Your coca-cola bottler presents Claudia Claudia based on the original stories by Rose Franken Brought to you transcribed Monday through Friday by your friendly neighbor who bottles coca-cola Relax and while you're listening refresh yourself Have a coke and now Claudia come for the seats to wash. I've got them all ready for you in the bedroom Good, I'll get them. I'll get them you do enough running you sit down here a minute and rest No all day long. I have been sitting Brits had an old sweater and I had to make new sleeves for it You can do everything can't you just like mama your mama and I we had a long time to learn no no noise It's more than that. I know because I'm just mastering buttons and it's for darling never never never I don't like darling either. You don't mm-hmm. Did your husband like the salmon moose? Oh, I knew I had something to tell you He loved it good and so did mama and so did Fritz did he know it was salmon when he finished Then I told him and he didn't like then but already has enjoyed it all mom and David knew right off and they liked it anyway You know, that's really a compliment to your birthday and dad is some left to I'll have it for lunch tomorrow myself Put it in some tomatoes is my only though, but we'll be like a new dish. I'll do just that I'd use it tonight, but mr. Norton and I going out to dinner says he's not mad about leftovers I noticed y'all dressed up nicely. You look so pretty. Is it the birthday? No, no occasion. Mr. Norton Just wanted to take me out to dinner.  Hey, I told you never to leave this door open I want to wake up murdered someday. Oh David your home. Wait a minute. I'll ask myself What about the door? All right. I'm not alone. Oh, hello, Bertha. Good evening, Mr. Norton I go now and get the laundry. It's all right Bertha No, no for me not one a kiss is better than a hello any day. You're home late. I'm home early Seems late to me. Should The matter of fact, it seemed late to me all day today. How about you? Since we've been married time seems to slow up around four o'clock be awful if it were lopsided wouldn't it? If what where us I mean if the day seemed long to me and short to you Well, we could always buy a couple of watches Good night, Mr. Norton. Good night.  I go now. I bring the sheets back soon. No hurry. We've got lots more We have it really but I can borrow some a mama say listen you're gonna have to get over that habit What do you mean?  I could love you for that alone. That's no good reason. I didn't say was the only reason You'd better not See her today. She all right. She says she's fine. I called master to come over but She says she's glad to be rid of us good for mama. I'll call her before we go out to dinner. Oh, would you David? Mm-hmm? What are you doing sitting down on the sofa? Why? So you can sit down beside me. I thought we were going out for dinner. Oh, we can eat a little later tonight, hmm? I'll be starved later. That's good. I'm all ready to go. So I won't hold you up Say David guess what I did today. Mm-hmm. Let's see. He washed your hair. How could you tell? Does it look awful? Not bad. It smells pretty. You know men are lucky all they ever need is a haircut every six months I'd look lovely with a haircut every six months Hey, you know, I'm sorry. I set down why now. I don't want to get up. That's good Well, I feel like a new man after my shave. Come on. Let's go wait a minute I thought you said you were already half an hour ago. I was but I'm not anymore What happened to unready you?  It's all over the place. You called mama. I did that too Well, there was no answer. That's funny. What do you suppose?  She's a big girl now. She can go where she wants. That's true Well, I'm ready. We're on our way at last.  Would you get out of my way now? What are you gonna do? Oh So you're the burglar mama, what do you know? You look so surprised.  Claudia gave me this key just yesterday I did in case you were in the bathtub and I came over. Oh, yes, so I did come on in mama Claudia who else have you been giving keys to? Oh, I give them to just anybody Well, I thought you'd be out for dinner. Claudia. Didn't you say you'll go? I suppose that's why you came over exactly Sorry to disappoint you, but we're a little off schedule.  Thank you. No, I'll wait here for you Now run along not without you and I'll turn around to go home again. Oh, that's silly. What'll you do home? Have you had dinner David tell her I did not come to eat You know look as if you'd have dinner have you none of your business see David. She didn't buy what secret code Did you figure that out when you've known mama as long as I have you'll know not to believe a word? She said they know attention David She doesn't know what she's talking about and the funny part about it Mother is that she makes the most sense when she doesn't know what she's talking about Mommy you stay here. We're about to have a family conference in the souls the two of you Darling, what have you got in the icebox practically nothing. Why?  I know that much.  She came over. That's what I think Mrs. Brown How about joining us and potluck for dinner? I wouldn't hear of it.  It's the door. I'll go who know no, let me let me I'm already there. You think this was your house David, who do you think? Oh? Fine. Good evening. Oh, lady. I make you that time you did for the rest of the schedules off and what's that? It's him again David the moving man. Fine. Ask him to dinner. Did you call him? No, did you? You can bring those things right in our wedding presents. I bet that David I'll bet she did that lady these barrels the butterflies and we're supposed to be out while she unpacked them all She is the limit that I am Mr.. Catch a barrel. Mama. You're terrible. I decided to let you sleep with him for a while Right here in the hall. Yeah, I'll give you a hand. Mama.  You don't I wish you'd have been out of the way The way was supposed to be I'd have had everything arranged for you. I'm glad we're not I want to arrange it myself Sorry girls. You'll have to move over Let's take it sideways There we are That's one two more to go wait till I take off my coat I kept trying rolling the other one. Hey mama. Let's open this one up and start taking things out. Shall we what about your dinner? Oh, I can wait come on. All right Come on Second one Yeah, yeah Let's try rolling them. I missed it right right Make an awful mess. Oh who cares home isn't home without one's wedding presents around watch out. Here's another one Get out of the way mama. You'll get out of the way yourself Another one I can't ladies watch your toes That's that We'll have to move the sofa out of here Claudia. Oh, no, they won't take half as much room when they're unpacked Of course, no, they'll shrink when they're out of the barrel. It is bonbon dish. It's beautiful, isn't it? Didn't like it so much at my house and looks different here somehow. Oh, that's the last of them Sure got a lot of wedding presents lady Well, that's all I'll be running home to the missus awfully late. I hope she won't be worried We've been married ten years almost Don't you worry after ten years? nah, oh Well, you've been awfully nice if we ever move again, we'll have we'll call you to help it I'll be looking forward to that lady. Oh Oh, yeah, I left that extra box in the hall what extra box when I put those two silver pots in At louisa's soup to range. I was hoping you'd lost them on the way. We don't lose nothing mister. No tonight. Good night Sticks look the first of a long line of candlesticks. Claudia. Let's have been a while. We do this. I'm hungry. That gives me an idea We'll use our wedding presents tonight. You go out. I'll unpack nothing doing. I'll do it faster alone Conceded to well Yeah, what is this? Oh the fish knives that look like little fish say we can use them tonight to my one silver plate coming up Hey, did what can we put on that? Peanut I've forgotten that plate. It's not bad at all peanuts for it. Yes, I'm what just unpacked don't ask questions I never begun this But you got my silver cigarette box. Oh good. We'll put cigarettes in it here Hand me three bonbon dishes mama. Wait a minute. I'll find them. I'll be back That won't be room in here to turn around. I'll put some of this stuff on the dining room table David Really, I didn't mean to walk in and upset everything. I was going to find no mother. We're glad you're here I have an idea. Claudia was pretty lonesome all day. She's got to get used to filling our own life She will just a little time here the napkins and glasses one for each of us. What are you cooking up, Claudia? You'll see I'm not sure. I want to see keep unpacking you two. I'll be right back. You heard her David Unpacked I am working like a beaver Almost ready for another barrel.  Claudia. What is this? What do you think it is? The salmon mousse of course dressed up in a tomato Very elegantly. Claudia you're mad.  Why not? That's right mother Why not and we can christen the little fish knives, too? Excellent excellent, you know, I thought since it was a cold leftover supper. We might just as well picnic in here That's very sensible. Claudia Well, aren't you going to eat mama? I might as well David if we lean against each other its backs will be much more comfortable, you know, I believe that's a good idea This is much nicer than eating out This is wonderful. I'm beginning to like this stuff mama you All our own things around and our old friend the salmon Well, I don't like to interrupt this but if you two would move over there'd be room on the floor for me All story material used on this broadcast of Claudia was under the supervision of rose franken and william brown maloney You know, it gives a hostess a mighty nice feeling to know she has a supply of frosty coke in the refrigerator Now that coca-cola is more plentiful. It's so easy to be prepared for hurry up hospitality Next time you shop ask your dealer for a comfortable supply a carton or a case Every day monday through friday Claudia comes to you transcribed with the best wishes of your friendly neighbor who bottles coca-cola So listen again tomorrow at the same time And now this is joe king saying or if war and remember Whoever you are whatever you do wherever you may be When you think of refreshment think of coca-cola For ice cold coca-cola makes any pause the pause that refreshes


In this paper, we explore the potential of using pre-trained convolutional neural networks, CNNs, for high-resolution remote sensing, HRS scene classification. We propose two scenarios for generating image features from these CNNs, one involving extracting activations from fully connected layers and another involving dense features from the last convolutional layer at multiple scales. Experiments show that both scenarios outperform traditional low and mid-level features, resulting in improved accuracy and performance. Furthermore, combining features from different CNN models further improves the performance. This study demonstrates the effectiveness of using pre-trained CNNs for HRS scene classification and provides a promising direction for future research. This article was authored by Fan Hu, Gui Song Xia, Jing Wen Hu, and others.


So roughly speaking, there are two methods of random metric theory, the way we are looking at the problems. So of course, there are many other different ways to approach random metrics, but what we have been doing is basically just two objects. One is the green functions, and the other one is some dynamic ideas. So the question is always you want to handle some local properties. So if we combine these two, we have some quite effective method to deal with many problems in the matrices. And so because RDDG was actually a lot of these green functions, estimate and also also part of the dynamic method. So I thought that one idea is maybe I just collect the dynamic idea using various parts of random metric theory, and then just look at all the problems so that we try to solve, we use some idea, some dynamic ideas. And now, so one is you can do this metric. So here are at least a few of them. So one is this match of Brownian motions and this convergence speed. And the other one is sometimes you can also use dynamic idea and green function estimate. And then it's also, I find it also amusing that you can also, in the green function comparison theorem, you can also, the result, you can also try to use some dynamic ideas. So I will give one example of this as well. And then also for the eigenvector, you can also look at eigenvector from a dynamical point of view. And last thing I will say is just some related result in all the dynamics. So this is the plane. And many of these results in this lecture hold on the four moments assumption. And so by Terry Tau and Ben Bu. But so I may not, so I will try to aside it, but sometimes I might forget. So all right, so the magic Brownian motion. So you take a, you take a matrix H of t, H is just, it's a symmetric matrix. And now you, you look at the, this is basically the magic Einstein process. This is db of t, each component bij is independent Brownian motion, but subject to the symmetry assumptions. So, so the, one, one can solve the, one can process. And so you find the H of t at time t is exactly the linear combination of initial data. So this H is H0, initial data. And plus the GOE piece. And so, so anything you say about eigenvalue of this flow, you will tend, you will tell you about, about the, the behavior spectrum property of this H of t. So, so the, if you look at eigenvalue equations, then this is the, the standard Dyson Brownian motions. So these eigenvalue equations. And then actually the eigenvector also has a equation. The eigenvector equation is, is very complicated. But you will see that, you even see that lambda k minus lambda l square term here. And then the, also in Tachington this Brownian motion is actually, there's, there's another match, there's another Brownian motions, independent Brownian motion here. So, so the, the dynamics looks very complicated. And then, but they are two, they are, the key important thing here is, if you look at the Dyson Brownian motion, it actually has the GOE. So here I take the GOE, but of course there's more general result, but, but let's take a real, real matrix, and then the GOE is invariant measure. GOE's eigenvalue distribution is invariant measure for these dynamics. And also, if you look at the, the eigenvector equations, then you find that the, the harm measure on the orthogonal group is actually invariant under these dynamics. So these are the two, two things we keep in mind. And then the whole thing, everything we, we are trying to do is really just to study these two dynamics and study the time to local equilibrium. You don't want to study the time to global equilibrium because the global equilibrium is very, I mean, the time to global equilibrium is very easy and it's not very useful. It typically takes a long time to reach it. And so you only study the time to local equilibrium. And now, if you understand this time to local equilibrium well, and you combine with Green function comparison method, and then you can prove the universality of random matrices in many cases. So this is the general strategy. Now, so let's start with the, the case of the eigenvalue piece. So the case of eigenvalues, because we are only, we only care about eigenvalues, so you can say the initial data is V of i, and V is, you just, you just give a different name of the initial data of eigenvalues that is V of i. So, so for the DBM, now the only thing matter is really the probability distribution of this V of i. A standard object is, is computer seizure transform is V. And this goes in V of E. And then you put a complex formula Z and E plus i eta is equal to this guy. All right. So the, the, the thing we are, we are, we are especially interested in major part of this because it tells you the, the distributions of, of this V, of this V i. Then, so the, the results is the following. It's, if you take, if you have certain range, that this imaginary part of this M of V is, is in between two constant. So you need to know that it's in between two constant. And, and then for range of Z in a fixed energy, you fix one energy E naught. And then there's some, there's some neighborhood of the size eta star. The eta star is, oh, by the way, I should, I should mention that in, in my convention, I always choose the, the eigenvalues. It's from minus two to two. And so, so eigenvalue is a space in this, a typical eigenvalue space in this, is one over N. So this is, maybe it's, it's good for you to make the comparison. All right. So you fix the, you fix the energy, energy E naught. And then you take a size eta star. And this N to 2 F is just slightly bigger. It takes a eta star. Then you find that, you find this, the, the, if you compute the, or you compute the correlation functions of the eigen, eigenvalue distribution. So here I, I, I suppose, I'm assuming that the, this standard notation of eigenvalue probability distribution of K point correlation function of eigenvalue correlation function. So I call this Pt of K at time t. And then you look at one fixed energy E, and then you, you shift by one over N. And then times test function, subtract the same quantity of the GOE is, is less than n to the minus epsilon, minus some spatial currency. So this gives you the, the fixed energy universality. And under, under the assumption that the, the initial data is the, the Stigio-Chainswell initial data is bounded for a certain range of, for certain range of energy. And also for the, for the complex parameter eta, is only bigger than eta star. It means you are trying, your assumption is initially, you don't assume the, the fine structure of eigenvalues. You only assume the, you only assume that it's bigger than, assume it's probably bigger than eta star. On the other hand, this is good enough to tell you as long as you know information up to the, the scale eta star.  And this energy E has to be in this interval, I forgot to say it. The energy E has to be in this interval actually has to be slightly smaller because you're assuming something in this interval. But your universality has to be slightly smaller. All right. So, so this is the fixed energy for quite some time. The, it's considered as one difficult question in the, if you, I mean you can ask the university in many, many different sense. And this is one of the hardest case. And this was done for the Wigner matrix, it was done by some of my coworkers. And in some permission case was done by Taowoo and some of my coworkers. And they are actually quite a few people here. So I didn't, these are all of them. And now, now all these, the, the result we have here is some sort of local convergence to equilibrium. But this idea of local convergence to equilibrium was, was first tried and by, by Erdisch, Schrodinger myself. And, but over there we assume that eigenvalue has satisfied some rigidity of eigenvalues. And, and then, and then also there's another sense of eigenvalue universality called gap universalities. And so, so I will not talk much about that. So, and the local version of gap universality was also, was also proved. And so we just did some, of heuristic, some previous result. And now, because today what I want to do is something a bit, a bit different.  I'll just do some, some elements of ideas. And in later on maybe I'll show you one or two pieces because, because what I'm trying to do is just show you that the dynamical ideas, it's actually useful in many places. So, so the elements of ideas we use the couplings and we also use homogenizations. And in some places we, we use maximum principle and finite speed of propagation in the, in some, in some sort of particle system method. And also we use essentially material for linear statistics but for non-compact functions. So, all right. So now, so this is the, the fixed energy in the, in the bulk. And then if you look at the question at the edge, then you have to assume something slightly, you have to assume something stronger is, you have to assume that at the edge there's a, there's a point because you don't really know, you don't really know what's where's the edge. And you know that you need to assume that the initial data, the fixed edge, and with this fixed edge, the imaginary part, behavior is the same as, this MSC is a semicircle. So here I have written this MSC, this is a semicircle, a stereo transform, this is for semicircle. And it satisfies the equation m star plus m z. Well, I think it's zero. So, so this is the semicircle, the stereo transform. And we assume that because we need a square root of behavior, so it's just, it's bounded below and above by the semicircle behavior. And then the, the time to equilibrium is, you need this eta star is bigger than n times two-third. This is the smallest time scale you can deal with, you can do it. And now if you take time, it's slightly bigger than the square root of eta star, then you can show that the statistics were converging to the GOE statistics. And I mean it's the same as previous theorem, it's just, it's just the eigenvalues at the edge. So converging to be here is the square root of eta star near the edge and the eta star in the box. I explained this eta star square root, where it comes from later. Okay, so, so you may, you may ask that why do we bother to prove all this theorem after spending so many years. And there are many theorems already approved and why do we go back to study all these local versions. So the reason is, we study a local version because for various reasons. One is the so-called rigidity of eigenvalues. So which was, I mean for those who you know, there's a concept which means eigenvalues has to be close to what is classical locations, quite close. But this is not an easy result. I mean this concept, it's not easy to prove eigenvalue has to be quite close to its classical location. So this is not easy. And the second thing is, there's some estimate called level repulsion. So level repulsion is always assumed in the previous work. And now here there's no assumption of level repulsion. And the next point is that the assumption is always local because in the beginning of random metric theory, if you want to talk about the box universality, you also have to understand the edge because the edge will influence the box. So now this is completely separate. If you want to know the edge, you just, you have to study the edge. If you want to know the box, you study the box. Everything is local. And also the assumption is, depending only on the boundary, it's a real transfer initial data. And the scale is the only bigger data stock. You don't have to go to the smallest scale. Then it works. There's a tiny, I was cheating a little bit. There was some tiny technical assumption about this V. I mean here, at least, there's only this assumption, but actually there's another assumption this V cannot be too big. You can imagine there's some V very far away, experiential, experiential, large. It's going to change many things. So when there's some sort of some, pretty still be bound on that. But otherwise, these are all assumptions. All right. So then I want to use random graph and to see the application of this method. And it can be the Erdisch-Renier or random deregular graph. So now, the random deregular graph, so the idea is now, in this case, we want to see what we can use for this method tell us. So now, so you can look at this Erdisch-Renier graph with probability P to select the edge. And so the typical number of edges, per vertex is MP. We write this as equal to Q squared. So Q squared, it's, Q turned out to be a quite good parameter. And then if you look at the sparse random matrix, I mean, here we use a center one, but you can also remove the center condition. Then you find it's minimally zero. The variance is one over N. Or by the way, the variance one over N is convention. We always follow, which is different from Terry's talk, which is, the variance is one. And now you look at higher moments. The higher moment is, it's not decayed like one over square root of N. It's decayed like one over Q to the K much, too. So higher moment decay is slower. And then this Q, you can choose, choose Q to be as big as the square root of N. And then it becomes the Bernoulli random matrices. But the point Q is very small. It's very sparse random matrices. So we prove that at some point, for some time that the Q is bigger than entry one-third. This means entry two-third of the typical number of edges is the bucket edge universally holds. And for a bulk universality later on, we show that for Q is bigger than entry epsilon, then the bulk universality holds. And this proof is completely, this proof is correct because we used the previous result I just mentioned about this local converging speed. And the reason is because for the sparse random matrices, this rigidity of eigenvalues is actually false. It's not just that you cannot prove it. It's even wrong. So that's why this actually is a lot of motivation is to try to prove things without this rigidity assumption here. This excellent result is, I think when we did this Q to the N to the one-third, of course I mean one can improve this. But then the D and the Kevin Schneidery, they were able to show that at the edge they can find a deterministic L so that with respect to this deterministic, there's no more, I don't understand. But where is the, did you see? Okay. Oh, there's no IC, there's no battery. There's no battery here. I can use it for quick. Okay, all right. So you just imagine where I talk. So they find a deterministic L and then find the lambda one minus L is Tracy Whedon. And the way they did it is they did not use any of these, the dynamic method I talk about. So what they were able to do is they just compute the green functions and compute the higher order expansion of green functions and find a self-consistent equation to higher order in Q. And then they were able to identify the edge eigenvalues for Q bigger than N to the one-sixth. It's a spectacular expansion because it's expanding to all the in Q. Yes? Why is Q bigger than equal to zero means? Oh, it's one, okay? So, sorry, sorry. It's, oh, okay, so it's, anyway. I mean, you need some edges, so it really means one, it's not there. Okay. All right, so this was the sparse result. And now we combined his method and this edge universality, this local converging spirit, Dyson Brown's emotions. And what we can do is we can do slightly better. It says there's a quantity called H-I-H-S square minus one over N, which is the fluctuation of the, so you look at this chi, which is the fluctuation of the edge, the total edge, and then you find out if you take your eigenvalues, subtract this quantity, L, and also in addition, you subtract the chi, then it will converge to Tracy Whedon in the time in the Q bigger than one-ninth. So what it means is at the point of N to the one's, Q equal to N to the one-sixth, yes, N to the one-sixth. The edge eigenvalue distribution is a sum of, is a summation of Tracy Whedon and plus a Gaussian chi. You can check that for Q bigger than N to the one-sixth, the Tracy Whedon wins over the Gaussian. And for Q below that, the Gaussian wins over the Tracy Whedon. So this is the, but once it below N to the one-ninth, our result become weaker and then we didn't identify, we didn't identify lambda i completely for below N to the one-ninth. And for that, you had to do expansion even higher order in Q. And so, well, I don't know, maybe some brave young people would be able to do it. So, and then, so this was the, and now, so let me go back to green function. So I just remind you the green function is written in this way, you take these eigenvectors and then you look at this lambda GMI, the eigenvalues, and then the imaginary part is typically look like this. So now, the green function estimates, you look at everything we have done and actually, essentially most people, whenever you look at green function, it's always some self-consistent equations. Now, you analyze it as self-consistent equations in various way and, but what I'm trying to mention is that sometimes you can integrate some dynamic idea into the problems that are doable. And so, so this is a case of D-regular graph. So for D-regular graph and as long as the edge D is bigger than log to some power, we prove the local law is correct and this means the green function is up to some almost entry minus one in the imaginary part is a good estimate and also the eigenvector is delocalized. And then, and then for the finite D case, the theorem becomes much more complicated because it becomes the custom K law and it also has a convergence and we also prove the eigenvector is delocalized but here we only do this in the bulk. There are some problems at the edge. Now, now the idea here is if you look at random D-regular graph, so besides the self-consistent equation ideas, the only thing we the only additional element we play with of course there's the real implementation is quite complicated including various multi-scale estimates. But roughly speaking is that if you look at point and then you look at it's a neighborhood and now you can see that as long as this radius is not too big it's basically like a tree with a few loops. And then the idea is now if you look at the edge you look at you look at the edge near the boundary then you re-sample then you re-sample the edge the edge of the boundary and this actually provides enough new elements to analyze the D-regular graph. I mean this is a bit hard to understand but roughly speaking random D-regular graph has some sort of because you have lots of conditions so your magic lasts a certain independence and the typical way we are dealing with a problem is always you assume the magic element is independent or maybe it's dependent on your neighborhood but rather than the D-regular graph the dependence is quite complicated so sometimes whenever you write down this short complementary formula it just completely there's nothing I mean it becomes completely dependent so you rely on this re-sampling ideas and to provide some additional randomness into the problem and so that's alright so I will just say this is some of the previous works and they are quite a lot and so I don't have time to go over all of them and how they prove it and so some of this comparison is really unfair because the deal is for fix the deal is for the deterministic whenever I put a star the deal is for deterministic ensemble so it's not a really fair comparison alright now the next thing I want to say is these switching ideas actually one can implement these switching ideas and provide some comparison result and I find this part is quite interesting the switching is you take a metric like this you just switch these two and then you find that the D-regular condition is completely invariant so this is invariant and then we call this the Q this generator of these switching dynamics so the switching dynamics has operated Q associated with these dynamics and on the other hand now you look at the Dyson Brownian motions and then because this number of edges are constrained in one one one direction and then then you denote as generators by L so then you find that if you compute the dynamics under two different dynamics one is under these switching dynamics the other one is you compute the same thing under the Dyson Brownian motions and then you this function I write a function of this A divided by square root of D minus 1 but actually this is really just this function is either the eigenvalue or eigenfunctions or typically what we took was a green function then you find these two dynamics actually are closely each other and with some error terms and then there are some error estimates so the regularity of F so roughly speaking what it says is eigenvector distribution of Dyson Brownian motions switching dynamics are closely each other so instead of trying to prove that our typical way to study the universality is to study the flow by Dyson Brownian motions and then show that eigenvalue didn't change very much under this flow but here we did it differently here what we say is you look at the Dyson Brownian motion and you look at the switching dynamics but actually they follow each other very closely for some certain interval and in this interval you find the Dyson Brownian motion already reached equilibrium and on the other hand already reached local equilibrium it's already reached local equilibrium and on the other hand you still know that these two dynamics are not far from each other and then because the switching dynamics will keep the probability distribution of eigenvalue for this Dirichlet graph invariant then you know that the Dirichlet graph satisfies this bulk set by the bulk universality so that's the idea so here you find that a different way of play with the dynamics it's not just the study is local equilibrium you actually compare two dynamics and then you find that one of them is invariant dynamics for your probability measure and the other is Dyson Brownian motion so this is a different way of play with the dynamics ideas alright so next I want to talk about the eigenvector yes yes what you are doing is for example you just compute a green think about this as just a green function so if you you can I mean a typical object we are computing is you take HMRZ and then you can compute the trace so this kind of quantity and this kind of quantity can tell you where the location of eigenvalue is so it's always play directly on the green function because if you look at the eigen if you really write down the eigenvalues then of course you still can differentiate it's not a problem but derivative eigenvalues were produced you would need this level repulsion to understand the dynamics the eigenvalue derivatives and so but once you use the green function so this problem is sort of avoided alright so next thing is about eigenvector now about eigenvector there was two kinds of eigenvectors one is dedocations and this dedocations this was because we we all start from Rödinger Schrodinger operators so this is the key concept we're trying to understand is the eigenvector it's sort of fairly flat and the other one is some stronger notion called quantum unicardicities and there are many people did that and roughly speaking it just says the eigenvector is basically flat but when they say flat it's really really flat it's really in terms of eigenvector it means if you average the eigenvector over a set then it's really completely flat it just becomes one and so so we will call this probabilistic QE because we are dealing with the matrix and then you take the set has to be large enough and then you take an average then you want to say it's completely flat alright


Hi, in this tutorial, we're going to talk about that, how to download and install Paint.net on your machine. First, what we need to do is to go to the website GetPaint.net, click on the download section, scroll a little bit down and click on this link and then click on this one. Now we want to save this zipped file to the R-computer, okay. Okay, now I'm going to open this file and I'm going to extract it to my computer, okay. Then I'm going to click to this executable file, okay. I want to install the Express version of Paint.net, I will click to the next. I will accept the agreement and now we're going to install Paint.net. Now installation process optimizing Paint.net's performance for our system, we're gonna wait maybe one minute and that's it. Now we're going to start Paint.net program and that's it, Paint.net program has been successfully installed on our machine. That's all in this tutorial, thanks for watching, don't forget to subscribe to my channel. See you in the next video, bye.


No, you don't have to niche down and constrict your passions and skills and experiences down to one specific problem that you solve or one specific goal that you help people achieve or using one specific niche, you don't have to niche down. So that's what I want to talk about in this video. And where this comes from is that I have been coaching thousands of solopreneurs and really multi-passionate people over the past 13 years. And this is one of the most common pains that they come to me with. They tell me, George, I have heard from other marketing experts that even though I have these different passions and interests, I have to just choose one of them. I have to just get down to that specific pain point that I help or specific set of pain points within a certain niche, within a certain type of person or within a certain stage of their journey. I have to get down to that specificity in order to have a successful business, in order to have people buy from me. And well, I have seen many people try to niche down, right? So what has happened? Maybe some of you have had that experience too, and I really encourage you to comment below and share with us what your experience has been of niching down. I've seen so many people do it, and yes, some people are able to make it work. People buy from their narrow niche, and they're happy with how it went. So yes, that's true. And yet many people, especially the kinds of people that I tend to work with who are multi-talented, multi-passionate, maybe very holistically minded where they see connections for everything, and so it's hard for them. It's painful, literally, for them to kill all the connections and just go, I'm only doing one specific thing. So that's what happens is they try to niche down, either they can't and then they think they're a failure, or they niche down and then before long they realize they're not happy because it doesn't really speak to their authentic business, the whole of what they can bring into the service that they provide for their clients. So here's the main message I want to share in this video. It's not really about niching down if you want a successful business. It's about credibility.  So when they say, hey, today let's go and try this thing that we've never done before or that you've never done before. I enjoy it sometimes. Let's go try some hobbies, some activity. They're like, cool. Yeah, OK. Because you just enjoy hanging out with them.   But hey, I want you to read this book. It's really good. And you don't think they're an expert in this topic, but you like what they're, what you generally like their values and their way of thinking. And so you're like, sure, I'll go check out this book on a topic that we've never talked about. And I don't know that you're an expert at it. OK. This is what can happen when you have an authentic business. This is what I've been talking about for years. When you have credibility with people, you can offer just about anything. And if they want it, that's the key, obviously. You know, let's, your friend, they introduced a book to you. If you don't care about the topic, no matter how expert they are in it, you're like, oh, thanks. But actually, I'm not interested in that topic. You know, that's fine. You don't have to, you know, but you still like your friend. So, so if you have credibility with an audience, you can sell anything. And if they want it, they'll buy it. Today, a lot of you, gratefully, I have credibility with a lot of you.  So today, if I launched a how to make origami course, I've never talked about this ever. Right. None of my contents about origami. You've never seen me do it. You know, you don't see any origami in my office. But if I launched how to how to make origami, I bet some of you are going to buy it. Right? If you want it, right? Like if, if especially something you've been thinking about, or let's take a totally different topic, let's say today, I want to launch, I'm going to launch an online course about how to gracefully set boundaries with family members, let's say. Now, I've not, I've never really talked about this. I mean, you could say that I've had some adjacent, you know, in my joyful productivity, maybe there's some adjacent, you know, connection to setting boundaries with family members, but I've never really thought about it. You don't see me as an expert on family relationships or family dynamics. Right? You don't. But you like the way I teach and you like my, we are aligned in our values. And so if I sold you a course on how to deal with family dynamics and of course on that sales page, I'll explain what my, what I think about it, you know, what, what my perspective is on it and what might, maybe my experience on it, why I think I am credible to teach this class, I will explain on that page and what kind of things we're going to cover in the course. Some of you are going to buy it. I'm pretty sure because I, I have generally, I have quite a bit of credibility with some of you. You see what I mean? So it doesn't matter what I sell. You, I have credibility with you. So you're going to buy from me generally all kinds of different things. I could sell you, I don't know. I mean, I could sell you how to, how to, how to garden. I'm pointing at a plant I have here in the office.  So you might say, how to have a thriving home plant, you know, situation. I don't even know what it's called. You see, it doesn't matter if I have credibility with you. You're going to buy it. So this is true with your audience, too. If you have credibility with them, if they trust your values, if they like your presence and your style and the way that you talk, teach, write, whatever it is you do, you can sell them anything. And if they want it, they will buy it. You can sell them one thing today and a completely different thing next month. Today I will sell you my house plant course.  You didn't know I was a downhill skier, did you? You didn't know that. I'm not, I'm bunny slope right over here. But, but if I were, if I had the secret hobby, you didn't know I was a professional. I was really good at downhill skiing or whatever. I just are uphill skiing. I'm not sure what I'm saying downhill. But you see what I mean? I don't know what I'm talking about. But if I, if I, if I did it and I never talked to you about it, chances are if I, if I do it often, I'll probably actually mention every now and then. But see, this is what happens. Like chances are you're not going to sell something completely out of the blue that you've never talked about because if it's a passion of yours, if it's part of your life, if it's, if you give energy to it on a regular basis, it's going to leak out in your content. I mean, this is why I talk all the time about why it's so helpful to create authentic content consistently, to show up as yourself, as your whole self, talking about the various things that you do that energize you, that you may want to provide a service about or sell something on one day. You see, like if I were a skiing person, skier, I don't know what it's called. Again, bunny slope guy over here, right. But if I did that on a regular, on a somewhat regular basis where I could like teach a course on it or sell some product about it. You're going to hear me occasionally talk. Oh, I just came back from skiing or, you know, whatever. Like that would be just part of my content because that's just part of who I am.  If George recommended it, I'm going to buy it or George is teaching. Do you see what I mean? So it really, you can, you can be your full self. You can be a reiki master, astrologer and an animal communicator. And you could teach flower arrangement and you could teach dog training and you could be a, you know, software tech wizard and you could be a career coach and you could teach people how to have a better marriage. All of that on your website. Yes, all of it, all of it.  George, I, I, you know, that's really what I care about is I don't know how to say, do I have to restrict something? No, the homepage is just your values. Hopefully I've just changed their life right now. Oh my God, really? The homepage doesn't have to be one specific niche. No, your website's homepage. When they come there, by the way, very few people go there. This is important when you are a multi-passionate or authentic business. You are selling one thing at a time. You're not going to, okay. So this is what not to do in a single social media post or single newsletter. You're not going to say, Hey, everybody, I would like to sell you my flower arrangement course, but also my downhill skiing course and also my how to have a better marriage course. I mean, why would you do that in one message? I mean, maybe some of you didn't realize you shouldn't do that in one message because it's just like, oh, okay. And some of you, some of your audience will have so much, you'll have so much credibility with some of your audience that maybe one or two or three people will buy all three of your, those courses that are completely different. Fine. But generally what you do is you do one message at a time, right? Today I'm selling you my skiing course. Okay. Two weeks later, I'm selling you my flower arrangement course. You know, four weeks later, I'm selling you my how to make origami course.   And if I have credibility with you, you'll take a look. If you're actually interested, certainly you'll definitely take a look.  So I trust you, I'm going to buy it. It's, it's, life is more fun this way. And you know it, life is more holistic this way. It's more fun when you can sell anything you want and talk about anything you want, because that's the fact it's true. Who, who is saying you have to focus on one thing? Here's where, here's where I can see why some, you know, this is, this is, and you could tell I have a, I have a rant coming on here. It's already been a rant, you might say. My rant is about this. So many of you are held back by this one line. Don't confuse your audience. It could confuse your buyers and a confused mind says no. Who the F is saying this? Please tell me who the F is saying this because it's not true. I mean, okay. Like I said, you probably aren't going to sell three randomly different things in one message. I get it. That's a little bit too random, you know, but you could sell one thing and then the next week sell a completely different thing. And your email newsletter subscriber, your social media subscriber, if you have credibility with them, they're not going to say, I bought your flower arrangement class the other day. And now you're selling me your skiing class. Suddenly I don't trust you anymore. But why would they say that? If they have, if they enjoyed your flower arrangement class, then they enjoyed your flower arrangement class. And now they're selling, you're selling your, your skiing class.  Oh, I didn't know you. So that's cool. Okay. Well, what, where's the confusion? I don't get it. I'm confused by this idea of the confused mind says no. I'm confused. Okay. Now I, I, I get that on a sales page, you know, when you're, when you're describing a service or product, I understand that if you're not describing it well, then people are confused about what the service and product is. Yes, I get it.  And my audience might get confused. If I sell for eight completely different things, my audience might get confused. Who is the person who is confused? No way. Your audience is smarter than that. Why would they, if they're not interested, they'll just keep scrolling. And they're not interested in your skiing class. You're not interested in your knitting class. They'll just keep scrolling. Oh, I like your dog training. Oh, your dog training class. Great. I have a dog.  Why wouldn't I buy from you? It's about credibility. It's forget the confused. Like, oh, my website has 12 different niches. Great. That means somebody who goes to your website and go, I'm not interested in your niche number three and seven and five and four, but I'm interested in your niche one and 11. Those are interesting to me. I'll check those out. Your audience is smart. Who is confused? They might see the search engine might be confused. And that might be true. We're not talking SEO here, search engine optimization. There is a, it is true. When it comes to search engine optimization, Google prefers that you are, like each website is a deep expert on one, the narrowest topic possible. Right. And yet, get this. Google loves Wikipedia. Might drop. Wikipedia is the most multi-passionate, multi-pronged website in the whole world. You go on there, you see everything. Why come Google SEO loves Wikipedia? Why? Because Wikipedia, even though it has a million different topics, each page is quite in depth. And people link to that page. But people don't go, Wikipedia. I don't trust Wikipedia because, you know, I, no, people trust Wikipedia. And there may be one, well, Wikipedia is written by lots of people. Yes, that's true. But one, have you, I mean, I've heard of people who volunteer to write Wikipedia articles and they write on all kinds of things. I mean, would they say they're an expert in, you know, photography and in plants and in, you know, 16th century British history? They probably wouldn't say they're an expert, but they've written Wikipedia articles on it, on Easter, right?  No, it's about credibility. So let's talk a bit about that before I finish this video. It's going a bit longer. So credibility comes from a couple of things. It comes from connection, caring connection. It comes from content. It comes from collaboration, right? And it comes from, I'm trying to think of another C, but it comes from the experience they have with you on a particular topic. So if you find another C for me, that'd be great. So let me collaborate. So caring connection. This is that example where you have a friend that you like hanging out with your friend and they suggest you all kinds of random activities and random topics to research.  I like the way you think. I like your values. Sure, I'll check it out, even though I don't think you're an expert in this. You recommended that to me. I'm looking at this topic right now, so I'll check it out. So that's caring connection, meaning and that's how that translates to your marketing and your audience. Well, if you have been messaging with your audience, maybe they comment on your things every now and then and you reply to their comments or you message them and maybe you talk to some of your audience members, they like you. They feel that you really care about them as a service provider. Your clients, of course, feel your caring connection too. Think about your clients, right?  If you were today, you find out, let's say you are a relationship coach, great.   Let's say today, they told you that they're interested in visiting Italy or whatever. And you say, oh my gosh, I've been thinking about visiting Italy. Let me give you some resources on what I found on Italy. They're not going to go, yeah, but you're not an Italian expert. I'm not going to trust you.  Now, they might not trust some random other person who sends them. You see what I mean? They're not going to trust some random person who sends them links on Italy. They're like, what? I don't know who you are. I don't trust you. But somebody that I... So caring connection creates credibility that they're going to be willing to take a look at your whatever new product or new resource that you're suggesting. And if they want it, they will buy it or check it out. They will consume it. They will buy it. They'll sign up. Caring connection and content number two. If you have seen me talk about how to breed rabbits. I think rabbits breed on their own. But if you've seen me talk about how to breed rabbits again and again on my content. And now I'm going to sell you a rabbit breeding course or rabbit breeding supplies. You're going to say, George, I've seen you talk about rabbit breeding for months now. You're going to sell me this rabbit. Sure, of course I trust you on it. You see, that's where credibility comes from as well, content. Collaboration. If I...  And I... You've never heard of me. I have no credibility with you. But suddenly your horse trainer, your horse mentor, introduces me as a fellow horse trainer who works with... I don't know, some aspect of horse relationship. I don't know much about horses. But you're going to say, I don't know who you are, George, but my mentor recommended you and you did a webinar together. You had a collaboration and now I trust you as well. I'm going to buy your course. I don't know who you are. You don't even have to realize that I am on the side. Most of my time I talk about Facebook ads or I talk about household plants. But in this collaboration, this webinar you attended, I talked about my horse product. And so you're not going to go, well, before I buy your horse product, I'm going to go and research everything about you. Oh, you also sell houseplants. Now I don't trust you anymore. I'm confused. Who's going to say that? You're on the webinar. I sell you the horse product. You buy the horse product, end of story. Or your horse mentor sends you a link to my sales page, specifically about my horse product and why I'm credible to sell this product. And there's no confusion. You just, you're on that webpage or you're on that webinar or you're on the social media posts. That's all you're seeing. If you're looking for it and if there's credibility, you buy. So be everything that you want to be with your audience. Don't worry about confusion. There's no such thing if you have credibility. Now I understand people say niching down because I'll just give my final kowtow to the people who are all about niching down. They're giving you a shortcut because if you niche down and focus on only one thing, then naturally you're going to create a lot more content on that thing. So you get credibility there. You're going to do a lot more collaborations just about that one thing. So more chances of people buying from you and you'll just have more caring. Every time you connect with someone, you're talking about that one very specific thing. So yes, I get it. But it doesn't feel quite right of all of who you are. Be all of who you are and sell one thing at a time. Yes, on the sales page, just talk about one thing. But you can have 28 different products, 28 completely different niches on your sales page, on your website. Just one sales page at a time and you promote one thing at a time. You could promote something different every month. For sure, I could promote rabbit breeding this month and I could promote how to install lights in your home the very next month. And if I have credibility with you, you're going to buy both. And you're interested, obviously. You might be interested in installing lights and one month you saw me selling my rabbit breeding course. You're like, whatever, scroll past it. I don't care. Oh, you want to learn how to install lights? And I like George, great. So I hope this is liberating to help you develop your authentic business and thank you for joining me. I look forward to your comments and your examples below. Thanks.


Financial experts say sailors who make fiscal goals for 2013 should create a plan now and write down their goals to stay on track throughout the year. The new year is an opportune time for service members and their families to evaluate where they are financially today and to make plans for where they would like to be both in the short and in the long term. Sailors can meet with their command financial specialists to review their fiscal goals and develop a budget. It's important that we stay consistent and develop good habits when it comes to financial planning. Plan now, build your cash reserves, eliminate debt, go see someone, ask for help when you need it. Sailors experiencing financial hardship should notify their chain of command. For the Defense Media Activity, I'm Petty Officer James Stolpec.


So, thank you very much so, for those who have survived the school pictures, so I, yeah, so, so. So, So, Remember, these things have an obstruction theory of expected dimension zero. And addition over there, not a contact. They have a C star action with five points and not every C star action such that it is compact. So, this formally applies localization. And as I said, one justice. So we have two special components. So, And we have seen that this is equal to minus one to the virtual dimension of the modern life space. Which is the same as the answer. I mentioned this model, so M was a monolid space, which. It's a single, but it also carries an obstruction theory expected dimension, which was just this number. And so, in particular, that the horizontal, but with an event just the virtual one that we see. So now we have last time. So, And we use for this localization. So, The abstraction theory to rather the dual of the abstract. So, what we have seen was just so we have. I assume that it does not exist, one can still work, but it's easier to take this case, then the dual of the obstruction theory is. And trace the X. So, that's, we want to compute this virtual intersection number on this modern life space on this virtual. And for this, we want to use this. And that's precisely that. The general. Now, which allows you to compute basically any intersection number, any version of the section on this model is based on the section that was able to section on this space. This is the right within range of them. Maybe I should. Maybe I can first start. So what kind of classes, what can we compute classes on what life space. So, for simplicity, we stick to the case of the records to is a formula for any range, which is similar but it gets more and more complicated rain goes up. But it's always. It's always more complicated way it's complicated up as is. So on end. So you want to see what classes do you want to stay here. If I write it just see what's going in the ranks to. And we have here, our universe sheet. So that's it for classes so far. For more. See how I push forward. The one life space. The ice journey class. The universe sheet. And the multiplying make complete sense of it anyway. This will be then class. So here maybe I always take, but don't write positions and say always keep. So we have these classes. And then the claim is we can take any polynomial in these classes. But. It's very well possible that this actually is the whole format of one life space anyway. And so, and so we want to come to this. We have to similar picture. We basically want to write out any polynomial. Write down the polynomial in some classes. It's such that you get the same. So we have. Similar picture. We have things that universal chiefs which we had seen before. And we have the universal keys. We were considering. With the best. This is just in points. So that this is the idea. It's an idea. And then for. This picture but not these things. So I think in this same day. I first take the new sheet. So there's one such thing here. So. We push it. Yeah. Which would like to see it. And basically put it back for three of them. And this was. So this is the sheet. On. Yes. Wow. The other things that we have, which we may not have. We have again. It's asking. Which has its projection. This is a second month. So in the same way. Oh, yeah. This is the full back corresponding bundle from the factors. This is actually this one. rotation. We have a. That we took. In such classes. We have to replace it by something. So we're not also. Yes. One. She says. All of them. Minus. You know. This is. Minus. One more. So we already had this. So. It's a very good. It's very good. You're going to ask. Equal to. So. So what it really means is that S is equal to the first term class. First term class. S. Which is a model. So it's a trigger line model. With a. With a non-trivial C star action. And S is the first term class. And then. And then. This is your best. So. CK. Your best. It's just. Corresponding. And so. Calling. For. A. One. S. Is. The line. Of the one. Is. This one on the. First time. This. Is. This one on the T. First idea. She. A1. Minus S. This means. First. By a. That. My. Plus. These. So here we have these. I do. And. We have. We have. We have. We have. We have. We have. We have. We have. P. And then this divided. The. S. S. And. In the end. We have. This is second term. Is. And then. It's a. Actually the world. So what does this pee mean. So. Just. So, he just needs this we do the same thing here. So, now, instead of. So this was. You know, which was you take the, on S times M, take the universal sheet. And you might like to take the ice turn class of the universe chief, and you might play with the pullback of S, and then you push forward. And here. Now, he was the same. So you have one. Yes. You might play with the pullback of alpha and you push forward to the part of it. So you do the same construction. And you place every instance of time I saw alpha in this polynomial by the corresponding expression that you get here. So alpha was only. Like this. The same. The same thing. The integral of this thing or there was points. This is some over all. One such and then such and one person. And take the integral for Chizuki's. I don't say that this. The whole more than zero. It's up to you. And then. He is again to see. Use this phenomenon. Is equal to one is a sum of classes. See one. A one times each smaller class. This is the form. Maybe I can't say. This is a, as you see it's rather complicated, but completely explicit. So you have this simple expression. So that's maybe some simple thing you want to compute here. So it becomes something very complicated in terms of the universe she's on the liver's teams, but it's completely explicit, and one can compute. So, but this condition is not very serious. Because they know when you if you twist. So if you have the same line, then the invariant will not change, but this condition will change. So you can. So, normally, this will only fix. See one. Fix see one. This will only hold up to a certain seat too. But if you twist by bigger and bigger powers of an empty line bundle. This tool you can that will continue. So this is somehow something one can see up. So this will therefore, always. And maybe one should see. So maybe one word. I mean, the proof itself is, you know, is one. pages, but the basic idea. It's really not so terrible. So one considers. Another life space. I think it's a she together with this section. This she, this one life space now has a kind of star action. This point locus of this C star actually consists of certain parts on and stands. One is something related to sort of written in the answer to this point. So something's one locus. I look like this. Other things. The other things. This is my life space. And now you integrate something over the one life space. We integrate something over this master space. Which of which you know it is zero. I'm just going to be zero. But when you compute it, you get that the fixed points, the fixed points corresponding to this will give you this. The fixed points will give you this contribution. And so therefore zero is equal to the difference of these. But it's normal to set up so that. In the trip is somehow not new, but. And then how I want to buy this. So in this case, we take the obviously to be the term class like this. Now, you know, the virtual tangible as a, as a complex. We can, and one can compute this by so we're again. Yeah. And so, and so that's what it is. The thing is, in some sense. Now, one is done because now everything works the same way is in the vertical case. So I can just catch. Because we are basically a thing we're indicating it looks very similar to what we were integrating in for the work with my friends. And so the methods will be precisely. The other round is the horizontal one. So how to modify. So that you just maybe for the C prime. This is. This is. Generating. We just like a, but the key is the one which corresponds with a one scheme. So, particularly can consider the part where n is equal to zero. So when we look at the zero points here. So this will be equal to the coefficient of q zero of the same. And something which I call our series start with one. And we have, so now we are looking at integrals with the systems of points. And this, we have to use some types of this is what is there. It says that if you have anything reasonable, then express the expression in terms of intersection numbers on the virtual surface instead of. You have a nice generating function. For such numbers will give you a map ticket in form where just is a product. So, so power series. Take it to the powers of the intersection numbers on the surface. And so you can do this program yet. And so, and if I have policies. So, we have this is a one. That's that I've sketched it. So, given this out. If we know these policies, we can compute the result of a bit later. For this power series, we only have to be able to compute this generated function here for certain examples of it. Some examples of that because only backup numbers. So in the same way, so this allows it to be. So, for example, we construct from these policies are some secret. The meaning is these are sort of services with an action of C. So, we can start with six points, you can take nine models on these, which are a brain for this action. And so then one can continue by the basic. You know, the fixed point on the action of civil schemes by the politicians, but to bits of petitions. And we have this fact. The other time, what the action is from the next group of these ideas she's, and it's even simpler to see what the action is on these top logical bundles. So you can just write down everything for the local contributions of the fixed points in terms of combinatorics partitions. And then, you know, this smart one could do something with it depends not so smart one in right program so you then so then. Yeah. So, so one has to be careful because one process to check off S. So the higher the key was the bigger this the ported S and one has to finally take the coefficient of s to zero so that makes this computation more difficult also for the computer. But it's not there. Anyway, we can compute this, and then one can look at it. And anyway, we find, let's consider this function. So we have the data function of the artist, the intersection form was given by this. And then we have the matrix, which looks like two on the diagrams like this. Now, we can also look at the data. They are like this. They are we just replace. And then we have the following lecture. So we even get a zero. Exist a lot of structure theorem. So, you know, the idea, once more, one, the version of the case and efficient. much of the following expression. So very similar to what we did for this event on the difference that they have of theorem. And then it's better function function for the minus one times these things and zero times the sum or better. So we have a sign. So this is the intersection number of that one. And then. Remember, this is precisely the same formula that we had. The structure. Instead of the data. Instead of the sort of unity, we had this data. It's better. I was very similar to this difference and they have this universal five just called C zero. C zero and C IG. And the relation to this is this is basically the formulation of this context. So, as we said, the ordinary function of all that shopper written invariance. It's nicely and the power minus one of a tower. So with the same thing, it says that C zero, C zero minus one of the top. This is because for the work for the vertical of the difference, and in the same way. And so it means the generating functions for horizontal and the vertical of the difference determine each other. And don't these DIGs and CIGs. We identified them here as modular. And so it makes sense to do this. And so we have shown this, that the range is two or three for four and five. This gives us this tells us this gives us predictions for these policies. It's a formula. There's actually very simple way how you transform the formula for the vertical of the difference into the horizontal by just like this.  Okay, so maybe I will. I wanted to say, so that's Yeah, questions. I'm still


Hey Gems welcome back to my channel so I don't know what I'm gonna title this video yet but I did want to record a video pretty much introducing you all to what I am currently creating of course except for life but my relaunch of gem stylistic is coming up very soon and I'm relaunching by having a ceremony that is very dear to my heart like I actually came up with the idea when I was still in Ghana I kept thinking of something to do to honor mothers to honor women and the idea came to me while I was in a deep silence like I was still living with my children's father of course but we had recently have a big argument so I kind of just went into a shell I wasn't speaking to him he wasn't really speaking to me and we're just coexisting at the time and all I did was journal every day I was journaling just my thoughts just everything that was coming to my head just based on everything that was happening to me then and for the past year I was just writing it down writing writing writing every day I was writing by the end of like a week and a half I have like 20 something pages in my journal just written out but throughout that time of silence I kept thinking of this space for women or mothers to each other but to feel safe and secure because at the time I did not feel safe at the time I was not feeling secure and I just got the urge like it the thought just wouldn't go away but every time I thought about it through all of that I gave pretty much vert to this idea of having an honor our mother's ceremony and this is pretty much the start of the next part of gem solistic I've always know I would be creating so I'm actually gonna insert my new logo somewhere here on the screen and it is an invite-only event but this is just the beginning of something way bigger that I want to create however I have to start small I have to start where I am and I'm okay with that so pretty much my new brand well my new logo as you can see it is a picture of a woman which kind of is a representation of myself because I actually used a picture that I took and I'll probably insert that picture here too a picture that I took when I was pregnant with my second son and the posture and the way I was posing in the picture I used that to as a as an example of how I wanted the woman in my logo to be seating and the the woman that designed my logo she actually did my first logo and she pretty much blew me away with what she came up with based on the ideas I had in my head she literally brought it to life all I did was send her the pictures I told her what I wanted I want the woman sitting in a lotus flower I told her my colors and I told her what I want it to say and she pretty much created exactly what I I had in my mind like the vision I had in my mind and I'm so thankful for her because she don't even know how much I love that logo and she was actually pregnant when she was creating it which makes it even more special to me but it's now going to be called gem stylistic lifestyle and wellness and the lifestyle and wound wellness is added because when I created gem stylistic it was always supposed to be about a lifestyle it's not just about natural products it's about living a holistic and natural lifestyle in every aspect which is what the word holistic means in the first place you look at the whole you don't just look at one aspect of health so throughout the cup I will say in the past year and a half but so I did some trainings I did a woman intensive herb course to you know gain more knowledge in herbs that are great for womb health for women's health and just my path in becoming more of a herbalist I got certified in that I also did a fertile womb lifestyle specialist training so I'm now certified in that I just want to have all the tools under one room we're not wearing products but I'm also offering a service to actually help women one-on-one to get their wombs back into a state of health that would allow them to you know have a healthy moon cycle and concede without any interference with you know medical doctor so I'm not saying that those things aren't needed in terms of the medical world I'm just saying that sometimes or even most times like trying a holistic approach in terms of changing your whole lifestyle helps and you know sometimes people aren't willing to do that or don't have the tools and knowledge and resources they need to do that and I want to be that for those women who actually wants to do that so like I'm not I'm not looking to just change people I want women who wants to change for themselves to seek to come to me so that I can help guide them through that process so even from my own experiences and difficulties that I have I know that I know that the methods that I've learned and implemented in my own life works and I can get into our whole not a video in terms of certain things that I personally went through that I had to change in order to heal myself so I'm just excited to bring that aspect of my passion to my brand and so yes this is pretty much me rewriting my own story writing my narrative and putting my focus into something that I am passionate about that I'm sincere about that I genuinely want to grow into something bigger in the future so I'm starting with my honor honor our mother's ceremony where I invited women who have had something in my life and that's from Bert till now so I will be honoring those women and then I also invited women and their daughters and giving their daughters the opportunity to honor their mother so of course I'll be honoring my mother and then I'll be honoring the other women that I you know invited for that purpose being that maybe they did something for me at some point in my life that meant so much to me that I don't know how I maybe would have gotten through without them or whatever the case may be so that's the gist of the ceremony and you know I'm actually doing it I'm actually putting a lot into this like I wanted to be a nice intimate I did get a decorator you know I ordered some stuff I'm creating my own gift boxes what else I'm having my sister and her boyfriend has a food business where they'll be catering healthy alkaline vegan meals it's gonna be a free course meal like I'm going all out in the capacity that I can so it's gonna be actually a really nice event I'm encouraging everybody to dress like the goddess the empress the queen that they are look nice and just feel nice and feel great I am so excited for it like a part of me is overwhelmed because I'm like how am I gonna get all it is done by me but I'm really like it's something that I really want to do and I don't even know how to explain to people why I feel like I need to do this but it's literally something that just came to me and it came on so strong that I know I need to do it so I'm actually strong on my solar return May 21st and why my solar return not as a gift not as something like I'm having a party for myself but that the gift in this is me actually honoring the mothers like I am really like I genuinely and I really am excited to do this so like what other day best to do this than my solar return as a gift to myself and I don't even mean that in a selfish way because the event is not for me it's for mothers like it's for all of us so and in fact like I don't have anyone at that event honoring me like I'm literally honoring everybody I've been through I feel like times we just go unnoticed like people expect us to be strong people expect us to do this to me take care of the children when they're sick and go to the school events and do like people we're automatically put in that category to just have it together and sometimes it gets hard sometimes we need a break sometimes we need help sometimes we need support and sometimes you just want to hear that we're doing a good job and if I can create a space where for one night mothers just feel loved on and they feel honored and they're happy and that's a great feeling to me excited to put that event on I hope that everything goes as planned because I've literally been planning everything from since I got down here because I don't want anything to fall off I don't want anything to be too last-minute so I've been writing down everything I'm making sure I'm as organized as possible and there's so much more to it that I probably can't just go into all the details right now in this video but I created I had already started creating so much to put into this even when I was in Ghana I created this whole e-book called kitchen sweep challenge where I write down step by step how to transform your kitchen into a healthy laboratory so instead of just being a kitchen your that that space turns into your medicine cabinet pretty much so you're eaten to live instead of living to eat if that makes sense so I wrote that I also have a healthy moon cycle template for women to on tips and things that they can do to you know create a healthier moon cycle for themselves and then I'm also creating a meal plan for people who just don't know what to eat I may not be able to accomplish everything before I have the baby but trust me I'm not stopping so yes this is the start of gem stylistic lifestyle and oh I'm also adding you know there's a whole other aspect of self-care that I want women to start to embrace more so yes that's it for this video I'm just excited for this new journey I'm on I've been feeling so much better and yes I have my bad days I do I really do I don't want anyone to think that I don't have my bad days because I am still processing I am still healing and I am still trying to come to terms with my new reality but I am still embracing it as well and I'm just excited to even have the opportunity to create something better to create something new and to provide for myself and my babies and live life really like it doesn't really matter where you are in the world but that state of mind your environment is what's most important if you're not happy if you're not at peace if you're not loved like those things take a toll on you if you are miserable it doesn't matter your location does not matter so I'm just happy to be here I'm happy to be around people that love and respect me that's all anyways that is all for this video I hope I made sense in all of this but yeah a good friend of mine actually reminded me of my video but I think I should start saying it again so to all of my copper color women watching protect your womb by any means necessary and to everybody watching protect your peace by any means necessary I wish you all love like and prosperity and you'll see me in my next video