319,
264,
34011,
293,
652,
309,
364,
39914,
34011,
550,
276,
16,
293,
276,
17,
1293,
486,
1813,
264
],
[
1379,
295,
367,
17,
370,
498,
286,
7406,
341,
538,
364,
39914,
34011,
550,
264,
24653,
295,
2031,
1804,
264,
34011
],
[
486,
312,
1203,
11,
486,
312,
264,
1379,
295,
367,
17,
13
],
[
407,
797,
3449,
300,
341,
6916,
307,
5361,
510,
570,
341,
307,
264,
912,
382,
276,
16,
498,
291
],
[
528,
710,
3732,
293,
341,
307,
264,
912,
1293,
295,
552,
366,
15090,
365,
3104,
281,
264,
34011
],
[
710,
3732,
13
],
[
1033,
1150,
7942,
13
],
[
5135,
4523,
484,
456,
311,
257,
20904,
411,
341,
8198,
294,
264,
10394,
295,
30468
],
[
5261,
295,
731,
544,
420,
1570,
456,
311,
257,
20904,
2597,
295,
21159,
490,
22736,
767,
1310,
24269,
1193
],
[
307,
406,
22736,
1310,
4748,
3071,
23952,
597,
307,
544,
2674,
813,
437,
286,
576,
584,
586,
457
],
[
337,
505,
309,
486,
312,
1547,
13
],
[
634,
630,
406,
754,
643,
281,
584,
300,
460,
307,
517,
647,
310,
317,
13
],
[
6281,
2540,
300,
291,
362,
280,
295,
2031,
257,
26110,
4471,
294,
512,
7006,
2031,
67,
257,
957,
26110,
4471
],
[
293,
291,
747,
2031,
281,
312,
2293,
264,
3256,
295,
264,
26110,
4471,
13
],
[
497,
17,
38,
2086,
1309,
291,
370,
309,
311,
26110,
4471,
294,
294,
309,
311,
406,
294,
497,
2597,
309,
311,
2940,
9102
],
[
370,
286,
820,
584,
280,
1718,
293,
1310,
2464,
280,
1718,
257,
4471,
490,
497,
67,
721,
666,
460,
13,
6281,
2540,
300,
264,
3256
],
[
295,
264,
26110,
2117,
1854,
460,
13,
467,
311,
665,
1580,
307,
13251,
337,
309,
13
],
[
883,
291,
434,
558,
291,
434,
558,
291,
434,
558,
13,
1033,
550,
456,
311,
257,
588,
2068,
3037,
295
],
[
341,
1874,
570,
437,
291,
362,
281,
360,
307,
264,
3480,
550,
718,
33937,
312,
264,
16998
],
[
992,
295,
257,
957,
21989,
299,
1422,
17377,
1422,
17377,
389,
295,
460,
19273,
2031,
13,
407,
445,
747,
264,
15236
],
[
295,
439,
3792,
1385,
295,
957,
21989,
299,
3935,
597,
5304,
2031,
550,
337,
633,
34011,
337,
1392
],
[
370,
718,
385,
445,
584,
5699,
286,
2464,
15546,
286,
914,
257,
34011,
294,
460,
264,
24653,
295,
2031,
15546
],
[
307,
2293,
264,
24653,
295,
33937,
15546,
597,
307,
445,
33937,
15546,
15546,
13
],
[
407,
767,
562,
291,
747,
264,
3256,
295,
257,
26110,
4471,
382,
428,
1561,
712,
992,
597
],
[
307,
2745,
1561,
712,
550,
439,
291,
643,
307,
472,
3792,
302,
293,
309,
27986,
264,
1379,
24653,
13
],
[
400,
498,
286,
362,
565,
321,
603,
808,
646,
293,
536,
577,
321,
393,
4172,
4176,
341,
1874,
490,
527,
589,
13
],
[
583,
286,
820,
584,
300,
1080,
20904,
630,
406,
754,
6552,
517,
647,
310,
317,
457,
550,
264,
10710
],
[
295,
257,
26110,
4471,
291,
362,
281,
312,
4748,
544,
5026,
466,
13,
4919,
1338,
439,
452,
8019,
3799
],
[
586,
382,
286,
848,
309,
439,
452,
8019,
3799,
486,
312,
34011,
13,
12548,
7942,
13,
639,
390,
732,
286,
2041,
13,
639,
486
],
[
312,
1045,
293,
286,
603,
584,
309,
294,
2283,
1310,
293,
550,
286,
603,
584,
746,
544,
13600,
13,
1018,
286
],
[
848,
264,
20904,
490,
257,
1265,
67,
618,
5261,
1293,
295,
2065,
64,
293,
295,
24269,
1193,
366,
406,
48936,
370
],
[
709,
294,
2115,
295,
24653,
767,
294,
2065,
64,
291,
362,
281,
8947,
264,
24653,
457,
309,
311,
48936
],
[
294,
2115,
295,
257,
1265,
67,
618,
5261,
293,
294,
2115,
295,
32181,
295,
3481,
420,
437,
307,
1219
],
[
1267,
327,
468,
30783,
3542,
13,
400,
415,
1619,
510,
300,
415,
1669,
2020,
295,
437,
309,
1355,
337
],
[
1783,
281,
312,
1267,
327,
468,
2024,
4866,
293,
550,
415,
1619,
300,
1783,
307,
767,
12631,
1854,
33937,
15546,
13
],
[
286,
603,
445,
829,
286,
500,
380,
528,
281,
6964,
1267,
327,
468,
30783,
457,
920,
286,
528,
281,
751,
466,
309,
13,
286,
458,
286
],
[
458,
321,
500,
380,
411,
300,
13,
467,
4523,
484,
300,
577,
4393,
286,
584,
300,
300,
337,
1561,
712,
6352
],
[
294,
422,
6,
33,
27378,
311,
3877,
286,
603,
976,
5110,
754,
1553,
17827,
24653,
286,
478,
2597,
300
],
[
286,
478,
885,
24247,
293,
1267,
327,
468,
2024,
3666,
366,
406,
264,
912,
13,
413,
12612,
295,
1164,
291,
500,
380,
458
],
[
437,
309,
307,
370,
437,
360,
291,
1127,
498,
309,
743,
13,
583,
718,
385,
976,
364,
1365,
797,
1553,
17827,
13
],
[
1079,
2086,
2086,
286,
994,
380,
584,
1267,
327,
468,
30783,
294,
264,
24653,
767,
1267,
327,
468,
30783,
294
],
[
264,
24653,
341,
307,
437,
286,
820,
584,
13,
583,
718,
385,
976,
364,
1365,
293,
2035,
718,
311
],
[
574,
412,
497,
17,
3732,
365,
15546,
885,
1176,
3732,
293,
1783,
885,
264,
7605,
295,
932,
282,
314,
44166,
314,
5044
],
[
813,
1958,
13,
407,
341,
7605,
295,
1164,
307,
257,
12956,
81,
984,
257,
588,
2199,
7605,
307,
1561,
712,
294,
497,
87,
293
],
[
309,
10002,
490,
437,
321,
366,
884,
300,
264,
24653,
295,
1783,
1804,
1176,
3732,
307,
497,
3732,
457
],
[
1783,
307,
406,
1267,
327,
468,
2024,
4866,
294,
2035,
2856,
2035,
309,
1355,
1553,
516,
666,
309,
457
],
[
1783,
406,
1267,
327,
468,
2024,
4866,
294,
497,
3732,
13,
3764,
286,
603,
829,
309,
293,
767,
309,
390,
1880,
286
],
[
914,
1553,
754,
5276,
321,
645,
294,
257,
3440,
294,
24786,
293,
5202,
486,
415,
4418,
264,
700
],
[
644,
295,
702,
751,
1417,
466,
1267,
327,
468,
2024,
3666,
293,
4258,
10932,
484,
300,
341,
307,
406
],
[
1267,
327,
468,
2024,
4866,
370,
613,
366,
406,
6179,
12363,
437,
321,
7081,
307,
731,
412,
341,
935
],
[
797,
309,
311,
257,
4074,
281,
584,
300,
20904,
562,
286,
994,
380,
6964,
309,
457,
718,
385,
445,
584,
1553
],
[
257,
20904,
1080,
14816,
294,
22560,
2270,
37498,
1389,
412,
1935,
337,
497,
77,
613,
366,
264
],
[
912,
370,
382,
938,
382,
264,
992,
420,
264,
7605,
307,
7642,
294,
257,
22560,
2270,
37498,
3287
],
[
550,
24653,
498,
264,
24653,
295,
1783,
307,
497,
3732,
550,
1783,
307,
1267,
327,
468,
2024,
4866,
370,
286,
486,
445,
584
],
[
588,
13501,
48863,
457,
912,
337,
22560,
2270,
37498,
370,
613,
366,
439,
7285,
9227,
294,
597
],
[
2445,
307,
4728,
37498,
538,
257,
26110,
13
],
[
1033,
370,
286,
528,
281,
3496,
264,
1036,
286,
2041,
2119,
2077,
281,
584,
512,
721,
466,
264,
8177
],
[
293,
370,
286,
486,
1856,
264,
1874,
293,
412,
1935,
1310,
976,
472,
1558,
300,
3665,
281,
6786
],
[
264,
24653,
13
],
[
1033,
370,
718,
385,
829,
512,
661,
5261,
370,
1400,
456,
307,
406,
709,
718,
311,
747,
497,
281,
312,
16429
],
[
10320,
295,
497,
457,
281,
385,
16429,
10320,
365,
3104,
281,
1203,
370,
498,
291,
528
],
[
257,
955,
14808,
1674,
365,
3104,
281,
1577,
3877,
406,
787,
264,
7285,
3877,
286,
445,
643
],
[
309,
294,
1668,
281,
751,
466,
264,
34011,
294,
16429,
10320,
293,
294,
1668,
281,
570,
286,
486,
1286
],
[
490,
497,
281,
16429,
10320,
370,
718,
385,
584,
829,
257,
24657,
337,
1783,
294,
497,
77,
718,
1783,
8199,
45708
],
[
264,
25138,
295,
1783,
294,
264,
955,
3877,
293,
586,
382,
7713,
321,
362,
264,
38546,
295
],
[
439,
264,
8961,
294,
497,
1270,
300,
307,
37498,
538,
512,
297,
456,
307,
297,
294,
297,
293,
321,
718,
2992,
312,
264
],
[
4875,
295,
264,
7157,
295,
7193,
3324,
10650,
370,
264,
6674,
7157,
8961,
294,
497,
1270,
300,
337,
633
],
[
8961,
2597,
337,
633,
297,
13
],
[
583,
767,
286,
669,
3102,
294,
422,
293,
624,
322,
460,
322,
264,
1594,
460,
406,
370,
709,
294,
497,
293,
437
],
[
3665,
505,
321,
727,
362,
6453,
1553,
309,
457,
437,
3665,
505,
307,
300,
562,
3449,
300
],
[
562,
460,
307,
294,
624,
11,
314,
293,
497,
550,
767,
460,
307,
5395,
294,
497,
77,
3732,
376,
293,
497,
558,
570
],
[
264,
21539,
307,
472,
321,
2644,
3109,
264,
41296,
307,
472,
321,
2644,
3109,
4959
],
[
365,
41296,
4018,
293,
550,
718,
385,
2464,
422,
38,
445,
281,
312,
422,
15236,
460,
293,
2992,
460,
281
],
[
312,
2992,
1804,
286,
11,
286,
307,
264,
6575,
8141,
15236,
365,
460,
13,
865,
1338,
422,
281,
264,
297,
3732,
1309,
291
],
[
422,
281,
264,
297,
3732,
293,
2992,
281,
264,
297,
3732,
3231,
13
],
[
400,
586,
321,
362,
264,
3832,
644,
4471,
516,
490,
422,
38,
666,
460,
370,
460,
307,
264,
957,
2793,
286
],
[
4160,
291,
286,
669,
5141,
460,
8199,
264,
10320,
370,
321,
362,
257,
3832,
644,
452,
516,
490,
510
],
[
281,
510,
293,
264,
28256,
307,
2293,
2992,
295,
460,
293,
294,
1186,
422,
295,
460,
307,
731,
309,
307,
257,
1594,
2992
],
[
295,
460,
307,
5646,
294,
422,
295,
460,
293,
422,
295,
460,
307,
264,
12909,
2047,
1674,
295,
2992,
460,
293,
460,
293,
264
],
[
497,
2793,
341,
307,
264,
497,
2793,
13
],
[
407,
3673,
309,
820,
312,
460,
8199,
13
],
[
1692,
291,
914,
30
],
[
865,
13
],
[
286,
11,
286,
11,
1338,
341,
820,
312,
460,
8199,
341,
307,
428,
1867,
341,
820,
312,
460,
8199,
286,
4382
],
[
264,
24657,
281,
1856,
309,
370,
460,
457,
291,
366,
558,
341,
820,
312,
460,
8199,
3231,
13
],
[
865,
1338,
13
],
[
400,
445,
281,
1856,
341,
281,
829,
341,
24657,
286,
669,
337,
983,
604,
992,
1854,
460,
8199,
286,
486
],
[
45708,
294,
9852,
295,
24657,
3832,
644,
295,
398,
3832,
644,
307,
257,
14641,
4471,
457,
286,
486
],
[
2464,
3832,
644,
295,
4471,
3832,
644,
295,
398,
281,
914,
264,
3832,
644,
295,
398,
15236
],
[
422,
38,
13
],
[
1033,
370,
286,
820,
406,
2464,
341,
570,
3832,
644,
307,
406,
257,
1577,
4471,
457,
286,
486,
2464,
341
],
[
293,
2199,
14816,
307,
370,
341,
307,
445,
2199,
9130,
300,
321,
360,
2049,
562,
321,
2924
],
[
341,
1507,
300,
337,
604,
1783,
294,
460,
472,
636,
281,
483,
264,
24653,
295,
1783,
307,
281,
352,
281,
16429
],
[
10320,
293,
747,
264,
3832,
644,
13
],
[
639,
307,
588,
1858,
281,
536,
341,
307,
264,
1389,
13
],
[
407,
586,
439,
295,
341,
390,
281,
352,
646,
281,
341,
1154,
321,
366,
1382,
281,
1223,
264,
24653
],
[
295,
1783,
15546,
370,
586,
321,
366,
646,
281,
264,
1154,
370,
6552,
586,
797,
1783,
6552,
1783,
294,
460,
307,
497,
1561,
712
],
[
293,
264,
24653,
295,
1783,
15546,
393,
312,
3720,
382,
264,
3832,
644,
295,
1783,
15546,
8199,
457
],
[
341,
307,
264,
912,
382,
264,
3832,
644,
295,
1783,
8199,
1413,
15546,
8199
],
[
293,
437,
321,
528,
281,
360,
293,
309,
1487,
484,
281,
362,
534,
588,
1481,
33246,
3620,
307
],
[
281,
586,
321,
12701,
264,
24653,
382,
364,
3256,
295,
257,
4471,
293,
437,
321,
528,
307,
281,
6964,
281,
9845
],
[
264,
9274,
295,
264,
4471,
4650,
281,
264,
3566,
3467,
322,
1783,
370,
286,
486,
2464,
309,
411,
341,
670
],
[
439,
3566,
3467,
294,
264,
422,
13206,
2856,
586,
294,
264,
422,
13206,
2856,
295,
11568,
1413
],
[
15546,
8199,
13
],
[
286,
669,
516,
281,
747,
264,
3832,
644,
295,
1783,
8199,
15546,
8199,
2010,
538,
2010,
293,
337,
1184
],
[
2010,
321,
486,
853,
281,
1223,
437,
341,
307,
293,
3449,
498,
321,
1223,
337,
1184,
2010
],
[
437,
307,
264,
3832,
644,
550,
321,
486,
483,
437,
321,
528,
370,
264,
1154,
264,
1917,
295,
264
],
[
1154,
307,
281,
1223,
437,
307,
264,
3832,
644,
295,
472,
2010,
1413,
15546,
8199,
370,
437
],
[
307,
295,
430,
497,
1413,
15546,
8199,
293,
286,
486,
360,
257,
588,
588,
2199,
1365,
700,
295,
439,
597
],
[
295,
1164,
321,
486,
406,
483,
1340,
457,
6552,
300,
430,
307,
264,
2010,
295,
364,
4478,
597,
307
],
[
37498,
337,
8961,
1854,
422,
295,
460,
370,
309,
307,
7193,
3324,
332,
379,
1998,
281,
512,
4478,
295,
460
],
[
341,
307,
588,
1858,
281,
536,
550,
321,
439,
437,
321,
483,
510,
307,
411,
264,
1108,
345,
7630
],
[
295,
264,
3832,
644,
295,
8961,
420,
406,
754,
1577,
1108,
345,
7630,
321,
486,
483,
644,
295
],
[
264,
1108,
345,
7630,
457,
321,
486,
406,
483,
1340,
544,
813,
300,
562,
321,
747,
264,
3832
],
[
644,
295,
341,
11568,
15546,
8199,
307,
2293,
264,
3832,
644,
295,
8961,
597,
307,
4478,
294
],
[
264,
24653,
295,
1783,
457,
321,
848,
1783,
307,
1998,
370,
309,
307,
4478,
295,
1783,
1413,
15546,
370,
341,
307
],
[
6322,
570,
562,
321,
747,
264,
24653,
295,
1783,
15546,
321,
294,
1729,
483,
370,
341,
307,
644
],
[
295,
1783,
15546,
370,
510,
295,
1164,
341,
307,
2293,
264,
13150,
295,
510,
341,
644,
370,
264
],
[
37498,
3467,
264,
3467,
597,
1856,
1854,
264,
3832,
644,
486,
808,
510,
486,
10586
],
[
1783,
2564,
597,
307,
295,
1164,
321,
362,
281,
362,
1783,
370,
264,
1880,
644,
307,
264,
517,
18767,
292
],
[
644,
264,
517,
18767,
292,
3467,
3467,
597,
1856,
412,
13202,
293,
510,
321,
5366,
264,
3480
],
[
10710,
321,
486,
818,
309,
23831,
3792,
302,
281,
257,
2010,
7123,
337,
604,
8961,
294,
460,
8199,
370
],
[
294,
16429,
10320,
293,
337,
460,
294,
264,
957,
1002,
293,
389,
294,
264,
957,
1002,
957,
21989,
299
],
[
321,
584,
300,
460,
389,
307,
2651,
8961,
498,
493,
281,
13202,
26601,
322,
264,
1411,
291,
483,
456,
370,
498,
8961
],
[
307,
9134,
294,
2992,
460,
460,
389,
420,
321,
727,
829,
2992,
460,
510,
498,
2992,
460,
8961,
27815,
460,
389,
337,
1365,
498
],
[
321,
747,
264,
7605,
27050,
3732,
1783,
584,
502,
670,
1783,
293,
321,
747,
2107,
12,
1115,
515,
4478,
510,
8961
],
[
550,
295,
1164,
264,
1783,
10298,
307,
2651,
341,
935,
572,
1413,
264,
389,
3543,
2086,
1309,
291,
2086,
309,
727
],
[
312,
412,
13202,
1309,
291,
2086,
460,
8199,
293,
264,
1874,
321,
7081,
700
],
[
300,
264,
15236,
3133,
286,
603,
584,
309,
411,
341,
370,
472,
20904,
307,
300,
6451,
456,
307
],
[
257,
23831,
337,
604,
8961,
294,
460,
8199,
456,
8198,
257,
16998,
3792,
302,
460,
389,
2651,
8961,
3449,
300
],
[
264,
1379,
1594,
307,
2651,
8961,
3133,
370,
633,
4478,
575,
512,
3792,
302,
2651,
8961,
2651,
309
],
[
264,
787,
2734,
307,
437,
562,
393,
291,
483,
1570,
813,
264,
1379,
1594,
3133,
370,
456,
307,
472,
3792,
302
],
[
460,
389,
597,
307,
16212,
294,
439,
661,
3792,
1385,
597,
366,
2651,
8961,
293,
321,
393,
45708,
309
],
[
718,
311,
818,
309,
460,
8961,
389,
8961,
295,
1164,
460,
8961,
307,
406,
3845,
457,
389,
8961,
307,
3845
],
[
633,
12424,
393,
312,
8614,
293,
309,
311,
1858,
281,
536,
300,
498,
8961,
307,
10344,
670
],
[
497,
294,
264,
422,
13206,
2856,
767,
754,
264,
12909,
21989,
984,
12909,
21989,
299,
2856
],
[
550,
460,
9861,
389,
9861,
6915,
281,
460,
8961,
389,
8961,
370,
767,
460,
8961,
389,
8961,
307,
264,
4707
],
[
295,
264,
2010,
295,
8961,
370,
321,
486,
45708,
309,
490,
586,
322,
337,
264,
958,
732,
2077,
538,
460
],
[
430,
389,
430,
689,
430,
307,
264,
2010,
295,
8961,
670,
497,
370,
309,
311,
534,
264,
23831,
3792,
302,
281,
264,
2010
],
[
286,
820,
584,
337,
1365,
341,
307,
406,
2074,
294,
22999,
17,
49,
498,
291,
2089,
604,
3935,
604,
21989,
299
],
[
1594,
370,
291,
393,
362,
732,
3792,
1385,
295,
21989,
299,
3935,
597,
366,
2651,
364,
4478,
457,
264,
15236
],
[
307,
6707,
457,
406,
1939,
294,
257,
1150,
294,
257,
1150,
370,
264,
20904,
300,
321,
7081,
293,
286,
486,
2413
],
[
365,
300,
300,
337,
633,
3566,
2010,
420,
422,
13206,
2010,
430,
87,
294,
430,
293,
460,
264,
3832,
644
],
[
295,
11568,
15546,
8199,
307,
2293,
731,
286,
360,
309,
294,
732,
4439,
264,
24653,
295,
26039,
389,
430,
15546,
597
],
[
307,
264,
912,
382,
26039,
389,
430,
15546,
15546,
370,
412,
13202,
437,
7001,
307,
264,
23831,
3792,
302,
293,
264
],
[
23831,
3792,
302,
309,
437,
307,
437,
24799,
264,
24653,
295,
11568,
15546,
293,
382,
257,
1874,
445,
516
],
[
646,
281,
689,
321,
1411,
309,
311,
644,
295,
264,
589,
307,
264,
912,
382,
264,
24653,
295,
1783,
15546,
307,
445
],
[
264,
11671,
364,
13785,
11671,
412,
341,
935,
15546,
430,
16734,
430,
2010,
322,
1783,
370,
497,
264,
25138
],
[
295,
430,
294,
497,
430,
16734,
1783,
430,
2010,
322,
1783,
718,
385,
2464,
309,
293,
718,
385,
1223,
691,
8240,
430,
691
],
[
8240,
1783,
430,
18779,
1783,
1392,
286,
603,
1590,
370,
412,
264,
917,
321,
362,
281,
352,
295,
1164,
281,
510,
293,
558
],
[
586,
309,
311,
257,
11671,
597,
307,
406,
1850,
577,
291,
4813,
309,
309,
311,
257,
11671,
670,
3467,
370,
456
],
[
50364,
307,
544,
589,
281,
360,
293,
544,
5245,
510,
321,
767,
281,
483,
341,
5629,
13,
51086
]
],
"temperature": [
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"avg_logprob": [
-0.23892179131507874,
-0.23892179131507874,
-0.23892179131507874,
-0.23892179131507874,
-0.23892179131507874,
-0.2876673936843872,
-0.2876673936843872,
-0.2876673936843872,
-0.2188374400138855,
-0.2188374400138855,
-0.2188374400138855,
-0.2188374400138855,
-0.11643736809492111,
-0.26863762736320496,
-0.26863762736320496,
-0.26863762736320496,
-0.26863762736320496,
-0.3644239008426666,
-0.3644239008426666,
-0.3644239008426666,
-0.29864501953125,
-0.29864501953125,
-0.29864501953125,
-0.29864501953125,
-0.29864501953125,
-0.25731807947158813,
-0.25731807947158813,
-0.20224587619304657,
-0.20224587619304657,
-0.20224587619304657,
-0.20224587619304657,
-0.20224587619304657,
-0.19509239494800568,
-0.19509239494800568,
-0.19509239494800568,
-0.19509239494800568,
-0.19509239494800568,
-0.19509239494800568,
-0.2451276332139969,
-0.2451276332139969,
-0.2451276332139969,
-0.2451276332139969,
-0.2451276332139969,
-0.2451276332139969,
-0.2451276332139969,
-0.1730111837387085,
-0.1730111837387085,
-0.1730111837387085,
-0.11872217059135437,
-0.11872217059135437,
-0.3048381209373474,
-0.3048381209373474,
-0.3048381209373474,
-0.3048381209373474,
-0.3048381209373474,
-0.23972046375274658,
-0.23972046375274658,
-0.23972046375274658,
-0.23972046375274658,
-0.23972046375274658,
-0.23972046375274658,
-0.2644599378108978,
-0.2644599378108978,
-0.2644599378108978,
-0.1888071894645691,
-0.1888071894645691,
-0.31841611862182617,
-0.31841611862182617,
-0.2059958577156067,
-0.2059958577156067,
-0.25490882992744446,
-0.25490882992744446,
-0.25490882992744446,
-0.2916225492954254,
-0.2916225492954254,
-0.2916225492954254,
-0.2403116673231125,
-0.2403116673231125,
-0.2403116673231125,
-0.2403116673231125,
-0.23145291209220886,
-0.23145291209220886,
-0.15922093391418457,
-0.15922093391418457,
-0.15922093391418457,
-0.18951472640037537,
-0.18951472640037537,
-0.18951472640037537,
-0.18951472640037537,
-0.18951472640037537,
-0.18951472640037537,
-0.18951472640037537,
-0.2149956077337265,
-0.2149956077337265,
-0.2149956077337265,
-0.2149956077337265,
-0.2149956077337265,
-0.19581840932369232,
-0.19581840932369232,
-0.19581840932369232,
-0.19581840932369232,
-0.19581840932369232,
-0.1974910944700241,
-0.1974910944700241,
-0.1974910944700241,
-0.1974910944700241,
-0.1974910944700241,
-0.1974910944700241,
-0.23772326111793518,
-0.23772326111793518,
-0.23772326111793518,
-0.23772326111793518,
-0.42580482363700867,
-0.42580482363700867,
-0.42580482363700867,
-0.42580482363700867,
-0.42580482363700867,
-0.42580482363700867,
-0.15019652247428894,
-0.15019652247428894,
-0.15019652247428894,
-0.15019652247428894,
-0.15019652247428894,
-0.15019652247428894,
-0.1503603756427765,
-0.1503603756427765,
-0.1503603756427765,
-0.1503603756427765,
-0.24401545524597168,
-0.24401545524597168,
-0.24401545524597168,
-0.27222275733947754,
-0.27222275733947754,
-0.27222275733947754,
-0.2224542498588562,
-0.2224542498588562,
-0.4534681737422943,
-0.4534681737422943,
-0.4534681737422943,
-0.4534681737422943,
-0.4534681737422943,
-0.4534681737422943,
-0.4534681737422943,
-0.4534681737422943,
-0.389909029006958,
-0.389909029006958,
-0.389909029006958,
-0.19190062582492828,
-0.19190062582492828,
-0.19190062582492828,
-0.19190062582492828,
-0.19190062582492828,
-0.2360607534646988,
-0.2360607534646988,
-0.2360607534646988,
-0.2360607534646988,
-0.46865934133529663,
-0.46865934133529663,
-0.46865934133529663,
-0.46865934133529663,
-0.23267067968845367,
-0.23267067968845367,
-0.2164677232503891,
-0.2164677232503891,
-0.2164677232503891,
-0.2164677232503891,
-0.2164677232503891,
-0.2914755940437317,
-0.2914755940437317,
-0.2914755940437317,
-0.2914755940437317,
-0.3777112364768982,
-0.3777112364768982,
-0.3777112364768982,
-0.3777112364768982,
-0.3777112364768982,
-0.1964973360300064,
-0.1964973360300064,
-0.1964973360300064,
-0.1964973360300064,
-0.1964973360300064,
-0.2170942723751068,
-0.2170942723751068,
-0.2170942723751068,
-0.2170942723751068,
-0.17494019865989685,
-0.17494019865989685,
-0.17494019865989685,
-0.17494019865989685,
-0.4481682777404785,
-0.4481682777404785,
-0.4481682777404785,
-0.4481682777404785,
-0.2563857436180115,
-0.2563857436180115,
-0.2563857436180115,
-0.1793580800294876,
-0.1793580800294876,
-0.1793580800294876,
-0.1793580800294876,
-0.1793580800294876,
-0.40309208631515503,
-0.40309208631515503,
-0.40309208631515503,
-0.40309208631515503,
-0.40309208631515503,
-0.2121174931526184,
-0.2121174931526184,
-0.2121174931526184,
-0.2121174931526184,
-0.2121174931526184,
-0.23810674250125885,
-0.23810674250125885,
-0.23810674250125885,
-0.2102154642343521,
-0.2102154642343521,
-0.2102154642343521,
-0.2102154642343521,
-0.2102154642343521,
-0.2102154642343521,
-0.2010076493024826,
-0.2010076493024826,
-0.2010076493024826,
-0.17449665069580078,
-0.17449665069580078,
-0.17449665069580078,
-0.23549312353134155,
-0.23549312353134155,
-0.23549312353134155,
-0.23549312353134155,
-0.16408735513687134,
-0.16408735513687134,
-0.16408735513687134,
-0.2435932606458664,
-0.2435932606458664,
-0.31156378984451294,
-0.31156378984451294,
-0.151028111577034,
-0.151028111577034,
-0.151028111577034,
-0.151028111577034,
-0.11935413628816605,
-0.11935413628816605,
-0.11935413628816605,
-0.11935413628816605,
-0.27325770258903503,
-0.27325770258903503,
-0.3798418343067169,
-0.3798418343067169,
-0.3798418343067169,
-0.24042606353759766,
-0.24042606353759766,
-0.24042606353759766,
-0.21192394196987152,
-0.38907667994499207,
-0.38907667994499207,
-0.3583098351955414,
-0.3583098351955414,
-0.24134664237499237,
-0.24134664237499237,
-0.17328305542469025,
-0.17328305542469025,
-0.12573793530464172,
-0.12573793530464172,
-0.12573793530464172,
-0.09236020594835281,
-0.09236020594835281,
-0.09236020594835281,
-0.28382134437561035,
-0.28382134437561035,
-0.193727046251297,
-0.193727046251297,
-0.193727046251297,
-0.193727046251297,
-0.1862371861934662,
-0.1862371861934662,
-0.1862371861934662,
-0.3955586850643158,
-0.3955586850643158,
-0.2700965702533722,
-0.2700965702533722,
-0.2700965702533722,
-0.24620696902275085,
-0.24620696902275085,
-0.15558657050132751,
-0.15558657050132751,
-0.12051159143447876,
-0.12051159143447876,
-0.23935070633888245,
-0.23935070633888245,
-0.23935070633888245,
-0.23935070633888245,
-0.16685666143894196,
-0.16685666143894196,
-0.16685666143894196,
-0.2872607111930847,
-0.2872607111930847,
-0.16005627810955048,
-0.294393390417099,
-0.294393390417099,
-0.20374353229999542,
-0.20374353229999542,
-0.20374353229999542,
-0.1447821855545044,
-0.1447821855545044,
-0.19773179292678833,
-0.16724790632724762,
-0.34860774874687195,
-0.34860774874687195,
-0.19725480675697327,
-0.19725480675697327,
-0.19725480675697327,
-0.18211722373962402,
-0.18211722373962402,
-0.2905329167842865,
-0.2905329167842865,
-0.17074169218540192,
-0.17074169218540192,
-0.17074169218540192,
-0.32694724202156067,
-0.32694724202156067,
-0.32694724202156067,
-0.32694724202156067,
-0.32694724202156067,
-0.32694724202156067,
-0.32694724202156067,
-0.22037239372730255,
-0.22037239372730255,
-0.22037239372730255,
-0.1787131428718567,
-0.1787131428718567,
-0.1787131428718567,
-0.23000453412532806,
-0.14208434522151947,
-0.16224122047424316,
-0.16224122047424316,
-0.16224122047424316,
-0.2162160724401474,
-0.21755337715148926,
-0.13580474257469177,
-0.13580474257469177,
-0.13580474257469177,
-0.195076584815979,
-0.195076584815979,
-0.195076584815979,
-0.14910943806171417,
-0.14910943806171417,
-0.14910943806171417,
-0.15140052139759064,
-0.1828150898218155,
-0.1828150898218155,
-0.13586027920246124,
-0.13586027920246124,
-0.13586027920246124,
-0.13586027920246124,
-0.15885455906391144,
-0.15885455906391144,
-0.15885455906391144,
-0.1659436821937561,
-0.1659436821937561,
-0.1622745245695114,
-0.1622745245695114,
-0.23704826831817627,
-0.26861825585365295,
-0.26861825585365295,
-0.20032291114330292,
-0.20032291114330292,
-0.3059505224227905,
-0.3059505224227905,
-0.260004460811615,
-0.20960955321788788,
-0.13833627104759216,
-0.13833627104759216,
-0.13833627104759216,
-0.1590326726436615,
-0.1590326726436615,
-0.23684540390968323,
-0.23684540390968323,
-0.23684540390968323,
-0.09072718769311905,
-0.09072718769311905,
-0.09072718769311905,
-0.20805858075618744,
-0.20309405028820038,
-0.319614440202713,
-0.12335638701915741,
-0.12335638701915741,
-0.16713570058345795,
-0.19736996293067932,
-0.37687593698501587,
-0.10882056504487991,
-0.10882056504487991,
-0.21019132435321808
],
"compression_ratio": [
1.4822335243225098,
1.4822335243225098,
1.4822335243225098,
1.4822335243225098,
1.4822335243225098,
1.2190476655960083,
1.2190476655960083,
1.2190476655960083,
1.5032680034637451,
1.5032680034637451,
1.5032680034637451,
1.5032680034637451,
1.0506329536437988,
1.3095238208770752,
1.3095238208770752,
1.3095238208770752,
1.3095238208770752,
1,
1,
1,
1.4615384340286255,
1.4615384340286255,
1.4615384340286255,
1.4615384340286255,
1.4615384340286255,
1.174311876296997,
1.174311876296997,
1.4871795177459717,
1.4871795177459717,
1.4871795177459717,
1.4871795177459717,
1.4871795177459717,
1.774319052696228,
1.774319052696228,
1.774319052696228,
1.774319052696228,
1.774319052696228,
1.774319052696228,
1.5464481115341187,
1.5464481115341187,
1.5464481115341187,
1.5464481115341187,
1.5464481115341187,
1.5464481115341187,
1.5464481115341187,
1.2898551225662231,
1.2898551225662231,
1.2898551225662231,
1.4341084957122803,
1.4341084957122803,
1.470588207244873,
1.470588207244873,
1.470588207244873,
1.470588207244873,
1.470588207244873,
1.7062146663665771,
1.7062146663665771,
1.7062146663665771,
1.7062146663665771,
1.7062146663665771,
1.7062146663665771,
1.3706294298171997,
1.3706294298171997,
1.3706294298171997,
1.3196721076965332,
1.3196721076965332,
1.6146788597106934,
1.6146788597106934,
1.3113207817077637,
1.3113207817077637,
1.3719007968902588,
1.3719007968902588,
1.3719007968902588,
1.3941606283187866,
1.3941606283187866,
1.3941606283187866,
1.4864864349365234,
1.4864864349365234,
1.4864864349365234,
1.4864864349365234,
1.384615421295166,
1.384615421295166,
1.6293705701828003,
1.6293705701828003,
1.6293705701828003,
1.703196406364441,
1.703196406364441,
1.703196406364441,
1.703196406364441,
1.703196406364441,
1.703196406364441,
1.703196406364441,
1.6100000143051147,
1.6100000143051147,
1.6100000143051147,
1.6100000143051147,
1.6100000143051147,
1.5871559381484985,
1.5871559381484985,
1.5871559381484985,
1.5871559381484985,
1.5871559381484985,
1.6234817504882812,
1.6234817504882812,
1.6234817504882812,
1.6234817504882812,
1.6234817504882812,
1.6234817504882812,
1.1954022645950317,
1.1954022645950317,
1.1954022645950317,
1.1954022645950317,
1.4335260391235352,
1.4335260391235352,
1.4335260391235352,
1.4335260391235352,
1.4335260391235352,
1.4335260391235352,
1.752293586730957,
1.752293586730957,
1.752293586730957,
1.752293586730957,
1.752293586730957,
1.752293586730957,
1.5098038911819458,
1.5098038911819458,
1.5098038911819458,
1.5098038911819458,
1.4658385515213013,
1.4658385515213013,
1.4658385515213013,
1.3333333730697632,
1.3333333730697632,
1.3333333730697632,
1.4320000410079956,
1.4320000410079956,
1.5087718963623047,
1.5087718963623047,
1.5087718963623047,
1.5087718963623047,
1.5087718963623047,
1.5087718963623047,
1.5087718963623047,
1.5087718963623047,
1.2358490228652954,
1.2358490228652954,
1.2358490228652954,
1.5402299165725708,
1.5402299165725708,
1.5402299165725708,
1.5402299165725708,
1.5402299165725708,
1.3178807497024536,
1.3178807497024536,
1.3178807497024536,
1.3178807497024536,
1.1170213222503662,
1.1170213222503662,
1.1170213222503662,
1.1170213222503662,
1.2678571939468384,
1.2678571939468384,
1.5497381687164307,
1.5497381687164307,
1.5497381687164307,
1.5497381687164307,
1.5497381687164307,
1.454545497894287,
1.454545497894287,
1.454545497894287,
1.454545497894287,
1.3939393758773804,
1.3939393758773804,
1.3939393758773804,
1.3939393758773804,
1.3939393758773804,
1.80978262424469,
1.80978262424469,
1.80978262424469,
1.80978262424469,
1.80978262424469,
1.6481481790542603,
1.6481481790542603,
1.6481481790542603,
1.6481481790542603,
1.541176438331604,
1.541176438331604,
1.541176438331604,
1.541176438331604,
1.2342342138290405,
1.2342342138290405,
1.2342342138290405,
1.2342342138290405,
1.399999976158142,
1.399999976158142,
1.399999976158142,
1.6749999523162842,
1.6749999523162842,
1.6749999523162842,
1.6749999523162842,
1.6749999523162842,
1.4133332967758179,
1.4133332967758179,
1.4133332967758179,
1.4133332967758179,
1.4133332967758179,
1.7040358781814575,
1.7040358781814575,
1.7040358781814575,
1.7040358781814575,
1.7040358781814575,
1.4620689153671265,
1.4620689153671265,
1.4620689153671265,
1.606217622756958,
1.606217622756958,
1.606217622756958,
1.606217622756958,
1.606217622756958,
1.606217622756958,
1.5149253606796265,
1.5149253606796265,
1.5149253606796265,
1.5165562629699707,
1.5165562629699707,
1.5165562629699707,
1.4912281036376953,
1.4912281036376953,
1.4912281036376953,
1.4912281036376953,
1.465753436088562,
1.465753436088562,
1.465753436088562,
1.3671875,
1.3671875,
1.0980392694473267,
1.0980392694473267,
1.7857142686843872,
1.7857142686843872,
1.7857142686843872,
1.7857142686843872,
1.8156424760818481,
1.8156424760818481,
1.8156424760818481,
1.8156424760818481,
1.209876537322998,
1.209876537322998,
1.4586466550827026,
1.4586466550827026,
1.4586466550827026,
1.296875,
1.296875,
1.296875,
1.1944444179534912,
1.3482142686843872,
1.3482142686843872,
1.2777777910232544,
1.2777777910232544,
1.47826087474823,
1.47826087474823,
1.4672131538391113,
1.4672131538391113,
1.5666667222976685,
1.5666667222976685,
1.5666667222976685,
1.4772727489471436,
1.4772727489471436,
1.4772727489471436,
1.421875,
1.421875,
1.7184466123580933,
1.7184466123580933,
1.7184466123580933,
1.7184466123580933,
1.6167664527893066,
1.6167664527893066,
1.6167664527893066,
1.359375,
1.359375,
1.5730993747711182,
1.5730993747711182,
1.5730993747711182,
1.29104483127594,
1.29104483127594,
1.3358209133148193,
1.3358209133148193,
1.454545497894287,
1.454545497894287,
1.6412556171417236,
1.6412556171417236,
1.6412556171417236,
1.6412556171417236,
1.6392405033111572,
1.6392405033111572,
1.6392405033111572,
1.3762376308441162,
1.3762376308441162,
1.0886075496673584,
1.1785714626312256,
1.1785714626312256,
1.6445783376693726,
1.6445783376693726,
1.6445783376693726,
1.5537190437316895,
1.5537190437316895,
1.1216216087341309,
1.1780822277069092,
1.3058823347091675,
1.3058823347091675,
1.545454502105713,
1.545454502105713,
1.545454502105713,
1.585585594177246,
1.585585594177246,
1.4600000381469727,
1.4600000381469727,
1.6172839403152466,
1.6172839403152466,
1.6172839403152466,
1.8453038930892944,
1.8453038930892944,
1.8453038930892944,
1.8453038930892944,
1.8453038930892944,
1.8453038930892944,
1.8453038930892944,
1.453125,
1.453125,
1.453125,
1.6403508186340332,
1.6403508186340332,
1.6403508186340332,
1.1643835306167603,
1.1184210777282715,
1.404958724975586,
1.404958724975586,
1.404958724975586,
1.1728395223617554,
1.1857142448425293,
1.668965458869934,
1.668965458869934,
1.668965458869934,
1.5360000133514404,
1.5360000133514404,
1.5360000133514404,
1.811188817024231,
1.811188817024231,
1.811188817024231,
1.1891891956329346,
1.399999976158142,
1.399999976158142,
1.8457447290420532,
1.8457447290420532,
1.8457447290420532,
1.8457447290420532,
1.7039474248886108,
1.7039474248886108,
1.7039474248886108,
1.508474588394165,
1.508474588394165,
1.5258620977401733,
1.5258620977401733,
1.1486486196517944,
1.5333333015441895,
1.5333333015441895,
1.4112902879714966,
1.4112902879714966,
1.423423409461975,
1.423423409461975,
1.1486486196517944,
1.1818181276321411,
1.7382550239562988,
1.7382550239562988,
1.7382550239562988,
1.5087718963623047,
1.5087718963623047,
1.7516778707504272,
1.7516778707504272,
1.7516778707504272,
1.5393258333206177,
1.5393258333206177,
1.5393258333206177,
1.2133333683013916,
1.1282051801681519,
1.100000023841858,
1.5423728227615356,
1.5423728227615356,
1.1558442115783691,
1.1948051452636719,
1.0909091234207153,
1.385826826095581,
1.385826826095581,
1.0724637508392334
],
"no_speech_prob": [
0.08491707593202591,
0.08491707593202591,
0.08491707593202591,
0.08491707593202591,
0.08491707593202591,
0.0002541036519687623,
0.0002541036519687623,
0.0002541036519687623,
0.0003020369913429022,
0.0003020369913429022,
0.0003020369913429022,
0.0003020369913429022,
0.0005087549798190594,
0.00004440361954038963,
0.00004440361954038963,
0.00004440361954038963,
0.00004440361954038963,
0.00008137224358506501,
0.00008137224358506501,
0.00008137224358506501,
0.0001431756973033771,
0.0001431756973033771,
0.0001431756973033771,
0.0001431756973033771,
0.0001431756973033771,
0.00015015607641544193,
0.00015015607641544193,
0.00012024644092889503,
0.00012024644092889503,
0.00012024644092889503,
0.00012024644092889503,
0.00012024644092889503,
0.00003714456033776514,
0.00003714456033776514,
0.00003714456033776514,
0.00003714456033776514,
0.00003714456033776514,
0.00003714456033776514,
0.00011121798161184415,
0.00011121798161184415,
0.00011121798161184415,
0.00011121798161184415,
0.00011121798161184415,
0.00011121798161184415,
0.00011121798161184415,
0.000030900097044650465,
0.000030900097044650465,
0.000030900097044650465,
0.00004294764221413061,
0.00004294764221413061,
0.00005642643736791797,
0.00005642643736791797,
0.00005642643736791797,
0.00005642643736791797,
0.00005642643736791797,
0.00003064936026930809,
0.00003064936026930809,
0.00003064936026930809,
0.00003064936026930809,
0.00003064936026930809,
0.00003064936026930809,
0.000025030198230524547,
0.000025030198230524547,
0.000025030198230524547,
0.0001554269256303087,
0.0001554269256303087,
0.0000780250848038122,
0.0000780250848038122,
0.00007354107947321609,
0.00007354107947321609,
0.00021589378593489528,
0.00021589378593489528,
0.00021589378593489528,
0.001071352744475007,
0.001071352744475007,
0.001071352744475007,
0.00009436134860152379,
0.00009436134860152379,
0.00009436134860152379,
0.00009436134860152379,
0.0000370459420082625,
0.0000370459420082625,
0.000021077268684166484,
0.000021077268684166484,
0.000021077268684166484,
0.00033759375219233334,
0.00033759375219233334,
0.00033759375219233334,
0.00033759375219233334,
0.00033759375219233334,
0.00033759375219233334,
0.00033759375219233334,
0.00015398956020362675,
0.00015398956020362675,
0.00015398956020362675,
0.00015398956020362675,
0.00015398956020362675,
0.00006737665535183623,
0.00006737665535183623,
0.00006737665535183623,
0.00006737665535183623,
0.00006737665535183623,
0.000019901091945939697,
0.000019901091945939697,
0.000019901091945939697,
0.000019901091945939697,
0.000019901091945939697,
0.000019901091945939697,
0.000854671117849648,
0.000854671117849648,
0.000854671117849648,
0.000854671117849648,
0.000538768304977566,
0.000538768304977566,
0.000538768304977566,
0.000538768304977566,
0.000538768304977566,
0.000538768304977566,
0.00009298127406509593,
0.00009298127406509593,
0.00009298127406509593,
0.00009298127406509593,
0.00009298127406509593,
0.00009298127406509593,
0.000022437801817432046,
0.000022437801817432046,
0.000022437801817432046,
0.000022437801817432046,
0.0001033994703902863,
0.0001033994703902863,
0.0001033994703902863,
0.00009942382166627795,
0.00009942382166627795,
0.00009942382166627795,
0.0001257304975297302,
0.0001257304975297302,
0.00035803107311949134,
0.00035803107311949134,
0.00035803107311949134,
0.00035803107311949134,
0.00035803107311949134,
0.00035803107311949134,
0.00035803107311949134,
0.00035803107311949134,
0.0007505517569370568,
0.0007505517569370568,
0.0007505517569370568,
0.000003843042577500455,
0.000003843042577500455,
0.000003843042577500455,
0.000003843042577500455,
0.000003843042577500455,
0.0001056110268109478,
0.0001056110268109478,
0.0001056110268109478,
0.0001056110268109478,
0.00018185839871875942,
0.00018185839871875942,
0.00018185839871875942,
0.00018185839871875942,
0.00004547834032564424,
0.00004547834032564424,
0.00003817649485426955,
0.00003817649485426955,
0.00003817649485426955,
0.00003817649485426955,
0.00003817649485426955,
0.000026623058147379197,
0.000026623058147379197,
0.000026623058147379197,
0.000026623058147379197,
0.0001934203173732385,
0.0001934203173732385,
0.0001934203173732385,
0.0001934203173732385,
0.0001934203173732385,
0.00001638865978748072,
0.00001638865978748072,
0.00001638865978748072,
0.00001638865978748072,
0.00001638865978748072,
0.00004603600245900452,
0.00004603600245900452,
0.00004603600245900452,
0.00004603600245900452,
0.000202825729502365,
0.000202825729502365,
0.000202825729502365,
0.000202825729502365,
0.001889759791083634,
0.001889759791083634,
0.001889759791083634,
0.001889759791083634,
0.000343816151143983,
0.000343816151143983,
0.000343816151143983,
0.000014293858839664608,
0.000014293858839664608,
0.000014293858839664608,
0.000014293858839664608,
0.000014293858839664608,
0.0012546004727482796,
0.0012546004727482796,
0.0012546004727482796,
0.0012546004727482796,
0.0012546004727482796,
0.00055380177218467,
0.00055380177218467,
0.00055380177218467,
0.00055380177218467,
0.00055380177218467,
0.00007934811583254486,
0.00007934811583254486,
0.00007934811583254486,
0.00001574136331328191,
0.00001574136331328191,
0.00001574136331328191,
0.00001574136331328191,
0.00001574136331328191,
0.00001574136331328191,
0.00004059927596244961,
0.00004059927596244961,
0.00004059927596244961,
0.00001541612800792791,
0.00001541612800792791,
0.00001541612800792791,
0.00006816791574237868,
0.00006816791574237868,
0.00006816791574237868,
0.00006816791574237868,
0.00006944368942640722,
0.00006944368942640722,
0.00006944368942640722,
0.000030189523386070505,
0.000030189523386070505,
0.00003215096512576565,
0.00003215096512576565,
0.00002493258216418326,
0.00002493258216418326,
0.00002493258216418326,
0.00002493258216418326,
0.000011114386325061787,
0.000011114386325061787,
0.000011114386325061787,
0.000011114386325061787,
0.00017961977573577315,
0.00017961977573577315,
0.000443645054474473,
0.000443645054474473,
0.000443645054474473,
0.00007454913429683074,
0.00007454913429683074,
0.00007454913429683074,
0.00004173668276052922,
0.00009419567504664883,
0.00009419567504664883,
0.00010383486369391903,
0.00010383486369391903,
0.00005389045691117644,
0.00005389045691117644,
0.000006577320164069533,
0.000006577320164069533,
0.00001628970494493842,
0.00001628970494493842,
0.00001628970494493842,
0.00008212214015657082,
0.00008212214015657082,
0.00008212214015657082,
0.00023921883257571608,
0.00023921883257571608,
0.00016101004439406097,
0.00016101004439406097,
0.00016101004439406097,
0.00016101004439406097,
0.00013454871077556163,
0.00013454871077556163,
0.00013454871077556163,
0.000024272711016237736,
0.000024272711016237736,
0.00010334340913686901,
0.00010334340913686901,
0.00010334340913686901,
0.00013934620073996484,
0.00013934620073996484,
0.000038639369449811056,
0.000038639369449811056,
0.00007191096665337682,
0.00007191096665337682,
0.0003418594424147159,
0.0003418594424147159,
0.0003418594424147159,
0.0003418594424147159,
0.00007753379759378731,
0.00007753379759378731,
0.00007753379759378731,
0.00008770907152211294,
0.00008770907152211294,
0.0005673852283507586,
0.00016451036208309233,
0.00016451036208309233,
0.00007578635268146172,
0.00007578635268146172,
0.00007578635268146172,
0.00007137219654396176,
0.00007137219654396176,
0.00003284507693024352,
0.00002841345121851191,
0.00011319610348436981,
0.00011319610348436981,
0.000056812157708918676,
0.000056812157708918676,
0.000056812157708918676,
0.000248857366386801,
0.000248857366386801,
0.000042133411625400186,
0.000042133411625400186,
0.00004974904004484415,
0.00004974904004484415,
0.00004974904004484415,
0.0005048692692071199,
0.0005048692692071199,
0.0005048692692071199,
0.0005048692692071199,
0.0005048692692071199,
0.0005048692692071199,
0.0005048692692071199,
0.0008770241984166205,
0.0008770241984166205,
0.0008770241984166205,
0.0001341777970083058,
0.0001341777970083058,
0.0001341777970083058,
0.000013292602488945704,
0.000009703760042611975,
0.00001529300061520189,
0.00001529300061520189,
0.00001529300061520189,
0.000049984704673988745,
0.000014280830328061711,
0.00007110098522389308,
0.00007110098522389308,
0.00007110098522389308,
0.00008734034781809896,
0.00008734034781809896,
0.00008734034781809896,
0.00026797442114911973,
0.00026797442114911973,
0.00026797442114911973,
0.00010883247887250036,
0.00007901559729361907,
0.00007901559729361907,
0.00002142578159691766,
0.00002142578159691766,
0.00002142578159691766,
0.00002142578159691766,
0.00003368268517078832,
0.00003368268517078832,
0.00003368268517078832,
0.000021436235329019837,
0.000021436235329019837,
0.00001768729998730123,
0.00001768729998730123,
0.0004103727696929127,
0.00045585958287119865,
0.00045585958287119865,
0.00002775790198938921,
0.00002775790198938921,
0.0003667715936899185,
0.0003667715936899185,
0.000205906544579193,
0.00016922442591749132,
0.0001591277978150174,
0.0001591277978150174,
0.0001591277978150174,
0.00006489215593319386,
0.00006489215593319386,
0.00008559448178857565,
0.00008559448178857565,
0.00008559448178857565,
0.00016371882520616055,
0.00016371882520616055,
0.00016371882520616055,
0.0004455760936252773,
0.000018181994164478965,
0.00011426350101828575,
0.00003539660247042775,
0.00003539660247042775,
0.000055921529565239325,
0.00003164814188494347,
0.0008338029147125781,
0.001567757222801447,
0.001567757222801447,
0.0013744891621172428
]
} | {
"slide": [
"201810150900-Peterzil_lrv_0_215.png",
"201810150900-Peterzil_lrv_216_1259.png",
"201810150900-Peterzil_lrv_1260_3479.png",
"201810150900-Peterzil_lrv_3480_3683.png",
"201810150900-Peterzil_lrv_3684_5111.png",
"201810150900-Peterzil_lrv_5112_12683.png",
"201810150900-Peterzil_lrv_12684_13451.png",
"201810150900-Peterzil_lrv_13452_14243.png",
"201810150900-Peterzil_lrv_14244_15071.png",
"201810150900-Peterzil_lrv_15072_16463.png",
"201810150900-Peterzil_lrv_16464_17039.png",
"201810150900-Peterzil_lrv_17040_17171.png",
"201810150900-Peterzil_lrv_17172_17231.png",
"201810150900-Peterzil_lrv_17232_21395.png",
"201810150900-Peterzil_lrv_21396_22367.png",
"201810150900-Peterzil_lrv_22368_23639.png",
"201810150900-Peterzil_lrv_23640_23831.png",
"201810150900-Peterzil_lrv_23832_24047.png",
"201810150900-Peterzil_lrv_24048_24779.png",
"201810150900-Peterzil_lrv_24780_26351.png",
"201810150900-Peterzil_lrv_26352_27203.png",
"201810150900-Peterzil_lrv_27204_28907.png",
"201810150900-Peterzil_lrv_28908_28967.png",
"201810150900-Peterzil_lrv_28968_31079.png",
"201810150900-Peterzil_lrv_31080_32843.png",
"201810150900-Peterzil_lrv_32844_32927.png",
"201810150900-Peterzil_lrv_32928_33803.png",
"201810150900-Peterzil_lrv_33804_34091.png",
"201810150900-Peterzil_lrv_34092_34247.png",
"201810150900-Peterzil_lrv_34248_35567.png",
"201810150900-Peterzil_lrv_35568_38951.png",
"201810150900-Peterzil_lrv_38952_39107.png",
"201810150900-Peterzil_lrv_39108_40883.png",
"201810150900-Peterzil_lrv_40884_41831.png",
"201810150900-Peterzil_lrv_41832_42083.png",
"201810150900-Peterzil_lrv_42084_43847.png",
"201810150900-Peterzil_lrv_43848_46391.png",
"201810150900-Peterzil_lrv_46392_46535.png",
"201810150900-Peterzil_lrv_46536_49463.png",
"201810150900-Peterzil_lrv_49464_50771.png",
"201810150900-Peterzil_lrv_50772_54611.png",
"201810150900-Peterzil_lrv_54612_56783.png",
"201810150900-Peterzil_lrv_56784_57107.png",
"201810150900-Peterzil_lrv_57108_59711.png",
"201810150900-Peterzil_lrv_59712_61271.png",
"201810150900-Peterzil_lrv_61272_61379.png",
"201810150900-Peterzil_lrv_61380_65999.png",
"201810150900-Peterzil_lrv_66000_68543.png",
"201810150900-Peterzil_lrv_68544_69095.png",
"201810150900-Peterzil_lrv_69096_71987.png",
"201810150900-Peterzil_lrv_71988_77039.png",
"201810150900-Peterzil_lrv_77040_77915.png",
"201810150900-Peterzil_lrv_77916_77992.png"
],
"frames": [
[
0,
215
],
[
216,
1259
],
[
1260,
3479
],
[
3480,
3683
],
[
3684,
5111
],
[
5112,
12683
],
[
12684,
13451
],
[
13452,
14243
],
[
14244,
15071
],
[
15072,
16463
],
[
16464,
17039
],
[
17040,
17171
],
[
17172,
17231
],
[
17232,
21395
],
[
21396,
22367
],
[
22368,
23639
],
[
23640,
23831
],
[
23832,
24047
],
[
24048,
24779
],
[
24780,
26351
],
[
26352,
27203
],
[
27204,
28907
],
[
28908,
28967
],
[
28968,
31079
],
[
31080,
32843
],
[
32844,
32927
],
[
32928,
33803
],
[
33804,
34091
],
[
34092,
34247
],
[
34248,
35567
],
[
35568,
38951
],
[
38952,
39107
],
[
39108,
40883
],
[
40884,
41831
],
[
41832,
42083
],
[
42084,
43847
],
[
43848,
46391
],
[
46392,
46535
],
[
46536,
49463
],
[
49464,
50771
],
[
50772,
54611
],
[
54612,
56783
],
[
56784,
57107
],
[
57108,
59711
],
[
59712,
61271
],
[
61272,
61379
],
[
61380,
65999
],
[
66000,
68543
],
[
68544,
69095
],
[
69096,
71987
],
[
71988,
77039
],
[
77040,
77915
],
[
77916,
77992
]
],
"timestamp": [
[
0,
8.640000343322754
],
[
8.640000343322754,
50.400001525878906
],
[
50.400001525878906,
139.1999969482422
],
[
139.1999969482422,
147.36000061035156
],
[
147.36000061035156,
204.47999572753906
],
[
204.47999572753906,
507.3599853515625
],
[
507.3599853515625,
538.0800170898438
],
[
538.0800170898438,
569.760009765625
],
[
569.760009765625,
602.8800048828125
],
[
602.8800048828125,
658.5599975585938
],
[
658.5599975585938,
681.5999755859375
],
[
681.5999755859375,
686.8800048828125
],
[
686.8800048828125,
689.280029296875
],
[
689.280029296875,
855.8400268554688
],
[
855.8400268554688,
894.719970703125
],
[
894.719970703125,
945.5999755859375
],
[
945.5999755859375,
953.280029296875
],
[
953.280029296875,
961.9199829101562
],
[
961.9199829101562,
991.2000122070312
],
[
991.2000122070312,
1054.0799560546875
],
[
1054.0799560546875,
1088.1600341796875
],
[
1088.1600341796875,
1156.3199462890625
],
[
1156.3199462890625,
1158.719970703125
],
[
1158.719970703125,
1243.199951171875
],
[
1243.199951171875,
1313.760009765625
],
[
1313.760009765625,
1317.1199951171875
],
[
1317.1199951171875,
1352.1600341796875
],
[
1352.1600341796875,
1363.6800537109375
],
[
1363.6800537109375,
1369.9200439453125
],
[
1369.9200439453125,
1422.719970703125
],
[
1422.719970703125,
1558.0799560546875
],
[
1558.0799560546875,
1564.3199462890625
],
[
1564.3199462890625,
1635.3599853515625
],
[
1635.3599853515625,
1673.280029296875
],
[
1673.280029296875,
1683.3599853515625
],
[
1683.3599853515625,
1753.9200439453125
],
[
1753.9200439453125,
1855.6800537109375
],
[
1855.6800537109375,
1861.43994140625
],
[
1861.43994140625,
1978.56005859375
],
[
1978.56005859375,
2030.8800048828125
],
[
2030.8800048828125,
2184.47998046875
],
[
2184.47998046875,
2271.360107421875
],
[
2271.360107421875,
2284.320068359375
],
[
2284.320068359375,
2388.47998046875
],
[
2388.47998046875,
2450.8798828125
],
[
2450.8798828125,
2455.199951171875
],
[
2455.199951171875,
2640
],
[
2640,
2741.760009765625
],
[
2741.760009765625,
2763.840087890625
],
[
2763.840087890625,
2879.52001953125
],
[
2879.52001953125,
3081.60009765625
],
[
3081.60009765625,
3116.639892578125
],
[
3116.639892578125,
3119.719970703125
]
]
} | en |
10.5446/59321 (DOI) | Non-forking and preservation of NIP types | https://av.tib.eu/media/59321 | "https://tib.flowcenter.de/mfc/medialink/3/de45d4f713ce75c7325b3db6bdeb44ea6b49a511db6e3d85eac7c8323(...TRUNCATED) | "CC Attribution - NonCommercial - NoDerivatives 4.0 International:\nYou are free to use, copy, distr(...TRUNCATED) | Mathematics | Workshop/Interactive Format
Lecture | 2018 | Kaplan, Itay | Estevan, Pedro Andres | "Adler, Casanovas and Pillay proved that if p is a complete stable type over a set B which does not (...TRUNCATED) | " Joint work with Pedro Andrés Estevan. I think he has another name, but I think I put only three b(...TRUNCATED) | {"id":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,3(...TRUNCATED) | {"slide":["201810151002-Kaplan_lrv_0_4571.png","201810151002-Kaplan_lrv_4572_5495.png","201810151002(...TRUNCATED) | en |
10.5446/59322 (DOI) | Strong type spaces as quotients of Polish groups | https://av.tib.eu/media/59322 | "https://tib.flowcenter.de/mfc/medialink/3/deaae7f6a805c5d520ca8f38a30fbe3d916cd938a597f3fc50f075be1(...TRUNCATED) | "CC Attribution - NonCommercial - NoDerivatives 4.0 International:\nYou are free to use, copy, distr(...TRUNCATED) | Mathematics | Workshop/Interactive Format
Lecture | 2018 | Rzepecki, Tomasz | Krupiński, Krzysztof | "In recent work with Krupiński, we showed that strong type spaces can be seen (in a strong sense) a(...TRUNCATED) | " So this is joint work with Christoph Kruppin's, get at least most of it. Pretty much all of the co(...TRUNCATED) | {"id":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,3(...TRUNCATED) | {"slide":["201810151140-Rzepecki_lrv_0_131.png","201810151140-Rzepecki_lrv_132_4007.png","2018101511(...TRUNCATED) | en |
10.5446/59323 (DOI) | Metastability | https://av.tib.eu/media/59323 | "https://tib.flowcenter.de/mfc/medialink/3/dec7dd6b2dc7b3916b31e13a3da3d5e30e9d2d1ab6f95c300ac0ed105(...TRUNCATED) | "CC Attribution - NonCommercial - NoDerivatives 4.0 International:\nYou are free to use, copy, distr(...TRUNCATED) | Mathematics | Workshop/Interactive Format
Lecture | 2018 | Rideau, Silvain | null | "In their work on the model theory of algebraically closed valued fields, Haskell, Hrushovski and Ma(...TRUNCATED) | " But to be fair, as far as I know, there is just one unique paper in the world that deduces stuff f(...TRUNCATED) | {"id":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,3(...TRUNCATED) | {"slide":["201810151230-Rideau_lrv_0_935.png","201810151230-Rideau_lrv_936_9323.png","201810151230-R(...TRUNCATED) | en |
10.5446/59325 (DOI) | Linear orders in NIP theories | https://av.tib.eu/media/59325 | "https://tib.flowcenter.de/mfc/medialink/3/def0321d5e893b72ddff365fd56b60ea8a799f033a534ea3b8f1187f9(...TRUNCATED) | "CC Attribution - NonCommercial - NoDerivatives 4.0 International:\nYou are free to use, copy, distr(...TRUNCATED) | Mathematics | Workshop/Interactive Format
Lecture | 2018 | Simon, Pierre | null | "A longstanding open questions asks whether an unstable NIP theory interprets an infinite linear ord(...TRUNCATED) | " that says that it says something a little bit weaker than saying that in any unstable NIP theory, (...TRUNCATED) | {"id":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,3(...TRUNCATED) | {"slide":["201810151801-Simon_lrv_0_1235.png","201810151801-Simon_lrv_1236_2291.png","201810151801-S(...TRUNCATED) | en |
10.5446/59327 (DOI) | Localized Lascar group | https://av.tib.eu/media/59327 | "https://tib.flowcenter.de/mfc/medialink/3/de34744ea8c63748c8a3f36161a2ab5b7e615b03883c89461da13c7b4(...TRUNCATED) | "CC Attribution - NonCommercial - NoDerivatives 4.0 International:\nYou are free to use, copy, distr(...TRUNCATED) | Mathematics | Workshop/Interactive Format
Lecture | 2018 | Dobrowolski, Jan | null | "The notion of the localized Lascar-Galois group GalL(p) of a type p appeared recently in the contex(...TRUNCATED) | " Pyong Han Kim, Alexi Kolesnikov, and Cheong-Buk Lee. And they will start from some motivations. So(...TRUNCATED) | {"id":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,3(...TRUNCATED) | {"slide":["201810161006-Dobrowolski_lrv_0_1703.png","201810161006-Dobrowolski_lrv_1704_2591.png","20(...TRUNCATED) | en |
10.5446/59329 (DOI) | NSOP_1 theories | https://av.tib.eu/media/59329 | "https://tib.flowcenter.de/mfc/medialink/3/deb178bec21cb28e3aba0e387661dd1ee3470e64356ed9bf8b30e9943(...TRUNCATED) | "CC Attribution - NonCommercial - NoDerivatives 4.0 International:\nYou are free to use, copy, distr(...TRUNCATED) | Mathematics | Workshop/Interactive Format
Lecture | 2018 | Kim, Byunghan | null | "Let T be an NSOP1 theory. Recently I. Kaplan and N. Ramsey proved that in T, the so-called Kim-inde(...TRUNCATED) | " for Alex's previous talk and then also preparatory talk for the next speaker. Believe it or not, I(...TRUNCATED) | {"id":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,3(...TRUNCATED) | {"slide":["201810161231-Kim_lrv_0_59.png","201810161231-Kim_lrv_60_827.png","201810161231-Kim_lrv_82(...TRUNCATED) | en |
10.5446/59330 (DOI) | Equivalence query learning and the negation of the finite cover property | https://av.tib.eu/media/59330 | "https://tib.flowcenter.de/mfc/medialink/3/de895066b5ee51c83b3003fc1afd9bde2ecf315fb06e94d6776f1d24c(...TRUNCATED) | "CC Attribution - NonCommercial - NoDerivatives 4.0 International:\nYou are free to use, copy, distr(...TRUNCATED) | Mathematics | Workshop/Interactive Format
Lecture | 2018 | Chase, Hunter | null | "There are multiple connections between model-theoretic notions of complexity and machine learning. (...TRUNCATED) | " the general setup of what machine learning looks like in general and what it looks like in the spe(...TRUNCATED) | {"id":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,3(...TRUNCATED) | {"slide":["201810161612-Chase_lrv_0_119.png","201810161612-Chase_lrv_120_191.png","201810161612-Chas(...TRUNCATED) | en |
10.5446/59331 (DOI) | Retro-stability: The fine structure of classifiable theories | https://av.tib.eu/media/59331 | "https://tib.flowcenter.de/mfc/medialink/3/de605731f67f306b2d16d35deda830f0fc2dacda3837fd5783bf8ad5e(...TRUNCATED) | "CC Attribution - NonCommercial - NoDerivatives 4.0 International:\nYou are free to use, copy, distr(...TRUNCATED) | Mathematics | Workshop/Interactive Format
Lecture | 2018 | Laskowski, Chris | null | "We give (equivalent) friendlier definitions of classifiable theories strengthen known results about(...TRUNCATED) | " I want to thank the organizers both for inviting me to this and for allowing this talk because I s(...TRUNCATED) | {"id":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,3(...TRUNCATED) | {"slide":["201810161801-Laskowski_lrv_0_59.png","201810161801-Laskowski_lrv_60_575.png","20181016180(...TRUNCATED) | en |
10.5446/59332 (DOI) | "An \"Ahlbrandt-Ziegler Reconstruction\" for theories which are not necessarily countably categorica(...TRUNCATED) | https://av.tib.eu/media/59332 | "https://tib.flowcenter.de/mfc/medialink/3/de2192d172408a6d128fc523afc8ff68b30eabfad595cd1233e46abf6(...TRUNCATED) | "CC Attribution - NonCommercial - NoDerivatives 4.0 International:\nYou are free to use, copy, distr(...TRUNCATED) | Mathematics | Workshop/Interactive Format
Lecture | 2018 | Yaacov, Itaï Ben | null | "It is by now almost folklore that if T is a countably categorical theory, and M its unique countabl(...TRUNCATED) | " But I'll talk about it nonetheless. It's actually nothing particularly deep. It's a personal obses(...TRUNCATED) | {"id":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,3(...TRUNCATED) | {"slide":["201810170902-BenYaacov_lrv_0_3155.png","201810170902-BenYaacov_lrv_3156_3215.png","201810(...TRUNCATED) | en |
End of preview. Expand
in Dataset Viewer.
- Downloads last month
- 0