[00:00:06]
>> Ok so this is some some joint work with my P.T.'s and Victor. And. The the movie but the motivation for much of this work is coming from beautiful result by a bad since Be a man and serve us a lot from about a decade ago and. What they what they showed is the following they showed that if you take any graph any other active graph.

[00:00:37]
Then it's possible to 2 sponsored by the graph so particular you can you can find we it's that you can put on. A limited number of the edges so that. The following happens. Whenever you take a cut and you check how many edges did you cut in the original graph and how many.

[00:01:04]
How many how much weight do you cut in the new Spotify prof you get about the say. Ok. Ok so there is a. You can reduce the setting to something that. Are purely in the edge of the terms and. Then looks like this. You are you have to have and the actor is in the end that mentions and there so it may rise so that the sum of the odd upper axis the anything.

[00:01:41]
Can sponsor 5 the sum and you can specify the some meaning that you can put weights in front of the back doors. So that you're still getting approximately their identity. And you only need to leave a number of the vector so we immediately number of the vectors have a weight that's positive Ok.

[00:02:04]
So the original statement that I showed you can be reduced to this by you do you do you start with the let caution and kind of a little transformation of that you get the identity and every adds kind of becomes one act. Ok. And if you if you're not familiar with with the sign this is the Ph d. ordering So what this means is that the eigenvalues of this new tricks years and by Matrix they're going to be between one man is absent on and on plus absolute Ok.

[00:02:38]
Ok so. I read that you understand this result a little bit better and what we can go do is I can give you a different a rhythm. For that the sort of slightly more general problem. And I'm going to explain you better grab them and then we'll spend the rest of the talk and kind of understand why they are the make your words and what we're actually comes from Ok so let's say let's see the following You are you have some positive sim if any traces these are in by any traces and they they sum up to date in d.t. and you would like to specify the sum so you would like to throw out some of these of these matrices and others they get a they get some positive way that's bigger better than Ok.

[00:03:28]
And. So. You can do the following. You can define. Said coolly which is. Ok the set of weights. Negative way it's the set of weights that you can put in front of to the meter is is so that. There's some how. As operator and on bounded by something Ok so this means that Ok This means that this me 2 weeks has eigenvalues that are between minus this and plus this Ok you operate on and this is the other maxims in your values of the this is the maximum absolute value of one of the I place.

[00:04:16]
Ok so if you're if you if you think about what is the set actually so. This is a it's a comic set it's a symmetrical next set. It's so major convex set off sort of fraction where values they could you could and then you can put in front so that you kind of get something that screws to the meetings in Inspector terms Ok good.

[00:04:42]
Now you know you do the following if you take. Them with you kind of knows what's coming next. You take a random question. And you get the feeling you compute the nearest Perry and in that body key intersected with the Hi Becky. Ok but if you think about how this thing is.

[00:05:09]
This played by it's going to be it's going to have entries between might have been run and if you put the bag in front of the. Then you get something that's gross of the 0 matrix and the spectrum tops Ok it's going to very small investors Ok and then you do the favorite game.

[00:05:31]
Of The Matrix this. And you replace the matrix n.a.i.a. with the one plus the sign that you got for the for the i matrix Ok. And that's the one that. If you see this you will probably see this is we this coming from and why why on earth should this be working.

[00:05:56]
Ok so. The. The key thing that you have to problem is that actually you make progress in every step in the sense that this said this isn't just the origin but definitely isn't there this said it's going to it's going to contain a lot of good positive colorings.

[00:06:16]
So it's going to contain a concert a fraction of the entries actually passed on by this one bad concert fracture and the vector is a constant fraction of the entries. 0 so you say bit of a metre he says at the beginning of the next iteration it's going to be as you're paid 9 times.

[00:06:39]
Maybe it was Ok. If the ones you. Like because one minus ones are Ok you can you can flip the science and you get that exponential it's measured. Feed a pro believe or you can just keep it and then say that the symmetry argument that you're Abbott having a lot of money is one of the reasons like every one's.

[00:07:06]
Ok. Ok so. Ok. Take some time to digest this and. Come back to the states Ok so. The technique that's behind this and kind of the understanding behind this comes from a different field of. Just because it's a water space or this. Ok this is this is kind of the apps are on that is basically the epsilon here if you think of this is being and this is the end if you set me on square rearrange what is my absolute and then the absolute is this.

[00:07:53]
Ok And so the question. Ok so. There's a there. Never fear of going to Tory it's that I like very much and that's called the discrepancy theory. And usually this is a phase of coming true terms so there is some. Member and elements and you are asked to find the good coloring and the coming it's.

[00:08:20]
A kind of moon so you're covering every element with the the minus one approx one order like red or blue like and the grey we use to color the sets evenly Ok evenly meaning every set ideally should have the same number of past ones at minus ones and if that's all possible at least minimize the in dots Ok.

[00:08:43]
And the discrepancy of the sexes it's kind of the minimal imbalance that you need so you minimize or grow the colorings and then what counts is the. Color set Ok. Just to give you some intuition. If you have n. sets and the lens and Torrie sets than one can prove that the discrepancy is only true then and this is a beautiful result by Spencer from these.

[00:09:22]
As the different setting that's popular where every I remember is and that was he said it's then you can prove that that is discrepancy of the sets of it's less than 2 t.. And the conjecture is that accounts of them through t. is enough but this has been open since the eighty's.

[00:09:41]
Ok. If you if you look into the literature then said of the main method how you how do you find a good good how doing this you find a partial coloring Ok so a partial coloring can mean that you're covering only a fraction of the other men's and the other 2 sue the colors you believe 0 or you can leave them anything that between minus one it was.

[00:10:11]
Ok. Ok so. There's some concept of math as how you can find partial colorings and then I'm going to show you and that's an algorithm and. Shamus me this is a do to myself. Ok. And Pure So those were left exactly the same picture. It was like that with the with the except it's not a political effect and drawing to it so again is it good sense Ok so.

[00:10:44]
Ok so this is from myself and 5 years ago and all Ok if you do the following you're have a comeback so music set came and it's kind of. That originated in the sense of value in the sense I got as you measure it which is set up isn't that different always on volume but we get so many moves Is that Ok if you take a random guy as you and just standard variants.

[00:11:10]
And expectations here and then you have at least an exponential. To being king so he's not too tight then the famous through then well you can find a vector y. that even came to a sect at the hypercube and a constant fraction of the entries are going to be impacted by this one and you can do this simply no time.

[00:11:36]
Ok. Good Ok so. How does the algorithm work again but I cue up by pure coincidence I lead with as follows that you pick a random gosh I'm. Ok I didn't see it coming but you compute the clue since. The hypercube in terms of the o 2 Ok and that that is your point why Ok good.

[00:12:08]
Ok so it was kind of bad to them Don's work for 2 Gad. Companies But that's Ok. And a question for the side. Ok what does it have to do you coud you use this a good them for example to reprove I spent the r.m.. To set. Your colleague's body it would be the set of x. kind of like or the set of car with this quantity is bad.

[00:12:53]
And then you sure there are this is a this is a kind of body it's a metric and there's some lemma that is going to tell you that because when we had any constrains the guys who measure it's going to be at that age Ok but I'm cause.

[00:13:09]
Ok. So this is the I've got them and. I'm going to give you a run saw it off and I was of the algorithm 2 reasons for several It's it's not very often that I haven't added that fits I once lived with analysis puts on one slide and secondly is that I will need some observation from there going that will that help us later Ok.

[00:13:36]
Ok so I wrote this I didn't work and Ok so this body this is just something come back to me to be glad you know this is not the set Fromm that spectra sponsored by. Ok. Good so if you if you think about what is actually the distance of a random Gaussian to kill intersect at the hypercube then 1st of all it's a it's a house for sure that it has to be at least some fixed come sometimes rude and the reason is that I mean this is already the this is I've read I'm going to the hybrid group going to calculate this by checking like in every quarter what's my expected distance.

[00:14:15]
Ok. This is. Where them greedy and that we need is. Coming from measure concentration. And the fame in words you can take and use it any measurable said Cuchulainn mentions and if the gosh a measure of the Sat is at least Ok at least exponentially small and imagine there's some constant here.

[00:14:42]
Then the expected distance of a random Gaussian to the set of narrowest a constant times route n. and what's happening is that Ok if you pick the constant here small enough you can get any small constant here that you like Ok Ok but so what this means is they have to have a large enough set in terms of the guys you measure then any read and read them guys will actually be quite close to it.

[00:15:10]
Ok good. So now she has his the kids of Asian if you if you get instead 100 is distance if you read them gosh into that Christmas point and came to say Could I pick you I guess I should be determined that this is this is a it's a kind of spooky but it's actually rather with misspelling them in the 1st place.

[00:15:34]
Where the brother is being picked in in k. and it has to satisfy these new constraints. Ok And maybe just a family in fact about can't expect Rose. Any construe that's not tied to the optimum you can throw it away and the value of the afternoon doesn't change Ok.

[00:15:56]
Since appears for the sake of contradiction that this play and why stab me have very few constraints that I passed minus one. Then actually. I can drop them as they're at the constrains Ok I want to keep the Titans. And now you are proud of the 1st. The 1st fact.

[00:16:24]
And you're part of this to this said came to sector with Radice I quoted it strips that to me Ok. But there. Again there is some level that I'm going to skip here but you can you can I do that the gotcha measure of keen interest. That's the one percent of the that mentioned many coordinate strips it's still pretty large and what you use here is that he was not stupid to begin with Ok.

[00:16:56]
Ok. Well now that you're done because you know that the expectation is actually there and there was no way I was pretty close to. It these. Are a few strips Ok and this got some to you it's going to be you can you can pick it so that it's less than than one or.

[00:17:15]
Ok this is a contradiction. I did get a couple of letters that he would need and I wanted to make it for me but this is the intuition Ok. Ok so. I write that this type of a good works to find. And specify in a graph. Ok so what.

[00:17:40]
Would you need it is you would need that this body key that comes out. Of the graph setting So let's say again now this is these are all the x. so that if you put the the exercise in from bad to Major says and you dig operate and on that this is small.

[00:17:59]
I would like. I would love to prove that this is the gotcha measure of this that is large not enough that it would be done. By the group work Ok. Ok. And we have to observe vision that this is indeed a it's a Semitic set and it's going to x..

[00:18:22]
The problem is. I actually there whether this is true so I have no idea what's what's the actually measure of the set that's coming out of the that you get by taking operated on by something. You can maybe. If you're determined to Asian how actually does it said Ruth Reichl look rag that has such a bad as you measure here some examples you can take you can take the Q You can to a bare radius approximately rude.

[00:18:55]
You can also take something different you can also take. Let's see. You can dig a body that's in for the mistake sions it's it's let's. Say aha off in one percent of the directions one percent of all the other directions so this is what is Abijah object anyway it is my set piece of this type I I have no idea.

[00:19:24]
Ok so this is a bit of a problem. It's kind of these yours that that that many tools actually sure that sad connects have a large measure. Ok Ok so I know on the other hand. If you get what if we were back to the algorithm. Then if you Chad was actually bad about humans then.

[00:19:54]
You use that key was enough. You used to do ra if that actually this is true Ok. So you used to do rather bad. If I take a sec to it with a couple of quid in it strips and that's I could gather she expected that guy is going to be close to it so this is a property that I makes using I do rather bad for the garage in the 1st place but I don't really have to I really need that probably.

[00:20:27]
Ok we need a smattering but maybe you have questions. Ok. Ok so the message is that. Well. There doing that I remembered I don't necessarily have to prove that Kate's cell is large in terms of the guys you measure it suffices to prove something bad's recur that's kind of created usually weaker.

[00:20:58]
And this is the statement that that we're proving is that if you 2. Streams come back said you. Scale it back I think I do says being a large constant and you took the McCoskey side with a small can sometimes. Rule. Die meter ball and miss I forgot to mention Ok.

[00:21:24]
And he can give you picture what it means is that so you have that said King slightly and now I read the distance of I 5 times through them you have Bruce up to Goshen measure. They are. Ok. This is strictly we could. You could imagine imagine. Stress and minus run by this man I mention a high popping was 0 bad actually if you took a random Gaussian the distance to my plane as it's a constant.

[00:22:00]
Expectation Ok. A case of this type here this case. This. Question. Ok. Good. Well actually this condition here implies something that's there's a concept in comics geometry that's called the mean way to have everybody remember ruthless the failure when you take a random direction and you check what is the advice true but I bet this direction but that you just speak with benefits and you took the average over there was with and that's the mean root of a body and from this you can actually just in the regulation derive that actually.

[00:22:56]
Offsets baddies this by the key has to be pretty large with some constant Times wrote it. Ok. And that actually this type of discrepancy led to them so that they have something to do or. Better I guess 1st through 2 more heat and ruin another. Ok Ok so we can prove something.

[00:23:24]
For these types of comics bodies that come out after these. Specifiers we can prove a paper that will make the descriptions of good work we cannot fight only prove that the measures of Life Ok. That the government does not work. So the meat of their goods I didn't get to that is yes so I'm I still have to prove this and we really need some people as he is about like how this thing is defined so something more problems Pacific.

[00:24:07]
Ok. Ok. Ok so. He has a very responsible way to run them Galaxian you to take. You to really really bad actions independently and you add them up are pretty pretty good Ok so you can imagine you do some kind of discrete battery motion. Better are you going to rate or any of those.

[00:24:37]
We do the same sort of. But it would give us some control over the process. So the following let's see that is a tiny step size and then maintaining. Back to us x. and said. We started 0 an hour for me with the steps I squared many to ration as we do the following.

[00:25:07]
We pick. Ok we have to bend x. and set. By a ed step size times a random Gaussian Ok and we do it so bad the cravings matrices of the earth. Add up to the identity Ok so I think I'll let you know bad I did the. Experts said is going to be a better bad bet at times Gulshan static goes Ok.

[00:25:39]
And I pick one of the standard one of the. I picked the tricks they used to drive the guys in that augments x. as favorites it's it should not exceed their identity in in any direction but each of the trees is going to be clues to the trace of the identity so I'm already like a comes in doing Ok so in that other words this is going to be gosh I'm bats it's less spread out than the standard garden but not much like a grip Ok so now do the following I have my back to us x. and Except that I'm still red hot x x p I said I'm moving this is going to be a bad invention and I have an appendix and I'm going to do it as I read.

[00:26:29]
I will make sure that x. is in the body k.. Ok. So. This is just so it doesn't I'm constrained by an emotion Ok good now at the end. Good you have. Your. Gosh in your sample got action which is x. plus said you have to back to x. which you may have actually x. Rose risen very far from from the.

[00:27:04]
Brownian motion so you actually know that the distance and expectation is small and it's going to be at most our product or it Ok. Ok. Good. Ok you. Know this is this is a. Good image of this is really impressive as the. Guy ocean and with the probability you can move this planet x. which is going to be in case this proves that it actually let go as you're tempted to be too far from Kate you have your witness x. is the witness that you're not too far.

[00:27:43]
Ok. Ok good. Ok Ok so. For me here are experts on the go right yes and you're trying to get that word. No thanks I have to be uniform. And it would probably it really be very likely to be kind of on the boundaries something this is not x. is not going to be uniform and it doesn't have to be expressed said this uniform this is a Gaussian random data set with with I believe that you're going to be close to a cursor Kate where you.

[00:28:28]
Are probably. I am trying to prove this thing here. I really need a bad my place to succeed with ability Aha off the bat my gosh it's. Too bad too x. and knowing that axis and k.. Ok. Ok. Ok so. The question Ok so here are the rules of the game is that I'm going to have magic to x. these I'm fractured signs.

[00:29:06]
I read to up to this roof letter gosh and it can be a guy action that's restricted in some directions and I have great times at any rate tunes but I need to get that. Somebody said I know this is as good as going to be. A Babbitt Aboriginal Ok so that so that x. is going to stink.

[00:29:32]
Yes here. Yes. Ok so. The process is going to succeed with a good probability but things could go wrong and it could happen that I'm not able to. To promise you that by x. days and k. But that's Ok I'm I-Am I want to feel with the ability are.

[00:30:03]
Ok. Ok I mean the picture. Could be like infinitely far away. No idea whether they were right as well. Or. Maybe you could see this this way Ok so what we're doing is. Ok so I have my pay day x. and I look at a potential function. And my potential function is like this taken me 2 weeks.

[00:30:45]
That will depend on x. and this is going to be Ok what you're doing is you take their identity you put a skier in front group of high rolling axis and you subtract. The summer breaks items they're Ok. The potential fountain is going to be the trees a bit of that meetings.

[00:31:11]
Ok. If you haven't seen this type of Fame since. Let's see what we can actually say about it. Ok so if if you can keep the potential function and it just less than infinite length then you know that actually the eigenvalues are going to be positive and if the eigenvalues of this thing I'm going to be positive then in particular you know the maximum eigenvalue.

[00:31:41]
Is less than this member here Ok there's something I need to see that would be needed to disappoint me to do that would pick later and this is the thing of x.. And. And you maybe you can take my word for it that you can actually I read it was the proper way of keeping the scene with a value beyond the 2 keeping the max alive you've got it we don't have to look at book Dark Times.

[00:32:07]
Ok good. So what's the intuition behind this potential faction. Every step you kind of operating x. and maybe some of the I could barely move cruiser to 0 then. Kind of a did this trees might actually grow but on the other hand if you kind of never ending right then this thing was getting regular So you kind of get a little bit extra every action and you hope that the trees kind of can be killed by the spectator again.

[00:32:38]
Ok Mark. Where you're the one. Where we. Were. This could be like end. Of. A good really I suppose it depends how you will come to that Ok Ok this is Ok. Ok so the main take it this is the until you let me out and then I'll tell you a question Ok so the men didn't go around.

[00:33:17]
As fair as that was my potential function so the trees are be in various of these major makes as Raymond as I am about doing this that's good enough. I can see that yes there's a there's some. Positives I mean a different matrix x. bad snap spread out there.

[00:33:40]
In the trees is going to be crazy to the lead entity and if I sample a lot to stab for this gosh I'm from this restrictive Gaussian that in expecting my potential function doesn't go up Ok good. Ok so how do. You have to say let's with Ok.

[00:34:05]
Ok how do you pick the Panthers. Ok the we do it this is a c. is going to be the absolute that's going to be right inside and you have to scale down her body k. by one rather 5 so this is this is this is the this is the car drive inside that you want and d. is going to be.

[00:34:28]
Kind of the sea bed divided by m.. And now you can check that rock at the beginning my potential Fanchon. It's going to be small enough it's going to be badly by the band that I need and at the end. Of the Abbey I go back up this matrix to that I'm getting by putting the sides and in front is bad about as the times that way with the left is going to be m. And so you get this is a different kind something yeah this is some kind sometimes apps are under about it by by off so this is basically going to be in a constant scalar of Ok.

[00:35:13]
Ok. Good Ok. Ok I really do feel some intuition of how this type of battery of Hunch actually works. Ok. Ok so let's say for the sake of argument let's a really random engine Ok this is the this is the minister of action. Depended on and I want to measure number x. the type of fan run divided by some barrier.

[00:35:44]
Let's say let's see some numbers. What actually happens if I do the update so I replace x. through it a small step Sat is times y. and let's say Why is this kind of like a galaxy and say expectational 0 and you have some handle on what to expect at Square.

[00:36:08]
Ok so what is actually the potential function change. We need. So the problem is this potential function and I always get confused but I think this is convex. Ok. That is the big. Bad Let's say you could have given that this is called Max Ok so the problem is that Ok it if you opted eggs and you go to the left or you read to the right Let's say that an expectation of staying the same the problem is that actually the expected value of the expected value of the potential fancy goes up.

[00:36:54]
You can you can stay a little bit ahead the 2nd degree day but then you can figure out that it will actually go up by the 2nd Do or do if you're very much Ok with this for this general use would be basically inspirational to the slack group Ok.

[00:37:14]
Ok good but this isn't I think we're doing is if you remember then we have. We have this I mean this thing is going it's grueling So I'm getting some slagging every every to ration. And Ok so the business like mean so let's you take the battery life I'm going and your trees the battery the power meter Ok So instead of as you have espresso a little bit so it's kind of right you don't you're fired you can get shifted to the right.

[00:37:50]
And again you can you can check the homosexually of the financial value change now it's going to be down. The dependence I bet is the same it's dead square that's good but this is proportional to the 1st throughput and the 1st $2.00 to $2.00 of his or. Her passion or 2 of the.

[00:38:12]
Legs squared Ok so all. We will have to do is we have to make sure that we can use the negative value to care for Spencer to value Ok is this always going to work. You can see that there's a problem if the sled gets very very small this business is going to have through this I mean we have to feel bad that I probably don't are that there will be a problem if you're getting too close to that as I'm told Ok.

[00:38:51]
Ok And this is actually the reason rather. To prove some assumption that if a producer package is better and better I can do the update stuff because if not then you can see that you meant that the time credits are not too fast or. Too Close to be due to the as I'm told Ok.

[00:39:17]
So how much time. To put in that Ok Ok Ok so this is some intuition. And a number of times now is some real trick schedules so where. I was. Expecting to get all the. Probably be a little bit big Ok so yeah I have to says Ok.

[00:39:46]
Ok so what's what's what's the basic idea here Ok so we have to say we have this new troops that we stuff into. It with then we take the trays of the inverse of it and again Dimitri it's going to be there's the in groups we take that into the matrix times high level where they could take the axes and then I'm subtracting the sum All right side up there Ok.

[00:40:15]
Good. And I have to pick and not go back to kind of a guy I shouldn't but I can I can say I can take some actions and see my gosh an is going to have that with that direction I kind of block out their rights and I can do that for after all if I spread Holmes comes in many directions.

[00:40:37]
Ok so particular. The 1st trick is you hear people that are thought to to you could point to x. then you know deterministically the this is this lens that's not going to go down. And you can actually do that argument that if in fact every little term that you see and that is and that you don't want to see No.

[00:40:57]
Ok Ok good. Ok so. Let's save them a bit about when to expect the rose. And so what we have is that we remain here said harshly at the trains of the interviews behaves as you do an update. So let's say we do find b. is the each week's.

[00:41:19]
Major weeks so this is sad to me too except that Rex and subtract the times being so this is this is my God. That I put in front of the eyes and. The new vector is going to get a little longer Ok. Ok good Ok so how does this thing actually behave if.

[00:41:42]
You do some kind of Taylor expansion and this is. The 2nd degree polynomial where you are in the matrix worlds so you have to be a little bit careful. So for example since I meant to screw a guy like Drew do you had to rearrange identical terms that's a really really have to resist like mad move to one side and you still that is sad.

[00:42:12]
Because. They do know it's commute they do not like you going to other things like the trees and cyclic so you could move like this as man to the end but you know Ok well you careful. If you read these papers anyway Ok good. And you can you can see where they actually were very you should you actually be to be taken care of.

[00:42:41]
Ok if you were to have to move to be you take the train to the might as well and be on the right has won this. This I mean here but this one was a lean year. You could just 1st that you're going to fight it out to the limit time that you're getting this is the same.

[00:42:59]
Saying that actually kind of an expectation the time you would get to getting would be easier anyway but you can do it in a stick Ok So actually the only thing we're getting here is that this is a to the minus one times this is something that's Perpetual to the identity of that's that's nice that's the only matrix that commutes let me through here so you get the trees up into the minus 2 Ok.

[00:43:23]
Ok so you can you can look at you can still. Get the 2nd. And you can see that there is a better scrape here the b. has a bad side in there I say everything that comes here actually what we're doing to give you a term that's proportional to Delta cupe.

[00:43:43]
That you're going to need as one of the little square many steps so there's a cube as a rule about 2 arms to throw it out. To some appropriate analysis but anyway so the only thing that remains is a strange looking to where you are this is a random guy was here and which are pretty friendly eyes.

[00:44:06]
This is the scene gosh and then you have to pry this from the left from the red and from the middle with it to the minus one it's a straight star and we sort of have to show that this is at most 0 or doesn't look very nice Ok.

[00:44:22]
Ok so this was some psychological recent Let's look at this weird. That kind of crisp answer the 2nd derivative from my right eye Mitchell picture if you think about it Ok. Ok. Could you can. I get you get these matrices do not commute a bat with a little bit care you can be convinced that you can group things a little bit together you can said w.i. to be the matrix where you take your magic played from the ride with 8 of the minus rod from the left with the might as a half now this is not even a symmetric matrix anymore but if you can stand group a little bit together Ok.

[00:45:09]
And then you can again group them together and actually see that if you're looking at is the expect it may. Have to some other produce I got tired of these major it is mages are Ok. This is the for being a snare. And the remember the for Businessman of the Matrix Matrix who get by.

[00:45:37]
The entry is summing it up and taking the square Ok good Ok so he has run too bad. You remember about Gaussians that if you have to expect the expected the. Expected square length of. Let's say. A guy from time some entries this is actually. Just. The length of whatever it is this thing Ok.

[00:46:11]
Ok so use that this is this is a paper this is good you can get rid of the expectation that now you just have to have some length here which you can rearrange that if you look at a little bit differently. So now you know where to him it's gonna buy you something you saw the truth I may do that I just threw it to the man is.

[00:46:36]
All I care Smedley my friendly. To summon a gravity you can do you can separate things. Ok. And that. And Ok. So. If you look at the 1st paragraph then you have to train it to the minus 2 this is something that could dispense to the 1st 2 about this Ok And this is a term that we.

[00:47:09]
Sort of half year that's kind of good. On the other hand we have to train somebody into the might as well and time say I The problem is that this factor could be arbitrarily large. And that's a that's an issue. It's an issue that we've expected because we live it you can't really do that it's that and that have the potential back in Cruse if somebody has a kind of truce with the asymptote and this may have to happen so some of the AI's may be kind of.

[00:47:43]
Leverage if you take the trays with the Muslims Ok. But not ripping sunshine that move the potential factors behind it so that we do the favoring not really keep the entries I hate. The trains up. So basically the contribution of the ice blue tricks to the potential faction it's bound it it's bad I suspect I have the average of over the it was a I this is just like Ok.

[00:48:20]
Ok so the point is that you actually know there are some of these entries. This is the potential of our. Journey to the entries that are not too bad compared to the average Ok and if you got it just doesn't break into. Into the credits where those values to watch Ok after this modification you have some bad on this and.

[00:48:51]
Now you can. Group things together and you use that assumption that the the sound of my meter he says this is sort of up about it I did indeed do that so you can kind of cancer this I had and you just get the trace of it to the money still with that factoid front Ok so this was a little this is a little quick but the intuition that I gave on the one eventual slide that kind of is the idea.

[00:49:18]
Ok. Good now you just read packing everything that we had and you see that there is some to that's got a term that's obviously professional too that are square. There was the trace of it to the minus 2 is a big investor is that begat What are the terms with negative.

[00:49:41]
That corresponds to the shift letter was positive that corresponds to the convexity and not. This is the put into a foundation and the potential action was bounded so you Ken. Ballen precisely by the best use of that but one thing is that. Ok. Ok Ok. Good to. Be with us yes yes yes there's a space is there was where.

[00:50:15]
Or. Yes yes yes. Ok good Ok so that's sort of. It I do have some problems if you like to hear. So there's the obvious one that's coming out of this paper is that. Ok let's see if I give you some pieces the major reasons they are let's see they're so that if you some are bad that they are powered by the identity in the p. is the ordering you make at this kind of body k. that we have been discussing at the time where.

[00:51:00]
You look at all the x. that I can put in front of their eyes so that the operator norm is bounded by the term that's kind of your absolute on. Is actually the government out so we have seen that basically the mean roots of this body is large but can you prove something that's created that if the stronger that would be nice it would be cleaner is so we do not know this.

[00:51:27]
So there's a sort of a bigger problem that also we had Saturday our working on was. Was a conjecture but I will make I mean only that if you we if you have matrices Iran to a end it's a in then that mentions and I promise you that the operate a lot of area of the major east.

[00:51:49]
About if I want. Is it true that there are signs that you can put in front so that the some of these signs need to says they operate on them is bounded by a concert times through there. This is true if the Beatrice's are by going to Major says this is basically spaces theory or if the edges are diagonal metres is and its prayers are very bad it's true.

[00:52:18]
Even from the dichotomy traces we have no idea. And so finally. There's another. Set off from a few years ago. Which is that which are resolved that the cat isn't seeing the problem. And that they should it is that actually if you take if you give me a very high so that you sum up that you get the identity and every vector is short.

[00:52:54]
Then this should actually it's possible to find Saturn so that the some of these vectors and this rip in m.. As well as I do believe between physical mind is absolute and process on. The method is never constructive there's no room behind this and there's too much to the moon to find those signs and this isn't as bad.

[00:53:18]
So whether you can find the signs are pointing in the time. Thank you for listening and until you get. Things No. This is this is right and then my understanding of the. So there is a cruiser the right one and this is right at. 100 percent so small but let.

[00:54:01]
Them walk on to the holes who will just use it to record. Yes but I could play with that doesn't I'll give them that I guess. You could answer another but that's no read through them maybe this is the future of them so fast that it really is a simple it's obvious that me so in that.

[00:54:27]
Small business you're saying just because your body just. Might. Make a plea do we through a few phrases but it does say. Yes maybe. But if you. Don't do it. With your we have already would be different for the. Yes that is true so that we treat Skype legs.

[00:55:00]
And want to brazen you off yeah yeah that's right my. Problem. Ok let's not opting out.