Okay, great. So I was originally here to give a talk specifically about what was called maximally verifying like clients and charting, and I will end up touching on a lot of that in this talk in certain ways, but I also wanted to make it broader a bit and basically make it be not just about scalability and just about charting, but also a kind of general outline at a proposal for what I think a sensible path for Ethereum protocol development in general to be going forward over maybe the next two to four years or so, right? So I'll start off by just let's take a look at what's been going on in Ethereum so far, right? So in Ethereum, we've first of all had a lot of successes. So first of all, the Ethereum blockchain and Ethereum protocol work. You may have noted the total lack of a crippling DDoS attack on the network this year. Many applications and we've gone to somewhere over a thousand applications at this point and I'm pretty sure people aren't even really counting anymore. And it shows in all the charts, right? So this is the Ethereum transaction chart and we just recently broke above 500,000 transactions per day for the first time, which in case you can't divide is something like over seven transactions per second. And so it's, well, it's like, or almost exactly seven transactions per second at the peak actually. So the skill, the amount of activity on the Ethereum blockchain is orders of magnitudes higher than it was one or two years ago. And this includes not just Ether transfers, it includes ERC20s, it includes Ether Delta, it includes various different DApps. And it just, it includes just a whole bunch of different smart contracts of various types. This is another diagram I'm fairly proud of. This is the global Ethereum network with more than 20,000 nodes worldwide with hubs existing all over North America, including East Coast and West Coast, hubs all over Europe, a bunch in Russia, a smaller but still very significant and growing amount scattered across the various hubs in Asia and increasingly nodes starting to show up in places like South America and Africa as well. According to the data that got scraped from either this site or another site a couple of days ago, we seem to have one node in North Korea, but not sure if that's actually in North Korea. Yeah, so in general, a lot of progress, right? And if you look at just by ZNTM as well, by ZNTM hard fork basically has gone through and you can now take advantage of cryptographic primitives that are available in accelerated form on top of the Ethereum blockchain in order to build privacy-preserving applications and people already are. So you just saw Zockerty's in the last presentation. Just yesterday, I saw on Reddit someone announced a protocol that was doing a reputation market based on ZK-Synarx. I know of one project that's trying to do things with ring signatures, actually multiple projects. So I also know of at least a couple of projects that are taking advantage of RSA verification to do things like better communicate between Ethereum and existing DNS between a custodian e-residency and just various other things. So the privacy part of the three kind of major challenges to Ethereum success that I've outlined, privacy, security, and scalability is actually quite well on the way to being solved. Now there is still a lot of work to be done. Someone actually has to build the ring signature stuff. Someone actually has to build Zockerty's. People actually have to use it. People have to actually make using their own knowledge-proof practical, but we've made progress. So what are the challenges? So scalability is probably challenge number one, right? So in, this is a quick excerpt from the Sharding FAQ where I talk about this challenge called the scalability trial lava. And I basically state that it seems likely that blockchain systems can only have at most two of these three properties at the same time. Number one decentralization, which I define as basically being able to run without trusting super nodes, scalability, so being able to process many more transactions than a regular laptop, and security. So basically, it's okay to be vulnerable to 51% attacks, but you definitely cannot be vulnerable to 0.51% attacks. There is a very, very big and growing graveyard of systems that claim to solve the scalability triangle, but actually completely don't. So I actually think that the scalability triangle is solvable, but it basically requires significantly more kind of complex technologies than the kinds of technologies that we have available. Part of this is something called the data availability problem, which I'm not gonna talk about today because it really does require a one hour presentation to go through. But basically, it gets this problem that a lot of people don't even think about, but basically not only do you need to verify that the blockchain is valid, you also need to verify that everyone can access the blockchain. And even ZK Snarks and Starks cannot solve that part of the puzzle. And there's mathematical proofs based on information theory that say that if you cannot solve that part of the puzzle, your protocol is basically cannot be secure. The relevant paper here is one, basically one that says that batch updates for cryptographic accumulators are impossible. So I'm gonna head to Justin Drake for that one. So it is a very significant and hard challenge. So basically right now, every node processes every transaction, the capacity of a blockchain is limited to the capacity of one node, and it's actually even worse because we need to have safety margins for anti-DDoS purposes, the limiting factor in Ethereum scalability, and Ethereum's likely worst point of DDoS vulnerability is likely to be discreet. And also Ethereum VM execution is currently absolutely not paralyzable. So these are just known facts and the severe limit, the extent to which Ethereum in its present form can scale. So the likely solution to a lot of this, there are more incremental solutions that we can do. So for example, you can use EIP648 to greatly improve paralyzability, but if you wanna really solve the fundamental problem, every node processes every transaction, basically sharding. And the way that you think about sharding is, well, basically you can split the blockchain state, so split the set of information that everyone needs to keep track of into N universes, and we call each one of these universes a shard. So each universe has its own account space, its own account, its own contracts. You can have transactions within each shard, and you only allow asynchronous communication between shards. So if you wanna send a coin from shard A to shard B, you actually have to have two transactions, the first transaction on shard A, and the second transaction on shard B. And contracts on shard A cannot synchronously call contracts on shard B. So you cannot call a contract on shard B and get a response and do something with that response in one transaction. And the benefit of this is that each client only processes a small portion of all the activity on the network. So the reason why we call it quadratic sharding is because if you imagine, though a processing capacity of one computer to be N, then you can imagine N shards, and each shard has N worth of activity. Each node would have to process the activity on basically just the one shard it's interested in, and every node would have to process the activity of kind of the entire system of shards. So the amount of activity of each node would basically be N at the top and N at the bottom, so a total of two N, but the capacity of the entire system would be N squared. And theoretically it can make the sharding be exponential, but quadratic is simpler, and there might be good reasons to keep it quadratic. And it may well be a reasonable approach to say, if we only do sharding up to quadratic, and if you wanna go up to exponential, then basically have to stick with things like sharding in plasma, or sorry, state channels in plasma. Another challenge that we've had is this kind of challenge of governance and protocol evolution. So basically, we have this trade-off where on the one hand we want progress, but on the other hand there is demand for protocol stability. And hard forks making deep changes are hard, right? So we've finally gotten Byzantium out on October 16th, and Byzantium is the result of over a year of work, a large amount of testing, and even after a year of testing, we still required some pretty epic work between the Go team and the Parity team doing super advanced forms of fuss testing that I don't even fully understand in order to catch a fairly large number of bugs that appeared pretty close to launch. So hard forks making deep changes take a long time to code, take a long time to agree on a long time to test, and there is a high risk of consensus bugs, especially since the Ethereum network has basically seven different clients that all have to implement the protocol and agree on it. But even with all these challenges, we've done Byzantium and we've done it successfully. So, but the problem is that on the one hand, hard forks making deep changes are hard. On the other hand, to get to what we want to see in Ethereum 2.0, so to get to this kind of highly sharded, highly scalable network, as well as all of the EVM upgrades we want to do, all of the parallelizability upgrades we want to do, EVM 1.5, EWASM, potentially more pre-compiles, in order to get to this ideal end state, very deep changes are exact to what we need. So what do we do? How do we handle the trade-off? So intuition, one blockchain, two systems. And I'll make this clear with the diagram that I'll show in the next slide. So here is what a sharding system as we're already starting to build might as could possibly look like, right? So you have the main chain and in the main chain, it's basically everything is basically as it was before.  Now, what we're going to do is in the main chain, we're going to add something or a called a Validator Manager contract. Now, what does the Validator Manager contract do? It basically runs a proof-of-stake system that maintains the consensus for a kind of two-layer sharding system that exists kind of on top of or even inside the main chain in a certain sense. So the Validator Manager contract would keep track of validators and would allow anyone to join and leave the system in the service of Validator and basically assign the right to create blocks on in each of these end shards that the Validator Manager contract keeps track of. The VMC would also keep track of block headers for each of these shards, but what the VMC would not do is it would not verify the full blocks of the shard and the VMC would not contain a copy of all the new consensus rules. Instead, basically, actually creating valid blocks, actually enforcing that the blocks are valid in the shards, actually enforcing that the data in the shards is entirely available and follows the desired consensus rules, basically would be the responsibility of the proof-of-stake that exists inside the Validator Manager contract. So the Validator Manager contract would initially be this kind of very low-risk thing to participate in. There would actually not even be a risk of total slashing in the same way that there is in CASPER and the goal would be to basically just get a fairly significant portion of ether-stake participating in this and basically if you are part of the Validator Manager contract, then you just get assigned randomly the right to create blocks in each and every one of these shards at periodic intervals. I should probably use more correct terminology.  So if you have a transaction, then transactions get grouped into collations and the collation headers then get put into the Validator Manager contract and they get put into transactions that get included into blocks. So we have this kind of two-layer structure where at the top layer, you would have the blockchain and at the bottom layer, you would have all of these different shards and each shard would be like its own universe and connecting the two universes, you have this Validator Manager contract which enforces the proof-of-stake, basically maintains an internal light client for each of the shards and processes and facilitates things like moving ether from the main chain to the shards and possibly from one shard to the other. So you can think of the differences between the two systems between the main shard world and the new shard world so somewhat like this. So scalability, so main shard is OFC. So C is the letter I normally use to refer to the computational capacity of one node. So the main shard would have a scalability of just OFC which is exactly what it does now and the new shards would have a capacity of OFC squared and the reason for this is you have OFC shards and each of the OFC shards itself has OFC capacity. By the way, if you're not familiar with big O notation just like don't think about the O's, main chain scalability of C, new shards scalability of C squared because there's C shards and each shard has C transactions per block. Consensus algorithm, so on the main shard the consensus algorithm is currently proof of work. In the very short to medium term we hope to transition it into hybrid proof of stake in the form of a Casper friendly finality gadget, a paper for which was released on archive something like five days ago and which is actually I would say nearing the kind of final stages of a full specification and eventually we will transition the main chain consensus into full proof of stake. Now the consensus switch itself is designed to be fairly conservative and so possible to implement in this kind of fairly slow and careful way, right? So for example, the first part of the switch to hybrid proof of stake actually doesn't require a hard fork. Technically it requires this weird kind of soft fork where basically it's not even like a real soft fork instead it's just an optional change to the forksuous rule that's used inside of a client. The only thing you would need a hard fork for is when we want to basically stop providing rewards to proof of work miners or at least reduce the words to proof of work miners and switch the rewards to validators. So the switch to proof of stake I'm fairly confident can be done in this kind of gradual fairly conservative way. Main shard has full security basically the same security guarantees we can expect to have now and the governance of the main shard basically in order to kind of satisfy the demand of people that really wants to have basically stronger immutability and less risk on the main chain we can have norms of governance that emphasize conservatism and strong immutability more than today. Now what happens to the new shards? So scalability OFC squared consensus is this proof of stake where in order to vote in the proof of stake you would have to deposit ether into the validator manager contract and the validator manager contract would randomly assign to you the right to create blocks on all of these shards. The security would be 50% on its majority of the on chain proof of stake holders now and basically what we want to do with Ethereum 2.0 and sharding solidify enough then eventually the shards would be kind of tightly coupled and the two would have roughly the same level of security. Now the governance for the new shards is basically just going to be on chain ether voting. So basically for these new shards because you have this proof of stake we might as well have this proof of stake be used in order to basically vote on what the protocol changes are within this sharded universe. So basically validators themselves already are going to be the ones that enforce the consensus rules so why not just have them be just the kind of are the ones that decide on the rules already so we might as well accept this and we basically actually take advantage of this because this kind of on chain ether voting governance would give us the ability to basically emphasize fast evolution in the new shards initially and then when we have this tight coupling and when every client becomes basically a fully validating client for all the shards then the kind of governance would become much more conservative. So this is kind of the dichotomy that I'm suggesting where we can have these two universes at the same time where the universe on the left, the main shard is just basically continuous working as Ethereum. We do not need to negotiate EVM 1.5 negotiate E-wasm between seven different Ethereum implementations and we can focus on relatively milder things that do things like state size control that do things like just improving security, possibly improving parallelizability a bit, heavy focus on the proof of stake switch and we have this other new universe or really these new end universes in which basically the implementation of all of these things that we've already been working on over the last one or two years can basically be rolled out onto the main net much, much faster. So here's what an implementation roadmap might look like, right? So sharding road, basically step one is you would implement this kind of sharding universe as a proof of stake sidechain and in a proof of stake sidechain in each shard you would have collations and the collation headers would be verified and processed by this on main chain validator manager contract. Block creation rights are assigned by a simple proof of stake in the VMC. Clients would just get randomly assigned the right to create blocks and randomly shuffle between shards and in the first step you could have one way ETH convertibility. In the second stage you could make the ETH convertibility two way. Now this is what a node for the Ethereum sharded network might look like, right? So you already have the existing Ethereum nodes that talk to the existing Ethereum network. Everything on the left does not need to be changed one single bit. What you have on the right is you could have a sharded chain node. Sharded chain nodes would be responsible for basically broadcasting full collations, right? So collation headers would go into the validator manager contract. Full collations would only be broadcasted in the sharded chain network between all of the nodes that are participating in the sharding system. So sharded chain node would be written in Python initially and it could be designed in such a way that it can talk to any Ethereum node by RPC. Now if we want fast evolution we could have like basically one or two clients for the sharded chain node initially and then over time ramp it up to the full seven as some more teams become ready. So basically you would have this kind of fairly partial split between what happens on the Ethereum side and what happens in these sharded networks where basically you would really have a sharded network for basically every single shard, right? So after this you would add two way convertibility. You would move the collation headers from the VMC into let's say being main chain Ethereum uncles and then step four would be tight coupling. So tight coupling basically means if you have a block in the main chain that contains a collation header which is invalid then the entire main chain block is invalid. So invalidity on the shard side will be able to cause invalidity on the main chain side. And this would be basically when the main chain and the shards would both switch into this kind of tighter security mode. And stage four is something that theoretically could happen later. Ideally it should happen basically when things on the sharding side are actually reasonably stabilized. So remember the shards that I'm suggesting are creating new address space. They are not affecting existing address space. This gives us a unique opportunity to make many very important, many efficiency improving but backwards incompatible changes to the Ethereum protocol. So what kind of changes do we wanna make? Well, we can just list through a few, right? So number one, changing merkle trees from hexary trees to binary trees. This is a no-brainer. It makes the merkle proofs four times shorter. Account state tree redesigns. So you might want to get rid of contract storage trees. You might wanna have multi-level nested trees. You might wanna have a different account system. These are things that we can brainstorm. These are things that we can think about and optimize. EVM upgrades. So basically, can we make the EVM efficient enough that pre-compiles are no longer necessary? It seems like it's very likely that we can. So in the Ethereum community, we have had these two paths for optimizing the EVM where one of them is the second of Greg Colvin's approach of EVM 1.5, which is more moderate upgrades in addition to the existing EVM. And the second one is this full-scale replacement with EWASM. Well, basically the idea would be let's work together, let's come up with what combination of these two models seems most reasonable and let's just immediately apply it as the only EVM available in the shards. And so this way, you would have much more efficiency without having to worry about as many backwards compatibility issues. Paralyzability. So there are fairly simple Ethereum upgrades that we can do. So EIP648 is one example. There's also a sharding EIP, which has a stricter version of EIP648, which basically lets you execute Ethereum transactions in parallel. If you can do that, then we basically have, if we really, if we want it, kind of unlimited access to the kind of big block scaling route. So if we have parallelizability, if we want to, we could always make the blocks bigger. And all that would happen is that the kind of resource requirements of running a node would just be a bit higher. But if you have enough cores, you would still be able to process a block fairly quickly. So parallelizability is something that if we tried to add it to the existing Ethereum, I mean, it would require some fairly deep changes. But if we can make this kind of new system where we have parallelizability right from the start, then it might be easier. Stateless clients. So stateless clients are a topic that I've written about slightly, but they're a topic that you might end up hearing about more and more over the next few years. While or so, the basic idea behind a stateless client is this, right? So, I mean, as you might have seen from the diagram, stateless clients have something to do with a Merkle tree. Everyone bow down to Ralph Berkel again. Yeah, okay. So consensus nodes do not hold the entire state, right? So consensus nodes, so basically full nodes, minors, anyone, do not need to hold the entire state. They only need to hold the root hash of the state. People who want to send transactions would apply attached Merkle branches, so witnesses, basically proving the correctness of the specific portions of the state they want to access. So if you have a state root, and if I want to send a transaction that sends 10 coins from address A to address B, I would provide two Merkle branches that prove the existing balances of address A and address B. Then the transaction, the blocks would get passed around with the witness, and that's all the information that a node needs in order to execute the block, execute every transaction in the block, and figure out what the new state root is going to be, and what the new Merkle branches are, and what the portions of the state are that got modified by that transaction. So the idea here is that we can substitute each individual instance of a disc read. So basically each individual instance of reading like a storage key or an account from disc, which takes possibly something like a millisecond with about one kilobyte of bandwidth, assuming we optimize the Merkle tree and assuming a billion accounts. If we go up to a trillion accounts, then we'll only need shock core 1,300 bytes of bandwidth. Now the benefits of this are that, first of all, it makes it much easier to reshuffle validators when charting. Basically the idea is that if you're in a stateless client's architecture and used yet suddenly reassigns to a new shard, you do not have to download the state of the shard. In a stateless client paradigm, fast syncing is literally pretty much instant. And this is true regardless of what the state size is. Stateless clients allow us to care much less about the state size. So even if the state was like one petabyte, nobody really needs to store that one petabyte and they massively increase parallelizability kind of as a side effect. This is one other secondary scalability path that basically it might even just make sense to do it in the shorted universe from day one. So regardless of each individual proposal, the general idea is that we could have ongoing development of the Ethereum protocol happen in two layers, where the first layer basically involves relatively small changes, efficiency improvements, getting Casper rolled out. And then the second layer is the one where rapid development and experimentation happens. And it's the one where we could get to thousands of transactions per second, actually fairly quickly, and on a significantly accelerated schedule if we want to. So if there are applications that do not need super-duper high security, then they can just go on the layer too fairly quickly. So this gives us the benefits of basically both approaches at the same time. So safe and conservative layer one and all this rapid development in layer two and the ability to use layer two in order to do all of these protocol improvements that developers have been dreaming about the whole time without having to go through this kind of political process of getting all seven clients to agree on it and write a million tests for it and fuzz it for six months in the short term. In the longer term, even as the sharding system solidifies, of course the sharding system will end up focusing more and more on conservatism and security and eventually the two will be kind of merged in some nice and clean way. But look, this is something that if we want to can wait three or four years. So the good news is that along with the initial work of unsharding, having a proceeding very quickly to the point where we are basically just inches away from having a working CASPER proof of concept testnet in Python, we also have initial work on sharding that's being done in this repo. So we have some developers that are working on a sharding capable client and that are starting to work on some of these stateless client improvements, starting to experiment with some of these optimizations. And the hope would be to also, if there are other client developers that wants to participate in this as well, then look, this is something that should happen totally openly and this is a process that should eventually go beyond just happening inside of one client. And basically in the short term, the sharding stuff would continue as kind of proof of concept in test mode. And the nice thing is that because this is not being done as a hard fork, because this is being rolled in gradually as this kind of loosely coupled overlay at first and tightly coupled over time, basically there is no sharp testnet to mainnet cutoff. So the extents to which the shards are attested and the extents to which the shards are mainnet actually kind of flows fluidly from one to the other over time. So basically this way we can get development of both scalability and a large number of other needed improvements in both quickly and safely with respect to the existing Ethereum protocol. Thank you.


In this lecture we are going to consider second order partial differential equations linear which are identified as elliptic equations and try to obtain a canonical form for the same. Recall we have seen this example or one example of an elliptic equation namely the Laplace equation. So the canonical form for an elliptic equation we would like to resemble this. If you observe here in this equation u x y does not appear and u x x and u y appear with coefficients 1. So this is going to be the model for us for an equation of elliptic type. So let us consider the second order linear equation 12 which is elliptic, assume to be elliptic in a region omega, assume further that the coefficient a, b, c are real analytic functions. This is a requirement for the proof of our theorem. In the examples that you want to do you may simply follow the procedure, you may you will still be successful. So let x naught y naught belongs to omega. Now there is an open set containing the point x 0 by 0 and a change of coordinates such that the 2L gets transformed into this equation observe this part where the second order partial derivatives appear, w xi xi plus w eta eta looks like u x x plus u y y and no mixed partial derivative w xi eta does not appear in this equation. So this is what is called a canonical form for elliptic equations. So equation is given to be elliptic, it means b square minus a c is negative less than 0 on omega. Since the equation is elliptic we cannot have a to be 0, a of x 0 y 0 if it is 0 what we have is b square, b square strictly less than 0 is not correct because b square is always greater than or equal to 0, square of a real number is always greater than or equal to 0. Therefore necessarily a should be non-zero and of course c also cannot be 0 for the same reason. So neither a nor c can be 0 at the point x naught at any point in omega in particular at x naught y naught and we are assuming they are real analytic definitely continuous. So by continuity of the function a, the function a will be not 0 in some open set u which contains x 0 y 0 by continuity. Recall the change of variables xi eta are given by two functions phi and psi if there is if this gives rise to change of variables you can invert back x and y you can write in terms of xi eta we use capital phi and capital psi for that and a function of x y can be identified with a function of xi eta and they satisfy these two relations and the 2L equation second order linear equation gets transformed to this equation where the coefficients a, b, c alone are listed here because they are the only things which are important as far as the type of an equation is concerned. So what do we need for proving the theorem? We want the w xi xi and w eta eta should appear with coefficients 1 in particular a must be equal to c and equal to 1 and b is 0 once b is 0 and a equal to c we can divide with a then I will get w xi xi plus w eta eta anyway. So the condition is a equal to c and b equal to 0 that is enough. So a equal to c this is a and this is c. So a equal to c means this equation must be satisfied and b equal to 0 means this equation is satisfied. The system of equations for finding phi and psi a remark on that recall that in the hyperbolic case when we want to find phi and psi the equations were decoupled right they were decoupled phi and psi could be solved separately. In the parabolic case they are weakly coupled the equations for phi and psi are weakly coupled in the sense that the equation for phi in did not involve psi at all. Unfortunately once we solve for phi and when we go and substitute in the second equation the equation reduces to the identity 0 equal to 0. Maybe it is a good thing because now we can choose psi arbitrarily as long as a Jacobian of phi and psi is non-zero. This can also be explained in following way. Once you solve for phi which is solution of the equation a of psi, eta equal to 0 we need not even solve for b psi eta equal to 0 because it is automatically satisfied by the invariance of the classification type under change of coordinates. Not only that it is satisfied by any function psi b of psi eta equal to 0 is satisfied by any function psi. Therefore we had lots of freedom in choosing psi the only requirement was that the Jacobian of phi and psi is non-zero. But for elliptic equation what is happening in the here also you have phi and psi mixed here also you have phi and psi mixed therefore it is a strongly coupled system for phi and psi of first order non-linear PDEs. We can overcome this difficulty by using the assumption of the real analytic city of ABC and some complex variable techniques. You will see that a crucial part of the proof we will skip we will not do the proof. This system may be rewritten as this right the first equation I simply transfers all these terms to the left hand side and I take a common so that I get phi x square psi x square to be common so that I get phi x phi y minus psi x psi y and C common so that phi y square minus psi y square it is exactly the same equation rewritten. Second equation I am just multiplying with I with because I have some idea what I want to do in the next slide that is why I put I otherwise the same equation I multiplied with I it tells us that some complex things are going to enter. So define a complex valued function capital phi by phi of x y plus I psi of x phi. So this system the system that we want to solve for phi and psi is equivalent to this one single equation now a phi x square plus 2 b phi x phi y plus C phi y square equal to 0. This equation we have seen earlier this is the same equation that we solved while determining canonical form for hyperbolic equations of course the difference was there that we could factorize as real equations factors were real but here it will not happen it leads to PDGs with complex coefficients given by this equation and this equation. In the case of hyperbolic equation there was no I first of all and inside thing was b square minus ac. So these are the 2 equations we have now a fact the system of equations has solutions near x 0 y 0 this is where a b c are real analytic functions is used this we can observe if phi is a solution of the first equation remember a b c they are all real valued functions therefore if phi is the solution of the first equation phi bar the conjugate of phi that let us call it is capital psi that is a solution to the second equation therefore it is enough to solve one equation essentially there is only one equation right but that will give us what we want because the phi is proposed as small phi plus I times psi. So you can identify a real part and imaginary part and hopefully that forms a coordinate system that defines a coordinate system therefore phi and psi are constant on the 2 complex characteristics these are right hand side is a complex valued function okay we have not studied in our Picard's theorem how to solve such equations okay this is where it is important that we use our hypothesis and show that solutions exist define small phi of x y equal to real part of capital phi small because we have obtained phi after this right this phi exists is what somebody told us this fact so I have phi and then real part I will call a small phi and imaginary part I call psi and this defines a coordinate transformation near the point that is left as an exercise very easy and the transformer equation has a required form because we made what we want right A equal to C and B equal to 0.  So B square minus AC is negative strictly less than 0 at every point therefore the equation is of elliptic type everywhere in the plane or 2. So let us transform now the given equation into it is a canonical form. So we need to solve these ODE's dy by dx equal to plus or minus i into 1 plus y square. So we need to solve this ODE i times this because minus i times does not matter it will be the conjugate. So this solution is given by tan inverse y minus ix equal to constant okay please accept this that this is a solution okay and then real part is x imaginary part tan inverse y or y square star here of course if this is constant I can multiply this with i also again right. So therefore it does not matter what I call the variables phi and psi in the theorem that we presented this was called phi and this was called psi but we are interchanging here it does not matter because there is no preference for the variable x or y or psi or eta. So therefore you propose u of x y equal to this w of x comma tan inverse y differentiate ux is x appears only here so it is w psi and derivative of x is 1. Uy will be w eta and derivative of this is 1 by 1 plus y square. So you can continue like that compute the derivatives go back and substitute in the given equation we get this. So that is a canonical form of the given PDE. So summarizing what we did is that we have presented a method to reduce a second order linear PDE which is of elliptic type to its canonical form we have of course assumed very high assumptions on the coefficients ABC but you may generally if you are given a partial differential equation of elliptic type you want to find its canonical form you may simply follow the procedure do not bother about the hypothesis checking for the theorem and you will still be successful if you are able to solve the ODEs which are coming on the way and we have seen the method is successfully implemented in the example. Thank you.


north region. I will be making two statements and then I will take questions. So whenever you're ready. Thank you very much ladies and gentlemen. Police are currently investigating an incident which commenced at Zilmia Road, Zilmia at approximately 9.45 last evening. Investigations have established that four male persons in a stolen vehicle left an address in Zilmia. A short time later police attempted to intercept the vehicle at a roadblock. As a result of the incident at the roadblock, a man received a gunshot wound to his right thigh and is currently under guard at the Royal Brisbane Hospital. The vehicle was then pursued at slow speed and was stopped by police on Gimpy Road at Kedrin. The three other male persons were taken into custody. They have been arrested and charged with a number of offences and appeared in Pine Rivers Magistrate Court this morning. Additional charges have also been preferred this afternoon and investigations are continuing. As you would be aware from our current operational procedure, police from ethical standards command are over-viewing the investigation, particularly in relation to the shooting. Second statement. In early December 2011, a series of ram raid-style offences were committed in southeast Queensland. This led to operation kilo-perpetual commencing on the 2nd of February by Crime Operations Branch property crime squad. They coordinated the investigation of these offences across all regions. The operation widened to include offenders that had been targeting business premises by using vehicles or weapons such as sledgehammers to gain entry. As a result of last evening's events, this operation has been upgraded to a task force named Task Force Kilo Perpetual and it comprises members of all regions involved. The task force has been headed by State Crime Operations Command. Offences have occurred across five police regions, including Metropolitan North, Metropolitan South, Southeastern Region, Southern Region and the North Coast Region and that comprises 35 police divisions. There are currently in excess of 30 persons of interest and included in that 30 persons is a number of juveniles. To date, 190 offences have been identified. These include burglaries, enter premises, unlawful use motor vehicle, stealing, armed robbery, dangerous operation of motor vehicles and evade police. I would ask any member of the public who has any information that may assist the police in locating the offenders to please contact Crime Stoppers on 1800 333 00. Thank you. I'll take questions. Well as I said, there's in excess of 30 persons of interest and these are four of those persons. Are they all linked? No, they're not all linked. There's different groups in different areas that are operating but they're using the same MO and quite probably some are copycats. Those investigations will be undertaken by the task force but early indications are that yes that is quite correct. Have you ever seen a spate of such a condensed number? I don't think we've seen such a condensed amount in such a short space of time at the amount that have been committed, which is why we're asking members of the public to come forward and assist us. It's quite obvious that a number of people know what's gone on and they know who the offenders are but they're not telling us so we would like their assistance to do that. But with this task force that's now been, is going to take over the whole of the investigation, comprising members of each of the regions, I'm hoping that we will get some excellent results quite soon. At this stage I can't give you their names, I'm sorry. I'm sorry, I don't know their ages at this stage. I think it's very concerning for all of us that we do have not only adults but juveniles involved in these types of offences because I think we've seen from the ram raids, they are quite violent in nature the offences that have been committed and the property damage that has occurred. So it is very concerning to think that some of the younger members of our community are being involved in these types of offences at such early ages. Those investigations are still continuing. They've been arrested on incidents that occurred in Metro North. Their further investigations have been conducted and there will be charges laid against these people from I would think Ipswich and Oxley and other areas as well. They've had an excessive about 20 charges laid on them today with a lot more to follow. No, he's only just got out of surgery. So he will be interviewed and charged at a later day. Sledge Hammers or Ram Raids, yes, with vehicles. I don't know whether they're getting better at them. I think that their style of entry is just smash and grab. It's use whatever violence you can possibly use to get into the building and then take whatever you can. I don't think there's any sophistication with them or that they're getting more professional. I just think it's just whatever means you can get in use that. To my knowledge, no. Yes, I think it's like what we've been doing with all the business premises is that we do have the crime prevention people that can go out and ensure that their security is as good as it can possibly be with CCTV, footage, etc. What we are asking that particularly the business owners, if they're there, that please not try and interfere and stop these people because you don't know what they will do. It's best just to let them take what they want and your love be saved then you end up being hurt and hospitalised. I think it's cash cigarettes, alcohol, anything they can get their hands on really. That's one of the things that our task force will be investigating and I'll be looking very closely at that. No, I'm sorry, I can't at this stage. That's being investigated by Ethical Standards Command and when their investigation is fully completed, I've no doubt that that will be released but at the moment their investigations are continuing and I would not like to compromise any part of that. The officers are coping well. They have a lot of support as you can imagine and they are cooperating with members of Ethical Standards Command and as is always the Crime and Misconduct Commissioner over viewing the investigation. They were known to us from as suspects, yes, but whether they're actually known for a number of different incidents that have occurred, I don't know, but they were known to police, yes. I would think it was excellent police work by the off-duty officers that, of all the BOLF spolos that we put out on a daily basis, that their night work had been on night work the night before and that they recognised this vehicle as being a stolen vehicle and took the appropriate action. Well, the five regions that I've nominated, they will have officers involved in it as well as a number of officers from Crime Operations Branch and the Property Crime Squad in particular. Oh, surveillance will be involved in it. All the areas that we need to join part of the task force, they will become a part of it, yes. No, about all of the offences, etc. that have been committed. I can't confirm that, no. No, no, that is between the three so far. They've been to court today and we've preferred further charges on them again this afternoon and there'll be further charges laid. So we'll be able to give you an update on the exact amount of charges within the next couple of days. Oh, yes, Baal was opposed this morning. No, no, he's been under guard at the hospital and has been in surgery on two occasions this morning, so police have not actually interviewed him at any time. We will wait until he has recovered sufficiently that he is able to be interviewed. It would have to be in its hundreds because it was, I mean, so far we've identified 190 offences and we also have 30, in excess of 30 persons of interest. So you could have five or six involved in in half of those offences. So I would think that the charges will run into the hundreds most definitely. As police officers, we don't normally give out our own personal opinions as to why things do occur. I think that we've seen this type of offences being committed over a long period of time. It's just that these all happen to to be connected and which is why we have such a massive amount at this point. No, no, no, the task force will be set up at crime operations branch and officers from each of the regions will be working out of the task force. So any offences that were committed at the Gold Coast, if there is any on the on out of the 190 Gold Coast officers will be working out of the task force in Brisbane. I would like to think that the arrests we've made overnight may make some of the offenders think twice about continuing to do that the offences are ram-raised, etc. Whether it slows them down, any time we'll tell on that. I can't make any comment in relation to that. I'm sorry, that's a part of ethical standards commands investigation. It does make it difficult, but with the amount of scientific evidence that we can glean from scenes this day, it does help us greatly. The CCTV footage is is one of our best tools to assist us. But it is they are very, very difficult investigations to do. I mean, we would like to think that we could have them all solved in a very short space of time, but we can't say that that will occur, because the investigations have to run their course. And as you know, we have to be able to prove all the elements of all the offences before we actually charge someone with them. I think it's, it is a good start for us in that we have been able to arrest four of the people that have been involved. And hopefully our investigations with these four will lead us to other offenders as well. No, they're a real passenger. Yes, that is correct. Look, that would be something that we'll have to go through all for with the task force and look at all the offenses that have been committed. It's but it is several years. But there has been a lot of stolen vehicles involved in these offenses. So it'll take a little bit to put which vehicles been used in which offense. Yes, with that's probably another thing that for the general public that we all seem to forget. We get home every afternoon. We put the keys on the table beside the front door so we know where they are tomorrow morning. But if your home is broken into and that's where your keys are when you get home, if you've gone out, your keys and your car will be gone. So with a lot of the more modern vehicles that have the locking mechanisms, etc. that you can't get through, they're actually doing breaks on residents to get the keys. Or some people are leaving their car parked in their garage with the garage unlocked or in the driveway with the vehicle unlocked. And of course the keys are in the ignition. So it does make it a little bit difficult. So we do urge people to think of somewhere else to put the keys except in the place where it's first to grab in the morning. And that is a growing trend and it is quite worrying. I don't think you can elaborate on why they're only targeting those sorts. You think the four wheel drives and the use perhaps for the ram rate effect that maybe they're stronger. I really don't know. I'm not a mechanic or a car enthusiast. But I'd suggest that's what their appeal is. No, because they're still in custody at this point in time. No, two were pleaded guilty this morning and were given probation. And one was remanded in custody. The two that were given probation were arrested by detectives from Castle Dine on further charges. And Ipswich detectives will be with them at the moment, interviewing them in relation to those. So they will reappear again tomorrow in court. I would suggest that it'll be at the Pine Rivers Court. But that would be something you would need to follow up in the morning. They're all still in custody, yes. Yeah, have a


Alright, baby.  How you guys doing? So today I decided to do a wash and go I clarified my hair. I made sure my hair was very clean Not going to details of what I use I will be putting everything in bow in the description box to set my hair in the wash and go I used the curls cream brulee whipped cream and I used two different type of gels I wouldn't recommend doing this But I only had half a jar of the Eccostyler Olive oil gel and then I went in with the wonder curl Get set ready hair jelly. So I'm gonna be using a diffuser to dry my hair because we have somewhere to go I know that my hair is not gonna dry all the way through, but I need to get it to at least 75% dry before we get out so This particular dryer is the red Conair Ceramic 2200 fro.  I Have not done a wash and go Since we've been in Texas. Have I? No, I haven't have a girl. I don't know. I think there's another clip somewhere. Yeah, she got All right, you guys so This is not a tutorial Head this is just showing y'all how I use a diffuser for my hair So one thing I like to do is keep my hands out of my hair because that causes frizz like I want to fix that But instead I'm going to shake my hair And let the curls do what the curls do I'm going to be using this diffuser on look I can't say I'm gonna be using this diffuser on high fan, but medium heat If that makes sense. Okay, so here we go I'm gonna my hair up to diffuse the hair that's underneath. Oh, yeah This is put up and half my hair is down. So I'm gonna go ahead and I'm trading 45 minutes later, and it's still wet in the back. You know what?  Sorry do my makeup Because because yeah, we don't we don't Don't not dry you can see some big pieces of this white residue will dry once it'll clear excuse me There's some bits of pieces in the back So for me personally like a lot of curly thick girls I get the best wash and go style by the third day in the middle. It is still wet you guys Like I said, I've been around my hair for an hour. We gotta go. I gotta go get some chicken and waffles We ain't got time for this So that is it everyone. Tell me how do you like to dry your wash and go if you do wear wash and go I probably have used a diffuser less than five times the entire time. I've been natural It's just it's what it is. So I'm gonna put a bobby pin up here to get it out of my face some see that and Then it'll get some length as I said once it starts to dry. So that is it everyone. Thank you so much for watching. Take care


First social of the semester. I am Sushi today. Guys, we're on our way to Yastarala! Yay! I can't even talk about it. It's Thursday and I want to start my weekend in the life right now on January 26th because I am Sushi today. I'm literally dressed up like Nagiri because I'm going to a social and the socials theme is dressed up to the first letter of your name. So, I literally made this Sushi costume. It makes me look chunky but you know, I wanted to do it because I thought it would be funny and it's super easy to make. Isn't it so cool? The lines literally look like the tuna slash salmon. Then I got this black ribbon that I just tied in the back. This comfy t-shirt is literally $5 at Walmart so it's like hell yeah.  I tried something different and it definitely didn't work out so that kind of sucks. I feel like I normally don't wear something funny like I never do. I always am like something like, I don't know, cute. This is my first time, you know, going out of my comfort zone. So I'm back home for my first social of the semester and it was so fun. I feel like when you're in a sorority you definitely should take advantage of everything that they give you and like all the opportunities and all the fun things. So, I'm definitely going to try to make it just every social but we'll see, I don't know.  Like I thought the sushi thing was so cute and like so funny. I think our next social squad goals and my whole friend group actually got cram shirts from our last squad goal themed event. I think we might just wear that again because it was $15 on Amazon so it wasn't very cheap. If we can get like two wears out of it that would be like prime because I don't think the first time we even took pictures in it. Hey guys, it's the next morning and it's actually Friday. I woke up thinking it was Saturday and that's why I like don't really like to go out on the weekends because or weekdays because I messes with my mind. And it makes me think that tomorrow is like Saturday or Sunday. I don't know, it's just weird.  I was like, oh my god, I totally forgot. It was even Friday and that I have meal plans so I'm kind of annoyed because I could have like went and got food if I had known her earlier but I totally forgot.  Right now it is almost 1 p.m. and I'm going to the gym with my friend Maricid and we are going to do the Kayla It's Suns pre-training week three. I don't think she's ever done it before but she's just going to do it with me because Charlotte's at work and she doesn't get off until six and I don't really want to work out at 6 p.m. like that late in the day, especially on like a Friday.   With the weekend vlog without me taking a nap I'm so tired. Went to Forever 21, American Eagle and TJ Maxx I didn't find anything for Gasparilla and I really wanted like black ripped jeans but I had these black moto jeans and I just felt like it was just a waste of money just to get the rips in them.  So I didn't get anything which I guess is good because then I can just like try to find something in my closet and not spend money. I'm at Fujiyama with my little.  Rally. Good morning guys, it's January 28th and today is Gasparilla. I'm on my way to pick up my friend Hannah. It's 7.24 and we were going to go get Chick-fil-A like after we got ready but then I was thinking like you know it needs to settle my stomach before like I get on the bus to go to Gasparilla so we're going to go get Chick-fil-A now. So I had to sleep over with my little and Jenny last night and we had to wake up at like 6.50 to start getting ready. So I just did my hair really quick so I could set and then I'm going to go do my makeup and I think I'll have time because I don't really take an hour to do my makeup anymore which is good and we have to be there at 9. So I feel like all of you are probably wondering like what the heck is Gasparilla? It's basically just a pirate parade and I don't really know why it's such a big deal but last year I went and I don't know I guess it was fun like it was a different experience it's kind of cool because people get buses to take us there and back and that's like a 2 hour drive one way and all you have to do is pay $20 which is not bad at all like a round trip to Tampa is $20 on a bus that's so cool. It's just cool to see like a different like city and I've never been to Tampa before this Gasparilla event last year so it was really pretty and I really want to go back actually like a whole weekend with like Ryan that would be so fun because I feel like there's still a lot to explore but like all the roads are blocked off so like if you try to Uber anywhere like it's so expensive I heard and there's no cell service anywhere because there's so many people there it's kind of like Mardi Gras I guess I've never really been to Mardi Gras either so I don't really know but the people on the parade floats they throw beads to like the people in the crowd ready for Gasparilla round 2 for us we are on the bus cause we got on the bus but we weren't going on the bus cause we heard it was like but like literally we got on and then the bus driver was like like there's no thing yeah yeah it was like that of course I just want to be there already so the parade is over and there's so many bees on the road we'll climb in the trees so it is 9.21 and I got home at 9 and we left Gasparilla at like 6 and one and a half hour car ride took 3 hours when the bus started smoking it smelled so so bad and then I felt like super nauseous other than that I think that this Gasparilla would be deemed successful I am just happy to be back like I'm about to take on my makeup and just get snuggled in bed I feel like nothing's going on right now so I'll see you guys in the morning good morning guys it's Sunday and it's really gloomy outside so I'm guessing it's super cold so I have this like big giant sweater I got from Goodwill on and I'm about to go to my sorority house for like a lot of meetings today so if I don't vlog a lot like I'm sorry but I feel like Sundays are just like my whole day of meetings so it's not gonna be like incredibly fun so I'm done with one meeting two more to go actually like three look how cute you look I love that camera I wish I would have gotten that one instead of my thing yeah we're at Brooklyn Water Bagels and it's my first time here so I'm excited to try it this is what it looks like I got the Brooklyn it's like the original I guess and I got a blueberry muffin for you know in my meeting so I'll probably get hungry so I got that for later I'm back home and I'm done with all my meetings it was just for my sorority and now I totally forgot that the Miss Universe is on tonight but my sister just texted me and reminded me so I like grabbed my remote and turned it on and if you guys didn't know I did two pageants it was Miss India, Georgia Teen pageant but it was a cool experience because I met so many other like Indian girls and not like my community because it was actually kind of far but I still like talk to those girls sometimes and like you know I'm friends with them on Facebook and it was just cool to see like where everyone is now but I mean if you're looking to do a pageant like I totally recommend it it's a really cool experience like yeah people are against it like it's so girly it's like whatever and I feel like you gain a lot of confidence like you are going up on stage in front of like hundreds or thousands of people you know


All right, the new trailer for frozen 2 has come out and it's fantastic and I want to talk about it Of course, let's go all the way back here and start from the beginning. So right off the bat I think this is awesome. It's gonna be super weird. Why because of those bars. I can draw a fantastic straight lines Kent. I this is a plus here But anyway in Switzerland, we didn't have the pan and scan thing where it was cropping With that resolution and then pan arounds to give us, you know, the different edges So if you watch ghost bosses for instance, it's told weird tangent But at the beginning when they walk in for the first thing, you only see a few of those Ghostbusters walking in I think Winston is cut out in the pan and scan version But anyway in Switzerland, it was always like this So for me the letterbox the black bars always means better quality So this is a weird weird way to open this but I wanted to say that it's awesome that I see those bars But let's continue here with this and one of the things that that struck me in this trailer was that in the original in the first One original in the first one It's basically this is kind of the world and then where they go towards at the end and the world felt kind of small It's still very cool. Actually. I like the movie a lot, but it felt kind of small this Far away far away. Not the sources more. It's felt more like what we saw before but the world has expanded I think this is very very cool as north as we can go I'm gonna bring this sound down a bit who knows copyrights and all those things that might even stick that sound completely Off maybe so here is the world. We are familiar with this I'm gonna turn down the volume a little bit for myself here because a copyright things YouTube stuff of music But so I can hear myself But alright, so I was already with animation and a super cute moment here But this is cool This actually reminds me something that we talk about in class a lot in terms of cycles Is it okay to have cycles on your reel? Anything is my answer is always why not? It really depends what you're shooting for Especially if it's for uh game stuff, you know depending on your gameplay animation all kinds of things But what's cool to me is that if you look at her arm, right you got your walk through here But she has that moment of bringing out that arm a bit more stretched out here on the side and then getting ready For a jump and kind of looking at this here So what I mean by this is that if you do have a cycle I think it always behooves you for some reason I would say behooves lately I like that word To tweak it a bit, right? You have a little bit of a change for contrast It sounds so cycling and it adds something that's a bit more of interest and a bit more contrast I think this is a great opportunity and you know, if you have other cycles And you want to populate your shot with something then you can have something like this and that's Absolutely that lets and again. I'm looking at armets and costumes. It's already printed. It's kind of cool. I like it It's so cute A lot of the shots are very short in this trailer, but it is still cool to see Look cutie pie, disney, uh, younglings That's like how they're animated. They're also super cute It's like the focus there Is that character moves the most even though this character has the arm going back I'm not really distracted by anything. It's pretty clear where to look I'm just already massively smitten by everything That's cool. And I like that this goes into as it says here a magical place And I'm totally on board with this having this getting into the more fantasy magical realm I mean, we had trolls and things before but still Cool to look at that cool look too with the lining I don't we're nerding out on things that you're seeing as well, but That's like the expansion of the characters or where we are And a lot of effects in this trailer too cool stuff there. I'm looking at Cycles is where you're gonna another thing of where if you have cycles, right one way to show them off could be something where Now you might not want to do effects, but you could have it's a bigger creature Some whatever creature that you want to do or a dragon or something And then all those cycles you did you can put in As they're you know fleeing as they should from this passive creature And that will be one way to show off your cycle But it's not just a cycle that we just show from you know front and side on your reel But it's part of the shot and it's just part of something where they're fleeing There's a purpose to them running or walking off That's cool. That reminds me of something that I just saw that I can't talk about That's cool This to me would be an interesting thing of a it's almost like a weight assignment Like you're pushing against something weighty And you're showing the force of the wind. It's cool Also, you're interesting to see contrast right one arm here facing like this Here's one arm with the shield like that both arms like this So kind of the same ideas but all contrasting and different It's very cool And you can see also the timing this how this character goes Has that bigger drop there.  Oh, I love this. This is the first thing I went. Oh, this is so cute I love this. I love that You know, you can tell the loving sisters and they're they're used to contact and there's Just the the friendship in the sisterhood there. She's so cute in the cute designs I feel offset asymmetry on that mouth, but I love that little thing here So cute Again, very short shots logo This is cute too. Anyway, we're I'm just reading the things here, but you have hold on. I can scrub it faster where She is resting on the shoulder and then as we go older here Now it's the opposite But that's always something I always recommend to the students contact points A lot of times contact points or just people who interact that it's like a fight or something where someone punches That could be so weird Tengen here, but not that you should punch her But it would be like a one or two frame contract contact if at all depending But to me interaction like this is much more impressive to see because it's very hard to do and it's depending on your tools It might be or a huge pain to do that, but it's always interesting to see and Potentially more interesting in terms of skill sets to show off Again very short shots, but I will look at this as a student or just for myself as well When when you start off, right and you're still looking at how how are my blinks working? How should I blink? I would always go and look at shots like these and Track like how is the ease in how long do they compress? How much do we see the eyes the pupils though the white in the eyes? How long is the ease in here? You can see how long this is With a little bit of relaxation at the end a little bit of eye dark there just just cool reference for a normal blink and this could be Loving You know, this is not something where someone is panicky So you can kind of write that all right loving or admiring or comforting These are kind of the blinks not saying that you should copy this and frame by frame And you put this always in your shots But to me this is a good study where you can write down how did they do how do professionals do this What what is the look for this and then go from there and that could be your starting point And of course you tweak it depending on your shot, of course Again very very short shots This would be your kid up assignment Yeah, it's not much to see here shorts Same thing here like this would be a tricky thing for your reel right not saying that this is not Well animated other stuff and good offset pose and everything but this on your reel is tricky because there's not much going on I like that character too This is a cool example here for your the accents watches This is also something that I constantly harp Uh in class and I annoy my students Just if you do your you know the classic thing is you do your body animation And at the end of your lip sync and a lot of times you can feel like your body's doing one thing And the lip sync is just copy pasted and I'm always a massive fan of head accents If you watch my channel, you know, you must be sick of this right now But I love head accents and I like this watch watch the amount of head accents just during the short moment It's just Where it feels like the words she's saying coming out of her mouth the jaw everything is connected There is just it feels like she is actually saying those words. It doesn't feel just the copy pasted Lip sync school Love all this here. This is cool This is something interesting Uh as a test shot or as an assignment in class where you have a character, but then you have outside forces Interacting and pushing and influencing the character and it could be something like even a bouncy ball with little wings or something Or something very easy to animate doesn't have to be all the crazy particles and you can't do effects But it's interesting to see this here. That could be something that you could do for your shots as well That's cool. Ah shot. Come on That looks cool I also love that little Reflection here of the moon just giving that nice silhouette because of the the darkness with the brightness behind this Well again lots of effects in the shot in this in this trail already to see here if you can see a bit of a It cuts out before it's going to say an interesting look at if you have something where a character reacts your arcs going back You have a bit of a moving hold body cuts out. I'm going to see that and look at the frames for moving holds This would be a good reference for your Your head turn a bit of a head squash. You can see a certain shape here and that definitely a squash Into a stretch here. You can see the difference here It's almost like she gets hit over the head here Well, you can see you can track arcs squash and stretch Reactions just on a head turn how there are changes in the fingers shoulders arms Same thing here. It's not just the head going down, but there are changes even though it's minimal You can see it in the fingers shoulders as well in the hands same on this guy You just can see that every frame counts here again as a students all important to look at and to study more effects Always a big fan and you see parts of cities with lots of people Where it's it just feels like the world is populated. It's big and the people actually there Same thing here. This would be your your influence assignment or something Pushing the other creature. You can see your stretchy legs And creatures you can never discount creatures. This is also cool to see how There's the drag in the head even though the force is here It is a squishy animal to some degree and you can see how much the head waits until it goes forward And in the react same thing here. This goes down and this has with the drag and then the drop It's cool to see. So it's just the whole thing here I like that too little thing at the end and Oh nice, you can see that little change in the face right there before the cut I'll see that to me. It's cool too. Again. You have walk cycles like your shot could be this Character concerned a couple steps turning around so all very complex animations. It's not easy to do This could be you could have fire in the foreground or whatever you have where the character goes away It's kind of like your panicky body mechanics pantomime shock But if you have other stuff like I said walk cycles populate your shot with these guys and you can even Kind of play with depth of field and blur it out so you don't scrutinize foot plans and things But this would then make it more Of a interesting shot to me where it's more shot related in terms of it's not an exercise But it's mainly about this character and you add some other interest and you add Your walk cycles or run cycles or whatever to populate the shot It's giving this a bit more of a cinematic look and to me also if you apply for company showing I can think in terms of you know a production shot And not just an exercise where it's an empty room in a character name to seem doing something Which of course has its place. It's not saying it's completely horrible and useless But I think this could be a cool example here How to use that again outside force It's cool. I love all that stuff here again See how it influences creatures differently humans and creatures love him It's cool Again lots of short shots. Cute This will be a good reference frame for Contrast of posing they all have to be somewhat different Yeah, we're in trailer land where everything is very short. But at this point I'm going into Cool textures the lighting the look of it Very cool interesting three fingers Still can see the contrast and the finger poses From Disney Yeah, again very short. This is cool. Though. I'm always a bit kind of switching into 2d or fake 2d Interesting the up and down and the arcs in the head again. It's cool to study how just a step How you have the accents in the head how the body works Look at that even the shoulders you will always forget the shoulders to some degree Love that a little up in her and she goes back here with the shoulders. It's cool to see all the details Also reacting right you have other characters in your scene. She says something boom right away reaction Darts processing what she is doing and then you got this patient with the eyebrows before she goes. Haha wait a minute. No, no wall Getting dark crystal vibes I love that show so everything now where I see fantasy style stuff with lorda ringsy things. I love it Whoa also cool frame again dark and the light silhouettes just for staging interesting to see That's cool. Sorry. This whole trail is going to be me going Love the colors, but I love this I know like I said I was looking at in terms of animation interesting things of contact points change of posing How she leans in it's just interesting all that stuff. I just I don't I just have to say I love animation. All that stuff excites me the squishiness The jaw because you have the shapes. I love all that stuff It's so cool That's cool too more effects cool reveal Again, all the changes even on something like this where technically the focus is on this Forget these guys Change the arms the steps short shots wall The silent framing through things I'm just looking at her steps Always details hand poses. It looks focus Always a fan that change of costume we saw these guys before But it's good to see for Full extension on the jump and on the land. You don't want to land with your legs Bent like this. It's going to be brutal for your knees. You want to cushion this stretch frame? Bam and you can see here as the root comes down That's the legs are stopping the root from from going further down But the momentum is still there and though the rest goes down finishing in the head and then the recovery and because she is somewhat off-balance Taking a step forward All seems simple if you just frame it out like that, but it's cool It's essential to make the weight work And the mechanics different type of landing there too This is q2 again I'm looking always in terms of a shot that you could do right where let's say you have A character and it's totally different from the other character. This could be a creature Or something we're like a snowman or something and the other character and you animate this now I'm a big fan of cutting things off like this where it's just there to support this and by support I mean he's scared and wants to go behind her for protection that to me is the main focus But you can still have another character in your scene That is just slightly animated again, not too much there. You still have post-changers in the fingers You don't need to see the face But again, they just add something a little bit of a camera move Just makes this an interesting shot and it's all about the face and the emotions So this could be your pants, but I'm shot and it's not just one character You have all of this to support that school Through back runs up. I'm looking as always on hand poses Always my eye on ik arms Looking at how much elbow change you have and how much does that influence hands That's a tricky thing you're getting into ik I'm with a soup ik where you have the wrist like this and as it goes down the wrist is still Bent like that. This is always very difficult to animate where you would have to Rotate that wrist down and then adjust the fingers frame by frame to adjust that grip It's a tricky thing to do Of course, I'm not going to be sitting here retaking disney animators It's just there I love this Effects here This would be my shot meaning I would animate something But he would never get out of this the effects would always hide everything that he animated That'd be so much out Also tricky to do a stand to a walk just the changes to it can see that big twist in the hips in that turn Do a very nice clear hand pose where something like this on the further away shot turns into I don't know scribbles are telling me. It's this triangle pose You have so nicely silhouetted by color here Very cool Getting very lord rings vibes here But cool again the expansion of characters in the world That's a cool model too. Yeah, it's a cool new set of characters and the looks and the costumes Back to short shots. Oh and then here Love this this was so basically when I was watching this.  Oh, I love this. This is so cool Love that that the rhythm and the swing of this. I love how she does Look at that. Oh Line of action and that holds the hold on that pose Boom coming down. Of course. I'm looking at effects and the lining too. It's so cool the drag on the hair And this is super clear silhouette I love the almost the relaxness and the confidence of those steps. I'm so in So November that's way too long way too long. We need it now Cool. Oh, yeah, and then this was the other part where it's holy moly. I love this really giving me a lord rings Why I'm here, but look at this Love that just the look of this like oh, that's cool with the water. I love all this definitely lord rings You would have a ginormous foot here But it's a cool look and I love the color contrast to between this Then we intercut with this. Oh goes back to you should always have creatures in your demo rail I'll be doing a Lecture potentially a lecture about creatures at ctn spoilers Maybe not announced yet. Maybe maybe not happening, but I'll be talking about stuff like this. Why that's vital cute design too cute Yeah, something else where the creature is on a hand would slightly animate this but then the focus is obviously on this That's a cute design too Look at that the squishiness of the face Speaking of face look at that. Oh stretch there. Let's see how the eyebrows drag this face gets stretched just a bit Got that Brows go down the hair. Yeah trailer land towards the end. Now you get only short shots Ooh, but come on better frame through this Whoa We'll see the effect interaction there all that By all that that means nothing someone's listening to is all that has like all of this here You can see here again the squash and the stretch into the Then I pose with the stretch on the on the shoulders going down coming back up Very cool. Even here you can see the stretch here The face and how the nose comes back up relaxing the face is there and the structure of the face a bit Land is the patient there food going down and pull The drag there. It's cool too. You can really feel it because of the extension of his arms All and then bring that thing over the thing. So I mean This was in a previous trailer. This always cracked me up to how quickly she gets grabbed here That should be dragging eyeballs here Crush rib cage to here looking at the details for Facial shapes asymmetry the squishiness of corners like don't go from a From an in their mouth shape to an extension where it just kind of locks You can see a bit of a curvature and then an arc and the squishiness there Very detailed stuff there holding your f's, you know, your f's over one frame Fire long long take on fire here It's very cool. Don't forget your secondary characters Let's have some long eyelashes. Anyway, let's go the past Oh more effects I just love the look of this Yes, that would be me but my shot wouldn't be this it would cut in here And it would be maybe maybe on a generous day that would be me But that's the reality of working for a movie like this or you got animate everything All of this everything is probably animated really nicely finger stuff everything facial But then you put the effects over it speaking of which more Oh, that's cool. I love interaction like that. Look at that. Where the smoke Parts first and then envelopes the character behind her That's cool Okay, this is going to be horrible, but No political agenda here, but I see a face like this for some reason it reminded me of this But anyway, look at that. That's a lot of work I feel bad for whoever had to animate this But again, cool to see when you think in terms of cycle Oh, let's squash there in the face where you do have to have variation even if it's just a little head turn Okay, just stick in a cycle. You got to have some changes there Oh, yeah, yeah, we are at the end of the trailer lots of short shots Oh, yes, this was the thing on the teaser Cool hand poses That I love this That is so cool. That was super cool in the teaser too. Just posing that's a nice line there Yes, I'm wondering what that is. That was always my thing though Is she going to go underwater to find the boats looking at facial features and mouth openings and reactions with they walk back So that's a nice look too lighting there And this was the other one like oh, they don't just sit around Now we're headed towards awesome lands That is cool. Come on. Look at this. I love that. It's also obscured That would be your more complicated cycle where you have direction change Influences behind it. That's a cool shot. So in I mean, this is absolutely fantastic Back to the shot continuation of that one.  Look at that. Look at those poses Very cool More effects. It's like this trailer is made for me. Oh nice So it's cool to actually frame through and see what's happening like glow You know, always think about silhouettes. You got glowy outline there You're speaking again of silhouette and hand poses and you can see the impact there again. Look at that Stops here. So has to push up shoulder goes up. Boom. You can see how low that head is compared to the shoulder Oh cool holding breath pushing of shapes That's cool. Look at that Sorry, this whole trailer has now devolved into me just going cool. Look at that Oh, nice.  I'm so sold So cool she gets on it too This movie is bananas things are introducing this movie Frozen two Nice Look at those colors. It's a beautiful frame too with that frame. It's just beautifully done I like that with this character. You can really play with this connect the shapes And the end shot is cool too where you got that locked on camera.  And then that's it, right? Yeah, remember that is it's holy moly a lot of stuff a fantastic trailer Beautiful colors. I'm gonna end this on This here actually the other shot with a beautiful shot Well, all right. Well, that's it. I'll leave it at that. That was my nerding out of this trailer Um, and thanks for watching if you're still here and watching this and I will see you in my next clip


I'm going to present a case series on intramarticular hemorrhage. I'm from the US Indian Department of Radiology from S.T. Numbangwai. Astroliter intramarticular hemorrhage is rare in adults and seldom described. After ruling out the history of trauma in US, etiologies remain possible. For most frequently found underlying reasons are atrovinous malformations and organisms. But other vascular causes we need to rule out are cavernous malformations and myemorrhages. Intramarticular tumors can also present with IVH. Atrial hypertension, anticoagulant use, coagulopathy, sensorine toxin, substance absorption with IVH. Finally, the IVHs can be idiopathic. And the necessary investigations, CT angiography, MRI angiography needed to be sought out for finding the cause. This is the aim of the pictorial assay of astroliter with the etiology of intramarticular hemorrhage. Vascular pathologies, AVM, aneurysm, cavernous malformations, voiomboid disease, duolefistry, and vascular disease of the avascular pathologies tumor, intramarticular tumor, producing factors like atrial hypertension, coagulopathy, anticoagulation symptoms, trauma, surgery, and the circulation of subarachneuritis. In case of bilateral intramarticular hemorrhage in the lateral matricle, the cause was found to be moiomboid disease. There is a narrowing of the internal carotid artery with complete occlusion of the supra-clinid portion of the internal carotid artery and multiple collaterals are seen. So, multiple collaterals are noted around the circle of the village supplying the anterior circulation from branches of posterior cerebral arteries. Likely suggested of moiomboid disease and there was complete occlusion of bilateral supra-clinid segment with narrowing of bilateral cervical and intravenial portions. In the case of moiomboid disease, which presented with isolated bilateral intramarticular hemorrhage, in case of intramarticular hemorrhage, fourth ventricle and some subtle subarachneuritis hemorrhage, in this case it was found to have thrombosis of the internal carotid artery, complete thrombosis of extra-clinial internal intra-clinic internal carotid artery. In this case of the bilateral intramarticular hemorrhage with subarachneuritis hemorrhage and the cause was acohomortitism, which is present, measuring 8.8 into 7.2 into 9.6 m. Next, in case of quantifier old female with sudden onset and responsiveness, isolated ventramarticular hemorrhage, third ventricle and lateral ventricle, third ventricle and fourth ventricle and the cause was found to be the multiple collateral rising from the posterior cerebral arteries. There is an idus here present and the draining vein into the internal cerebral veins. So, this is the arterovineal smart formation. Fifth-year female with cerebral vascular accident, intra-ventral hemorrhage noted in the left lateral ventricle and there was found to have ABM in the spleenium of corpus callosum, posterior to the spleenium bilateral middle element with the cyclone and resumon dilatation, nidus measuring 11 into 13 to 10 mm, feeding artery from right posterior cerebral artery, F3 here from the right posterior cerebral artery and the draining veins into bilateral internal cerebral veins and middle ventricular veins, which in turn was drained in the spleen-draining-performed-prepared internal straight sinus, straight venous sinus, again into supracellatoid. So, this case of complex ABM formation with aneurysm. So, ABM formation can lead to hemorrhage restricted to the ventricular system and have been identified as causus of intramarticular hemorrhage. Another case of 35-year-old with headache, blood hemorrhage, right lateral ventricular hemorrhage and pitifiable angiogram was within the normal limit. So, this is one of the cause, where we don't know. Intramarticular bleed, Z-ary homing, there was on T2 major, there was a tumor, large loblated myelianus interventory lesion was present with normal resistance with tumor bleed and intramarticular extension was also present. So, conclusion is a variety of pathologies are responsible for isolated IVs. So, we have to do CT angiography, MR angiography and catheter angiography is must. It provides complement information for the concurrent evaluation and a potential treatment access. Thank you.