Goedemorgen. Het is oké? Ik oog veel van jullie een referentie. Het werkt niet, hè? Het werkt niet, maar ik denk dat dit de positie is. Hallo. Het is beter? Misschien is er een gevolg van dit. Nee, dat is het. Alles loopt, oké? Er is een red button hier. Hallo? Dat klopt, hè? Hallo. Gaat het goed? Ik zou zeggen iets, hè? Goedemorgen. Maar kan je me begrijpen? Oké. Laten we gaan. Veel van jullie vragen een referentie over de proof van hyperboliciteit van renabilisatie voor circlediffiummorphismen. Laten we het opnemen. Het is ook op de noten van vandaag. Laten we het opnemen. Referentie voor hyperboliciteit van circlediffiumrenormalisatie. En het is door G. Stark. Het is een lange titel. Maar wat is belangrijk? Ja, je kunt het uitkijken. Het is heel lang. Het is in non-lineariteit. Non-lineariteit. Nr. 1. En dat was in 1988. En de pages zijn van 541 tot 575. Als dit niet genoeg is, dan kan je gewoon naar de noten van vandaag, dag 8 en het hele titel zal in de top zijn. Dus je moet het kunnen vinden. Oké. Dus we hebben de circlediffiummorphismen afgemaakt. En de laatste tijd gisteren begon we te renormaliseren. Unimodal maps. En wat was de situatie? We bekijken de spas van unimodal maps. Het ziet er iets als dat. Waar dit iets is. C3. En waar de tweede derivative in de kritische punt is. Dit is non-zero. Dus er is echt wat kritische shape daar. Dus de spas van deze is de spas van unimodal maps. Maar dan zijn er renormalisable ones. En de renormalisable ones hebben de property dat er twee intervallen zijn, die worden geïnterveld. En er is een intervall u en een intervall v. En dit is geïnterveld met f. Dus als je een systeem hebt zoals dit, dan call het renormalisable. Dus dan zie je, als je renormalisaties doet, je moet de domain waar je zoekt. En we gaan deze intervallen gebruiken. En je ziet, de return map is gewoon de tweede intervallen. Je neemt de intervallen, het gaat hier en dat komt terug. Dus als je de graf van de tweede intervallen ziet je iets zoals hier, we hebben u en hier hebben we v. En het ziet er iets als iets zoals pad. Het ziet er iets als dat. En dan doen we, hier zie je weer een kleine unimodal map. Upsite-down. Als je dit in een f-niveau kan rescalen. En dan zie je weer een unimodal map. En we callen het renormalisatie van f. Oké. En dan hebben we gisteren de meeste theorem. Laten we gewoon het brengen. Dus de foto van dit renormalisatie-operator is precies wat je begint te zijn.  Het is gewoon een curve, een 1-dimensionale object.  Dus en wat is belangrijk is dat de eigenvalue hier, de ongeving eigenvalue is de nummer dat we nog vroeg hebben. Dat is deze nummer 4.669. Ik remember gisteren we were looking at parameter universality en er was deze nummer 4.669 showing up. Dus dit is briefly de statement maar wat is belangrijk voor ons is dat als je deze stabele manifold deze manifold dat is exact de klas van infinitie renormalisable maps. En dit is een topologische klas.  En nu zie je van deze laatste deur van het hoofdtheorem dat de set van infinitie renormalisable maps die een topologische klas is gewoon een manifold. Het is een manifold een stabele manifold van deze operator. Dus dit explicent al wat over de bifurcatiepattern. De topologische klas van infinitie renormalisable maps is een manifold. En oké dus laten we zeggen een beetje meer over de dynamiek van deze dingen. Nou, misschien kan ik... Laten we deze eigen waal 4.669 gebruiken en explain waar de parameterdependant uit komt. Dus laten we een foto maken van alle unimodal maps. Deze zijn 3 maps met een non-degenerate kritische punt. En binnen daar er is een strip van de maps die een renormalisable is. Dus deze zijn de een renormalisable maps. En nu de foto van renormalisatie die hier is op een infinitie de foto. Dus wat is renormalisatie? Dus het zet de set van renormalisable maps en het maps naar iets zoals dat. Dus je begint te zien een hyperbolic foto. En daarnaast hier hebben we onze fixe punt en daarnaast hebben we onze onstable manifolds en daarnaast hebben we onze stable manifolds. En we weten dat de renormalisatie in de fixe punt een eigen directie en de eigen value is deze nummer 4,6 6,9. En laten we zien hoe dat nummer in deze parameter een universale foto is. Dus let's recall wat er gebeurt. En remember of you take your family if you take a family of unimodal maps then you can create kales, which we discussed yesterday and it goes according to a period doubling cascade. En we do something like this. First your dynamics is controlled by an attracting fix point. Then at some moment you get a period 2 point and then a little later you get a period 4 point. So it continues like this and then there will be an infinity value where the chaos begins. And then yesterday we discussed that these bifurcation moments where the period doubling happens converges to this boundary of chaos and it converts exponentially fast and the rate is 1 over 4.669. En so now let's use this picture to explain this parameter universality and you probably you see already how it goes. Let's do it with a little color orange. So this is our family here and this family, I don't know where it is but this may be sitting somewhere here. Some curve. En now we see that this strip are the once renormalizable maps but then there will be a thinner strip where you have maps which are twice renormalizable. And then there will be and then where the family enters this strip you have to check that is exactly this moment. So this corresponds to the moment t1 but then there will be a thinner strip where the maps are three times renormalizable and where the family enters this third strip it will be t2 and now you see the picture happening each time you go, if you have more renormalizations you get exactly these intersections converging to the infinity which is the intersection of this stable manifold. Ja, that is a big part of the theorem.  That is part of this theory ok But now you see if you have a saddle point in hyperbolic saddle point then you know that these pre-images they accumulate everywhere at the stable manifold is the item value of the fixed point you know from saddle point. So here you see explained why these bifurcation moments convert this is the item value of 4.6 Ok, so now this really explains the bifurcation picture and this is sort of where where the history began this 4.6 was the first thing they observed the tracer in Coulet and Weikenbaum and this is historically the beginning point. Ok, so but now let's go on to the rest of the dynamics 0 0 So you might think that this curve is the logistic family and here you have the infinite renormalizable map in the accumulation of period doubling and now we saw part of the theorem that the stable manifold is exactly the topological clans and so indeed these maps are conjugated and yeah and so this is like the third part Ok So you were saying that at some point you would describe fixed point the sum information we have about a fixed point and we'll give you some this fixed point is a complicated object and you cannot write down a formula it is some analytic function and it is something don't even try to write down a formula and hopefully today I can show you why it is something complicated and you cannot make this by hand Ok, so remember when we did the circle renormalization we had towers and we built a whole dynamical picture out of these towers of what are the towers for our period doubling renormalization how was it again so we start with this interval so u1 which goes here and comes back under two iterates within this map is again renormalizable so inside here you will again find two intervals which are exchanged by the second iterate and let's say this is interval u2 and that will be returning by f2.2 times f2.2 is f2.4 and then inside here there will be another interval this will be exchanged and it will be interval u3 and that will be returning into itself by f2.8 and so there are these intervals they are shrinking down that's a little theorem and they return by powers of the time return times of 2 to the n so you see that the tower from this you get the tower and the tower is extremely simple so here you have so here we have a picture of the nth renormalization like what we did in the circles and you see there is a bottom we call un and then there are all the images going up in the tower and the height of the tower is 2 to the n and when you come to the top you just return back so you see the nth renormalization is just a rescaling of 2 to the nth iterate ok, so remember when we did the circles the big tool was how does renormalization act on these towers so let's see how this works we do the same program as for the circle case and so how do how does renormalization act on towers and so let's see so here we have our little map and we have to take a u and we have to take a v and then we have to take the return to u now let's say that this is actually the nth renormalization and we have un here and we have vn so let's see how the corresponding tower looks like and it is something like this here we have un and we have 2 to the n layers and now inside here oh, I shoot this is un and this will be un plus 1 and vn plus 1 for the next renormalization no, it's a unimodal map and because it's a unimodal map so you start here which is this interval and there is a critical point and now from here to there you get something fold and then you just go up morphically and then you return the first step is the fold ok but now let's see where we have to cut so we have a picture of the nth renormalization and we want to make a picture of the next renormalization so apparently we have to take un plus 1 here en we have to take vn plus 1 en we have to cut along this the orbit of un and we have to cut along the orbit of vn and then we have to paste them together so we have here so here we have the part see the first part and here we have the part on top so you just take 2 substrips of the tower and you stack them in a different way and so now you get the tower of length 2 to the n and the bottom is un plus 1 and so renormalization is again if you look on the level of towers you cut off pieces and you reorganize them it's really the same as in the circle case in the long sense it is simpler because the tower is simple it's just one building so i think now we have all the tools to oh no, one more thing so let's let's start to discuss the dynamics of these maps and as in the circle case we get the layers of the tower which form our pn which is the dynamical partition is exactly the same as in the circle case we know there is a little problem that if you look at the intervals at the nth tower the intervals are pairwise designed but unfortunately the union of all those pieces is not the whole interval and you can see why that is true because when we cut off the pieces to construct an extrinormalization we forget the middle part so you cut off, you lose something so but at the same moment it is just an interval which runs around and then it is back so instead of calling it the dynamical partition we call this cn which is just the same as this partition is beside the nth renormalization cycle cycle just two to the n intervals which are running around through the interval in the case of the circle difgeomorphism it was a little bit complicated to draw the picture in the circle where all these intervals in the circle case it is a little bit complicated but in this unimodal case it is easy it is really easy and it is just constructing a counter set so let me make a picture for that let's see, let's first start with our interval en let's make the first the first cycle and now you know what it is, it is this interval and it is this interval and this one goes here and that one goes back so these two intervals form the first cycle but now inside here you are renormalizable so inside here there are two intervals and there is one here en they are exchanged by f to the fourth f to the two so that means if you take f they will get copies here taking this picture and you iterate by f and you get it here and now you see four intervals and that is the second cycle and now you know so inside here you will see again because it is renormalizable things which are exchanging under f to the eighth and this picture will be transported through through the cycle and it comes back here and it will be back and that is your third cycle and of course this process continues and in the end you will have some set here which is called B en which is just the intersection of all those cycles and of course it is going to be a counter set let's make a precise statement let me make a precise statement about the dynamics so a theorem so f is infinitely renormalizable and then let's see it will be indeed an intersection of all those cycles and then the C is a counter set and this is not completely obvious because you see how this is exactly the same as the construction of the middle third counter set you cut and you cut and you get two intervals it's the same thing but you are not sure that each time you cut off a definite piece so it's not like in the middle third counter set you always cut off a third and you cut off a third it could be that that you start off to cut out less and these intervals they actually stay large so you have to prove something that you always definitely cut off something and that is behind this part of the theorem and to do that you have to do again like you remember when we did circles with your morphisms we had to prove a key point was that the sizes of the intervals in the tower goes down to zero and to do that we needed nonlinearity and we needed that the map is C2 and so in this case we have a critical point and that complicates things and that's why we need C3 so there is some analysis behind here not difficult analysis in C2 probably a dust fill I think so Sarma knows I think it fills in C2 of course it fills in C1 but I think also in C2 another part of the statement about the dynamics is that this counter set is actually in a tractor so let me prepare that for every n there exists a periodic orbit and let me call it pn which is contained in the end cycle of period 2 to the n it is just some point which follows the tower and comes back so this is our periodic orbit so there are and the third point I cannot find that place so the next line should be under there but there is not so much space so the third point is that if you take an x which is not in all those periodic orbits if you take a point which is not periodic x is not periodic then the limit set of this point is exactly this counter set so every point which is not periodic will eventually get sucked in in deeper and deeper cycles and eventually it will be running around around this counter set ja ja ja ja ja, that is the same difficulty as proving that this is a counter set the existence is not so difficult and because you have an interval which runs around in 2 to the n steps and returns so you know the return map has a fixed point and these are disjoint intervals so the actual period is 2 to the n it is sort of simple and what will happen is I can make a proof you will see how it goes so you have your point and let's say after a couple of iterates you get into un so you get into un en nu let's look at the n plus first renormalization so what you have to do is you have to take the second iterate on this interval let me be brief let me not get very crazy about it so you get something like that so now your point lends in somewhere here but now this periodic point is expanding and now you see that once if you get in the gap you are not in the next cycle but now this guy is expanding and eventually you will get in the next cycle so you enter a certain renormalization level maybe in the gap but then inside this gap is this periodic orbit which is expanding this is like classical dynamics it's sort of this is not so difficult you need some tools you need to know what Swartzian derivatives are you need to know non-linearity things it's not difficult but you have to prove that ok so now we have a clear picture of of how the dynamics looks like and we see we see in our map this counter set and we know that this counter set is actually the attractor of the system so the dynamics is sort of clear so let's go and fill in the other parts of our diagram let's start with with the easy part the topological part topology and the theorem is it's the same as in the circle case so if f and g are infinitely renormalizable so they all have these towers they both have this counter set then f is topologically equivalent to g and that means that there exists a homeomorphism and this h of f is g of h so they are conjugated and remember in the case of the circle our towers filled the whole circle so the structure of the towers allowed us to build the conjugation as you know it is a little different our dynamical partition our cycles are not filling up the whole interval so that's a slight complication but the proof goes through and so for people who know needing theory and so it means that it says that there is a theorem by Milner and Thurston that says that if you know how the critical orbit how the critical point jumps around if you only know the combinatorics then f and g are the same and like from these cycles you can read off how the critical orbit jumps around en so this just follows from from an old theorem by Milner and Thurston in general you might have many pairs and the number of pieces might grow but the tower is always simple so we are done with this topology so let's go to the measure theory and so we see our orbits accumulating a discounter set en now let's see whether there is a measure which describes the distribution of our orbits so the measure theory and the theorem says that f has a unique invariant measure on our counter set and that means that indeed let's call it mu en dan if you take measurements and that will convert to our measure en you know the proof and by just looking at the towers and so remember that so the tower is a realistic representation of the dynamics we see intervals and intervals but we can do it more schematically by just replacing all the intervals and the maps by just a directed graph and so what you see is you get the end cycle and we just represent that by a directed graph of two to the end point and then there is the next cycle and of course that is included in the previous ones and you see you take the next one is included here and that will be represented by again a directed graph but now of to the end points you remember from when we did the circle it was more complicated we had two loops but now the situation is very simple so and the way this guy is included is just by winding twice around en so now you see remember we went this is the graph and then we went to the measure spaces which were just Euclidean spaces and we see R and we see 2 and we see R and in the inverse limit of these measure spaces we just see R and that means that the object here is uniquely ergodic so maybe somebody was not wrong last week so it might be a bit fast for that but we see and you use the combinatorial structure which you get from the renormalization schemes to produce the measure like in the circle case it's not just this that you also need to do if you take point outside the average is converged nah, that's okay yeah, but you still need to use the specific structure no, you know if you have something if you have a point which converges to something which is uniquely ergodic and then what you can do is you take your point and you put point masses on the orbit and then that measure on the piece of orbit will converge to some inverse measure on the limit of one, there's only one okay and this is the simplest situation you can imagine from a combinatorial measure here at the point of view nah, you know this is minimal because so it's minimal, you will get everywhere every point will accumulate we don't have the distance we don't have stable sense I think you do I think you do I think you do no, you don't it's complicated you don't you do it's the same thing yes and no I think you do I think you do and so what happens is I'm not 100% sure so if you take your point somewhere it's sitting here getting away from the fixed point and it gets in the cycle and then you start to jump around in this cycle so you could take any point and it would stay with a fixed distance to that point but then later you get to the deeper cycle this argument works I think what happens is I don't know I cannot really imagine probably if a point will get closer exercise, that's a good exercise it's not so difficult but I cannot come up with a definite answer you had a question say it again this is called an odometer it's like the limit of cyclic groups and if you go to other renormalisation schemes the limits there will always be a counter set and this counter set will always be the inverse limit of cyclic groups odometers ok, so ok, that's good so we are done with the measure stuff just a thing this measure is simple like this has a quarter this has a quarter this has a quarter so it's really a very simple measure and on the next scale this has an eighth so it's really the simple measure but the proof is the same as in the circle case so now let's go to I think it will be good let's go to the geometry where this excitement comes let's call it excitement now you will see where the complications are there is something going on here so let's there is something called scaling ratios if you take the middle third counter set then you take an interval and you cut off a third and this third is the scaling ratio but in our thing we don't know how this scaling goes so let's see what is going on so let's take the nth cycle and let's take one of our intervals there are many of them let's take like this one so then there will be exactly one sitting above so let's call this one I let's say this is this is our interval I and it is contained in one of the higher scale so let's call it T of I and then the scaling ratio of I I make it a big number it's important it's going to be nice it's simple, it's just the length of this I divided by something just how big these things sit inside here if you would be building the middle third counter set this would be just a third and now you look at at the set of a map F which suggests all these scaling ratios when you look in the nth cycle as you look at how these numbers tell you how the nth cycle is constructed out of the previous cycle so these numbers if you know all of them you know exactly the counter set and now you say sigma of F is the limit of this so how do you write that so you take the union of all those guys you take the closure and and you take the you know so this is the limit so all these numbers this will be something 0 en 1 en so this set and it is some subset of the unit interval and so all the limiting scaling ratios you can observe and this set describes like the deep the deep structure in this counter set and so how you this set tells you how asymptotically the construction is made I think I repeated myself ok and now there is a theorem let's first do an example if you do the middle third counter set you can define the same numbers and then you see that sigma n is just a third and you see that the limiting one is also just a third and so this set of the middle third counter set you see that this is a very simple scaling structure and this means very simple scaling structure you could call this a fractal and wherever you look you will see exactly the same picture now let's give you a thing about let's tell you something about how this set really looks like and one thing is that the set of scaling ratios is exactly the same as the set of scaling ratios for the renormalization fixed point r of f star is f star and so the way our counter set is constructed in an asymptotic scale is exactly the same as the counter set of the renormalization so f is infinite renormalizable and so our f is infinite renormalizable let's make a picture here we have our fixed point and here we have the stable manifold and our f is somewhere here and you know that these are the infinite renormalizable maps and so our f is infinite renormalizable we get an asymptotic scaling numbers and that is exactly the same as the scaling structure of the renormalization fixed point en of course that has something to do that renormalization converts there every renormalization remember is zooming in of course I only have one place but at least there should be something here which is inside there but actually they are the same so the asymptotic scaling structure of the attractor of an infinitely renormalizable map is exactly the same as the scaling structure of the fixed point so en nu the sort of the the mystery comes no, no, no no, no, no we are only doing pair doubling renormalization en remember we never talked about any other renormalization is everything is pair doubling dan is it not true no, no, it's not true ok so the fun part sort of the fascinating part is that this thing itself is a counter set so it is completely opposite to how you make the middle third counter set so in the middle third counter set you do everywhere the same and this counter set there is actually no two places where you do exactly the same so the scaling structure the way you build this counter set is highly extremely rich wherever you do something you have to do it according to that spot if you do it, if you zoom in somewhere here in a very different way and you can distinguish two places by looking how you are zooming in this is a counter set so it is an extremely rich scaling structure and this is what you observe what people measure in nature so you can imagine that in the middle third counter set you can make with your hands it's always a third, a third, a third in scale you have different numbers you will never be able to do that by hand but nature does it like that a consequence of this and you will not be surprised maybe that could be an exercise so if f and g are infinitely renormalizable and then you know that f is topologically conjugated to g and now if you look at the restriction of the conjugation to the counter set of f to the counter set of g then this thing is differentiable so we see rigidity again and you can imagine if these two counter sets have the same way of they are made the same way on an asymptotic scale and then you will see a fine relations between f and g so if you know this you can prove this ok, so let's continue tomorrow and then we start to discuss hand-on dynamics so the last two days will be hand-on dynamics and we will be mainly discussing the phenomena of sort of like today it will be difficult to squeeze out exercises things will get quickly complicated so let's keep this as an exercise for this afternoon so bye


Alright, moving to other parts of Africa, now, Zambia's President Edgar Longo has ordered the deployment of the military to curtail electoral violence ahead of the August 12th general elections amid criticism by the opposition and other groups. The president further stated that troops had already been deployed to some areas in the capital, Osaka, and would be sent to other areas in the country if the situation demands so. There have been incidents of violence by ruling party and opposition supporters across the country in the run-up to the vote.


How's it going guys? Today we're going to be doing another mold pour. And as you see, we have over 2,000 molds in here. And I've done a little bit with them before. I've showed you the greenware and I've kind of cleaned one out. But we're going to kind of pick up some random ones and see the kind of the conditions. And a lot of these, I don't even know what they are because they're technically not mine. They're my fiance's moms. But I'm able to use and kind of, you know, slip cast any of these that I want. So let's go ahead and pick some up and see all the different kind of things that are in here. Let's see how dirty they are too. So this one right here, you can see we have quite a bit of stuff falling out of these. These look like chili peppers. So you could cast those and make them green. Or red hay then. We have some kind of like little buoy or something like that. I'm not really too sure. Let's go back here. We have some bigger ones. What about this guy up here? We have some kind of like pot. We have like a bunch of spider webs in there. I don't know if you can see that. That's pretty gross. Put that back up here. These are pretty heavy too because it's all made out of plaster. We have one next to it. It says S2564. Whoa. Oh man, this one looks cool though. There's some kind of angel or something. It has all kinds of webs in there. I can't speak. There we go. Let's just pull some random ones out of here if we can. I don't want to move too many from in here because I'm not sure of the organization. But this one looks like a solid one honestly. I've never seen that. It's pretty cool. It's like a vase. Some small one next to it. It has some webs on the outside. And it says something. I can't really make that out. It's kind of hard to see. Yeah, it's all kind of weathered out. This is some kind of... Ooh, it's a teacup. Check that out. That's pretty cool. Look at it in the light right there. I like that. Sorry if the audio is not like super great. We're using the ASMR setting on my LG phone which is kind of... It's more intense and you can really hear everything. But sometimes I feel like it's too loud. And then we have this. It looks really new. This one right here. It's kind of like a snail. You can even see the ridges on this one right here. It still has like all of the details on it. This is actually really good. I'm going to set this one to the side actually over here. Because we can actually cast this on a different video. I'm going to put it right over here so I don't forget about it. And then these are all the ones right here that we're actually getting rid of because they're damaged. Or we have duplicates of them so we don't need to have the other. But if we see over here we have a ton of these stacked right here. Some of these are just not even good anymore. Like this one you can see the pitting has kind of ruined that portion of it a little bit. It has a lot of dirt in it. But you can see this is a fish. And you can still pour stuff in there and get the fish thing out. That's not technically broken but it's still a little bad. Let's get out of here. We have some kind of angel again. Looks like a cherub maybe. And we have flat ones too. You can see down here we have like a cactus.   Got some flowers. These are like little lily pads right there. That's pretty cool. And then I'm not sure that's like a leaf heart. It's going to be a lid. We have a bunch of stuff like that. And I'm going to get a close-up on that. I'm not really sure what that is. That's like a little angel I think. We have this one right here. It looks like it is a camel. There's some guys on a camel right there. Some kind of thing like that. I think this is a coaster honestly. Some of these could be used as coasters. A lot of the flat ones probably can be. And we have like a little kind of house villa kind of thing. And then a more you know defined one. This is like a church has a bell at the top. And then we have a dog with a fire hydrant right next to it. So that's pretty cool. And then I guess some little interesting ones like this. Not really sure what that is. Of course the bigger ones are the interesting ones because you can get some pretty cool stuff out of that. Oh look at that. That's a guava. It's a lot of spider webs in the bottom though. Might be able to clean that out pretty well. Let's set that down. But this whole entire shelving unit all the way there, all the way down. We need to move this middle section to the center so we can actually get to all of these. Some of these are a lot newer too. There's a lot of newer ones on this shelf. A lot better condition. This one looks like we'd be okay. I have no idea what this would be. It's kind of terrifying. I'm going to be honest. Like look at those eyes. And that's kind of creepy. And if you follow me back here. I turn this light on. Hopefully we'll actually see how we won't point it to the back wall right here. The lights are adjusted. We have a ton of different ones over here. So we have ones that have two different portions to it. Two different pour points. Right here it looks like there's some kind of little bow. Some kind of animal on the side as well. It's pretty cool. And then we have some flat ones. Well actually let's pull from the top because these are a lot of flat ones we'll keep at the top. So this is a lid.  Well not a plate because it has that thing right there. At the center of the whole looking doughnut. There's this flat one right here. It has multiple different pour points. Some kind of more chili peppers. I feel nervous sticking my finger in there. I never know if there's going to be something there that wants to bite. It's like that guy back down here. And then this thing is probably like a bowl of a plate. Because they're usually shaped like this. So you pour them right here.  It's like kind of scrunched in edges. Kind of like a pair of curtains or something. And then over here we have some more flat ones. This is another lid. But I want to check out the one that's below it right here. There we go. This is pretty cool. It's some kind of cow I think. Some grass in the middle here. Probably use that as some kind of plate maybe. And then in the back right here we have another one. Some of these are really dirty. Like I said they were left outside to the elements. So they're alright. So this is a bunny. Good thing for like Easter or something. It has like a little egg right there. All these need to be cleaned out too. And then if you want to see right here. This is kind of what we do to recycle the clay. Some of the things that are broken or they were too thick. I'm going to get a little piece right here. That's kind of terrifying. You can see that's kind of a creepy little head. And if you want to see if we get really close. Go ahead and move forward. Focus in. That's what happens. That's how brittle it is. That's why you fire it in the kiln. And it will cause it to get really hard. Like a mug or something. But we break all of this up. We mix it with water again. And it turns back into kind of like a clay slushy. And then that's what goes back into there. So we save all of it. So none of it's wasted. So even the parts that aren't good anymore. You know just break it up. And then mix it back in. And you can continue using it so nothing's wasted. So let's go ahead and go back. If you see these shells right here that we're moving through. Turn around real quick. We have these shells right over here. We have two of them stacked side by side. So we have ones on this side. And we have ones on this side over here. So that's pretty neat. Of course we have really large ones too. Like this. But I don't really want to take that one apart. It does look some of the bigger ones. But maybe not ones that are that big. So this one looks like maybe like a little girl with a dress. Now it's another cherub. The person we got this from, I believe she was pretty religious. So she had a lot of religious memorabilia in here. And we have a cute little bear right there. There's a little bow tie on. And then the back right here. We have some kind of cup. So this is a really nice cup. But look how dirty that is. Really dirty. There's a lot of silverfish that came with these guys. And they just started to clear out. But we had silverfish in here. And then the garage for a while. And actually probably still have to bug bomb it. Because there's just so many silverfish. It was crazy. And then all the way from here to the back. From the bottom all the way up here to the top. All full of molds. And I had to reinforce these shells too. Because there was some water damage in the back. As you can see, this is into the beam. But the actual ceiling is a little coming down right here. But it'll be fine. I'm not sure if I picked that one up yet. Let's see. Oh, this is a lid. Oh, look at all the dead spiders in there. Let's not zoom in on that. I don't want anybody with a ragged oboe again. I'm too freaked out. I'm going to grab that one next to it though. Maybe it has. Okay, check this out. So this is a thing for soap. Probably you put the soap on there. You put it on your bathroom counter. It's a soap dish. It's a soap dish, yeah. Soap dish. Yeah, what she said. And then some small ones too. This is like some kind of lid probably, or maybe a little jar. But yeah, there's like a ton of stuff in here. So this is just a really rough kind of view of some of the molds that we have in here and some of the things that we could create. We have a lot of tiny ones too. Like little tiny houses. And there's so many that we don't even have the pieces for yet. We need to find the matching pieces. But I really like these kind of things. Kind of terrifying. Looks like a little baby or something. And yeah, soft. This is a little small kind of tour of the molds. I want to be able to do another one kind of like this. Just like a general pickup and kind of check out, you know, like what we have kind of tour. Because a lot of people don't have things like this. Unless if we can find one right before we end, it has a date on it. So right here, this one says 2001. But we have ones that are really old in here, like really old. 40 plus years maybe even more. It's just finding the ones in here that actually have a date on them is kind of hard because they get worn out. 1980, right there. You can barely see it though, it's faded. But if you can see right there, focus in. And 1980. So on this right here is that little cup that I pulled out over there. So a lot of these are really old. But yeah, so there's so many in here. And even for me, this is kind of like an experience going through all these things. Because like I said, these are like one of a kind. And a lot of these don't exist anywhere else in the world unless, you know, people have been collecting these. So they're probably far and in between. And I think cleaning them up will be really fun. And I think I will do some videos about, you know, how to do that and whatnot. But for right now, I hope you guys enjoyed the vlog. I know that my last one got quite an amount of views. And I'm really happy about that. So if you enjoyed this, please share this with your friends. Don't forget to comment, rate, and subscribe. And a huge shout out to my patreons that you guys so much for your support. You guys are awesome. And as always, I'll see you in the next video.


Would you have picked these up or left them behind always a bolo? Check the inner pocket for cash when you find them. They almost always have money in them Hey bolo buddies, thanks for watching. All right in this video I'm gonna share with you some big money bolos and almost every single item in this video came from The Good Will Bins, which is where you pay By the pound if you have not been to the Good Will Bins you have got to check it out You can check out my Good Will Bins videos, but I also Want you to go over and check out our featured seller a Rural squirrel her Good Will Bins videos that is basically what her whole channel is about is thrifting at the Ben she brings you live footage from the Good Will Bins and She teaches at the same time. She is also so close to 10,000 subscribers. I Just featured the funky pickle thrifter. I'm not sure which video is gonna come out But both ladies are so close to 10,000 subscribers. So definitely go give them both a sub Totally different content, but both incredible YouTube channels. All right, so a rural squirrel Almost to 10,000 subscribers you guys go show her some love with a sub. Here is her eBay store Again lots and lots of great items a wide variety when you shop at the Ben's you never know what you're gonna find But she will pick up anything. She is not exclusive to just clothing She loves selling shoes and finds all kinds of amazing shoes at the Good Will Ben's But definitely sub her up check out those Ben's videos And I think you're gonna be pretty shocked at some of the amazing things that you can pull out of the Ben's I will have her link down below both her eBay store and her YouTube channel If you click on the heart that will save her as a seller You will get notifications when she posts new items and also if she runs a sale Let's get into these Bolo's Right So Let's talk about this She has been buying up quilts at the Ben's which I think is absolutely incredible that she found These amazing quilts that you're about to see at the Good Will Ben's Just incredible. I mean look at this. Look at it. Look at the detail. It's it's just beautiful And she put it on a bed. Oh my goodness. I just think this is such a great way to showcase your item I was saying I just did a video recently on quilts and I was like I'm too lazy to like wash it and put it on a bed and I'm usually too lazy to even buy the quilts and when I do buy them that we usually keep them because I love them so much. I Grew up with quilts from my grandmother That she had made so I just have this special like place in my heart where they just remind me of grandma And I want to keep them and my husband is kind of fond of them also So we have some nice quilts that were my grandmothers and also Some that I have picked up at garage sales that I just decided to keep instead of sell They're probably a Bolo, but that's okay. We're gonna hang on to them for now She says the quilts have become some of my favorite things each of these last three quilts came from the Denver Ben's where they have linen pricing at 69 cents a pound So I paid less than five for all of them three quilts and sold the vintage. Let's see. Let's do I've got them in the wrong order. So this one is a 1950s pink grandma's flower garden quilt for a best offer of 300 and it was a repeat buyer Here is this one that is just kind of a patchwork interesting patchwork looking at quilt It doesn't look as big. It's from the 1920s to 1940s. She says and it kind of has that horse Did you see the horse? So that's gonna catch the eye of maybe somebody special that is interested in horse Items and even the patchwork kind of has that country horse cowgirl theme to it So she sold this one for 149.99 and it was also a repeat buyer Look at the butterflies. Oh my goodness the Goodwill Ben's you guys The how did this make it to the Goodwill Ben's? That's all I want to know Just how how how how? Look at that. It's beautiful. It is absolutely awesome Alright, so I'm gonna just keep looking at all the pictures. It's so cool This one sold for a best offer of 170 which was also a repeat customer So she must have a lot of people that Come back and buy from her over and over again, especially her quilts Okay oboes Bridger trail hiking boots waterproof Shepra lined hike snow camp size seven and a half now This is a little different for her These are boots, but she did retail arbitrage on these she paid 24.99 and Took a best offer of a hundred dollars on these She said they came from a mom-and-pop sporting goods store in Arizona This is a vintage fallen hoodie sweater full zip up olive ivory striped jacket Medium and she used the word why 2k which is a fantastic word that People are searching for on eBay. So use it if you know, it's from that time frame. I Found this at the Denver Ben's and did a tick talk about it paid $1.69 for it I it felt like nice quality. So I did the comps and it blew me away I listed mine at the same price range and this was a full price sale of a hundred and thirty nine ninety nine So if you guys are on tick talk check out a rural squirrel over there as well Here is another quilt vintage 40s quilt block patch pattern hand quilted and Look at that This one is a cover let Bread bedspread. So I think it's like a topper. Maybe I'm not sure but it looks thinner This one came from the st. Louis Ben's a dollar twenty nine a pound there This made I'm sorry that made this quilt less than four dollars accepted a best offer of seventy to a repeat buyer Don Joy Defiance force point left knee brace size small This she said I always pick up these frame knee braces when I see them I had to wear one in volleyball and know how expensive they can be I accepted a really low best offer on this for 32 But it only cost me a dollar 69 a pound at the Ben's Yeah, and sometimes you just think maybe somebody really needs it Oh I'm sorry Onsen 100% Supa Mia Suppy ma Suppy Muah, I don't know what that SUP IMA some sort of cotton I guess is that the brand or is that a type of cotton? Let's see. It's a type of cotton Does anybody want to tell me how to pronounce that down in the comments like put the little Dashes and tell me how to say it because I have no idea what that even means But it's beautiful. It's kind of got that waffle look to it look at those towels Turkish towel set and She paid 69 cents a pound at the Denver Ben's because they are considered linens These waffle woven towels sold for a best offer of ninety eight dollars plus shipping for used towels That is incredible and definitely one you want to be on the lookout for there is the little tag This purse is a mid-century modern corrett brand purse came from the Jackson bill Ben's I think it's so funny that she knows where all of her items come from like she keeps track of which goodwill Ben's it comes from I absolutely love it and cost her a dollar 29 a pound She accepted a best offer of 85 on this I think probably that helps her keep track of what she paid for her items Because each goodwill Ben's has different pricing. I'm not sure if you're familiar with the goodwill Ben's and how that works, but My goodwill Ben's the last time I was there. I haven't been there for a while was $1.99 a pound For anything clothing or hard goods and then books. I think we're at 69 cents a pound So we don't have a linen rate at our goodwill Ben's so each goodwill Ben's location is different This is a Valerie Stevens evening 100% silk sequins jacket cardigan black gold and silver and She sold this for 4999 and it came from the goodwill Ben's This is a Tilly Durables hat. It's like a fishing hiking cap She said always a bolo check the inner pocket for cash when you find them. They almost always have money in them Ha, I love that tip was a great condition and sold for a best offer of 50. Where's this pocket at? Where's the pocket? Is it on the inside am I missing it? It's a hidden pocket. Maybe you Guys see a pocket. I Don't see a pocket. So look for a pocket if you see this brand. It's got to be here In her pocket. I can't find it. Do you guys see it? I want to know I want to know Okay, it's like a hidden pocket. I think or maybe it's on the outside It's on the outside Where is the pocket? Where's the pocket? Okay, I've spent way too much time on trying to find the pocket But that is a good-looking hat for sure and I would definitely keep the a pocket tip in mind This is a vintage Red Goose shoes advertising promotional cast iron and it says a still bank So it's like a piggy bank. I guess right still bank. Is that what that means? Let's see. Yeah. Yeah. I want to see the bottom. No Yeah So I think you put the money in and then you like open it like just crank it apart. Do you think I don't know Just recently sold best offer $42 now that was probably I don't know if her bins is always by weight Or for heavy items they give them a discount But that one could have cost her a little more vintage peach negligee body suit panties Outfit from the 70s or 80s. It is fancy. She got it at the bins took a best offer of sick I'm sorry 26 for that Look at this 1950s handy can steel flat top black cherry soda pop can What look at the graphics it's pretty bad condition guys and Yeah, black cherry soda Have you ever seen such a thing? 1950s Steel pop can best offer $50 and she got it at the bins so that probably weighed next to nothing She probably had less than a dollar in that would you have picked it up or left it behind? let me know in the comments and This are ultra men's sneakers running shoes Goodwill bins took a best offer of 20 on those and those are you guys she got those out of the Goodwill bins Look how good they look Those are in great condition. Oh, I lied They do have a little bit of an issue right there, but overall they look really nice. Maybe it's the vibrant blue Declab or to decala decalib decalib. I'm not sure how to say that, but that's the brand D. E. K. A. L. B Why did they have to have such hard names? It is an orange corn Detastaline hat from snap back cap sun flap mesh Screen so it's got a screen over it. Check it out. It's even a little dirty What do you guys think? Interesting, huh? I think came from Jacksonville, Illinois bins and sold for a best offer of $40 this is a Chanel quilted black eyeglass sunglasses case and She said came from a yard sale in Minnesota while visiting reseller friends up there I'd flip that 365 vintage hunter and Mick flippin ship It was in a free box with a few other cases sold for a best offer of 50 So those were the other youtubers that she was visiting definitely check out their channels as well And she sold this for a best offer of 50 Little pet shop tiny or I'm sorry teeniest tiniest These are itty-bitties kind of like a poly pocket, you know the real little ones But this is pet shop She said I let's see a goodwill bins fine. Honestly can't remember which location I remember shoving them in my cart, but I don't know remember where I was sold for a best offer of 45 I don't know how you could remember where you were ever.  She said the deer head retailed for over 200 She said I told I sold it to a repeat buyer for 48 99 how on earth did a paper mache item make it through the Goodwill Bins? That is crazy Did she hang it on her wall? Or did she just make it look like that? It looks like she was hold on a second. Let me go back here It looks like it's hanging on her wall. How did she did she actually hang it on her wall? Or is that like some fancy? Some fancy editing skills that might be some fancy editing skills. I think it is. Oh my goodness That's awesome. All right. Here we go oboes juniper trail hiking boots. These were also from that same Retail arbitrage mom-and-pop shop 24 99 each sold this pair for 65 plus shipping Ralph Lauren Denham and supply varsity Letterman jacket navy and white it is wool. She got this at the Goodwill Bins in Bridgetown After the let's do lunch podcast meetup and sold it for a best offer of 80 dollars. Yes, so a rural squirrel and Angie resells they have a Channel together called let's do lunch podcast. So definitely check that out if you have not ever seen it This is a smart wool Murano Is that my no merino merino and I don't know how to say it so anyway It's okay. It's alright. It's an ultralight vest full zip run hike sunset coral the MSRP on this was a hundred dollars and She sold this for a best offer of 45 dollars and she got it at the bends Edwardian Edwardian era women's shoes French heels dress white size six narrow and They are no longer white guys. These are old old old. I don't know would they be considered? Probably antique right Look how beat up. They are you guys would you have picked these up or left them behind? Let me know Down in the comments. I mean they are they're really old and They're collectible, but she got them at the st. Louis downtown bends for a best offer I'm sorry sold them for a best offer of 50 50 dollars for these. Look at them. Look at the damage $50 do things sell that are damaged. Yes all the time all the time It's just like when I talk about People scrapping jewelry, you know, so many things go to scrap or to the dump that people collect and love Kiplings large laptop backpack blue flowered bag with pockets and monkey fob and This one sold for a full asking price of $59.99 And the buyer paid shipping vintage 80s 90s bike shorts bright colorway pattern iconic 80s 90s these came from the Denver bends and she took an offer of $55 plus shipping on those Alright you guys that was a rural squirrel and again so close to 10,000 subscribers go show her some love 189 videos that's it and she has that many subscribers So that tells you she is doing something right go show her some love you guys with a sub and Check out her eBay store as well. Those items are linked down below. Thank you so much for being here Thank you for watching be sure to like comment subscribe and share and thanks for watching


Hello, this is Laptop Media, and today, we will show you how to open the Lenovo Idea Pad Gaming 3i 2022 and what's inside of it. Getting inside of this laptop is a bit unorthodox. First, you need to undo two Philips head screws located on the back and pull the plastic shroud. After that, undo all 12 Philips head screws from the bottom panel, and pry it with a plastic tool. Unplug from the back and work your way to the front of the laptop. Please remember, it would be of great help to us if you just hit the like button and subscribe to our channel. That would motivate us to make even more and better videos for you. Now that we're inside, we see a 45 watt hours battery pack. It lasts for about 3 hours and 30 minutes of either web browsing or video playback, which is not a great result. If you want to take it out, unplug the battery connector from the motherboard, and undo all four Philips head screws that keep it in place. Remove the metal shroud to access the two Sodom slots. They work with DDR4 modules and we are pretty confident that you can fit up to 64 gigabytes in total. Storage wise, there are 2M.2 PCIe X4 slots. The 42mm 1 supports Gen4 drives, while the 80mm 1 is limited to Gen3 SSDs. The cooling setup comprises two heat pipes which are shared between the CPU and GPU, a third one meant for the processor, a total of four heat sinks, and two fans. Furthermore, there are heat spreaders placed on top of the VRMs and the graphics memory. If you'd like to see more disassembly videos of the latest notebooks, we would highly appreciate if you hit the like button and subscribe to the channel.


So now that we've gone through the concepts of entropy and free energy, we're gonna be able to revisit amino acids but study them on a higher level. Let me start by drawing one for you. We have the C alpha, we have the NH2 group, and we have a COOH group. Then I have a hydrogen that I'm drawing this way to indicate that it's going out front and the R group that I'm drawing that way to indicate that it's going in the back. There are two things that I wanna share with you here that's important. This would be the way that amino acid might look like if we had it as a salt or something but it's not gonna be the way it looks like in solvent. What's gonna happen in solvent is that this hydrogen will be released to water. So we're gonna end up with a charge group here, COOH minus, and the NH2 group, in contrast that will add a hydrogen. So we're gonna end up with a plus charge here. So unless we have a charge on the amino acid side chain, the entire amino acid will still have neutral charge but we have one ion here that's positive and one ion here that's negative. There is a special name for that. It's called the so-called zwitter ion or they have zwitter ionic properties. That's true for all amino acids in solvent at neutral pH. At other pHs it might be different. The other thing worth considering is that this has a tetrahedral shape with the alpha carbon in the middle and then we have four different groups. If you remember your upper secondary schools chemistry, that's gonna be a so-called chiral center meaning that I can draw the mirror image of that molecule. NH3 plus and then COO minus are in the back and H in the front. There is no way to turn the left molecule here into the right one by mirror rotation. If you don't believe that, go ahead and build them with your small custom molecule building site, block and then see if you can take this one and rotate it into that one, it's impossible. The only way I can do that is by swapping two bonds on the central alpha carbon here. Now, knowing free energy and kinetics, you should say that it's only a matter of probability, right? At some point we will be able to cross that barrier. Technically that's true but for amino acids and chemistry and things that have to do with life science, these timescales are so long that it's effectively infinity. They can't take that amino acid and move to that one. In chemistry, you might remember that we occasionally use names as S and R enantiomers to refer to molecules that have the same chemical composition but different chirality. All the physical properties of these molecules are the same, apart from the fact that they tend to rotate polarized lines in opposite directions and that's usually the way we tell them apart. In chemistry, it's quite different though because remember, these are gonna be interacting with tons of other proteins, large molecules such as enzymes breaking down meat into individual amino acids in my stomach and the potentially antibodies interacting with viruses and everything. And if the fundamental molecule here has a property that has a built-in handedness, that's gonna pass on to the entire proteins. So a certain type of molecule here, if you use that to build an entire protein, will not necessarily be compatible with a protein that can bind the other form. And that is quite different from physics and even normal chemistry. The problem with that S and R enantiomers, there are very strict rules how to name them but when it comes to amino acid, if we strictly follow those rules, that would mean that some of the amino acids are S forms and other are R forms while in practice they all look like this. That has to do with naming rules and sulfur, for instance. What we do instead is that we always stick to definition and then we say that these are L amino acids and those are D amino acids. If we do that, it's gonna turn out that every single amino acid we're gonna stumble upon under normal circumstances, at least in nature, are L amino acids. My entire body is built using L amino acids. That again, are fundamentally slightly incompatible with D amino acids. My enzymes would likely not break down as an entire protein built from D amino acids. That's gonna be important later. Remember this handedness, it will come back but now I think it's time to take a look specifically at the side chains and how they create the diversity.


So, now what I want to do is that I want to see that if I introduce now diffusion, if I now say that these neighboring cells can all talk to each other by diffusing from one cell to the next, can I destabilize the steady state?  So, again if I now again write that equation in the matrix form. So, what I have is del w del t right is gamma times a the stability matrix times w ok a being whatever it was plus let me write down since I am writing it in matrix form plus let me write down a diffusion matrix times Laplacian w ok. And what is this diffusion matrix then? The diffusion matrix is simply 1 0 0 small d.  So, I have just written it in a matrix form nothing else. So, what I want to do is I want to look at this diffusion driven instability if I can generate a diffusion driven instability of this homogeneous steady state that I have written down. So, what do I do? I can start off by saying that let me define these space let me just define the spatial eigenvalue problem which is is equal to some k square times w with the same boundary conditions that n dot grad w. So, let me define. So, what am I doing basically just to summarize. So, I want to solve this equation what is the standard thing to do maybe I do a separation of variables right. So, let me say that I will write this let me say this is a small w I will write this as some capital W of space times some capital T of time right. I will plug it over here into I will plug it into this equation then I will get one term which depends only on the time terms I will get one term which depends only on the space terms right. What will the space term will look like? It will look like something as Laplacian of this capital W is equal to some k square times this w. So, you can find out the eigen let us say that w k are the eigen vectors of this equation. So, let me say eigen w k are the eigen functions of this equation satisfying this boundary condition whatever is the boundary condition and then for this actual w that I am interested in you can construct a general solution. Let me write a small w you can construct a general solution by doing a linear superposition of all of these eigen functions right. So, I can do a linear superposition of this w k of r the time part will we look like e to the power of some lambda t times some coefficients c k some lower all wave vectors k right. Is this clear? You do a separation of variables you get a set of basis functions from the expansion from this equation of this special part which are these w k which obey whatever boundary conditions that you have which means that you will have some allowed wave vectors right. For example, if you are doing in 1 d your eigen functions would be some cos let us say n pi x by l or sin n pi x by l right and your allowed wave vectors would be n pi over l right for various values of pi. The time part the time equation will look like something like del t del t is equal to some lambda t right if you if you use this separation of variables which will have some solution like e to the power of lambda t. The general solution you construct by doing a linear superposition of all of these allowed solutions and then you have to find out these coefficients the c k's by matching whatever initial condition is given right this u r comma 0 and v r comma 0. So, now, I take this I take this general solution and I put it back into this equation that I want to solve del w del t. So, what does this give me del w del t? So, the left hand side will del del t will bring me down a lambda a lambda times a w right. On the right hand side I have this times this is the capital W this make it clear. The left hand side on the right hand side I have gamma times the stability matrix times this w of k plus d. So, ideally you get this with a summation. So, if this holds true for every value of k then you know that the your original differential equation is going to be satisfied. What is this Laplacian of this capital W k that is nothing, but minus k square w because that is how this is defined right this is the. So, I can just write this is gamma times a times W k minus d k square W k. The Laplacian of W k is k square of W k these are the Eigen functions of this equation. So, I just use that. So, now what I need to do because I want to now evaluate these lambdas because these will now tell me that in the presence of diffusion how does this how do perturbations from this steady state behavior right. So, again I will solve the characteristic equation and the characteristic equation in this case is. So, again the characteristic equation in this case is lambda times i the identity matrix minus gamma times a the stability matrix plus d times k square this is equal to 0 lambda times i minus gamma times a plus d times k square ok. So, now I can write this down I know what all of this is.   So, this is now my two new equation that I have to solve I know what is my a I know what is my d I know what is the identity matrix I can just write this down and again I now need to solve it for lambda ok. So, let me write this down.   So, right I have not written the last term. So, what is that minus gamma square f v g u ok all right.   It is a horribly complicated well not horrible. So, it is a slightly complicated equation, but provided I have done everything correctly this is what you get. So, this is the equation for lambda again a quadratic equation, but at least it is a quadratic equation. So, I can just solve it.  So, I am not writing it out again minus 4 h of k square. So, it is just to save me writing this let me call this entire thing as h, h as a function of k square ok divided by 2. So, what I note is that in the absence of diffusion the lambdas are less than 0 or at least the real parts of lambdas are less than 0. Accents of diffusion basically means that k square is equal to 0 right. So, I know that the real parts of lambdas lambda at k square equal to 0 these are less than 0. So, because it is a stable steady state. What I want to find out is that is there any value of this wave number k such that this real part of lambda as a function of k this becomes greater than 0 right. If I can make this eigenvalue positive at least one of these eigenvalues positive then I can destabilize what was previously a stable steady state and I have this possibility of patterns arising ok. So, how can I do this? So, maybe let me see I can have this quantity itself whatever is in the bracket if this was positive right then this will come with a negative sign. So, then I would have at least one positive root ok. So, I could say that well what is the possible. So, if I want this thing that real part of lambda is greater than 0 if if this thing k square into 1 plus d minus gamma into f u plus g v if this thing was greater than 0 then I can have this that real part of lambda is greater than 0 or I have another possibility which is that if even if this was positive if if let us say this was positive, but h of k square is a negative quantity then this negative negative would become positive I would in this negative term I would get something which was ultimately negative right. So, another possibility is that this h of k square this term is less than 0. So, let us look at these possibilities let us look at this first one k square is the wave vector yes which one is what do I want that should be less than 0. So, if this is what I want remember k square is the wave vector. So, that is positive d is the ratio of the diffusion coefficients. So, that is positive. So, this is a positive thing f u plus g v I know is negative because that was a condition of my stable steady state which means that minus of a negative quantity is always positive. So, this is positive this is positive therefore, this cannot be true this can never be negative ok. So, this condition is out of these two possibilities this first one is never possible. So, all I am left with is this term that h of k square is less than 0 for some k for some for some k it may not be for all wave vectors, but at least for some k this should be less than 0 ok. Yes provided you have a stable steady state for the reaction diffusion if you did not then yes of course, I mean if this was if this was positive then in general yes, but then you would not even see a stable steady state even in the absence of diffusion. Yes, yes of course. So, for example, if I had an equation if I had a chemical reaction like this I have two species a and b let us say they both up regulate themselves and they mutually down regulate each other ok. So, neither is a clear activator or inhibitor right. Remember last day we were talking about a pattern in the equation that I saw in that we looked at where a activated itself and activated b whereas, b auto inhibited itself and inhibited a right.  So, therefore, this would not be true and this is a very common pattern you will see in lot of in lot of genetic networks when you talk about transcription factors mutually interacting this is a very common motive that you come across. I mean simplified no motive is just two chemical reactions, but roughly yes such motifs are possible. So, what is h of k square oh I have rubbed it off sorry let me write down one small what is h of k square.  So, this was my h k square and I need this to be less than 0 in order to have some wave vector where this lambdas are positive ok. So, in order to destabilize the system ok. So, now, let us look at this d k to the power of 4 is always positive ok.  So, therefore, this term is also necessarily always positive. So, the only way you can have and this comes with the negative sign remember. So, the only way you can have this to be less than 0 is if this term is greater than 0 right. So, that the negative sign will make it less than 0.   Now, let us think about what that means. So, I have this condition now d f u plus g v is greater than 0. Can you have f u and g v both to be of the same sign? Yes. Can you have f u and g v both to be of the same sign?    So, a is my activator and b is my inhibitor. So, which says that and this ratio being greater than 1 says that whatever is my inhibiting species that my inhibitor must diffuse faster than the activator faster than the activator. If I had reversed this if I had said that f u was less than 0 and g v was greater than 0 then I would have the less than 1 which again physically would mean the same thing that whichever was the inhibitor species that then in that case in this case b is the inhibitor if I took the other assumption then a would be the inhibitor and the what I this would say is that a would then need to diffuse faster. So, this inhibitor must always diffuse faster than the activator that continues to hold true ok. So, let me write this condition here as well the d f u plus g v that is greater than 0. So, this was 1, this was 2, this is 3 ok, but ok is there anything else? Even if I had said that this is greater than 0 I need this to be greater than 0 in order for this h of k square to be negative. So, it is a necessary condition it is not sufficient right because to get what to be sufficient this term d f u must overcome this this and this the contribute the positive contributions of this and this. So, this is a necessary condition, but in order for it to be sufficient condition. So, let me get rid of this anyway this condition does not matter. What I need to say is that let us say that I have this h let us say I plotted this h as a function of k square right some arbitrary whatever depending on what f u is some curve h of k square. What I need is that at least yes yes the first condition is still satisfied, but let us say that both are negative right. Can you ever get f a positive number into a negative plus another negative number giving greater than 0 will you cannot destabilize the steady state. So, that is what we are looking for right. So, in that case you cannot destabilize the steady state. So, a thing like this a b which is I think what you are saying that both sort of repress themselves it can be a valid chemical reaction, but it will never have a pattern in this sense. In order to have a pattern you need this that if one is activating itself the other needs to replace itself. So, that is what 1 and 3 together imply. So, coming back to this. So, I need this h k square to be less than 0 for some value of k at least. So, what I can say is that this is a necessary condition for it to be sufficient what I can look at is I can look at the minimum point of this h curve ok wherever this h of k square has a minimum. And if at the minimum point let us say this is my minimum point if at the minimum point the value is less than 0 then I am guaranteed that at least 1 k exists where I have this h of k square less than 0 there might be more. So, the curve might look something like this in which case this whole range of k's are k's that will destabilize the steady state, but at least if I have 1 then at least I am guaranteed that I have 1 wave vector that will be destabilized. Yes, k square the middle term has a k square. Thank you. Yes, I should not draw a random graph is that what you are saying? I can still draw a nice draw something like this yeah I should not draw a completely random graph ok. So, then what I want to do is that in order to have a sufficient condition that this h of k square is less than 0 what I will find out is that I will find out the minimum of this h k square curve. So, I will do del h del k square equal to 0 ok and find out. So, this I will solve this this will give me some value k m where this curve h of k square has a minimum. I will find out the value of h at this k m point and see whether that is less than 0 on so del h del k square. So, del h del k square is 2 d k square minus gamma d f u plus g v is equal to 0 which means that this k m square is equal to gamma into d f u plus g v divided by 2 d and the value of h at this point the value of h at this k m square the value of h at this wave vector at this minimum wave vector is d into k to the power 4.  So, this is actually the same term this d and so this is 4 d and that is 2 d so minus 4 d.  So, this is the value of this h this quantity h at this minimum point. So, what this so for this to be negative what this means then is that this term must be greater than this determinant of a obviously.  If this condition is also satisfied then at least I am guaranteed to have one wave vector at least one it can be more I am guaranteed to have at least one wave vector for which you can destabilize this stable steady state and of course, the critical wave vector will come when you equate this to 0 right. So, basically if I think about this h of k square plot. So, if I think about this h of k square plot h of k square versus k square plot some h looks like this for some parameter values. Let me say I am just tuning the diffusion coefficient for example, then at some value it will just hit the axis and then for some values like this. So, this is let us say some critical ratio of the diffusions this is some d greater than dc. So, act the critical whatever parameter let us say the diffusion ratio of diffusion coefficients you have this one single wave vector that will destabilize the steady state. If you increase this ratio even more you have this whole spread of wave vectors from this k 1 to this k 2 for which this stable steady state will become destabilized. So, correspondingly if you were to plot this lambda the actual eigenvalue you would get something like this and then something like this and then something like this. At the critical point there would be one wave vector for which this lambda is positive or just positive and then there will be a range of wave vectors. So, this is the fourth condition that I now have for pattern formation which is that whatever I wrote down. So, d fu plus gv whole square minus 4d into the determinant of a that must be greater than 0. So, provided so, these two conditions 1 and 2 are required to say that you have in fact, a stable steady state even in the absence of diffusion. Conditions 3 and 4 are what is required for this diffusion in order to destabilize this steady state and give rise to some spatially extended patterns. So, these four conditions and this is quite generic I have not recourse to any particular form of f and g or whatever. So, you can take any chemical kinetics that you wish and think about whether you can destabilize the steady state or what this says is that in order to form a pattern at least by this sort of a reaction diffusion mechanism you need to have one of the species which is an activator and another species which is an inhibitor and then something about here. So, you can try to construct patterns and see that what sort of chemical networks even if they are going into the rates and so on what sort of chemical what patterns of these chemical reactions will can give rise in principle to pattern formation through this reaction diffusion mechanism. For example, like you said if both of these were inhibiting auto inhibiting then you would never have a pattern.


The president's budget proposal eliminates public media funding. Learn how you can help at lptv.org slash action. After an extensive search, the board of directors for the Kuyuna Regional Medical Center selected its new chief executive officer earlier this week. Kyle Bauer was selected to lead the CRMC staff of $950 and manage a budget of $115 million. Bauer has served as the interim CEO for the past six months after previously serving as chief financial officer for the medical center. Bauer has also worked at hospitals and health clinics in Phoenix, Arizona and the Twin Cities. As for goals as CEO, Bauer hopes to keep medical care costs down and improve quality amid the talk in Washington regarding health care. We have a product that is becoming unaffordable for large segments of our population and we have to address that. And so our goal here is to reduce cost but also increase quality and enhance the customer's experience. Kuyuna Regional Medical Center is a non-profit organization with 40,000 physicians and specialists, 60 healthcare providers and 60 volunteers. If you've enjoyed this segment of Lakeland News, please consider making a tax-deductible contribution to Lakeland Public Television.


What's up y'all make sure y'all check out boss talk 101 with e ceo and mr. Jamaica she boy low DZ man Boss talk 101. See you need hustle. You feel me? Introduction when I when I turn the camera on him you niggas gonna really realize who it is man my guys here man One of my closest friends man low DZ is in the building. What's going on, baby? Man just just happy to have you back on the on the show. We've been doing a lot man We've been working over here. You've been seeing us. Yeah, I see you working man Commend you man since the last time I was here to to where it said now man. Yeah, I really done grown man Man, it's cause you salute you man.  You always there for me, bro So let's be real man Like I know I read a lot of times some of this stuff is new to me be honest with you and it that Lifeline right like I see like all of these ask him about it and you always pick up the phone So I appreciate that man. Oh god, bro. Yeah, the small contributions help. Yeah, like like like crazy. Yeah, we appreciate everything Yeah, you know, that's all the way we're gonna pull this thing together to see to keep us thriving and keep it That's having folks in front of these microphones man. Yeah, let's take together exactly Everybody have so many questions. I don't know what you don't want me to ask you But I'm first gonna ask you about yellow like is what's up with yellow like is he how's he doing? You know has how's how's there I seen that song came out yesterday. Yeah, I posted it on my page. I did too So just I've been seeing you like I said mean you talk briefly about the school just tell me a little bit about what was going on It's not necessarily a school. It's the foundation. I'm starting it. I was partnering with the school. Okay District, but what inspired it before you get into it. I'm glad you asked it. What inspired it was my son He he was pretty like a mediocre student first, you know, I'm saying and then No, he started taking these classes and it made him get into production and things of that nature It's great start skyrocket You know, everybody don't like to listen to my interviews all the time because I don't like you know, I ain't clickbait Listen listen people that's gonna hear what I'm saying over here on what I'm about That's the people that need to hear exactly because you know at the end of the day What we trying to do is make this shit last as long as it can Man, it just this stuff been feeding families around here. They've been changing lives. This music scene needs to Needs to grow more. You know saying it right now. We just turned it down. You know, I'm saying and I ain't with none of that Yeah, yeah, we gotta keep on eating right here. And the only way that's gonna happen is we thrive me say You ain't gonna have nobody behind this microphone As long as long as keep up people keep on doing what they doing They pull down everybody from Dallas to forward man. They pull out anybody that make it up to her.  Yeah, you start taking off and you got you know Me is in the bank and really getting the word.  I don't know what they talk about.  I get it.  I Just crazy it's just like the amount of respect that kids and even like it ain't even just that with people man It's like I saw it even like to say with rappers They don't respect the rappers that came before them. No, you know, I'm saying they're like I didn't need a nigga They're rapping they need to get out the way, you know, I said, you know, no, man It was like it was a time where you know, I necessarily was old enough to be listening to KRS one and rock him But I went did my homework and I respect him because they came for me And that was the niggas that everybody that my cousin was told by Yeah, no, I'm saying so I don't know that just the homework a nigga went back and did because I feel like You know, no, no, I agree and especially I always say specially So it was just like, you know, there's the respect I had for the generation that came before me. That's all We had a lack of bed now, I'm still saying Everybody just started out cool Just like how they won't go just they won't you know That's another Okay, it's up to people like you You did your generation has been between so we got to keep the bridge going We got to educate the people that's to put in, you know, in our presence You know, I want them to know about those people that I had to go do my homework on too That's how cool, but I would rather them even know about the people that just came right before them Yeah Yeah, you know, if you want to go back that far you can but just just do your homework on the people that came before you what scar faces in yeah And in the will of D's and a body of JG's and JZ and not all of them man Just do your homework on the people that came before you and see how they did it If you really just look at J and just study him man, you can actually be a better artist I'm just saying how he moved. Yeah He the first one that really just kind of didn't curve out with the label. Yeah. Yeah, I want to sign me I'm a sign myself. He the first one that did that. You tell my Jay. Yeah, well, you got to say Dame Dash when you say that He put up he was a third of it That's a whole collective movement, but I'm saying bro. He did sign the self Everybody be trying to discredit I'm just saying it's a group Yeah, we on boss talk