Translating sustainability principles into actionable items is a difficult challenge for the industry because they need to meet the standards of extended producer responsibility. In other words, it means that they have to analyze how the sustainable principles apply across the length of the supply chain. They need to do this without compromising the quality of the product, the promptness at which it will have to be delivered. They cannot compromise profitability and they cannot compromise the original characteristics of the products of services. In addition to that, standards need to be applied across different processes, across different types of industries and across cultural and geopolitical boundaries. Interestingly enough, the greening of the supply chain has been driven by market pressures. As consumers begin to be more aware of potential risks to their health, potential risks to the health of those who are providing and producing the services and goods that they are accessing to, they become more concerned about issues of fairness, about issues of trade equity, et cetera, et cetera. And they also become a lot more concerned about the degradation of the commons that occurs not only locally, but across national boundaries. And this becomes even more of a challenge when you're moving the taxing of the commons from places that are very well regulated, where we understand a number of the limitations that a particular chain will have to places in developing countries that may have looser environmental regulations. Consequently, the drivers for the industry to make these changes to green the supply chain are a response to attempting to maintain a customer base, they're a response to becoming more efficient in the use of resources, and this is of particular interest when it comes to waste management, because one of the major challenges that we face when we recycle is the closing of the recycling loop. And the incremental drive in using recycled goods as opposed to virgin resources might finally effectively close that particular loop. As you may imagine, there's significant overlap between the total of principles of sustainable engineering that we covered as part of your module one, and the best practices for greening the supply chain. They both come to address some of the same challenges and come from some of the same motivation. However, in this module, we're not just tackling environmental system concerns in a vacuum. We're doing an analysis that includes the economic drivers that will lead us to the use of limited environmental resources.


Hello everyone, welcome to the next lecture, we are discussing the topic of application of remote sensing in water resource management. In the last lecture, we just briefly defined what water resource management is and we also saw some of the tasks that or some of the sub domains that is encompassed within the term water resource management and also we have seen what all the different variables required in water resource management that can be retrieved from remote sensing. We started discussing about evapotranspiration and we defined what evapotranspiration is. We saw vegetation index based models, Penman-Mortetor-Pricetailer based models for it and we also discussed the surface energy balance equation. So today we will continue with the topic, we will see how the surface energy balance equation is solved using remote sensing data sets. So this is what we have seen in the last lecture like what surface energy balance equation is. So to simply write Rn like to simply write what normally we will do in remote sensing is, we are interested in getting this e term, e term is the evapotranspiration that is what we need to get it. So we will try to estimate each of the radiation components separately, we call it as Rn. So this Rn minus this Q, some people will call it Q, some people will call it G. So Rn minus Q is equal to H plus E. So this implies E is equal to Rn minus Q minus H. So this is like the generic representation of evapotranspiration using the surface energy balance equation. We try to estimate all other variables that are involved in the surface energy balance equation and estimate ET as a residual. This is one such way, we will see this in the next slide. Say as I told here the evapotranspiration ET is estimated as residual of the surface energy balance equation, lambda E is nothing but the ET term or E term whatever we can call is equal to Rn minus H minus G where G is nothing but the Q we have seen in the previous slide. Other than this we can also estimate evapotranspiration as a fraction of Rn minus G that is ET is equal to some fraction of Rn minus G. This is also possible to directly estimate. So if we can just get only this and if we can estimate this fraction somehow then ET can be estimated as a product of these two. Here we would not be worried about H, we would not be trying to calculate H in this method but we will try to estimate this the fraction of energy that is going out as evapotranspiration into Rn minus G. So we call this as EF evaporative fraction. If we estimate this EF using this Rn minus G we can estimate ET. So this is one way or the most common way is estimate ET as the residual of the surface energy balance equation. So this simple equation is now being expanded here. So that is this Rn term is now being expanded into it is like full-fledged thing. Here you can see that albedo term is there LST that is like the surface temperature what we get from thermal infrared remote sensing that is there and surface emissivity. So all these things are that can be or the variables that can be retrieved from remote sensing. We estimate LST or we retrieve LST, we retrieve albedo, we estimate surface emissivity and so on. Using this thing in addition to incoming solar radiation we can estimate this Rn. So this Rn is equal to this. Similarly H also can be estimated using like the heat transfer equation. So I am not going to expand in detail how this equation came in, what all the different assumptions behind it, how to estimate it and all that will be like that those things are out of the scope of this particular course. But there also to estimate H the land surface temperature that we estimate or that we retrieve from the thermal infrared remote sensing data sets is needed. G heat conduction again we will be needing LST NDVA like vegetation indices and so on. So effectively combining the surface temperature, albedo, vegetation indices, surface emissivity and all the variables we try to estimate each and every component of the surface energy balance equation and estimate it. So this is the basic principle of how remote sensing is useful for getting evapotranspiration using the surface energy balance principle. Again this is like a equation, physical equation that is present. We try to solve this equation by getting different inputs from remote sensing. So some of the common models that a few of you might have heard is Siebel. This is based on surface energy balance based model. Another close model is called metric, they are related with each other. There is another model called alexi-dyslexi family, there are like another model called sparse. There are plenty of different remote sensing based models, they are like even very good review papers on the topic. So basically this itself is like a very big field on its own to cover. But just as kind of like a summary, essentially surface energy balance based models require land surface temperature data, without that particular data we will not be in a position to solve surface energy balance based equation. So that is one thing, primary thing. So this is the primary backbone of solving this equation from remote sensing is thermal infrared remote sensing. In addition to this, we will also be needing vegetation variables like vegetation indices, emissivity, albedo and so on. So that is to deal with the basic concepts about how to retrieve aeroportranspiration from remote sensing data sets. So there are like plenty of global data sets available. Say MODIS provides its own aeroportranspiration product, there is another product called GLEAN which is which again provides like global aeroportranspiration, then the PTGPL product is there. So there are like plenty of available ET products at global scale which we can download and use for our applications, even like a simple internet based search will take us to the product page of these different products. So next we are going to talk about soil moisture and how remote sensing is helpful in retrieving soil moisture. So to simply define soil moisture is the amount of water contained within a soil whether it is a volume. Say you have a box, it contains some amount of soil. Say what is the amount of water like the volume based, what is the volume of the soil and what is the volume of water, if you calculate it that is one way of representing soil moisture or if you calculate this box contains this much mass of soil and this much mass of water, if you calculate the mass that is another way of representing soil moisture. That is we can these 2 definitions are basically called volumetric definition or gravimetric definition. Volumetric definition is volume of water divided by volume of soil. So gravimetric is mass of water divided by mass of soil that is essentially if you take like a small box filled with soil it will be kind of like a, it would not fill all the entire thing, there will be some gaps in between it and when you pour water over it the gaps will be filled. So we can either estimate the amount of water with respect to the volume or with respect to the mass content. So calculation of or estimation of soil moisture is of paramount importance to agricultural applications, drought monitoring, meteorological applications and so on. It is one of like the major, we call it a state variable. So it is like one of the important variables in the earth system which is required for several applications. So estimating the soil moisture again is not like a straightforward task. Normally we would not observe soil moisture also directly, we have to do some other measurements and get soil moisture out of it. So the soil moisture can actually be defined as a surface or skin soil moisture like only at the surface. Say you are just touching the surface without going in, how wet you feel? It is an indicator of surface soil moisture like the skin soil moisture you can call or you can estimate soil moisture at the near surface with respect to say 0 to 5 centimeter or 0 to 10 centimeter depth. Some small depth of soil say this much 5 centimeter depth, 10 centimeter depth and so on. So that we call near surface soil moisture or we can also estimate root zone soil moisture. Say some vegetation, they will extend their roots under the soil. It may extend to some distance say half a meter, 1 meter or some large trees will have tens of meters of roots. So those vegetation can actually extract water from soil up to the depth to which they have the roots basically. So we can define the soil moisture present within the root zone or even we can define soil moisture all the way up to the groundwater table. Say this is like the groundwater table. So after this all the pores or all the gaps in the solid earth will be filled with water, we call it a saturated zone. Above this zone there will be a mix of soil air gap plus water filling within the gap. So we can also somehow estimate soil available, water available in the soil column all the way up to this saturated zone. So that is also possible. So for different applications people will normally define soil moisture in different sense. Surface soil moisture, near surface soil moisture or root zone, battle zone etc. So the depth of measurement basically will vary and as the depth changes the estimates that we make also changes because the surface may appear really wet. But the while water may not have percolated down say after say 15 centimeters or 20 centimeters. So if someone else calculates the roots on soil moisture up to 1 meter depth the estimate will be totally different. Say to conceptually still let us say this is depth, this is the surface ok. Let us say this is soil moisture. Let us say in the surface you have just watered the soil. So the soil moisture will be very high. Normally soil moisture with respect to 0 to 1 ok, some sense. So as soon as you wetted the soil using some water the soil, the soil moisture will suddenly increase in the surface then slowly it will go down. Say this is like the root zone say 1 meter depth. If I calculate the soil moisture within the first 5 centimeters I may calculate it say 0.9 very close to 1. It is just watered. If someone else calculates soil moisture up to 1 meter depth it can be something like this. It can be kind of like an average which will tell the person may calculate let us say 0.5. So there is like a huge difference. The depth of measurement is going to change the way how soil is distributed within the, how moisture is distributed within the soil column. So this is really important to understand. So how remote sensing is helpful to retrieve soil moisture? Almost the different kinds of remote sensing that we have seen optical, thermal, active microwave, passive microwave, combination of any of these are all helpful to retrieve soil moisture in some sense we will quickly see certain examples of how these different kinds of remote sensing is helpful to retrieve soil moisture. First we will start with the optical remote sensing for soil moisture. Optical here I mean using surface reflectance in the visible NIR and STI air bands. So we have already seen that whenever water is added to soil its reflectance will go down and the reflectance is more pronounced in SWIR bands. So this we have discussed it in detail when we discussed about the spectral reflectance characteristics of different land surface features. So reflectance is a indicator or soil moisture is one of the primary control of reflectance as soil moisture increases reflectance will decrease. Using this property people have estimated the moisture content of the surface in some studies. But almost all the studies that try to relate the change in surface reflectance with soil moisture or empirical in nature, empirical in the sense they are either done in like lab controlled lab conditions or like in field conditions which are well known normally. Because reflectance does not only depend on soil moisture but also depend on several other factors say for soil itself if you take the soil organic content will reduce a reflectance. So when there is organic soil and there is like a normal soil inorganic soil let us assume both of them have same moisture content but organic soil may still appear darker because of its organic content. So these are all some other variables which will change the reflectance of the soil. So there is no directly we know that soil moisture will influence reflectance but there is no direct relationship that we can establish and that we can apply to various regions to estimate soil moisture. But there are some studies which used this reflectance with soil moisture to do it but not often used. However recently there was like a model developed called Optram optical crepezoidal model which showed that with using the reflectance data it is possible to retrieve soil moisture. So there the authors have used what is known as the STR that is SWER transform reflectance that is we have the reflectance surface reflectance data in SWER band typically like lying between say 1.4 to say 2.5 micrometer within this range if we measure the surface reflectance then transform the surface reflectance into like some sort of like a scaled variable say 1 minus reflectance the whole square divided by 2 times the reflectance. So this is the STR the transform reflectance. So for each pixel within an image for each pixel we will have this STR value and correspondingly we will have NDVI value for that particular pixel right we can estimate it NDVI can be easily obtained from red and NAR bands. So just for a given steady region try to plot the values of NDVI and this STR for all the pixels. For one pixel let us say the NDVI is 0.2 STR is let us say it is like 0.5. So this is like a coordinate so the x coordinate is 0.2 y coordinate is 0.5. So somehow plot it. So like this if we plot it for all the pixels then it will appear in form of like a scatter plot like this and within this scatter plot we can define 2 edges one is called the dry edge and the top end is called the wet edge where the wet end signifies the soil that are full of water and the dry edge signifies that are like dry soils. So once we fit this envelope then for any pixel having a corresponding value of NDVI and this STR we can calculate the moisture content. So this is like a very simple expression I am not going to discuss in detail about how the model works what are the physical assumptions behind it and all but simply put reflectance based soil moisture earlier like many people were not able to successfully retry but in the reason past a study demonstrated few studies have demonstrated that retrieval of soil moisture is possible from optical remote sensing data sets but again there will be high influence of vegetation other variables which can change a reflectance. So all these things we have to keep in mind but it is possible to retry reflectance based soil moisture. Next we are going to see is using thermal infrared remote sensing for soil moisture estimation. So the surface temperature will change when a soil is dry or wet if the soil is dry most of the incoming radiation will be used up to heat up the surface. So the surface will be at a higher temperature whereas if we add water to the surface then the incoming radiation will be used up by the water to evaporate and then the energy will be used up by the water itself for this phase transformation. So the surface temperature would not have increased to a large extent. So wet soils will be cooler dry soils will be warmer. Using this technique people have estimated soil moisture. In addition to this the thermal inertia of the soil also will vary like wet soils have high thermal inertia like the temperature will not change much within a day whereas for dry soils the temperature will change drastically within a day. So observing multi temporal surface temperature measurements like what is possible from geostationary satellites it is possible to retry soil moisture. Say an example for thermal inertia is given here. So this is over like the a particular area or a field in Karnataka state this is from ground measurements basically. So this is obtained over like a parcel of like soil. Here you can see the surface temperature measurements during when the soil is dry is actually shows very large difference in diurnal cycle like it is low at nights then it suddenly increases and reaches like a peak value during daytime then it comes down whereas when the soil is wet the diurnal cycle is more sub cured the temperature difference between the maximum minimum is pretty low in the order of like few kelvins whereas it is almost in order of the 10 Kelvin when the soil is dry. So the thermal inertia property can be used when or to estimate soil moisture but again this is not straightforward task because soil moisture also can change within a day someone may irrigate or some high kind of like drying can happen anything can happen so. But some studies have utilized these properties of thermal infrared remote sensing data sets like that is the land surface temperature to estimate soil moisture. So the combination of optical remote sensing and thermal remote sensing is again useful for the estimation of soil moisture. So one model is called like the classical triangle model which is useful for estimation of soil moisture. So again we will just see like the basic principle of the working. Say we have already seen this OPTRAM model right. So similar to OPTRAM model where we will plot STR and NDVI here we plot LST and NDVI where LST is the land surface temperature that you get from thermal imagery. If you plot it it will form in kind of like a triangle like this like all the scatter will fall somewhere like this where this top edge is called like the dry edge which represents low soil moisture conditions and this bottom edge is called the wet edge which represents high soil moisture conditions ok. So over a region over a steady region if we can estimate this LST and NDVI if we can plot them together and from that scatter plot we will be defining this dry edge and wet edge after defining them we can estimate soil moisture. This is again one of the widely used model for estimating soil moisture from by combining thermal infrared and optical datasets that is NDVI. Here we are primarily using NDVI because NDVI is form is the x axis and LST is forming like the y axis. So these are all very simple models for estimation of soil moisture from optical or thermal or combination of optical and thermal remote sensing. But what all the pros and cons of this optical remote sensing basically say first thing reflectance based methods. We have very good spatial resolution nowadays we have data available at say 10 meter resolution freely available to us. So we can theoretically estimate soil moisture at 10 meter right multiple satellites are available and for very long time say from 70s onwards we have like a good time series of data. Hyperspectral sensors are promising yes. So all these things are advantages very high spatial resolution multiple satellites hyperspectral sensor can still provide very good information and so on. But the major disadvantages are the reflectance is weakly related to soil moisture content because the reflectance can change not only with respect to soil moisture content but also with other things. And also reflectance based models cannot be applied during cloudy conditions because during cloudy conditions we will not be in a position to observe the earth we can see only the clouds. At night time we cannot do it because we need so sunlight to calculate reflectance and sunlight will not be available at night. And this is we can like neglect poor temporal resolution now we are having data maybe once every 5 days or 6 days still which is okay. Then the thermal based methods again they provide good spatial resolution multiple satellites and all. But again here also cloudy conditions is a major thing we will not have thermal data during cloudy conditions earth's atmosphere will change the LST. Atmospheric effect plays a major role and also there are like in the models that use temperature and reflectance together or any way together there are like lot of assumptions involved if those assumptions are not satisfied the model may poorly perform. So effectively optical and thermal remote sensing can estimate soil moisture but soil moisture is not highly correlated with this with what we measure in optical remote sensing or thermal remote sensing. So this is not one of the widely used technique. So for estimating soil moisture microwave remote sensing is often used and we will discuss how microwave remote sensing is useful for soil moisture estimation in the next lecture. Thank you very much.


An ECG or electrocardiogram is a simple, non-invasive procedure that records your heart rate and its rhythm and is prescribed to assess the heart health. For this procedure, you are given a hospital gown to change into and settled comfortably in a private area throughout the procedure. You will need to remove your ornaments, accessories, purse, belt etc. The technician then attaches electrodes on your chest, hands and legs. These are connected to a machine which records your heart's electrical activity and prints out the result. Once that's done, the technician removes all the electrodes and your report will be issued soon after the procedure at the reception. The procedure is completed painless and takes only 10-15 minutes, post which you may proceed with your daily routine and activities.


Porky Games and Avalanche software have added a trans character to Hogwarts Legacy. This has received a mixed reaction due to what JK Ryan has said about the trans community. A question many people have had and I've even thought about it myself. And be sure to let me know what you think. Would they have added this character if the JK stuff didn't happen? Or was this character added on purpose to demonstrate Avalanche Porky's stance on the situation and to make it clear they do not agree with what JK did? What do you guys think? Guys check out my channel for more Hogwarts Legacy.


I just met Dominic about 10 years ago, and he's working on this very beautiful, hydrodynamic behavior, the dub, the Cheerios effect. He has worked on a number of very interesting problems, and I'm looking forward to hear his presentation on elastic deformation, so take it away, Dominic. Okay, thanks, Mike, and thanks to the organizers of the school for everything they've done and for the invitation to be here. I've really enjoyed it. So what I'm gonna tell you a little bit about is some of the work we've been doing on elastic deformations and really thinking about instabilities. So there's sort of a classic area of research in mechanical engineering and physics and applied math, but generally what people are interested in is understanding when things go wrong. So if you're designing a water tower, then you don't want this to happen, okay? So this is a water tower that was built in Belgium in 1972, and it was a very good water tower until they filled it with water, at which point it promptly collapsed, okay? But you see other examples of elastic instabilities, for example in numbers that you use in your house, as they sort of heat and cool differentially, that you get this delamination instability, and if you have rugs or carpets, then you quite often get this little bump, okay? So these are examples of instabilities that you may want to stop or prevent, and that sort of mindset gives you a sort of a set of questions that you're interested in. So for example, you're interested in understanding when your water tower is gonna fall down, and you don't really care how it falls down, right? You just wanna make sure that it doesn't, so you work out when it will happen, you add some safety factor, and then you're done, and maybe you wanna worry a bit about imperfections. But more recently, people have started to think a little bit about how instability can be a good thing. So there's a paper by Pedro Rice, who did a post-doc with Mark, I think, recently, so from 2015, where he says, okay, actually, there are some features of instability that are very useful, and nature knows about this, okay? So for example, the Venus flytrap uses fast motions associated with snap-through instabilities to catch this poor fly, okay? And similarly, there's a recent paper in PNES a few months back, looking at how a ladybug that deploys its wings, and again there, it's kind of an elastic instability. This is videoed at 4,000 frames a second, and then played back at 25, so you can see that it's very quickly deploying the actual wings underneath the case. So people have started to use these fast motions to generate, for example, here, there's some lenses that are gonna suddenly change their curvature, and so sort of de-focus, and the idea is that you can have very fast motions just by using elasticity. So nature here says, okay, actually, elasticity is a good way to make very slow changes and to store energy slowly, and then to release it very quickly, that's why the Venus flytrap is able to move so quickly, it couldn't do that otherwise. But there are other features of instability, so for example, you can quite often get some very regular patterns, so what I'm gonna talk a lot about here is some wrinkling, and this is useful in photovoltaics because people wanna make larger surface areas, capture more light, there are other applications in flexible electronics and making lock and key colloids. But the point I want to make is that if you're asking these kinds of questions about how do you actually use instability, then the kinds of questions you ask are a little bit different to just understanding when does it happen so that I can avoid it. Okay, so I don't know how much people know about elasticity, so I wanna sort of take a step back and say, well, how do thin elastic objects deform? So what you should be thinking of is something like a piece of paper. A piece of paper is very long and very, very thin.  So I can accommodate this by just changing my length a little bit. I squash the two ends together by some distance delta L. But you also know that if I do this, in reality, what it's gonna do is not change its length, but instead buckle out of the plane. Okay, so that's what we see, what we call bending. And if you think about the very, even though the thickness is very small, you can think that maybe along the centerline there's no stretching and on the outside there's gonna be a bit of stretching, on the inside there's gonna be a bit of compression. Okay, so the question is when will it do each of these two things? So if we look at the energy, there's basically an elastic energy from changing your length and here there's an elastic energy not from changing the length, but rather from stretching the outside and compressing the inside. And it turns out that the cost of changing the length compared to the cost of bending scales with the length times how far you push the two ends together divided by the thickness squared, okay? So the important thing here is that this is thin, okay? So if this is thin then the T squared on the bottom line is very, very tiny and this energy for changing my length is very, very high compared to the energy of just bending out of the way, okay? So as the thickness goes to zero you expect objects to deform without changing their length. That's a sort of class of deformations that are called isometries, okay? So if you look in a dictionary an isometry is a map that or it means of or having equal dimensions, but in mathematics it means a transformation without changing the size or the shape, okay? So what we're doing when I bend this piece of paper the length of the piece of paper does not change appreciably. Okay, so why is this really useful? Well, it's useful because there's a theorem due to Gauss called Gauss's remarkable theorem that says that the Gaussian curvature of a sheet doesn't change when you do an isometric deformation. So what's the Gaussian curvature? So the Gaussian curvature is if I do this there I can write down two curvatures there's one in this direction and there's one in this direction, okay? And the Gaussian curvature is just the product of those two curvatures. Why is this important? Well, it's important because it tells you that if you take a piece of paper and you try to sort of stick it onto a sphere that's the closest sphere I have then you cannot do it without stretching, okay? So if you do you can't just sort of nicely map it onto that, onto a sphere. And obviously that's a big problem for map makers. You want to make a two-dimensional map of a spherical earth. And so you cannot do that keeping the distances all the same. So you have to stretch some distances and there are different projections depending on which distances you want to stretch. I've got two kids.  So this is really a consequence of Gauss's theorem and the fact that very thin objects don't want to stretch. But it's also quite useful. So in our group the Gauss's theorem is actually called the theorem pizzeria, okay? And the reason for that is that when you eat a slice of pizza you sort of pick it up and you know that it's just gonna droop under its weight, okay? So that's bad because it's very hard to eat this piece of pizza. But we also know that what you do is you say, okay, if I bend the edge so I curve the edge with my hand then this direction stays straight and I can eat it very happily. So why is that? Well, it's because of Gauss's theorem. So Gauss's theorem tells us that this thing is flat. It's got to keep the same Gaussian curvature. The Gaussian curvature here is zero because it's not, it's straight in both directions, okay? And so if I impose a curvature in this direction, in this direction, I know that the product of this curvature and this curvature has got to stay zero and the only way I can do that is to keep this curvature zero, okay? So then this direction stays flat, okay? And I can eat my pizza, okay? And this is the same idea about that's used to make corrugated roofs, okay? So the idea is that if you induce some curvature in one direction, it becomes very difficult to bend in the other direction and that gives you very strong materials with very low weight. Okay, so I've shown you that a flat object can't be mapped to a spherical one, okay? And I've also sort of hinted that if you turn it into a cylinder like this or if I roll it up into a cylinder, that's fine because although I've got one curvature in this direction, I don't have any curvature in this direction, okay? And so the product of the curvature, so the Gaussian curvature is still zero. But that's, okay, so mapping to a cylinder is fine, but that's a bit boring. If that's the only thing we can do with a piece of paper, then life is a bit boring. And the question is, are there any non-trivial isometries, okay? So I'll come back to this later, but one example is what's called the decode, okay? So this is what you get if you take a piece of paper and put it on a cup and then poke. You get this little bump. And what you see is you sort, if you look at this very carefully, you can see that although it's curved in this direction, in the sort of perpendicular direction, it stays straight. So that's why it's called a cone, okay? And the D in decone is because it's developable. It still has no stretching and it still satisfies Gauss's theorem. I've mentioned here a few papers from the late 90s, partly to convince you, well partly to sort of in the interest of academic honesty, but also partly to try and convince you that although this is quite a simple observation, it's actually led to a number of papers in some pretty high profile journals. And so if you have the right question and are able to sort of find a simple demonstration of it, then there is an option for having high impact as well. Okay, so the decone is one example of a non-trivial isometry for a flat sheet. What about a curved object? So if you take a tennis ball, okay? So a tennis ball has non-zero Gaussian curvature, okay? So it's got a curvature in this direction and a curvature in this direction. So the Gaussian curvature is one over R squared. So clearly we know that I can't make it flat because that would mean you'd have to change the Gaussian curvature to zero. But what you can do is you can turn it inside out, okay? And when you turn it inside out, then okay, it's not quite true at the edges, but the main part of the sphere has exactly the same radii of curvature as it did in the other state. And so again, that's an example of an isometry and it's what's called mirror buckling because it's as if you reflect the sphere or the spherical cap in a mirror. Okay, so what I want to sort of spend the rest of the talk on is thinking about some poking induced deformations, okay? So this is how I sort of came into this area and I was originally interested in this because we wanted to use poking to measure things about plant cells. And I was talking to Toyota about it yesterday. So there's definitely potential application but in everyday life, you want to check before you get on your bicycle, you want to check whether the tires have the right pressure. So what you do is you kind of push on it and if it's too soft, you know that you need to inflate the tires a bit more. A bit of Google science revealed that there's something called the finger test for testing whether meat is well done. So the idea is that you cook your meat in the oven and you want to know is it cooked to the way I like it and the idea is that you poke it and see how soft it is and then you need to compare it to something.  Okay, so that's the sort of consistency you should be looking for in well done meat. And if instead you like it to be rare, then you need to take your thumb and your index finger and push that bit there and you see that you feel that it's much softer. Okay, and that's apparently called the finger test. But obviously this is a bit quantitative. So people also want to make this quantitative. So there's some experiments on polymer zones which are basically big, well, 100 micron shells, capsules, okay, filled with some solute and that gives them a sort of turgor pressure. And the question is how can you measure that turgor pressure without poking a hole in it? And the idea is that you basically come along with an AFM tip and you just kind of push it and you measure the indentation for a given force or the force for a given indentation. And then if you do the calculations properly, you can sort of work out what the pressure is from that. So we started off doing this and the basic idea is that when you push a shell like this, there's this deformation that I told you about that's isometric and it's called mirror buckling. So what you read in the textbooks is that if you take, I won't be able to do this now. So if you take a sphere and you push on it, then at some point it is able to deform as if, so not exactly like this, but as if I turned a bit of it inside out and then glued it back on. So if I poke this, it accommodates it by kind of turning a bit of it inside out and being glued back on. And that's what's called, so that's mirror buckling. So you can just do some simple geometry that says, if I poke by a distance delta, then because it's a parabolic geometry, the radius of the cap that I need to turn inside out is a square root of delta times the radius. And the point is what I told you earlier that this mirror buckling is an isometry. It doesn't cost you any elastic energy to do that. There's a bit of energy where I have to do the gluing because there I've kind of changing the curvature very quickly, but everywhere else, it's basically isometric and so it's free. So if you look in the textbooks in Landau and so on, it tells you that a shell will try to do this. Okay? And the point is that as the thickness of the shell goes to zero, this, the Pogorelov ridge, that's a name for the bit where you do the gluing, becomes very small and the energy in it becomes very small as well. And so this poking is really a kind of free lunch. It doesn't cost you any energy. Okay? So the idea is that mirror buckling is a free lunch. So, you know, we know that in physics, everything likes to go to the smallest possible energy. So it should do this, right? The problem is that when we started doing simulations of this, this is an abacus simulation of a pressurized shell. So just like the polymer zone, you basically poke on this shell and everything proceeds very nicely. You can convince yourself that it's gonna do this mirror buckling. And then at some point, everything goes horribly wrong. Okay? And it starts to wrinkle. So we had a long back and forth. So this was done by some colleagues in the US, the simulations. And I said, oh, well, maybe it's wrinkling. And they said, oh, no, I don't think so. And I think the thing that convinced them in the end was to take this sort of packaging that you get in your Amazon boxes or whatever. So it's a kind of some plastic inflated. And then if you poke on it, then you do indeed see, the native guys in the front row, maybe you can see that there is indeed some wrinkling. Okay? So, okay, we did an experiment. We sort of characterized some of the properties of this and we used a beach ball just to make a pretty picture. But it's basically the same mechanics. The same thing happens if you don't have an internal pressure. So if you take a water bottle and then just poke on the edge where it's duck curved in two directions, then you again see that instead of being axi-symmetric and doing the mirror buckling thing, it prefers to make a triangle than a square and then this crazy pentagon star thing, okay? And so this poses the question of, well, you know, it might be, it's all very well to say that mirror buckling is a solution, okay? But it's not the solution that you see when you do an experiment. And so the question then is, well, why? We said that mirror buckling is a free lunch. Why is the system not taking this free lunch? Okay, it's meant to be free with a bottle. Oh yeah, okay, so, okay. So you can, this is just meant to be a demonstration. Yeah, you can worry about the fact that the curvature is changing in space, but even if you do it with a ping-pong ball or something with a constant curvature, you see the same thing. So it's not a feature of changing in the curvature. Okay, so what I need to do is I need to understand a little bit about this wrinkling instability, right? And the first thing I need to do is I need to understand whether or why it's happening. So the sort of simple model we have is that you've got a beach ball, so you have a sphere with some pressure inside. It's got some radius. It's got some thickness, and then you poke. And there's some calculations you can do to sort of work out what is the stress within the shell as you poke. So before you do any poking, okay, the thing is stressed because of Laplace's law. You wanna have, you've got a pressure difference between the inside and the outside. You need to have some tension in the sphere. And then as you poke, the stress near where you're poking is very, very large. And then far away from where you're poking, it goes back to the value it had originally. But the point is that as you, so this is how much I've poked. So one 10, 15 in some dimension of this unit is just how much I've poked. So for small poking, it kind of stress decreases, it dips, and then it goes up again. And the reason for that is that as you poke, you pull material in, okay? And you're doing that in a circular geometry, so you're taking a circle that was here and bringing it closer in, so it becomes relatively compressed. If you poke enough, then what you find is that that compression becomes absolute, not just relative, and that gives you, that's basically the same as taking this piece of paper and squashing it, okay? So if it's thin enough, it will just buckle and that's what causes the wrinkling. Okay, so as you increase the indentation depth, you pull material in, and then the key thing is that this hoop stress becomes compressive, okay? And the thing that you can get from this calculation is that near the indenter, the stress is still tensile, still positive, so there's no wrinkling. Then there's a little annulus where the stress is negative, and so I expect that to be wrinkling, and then outside, it's all positive again, and so I don't expect to have wrinkling. And if you look at this picture, sorry, the small r is, yeah, okay, thank you. So the small r is the distance from the poker, so the poker is at r equals zero. But as you go, if you look at this picture, as you go from the indenter, there's no wrinkling, then there's wrinkling, and then there's no wrinkling again. Okay, so what? Okay, so this is the fourth line of your abstract for those people who write abstracts with me.  I mean, it might just be that it's a small perturbation of mirror buckling. So actually if you look, kind of you get yourself a transparent beach ball or also in these experiments on polymerosomes, if you look through the side, then what you see is it's got exactly the wrong curvature.  Okay, and so we see the same thing in our simulations. Here's what the black curve here is, mirror buckling would suggest you should have, and our simulations have exactly the wrong curvature. So again, I'm sorry, and then because we're really interested in indentation experiments, there's a sort of quantitative thing that this force you get from mirror buckling is too large by about 50%. So you need to understand why the wrinkled solution has a smaller wrink, yep. Okay, so it is mad, but the way that it shows it's mad is by making the stress negative. So if you think about Laplace's law to still have a positive jump from the inside to the outside, you've got to get this compression. You've got to turn the sign of your tension into a compression, and that's why it's doing that. So I'm just saying, well, no, but it's just, I'm just saying that if you change the sign of the curvature, which is what you're complaining about, if you change the sign of the curvature, then you've got to change the sign of the tension as well, right? That's all I'm saying. To get the same pressure difference, the pressure difference is definitely positive, right? So if you, and you know that the tension times the curvature equals the pressure difference. So if you change the sign of the curvature, which is what I do by doing this, then you've got to change the sign of the tension to keep the same product. So, no, I only had to cut my tennis ball in half to show you that. That was, so here, sorry, in this case, the pressure is inside. Yeah. There's not much pressure in a tennis ball.  That's not because of, that's just because of this breaking of the axi-symmetry. I'm not sure I've answered that. But Lorette, are you happier? So. No. Okay. Okay. This is coming in different colors. Okay, so sorry, I haven't explained it. So different colors here are increasing, changing the indentation depth, and I've rescaled everything so that it should collapse. And the pressure? So here, so okay, so everything I'm doing is at constant pressure inside, and you might worry if I have a finite volume or number of moles of gas, then the pressure should increase, but you can still, depending on exactly how big your shell is and so on, you can still be in this limit that the pressure is effectively constant. Yeah. Okay, so. Yeah. Wait, I'm gonna. Which one? This one? This one? This one. Well, this is meant to be the same as this, or that's what I'm trying to show you is that it looks similar. Wow, but hang on, it had curvature. Well, okay, I'm gonna talk a little bit about, it's not, okay, you need to calculate, it's not just, is it able to stretch a little bit? So you need to calculate the elastic energy carefully, and I will talk about that in a second, okay. What am I doing? Okay, so what should we do? Say, when you assume that, sorry, when you sort of solve things in elasticity normally, and fluid mechanics and everything, you sort of assume that the stresses in a different direction are kind of in the same order of magnitude, so when you solve the equilibrium equation, so div sigma equals zero, you're assuming that both stresses kind of play a role, or in general, you do that. So for a sort of axisymmetric geometry, with some tensions applied, it turns out that if you do that, you get what's called the lame solutions, you get that the stress is some constant, plus or minus, depending on whether you're RR or theta theta, radial or hoop, divided by R squared. But the point is that wrinkling is really caused by compression, right? So if I take this piece of paper, and I, okay, it has a high stress, and then once it buckles, it has a much smaller compressive force, effectively. So what's happening there is that the wrinkling is able to sort of relax the compressive stress, so instead of saying that these two things are the same order of magnitude, you have to say sigma theta theta is basically zero, and then when you go back to div sigma equals zero, instead of getting this solution, you find that sigma RR is one over R. So, okay, maybe that's a bit technical. What I'm really trying to say is that instead of thinking about a membrane, a wrinkled object is a bit more like a spider's web, okay, so in a spider's web, you have these radial threads, okay, and there, you know, you kind of push on the spider's web, those radial threads get stretched, but the ones in the hoop direction, the kind of circumferential threads can just buckle out of the way. They don't do anything. It's not that there's a big compressive stress in the hoop direction, okay? So this actually changes the behavior of the system qualitatively, and so now I can actually calculate the shape that my pressurized shell should have when I get to very, very large indentation depths, and that gives me this dashed black curve here. Again, I can tell you what these prefactors are in terms of the square root of five, but what really matters is that you've got a kind of universal deformation. So I've rescaled the indentation depth, but the shape, by how far I've pushed, and then I've rescaled the radial distance by this geometrical length scale, the square root of delta times R, and this shape then becomes universal, and you can see that the numerics are sort of converging onto that shape as the indentation depth increases. So the shape has got a geometrical character, just like mirror buckling had. Mirror buckling was just taking a spherical cap, turning it inside out. I found something else that has a similar geometrical characteristic, okay? But we don't understand why this, or in what sense, this wrinkly shape is better than mirror buckling, if it is, okay? So normally, as I said, better means lower energy.  So what I need to do is I need to understand how the system chooses the number of wrinkles. So to do this, it makes sense to sort of take a step back and to think about the sort of simplest problem of wrinkling, which is essentially to have my piece of paper now floating on the surface of water, okay? And then I do the same thing. So in the absence of water, we already said that it's just gonna do a single buckle like this, right? But if I do that when I'm floating on water, then I have to lift up a lot of water, and that's gonna cost me a lot of gravitational potential energy, okay? So instead, what I could do is I could do lots of really tiny wrinkles, and I could accommodate the same compression with very small wrinkles, but that will cost me a lot of bending energy, okay? So somewhere in between those two is kind of an optimum, okay? So it has sort of conditioned because I've compressed, which tells me what the amplitude of the wrinkle compared to its wavelength has to be in terms of this squashing distance. And then I can calculate the energy. So there's an energy because of the curvature of the sheet and there's an energy because of the gravitational energy of the liquid that I displace, okay? And what I find is that, as I expect, so bending would like me to have as large a wavelength as possible, while gravity would like me to have as small a wavelength as possible, okay? So somewhere I've gotta kind of go between these two things, and it turns out that if you sort of minimize this energy by changing the wavelength, then you find that you wanna have this particular wavelength at the fourth route of the bending stiffness compared to gravity. Okay, so this is one particular kind of wrinkle length, okay? This maybe what happens if you have a substrate stiffness, but there are other options for how these well-developed wrinkle patterns can be determined, okay? So one is if you have a lot of tension along the wrinkle length, okay? So this is, again, sort of in the early 2000s, thought about what happens when you pull on a sheet and there you get an effective stiffness because of the tension that you're pulling with. And then more recently, we've thought about what happens if you have a curvature, okay? So you have, it turns out that you have a stiffness because of the radius of curvature along the wrinkles. So again, I'm not going into much detail here, but the important thing is that you can sort of add these different stiffnesses up, and you can generalize this wavelength here. Okay, so in my problem, I've got a curvature, I've got this sort of universal shape that I know, and I can, and it turns out that it's this curvature that really matters. So you can write down what is sort of a relatively simple law for what the wrinkle number is as a function of the distance from the indenta and the indentation depth and the thickness and all of these things. But there are a few interesting features of this law, okay? One, I think is that, you know, we've got a lot of things in this problem we've got the Young's modulus, we've got the pressure, the thickness, indentation depth, all of these things. But in this law, what we see is not the Young's modulus or the pressure, rather the just the thickness, the indentation depth and the radius of the shell. And then the other thing that I think is interesting is that there's this kind of curious r to the three halves, so radial position to the three halves law. So you can sort of, well, I can, these shows you some simulations at different indentation depths and you may or may not be convinced that as I increase the indentation depth I'm getting different numbers of wrinkles. But also if you look very close to the center there's a small number of wrinkles and at the edge there's a much larger number of wrinkles. Okay, and it's not just that the system likes to keep a constant wavelength because this, that would need you to be your wrinkle number to be linear in r. Okay, so there's something a little bit complicated or non-trivial that we're able to find out here. But I still haven't told you why it's doing this. Okay, so I've calculated the number of wrinkles and now I can calculate the energy and it's again a little bit technical but if I, I've got to allow the thickness to go to zero but to really understand or for my approach to be sort of valid I also need to let the pressure go to zero and I have to let the pressure and the thickness go to zero in the right sort of way. Okay, so I need the pressure divided by thickness to go to zero but pressure over thickness squared to go to infinity. So it's maybe a little bit of a mathematicians thing but the important thing is that now it's not really an isometry in a sense of Gauss's theorem, okay? Because what we have, but if I calculate the elastic energy it's essentially not the important energy in the problem anymore. What instead I'm finding is a very, so maybe a very boring answer which is that when I look at my new shape the dashed black curve here I'm displacing less gas than the mirror buckled solution. Okay, so the mirror buckled solution would take all of this gas, push it out of the way and that's Loretz complaint. Okay, this wrinkled solution displaces less and it has a kind of a very small elastic energy. Okay, so well that's basically the answer to the problem is this is allowing you to displace less gas and so your PDV is much smaller than it would have been in the mirror buckled case. Yep. Okay, so here. Yeah. It's a pressure resistant test. Okay, you're displacing less, but why doesn't the number of wrinkles does not depend on the pressure? Where does that get the most? How do you do that? Well, it's essentially because the wrinkling allows you to get to this universal shape and that universal shape is just some geometry, it's a non-trivial geometry, but it's in geometry. So that doesn't depend on the pressure and then it's this thing about the curvature along the wrinkles is what matters. So you sort of lose the information. I mean, this is probably not of much practical interest in the sense that you have to go to very large, you have to poke a very long way to get to this regime. Okay. Okay, so I want to just give you another example of the same thing. So if we take a very thin polystyrene membrane, I'm not sure you can see it, but it's maybe a couple of centimeters in diameter and then a hundred nanometers or so in thickness and it's floating on water. So because it's floating on water, the surface tension at the edge is pulling on it. So it's again tense. That's a little bit like the Laplace tension or the tension from Laplace's law in the shell case. And then I'm gonna poke on it. And we can sort of do a comparison between the pressurized shell and the floating membrane case. So again, we had a tension because of Laplace pressure. Here we've got a surface tension. In the case of the pressurized shell, we had mirror buckling for the floating membrane. The only thing I can think of is this decone. Okay, it might not be relevant, but that's the only thing I can think of. So these are some experiments. So on the top, you're gonna see a sort of top view of this poking. And in the bottom, you can just see a sort of cross section through there. So what we're doing is just increasing the indentation depth. We're pushing from underneath. And what you see is that wrinkles very quickly reach the edge of the film, okay? So that turns out to be what's important in this problem is that it's a little bit technical, but you just need to sort of solve the problem and you find that basically as soon as you touch the film, the wrinkles are everywhere because the film is finite. Okay, so the other thing to take away is that I don't see any sign of a decone here, okay? It's just more or less looks axi-symmetric. Yes, there are some very fine wrinkles all the way around. So we can do experiments of different indentation depth and with different thicknesses and different radius sheets. And then if we do the calculation based on what's called tension field theory, this idea that the hoop stress is compressed or is relaxed by wrinkling, then what we find is a sort of universal shape, this black dashed curve, which I get very excited about because it's just an airy function, okay? And I can calculate it analytically. But the important thing, again, I'm not going into the details, but the important thing here is that the elastic energy doesn't really enter into the problem. So the force that I'm using to do this indentation is more or less independent of the elasticity of the sheet. What it does depend on is the density of the liquid, the surface tension that I'm applying and the radius of the film. So again, it's just a sort of geometric thing. So I wanted to sort of bring this back and explain a little bit better than I did in the snapshot on Monday morning or what the purpose of the hands-on session is. So the key thing in both of the problems I've just shown you, the shell and the floating sheet is that there's a small tension at the edge, right? So if there's a pressurized shell, that's the tension from Laplace's law. In the floating sheet, that's just a surface tension, okay? And so the idea here was to think about whether you can sort of do the same thing for a decone, okay? So this is an experiment with G pointing upwards, okay? So what I've done is I've sort of, so this is my, an experiment we took, photo we took yesterday with a green sheet and we've got the pen pointing up this way, the sheet is here and then we bring the cylinder down. But the image I showed you of the decone earlier had it kind of the other way around. So I just showed you that gravity is now effectively pointing upwards in this picture I've rotated it. Again, you get the nice decone, but if you do it the exactly the same cylinder size, exactly the same indenture and exactly the same sheet, but with gravity pointing downwards, then instead of the decone, you see something a little bit different. You see, you again see this big region where it's lifted up, but on the outside it kind of points down. And that's what the groups who've been working with us, I think Monday, Tuesday and Thursday did a lot of, we did lots of different sheets, lots of different cylinders, lots of different thicknesses, and we just observed whether it goes up, sorry, down like this or up like this. Last night I had to think of a snazzy name for this and the best I could do was to call this the G cone, so the gravity cone if you like, but the results from so far in the hands on school is to sort of map out this phase diagram. So we've got two parameters, we've got the diameter of the sheet, the diameter of the cylinder in which you're sort of poking the sheet, and there's a region where the cylinder is bigger than the sheet, so that's not allowed because the sheet will just fall through. But what we see is that there's a very narrow band where there's a decone and there's a very large band, roughly. We don't understand the transition between these two regimes yet, but there's a much larger regime where you see this G cone. Again, what we're trying to do now or what we'll be trying to do today is to try and understand what's going on in this regime. Okay, so I just wanna spend a few minutes talking about a different kind of elastic instability. So if you've fallen asleep, got bored of wrinkling, maybe now's a good time to wake up and, because I'm gonna talk a little bit about snap three. Okay, so I'm gonna sort of return to this mirror buckling that I told you about. Okay, and I'm gonna say that, remember I told you that I could turn, I cut the tennis ball in half, I can turn it inside out and everything is fine. Okay, but that's not quite true.  It will just straight away return to its flat geometry. Okay, or its normal geometry. Okay, so actually if you kind of calculate what the, if you sort of look at the equations, it turns out that the important parameter is the thickness of this cap that you've cut off compared, sorry, the thickness of the shell compared to how much you've cut off.  Okay, so this is an example of what's called a snap through transition yet. You go from a region where there are two stable solutions to another region where there's just one. Okay, and I forgot to bring it with me but there's this famous toy called the hopper popper, okay? Where you turn the sphere inside out and then you put it on the table and you let go and it jumps into the air. People seen this, okay? So this toy is basically just beyond this transition, okay? So you might be interested in, how long does it take to do this snap? And the way to estimate it is to say, okay, I've got some bending stiffness because that's what's trying to turn it inside out. There's also the inertia of the thing because it happens very fast and so on. And what you find is that actually, it should snap through in about 10 milliseconds. But when you do it, you have time to put it on the table. I mean, 10 milliseconds is how fast it, you know, this just doesn't resist at all, whereas a toy takes a little bit longer, okay? So people have said, okay, well, if it's slower, then that means that it must be damped. It must not be an elastic object. There must be some viscous dissipation in the system. Seems like a sensible conclusion, but there's a lot of examples of this in other systems. So for example, in the Venus flytrap, again, if you estimate how long it should take, the sort of theory says it should take between 1, 1,000th and 100th of a second and actually it takes something more like a 10th of a second. And again, people say, okay, well, it's slow. So, you know, there must be some dissipation. But again, it's generic. So there's some work on what's called the pull-in instability in MEMS. And the thing that they see there is they measure how long it takes to do this snap-through as a function of the control parameter. So if you're out here, okay, far away from the transition, you see that it's very quick, which is just what you expect. But as you decrease the control parameter, which in this case in the pull-in transition is the applied voltage, okay? As you decrease the applied voltage, the time taken to transition goes up and up and up and up and it gradually diverges as you get closer to the transition. So again, this is people who know about the dynamics close to saddle node bifurcations. This is very well-known. If you look in the book of Steve Strogatz, he says that there's what's called a bottleneck caused by the ghost of the saddle node bifurcations. So if you don't know about saddle node bifurcations, just think about this ODE, okay? So dx by dt equals some control parameter plus x squared. If lambda is negative, I have two solutions, one stable, one's unstable. And on this side, because I've got an equilibrium, I can do linear stability analysis, everything is fine. If lambda becomes positive, then there's no equilibrium anymore and I can't do a linear stability analysis. But this system is so simple that I can just solve it, right? So I can find that if I start off at x equals zero, then eventually I'm gonna get kicked off to infinity and the time it takes me to get to infinity is gonna go like one over the square root of this control parameter, okay? So as I get closer and closer and closer to this transition, I get that the time diverges. This is the bottleneck that Strogatz talks about. So the question we asked was, well, could such a bottleneck be what's causing this slow snap-through in all of these different systems? And the question is not really whether it's in that, in the hopper-popper or whatever, because that probably does have some rheology, but just is it true that if you see slow dynamics, then there must be dissipation? So we wanted to find a problem that doesn't have any dissipation and that we can do the mathematics for. And the sort of problem that we came up with is not quite this, but it's easier to show you if I do this. So you take a piece of paper and you say, okay, we know about this situation, okay? We know about this case, but if I've got my hands close together, then I can also have this one, okay? So this is by stable. It can be in either this case or this case, okay? But this inverted case is a bit like the mirror-buckled solution as you turned inside out case. This one only exists if my hands are very close together. If I start to pull my hands apart, then it's not gonna do, it doesn't do it very excitingly, but it snaps up. Yeah, so automatically it just goes to the other case. Yeah? So that's more or less the system that we studied. We had to fiddle a little bit with the boundary conditions to make sure that it's exactly a saddle node bifurcation. But here the key parameter is really how much you've kind of squashed your hands together. And it turns out that there's a sort of geometrical parameter. There's only one dimensionless parameter, essentially governing this problem, which is the angle that you have the ends at compared to how much you've squashed them together. So if I squash them together less, then this mu becomes large and I snap through. Or if I have a smaller alpha, then again this mu is small and it's gonna be, sorry, yeah, I said it the wrong way around. The important thing is that there's a bifurcation at some critical value of mu and you can sort of calculate the diagram. So if mu is large, then there's only the case out here. And if mu is small enough, then there can be two equilibria. So it turns out that the key parameter here is how far beyond the transition are you? So what we do is we basically, we take our piece of paper or actually PET in this case and we kind of push it into the regime where there's only one scenario, one equilibrium and we hold it in the sort of wrong equilibrium. And this is a movie taken at, I think a thousand frames a second, I think it's playing. Okay, so you just watch how fast it snaps through and you watch it kind of oscillate like this. And then you try to quantify this and you measure, okay, this is a time trace of the center as a function of time. So you see it does nothing, nothing, nothing and then it suddenly accelerates and then it oscillates. Okay, so we can plot. We need to work out where we actually lost contact in here but we can plot that as a function of time. And what you see is that first of all, you see two things that I think are interesting. First of all, you see that the growth on a log, the log plot at small times is a straight line. So there's power law growth. It's not an exponential as you'd expect from linear stability analysis. And then also as I change the value of this parameter mu, then you slow down the dynamics. So we can actually sort of quantify this if we measure the time taken to snap. So the time from here to the peak there and we plot this as a function of where how much end compression I had. Again, we do this for different material properties. So again, we see that the snapping as I get closer and closer to the fold or to the bifurcation, the snapping takes longer and longer. Okay, so to choose a zero you have to do, yeah, so what you do is you have to, so basically you have to sort of, you know where the zero of w is, that's okay. And then you, I've forgotten exactly what you do actually. If you know what the answer is gonna be, then it's very easy. You kind of do w squared as a function of time, sorry, the square root of w is a function of time. But you can do that in a more rational way, just by looking at how things, by allowing different power law exponents is, so I mean if you do it in the supplementary information, we show you what it looks like on a semi log plot and it definitely is not correct. There's definitely not a straight line. And then the question is if it's not an exponential, what power law is it? And you can sort of work out what power law it is just by assuming it's some power law and then differentiating one over the log or something like that. I can show you, but there is a, yeah there's a slightly delicate thing to work out exactly where t zero is. Okay, so we can, I'm not gonna go into the details, but basically we can look at the equations for the dynamic motion of this arch and we can do an asymptotic analysis which allows us to take a PDE and make an ODE. Again, we can solve that ODE up to some quadrature analytically and that gives us this black solid curve and it also suggests how we should rescale our experiments and they more or less collapse, okay? And we can also look at how long it takes to snap as a function of this parameter delta mu and again our experimental data more or less collapse and then crucially we're able to get without any fitting parameters this black straight line which surprisingly as of our analysis is only really valid for this delta mu close to very small but it works out to order one delta mu as well. Okay, so maybe you guys know all about this because this is really just an example of critical slowing down near a phase transition it's maybe not so surprising, but I think in the sort of community on elastic instability people weren't really aware about the possibility of studying this sort of quantitatively. So that's all I wanted to say really. I've hopefully convinced you that there's several interesting physical phenomena that involve elastic instability. I started off by telling you that isometries are useful to understand the deformation of elastic objects but I also maybe told you that you need to be a bit careful about that because wrinkling may let you see something different or we uncover a new kind of isometry which we call wrinkly isometries and then also the structure that you get with these isometric solutions can be interesting and the way they disappear can give you this critical slowing down and then in the spirit of the school I think I hope I've convinced you that there's a lot of potential here for some relatively simple experiments. It needs a little bit of care to think of the right question but you can actually learn something new with some relatively simple ingredients. So with that I should thank the various people I've worked with on the three problems I've talked about and even though I'm British I'm gonna thank the EU for the money and then take any questions that you have.


hey everyone welcome back to figure fantasy so today we'll be having fun and let's look at the future skins that will be dropping to the game again the data for this one is based on the japanese version of figure fantasy so without further ado let's move on to the first skin that i'll be featuring okay so first on our list of skins is going to be ihinahara so ihinahara this is actually her skin with this is actually a a buddy theme that she's sitting on a roulette table so this is a 3d art and this is your 2d art so again basically similar next is going to be aketshi matsuhide so aketshi is actually an owner of a flower shop so these are actually i'm not sure yet on when these skins are going to drop but for future reference just in case you want to save for them you can save for them because there are some of the figures actually have multiple skins with them so this is aketshi with a 2d art next is this is actually one of the worst um skins that i've seen this is alfred with just a pumpkin head i bet this skin is going to be for halloween so one of the worst skins that i've seen so far so next again this is 2d art and we have here is awizo relaxed i think going to the beach um i like this one of her caring pop corner thing and juice and this is her 2d art 2d art actually looks better next is going to be asteroth so asteroth here um i've seen this arc that which has mvp on it i think this is part of an event so hopefully even free to play players can actually avail of some of the skins so this is what actually his sword is actually improvement and as for the 2d art it's more it's the same but with a different angle and we have here aiyane okada okay so aiyane is a volleyball star um the detail in the base we have there a i think that's a skipping rope i'm not sure what that is and that's a volleyball so this is her 3d skin and this is her 2d uh version okay next is that a masamune so he is um carrying i'm sure those are kendo sticks so he has two kendo sticks that's the first and here is his 2d art so almost the same of the 3d art so next is one of my favorite um skin so far is ireksha with her bike i'm actually a bike enthusiast by the way and i love i love this one i really love this one i'm gonna get this one once it goes out so again this is her 2d art i love the 3d more it the the bike pops out more in the road the detail of the road terrific okay so this actually is what i'm talking about guys so ireksha has two types of skin so this one is um her facing a mobile phone a big mobile phone so i like the other one better and this is the 2d art for that one so again um the bike is gonna be my choice so next is going to be harriet um our long range girl so she is actually carrying a flare gun plus a sniper rifle and plus there are a lot of cans on the base this is actually her 2d uh version and um i like i like the 3d version more so next is going to be hatsune miku i love her hair she kind of dyed the edges of her hair um those are i think beige or yellow or something um i like her hair in teal better so for this one i just like the detail with the lamp post and i think that's a manhole that she is uh uh standing on or that is just a floor so this is the 2d art so 3d looks better um the the lamp post pops out then this is also another situation where they have two skin so this is hatsune miku in another version of a skin so she has a i think that's a fox with on top of a macaroon and a lot of goodies there so again i prefer her hair teal no highlights please this one has a pink highlight so none of this for me um and this is the i think the the 2d rendering is much better this pops out more so next is going to be irons uh this is this is actually the skin that we have right now uh with her dragon carrying um the her camera and this is the 2d version but she has another version guys just in case you haven't bought this version she has another version with mvp uh arc mvp and that is her dragon there kind of angry and this is the 2d version of her skin so pretty much very nice better than the than the previous one next is going to be arena so arena has kind of a theme of iron man for this one uh um yeah similar to iron man so her i don't know what you call that one her mech or something is doesn't have a different design it's the same design and here is the 2d rendering of the design and also we have here is zoomie is zoomie with a bus stop and a fire hydrant so similar to the other one that we saw earlier so this one looks kind of better with the fire hydrant uh there at the back of her so this is the 2d art and the next is going to be kizuna kizuna by the way also has two versions of skins so she has this one which is based on tennis and she is on i like i like how it's set um with with her i think serving and the background she has their balls and um and the case for her racket so this is the 2d art a lot i like the 3d art better 2d art then this is the second one that she has so this is more or less a birthday cake honestly i like this better than the other one than the tennis version and this is the 2d rendering of her skin next is going to be loris so loris is has this again mvp skin with her mortar of course and this is the 2d version so this one i like this one than her regular outfit and the next one is going to be lubu which the base actually looks tremendous for a collector of tabletops um i like this version the the base is actually full of you know full of animation full of you know full of design and the waves fill out the the base itself so i like this one and this is the 2d rendering of his skin next is mako so we have this already in the store so mako with her i think that's a bunny computer bunny down down there so i also like this skin but i don't use mako enough um and here is her 2d rendering of the same skin next is going to be megan so megan tomorrow will be getting this skin of hers um i'm sure a lot of people are going to be getting this again i really like the design and i'm just gonna call it a design at this point and this is the 2d rendering and i like a 3d by the way really pops out okay next is going to be midori so midori has a sparring partner with a sparring bag so i don't want to really want to mess with her so this is actually a nice a nice one with a big base it's a square base i think this is the first time that i've seen a square base and this is the 2d rendering of midori okay next on our skins is going to be ophelia zoophilia i think we've had we already have this in the store so this is just i think a blue version of her current skin and this is actually the 2d rendering so um this is kind of very simple version of her so let's move to the next so we have usuki okada so usuki in our game has red hair so again this is his racing skin with a lot of tires in on the back i think the tires don't fit racing it should be it should be what they call this it should be spikeless or something so this is the 2d version and next is we have rafa el so rafa el this is i think she's on the beach without outfit well well well so anyway um this is a what is a good looking skin as well and here is the 2d rendering of that outfit okay next is going to be riko one of the most simplest skins that i've ever seen um she's using a bomb and this is the 2d rendering that the bomb is actually smaller here while this one it's actually bigger next is re one of i think the skins that i would like to purchase as well this nurse skin looks looks really really good and this is the 2d version of that skin next is going to be rinn so rinn is actually one of the future um figures that we're going to get so this one i like this one as well um 2d doesn't do just this a 3d is actually looking better next is going to be riyoma so riyoma is what is his sport i think this is going to be kickboxing or mix martial arts so this is his casual skin to the gym and this is the 2d rendering i like his hair here better with the 2d next is going to be sakura so sakura is going to be featuring i think this is a japanese kimono and this is the 2d rendering of sakura next is going to be sanada yukimura so what is he doing with a stop sign here nice nice detail by the way newspapers and the base and very casual i think he's using the stop sign as a weapon next is going to be his 2d rendering okay so we have here susumi so susumi has a pan as usual it's a green version and she has an apron on and this is her 2d art vasily here with his 3d skin um the colors here do not do not fit well i think i like the other you know the other version of him the original one they should have i think lighten this one go go all the way teal for his uh for his skin um they still left a little bit of blue in in in the hands of this mech so this is the 2d rendering of his skin next is going to be vivian so vivian is sitting on a cactus and i think this is the 3d version i hope she doesn't injure her back there and this is the 2d version as well then we have here yamazaki you've seen him in the thumbnail already one of the coolest skins i've ever seen um he actually has a teddy bear beside him i don't know what the meaning of that teddy bear but again this skin looks cool on him so here is the 2d rendering i really like the 3d one so next is going to be yoko kasumi one of i think the best skins i've seen with the detail obviously with the swimsuit um she's on the beach and she has there an oxygen tank uh i don't know what those two are she also has flippers but again good good skin i'm gonna get this once this is going to be out and here is the 2d rendering of that skin okay then we go to yulanda so yulanda one of the future figures that we have so this is i think she's selling what drinks she's a waitress and look at that outfit for a waitress and this is the 2d version of that skin and we have here yawina mezuki so yawina also has that mvp arc so and she carries a very big lollipop in gold i like the older one i think rather than this one and this is the 2d art then we have here yuna yuna is actually an existing this one is an existing skin in our store so if you are if you guys are using yuna this one this skin is also good i like this as well but i don't use yuna that much that's why i didn't buy this skin so this is the 2d rendering then you have here zarola so this one is still available um i think as a limited offer i already bought this one and more or less if you want to power up your zarola a little bit then you can buy this skin oh by the way guys the skin actually gives you a couple of stats stat boost if you if you get the skin so hopefully you can get this one as well and we also have zeffy or so zeffy here relaxing on the beach with her chair and with her i think that is just a bag at her back so and also a ball so do we i think she's about to do some beach volleyball for this one and here is the 2d rendering of this skin then zeffy here has another version so this is actually i think a wedding skin or a bridesmaid skin um i like this as well then here's the 2d version of this skin then we have zhaoyun zhaoyun is like a valley with his spear funny doing valet and having that spear and here is the 2d version by the way then you you have zhaoyun so looks like a cool professor going to class with ears popping out and with the long hair okay and here's the 2d version of zhaoyun okay and we are up to our last one which is zhongliang um here doing his assignment on a fence with a packet of milk very interesting very interesting nice art as well and here is the 2d rendering of that skin anyway guys um these skins will be dropping in the future and hopefully you could probably save up um if you wanted some of them i'll be putting the list of skins in a drive then sharing it in the description below so you can actually access that in the description okay guys so thank you very much for staying this far and hopefully you like this um video of all the skins um in the future that will be dropping in the game and um don't forget guys to if you consider subscribing don't forget to click that bell icon and also put a like to this video thank you guys stay safe take care this is the warden and i'm out of here


I recently discovered something that Google kept secret about the Chromebook. This is truly shocking when you really think about it. Known as the kill screen, this error appears on roughly one in about 100 of the total amount of Chromebooks. It doesn't matter which Chromebook you have, any of them, except really old models, can get this. Let me break it down for you. The Chromebook starts up like it should with the Chrome animation, and cuts to black like it normally does. If your Chromebook has the kill screen, then instead of going to the login screen like it should, it goes to a screen with a similar UI to the developer mode, or something you would get if the operating system was missing or damaged. The error screen states that the Chromebook encountered an unexpected error, and that normal use could not continue. The screen advises you to call 1-800-4-G-O-O-G-L-E-1-800-440-66453, no longer in service, to fix the issues. There is also an error code at the bottom of the screen. The Chromebook will shut off after a few seconds. If the Chromebook is powered on again and the issue persists, the animation plays again, but after a few seconds it glitches out, and strange glitching noises e-nick from the speakers. The screen goes black again, and then cuts to another error screen, stating that the computer could not be used, until a Google representative or employee fixed it. The error code at the bottom of the screen becomes just the number 6 and nothing else. The Chromebook, like last time, will power off after a few seconds. If it is powered on again and the issue is still present in the system, the animation will play again, and then glitch again after a few seconds. More glitching noises e-nick from the speakers, as the screen cuts to black again. After two seconds, another error screen appears, stating that the error cannot be fixed automatically, and must be manually fixed with the system recovery tool. The screen states that this is a last-ditch effort, and can result in serious damages to the internals of the Chromebook. If the user presses upon the T-arrow arrangement, the screen cuts to black, then cuts to another screen. The screen states that Google cannot be held accountable for errors that are caused using an authorized software present on the system. The user, if they decide to continue, must download the Chromebook Recovery app via the Chrome Web Store, to create a recovery media USB for their make-and-model. If the user decides to continue, then the user must plug in the USB into the USB port on the Chromebook. The screen then once again cuts to black, then the verification of the media begins. It is said to take up to 10 minutes to complete. Once verification is complete, the recovery process then begins. It can take from around 45 minutes to 6 hours to complete. If the recovery succeeds, the Chromebook begins the power wash cycle, putting the Chromebook into a factory state. However, this is an extremely rare occurrence, and the system recovery process is most likely to fail, within the first 25 to 30 minutes of the recovery process. If the recovery fails, the screen goes black for a second, then another screen suddenly appears with a big fatal error text in all capitals, accompanied with an extremely loud sound. The big text constantly flashes back and forth from red to black, which is likely to cause seizures in some people. Six seconds later, a message appears on the screen, stating that the recovery process had failed, and the internals have been permanently corrupted. A few seconds later, the screen glitches out and the sound completely phrases, then the Chromebook instantly shuts off. There's not even a white flash beforehand, it just instantly shuts off. At this point, your Chromebook is not only brick, but it quite literally dies. There's absolutely no power going to it. None of the lights on the front work anymore, and the charging port at this point is basically fried, and unable to charge the device. Our Chromebook cannot even be repaired at this point, you will have to buy a brand new Chromebook. There have not been any reports of successful repairs, so your Chromebook is basically useless now. Google set this whole thing up.


Thanks to the Prime Minister of the Democratic Republic of Congo says that people can start returning to the eastern city of Goulmar following last month's volcanic eruption. Jean-Michel Sama-Luconde, who is in Goulmar with about a dozen other ministers, spent a time to say that there to have been a sufficient reduction in seismic activity. About two-thirds of the city's population of 600,000 fled after the eruption of nearby byddwch chi'n mynd i'r gwneud hynny'n gwneud hynny'n gwneud hynny'n gwneud hynny.


go to our main camp. Doing two randoms right now for these two cards. We've got a Ja-Marant, Lamello Ball, Paninian War, first off the line, exclusive seven out of eleven dual card between the Grizzlies and the Horns, and then we have from 2022 Origins, Triple Auto, County Picket, Desmond Ritter, and Malik Willis. So I'm gonna go ahead and bring you over to our random screen. The way this is gonna work is I'm gonna roll the dice. We're gonna random both cards the same amount of times, whatever the dice is, and then whatever team is on top after the specified amount of dice, like random rounds, I guess the way to say it, will receive the card. So we'll start with the Ja-Marant, Lamello Ball, dual frame, Noir, first off the line card. Good luck to the Hornets and the Grizzlies. Grizzlies Hornets, perfect. We're gonna random both four times. That is it. So we're gonna start with the Ja and the Lamello. Four times. Good luck. One, two, three, four. Going to the Hornets. Hornets will receive that card, and then we've got the County Picket, Desmond Ritter, and Malik Willis, Triple. That's gonna be between the Steelers with County Picket, the Falcons with Desmond Ritter, and the Titans with Malik Willis. Four times on that. Good luck to these three teams. One, two, three, and last round, four. Going to the Falcons. Falcons picking up the Triple. Thank you guys again. I'll get this uploaded to YouTube.


Alex tab salutations. The problem with competition is humans are so suggestible it is more profitable to hijack human psychology through marketing than by providing excellence. Here's the kicker tab, right? That used to be the case. Now it's changing up a lot, right? Now it's changing up a lot. Like I didn't change, I didn't adjust what I was putting out as a creator to maximize my profits. On the contrary, I doubled down, tripled down quadrupled down on what I believe to be the truth, right? What I thought was important because I was gonna stay true to myself and to you guys, right? To my audience, right?   I think that the direction things are going is authenticity that matters now. And it has always mattered.


Another day, another shipment. Let's see what's in the box. Day in the life of the wine writer. This is the first time I've ever seen this. What a good idea for padding used corks. So difficult to do this one-handed. Oh, nice. Interesting use for corks. I'd love to know, what do you guys do using your leftover corks for?


Good morning and welcome to the 14th meeting in 2022 of the Local Government Housing and Planning Committee. I would ask all members and witnesses to ensure that their mobile phones are on silent and that all other notifications are turned off during the meeting. The first item on our agenda today is to decide whether to take item 3 in private. Are we all agreed? Agreed. Thank you. The next item on our agenda for today is to take evidence on the building Scotland amendment regulations 2022, and this is a negative instrument. There is no requirement for the committee to report on it. However, given the committee's long-standing interest in issues relating to fire safety in buildings and following on from the work of our predecessor committee, the intention is that this session will also provide an opportunity to discuss the topic more broadly. We will then take evidence on the regulations from the minister for zero carbon buildings, active travel and tenants rights at our next meeting. We are joined for this session by David Aitken, building standards team leader at Dundee City Council and past LAB SS chair, as the local authority building standards Scotland. Chris Ashhurst, who is the group coordinator of High Rise Scotland Action Group, Peter Drummond, who is the chair of practice at the Royal Incorporation of Architects in Scotland and chair of the most recent fire safety review panel, and George Edward Edwards, technical steering group manager of the fire protection association, and also Laura Hughes, who is the manager of the general insurance association. We also have Craig Ross, who is from the building standards specialist royal institution of chartered surveyors and apologies from Dr Paul Stollard, chair of the review panel building standards fire safety in Scotland. I welcome our witnesses to the meeting. There's a lot of you, which is fantastic. Before I get started, I wanted to correct a factual error in the meeting papers for the record. The paper states that Dr Stollard chaired the most recent fire safety review panel. This panel in fact was chaired by Peter Drummond of the Royal Incorporation of Architects in Scotland and I apologize for any confusion. Dr Edwards and Laura Hughes are, as I said, joining us remotely today. Given the number of witnesses for this session, I don't expect everyone to respond to every question. It would be helpful if members could direct their questions to a specific witness where possible, although I would be happy to bring in others who wish to contribute. If other witnesses do wish to comment, please indicate your desire to do so to me or to the clerk and I'll bring you in at an appropriate point. I'd be grateful if George and Laura you could indicate if you wish to come in by typing an R in the chat function in blue jeans. I'll now open up the session to questions for members. I would like to begin with the first question. Having said that, I'll direct it to you. I'll start with Peter Drummond. Are you satisfied with the requirement to use only non-combustible material in external wall cladding systems on buildings with a story 11 metres or more above ground? Is sufficient to protect occupants from the possibility of significant fire spread up the outside of such buildings? The panel was tasked with an evidence-led approach on a fairly wide-ranging brief to identify all the key areas of risk for building occupants in high-risk structures. That evidence indicated that the principal risk was above 18 metres with a still very significant risk above 11 metres. There was no substantial evidence made available to the panel that properties below 11 metres carried with it the same level of risk. Now, there's a number of reasons for that. As you know, residential buildings in Scotland and some other use classes now require sprinklers within them. There are very clear requirements in the Scottish building regulations for fire tender access and for five fighting facilities. What I would say, however, is that the panel's view on that was predicated on all these different parts of the regulations working together in a unified approach rather than just one in isolation. For that reason, my view is that we should continue to monitor the fire risk and emerging patterns in buildings below 11 metres to ensure that our guidance and, indeed, in some cases, regulatory bans remains robust. Thank you for that response. I just wonder if anyone else wanted to come in on that question if not, we can move on. No, okay, so I'm going to now bring in Miles Briggs with a couple of questions. Thank you, convener. Good morning to the panel. Thank you for joining us today and online. I wanted to carry on the line of questioning with regards to the regulations specifically, which would prevent the use of highly combustible metal composite material in the use of external cladding and insulation. Do you think that the definition of that material specifically is robust enough to ensure that any dangers posed by such materials has now been reduced to the most reasonably practical way of taking this forward with the regulations? Or is there other things we need to look at specifically with regard to that metal composite? Peter, you touched upon that, so maybe start with you. Thank you very much. That occupied a very significant part of the deliberations of the panel. The concern was that we had to provide something sufficiently tight to pick up the most flammable forms, but which wasn't open to abuse. So, for example, we've recommended a ban on paneling up to 10 millimetres, comprising a calorific value of 35 megajoules a kilogram. Apologies for that. The risk, therefore, is that someone will produce a 10.1 millimetre board or a 34.9 megajoule calorific value. Again, for that reason, our advice to the ministers and to yourself is that that will be kept under review. Lightly, there will be new construction product materials coming through. Because of the provisions of the building safety bill and the new product regulator, one would like to think that they will be tested more robustly, but I would suggest to you that we should not be taking risks in respect of people's health and wellbeing and their lives. Therefore, there is a need for the on-going work of building standards and review panels, such as the ones that both myself and Dr Stollard shared, to keep coming through the system and also feed in real-life intelligence. I am also going to take a slight diversion and point to the wealth of very detailed technical assessment done for the Grenfell inquiry, not all of which has been fully analysed by us yet, which may also cast light on where we need to make future changes to the regime in order that we continue to have a set of building regulations that put public safety front and centre. Does anyone else want to come in and online, George?  I understand the focus on the metal elements in the external wall system, which of course is significant, but as an owner and experiencing it myself and across Scotland, we're finding that if we're wanting to get guaranteed safety, it really has got to extend beyond that, and I think that that's been touched on, but to focus simply, and I can say of a building that I'm intimately involved with, the cladding there is class as A1, wonderful, but 14% of the wall area is rendered, and behind that apparently is some EPS system, so merely, if I can put it that way, focusing on metal and is perhaps unhelpful, and certainly some of our owners are beside themselves with that whole definition and can't understand why their building can't be deemed to be safe when it actually hasn't got this cladding on it. Technical difficulties, and Laura and George apparently cannot hear us, so I'm going to temporarily suspend until we sort that out, because I really want them to be part of this conversation, so apologies for that. Apologies, Miles.