Welcome to another edition of Barnes Takeout. My name is Kaylyn Jewel. I am senior instructor in adult education at the Barnes Foundation. And today I wanted to talk about some of the ancient objects that are on display in gallery number 15 and we'll go into gallery 16 for just a moment. But here we see gallery 15. We are looking at the east wall. Gallery 15 is filled with all kinds of different works of art. We've got lots of great paintings by Matisse, by Cézanne, some American painters like John Cain. But what I wanted to spend the next few minutes talking about are these ceramic pots that are on display in the glass case along the east wall of gallery 15. And we're gonna look more closely at this one here. So here it is. And it's an object that was made in the ancient Greek world. It was made sometime in the second half of the fourth century BCE. And it is made out of terracotta. It was thrown on a potter's wheel. We can get a sense of that manufacture if we look at it a little bit more closely. One of the things I love about this example is that the rim or the lip of the upper opening of the neck of this vessel is a little bit wobbly. And we can get a sense. Those of you who may have made objects on potter's wheels might know that sometimes it's hard to get it centered. And the fact that it's very soft and filled with the clay has a lot of water in it at that moment makes it sometimes a little bit difficult to deal with and to handle. And so we can see a little bit of that happening with this kind of asymmetrical wobbly opening at the top. So we're getting a sense of really the artist who made this. If we look down, you can see that we've got these really great, tall, thin handles which are called strap handles, which were applied to the vessel after it was finished on the potter's wheel. If we go down, we can see this really fabulous, pretty intricate and elaborate decoration, which we'll come back to in a second.  The overall shape of this object is known as an amphora. And an amphora in the ancient world, whether it's the ancient Greek world or the ancient Roman world, were really used as storage jars. They were, in many cases, kind of like the cardboard boxes of the ancient world. But what we see here is a very elaborate version. And it's in a remarkably good state of preservation. There aren't a lot of cracks in it. We get an indication, if we look at one of these handles, that maybe this handle had broken at some point and there's a little bit of a break here and was repaired. Again, we don't know when it was repaired, but at some point it had been. And these types of vessels, these amphora, have different proportions. And the version that we see here is very tall and slender. And we can get a sense of that tall cylindrical proportions by the decoration that is on the surface. So this sort of tall element here in the center of the object, the elongated thin handles that we already looked at, this delicate little foot down at the bottom sort of gives it a sort of stretched out quality to it. And those slender proportions have led art historians who specialize in this material to connect these types of vessels to the region of Apulia or Apulia in southern Italy. So we're talking about what corresponds to the boot of the Italian peninsula. And it's interesting to think about ancient Greek material appearing in Italian places. So Apulia in the ancient Greek world was one of the first colonies that the ancient Greeks formed after they left their kind of mainland and the islands in the Aegean. And so when we see these types of objects, these ones that are very elaborate, they are painted on all sides. This is the side that is facing us when we go into the gallery, but let's look at a side view of it. And here you can see that underneath those strap handles, we've got this decorative motif, which looks vegetal and it's a palmet. We see a similar palmet on the upper part of the neck, and there would be a palmet. If you looked at this object on the other side, you would see a palmet on the other side. So let's figure out this decoration because the decoration is sort of helping to explain what this object had been used for. We can see here, and I'm just gonna focus on this central scene. We can see here we have two human figures who are represented on either side of this tall rectangular monument that is set up on a plinth. You can see that the figures here are standing on the ground, but they're lifting their legs up to place them on top of this little plinth or this platform. You can see that the female figure is holding some kind of a fabric ribbon, and she seems to be walking towards the central element. While the male figure on the left is holding up a little bottle and sort of looks like he's pouring it onto this object. So what on earth is going on here? What we are seeing is a funerary scene. We are seeing a representation of something known as a funerary steely, or you can think of it kind of like an ancient tombstone. And these were set up in cemeteries or necropolis in the ancient Greek world, and they were often visited by family members of the deceased, and they were often decorated. So what you see here is the decorating and these sort of ritual libations that are being paid, these votive offerings that are being paid to this deceased person. We don't know who it is. We don't know the identity of either of these figures because we don't have any inscriptions, but we are generally seeing here a representation of a funeral scene or people coming back after a funeral had occurred. And we can also get a sense of the type design that is represented here. This type of amphora is often referred to as a pan-aphanatic amphora, and that is a reference to vessels that were given as prizes for different kinds of athletic competitions. So this would be like a trophy. These forms would be like trophies that were given. They were filled with oil, often perfumed, and they were given to victorious athletes. And so the one we are seeing here was not used for that purpose, but it sort of hearkens back to that idea, especially if we look at the other side. So let's look at the back. And what you see on the back, which you can't see when you're in the gallery, so it's nice that we've got these pictures of the back, what you see here are two male figures that are covered up in these big robes that are called hemations, and between them this curious little symbol. And it's not actually a symbol. What you're seeing are jumping weights, and they are weights that were used by athletes in ancient Greek athletic competitions. And we have several of them that survive in the archaeological record, and I'll show you a picture of a pair of them that are on display in the Archaeological Museum in Athens. So you can see here, this is where the athletes would hold them, and they would use them as a way of kind of propelling them forward in things like the long jump and other jumping competitions in the ancient world. So one other aspect that I wanna point to is this little object here that the male figure on the left is holding, and then it's a little, it's a bottle. It's a type of bottle known as a Lechithos, and it's an oil bottle. We have several, there are three Lechithoi that are on display in gallery 16, which is right next door to gallery 15. So I'm just gonna show you gallery 16 for a second. This is the North Wall, and you can see two of them on display in the glass case. So you can see the tall one here, and then a smaller one here, and you can see that distinctive handle there. So again, these are the types of bottles that would be used by people visiting funerary monuments. They would be filled with oil, and this oil would be poured onto the funerary steely. So when we return back to this object, what we are seeing are two people who are engaging in the kind of memory of the deceased person. We also know, and I'll sort of wrap up with this, we also know that this vessel itself is hollow. So the bottom here is actually open, which means we know that this vessel specifically was also used within a funerary context. So the Lechithoi that you see in the male figure here holding would have, if you imagine a person visiting one of these monuments, there would have been an amphora like this on display next to the big steely kind of on this plinth, and a Lechithoi could have been used to pour perfumed oil into this object. And by doing so, the fact that it is open at the bottom allowed the oil to seep onto the ground below it and into the ground below it. So it's a way of connecting the earthly realm with the realm of the deceased. So it's really an interesting object when we think about it in those terms. So when you come back to the barns, I would keep an eye out for all of these interesting vessels that are on display, because they really give us a sense of how ancient people, in this case how ancient Greeks, were dealing with their deceased loved ones and how they were using these objects that we see on display. So thanks for joining me today, and be sure to stay tuned for another Barns Takeout. Take care. Thanks for watching, and for your support of the Barns Foundation.


You know, I often hear folks talk about how concerned they are with freedom of speech, but they never actually address the real threats to freedom of speech, which is an actual crackdown by the government on your political speech. It happens all the time as it relates to BDS and increasingly as the Black Lives Matter movement gains more prominence rather than actually addressing the concerns of people out in the streets protesting against police brutality, Republican legislatures and dozens of states literally are choosing to ignore their concerns and just try to criminalize them from speaking their mind in the first place. The first state to sign an anti-protest bill into law is Florida. Ron DeSantis did that just this week, but I alluded to the fact when we talked about that that Florida isn't alone and this new article from the New York Times sheds a little bit more light on how prevalent this is and how egregious these laws are so as Reed Epstein and Patricia Mazae of the New York Times explains Republican legislatures in Oklahoma and Iowa have passed bills granting immunity to drivers whose vehicles strike and injure protesters in public streets. A Republican proposal in Indiana would bar anyone convicted of unlawful assembly from holding state employment included elected office. A Minnesota bill would prohibit those convicted of unlawful protesting from receiving student loans, unemployment benefits or housing assistance and in Florida Governor Ron DeSantis signed sweeping legislation this week that toughened existing laws governing public disorder and creating a harsh new level of infractions. A bill he's called the strongest anti-looting, anti-riding, pro-law enforcement piece of legislation in the country. The measures are part of a wave of new anti-protest legislation sponsored and supported by Republicans in the 11 months since Black Lives Matter protests swept the country following the death of George Floyd. The Minneapolis police officer who killed Mr. Floyd, Derek Chauvin was convicted on Tuesday on murder and manslaughter charges, a cathartic end to weeks of tension. But while Democrats seized on Mr. Floyd's death last May to highlight racism and policing and other forms of social injustice, Republicans have responded to a summer of protests by proposing a raft of punitive new measures governing the right to lawfully assemble. GOP lawmakers in 34 states have introduced 81 anti-protest bills during the 2021 legislative session, more than twice as many proposals as in any other year according to Ellie Page, a senior legal advisor at the International Center for Not-for-Profit Law, which tracks legislation limiting the right to protest. So I want to repeat that 81 anti-protest bills have been introduced in 34 states in the 2021 session. This is absolutely terrifying. Anyone who claims to care about freedom of speech, who's concerned about censorship in America, this is the time to sound the alarm right now. This is absolutely horrific to see. And you know, you hear these details, right? You hear, oh, well, it offers immunity in some of these bills and states to people who mow down protesters who drive their cars into crowds of people. But that can't be correct. That has to be hyperbole. There's no way that that's real, right? Actually it's literally real. It's included in Florida's bill as well, which has gone into law immediately. Now what these laws do and what the one that Rhonda Santis signed into law does is they grant immunity to people who drive into crowds of protesters and literally kill them by shielding them from lawsuits and criminal penalties. So long as the person who drives into the crowd of protesters, they claim that it was because of self-defense. That's literally how they're justifying this. So in other words, the neo-Nazi who drove into a crowd of protesters in Charlottesville that killed Heather Heier, all he had to do, if he lived in a state with one of these laws, is say it was self-defense. I know that there's video footage of me driving into a crowd of protesters, but you know what? I felt threatened. I felt like this mob was coming after me and I had no choice but to drive into them to stop them from attacking me. Now what's funny is that the lawmakers in Florida will defend this provision here and they'll say, no, no, no. That's different actually because the individual, the neo-Nazi specifically who drove his car into a crowd of protesters in Charlottesville and killed Heather Heier, he wasn't actually doing it out of self-defense. He was doing it because he was a bad person. He wanted to kill people. Okay, first of all, how exactly are you going to prove that? Second of all, do you not understand how you're incentivizing driving cars into protesters by Republicans who think that these folks are terrible? They're the mob? I mean, look at that gun-toting couple that threatened to shoot protesters that were marching to a governor's mansion. Folks around this country clutch their pearls as they see these protests and they want to see violence be done to folks. I mean, look at how many people defended Kyle Rittenhouse. So what you're doing is you're giving these folks an excuse, permission essentially, to kill protesters. It's unbelievable. It's insane. Folks claim to care about freedom of speech, and they're not talking about this, unreal. Now this is less controversial than it should be because, you know, they can point to all of the rowdy protests, the rioting, the looting, and they could say, look, we're just solving a problem. We're responding to an issue. But if you are buying into that frame, understand that you're being duped by a media that has an incentive to publish sensationalist content. Most of the Black Lives Matter protests are actually overwhelmingly peaceful. But which one do you think a news organization like CNN or Fox News is going to want to play? The one where there's violence, which will attract eyeballs, which will in turn attract advertisers, or this boring, peaceful march where people are just chanting justice. I think we know the answer to that. And don't take my word for it. Take the word of analysts at the Washington Post, who in 2020, they looked at more than 7,300 protests. And here's what they determined. The overall levels of violence and property destruction were low, and most of the violence that did take place was in fact directed against the Black Lives Matter protesters. In short, our data suggests 96.3% of events involved no property damage or police injuries, and in 97.7% of events no injuries were reported among participants by standards or police. In other words, because the media has led folks to believe that these Black Lives Matter protests are overwhelmingly violent, well, maybe this type of legislation is justified. Now how they try to justify giving immunity to people who drive their cars into crowds of protesters, that's a different story. But when folks are afraid and they think, oh my God, these protests really are violent and we've got to do something. I mean, maybe I support the First Amendment, but we've got to do something, right? They're giving the government permission to do things like this, and guess what? You might not like the Black Lives Matter protests, but this does cut both ways. Republicans that want to protest? If you live in a Democratic-controlled state that institutes more regulations on guns, for example, well, guess what? This can be applied to you just as easily. Now luckily for you, most cops are Republicans and conservatives, so they're probably not going to crack down on you so much as they crack down on Black Lives Matter protesters, but still understand the implications of this law and how it affects everyone equally. You might think that this is a gotcha to the Black Lives Matter protesters who have been demonized, but this is, make no mistake about it, further erosion of the First Amendment. And the folks who aren't talking about this but report to CARE about censorship and freedom of speech in America, they're frauds if they're not sounding the alarm right now about this issue.


Now, before I go further, I've been talking about religion or Christianity like it's one thing, and we cannot really understand conflict unless we get into more detail about religion. And again, I'm going to focus on Christianity because that's the dominant group in the public sphere, at least in the North American context, and I'm going to talk about these groups' official stance towards the knowledge produced by religion and science. Now I want to put official in sort of quotation marks here because while Catholicism has a close to official way of making claims, most of Protestantism doesn't. So I take official to mean the sort of preponderance of opinion within these groups. So let's start with Catholicism. There is an official teaching about fact claims here. So at least officially, Catholicism incorporates scientific discoveries into its theology, often with a time lag, and modern Catholic leaders have long claimed that they have no methodological conflict with science. So for example, every pope since the 1930s has affirmed the autonomy of science. Similarly, Catholic teaching holds the doctrine of two truths. The scientific knowledge cannot contradict with religious knowledge since they both come from the same source. So this has been long supported by papal statements. I'll just read one for you here. This is Pope Pius XI in 1936. Science which consists in true recognition of fact is never opposed to the truth of the Christian faith. And then 50 years later, John Paul II says, science can purify religion from error and superstition. Religion can purify science from idolatry and false absolutes. Of course, Catholicism believes in all sorts of fact claims that do not make sense with the contemporary science, such as the idea that Jesus rose from the dead. The best way to put this is that Catholicism is not in conflict with any fact claim that any scientist actually cares about. The resurrection does not matter to contemporary science. The only group I'm just going to lump together are conservative Protestants. In the US, these people are called evangelicals and fundamentalists. This is perhaps 25% of the American population. In the UK, you would use the term here conservative evangelical. In Scotland, these people are not quite clearly represented by a reform theology. The people who wouldn't be part of the Church of Scotland, but the we frees and people like that. It is extremely difficult to distinguish between an evangelical and a fundamentalist because the people in these communities are not even clear about the UC's terms themselves, so I'm going to lump them together. But in general, in this tradition, there would be support for the idea that the Bible is true in what it says about the natural world. It's extremely important in this world to defend the truths of the Bible. One way to identify Protestant fundamentalists, for example, is that they're more likely to not try to accommodate scientific research at all. They would say that since the Bible says the humans were created distinct from animals, that's true and the scientists are simply wrong, end of story. Another way would be that they want a super plain reading of scripture with as little apparent interpretation as possible. Evangelicals, who are sort of more liberal in their interpretation than a fundamentalist, would still want to say that the Bible is true, but also want to say that science is true. This has led to all sorts of efforts to make scientific claims compatible with a transparently read Bible. For example, some evangelicals say that what the Bible says about Adam and Eve is true, it just took a lot longer than one day. And the God who has caused these evolutionary changes all along. Or another example would be that if you look carefully at the Bible, you would see that it doesn't talk about days. Each day actually means length of time. But in general, if you find a religious leader saying that Darwin is wrong, they're most likely from this conservative Protestant tradition. Another group are liberal Protestants or mainline Protestant. In Britain, I would say many Anglicans would fit into this category. In Scotland, in general, the Church of Scotland would generally be in this category. In contrast to the conservative Protestants, this tradition has basically embraced each scientific claim as it has emerged and modified theology to make this all fit into one seamless piece. So there's a very nice interpretation of this by historian John Roberts who says that mainline Protestants have maintained that the progress of scientific investigation required Christians to make significant revisions in their apologetics, doctrine, and biblical interpretation. This is one of the defining features of the liberal Protestant tradition in the first place. So to return to the Genesis account, a liberal Protestant would just dismiss the factuality of it and simply say, this is an account of the biblical writer trying to understand the mystery of God's creation. But you're not supposed to take this literally. Like Catholics, it's extremely rare you'd find any mainline Protestant or liberal Protestant leader who disagrees about any scientific fact claim that any scientist cares about. Again, liberal Protestants would still believe in things like the resurrection, but since there's no scientific department of anti-resurrection studies, it doesn't really ever matter for any relationship between religion and science. So for religion and science, it's important to know that many of the divides we see among Protestants, I just labeled these different types of Protestants for you, these were essentially caused by debates about enlightenment knowledge in general and science in particular. So by the Enlightenment, I mean these ideas that emerged in the Enlightenment era in Europe, which emphasized the idea that we should be able to determine facts about the world through our own observation and reason. But if you go back to the 19th century, again, mostly in the North American context, Protestants were divided over things that are not really relevant today. There were Lutherans versus Methodists, Baptist versus Presbyterians. Nowadays, you really have to be an expert to know what those differences mean in terms of theology. But what eventually became to happen was that within each group, there was a debate over biblical knowledge. How do you know what the Bible really says? Well, one group, influenced by Enlightenment thought, said we should use our observation and reason to decide what the Bible really says. Another group rejected that view. So a classic debate at the time, this is like the end of the 19th century, was Jesus born of a virgin or not? Well, you can go back to the original Hebrew and decide, well, does that word mean virgin or does it mean young woman, et cetera? So the liberals were the ones who would look back to the original Hebrew and decide what was the word actually meant at the time. This divide caused a schism within Protestantism that exists to this day. The divide between Lutherans and Methodists and the like is largely gone in society. And you're split into these groups about debating over knowledge. And so essentially, this divided just described between conservatives and liberals is not only about knowledge itself, but also due to debates about science. For example, the liberals were the ones who were accommodating to Darwinian views and the conservatives were ones who were not. Let me make one other little bit of history here to clear out, finish up my story, which is, as I just said before, there was in the late 19th century, there was this unified conservative Protestantism splitting off from what is liberal Protestantism. In North America in the 1940s or so, these conservative Protestant split into what are now called fundamentalists and evangelicals. The evangelicals are sort of a compromised movement between what they saw at the time of the rigidities of fundamentalism and the wishy-washiness of the liberals. That here hits the cultural sweet spot in American context and evangelicalism becomes by far the dominant group. Fundamentalism remains a fairly small minority and liberals somewhere between evangelicals and fundamentalists in terms of size.


Okay, so we're moving to our second talk this afternoon. The speaker is Frank Paulman from Munich and he's going to tell us about quotient symmetry protected topological phenomena. Good. Thanks. And also thanks to the organizers for putting together this nice program and for giving me a hint to, for me to speak here. So, so I'm actually very happy about Nathan's talk because you already I'm showed that there are interesting phenomena that we find in symmetry protected topological phases. And what I want to show is that several of these phenomena can also survive even though the face itself is trivial. So so basically I want to show that certain topological phenomena, including edge modes and even topological phase transition can occur in trivial phases. So, so in the work that I'm presenting has been done in collaboration with Ruben Verizon, who was a student in my group in Munich, and Julian people who currently still is a student in my my group. So, so, so let me start slowly and I start basically reiterating a few things that Nathan already pointed out. So basically want to discuss or introduce the concept of different quantum phases and then show how we understand or distinguish different symmetry protected topological phases. So, so first of all, this is very general, but, but if we just talk about a gap quantum phase of matter. Well, we first of all talk about zero temperature. And, and we say that two Hamiltonians are in the same phase if there exists the path of gap Hamiltonians connecting them. In a, in a picture we would say that well we have now to Hamiltonian it's not, and age one, and we can now tune along some path given by by alpha and and the gap remains open so we can just do this without encountering any face transition, then we would say that these are in the same phase. And if there is an avoidable gap closing, then we would say that these are two different phases. So, of course, we might accidentally run across the face in a different position, but then we can find a different path, which, which is a gap then be on the same face but but if it's impossible to find any, any, any path without closing the gap or without encountering any kind of discontinuity, then, then we would say these are two different phases of matter. So and the kind of one of the prime examples for different topological phases is the the so called whole day in phase and while conventionally we find the whole day in phase in the spin one Heisenberg chain. And the, we can also think of a dimerized spin one half chain. And the spin one half chain or the dimerized been one half chain also has an SO three symmetry, where as the two cross the two is the subgroup of the symmetry time reversal symmetry translation symmetry so there's a bunch of symmetries in in in in this some showing here. And now, as we tune the parameter delta, in this case, we find that this system does undergo a phase transition at delta equals to zero. So, so really the translation variant Heisenberg chain is critical. So so so that's a critical point. And then we find that left and right of this critical point. There are two gap phases. So, and these gap phases cannot be distinguished by symmetry breaking, but instead they can be distinguished in terms of so called symmetry fractionization and this is something that Nathan already talked about, but let me just elaborate a little bit about how we can distinguish these two phases. And how, how, how, how basically, we can derive the topological invariant or label the two different phases. So, so, so we know that the Hamiltonian is gap, and we find now symmetric ground state. Which is now symmetric under certain symmetry operations. abstractly speaking this could be just G and age, but in practice this could be for example, global like spin rotations that are rotate all the spins by pi about the x axis or the axis. Right so so so have now a certain symmetry group under which the ground state and also the Hamiltonian is symmetric. And now what we are doing is we just now take a segment of our ground state and we apply the symmetry operation to it. Since it's a symmetry operation, it wouldn't change the the bulk, right because by definition if, if we just apply the symmetry operation to the state, it's invariant under the symmetry operation. However, it can act non trivially on on the edges. And what we can then do is we can then say that well, let us now represent the symmetry acting on the entire state in terms of operations acting only at the edges. And, and, and the assumption that we make now is that the symmetry acts, we were trivially on on the bike so we would say that inside like the the the local degrees of freedom have a have a linear representation of the of the symmetry. And if you stick to the example of spin rotations we would say that we can have like an integer spin rotation right so an integer spin rotation is a, and if you just take the, the, the algebra that's, that's a linear representation of the, of, for example, so the boundaries, in this case, however, might be a projective representation so productive representation is then a linear representation model or some face factor, and in particular sticking to the example of spin rotations, the, the boundary might transform in terms of half integer spin. Right so so and so in this gives us now a way how we can classify the different projective representation so we say that well, we have no certain symmetry group for symmetry group we can have certain different projective representations which are classified by sure multipliers or the second group homology, and then I can now classify can now just steal the labels that are used for classifying different projective representations and I can label different symmetry protected topological with it. So, so now we can apply exactly this to the aforementioned whole day model. So, so we have now here our dimerized Heisenberg chain, and we had this is the same face diagram that I showed before so we have the critical point where the model is a translation Heisenberg model, which is gapless and then we have these two gap phases. And if you now look at it, if we just look at the left hand side, we find that, well, at the, at the boundaries we have singlets and the singlets transform trivially under the symmetry and these are s equals to zero state at the boundary. So, so, so this would be the face factor is, is zero. So, so our x and our y commute. Right so so this is what we would say is now a trivial phase. So if we just take the other side of the face transition. Here we have the spin one half degrees of freedom localized at the edge right if you just take the extreme limit of dimerization. That's pretty clear that we have here just a single spin one half and the, the Z two cross Z two, or like the Z two rotations about the x or y axis, they're just represented in terms of the poly matrices the poly matrices anti commute. So we find that here, the face is, is pie. So now we have two different labels and these labels cannot change, unless we cross a face transition. So, and with these different labels or with the non trivial representation of the symmetry in terms of the edges, we get certain topological phenomena, in particular, based on these commutation relations we can directly deduce that we will have degenerate edge modes. And also, if we just look at the entanglement spectrum of the system, we find that they're necessarily degenerates. Good. So, so far, we have now, hopefully somewhat clean definition of what we mean by the, by an SPT phase. So now I want to just do the next step and I want to just destroy it. So, so let us now say that we have a Hamiltonian realizing a certain SPT phase. How can I trivialize it by trivializing I mean that I just can connect it to a trivial state without having to undergo a face transition. The way I mean one way on obvious way is that we just break the symmetry. So let's say we take the C2 C2 model, and we apply a staggered field to this, then the system becomes trivial, and all these nice features including the, the edge modes or degeneracies are immediately lifted right even if I apply a really tiny field. The edge modes are lifted and, and, and, and all these nice property go out of the window and if I look at these face diagram, and I just apply a staggered magnetic field. Then I would also point that there's no longer a phase transition. So, so the gap will be, we'll get small but we wouldn't hit an actual phase transition. So this is option number one, but there's another option which is maybe less obvious. So we can also extend the symmetry group. And if we extend the symmetry group, we can again find a path connecting the topological phase to the trivial phase. But the question is, if we do if we are doing this, to which extent to certain features that we learned from about from the topological phase, or certain how to which extent to certain topological phenomena actually survive. And this is what I want to discuss in in the following. I want to discuss it by by considering a very concrete model. And let me just walk you through the, the, the Hamiltonian. So, so I'm considering now a Hobart model. And so I have just a Hobart model, where we have now hopping between neighboring sites, but then on top of this, we have a staggered chemical potential. So the, so so the chemical potential is, is, is alternating so on on even sites say it's, it's it's lower than on on odd sites. And that's it's basically right so we have just a staggered chemical potential. And then we have a dimerized dimerized hopping to be allowed the hopping between neighboring sites to be the dimerized to be on say on on on even an odd bonds we have a different hopping strength. And at last we have the onsite Hobart interaction. So, so what I want to do now is I want to look at the phase diagram of this model, and I want to focus first on on a special limiting case. Let us now first consider the case where you, the Hobart interaction is is very large. The Hobart interaction is very large. We can apply perturbation theory, and we do get exactly the Hamiltonian that we had before, right so basically if you was infinitely large, the, the, the top of this phase diagram is exactly described by the Heisenberg Hamiltonian with a dimerized exchange interactions like so so so so for you infinity we exactly have the model that we had before. And so we know in the limit of you being infinity, that there is a phase transition between an SPT phase, and a trivial things. So but now if we just plot the entire phase diagram which we evaluated here using DMG, and it has also been evaluated by others before. What we actually find is that now, if we just lower you sufficiently that in that case, there is a path without actually crossing the phase transition. So, so there's a question in the chat, whether we can consider this extending of the symmetry as a central extension of the group, and maybe hold on for a bit I'm getting into the detail what we actually mean by extending the symmetry. So, so so now basically if we are considering now this, this, this full model model, we see that there is now a path that we can adiabatically connect them right so it's easiest to see even without doing numerics is we just start from the fully dimerized limits. And in that we can can see how we can reduce you to the non interacting limit. And once we are in the non interacting limit, this is basically just an SSH chain in the presence of a staggered chemical potential. And then we can actually directly show that there is a path where we never close the gap. So, so so now what we have shown that that there is actually a path between what used to be the SPT phase to what used to be the trivial phase so so it's, so so we have now actually trivialized to the, the SPT phase by extending the symmetry. And what basically happens in terms of the symmetry is that the, and how about model has an SU two symmetry, which in the limit of you goes to infinity becomes basically the quotient group of SU two and the fermion parity symmetry so so we have SU two mod the fermion parity. So so basically if you was infinitely large, then the, then the, the parity symmetry P becomes just a number which is is then same same minus one for for each side, and, and that's it. And if we have SU two for SU two, we know that there are no non trivial projective representation so the H two of SU two over you one is equals to zero, while H two of SO three is actually equal to Z two, so we get these two two phases. So, so, so what we basically have shown that is that the holding phase can be trivialized by adding charge fluctuations of the charge degrees of freedom. So, so, yes. So, so, so, so the way I mean I'm talking about how filling as you say, and so basically, if I just define my habit interaction display this ensures that the ground state is always at half filling. Yes. That's another question in the chat. Does color density show the parameter fire. No, no, no, thanks for this question. So, so, so the, the, the, the quantization of fire to having fire either. Zero or pie this only exists this distinction only the stick this distinction only exists for for you being infinitely large for any finite you this, this, this is no longer well defined as I'm showing in a, in a, in a moment, but let me just come to this so the, the, the, the color coding is really just really nice. So there's no no meaning to having these colors just to show that we can go without a discontinuity from one to the other phase. This is all all this means. Good. So, so, so, so but now we see that there is a path where we can go from one phase to the other without crossing a phase transition but we still we see that there is still a phase transition. In particular, we see that if we go down to a certain you, what what happened is we just have now the, the phase transition so this is a C equals to two to one critical point as we go here, down to some UC two. And if we just go below the C two and above UC one, then there is actually a first order transition between those two phases. So we're just commenting on what's going to happen or how we can explain this in a, in a moment. So, so let me let me know first discuss the stability of, of the edge modes. In particular, what we find is that the edge modes that we have for Delta, larger than than zero so so the these topologically protected edge modes, they are actually robust up to a certain certain point in this case diagram so basically if we look at the ground state degeneracy, we find that there is actually a four fold degenerate ground state degeneracy, all the way down to this dashed blue line. We have not a bike transition but a boundary transition to another state where where the, where the, where this feature, namely this 240 degeneracy disappears. And the way that we can understand it is that if we just look now how the symmetry acts on the on the boundary states that we find for the full Harvard model. The other relation is that our x times r y is equal to the fermion parity times our r y times times our x in the limit of you goes to infinity. The fermion parity is frozen to to minus one, and we get the r x times r y is equal to minus our y times r x, but, but the eigenvalue of PL cannot change immediately so we have a gap protecting. This this quantity and thus we find that the property of having the edge modes is actually so so so robust for a certain range of of you, a fairly easy way of seeing this is that if you just go to the completely limit if you go to the limit where delta is equals to one. So here we just have a completely like a completely decoupled spin on on on on the side, and then this you see that there is a level crossing exactly at you equals to to delta. Right so and, and if we now plot the low energy, the ground state and the low energy excitations of the full model for a finite delta so so now we are here at this, this dashed line. And what we see is that there is now this level crossing so for sufficiently large you, we find that there is a fourfold ground fourfold ground state degeneracy, and we can now identify the different ground states by having either kind of on the first and the last site and this is what this indicates that both up, both are down, one is up, one is down or the one is down or the other one is up. And if we, if we cross below this, then there's this kind of level crossing at the boundary states, and we see that the, the unique ground state is given where both, where both spins here are localized at the, at the left boundary So, so so so just to, to conclude so so so basically here, we have some, some cartoon state, which is a state that we would that this, this is a state that we would get for, for an infinite kind of a staggered field. Yes. Yes. No, no, no, no, if we have, and if you look at the Hamiltonian we have the staggered potential. This is very important because if we don't have this, if we don't have this, this delta. I mean, if we don't have this delta, then we basically have here some SSH chains and then we know that then then there are actually different topological phases and in that case, even if if if we, if we, if we wouldn't if we would set delta to zero. We have actually two distinct topological phases in terms of inversions like lettuce inversion symmetry. Yes. You say that it's clear that at some point there should be I think that you know, well, I think that I wanted to say that several years ago, I think that I think that's the point. Yes, exactly. So, so this you see to is an I think this is this has been looked at before.  I mean, what what what has been looked at before is this this line this line of delta equals to zero, this one has been looked at and it's known that there is this extended critical regime. Yes. But what we are just now basically arguing is that we are just transferring this to this exactly so so. We put these things together saying that well in the limit of you infinity that we do have SPT for you equals to zero we don't have SPT and now we are just putting this together and and one of the non trivial statements at least which wasn't obvious to me in the beginning was that that in fact the edge modes are robust right so we just find that this this property of these edge modes is in this is that these are protected for a finite range of parameters. But what do you mean. No, no, no, no. No, no, it's just the same so so so so in fact, this would maybe what you're referring to would be the case when delta is zero if you delta zero then you would have an SPT. And then you have a sushi for a model with fractional charges, but but but but we prevent this from happening by having this this ionic carbon model. Yes. And just because of the staggered chemical potential. So if you just, if you just were to make this potential very large then you find that the ground status is the one that the Dublin is on on the right on the left side. The, the, like, like you mean here on the right hand side. On the left side that it terminates with the at the icing transition. Probably I don't, I don't know any allergic I mean this is basically just reading off the results from from from from the numerics. And this argument, which which I just gave now for the degeneracy of the ground states we can directly apply also to the degeneracy of the entanglement spectrum in particular what we can do is we can now think of an infinite chain, which we cut at a given bond and put a partition into subsystem a everything left of this bond and subsystem be everything right of that bond, and we can then look at the spectrum of the reduced density matrix. And we find that that there is again a phase transition where are there basically the low energy states of the of the entanglement spectrum or equivalently the dominant eigenvectors of the reduced density matrix. Either have half integer or integer and spin. And then this is, I think also, at least intuitive, a nice understanding, because in the limit of large you there are no charge fluctuations, and the entanglement spectrum, either is only integer or half integer so we would find that either all energy states are degenerate, or all of them are non degeneracy. So, and, and if we if we allow charge fluctuations, then have like a half integers can can hop between the left and the right. And then at least the entanglement spectrum starts to mix and then there is again, just at a certain value of of of you, then there is a crossing of the integer and half integer levels. So as shown here. So, so so what I've shown so far is that the despite that those two phases can be now adiabatically connected that the edge modes remain robust over a certain range of parameters.  And in particular, there are a few steps that I just want to mention here. So first of all, there is a duality symmetry between delta and minus delta. And so that if there is, if there is a direct transition between the two phases that always has to be at, at delta equals to zero. And we can see this duality symmetry from some modified translations where we just have now some symmetry D acting now on our line operators, which is in minus one to the power of n times CN right recall that the regular translations are no longer symmetry because of the staggered field, but these modified translations are then still a symmetry of the model. And based on this, we can also deduce that based on these modified translations that in the limit of you goes to infinity that we, we have a guaranteed lead Schultz-Martus theory, which guarantees a face transition at delta equals to zero. In particular, what we, what we use is that, and that in, in this case we have an onsite projective representation of the spin rotations right so so so for the translation invariant case, we have one spin per unit cell. And then we find that on each side, the rx and ry anti-commute and thus we can just immediately apply the deep Schultz-Martus theory. We can also extend this argument to, to finite you, and particularly if we have finite you, we actually find an emergent deep Schultz-Martus and theory, which, which, which enforces the stability of the, of this critical line further down of this face transition and let me just briefly tell you how we, how we can come to, to this conclusion. So, so, so, so I just pointed out, sorry, let me just come back to I pointed out, if you goes to infinity and if you don't have any charge fluctuations that then parity is a six to two minus one so we can just replace it by a number and that's why we find that they, that's onsite projective representation. So now, we argue, however, that the fermion parity string has a long range order as long as a fermionic parity operator is is kept in particular. We can now calculate a string order parameter where we just apply parity to a consecutive number of spins. We find that this scales as some constant smaller than one and larger than zero times e to the theta times the n minus m. So, so something which alternates and this theta can either be the zero or, or pi right so that we call in the limit of u infinity p is minus one and then we find that theta has to be pi because it's plus or minus one for even odd length of strings. So, so and if we have data equals to pi, this implies now some emergent anomaly that forbids unique and gap ground state so we can, for this emergent analogy we can show that by contradiction that if we assume a unique and gap ground state that we are running into a contradiction assuming that we have a pi here and thus, we can then prove that if we have this emergent anomaly that they are that that this yields a leap Schultz-Martis theory. And, and, and now there's basically two ways out of this. So there are two ways out of the system for kind of changing the fermion parity, it's either at the fermion parity gap closes, or it spontaneously breaks a symmetry, which destroys the these modified translation symmetry. And in our case, the system opts for the second option so it spontaneously breaks the symmetry. And this is happening along, but this is in this plot hard to see between UC one and UC two. And what we, what we actually do, we are now plotting here at along this, this line, we plot the, the, the string order, like we just plot what are going to design string order, which, which are differentiated between those two cases. So if the string order s plus is non zero, it tells us that a theta is zero. And if the s minus is non zero, it tells us that we are in this pi phase. And we see that indeed at this regime, where the break the symmetry like this coexistence region where this theta is no longer well defined we see that this is changing. So so we see that this is, this is the part where, where the, where the firm in parity is, is, is changing and this is where our deep shows matters theory and no longer holds and we can actually have. We can have a continuous path without crossing a phase transition. And this I, the, the, these ideas now I explained in detail for this model for this anionic and how about model, but exactly the same ideas generalize to, to general SPT model so the recipe is that we can say that well, let us now pick some SPT faith protected g, and then we can now extending the group g to a larger group g tilde, such that g is g mod h, and then we can find that the, that quantum numbers of this additional symmetry which in, in our case that we just looked in was was fermion parity. Those symmetry group and labels distinct representation and edge modes and remain reg modes and also transition remain robust as long as the excitations are charged that are charged under under age are gacked. So so it's a very general recipe that applies to that we can just use for for all SPT phases. So the same concept that we had here for for one dimensions where the SPT phase are classified by the second homology. We can also just do this to higher symmetries for example we can take a Z to two dimensional SPT and extended to Z for which is trivial, and then again trivialize an SPT phase for for two dimensions. So a couple of days ago or a week ago, there was actually a paper by Drew Potter and, and when, and it's not so long when I think, but, and, and, and they actually showed that the same idea also allow to to to find so called quotient protect quotient symmetry enriched phases right so so so the same ideas I extend to to to a number of different cases. So, so let me just very briefly now show a case where where we just have a realization of this SPT phase using exactly a Hubbard model of this type and this was done together with a group of a manual block where they amazing. And it builds to to actually design now for me how about letters so so so they have digital mirrors so that they can specifically design a letter says such such for example the this, this, this letter, and then depending on on how we basically visualize the model, they can now either tune into a into a SPT phase or they can tune into a trivial phase right so so so basically using the setup they can now exactly model this is face diagram that we are looking at at before. And then we limit ourselves to the limit of sufficiently large use so that we are deep in the in this regime of an SPT phase, and then they can using using quantum gas microscopy, they can actually directly measure, for example, non local order parameter they can measure string order parameters to distinguish different SPT phases so so the way that this can be done because the string order is in for the spin rotations and Z direction so they can just measure in the Z basis, and then by doing many snapshots, they can have directly measure the string order, and the string order then can distinguish the two different topological phases right so we can design a string order which should be non zero in the trivial phase, we can design a string order that should be a non zero in the other phase so we can just really tailor make the string orders to detect the different phases and then they can now be measured in in the setup, and also they can directly see the edge modes that are present in the in the non trivial in using these quantum gas mic microscopy. So, so this was just a short interview and now, before I'm concluding, I want to show one other way how we can like use these idea of of getting out out out symmetries, in particular, if we take now chains of that have a Z n cross the end symmetry, then the low energy symmetry, and so is the quotient group, which can stabilize actually direct trans continuous transitions, where we might not have expected them. And let me just briefly show the main idea and the main idea said, let us demonstrate this for the for a Z four cross the four model, where we can for the Z four cross the four we, we have four different topological phases, as opposed to the two that we had for the two cross the two. And, and, and we can now write down sort of toy Hamiltonians, which we, which was was done by the work by two and all. So, so you, these are generalizations of the, of the Z two cross the two cluster model to to hire groups of the n cross the end. And it is D is now labeling the face so so so for D equals to zero. This would be in the trivial faith but equals to one it's an SPT phase one and SPT phase two and three, depending on how we choose a D in this this Hamiltonian so this is nice to play around with and the symmetry generated are now the products of access. Where the product is over some subletters a and subletters B. And what was known before, and this is a work by two and others is that there are actually direct transitions between neighboring SPT phases so if you have a SPT phase D and D plus one, then there's in between a face transition with a continuous transition with C equals to two. And what wasn't really understood is what happens if you now have transitions between phases which are not neighboring right let's see that you have a tradition between HD and D plus one. And what you can actually show using the formalism that I just discussed that there is actually indeed one direct transition between those two with C equals to two to one. So here is similar very similar to what we just discussed before. So H2 conserves now local Z two symmetry generated by X squared, and the symmetry flux has long range order so we have these string order parameters that we can design and the low energy sub space is actually Z two cross Z two. So we see that in the projected to the low energy subspace, the model reduces to a simple and cluster state model. And it's impossible to enter intermediate SPT phases, as long as we have this long range order which is protected by a gap. So we find it that there's now not a fine tuned critical point but again, a regime where we have the C equals to one critical point, which is an end which which which is an ending again in a spontaneous symmetry broken face so so the same idea that I discussed in some previous video for the eye on a cupboard model also applies to the CN cross CN. And in this case we can actually use it to argue that there should be direct continuous phase transitions between SPT phases which are not neighboring phases in this context. We can now come to the conclusion. So, so there are different ways of trivializing SPT phases, we can break the symmetry which immediately throws all the nice properties out of the window, or we can extend the symmetry. And if we just extending the symmetry group as shown for example for the eye on a cupboard model. We can actually find that certain features such as the edge modes, and also the critical point will survive for for for certain range of parameters. And this has, I think quite some experimental relevance because the, if even if you are thinking about a simple spin model in a real material it's probably realized in terms of some some effective low energy model for a Harvard model. Now, what it basically tells us that that certain of these features are relatively robust so which makes it more likely to actually observe it. And, and the second part is about the transitions and the work as I said I want to thank Ruben and Julian for this nice collaboration. The first part is already published and the second part about the CN cross CN is currently in preparation. So, so this I want to conclude and thank you for your attention. So, thank you, Frank for this nice talking for keeping time. We have time for questions. So, thank you. When you when you build the project symmetry for the for considering this edge states that you have. So you're large, the symmetry group of your chain. Yes, and you build this project symmetry group right. Well, no, first, I mean we have, we start off with a symmetry in this case, so three. Yeah, where we have projective representation so we have basically a projective representation with zero and pipe. So, and now, now we enlarge it so we just go from so free to as you to. And in this case, as you to has no non trivial projective representations. As you to have even number dimensional representations. So you are going. No, but we don't have projective representation so so my question is out of what you build the projective group. You have a original point group or some group right that's your system and then you out of the original group you build the projective symmetry group. Maybe this is the answer to your so so so let us say that you have an SPT phase protected by some group G. That could be the two cross the two. Okay. And now what you're doing is you just extending the group to a larger group, but you're extending in such a way that the original group G is your G tilde mod H. Yeah, I understand that. Okay, maybe the night. So my question is, so for instance, at the beginning you had this is all three group, right, then you enlarge it. Yes, and you get the SU two. Yes. Okay, so the SO three has, you know, all the number dimensional representations so they correspond to integer angular momentum if you want. And then the SU two is the symmetry group of the half integer angular momentum. And then it has even dimensional representation. This is what you get the genesis. Yes. Yeah, sure. So my question is, in that case, when you go from SO three to SU two that means that your system which originally has an angular momentum now has a half integer angular momentum. My question is, which is the source of that half angular momentum that you are adding to the original one. The source that we get this is for this how about more or less that we just add permanent parity into the game. But maybe we can talk in private with me I'm still not 100% sure I understand the question correctly. Frank, I think there's a question on the chat. Any other questions from the room. How can we deduce the Hubbard parameter value you which corresponds to the most stable phase.   It's not a phase anymore if you're asking about what, what is the you for which the topological phenomena are most robust that would be that you was infinity, because if you infinity we have a, we have an actual SPT face. So maybe this answers your question I don't know. The only question. Yeah, I mean the others I. Oh wait. Yes, right to the other thoroughly answered on the way. Yes, it's just a question about the dimensionality so here you're talking about one these systems. So do these ideas somehow generalized to say 2d I mean yes, so like we can the classification of 2d topological phases by enlarging local. It's quite slight right so so so the same concept you can generalize to to higher dimensions for example you can take a 2d SPT, which is classified by, by third comology, then you can just start from the two where you have a non trivial third you can extend it to the for and make it reveal. So so the same idea applies also to higher dimensions. And as shown by this paper by Drew Potter. Also, you can apply this to symmetry and rich topological phases. Thank you again.


Hi and how are you all today? The question says, find the value of the unknown interior angles X in the following figures and the figure which we need to deal with is this. We are given the exterior angle as 100 degrees and one of the interior opposite angle as 70 degree. We need to find out the other interior opposite angle. Now let's start with the solution. We know that sum of interior opposite angles is equal to the exterior angle that is according to exterior angle property of a triangle. Now here we are given the interior opposite angles as 70 degree and X respectively. The sum will be adding them up and it will be equal to the exterior angle which is 100 degree. What we need to do next is we just need to find the value of X that will be 100 degrees and 70 will be transposed from the left hand side to the right hand side. The sign will get changed and from positive it will be negative. This further implies X is equal to 30 degrees. So X is equal to 30 degrees is the required answer to the given question. Hope you understood the solution well and enjoyed it. Do have a nice day.


Hi everyone, welcome to basic science series. We are going to start a new lecture series on biochemistry the first topic that I'm Including is protein synthesis Before we start to the lecture part I want to tell you that which book I am following for my notes The book is principles of biochemistry written by David L. Nelson and Michael M. Cox. It's a fifth edition of the book. I do recommend that you read this book This is a fantastic book on biochemistry. All right. Let's start the topic Let's talk about proteins proteins are macromolecules They are the end product of most of the biological pathways they are produced inside the cell by cellular machinery and They perform various function to keep the cell alive for that. They must be Synthesized as well as they must be degraded in a tightly regulated manner To produce a polypeptide chain. We require 70 different ribosomal proteins Additionally, we need 20 different enzymes just to activate amino acid precursors We need approximately 100 enzymes just to process the protein after its manufacture and We need additional 40 rRNAs and tRNAs molecule total 300 molecules to make polypeptide change Nearly 90% of cell's chemical energy is required for protein synthesis 1500 ribosomes molecules and 200,000 tRNA molecules in addition to 100,000 protein work together to form 35% of cells trimase in a typical E. Coli cell Inside an E. Coli cell Protein molecules are synthesized Constantly and at a very high speed these molecules Synthesize 100 amino acid long polypeptide in just five seconds Therefore the regulation of protein synthesis is very important There are three major experimental evidences that were provided to understand the protein synthesis First one is 1950s ball Xemic experiment where he studied protein synthesis by using Redulable amino acids what he has done. He injected those amino acid inside animal rat After injection what he has done is he took the lever of the animal out He tested it for Redulable amino acid and what he has found is those amino acid were present in specialized area inside the cell Immediately after the injection minutes after the injection and those areas were known as ribosomes Here is a quiz question on this experiment While lever was selected as this study organ in this experiment to study protein synthesis Why not brain? Why not gut? Why not heart? Why these organs were excluded? Why only the lever was selected? Please answer in comment section Second major experimental evidence was provided by Mahiyan Hoglan and Xemic where they took the Redulable amino acid and Found that it binds to RNA molecule and gets activated later. This molecule was found as tRNA or transfer RNA Third important contribution was provided by Francis Crick He asked a question that how four letters of DNA can transfer the information in the form of 20 letters of proteins How is that even possible? He suggested a model for that. He said that amino acids had the ability to bind to adapter molecule That further binds to DNA molecule to transfer the information in the form of protein later This process was termed as translation Let's summarize the presentation We learned about the proteins what are proteins and why they are so important how they are synthesized inside the cell What are the different? Historical experiments that were performed to understand the protein synthesis. I hope you like the presentation Please stay tuned for the new videos that are going to come into this biochemistry series. Stay tuned. Please subscribe the channel Thank you and namaste


Happy Monday. Yes, it is the first day of school today and I don't know how I feel. I'm not as nervous as I have been in the past years. I guess maybe it's because I've been through it a couple times now and it's nothing new. But I'm very excited and I just keep praying to myself. Be confident, be strict, be kind, like all the things I wanna be. So I'll show you guys what I'm wearing today. I'm just wearing my staff shirt because we're required to wear our staff shirts today so that parents and people that are around campus know who to ask if they have any questions. And then I'm wearing these Chico Pixie Pants from Old Navy and my Black Teaks. And that's all I'm wearing. And I've already gotten started on my breakfast and I have a smoothie that's already made and I've been drinking all morning. But I'm just having a bagel for breakfast and then I also packed banana. So my fruit for the day is a cup of strawberries and a banana. And then I also have for lunch, I have an organic yogurt. I have some carrots with laughing cow cheese. I've got some almonds and I have a Chinese chicken salad from Trader Joe's. It's almost 6.30 now so I'm kind of ahead of schedule. I woke up at 5.45. My alarm went off at 5.15 and I snoozed it until then. But I didn't have to do my hair because my hair was already curled from yesterday so that's kind of a perk. Perhaps I should curl my hair every night and sleep on it. Are you guys gonna miss mama? It's school year time, huh? Come here. Come show me love. Yeah, I know. I'm gonna miss me all day. You've kind of gotten used to it though with all these meetings and stuff and setting up the classroom. Bite up, boy. Bye-bye. Hi guys, I'm at school and I'm just walking around and plugging in and flipping on all the lights. And I just turned my computer on and I put all their attendance buttons on the attendance board. And I'm just reminiscing from first day of school's past and just thinking about things I wanna be intentional about today, things I wanna make sure that I cover. They're all in my lesson plans so I'm gonna go ahead and print those. That way there's no way for me to forget anything because they're printed. So I'm gonna do that really quickly The room is ready. What are you doing in here? I gotta put my non still. I feel like I have a red mustache. What? You're gonna have a red mustache. I got a red mustache. Do you have powder? I do. There are my notes. Oh, I'm gonna use those real quick. I got them. Light beige or maybe I have something here. I think that's mine too. Ganey, we're all matching.  So I think, I literally think we're all wearing the same exact thing. But that's fine. I'm gonna tell people and be like, hey, follow me on Instagram because that's where I'm doing most of my vlogs this year. Yeah, yep. It's lunchtime and I'm already exhausted. I have 37 students and it is crowded in here. It's insane. And I hope you don't care about chewing because I'm gonna eat and talk. So far, the class has been really good. I can already tell who my difficult kiddos are gonna be but that's okay because they're letting themselves loose now. Someone's coming in. Like, what am I gonna do? You should ask the grader. She's commenting on my backpack. I think there's nothing we can do. We're gonna just eat more. Well, and they're all way fuller now than they will be. Yeah. Because they have all their supplies in it still. What have you done so far? Basically nothing. Yeah, me too. I'm gonna have to fit the read aloud. Yeah, I literally did a read aloud. I got there, I don't know. I'll just do it tomorrow. I'll move whatever I was doing tomorrow to the next day. I feel like I need more surfaces for my kids to sit at because they're all on the floor. Yeah. Because they're so many of them. I opened up that white table thing there for standing. Okay, I'll tell her right now. Hey guys, so Jennifer and I are sitting here and we're grading our writing assignments. It was a long day.  So I think that they plan on maybe getting another teacher for the fourth grade. Hopefully, fingers crossed. You were holding it anymore. Yeah, you guys would be the very first ones to get it, I think so. Let's see if this works at night. And then you'll have my size hope. So all of the kids that came from her class, I called Reese's pieces.  People, Sam today was like, so you pronounce it Reese, I'm like, yes I get it again. And he goes, can I ask why? I'm like, I don't know, my family just says it that way. It is Wednesday and I just wanted to come on and vlog a little bit today. No particular reason, just thought I would vlog a little bit. Yeah, so I just walked in the door. I did go to Starbucks and got a small coffee and some water. Basically, I'm in love with my class. I love every single child. I'm obsessed with them. There are 37 of them and it does not look like the number is changing, going down anyway. But there has been rumor that the district is looking into our numbers and it looks like they might be bringing in a fourth grade teacher. I have not heard that yet, but I've had people in my room both days now counting heads. And so I think that that might be part of their plans is to add another section of fourth grade. The fifth grade numbers are really high too. So I think that they might even consider doing a combo class, which would be smart because then we could get a couple taken from each class and then it would release the burden on us a little bit more. So we have a problem. I took my clips home yesterday to put their names on the clip chart and I totally forgot them on the coffee table, so that sucks. But that's okay, they're gonna forgive me because they know that I'm pregnant and forget stuff, so they'll forgive me. So I just basically, you guys saw everything that I just did.  And yesterday I changed the date. So I try to change the date the night before I leave. That way I don't have to remember the morning. But I am working on, I'm checking emails and checking over my plans for the day to make sure I have everything ready, everything copied. And then what I will do is I will start planning for like next week. And I'm pretty planned through the week, but I just need to finalize the plans and make sure everything's ready and copied.


Hi, I'm Zor. Welcome to Indizor Education. Today we will talk about derivatives of higher order, like second derivative, third derivative, etc. Well, first of all, let me just explain the concept behind this. What is the derivative? The definition of the derivative of the function f of x, which is defined at certain interval, which might be infinite in both sides. At point x0 is a limit of increment of the function. At point x0, given the argument increment of delta x, as this increment of the argument tends to 0, if this limit exists, obviously. So, let's just think about what this represents. We have to have the function and we have to have the point. And at every point where this limit exists, we have the definition of the derivative. So, basically, what I'm saying is that this whole thing represents a function, which obviously depends on the original function f, but it's defined at point x0. Now, let's consider this function and consider all these argument points within this interval, where this limit exists. So, for all these points x0, where this limit exists, function g of x0 is defined. So, basically, it's a function. And what's the domain of this function? Well, it's not greater than this, obviously, domain, because it's only for those x0 from this domain, where this limit might exist. But it might exist on a subset, on a smaller set of points, smaller than this interval, if there are certain points where this limit does not exist. So, in any case, this function is defined on some kind of a domain, which is a subset, or maybe exactly equal to AB. And it has certain values. The value is this particular limit. So, the derivative is a function, which is defined on maybe narrower, but maybe exactly the same interval as the original function f at x is defined. And obviously, it depends on the function itself. And traditionally, this is denoted as f and prime or something. Another definition, another notation is this. But now, it doesn't really make sense to use x0, because we're talking not about a particular point where this particular limit exists. We can talk about any point from this interval where this limit exists. And I can actually just scratch this 0 and talk about new function f prime of x, or df at x divided by dx, which denotes my derivative of the function f at point x from this interval. Obviously, with a node where this limit exists. Okay, now, remember a few samples which we have, like the derivative of a sine is a cosine, etc. It's a function. So, if this is a function, I can actually consider a second derivative, which is a derivative of this derivative. How is it defined? Now, we don't need this g anymore. Well, it's denoted as f second of x, which is limit f prime of x minus f plus delta x minus f prime of x divided by f, again, wrong, divided by delta x. Because delta x goes to zero. Again, obviously, if this limit exists. So, first requirement is that the derivative exists, which means this limit should exist at point x. And then, if the derivative of this function, which is this limit, exists at point x, then we have a second derivative. So, it might be defined on even narrower area. So, basically, if you forget about this if the limit exists node, which I'm trying to emphasize every time. Basically, we're talking about one function defined on some axis, which is dependent on original function. Then another function, which is dependent on this function. And we can continue this as long as these limits exist. We can continue this process of building the second, the third, the fourth derivative. It's a derivative of the derivative of the derivative as many times as you want as long as these limits exist. Well, sometimes there are cases, which you can infinite number of derivatives take. Derivative from derivative, infinite number. If this limit is defined, for instance, for every point and throughout the whole process, then here you go. Now, the third derivative is obviously f. It's limit of f second derivative as delta x goes to zero. And then, obviously, you can define f four. I'm using the Roman numerals, which is the derivative of the third derivative, f five, etc. So, usually, with these higher order derivatives, derivatives from derivatives, we can use Roman numerals. Another notation is, let me talk about the second derivative, this one. d square x d f at x. I'm sorry. Let me just have a little bit more real estate here. d second f at x divided by d x square. So, that's another notation for the second derivative of function f. So, this is one with second slashes or whatever you call it, Roman numeral. And this is another notation. And the third one and the fourth one, etc. So, these are regular integer numbers. d cube f at x divided by d x cube. That's the third derivative. d fourth, etc. So, these are the concepts behind taking the second, the third, the fourth derivatives and any other higher order derivatives. So, that's what they are. That's what higher order derivatives are. It's derivative from the derivative. And now, let's just have a couple of examples. And in my examples, functions have this limit everywhere. So, the first function is f at x is equal to a constant. So, the graph is, obviously, if this is a, so the graph is straight line. Okay. What's the derivative of the constant a? Well, we already actually spoke about this before.  So, the limit is zero. So, the first derivative is zero. Now, zero is also a constant, which means that the second derivative, the constant, the first derivative is the constant. So, the second derivative would be derivative from the constant. And we know this is zero again. And the third one and the fourth one up to infinity. So, this function is differentiable up to any degree. So, any order, the second, the third, the fourth order derivatives, they all exist. And starting from the first, they're all equal to zero. So, that's my first very, very trivial example. Now, how about function equals x to the power of n? Well, before, we actually made some calculations and we found that the first derivative is equal to n x to the n minus one, right? Now, it's very easy to prove, and I will probably talk about this separately, that if you have a multiplier, a factor times some function, then this multiplier can be taken out from the differentiation. Why? Primarily because, again, remember, limit f at x plus delta x minus f at x divided by delta x. If f at x can be represented as some multiplier times something, this multiplier will be here and there. So, it will be outside of the limit, which means that the derivative of the function of this type can be represented as the same factor times derivative of this. Now, what is derivative of x to the power of n minus one? Well, I mean, we know about this. So, obviously, this would be this times this, right? So, the power becomes a multiplier and the power is reduced by one always. So, if it's already n minus one, so it will be n minus one and this will be n minus two. So, what will be the third derivative? n n minus one and minus two x to the power of n minus three. How many times it will last? Well, it will last actually n times. So, you see, the third is minus three. So, the nth, I'll use n in parenthesis as an indicator that this is nth derivative, would be, well, obviously, it will be n factorial, right? n times n minus one, et cetera, times one. And here you will have x to the power of zero, right? n minus n, which is one. Now, this is a constant, right? So, the next one would be zero because this is now a constant. So, nth different derivatives as nth and after n starting from like n plus one, you will have zero, zero, zero, zero, zero as derivatives. Also, infinitely times differentiable, differentiable function, infinitely differentiable. Okay, next example. Next example is a to the power of x, where a is some kind of positive constant. Now, the first derivative we already log a rhythm of a times a to the power of x. So, we already calculated this in the previous lecture. So, what is the second derivative? Well, I was talking before that the multiplier can be just factored out, and then the derivative of the rest can be taken. Which means we will have logarithm of a and then again logarithm of a. So, it will be square. The third derivative would be cube. The nth derivative would be logarithm to the power of n of a and a to the power of x. And by the way, if my a is equal to e, remember what e is, right? Then this natural logarithm of a is actually equal to one, right? Because this is natural logarithm is logarithm with a base e. So, which power I should raise e to get e? Obviously, one. So, it will be one, one, one, one, one to the power of n is obviously the same one. So, e to the power of x is e to the power of x. So, any derivative from the function e to the power of x, the first derivative, the second derivative, the nth derivative, any derivative would be exactly the same as this function. And this is absolutely remarkable property of the function e to the power of x. That's why it's so special. I mean, all these exponential growth, etc., which we see observed in the nature, I mean, it's really very much related to a very special property of this function. It's a very important function e to the power of x. Okay, my last example is from trigonometry. If my function is equal to sin of x, okay, we already spoke about the first derivative is equal to cos sin of x. Now, the second derivative is minus sin of x, right? The derivative from the cosine. Already spoke about this. Now, the third derivative is minus, because this is a multiplier, right? And then derivative of sin, which is cosine. Fourth derivative is again minus, and derivative from the cosine, which is minus sin, which is minus and minus will be positive sin. So, we see that the fourth derivative is exactly the same as the original function. And then it repeats itself. The fifth derivative would be like the first, the sixth like the second, etc. So, it's periodically actually repeats itself. All right, these are just simple examples of different higher order derivatives of certain basic functions. And in reality, obviously, the functions which you will be dealing with will be a combination of these more or less. All right, so that's it for this particular item. So, you know what higher order derivatives actually is. I do recommend you to take a look at the notes for this lecture. They're very brief. But just to make sure that you basically inculcate this concept into your mind, it's important to read it on Unisor.com. That's it. Thank you very much and good luck.


Introducing Richter. The hero that lets you know by the first minion wave if you're going to run circles around him or spend the entire game scared shitless. Richter's passive is Neuroshock. This applies a mark to targets you hit with abilities that allows you and your buddies to deal extra energy damage on basic attack depending on Thickboy's max health and the target's current health. It's like having a stacking persistence thingar boost that also affects your allies baked into your kit. I know this seems strong, but it's not like he generates a constant pulsing AoE lightning storm around his body that can easily apply this. Richter's E ability allows him to generate a constant pulsing AoE lightning storm around his body. Richter's Q, electro whip, is a move straight out of pimp school.   Richter's right click is the iconic chain pull. Use your third eye to target the area you think your opponent will be in the next 5 minutes, then whip your chain forward at the speed of syrup to maybe grab them if you're really lucky. It's by far the hardest ability to hit in the game, as well as it should be.  In fault, landing the hook shaves time off the cooldown, meaning you can yank that carry across the lane and still run up to snag the support out of their tower.   Richter's ultimate is impalement.   While this was usually a self-destruct button in Paragon allowing everyone you didn't catch with it to hit you with the everything, strange matter changes so you take 50% less damage when using the ultimate.  Just hope your fully shielded team is smart enough to follow up after they see you soak up three enemy ultimates while two people are stunned out of position.  While Richter can be a force to be reckoned with, he can easily be countered by simply walking up to his face in a serpentine pattern.  Or maybe that's just how you counter me, I don't know. Go solo lane Rick and yoink overconfident graystones under your tower on Richter today. Teach people what the back of their tower looks like on Richter today. Claim that that was an intimidation hook on Richter today. Man goooo. Shout out to Foolish Blood Hunter and Farath for becoming members of the channel.


All right, I've been in Vietnam So I haven't had the chance to make videos, but it's here 2k16 and I promise I'm not going away during an NBA 2k release ever again, but here it is and In one of the vlogs I asked you guys to vote on a on a my GM team It ended up being the Los Angeles Lakers. So let's do this. Let's start. This is my GM with the LA Lakers Obviously, it's a team that I'm not going to relocate. So don't expect any relocations For this series, but eventually I might actually relocate a team We'll see what happens with that But yeah, definitely not for the Lakers. You can't move the Los Angeles Lakers to another city. That's just That's just not right. I mean Yes, it's not right, but you know, obviously they did move ones from Minneapolis to LA But uh, yeah, let's look at the roster real quick just to start out Obviously, you know got Kobe Bryant but I have some huge plans that I make to do that I make to do that I That I wanted to do. I don't know what the right way. It's not kind of a thing right now I'm just too high to get in to my GM But um, yeah, I've got some pretty big plans that I want to do for this team Lou Williams is actually rated pretty high 80 overall Over D'Angelo and Jordan Clarkson, but he's probably gonna come off the bed. This team is so weird like I Don't know how I'm gonna run this team where I run D'Angelo Russell Jordan Clarkson So what I want you guys to do is first of all if you're new hit that subscribe button for daily NBA 2k 16 content Mostly my GM content if you're into that kind of stuff Leave a like on the video if you're just watching it right now Please it really I really really appreciate it when you guys leave likes on the videos because it shares the videos around which means that other people who can watch it that don't normally watch this channel and Yeah, if you could please leave a like I really appreciate it. Let's see if we can reach 1000 likes on the video But let's do this Hi, I'm Eric bonus owner of the LA lake is welcome to the team Let me go over things that I want you to achieve this year. I want you to win at least 20 road games this season. I Don't know about that.  No doubt about that. They'll be really tough. I'm I might accomplish that I Want you to acquire two lottery pigs in next year's draft. That's In the next draft, that's a will accomplish. I'll do that for the owner two lottery pigs. All right, cool Um, I want no I expect this team to lose less than 37 games this season. I might accomplish that I'm not sure what I want to do D'Angelo Russell knees 26 minutes a game at a minimum. I Absolutely hate D'Angelo. No, it's kid. Yeah, I will accomplish that Let me go over my wishlist items Okay, we need to be in the top 10 in attendance. I'm putting it on you to make it happen I might accomplish Fair enough. I have your eyes. This wasn't a suggestion. I know you can do it.  Yeah, where the lake is we got all the money in the world, but we see we still can't seem to sign up a free agent I don't know what the lake is doing. Okay? Nobody has posted a 2016 draft class and I don't want to risk Getting a wrong the wrong one because I have seen some RGM series where they do that So I'm gonna order generate this the the rookies for this series But I have huge plans that I want to do with this with this my jam series now I haven't played my GM yet, but I have seen some videos on it agent zeros videos I've seen, you know Envy's videos. I've seen tgt v's my GM videos I've seen Funky's my jam crispy flags a whole bunch of my GM video so I kind of know I have an idea what What's gonna happen, but I do want to get into this the target list and How do I do that here target list add a player? The target list that's that's to get right and then the trade block untouchables. Well, what's the target list? That's who I want to get right? Target I'm pretty sure it's who I want to get like who I want to target. So let's do that I Want to target?  It isn't my GM series Not the real NBA, but it won't be super unrealistic now I'm learning how to cope with this. I'm learning how to Balance it out in the comment section. I got these people right here.  I got these people over here I like it unrealistic keep doing it unrealistic the more unrealistic the better and then I got these people over here saying It's alright make some trades that are unrealistic, but you know, don't make Superstar unrealistic trades if you know me just average it out a little bit So I'm going I'm leaning towards this side. Just doing what I feel is right so let's do some plays that um that are obviously Not obviously that are plays that are rumored to be going or who I think will not stay for the team For much longer in real life One of them has to be DeMarcus cousins obviously in this year rumored to be to be you know getting traded so DeMarcus one of the players that I would love to get on the Phoenix on the Not Phoenix sons on the Los Angeles Lakers. I'm used to the thing the other Phoenix sons my GM series. I was doing um All right Let's keep looking Let's keep rolling So I kind of need to sort out the team first but here here here here One of these three guys you already know they're gonna go one of them has to go like you kind of three centers That are all over about seven foot. I think they're all over seven foot on the well six eleven But oh they're okay, so they're all about seven foot six eleven six eleven seven feet tall, but you know that one of them is gonna go probably either gonna be Jalil or Joel has funny Jalil or Joel But because Noel he can play the four whereas MB and Jalil they can't really play the four. I don't think So I'll put both of them. I'll put both. I mean if one of them comes to me. I'll take it But yeah, I'll put both of them up next player is a player now. This is definitely unrealistic, but Nobody would understand what I want to do But maybe maybe someone might understand what I want to do All I'm saying is Wiggins and demarkers. That's all that's that's the only hint I'm giving you guys Who's another player that maybe Could join the Lakers who was some players that are on one-year contracts Durant's on a one-year contract But obviously they're not gonna offer me Durant So I'm gonna remove the player, but the thing is if Durant leaves the Thunder Westbrook might want to leave the next year because he he's from LA He's from UCLA. So I'd love to get Westbrook on this team. I could see some stuff happening with With the Celtics involving Isaiah Avery and obviously Marcus Smart, but I Guess we'll have to see what happens with that. Not really too keen on Another point guard on the squad. So I mean we already have DiAngelo Russell and we already have Jordan Clarkson So I don't really know what I want to do with that. I'd like to get Justice Winslow But I don't think the heat will ever give them off as of now Unless they got something for them Yeah, I'll just go after Justice Winslow. I am a heat fan. So yeah To well and beats only two stars. Maybe because he's injury prone. So that's the target list done trade block Yeah, so this is the trailer. This is our team. Who's on the trade block? Made a well piece first of all on the trade block.   I think It's not huge. It's only oh, no, it's not it's not a big contract at all But I'm it's not really a huge brand of best fan. I'd rather focus on Julius Randall in the starting lineup next player anyone Ryan Kelly nobody really has huge trade value and I like Robert Sackford to just to stay on the bench just because he has funny celebrations and I'll get rid of Anthony Brown, maybe Untouchables obviously Kobe And the Angela Russell. They're my untouchables and That's all set. All right, cool Let's get into this. Let's first of all go to free agents. Are we planning to tank? Or go all out. I'm thinking a bit of a tank I think we're gonna try and tank this not tank But not do super well if you know what I mean like Where the lake is we're not gonna do that. Well, let's be honest Unless some crazy thing happens, but I don't think we're gonna do that. Well contract extensions Nothing's going on here. All right Trade finder Let's just go off. Let's just look at some trade finders right now. I do want to go after some picks though actually Remember our owner said that we needed to first no two lottery picks. So let's do that Wow, we just oh, we don't even have our first round peak.  We don't have our first round pick.  That's terrible Huge fan of Harold love this man. I want to pick Harold up. You know what? Let's um, let's actually go to how I want to go to how I want to see what uh What is that so looking like first of all? Let's see. So what I think of Harold I think he's kind of like a Kenneth Reid Ben Wallace type of player. Well, that's what I hope he is in the end a Ben Walls shop-blocking machine His defense is actually not good like Ben Walls, but that's something I'll improve as as time progresses So he won't get any minutes And this is a perfect trade Anthony Brown don't want him on the team. Well, like I don't really mind him but I like Harold sir Unless I want to do something like this, which I get rid of meta and I get rid of Brandon Bass, but Or something like this This trade actually That's a perfect trade because KJ I like KJ McDaniels and I know he'd want to get traded from the Houston Rockets Because he didn't get any minutes for the Houston Rockets, but I didn't know he got minutes for the LA Lakers and Anthony Brown's one and a half stars, which means I can probably get a pick out of this I'll let's see what they say to this trade. I reckon this is a pretty fair trade They're only giving up three and a half. Let's see if they can give up another pick as well another one another So how do you do this? How do you? Do you want to make this trade? Yes We're not giving we're not getting enough back for what we're giving up However, I have a counter offer that can work nicely for both sides 2018 first round 2018 Lakers By the time 2018 comes, we're gonna be good, right? How about I won't give up that second round pick How about this? I have a counter offer No, I'll take the the other one. How about this? Counter offer. Okay, we'll just give them two second round picks a 2018 first round pick which in my opinion, we're gonna be good by 2018 anyway We're gonna give up Ryan Kelly Anthony Brown and metta well piece. We're gonna take Harold taking Daniels take Two first round picks one from 2017 one 2019 and also pick up two second round piece. I'll take that sweet They agree to the trade offer. I Appreciate the trade. I like that trade. We get rid of barmas metta well piece and the reason my guy looks like that is I didn't create Nick Smith I don't know if I've shown you guys this but my guys called Willie stoker as a joke Yeah, I might try to make it funny my career out of this rebuild rotations No, I don't like these rotations put Randall in the starting lineup in my opinion. Yeah, that's a bit better Nick young I guess I Mean Clarkson or Nick young I I probably rather have Clarkson in there How tall is the angel a 6 5 Clarkson 6 5 and Kobe 6 6 so Kobe's gonna be running some more forward this year Okay All right I'm just getting used to all these all them all the minnows and stuff like that because it's all new to me I haven't played this yet, and I do want to give a few minutes to To Harrell so let's do that Harrell can have I Mean Harrell was younger, so he deserves more minutes than Robert Sacra All right, that's not bad I like Harrell so you can have about four minutes per game, and that's episode number one done So I do hope you guys enjoy the video if you did leave a like Let's see if we can reach 1000 likes for the next episode, and I'll be sure to post it immediately Yeah, Los Angeles Lake is wrote to the championship. Here we go. I'm out. Peace


This study aimed to examine the relationship between electronic health record interoperability in patient safety or care quality in high-income settings, finding that EHR interoperability positively influenced medication safety, reduced patient safety events, and reduced costs. However, true measures of effect are difficult to determine due to heterogeneity in outcome measures and study quality. The authors suggest the need for common health information technology research outcome measures and future research efforts to explore both positive and negative impacts of interoperable EHR interventions, as well as patient perspectives. This article was authored by Edmund Lee, Jonathan Clark, Huthin Ashrafian, and others.