Hello, and welcome to this presentation of the STM32 hardware semaphore or HSEM module. It covers the main features of the module, which is used to manage the access permissions and synchronization of resources shared between multiple processes. The HSEM module integrated inside STM32 microcontrollers provides semaphores used to synchronize processes and manage access permissions for shared resources.  Applications benefit from process synchronization and shareable resources, the in non-blocking interrupt-based semaphores. The HSEM module is located on the AHB bus. It consists of an AHB interface containing the semaphores and an interrupt management block handling the interrupt distribution. A dedicated interrupt is provided for each CPU. Each CPU has its own enable, status, mask, and clear register for the semaphores. Each semaphore consists of two registers. A write-read register used to write lock the semaphore in the two-step procedure and read back to the semaphore's status. The write-read register is also used to free a semaphore. The read lock register is used to read lock the semaphore in the one-step procedure. The AHB busmaster ID is used to identify which CPU is accessing the semaphore. This ID is stored in the semaphore when locking and can be read back from the semaphore's status as core ID. In STM32H7 microcontrollers, CPU1 uses core ID 0x03 and CPU2 has core ID 0x01. In the two-step write lock procedure, a free semaphore will be locked by writing 1 to the lock bit in the semaphore write-read register. The core ID and process ID used during the write will be stored in the semaphore. A process has to check that the semaphore is locked by it by reading back the write-read register. If the read-back semaphore core ID and process ID matches the one written by the process locking the semaphore, it is locked by it. If the core ID or process ID doesn't match, the semaphore has been locked by another process. A locked semaphore can only be unlocked by writing the lock bit to 0 with the corresponding core ID and process ID. Writing the core ID or the process ID or writing the lock bit with 1 will keep the semaphore locked. In the one-step read lock procedure, a free semaphore will be locked by reading the semaphore's read lock register. The core ID used during the read will be stored in the semaphore during the read cycle. When the semaphore core ID value read by the CPU matches the one from the CPU and the process ID equals 0x000000, the semaphore is locked by the CPU. In the one-step read lock procedure, there is no process ID. The process ID when locked by a one-step read lock procedure will read 0x000000. If the core ID doesn't match or the process ID is different from 0x000000, the semaphore has been locked by another CPU or process. A locked semaphore can only be unlocked by writing the lock bit to 0 with the corresponding core ID and process ID. The two-step and one-step lock procedures can be used concurrently. In this case, the two-step lock procedure must not use process ID value 0x000000. In the case where locked semaphores from a malfunctioning CPU are to be cleared, this can be done by writing a key value in the HSM-KR register and writing the key value in the core ID in the HSM-CR register. It will clear all semaphores locked by the corresponding core ID. A semaphore get-free interrupt will be generated when enabled. Here is an overview of HSM interrupt events. When a semaphore is freed, an interrupt can be generated to a CPU. Each CPU has its own set of semaphore enable or IER and status or ISR states before the masking or MISR and clear or ICR registers. Here is an overview of the peripheral status at specific low-power configuration modes. The HSM is not able to change state in system sleep and stop modes. In standby mode, the content of the HSM is lost. The HSM can only be changed when the system is in run mode. The HSM is able to interrupt and wake up a CPU in C run, C sleep, and C stop modes. The HSM can be used to handle, one, access to shared resources. That is, this allows you to read, modify, and write registers in a protected way or to share functions like the AES encryption engine, random number generator or RNG, etc. Two, synchronization between processes. That is, used by a process to wait for input from another process. Here is a list of peripherals related to the HSM module. Users should be familiar with all the relationships between these peripherals to correctly configure and use the HSM module.


Okay, very good morning. It is Monday the 23rd of August. Hope you had a fantastic weekend. Of course, this week is going to be bookended on Friday with this guy Jerome Powell is going to be given his much anticipated Jackson Hole Symposium speech and obviously that's going to be one of the main focal points for this week, this month as we look out for the details around the commencement of tapering at the Federal Reserve. But busy week overall in store we've got the flash PMI data is coming out in just a short while for the UK and the Eurozone in the US today for Monday to kick off the week. We've also got German iPhone Tuesday and then you've got the ESP minutes coming out later on in the week and from a US perspective you've got things like the home sales data on Tuesday, durable goods Wednesday, Q2, second reading of GDP on Thursday then of course POW with personal income spending data and a final University of Michigan reading on Friday as well. So plenty to go out in the week ahead but let's just have a quick look as we always do at the charts this morning and actually we've got a fairly positive open to kick off the new week. So quick look across these different charts and both Eurodollar and Cable in the top left-hand corner are up about 20 and 36 pips respectively. So a bit of a reversal if you like of sorts of what we have been seeing across most of these charts. The dollar index this morning is a little bit softer It's down about two tenths of 1%. We have seen equity index futures trade higher. The DAX up already around 132 US indices following the overnight positive tone that we've seen in the Asia-Pac region of where all bosses traded on a firmer footing. Very quiet in fact in terms of major weekend news flow. I've only got two kind of smaller points that I want to go over in a moment before I go into that actual week ahead calendar. So very much just more positive sentiment overall So we have had Treasury yields tick up a little bit and the dollar dipping for the first day in six And so that's also helped WTI crude just claw back what had been some pretty persistent losses So during the Asia Pacific session and really since the open of electronic trade where we open pretty much at where we finished at the close on Friday We've just continued to move back higher. So having traded at the 61 handle We're now trading back up at 63.43 up about $1.27 this morning Elsewhere Tino is down two and a half gold Respecting a relative range for the time being again a little bit of injection of pace with some of the dollar weakness seen overnight To bring it up to the top end of a short-term trading range here from the last couple of sessions For the time being up around $3 So couple of weekend news stories. I said I would get you up to speed on starting with Fed speak from Robert Kaplan who is a non-voting member But I thought quite interesting because Kaplan as you know sits at the much more hawkish end of the spectrum And in fact he came out and said on late on Friday that he's open to adjusting his view that the Fed should Start tapering its asset purchase program sooner rather than later if the Delta variant strain persists and that hurts economic progress So just quite interesting of course timing going into the Jackson Hole speech on Friday that Kaplan's kind of just eased off The hawkish pedal a touch here Then the other thing that was of interest over the weekend as I said overall It's pretty quiet though on that front in terms of actual meaningful news for the open this morning Was this that the US Treasury Secretary Janet Yellen has told senior White House advisers that she supports Reappointing Jerome Powell as the Federal Reserve chair according to people familiar with the matter And the general interpretation of this is a couple of things for one her endorsement definitely Would increase the likelihood of him being reappointed obviously this is Quite different to the Trump era where obviously having the Treasury Secretary giving such a Stamp of approval is likely to be listened to Very much so by the president and so as far as markets take this It's not as if this is a short-term positive reaction that's created these moves this morning But indirectly the markets do like continuity Particularly around policy setting of something of such magnitude as monetary policy So generally speaking whenever you have a replacement there's a degree of uncertainty around that new person Are they going to be able to handle the kind of gravity of the situation are they going to change the direction of Fed policy? And all of that can have negative connotations as far as market price is concerned So overall markets tend to react more positively to continuity of which this would be the case So yeah, something just to be of note in terms of the timing of this So a Fed share essentially serves a four-year term and so Powell's four-year term comes up at the end of February and hence the reason why this discussion is happening right now Just before I move on don't forget. We've got 22 days left until the new amplify me Platform launches so for any students whether school, college, university or graduate This is going to be your kind of one-stop shop to take part in our latest free Simulation technology that we use all of the big banks and also our community where we'll have tons of exclusive content That the team will be putting out. So I'll drop the link into the video Don't forget to register for that if you definitely want to take part in that community We'd love to have you involved if you're watching on YouTube. Don't forget to like and subscribe Otherwise look at the calendar for the week ahead as I said Monday We really kick off in earnest with the flash Manufacturing service PMI so as ever definitely these are going to be potentially market moving for the respective currency markets so euro and Pound to be keeping an eye on today But overall the greater context of the week certainly is going to be more driven by the US dollar But short-term intraday keep an eye on those flash PMIs coming out this morning Then just quickly an overview of Europe first and then we'll talk about the US calendar on Wednesday you've got the German iPhone data coming out And that's going to be always quite interesting to watch and that comes out on Wednesday morning And then pretty much that is it as far as mainland Europe is concerned The only other thing of note is going to be the ECB minutes, which we're going to get on Thursday and as far as they are concerned It's going to be scrutinized for any clues over the ECB stimulus plans before a decision Within months on how economic support will evolve after the current end date of its pandemic purchase program The PEP comes to towards a conclusion at the end of Q1 of 2022 So much like what the Fed are doing with this kind of Front-running the communication of tapering Takes many months for this actually be implemented And so we start turning our attention to the ECB for similar signs given that predetermined end date of that Extraordinary stimulus program that they inaction during the pandemic crisis Otherwise for the US. Yeah, it's pretty busy as well. So this afternoon You get the relative flash PMI data as well coming out in the States today You've then got new home sales on Tuesday Durable goods Wednesday get the second reading of US Q2 GDP Expected to be unrevised at 6.6% on Thursday Alongside initial jobless claims expected to still be tracking down at around the 350 margin Jackson Hole does begin But remember power doesn't speak until Friday alongside the power speech You're gonna get personal income spending and core PC price index and the final University of Michigan reading as well So I'm not gonna really go into power too much at this point in time that obviously is the key event Analysts at ING have said and this is generally the market consensus that most are expecting the Fed to shed light on its tapering decision Which could be announced as earlier September and started in October That would be the much more aggressive on the hawkish side in terms of that timeline The other thing that's going to be in focus is the length of the tapering cycle as well So with a few FMC members recommending quicker fed tapering cycle Then was most recently seen back in 2014 the last time that this kind of sequence was was exercised by the Federal Reserve But as I said, I'll be going over this in way more detail when we get to the back end of the week The other thing as well is just keep an eye out for any other kind of Fed commentary any source commentary The tapering announcement and people's looking for for more details on that is a big event on Friday And so up into that point it wouldn't be surprising if we get some little more Details just kind of drip fed into various news agencies So something to be aware of and the final thing to mention was what's happening this week is that the House of Representatives in US politics returns from summer recess early to consider the US president spending plans including the multi trillion dollar budget What the FT was saying at the weekend is that Democratic congressional leaders in the White House trying to pass this huge budget And their one trillion dollar bipartisan infrastructure package in tandem In order to satisfy them both the progressive and the moderate wings of the Democratic Party Yet the tactic has exposed those some sharp rifts among Democrats, i.e. trying to push through both bills at the same time And has raised the possibility that neither the budget nor the infrastructure bill will make it over the line So again, just something to be aware of and keep an eye on I'm sure you're going to get more commentary out the likes of Nancy Pelosi and others as we go through the week But that is it. I'll let you guys get on with the session Feel free to drop me a comment on the video if there's any questions at all and have yourself a great great day You know great week ahead. Thanks very much


فبطّه آدك الأيام فتّى وتنهى الحمد لله رب العالمين والصلاة والسلام على أشرف الأنبياء والمرسلين سيدنا ونبينا محمد وعلى آله وصحبه والتابعين لهم بإحسان إلى يوم الدين أما بعد إن شاء الله تعالى تليس بانفت is going to be from the Kitab Sir Allah من نبلا بعى الإمام الذهب رحمه الله and the benefit that we will be speaking about is التزهد في الدنيا A student of knowledge and a scholar they need to come with or they should be those who are aesthetic who leave and let go of the pleasures of this world and turn towards the hereafter ما أقبح بالعالم الداعي إلى الله الحرص وجمع المال What is worse for a scholar who is calling to the path of Allah than being one who strives for the dunya and gathers wealth يحيبني يمان رحمه الله he said سمعت السوفيان I heard سوفيان say المال داع هذه أمة money and wealth is the illness of this أمة والعالم طبيب هذه الأمة and the scholar is the doctor of this أمة فإذا جر العالم أداء إلى نفسه فمتها يبرأ الناسة إذا كان سكولا يساعد هذا العالم إلى نفسه when are the people going to be cured because he is meant to be the doctor سكولا هو دكتر he cures the problem of the أمة and wealth is the illness of this أمة so when he starts to take that illness on when are the people going to be cured عبد الله يبلو ببارك also said سوفيان said ليس بفقيه he is not a faqi he is not a faqi من لم يعد البلاء نعمة والرخاء مصيبة he is not a faqi the one who does not consider an affliction from Allah a blessing والرخاء جي as a affliction he is not a faqi unless he reaches a point in his life where when Allah afflicts him with something that is harmful he sees it as a blessing and when he is going through times of ease and happiness he sees this as a مصيبة this reminds me of a story that I read one time there used to be a king and this king had one of his بطانة بطانة is a person who advises the leader who sits with the leader this individual who used to sit with the leader had an attitude or what was from his norms was every time something happened he would say maybe there is goodness even if it was something that was an affliction or a problem he would say maybe there is goodness so one day what happened was the king he got severely wounded on the face and this بطانة said to the king as he always did have these norms of saying this he said to the leader لعل فيه خير maybe there is goodness in it for you so the leader became excessively angry the leader, the king he became excessively angry and so he said to him the wound on my face the wound on my my face how on earth can it be a blessing how can there be good and good in it and so he ordered the gods to arrest this man and so he was thrown into prison he spent some time in prison and so the leader, the king one day the king one day he went out the king what he went out and when he went out alone he sneaked away from his guards he went out to hunt and he got caught by highway robbers the king the highway robbers they caught the king and when they caught the king they said to the king they looked at the king and they realized that he had a severe damage on his face but they didn't know it was the king they were highway robbers who wanted to sell him as a slave but when they saw that he had a big deficiency on him they knew he was not worth nothing and so he was freed by the highway robbers they let him go he came back to his palace and so he said to the people bring back the man who said to me that day that maybe there is good inside the wound there's good in the wound that happened to your face bring that man over and I need to speak to him he said to him والله you were right والله you were you were right because there was good in it if my face was perfect I would have been sold as a slave and the man who was brought out of prison he looked at the leader and he said I missed something in the story when the king said to the guards arrest him he said to the king when he was again being arrested he said to the king when the king ordered the guards to come and arrest him he said to the king maybe there is good in me being imprisoned as well and so when the king told him that there was good in it the wound that happened to him the man said to the king and there was also good in me going to prison why? because if I wasn't in prison I would have been with you and I would have been sold as a slave I would have been sold as a slave so what that tells us is that the person who truly believes in Allah he knows what that the and the sometimes you can be afflicted and you can be a blessing another story أبي قلابة الجرمي who is أبي قلابة الجرمي who is a student of أبي عبس he came to visit a man who lost his two hands he lost his two hands أبي قلابة came to him and he told him he said to him maybe there is good in it that Allah took your two hands away from you the man said to أبي قلابة عجب اللق amazing be to you what good can possibly be in me losing my two hands what good are you talking about the man said I couldn't understand what أبي قلابة was trying to say in the point that he said me losing my two hands is actually a blessing they could be hating that for me he said until he sent a messenger to me and this messenger came running and he said to me يزيد من معاوية has appointed you as the commander of the army who is going to fight against حسيني who has reached كوفا and so he said I sent the messenger or messenger you can see my situation on my hands I'm not able to fight and I cannot lead and so he said in my place or may Allah have mercy on those places and then he said I remember the statement of أبي قلابة that me losing my two hands saved me from killing the son of the daughter of the messenger فاطمة فاطمة رضي الله تعالى عنها حسيني I was saved from killing him because of the fact that I didn't have my hands so ليس بفقيه من لم يعد البلاء نعمة والرخاء مصيبة and we will stop there إن شاء الله فهو this benefit بإذن الله الكريم سبحانك اللهم بحمدك أشدوا الله إله إلا الله استغفرك وأتوبوا إلي


In this video, we're going to have a look at some tools that you can use to help with your note-taking. So there are many different reasons why I might take notes. I mean, take lecture notes, notes on your reading, some quick notes that you may just have to refer to later in your life, notes that are very closely connected to each other, but also may want to take audio notes. So let's have a look at five tools that may actually fit your needs in this area. And I want to talk about Notion, OneNote, Evernote, Google Keep, and AlterAI. And these are tools that actually I personally use all of them for different purposes. So let's have a look at them one by one. So first, Google Keep. Google Keep is a very simple note-taking app. And it is great for quick lists, shopping lists, also quick photos or pictures that you need to remind yourselves. For example, here is my reminder of the reading electricity meter. Here is a quick reminder to check out these tools. I often use it for my shopping list. If I see an interesting sign, I may take a picture of that as well. And so that's where I take my quick notes that I don't need to perhaps link together too much or use beyond a very short amount of time. But there are tools that actually give me much more ability to collect things. And Evernote is one of those. Evernote is one of the original note-taking apps, one of the old standards. And perhaps many people are moving away from it because it hasn't kept up with time, with the times. But one thing that is great for is collecting a long list of resources in various notebooks. So you can do that. And what is particularly good at is for integrations. Because it has been around for so long, there are many other tools that integrate with it and send data to it. And so I will use that Evernote for that very often. Now, one note is a competitor to Evernote and it is much more visual. And it's great for taking notes in lectures because it is possible to create multiple notebooks and then have pages and sub-pages. But you can have things together with each other side by side. And the one great feature that you can have with Evernote is that it's really great for taking photos off screens. So if you're in a lecture and you take a picture of something that's going to be presented, Evernote will do that really easily. So I often take notes in Evernote as I'm sitting in a lecture. And then I use my phone as well to take pictures and it will insert it directly into Evernote. And actually it will make really a good job of converting what's on the screen into text. And so often it's much better quality than just taking a picture with your phone camera. Now, there is a new favorite or a new exciting tool that lots of people are talking about. And that is Notion. Notion is really great if you need to combine a lot of tools. For example, even allows you to do some project management and collect things together in your research. And for example, one of the great features of Evernote is that you can create databases with it. So the database that lists all the tools that I have collected for these videos is made in Notion. And I can easily share it. So Notion is really great. And also it is now free for all students. So you can certainly take advantage of that. It doesn't have some of the same nice features around photos that Mono does and doesn't have some of the visuality. But it certainly is, again, very powerful. And it is one of the more higher investment tools because it will require a bit of time to get the most out of it, but it will perhaps repay you that investment. And finally, for audio notes, there are not as many tools. But the one that I would recommend that you look at is alter.ai. And it is really intended for transcribing of meetings and taking meeting notes. But it's also good for just making a little audio note. So you can get a mobile app or use the website. You can upload an audio to it or just directly record it. Then it will do a transcript of what you're saying. And then you can combine it with pictures and then use those notes to remind yourself of what you want to do. So again, alter is a tool that isn't primarily for note-taking, but it can be really good if what you want to use is audio for taking your notes. So thank you for watching. And in the next video, we're going to have a look at tools for writing.


I would first like to thank the organizers for this wonderful conference. It's the first time I'm here. So before I say anything about math, everything I will talk about is a joint work with Juhi Jiang, who is based at University of Southern California and this year at the IAS. So the good thing about this long title is that it basically tells you the statement of the main theorem. There are certain special solutions to the system of PDEs I will look at, Euler Poisson system. It comes from astrophysics. It's a model of a star. And we will be studying the nonlinear stability of that class of solutions. So let me give you an outline. Well, first, I intend to spend quite a bit of time introducing the system and talking about general properties, perhaps the expense of giving you some technical details in the second half of the talk. But I can promise that there will be two different stability results pertaining to two different types of special solutions that we will look at. And one of them, the first one is easier. This is the one where I will probably give you a pretty complete idea how the proof works. And hopefully there will be time to talk about a second type of expansion. So before any of that, let me just tell you that the gravitational Euler Poisson system is the model you would find in a typical astrophysics book describing a star. So it is a Newtonian model. The idea is that the star is a gas or a fluid kept together by gravitational forces. So let me tell you what this talk is not about. It's not about relativistic models of stars. But nevertheless, part of the motivation for studying this model indeed comes from relativity. And I will mention that hopefully at the end of the talk. So as I said, the star is idealized as a compact fluid body. And the word compact is crucial because it suggests that there is a boundary between the fluid and the vacuum. And if you want to model such a problem and if you want to track the behavior of the boundary, you have to allow for that boundary to move. So this causes this gives us a moving star vacuum interface. So what is the model? Well, first I'll tell you the unknowns. The first three are unsurprising, and you're used to them, the fluid density rho and the pressure p, the velocity field u, and the gravitational field phi that you need to couple to the Euler equations. The fourth unknown or the further unknown is the support of the fluid because we are trying to treat a free boundary problem, OK? And the equations, again, you have seen some fluid models this time around. Don't be scared. We will soon forget about this slide. The first equation is the continuity equation. The second one expresses the conservation of momentum. And on the right-hand side, you will see a term caused by the gravitational field phi. And the gravitational field phi is self-consistent. It's generated by the mass density here. So Laplace phi is 4 pi rho. This is the Poisson equation. And these are the equations of compressible fluid dynamics. On top of that, we have to supply this problem with some boundary conditions. And remember, this boundary is moving. So the kinematic boundary condition is the natural statement that the boundary is moved by the fluid particles. So the normal velocity of the boundary is, in fact, the normal component of the velocity field at the free boundary. And the pressure vanishes at the boundary, OK? Now, if you count the unknowns, you will notice that there is one superfluous unknown here, namely the pressure. There is no evolution equation for pressure. And you need to prescribe a further equation or a relation to actually close this system. There is one equation for rho, three for u, one for phi, but there is nothing for pressure. And this is typically done by prescribing the so-called equation of state. Now, equations of state is something that, in the physics literature, you would look at a particular model of a star. And you would hope to have a good understanding of what the equation of state is for that particular star. What is commonly done, people study the so-called polytropic equation of state where p of rho is rho to a certain power gamma. And this gamma is the so-called adiabatic index, and it's allowed to vary between 1 and 2. So the way I would like you to think about this model, I would like to think of this nonlinear problem as parameterized by gamma. OK? And as we vary gamma, the qualitative properties of the solutions of this system will change. OK. So let me comment a little bit on the literature. The vacuum interface itself is known to cause severe analytical difficulties in improving even local well-posedness. So the basic result in the theory of compressible fluids with vacuum interfaces are basically these two results. They came independently in 2010 by Couton and Scholar and Jang and Masmoudi. And so what they have to contend with is a certain degeneracy. I will explain it as we go through the talk that is caused by the presence of the vacuum. But one of the key insights is the use of the Lagrangian coordinates. So if you use Lagrangian coordinates and if you rephrase the compressible Euler equations surrounded by vacuum, you will discover that there is a wave equation, a quasi-linear wave equation lurking in the background. OK. So that's somehow one of the big revelations. It is a degenerate wave equation, degenerate in the sense that I will specify later. But the word degenerate has to do with the fact that some of the coefficients of the wave equation will vanish at the vacuum boundary. And another key assumption in these works is that of a physical vacuum boundary condition. So here I'm stating what it is. It's a statement that the fluid enthalpy, which is this quantity, rho to the power gamma minus 1, has to have its normal derivative at the vacuum boundary, has to have a sign. This is perhaps not a well-motivated condition. So physically, it expresses the fact that particles have to accelerate in the normal direction at the boundary. But I will soon actually motivate this. Sorry. I will soon motivate this condition by finding particular steady-state solutions of the Euler-Poisson system that satisfy this. And indeed, if you open an astrophysics textbook, the star models, the steady-state star models that they look at, all satisfy this type of condition, the compactly supported ones. OK, so there is a little bit of a discrepancy here. Here I'm citing works that pertain to a compressible Euler equation. You could ask me, what about Euler-Poisson? Because I'm adding a gravitational field. From the point of view of well-poseness theory, this is a lower-order perturbation. So if you understand how to prove well-poseness here, you will understand how to prove well-poseness for Euler-Poisson system. But nevertheless, there is a reference that I'm missing. Where is the chalk? Oh, fantastic. Thank you. So prior to this work, there's one result which deals only with a priori estimates, where with the Lienblad, Couton, Lienblad, and Scholar. So this is one missing. So on a priori, bounds, Couton, Lienblad, and Scholar. And for the Euler-Poisson system, this came much later entirely based on these ideas. There's a work of Luol, Schinn, and Zang. OK, this is for it. Not for the vacuum interface. So for the liquid case, when the density is not vanishing at the boundary. So this is a significantly less degenerate model. OK, so instead of I will now try to give you to do three things for you. I will give you an example of this famous class of steady states known as Lane-Emden-Stars. Then I'm expecting a psi of relief. I will produce an invariant rescaling for this problem and explain a certain notion of criticality that enters. And that will motivate the main results. So most famous class of special solutions are the steady states. And the way you produce them, you set rho to be only a function of x. And u to be 0. So you plug it in. And the whole equation reduces to this statement here, gradient of rho star gamma plus rho star gradient phi star equals 0. So if you use the fact that this satisfies the Poisson equation, and if you call this object here, this is the fluid enthalpy. If you call it w, then in spherical symmetry. And this is a very classical stuff. You will see this in astrophysics textbooks. You reduce the problem to solving this ordinary differential equation. And the key question is, for what powers of gamma can I find a compactly supported solution with finite mass? This is your star. So the answer is, if gamma lies in this magical range, then you can find profiles that are indeed compactly supported, say, if it supports 0, 1, and they look exactly like that. Once gamma hits 6-fifths and below, no such solutions exist. It's just the ODE analysis. And moreover, these special solutions in fact satisfy the physical vacuum conditions. So w prime at 1 is strictly less than 0. So the density profile is not smooth across the vacuum interface. And this is one of the critical points that works of Couton, Scholar, Jang, and Nassmoudi had to address, the fact that the non-smoothness is a generic part of the theory. So you have to find a way, what does it even mean to differentiate the equation in a proper sense? You can ask, however, and this is what physicists did, are these solutions stable? So now the numerology gets even more interesting. If gamma lies in this range here, the lane end and stars are linearly unstable. And if gamma, sorry, this is wrong. And if gamma is bigger, equal than 4-thirds, it's in fact linearly stable. So it's just some basic linear analysis. Of course, it begs a question about two questions. Are these results, can they be upgraded to a nonlinear statement? So the first result can, it is not terribly surprising. Once you have a growing mode, which is indeed the main source of instability in this regime, and once you have a good well-posedness theory, then you can prove nonlinear instability. And this was shown by Jang in 2014. Nonlinear stability is a much, much more contentious point. So there is a conditional nonlinear stability result in a very, sort of a fairly weak topology generated by basically just the energy by Ryan from 2003, which says, if the solution exists, it will be nonlinearly stable. However, there is absolutely no reason why the solution should exist. Why small perturbations of these steady states should. And it's in the same regime, 4-thirds to 2-thirds. Pardon? The result of Ryan, is it in that regime? Right. So the result of Ryan, the topology is so weak that it doesn't really see the free boundary. In particular, this type of stability result, if you think about it, will actually allow you to have support splitting up and things like that, the topology will not see it. So you really need something else. And in fact, I will talk about it at the very end. I will highlight this as one of the important open problems in this development. OK, so let me try to perhaps explain these exponents that show up in a different way. And this hopefully will help you appreciate them. There is an invariant rescaling of this problem. To each gamma, you can associate this invariant rescaling, where you just work it out, and you realize that if rho and u are solutions, then rho tilde and u tilde are also solutions. And the way you rescale the time depends on gamma. So really, the value of gamma has a lot to say about the structure of the solutions of the equation. In particular, the pressure and the potential, you can work it out from these how to rescale. OK, the second ingredient that I'm sure you will appreciate, what are the conserved quantities? Well, the mass, this is easy, the integral of the density, and the total energy. Let's just briefly talk about the total energy. This bit here is the kinetic energy. This object here, remember, rho is Laplace phi. So if you integrate by parts, this becomes negative gradient phi squared. So this is just the gravitational energy, which gives a negative contribution. And this thing here, rho to the power of gamma, is some sort of thermal energy of the star. OK, and the question that is quite natural in this context is, what is the behavior under the rescaling? And then you discover that the mass remains in variant exactly when gamma equals 4 thirds, and energy remains in variant exactly when gamma equals 6 fifths. And these are the two numbers that are popped up here. So this is to suggest that this numerology has a special meaning. All of the results for the remainder of the article of this presentation will pertain to the mass critical case gamma equals 4 thirds. I said article because the actual article is supposed to appear tomorrow on the preprint server on the archive, and there are several spots where we said and erased all the results for the remainder of this article pertain to this case. Anyway, but OK, this is the key point to remember. So gamma equal 4 thirds is the mass critical case. I'll tell you why physicists like this particular value of gamma. You see, it's a rescaling that keeps the mass preserved. So it's a very appealing to think of gravitational collapse or expansion, and some process that sort of cascades through these scales, lambda, but it preserves the mass. It's very natural. So the beautiful thing about this is that, in fact, there is a result of Goldreich and Weber, who were physicists in 1980, where they found explicit examples of collapsing and expanding solutions of examples of stars in this mass critical case. And now let me do some computation. So if gamma equals 4 thirds, and if my self-similar rescaling is this, so you can work out what this coefficient is here. So 2 minus 4 thirds is 2 thirds. 1 over that is 3 halves. So this becomes p over lambda to the power 3 halves. So to have a self-similar collapse or a self-similar rescaling would correspond to having a collapsing solution whose radius scales like t minus t to the power 2 thirds, or an expanding solutions that expands at the same rate. This would be the corresponding coefficient if you wanted to call this a self-similar collapse. However, if you follow through the paper of Goldreich and Weber, already implicitly there and then explicitly in the works of Makino and Lin, you will find a second family of expanding and collapsing solutions that expand and collapse at the linear rate. And this was part of the mystery for us to understand, first of all, the reasons why this is so. Yes? Which one corresponds to the explicit solution also? This one. This is not explicit. This is just, you see, I'm putting this sign here. So this is informal. In a few slides, I will give you the exact formula. There is a correction term. For this guy, there is no correction term. This is perfect. This is corrected. By a log? No, no, no. It's just something linear. You will see. You will see. No logs. Well, we'll come to that at the end of the talk as well. So there are two families of such. There are two families. No, no, no, no. When I say self-similar, I mean this. But there are two families of collapsing and expanding solutions. For whatever reasons, in these works, you will never read this off. They will never talk about the rates. They sort of analyze this and they say there exist these solutions. But if you really look at the rates, this is what will happen. I will give a more precise statement later on. So this is an informal statement. Informal statement. So don't ask me for any norms and so on at this moment. This will also come later. The basic result is that these self-similar expanding profiles, so I'm not talking about collapse, the ones that behave like 2 to the power 2 thirds, are co-dimension 1 stable. And the linearly expanding guys will be just stable. In this case, we can actually talk about asymptotic stability in this co-dimension 1 set of perturbations, whereas here, we do not characterize the asymptotic attractors. There are just some objects that do not necessarily belong to some sort of nearby linearly expanding profile. This is the gist of the results. I will get into the technical statement slightly later. In particular, these are, to my knowledge at least, first non-trivial examples of global solutions for this free boundary problem. And what is very important for us, we can actually say that the support of the star, of the nearby perturbed star, actually grows at approximately the same rate as the underlying background solution. So now let me take you back to what ideas go into the well-posed in this theory, and this will bring us then to the proofs of these results. As I mentioned, the fundamental idea in this work is to use the Lagrangian coordinates. So you've seen them, eta is simply the flow of particle trajectories. And because I will now make an assumption of spherical symmetry, it is actually not fundamental to these results. But I want to make it because it will display, in a very concise way, the fundamental structure that we need to worry about. OK, so assume everything is spherical symmetric. Now maybe it's slightly pompous to say structural miracle, but it is a beautiful structure of these equations. If you use Lagrangian coordinates in spherical symmetry, you discover that chi, which is now my Lagrangian map, satisfies a second order equation, chi tt plus fw of chi equals 0, where this fw is some nonlinear operator, which is of second order. And this is an effective quasi-linear wave equation on a compact domain. Now there's this thing here. I don't want to display fw because it's not beautiful. But I will tell you later how it linearizes. And you will be convinced that it is a second order. It's a second order self-adjoint operator in a certain space. Is this an artifact of the fact that this would be irrotational? Correct, absolutely. So now that you raised this question, so if it were not spherical symmetric or irrotational, if it were pure, if it were the full system, the flow map eta would satisfy an effective wave map type equation. The divergence of the flow map, which roughly corresponds, which captures the irrotational part, would satisfy any equation like this. So divergence of eta t of the velocity vector field. Whereas the curl would satisfy because, OK, so let me explain this, because remember in the Lagrangian coordinates rho dt u, where u is now pullback with respect to the t is my Lagrangian time and u is my Lagrangian velocity, plus the pullback of the pressure would be something. And your curl of this guy vanishes. So this would give you, if you wish, a transport equation for curl and the sort of estimates of curl of u for free. So the idea is that when you deal with the full problem, you estimate the divergence in the curl in two separate ways, OK? So you have a, but you can't decouple it. Absolutely, yeah. So this is in some sense at the heart of the approach of Couton, Scholar, Jeng, and Nussbaum. The framework, the functional framing is slightly different, but this is the main idea. OK, so there is a wave equation. That's my point.  So it's a wave equation on a compact domain. So let me now re-derive, really, those solutions of Goldreich and Weber and other people. I will call them homogeneous solutions because I will make the ansatz that chi is just lambda of t. It doesn't depend on r. Now this will tell you this ansatz will reduce the previous equation to this thing here, lambda squared, lambda double dot, plus something which doesn't depend on t equals 0. So you can separate variables, and if you do that, you discover that there is an ODE, satisfied by lambda. And there is another ODE in r, this time satisfied by w. You are interested in finding compactly supported solutions because that's what qualifies as a star with a vacuum interface. And you can do that. There is a magical value of delta star such that for any delta bigger than delta star, you can find compactly supported solutions to this equation. And this is just an ODE. You can really, for a range of values of delta and initial conditions, you can really classify and see what are the rates of collapse or expansion and so on. So I will do that for you in a second. As I said, you can solve this ODE, prescribe the initial conditions. There are three parameters in the problem, the delta that I mentioned, lambda 1, initial velocity, and lambda 0, the initial radius of the star. There is a conserved energy associated with this ODE. It's trivial. It's this quantity here, lambda dot squared plus 2 delta over lambda. In the Eulerian description, so you can always go back and recover the original solution in the Eulerian description, the density and the velocity will look like this. So the key point, the density is like lambda to the minus 3. So if lambda is shrinking, density is blowing up. If lambda is expanding, the density, the fluid wants to tear itself apart. It's decaying. And remember, as I mentioned, self-similarity would correspond to these numbers here, 2 thirds. So it turns out that self-similar solutions exist, but only when the energy is 0. If energy is not 0, they're not going to obey the 2 third law. And instead of solving the ODE, I also could not resist not putting this on the slides. This is a kind of a bifurcation diagram that explains what's going on. So this is a delta axis. This is the lambda 1 axis. And I'm fixing initial radius to be exactly 1. So what you see here, the light blue region, this is the range of parameters that will give you always a linearly expanding profile. So a profile that behaves asymptotically like some constant times t. These guys here, the thick blue line, this is exactly the self-similar expansion. And there is an explicit formula that I will give you for them on the next slide. But you have to insist that the energy is 0. So they're obviously unstable, because if you can go either way from the surface, the question is if you look at perturbations of energy 0, are they going to remain stable? And then you have the collapsing. So as you can see, most of the collapsing solutions in this range here exhibit self-similar collapse. And then there is this lower dimensional line where the collapse happens at a linear rate. So this is the structure of this space in this two-dimensional parameter space. And the basic results are this light blue region is stable, and this thick blue region is a co-dimension 1 stable. So for this line, you have an explicit formula, which is just a statement that E equals 0. Do you have a question? And below also. You mean here? Yes. Well, I can tell you a very weak statement. I can prove that these are non-linearly stable in the self-similar frame. But this is not a good statement. You really need to. I don't know whether. So for instance, I don't know if you perturb an element here, whether it will collapse. OK, so this is an open problem. I will state it at the end as well. Is a linear rate of expansion the physical one, the one with astrophysics, not interesting? So Goldreich and Weber completely discarded this for whatever reason. And they were interested in collapse and expansion at self-similar rate. If you look at other physics literature, they actually mostly don't want to deal with the free boundary. They are aware of the difficulties that come with that. So in their literature, they think of Goldreich and Weber as the only example where you have a sharp boundary. In most cases, they are worried about constructing some type of solution that exhibits either collapse or supernova or the expansion, where you have infinite support but some sort of decay at infinity. This would be, for them, good enough. And this also leads to good problems because most of it is also not rigorous. So part two, how to study the stability of the expanding guys. This is the formula that Frank, I believe, asked me about. This is the explicit formula for self-similar expansion, whereas you don't have explicit formulas for the linear expansion. They are sort of approximate. But they are to the leading order just this. So I will, from now on, focus only on the expanding profiles and perhaps pompously call it supernova, but let's just say expansion. And I want you to recall that they satisfy a wave equation of this type. And FW is some second-order operator. Now, how to study this object? Well, you adapt your unknown to what you believe to be stable by dividing by it. Basically, you define psi to be e divided by lambda bar. And then you expect psi to be a small perturbation of 1, which happens to be a steady state of this equation, in this case. So if you make this ansatz and plug it back in here, you get this equation, second-order PDE. Now, how to deal with this? Well, self-similar expansion really behaves well with respect to the scaling and variances of the problem. So the natural thing to do is to rescale time with respect to the corresponding self-similar rescaling. As you will see, the linear expansion doesn't sort of, the equations don't digest it very well. So it will cause certain difficulties that will not be present in the self-similar case. OK, so let's move on to the disk. Are there any questions about this? So is it absolutely clear what's going on? So let me tell you the basic structure that will allow you to understand the stability. So let's now focus purely on the self-similar profiles. So remember, let me write it down. Self-similar profile is something that expands like t to the 2 thirds. So this is the radius of the star. OK, xc is a steady state of the equation on this slide. So I will write xc as 1 plus a perturbation. xc equal 1 is a steady state. You do that. You demand that the initial energy is 0, that the energy of perturbation is 0, energy is conserved. So this is a conserved quantity. And of course, the key idea, as I mentioned, is to pass to the self-similar time. If you do that, so this roughly means that the new time behaves like log t, because lambda expands like 2 thirds. If you do that, you discover that this coefficient here is constant. I call it b to honor some of the notation that's popular in the literature. This quantity is a constant which is strictly less than 0. What about the nonlinearity? So you expand the nonlinearity around 1. And you discover, and this is now the point that will perhaps clarify one of the questions, you discover that in the linearization of this operator is precisely this second order elliptic operator here. But now I want you to stare at it for a second. Notice that when r is 0, this term vanishes. And when r is 1, remember w delta is the density profile, is a density profile, which roughly looks like this. And it also vanishes at r equal 1. OK, so in this sense, this operator is degenerate, because it has these vanishing weights, if you wish, at 0 and at 1. This is not surprising. This is the type of difficulty you will see in the study of just pure Euler. But it is suggestive. It tells you what sort of functional framework to use to, in fact, deal with this problem. It tells you that you need to use some type of weighted spaces. Now the cool thing about this, if you pass to the self-similar time and you express, you now try to find the equation for phi, this is what you discover.  OK, and you immediately see, realize what should be the mechanism for stability. It is precisely this blue term here, which acts as an effective damping. This is a consequence of the fact that we are expanding our profile around an expanding solution. If I did it around a collapsing solution, the sign here would change, and this would be sort of the opposite of damping. So it's a wave equation, quasi-linear wave equation, on a compact domain, but with a disdamping term. So the correct statement is that there is a high-order energy. If it is initially sufficiently small, and if the physical energy initially is exactly 0, then there is a global unique solution to the above equation, and in fact it decays exponentially fast in the s variable. In the t variable, this translates into some sort of algebraic decay. How? The high-order energy is bound to the cation theory, I guess, right? Precisely. This is what you need. This thing, it's you, the derivative, right? Thank you for the question, because I now, in next slide, I will write down the energy. The L delta is a self-adjoint operator with respect to this weight. Remember, this vanishes at 1. This vanishes at 0. Spectral gap, OK, 1 is in the null space of this operator. So if you're orthogonal to the null space, you're good. You have a coercivity of this operator. This per se is obvious. What I do want to pay attention to, this first term is just the L2 norm in the corresponding weighted space. If you want to control the first derivative, you have to raise the level of degeneracy in the weight W delta. So the derivative gets first? Correct. Correct. So this is a critical observation? Not in one, each other. No, it's only in this direction. Absolutely. Excuse me, the B you mentioned is a constant? It's a constant. It's a constant. B is a constant due to the because of the self-similar structure. Because I'm expanding around self-similar solutions. So it really captures that. This is the correct information about the self-similarity that you need to address this. Term is obviously the good guy. This term here, this is the bad guy. This will give you exactly one unstable mode. But this is not surprising. I told you that it has to be unstable. Remember, the self-similar expansion was a co-dimension one phenomenon. So clearly this has to be the case. But this unstable mode, which is caused by this negative term, is remember, we are starting with zero energy data. And so the unstable direction you can check this is transversal to this zero energy surface. So you can really control the instability, which translates in this case in controlling the inner product of phi and the generator of the null space. So this allows you to carry on the estimates. So think of this as the relationship that you get by linearizing energy equals zero. This is the quantity that you will control just by doing some simple energy identity. And this is the quantity that you want to control to get a spectral gap. Now answer to Pierre's question. Indeed, the energy is motivated in this case quite concretely in the second case more philosophically in the case of linear expansion by the works of these people. And now we come to your observation. Indeed, I don't want you to gain any intuition from this. All I want you to see is that each time I take a further normal derivative, I have to raise some index in my weight. And this is how this works. These delta k spaces are basically mean you add a k-fold power of w delta to your weight. And this is the only way you can do it. So more spatial derivatives implies more degenerate weights. And now you will trust me if I tell you how the energy method works. You try to prove something like this. This term here kind of comes from that damping term that I mentioned. Of course, you need to control the energy by this dissipative term here. You can do that because of the spectral gap property. It's a technical thing, but you can do it. And this, in principle, gives you the exponential decay and completes the proof. Perhaps I want to say, going back one slide, this is a mixed space time norm. In reality, you do the estimates only with the time vector field because it commutes with the equation entirely. And then you use elliptic estimate to control the higher space derivatives with the time derivatives. So this is what I just said. I used the time derivatives to build an energy, elliptic estimates. Now what are the technical tools that go into closing the estimates? Remember, there is a non-linear right-hand side, which actually looks quite dirty when you write it down. So the tools are hardy inequalities, which exactly allow you to deal with those weights that degenerate at 0 and 1 and embeddings between weighted solvable spaces. If you take enough derivatives, you can control lower norm by the L infinity norms of lower order terms by the energy. So I will not go into any of this. This is technical. I have eight minutes left. So let me then say just a few words about this second problem here, and maybe try to hint at least at why is it more difficult in the first one. The difficulty appears not to be conceptual, but more technical. But it is there, and it took us some time to understand. So the reason why this problem is harder conceptually is because you cannot, it doesn't honor the self-similar structure of the problem. It doesn't honor the scaling invariances. The solutions are there, but how do you prove that they are stable? So it will necessitate the use of a, so first of all, the fact that it is not honoring the invariance will disallow us from using time vector fields. As you will see, they will not commute well with the problem. Because now when I write down the equation for my perturbation phi, it will have time-dependent coefficients. So we have to come up with a correct spatial derivative that will capture the degeneracies at the vacuum. And this is precisely this vector field here, which just happens to be a five-dimensional Laplacian. But this is a coincidence. OK. So what we do, we rescale time so that tau, the new time, becomes log t again. So this means dividing by this linearly expanding profile. Let me write it down here. So lambda tilde of t, think of it as something that behaves like a constant times t. And if you write down the equation for the perturbation, now you discover that you have these tau-dependent factors in front. And lambda tilde is growing exponentially in this new variable. OK, so you have, let's say, e to the tau plus e to the tau plus this stuff equals right inside. This is still causing a certain damping effect. It is well known that the expansion produces a sort of a stabilizing effect. It's reminiscent of works, for instance, in the general relativity, when you have cosmological constant, for instance. And this is the statement. There is a high-order energy, e tilde, that will allow you that will give you global existence for small data. However, you cannot classify the asymptotic attractor. So all you can say is that the tau derivative of the solution decays, but you cannot say that in the limit, you belong to a nearby member of the linearly expanding family. This energy is constructed by using powers of this elliptic operator that I just wrote a second ago. And so to be honest, I can talk about it. I have a few more slides at the end, but let me not go into the technical aspects of this proof. Let me just say that this operator is designed to capture very precisely. So in other words, you cannot just blindly apply dr derivative or dx. If you do that, you will create singularities at zero and they are hard to handle. So you have to do it with a particular combination. And this operator S is a reflection of the structure of that weighted-generate operator L delta. And let's leave it at that. So I will not go into the technical details. Yeah, that's a good question. So it's just something that sits nearby, but it is not one of the proof. Yeah, you can prove that the time derivative of that is decaying. So that gives you convergence to something. Yeah, and it's nearby. It's actually reminiscent of a result I heard recently in a talk by Falker. Falker Schluhe. And sort of in these expanding space times. Who's here, I believe. Yeah, very good. Let me spend the remaining three minutes discussing, perhaps, questions that I find very interesting that are open. So here's this kind of a silly lemma.  Do there exist collapsing or expanding compactly supported stars? So this is the key for other values of gamma. They should exist in the supercritical range, which is between 6 fifths and 4 thirds. This is an open question. What you can show, however, is that self-similarity and a spherical symmetry cannot coexist for this type of collapse. Gamma being 4 thirds is absolutely critical for the two to be able to combine the two. And this is a very silly thing to show. It's just some conservation law type calculation. So the question is, can you construct axisymmetric self-similar collapse in this mass supercritical range? I think this is a very interesting problem. And now we come to the question that I believe Sergio asked. Are the Lane-Emden stars that I mentioned before in the subcritical range, where they are shown to be conditionally stable, are they, in fact, stable? So a possible mechanism, and you will see this discussed sometimes in the physics literature, is some type of shock formation. This is not known, and this is really, really important. So let me remind you of this picture. The big, the pink elephant in the room is, of course, this region here, the collapsing region. So as I said, it is not difficult to show that it's nonlinearly unstable. The moment you have a correct well-posed in this theory and you have a growing mode, it's very easy. So again, this should be a lemma. But it's really unsatisfactory, because it doesn't tell you anything about the real instability. In particular, it could it be that the nearby guys decay, collapse at perhaps a corrected rate? Do they collapse at all? So this is just in spherical symmetry. You don't have to go outside spherical symmetry to ask these questions. And finally, I would like to point out this viewpoint that I believe studying this family of problems is really the right thing, because you really want to have a robust understanding of how equations of state affect the collapse. This is partly motivated by general relativity. If you think of the most famous example of a collapsing star in general relativity, it's the Oppenheimer-Sneider solution, which corresponds to the case where you don't have any pressure. So by a result of Christodulo from 83, I believe, you can show that these are generically unstable solutions. If you go outside the homogeneity class, you generically form naked singularities. So you want to, well, you can devise the various conclusions out of that. One conclusion that I like to draw out of that is that studying non-trivial equations of state is an important problem. And this is just the Newtonian version of the problem. But realistic equations of state are, this is just a final comment, is in principle quite a hard problem, even for physicists. So this is my favorite relativistic astrophysics book by Zeldovich and Noviko, where I sort of learned this stuff about. And they discuss one of the sort of high-density regimes for the equations of state. And here's what Zeldovich says. So it reminds him of Averchenko's parody. And it is precisely like this. So if you look at the astrophysics literature, there's a huge disparity between what people think is the right equation of state. Anyway, yeah, so think of it as a teaser trailer for this wonderful textbook because it's a very serious book. But as I browse through it every now and then, I find a gem of this type. Thank you. Questions or comments? In case we need the answer. So it seems the non-year-expanding rate is unstable. Does it mean that there's a type of solutions which will converge to the stable one, which is a linear one? OK, so if you generically perturb a self-similar expanding rate, if you generically perturb it with perturbations that have positive energy, then they will converge to the linear guy. So it's generically unstable. And these guys are stable really only in the co-dimension one sense. The linear is the stable. The linear takes over, not the self-similar. Yeah, yeah, exactly. So do the faces know this part? It should be of great interest to them, I guess. Well, if you're really. The supermodel should be linear. Yeah, exactly. So if you read Goldreich and Weber, which was one of the starting points for this project, for some reason they really say we will not be interested in these linear rates. They drop a constant of integration in a second order ODE, which allows them to go down one level and get an explicit solution, which is this. And they literally say, ah, we kind of don't want to look at that. So but this is the finding. I mean, part of the mess was also to understand the exact structure of this parameter space. Yes? The non-linear unstable result covers the whole regime of gamma written there? Actually, it actually goes up to here. I did not tell you what happens at 6-fifths. At 6-fifths you still get a steady state, but it is supported all the way up to it's infinite. It has an infinite support. And it turns out to be, so in this case, the steady state is exactly the steady state of the critical Quintic wave equation. So it's one of these guys. I don't know, there's some constant here. And then I believe one over two, something like that. So this is the rate of decay of this guy. So it is the most complicated case before the last week? In this work, well, it's quite different, because here you are working on a compactly supported domain. So you really have to rely on the well-posed theory for vacuum interface problems. Here it's different. Here the weights are at infinity. So it has a flavor of it, but it's quite different. I'm not sure how to. So this result is also due to Jang. So this equality case, all of this is due to Jang. Sorry, in this range, I have to be careful. This is the instability range. What this result shows in a way is that the fourth-third case, which is linearly stable, is, in fact, widely unstable. So the result is radial from the stability? This is entirely radial. So the paper that will show up is entirely radial, but it holds also for non-radial perturbations. So I sort of alluded earlier to that. There's a reason why the radial case is simpler, not just for obvious reasons, but also because it reduces the whole problem to just a quasi-linear wave equation. When you work with a non-radial case, so the fluid is not irrotational anymore in general, so you have to account for the vorticity, which does not have this wave-like structure, but it satisfies a certain transport equation that allows you to control it. Well, what makes the vorticity better than in the standard order equation, where you know that you cannot control the vorticity? Is it because of the? Well, it's a local in-time result. I mean, but. No, this is drawback. Yeah, yeah, so all I'm saying, I'm not sure if I understand your question, but all I'm saying is that. Right, I'm talking about the stability, the long time. Ah, for the long time. For the long time result, you don't have a long time. Oh, if you haven't produced it yet, no, no. So for the irrotational case, it will be straightforward. And for the vorticity in the long-time result, I don't want to make the claim yet. I don't think. Because I don't see why it should be any better than. Than in the regular case, yeah, maybe. No, no, you have this damping effect, so it will be better. It will be better. But will affect the vorticity? No, no, you have, for instance, for Friedman, LeMette, Robertson, Walker space times, you have a stability without irrotational assumption. So when you have a cosmological constant, for instance, and you perturb away from that, you see this, you can control the full flow. Of course, this is not a free boundary problem, because this free boundary problem in GR is more. Okay, but the mechanism is not the mechanism.  All right, so thank you. So we start again at 11.30. Thank you. Thank you.


This paper proposes a novel approach to understanding the relationship between historic agriculture and soil erosion. By integrating landscape characterization data from archaeological sources with GIS-based models, it demonstrates how different types of historical farming activities can reduce soil erosion risk in response to current environmental conditions. This methodology has potential applications in landscape planning and management, as well as providing insights into the long-term effects of historic agricultural practices on soil health. This article was authored by Filippo Brandolini, Tim Seeker-Noud, or you Shrivostava, and others.


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Hello everyone, welcome to another episode of SugarMD. Today we are going to talk about glucose management while fasting. Especially now the Ramadan is kicking. There are a lot of Muslim audience we have that are watching and they have questions. What am I gonna do when I am fasting? And when they fast, believe me, they fast. They fast from dawn to dusk. So it can be up to 18 hours of no eating and no drinking, not even water. Not even water. Yeah, you heard me right. So I'm telling you, it's tough, tough, tough. Especially if you're not used to it, you're diabetic. That may be a little tough situation unless you know what you're doing. And today what I'm gonna talk about is not a medical advice, but the general idea to you to make sure you do not make a mistake on your own and always consult with your doctor before you attempt anything. So let's get started. Okay guys, so you want to go into a fasting regimen to lose weight or religiously during Ramadan, whatever it may be for you right now. When you're fasting, a lot of things happen in your body. Now number one, your body goes into stress reaction. What is stress reaction? So your body makes catecholamines, adrenaline, noradrenaline, stress hormones like cortisol. So all these things actually initially can give you a little boost of glucose, right? But then after a prolonged fasting, after eight hours or so, your body starts running out of glucose. Now when that happens, you have to burn fat. You have to burn fatty acids to turn them into ketones so that you can use these ketones to spare the glucose or to make a little glucose at least keep going to your brain because you have to have a certain amount of glucose in your body. Even when you're on a keto diet or when you're fasting, you have to maintain your blood sugars at a certain level. So you are fasting. After a while, you're in a keto mode. The problem is for a lot of you is the medications. So if you are a diabetic who is not taking any medication, trying to do anything with the lifestyle changes and is working for you, in this case, it's really not a big deal. When you're fasting, you will do well. Your blood sugars will improve.  If you have not practiced fasting and if you're jumping to an 18 hour fast, that can be really striking. So having said that, for people who are on, say, metformin, that may not be a big, huge deal.  I'm not trying to say it's good or bad for you in this video. We have videos about metformin. You can check that out. But if you're on metformin or agents that do not necessarily cause low blood sugar, even when you don't eat. So actually, there are some rare cases of metformin causing low blood sugar, but at large, it does not necessarily do that to majority of the people. So let's say you're on metformin or you are on a medication like Genuvia or Tragenta, or you're on a GLP-1 agonist, say, for example, Osempic, Trilocid, Victosa, Bayata, Biterian, all these things. They do not necessarily cause low blood sugar. I know they can, but it's not really common. So if you're on these drugs, you may still be safe to fast for a long time without causing a low blood sugar. And why I keep talking about a low blood sugar? Because it's a big deal. If you have a low blood sugar, you're breaking your fast, you may end up in the hospital, not a fun situation. It doesn't feel good, believe me. I have tried on myself before. It does not feel good. And most of you probably had experience at least once if you're on medication. So if you are on a medication group such as Sulfonylurea, and these are like Glypozide, Glymopride, Glyboride, the problem with those agents, they raise your blood insulin level and it stays up. And if you're eating only once or twice a day, and if you're not having anything in between, or like, say, you know, you're having one meal at, you know, 7, 38 o'clock in the evening, another one and very early in the morning, then you are looking for 16, 18 hours of fasting, you know, once the sugar carbohydrates, whatever you ate runs out of your system, then you are at the mercy of these drugs that are still in your system, and they will not really show mercy to you.  And if it is too much of a medicine, you can definitely go down. So I definitely do not recommend using Sulfonylurea for that purpose. Now, what is an alternative to Sulfonylureas? Now, of course, the GLP1 drugs we talked about, like here, are alternatives, or the DPP4 inhibitors like the genivia and trigenta, like here. So these are good ones to try when you're fasting, especially during Ramadan month. But on the other hand, if you cannot get your hands on these because they're expensive, you may not like the side effects or whatever the reason may be. There is a cheap alternative. They're called megalithonides, but you can call it short acting Sulfonylureas. So they raise your insulin level, but not too long at a time. So you have to take them before every meal. So my favorite is Repagilionide. There's another one called Natagilionide. But the Repagilionide, the brand name is Prondin, and the Natagilionide is Starlix. When you take these drugs before the meal, they increase your insulin a little and then they fade away, they disappear, they're eliminated, provided that you do not have major kidney problems. There's another factor for a lot of diabetics. If you have kidney problems, things tend to stay in your system longer. So you have to be extra careful with that. Of course, when we are talking about all these drugs and so forth, the most important thing for you to do, especially the first few days of fasting, is checking your blood sugar so many times to understand what's going on. Because if you don't really check your blood sugar, it may hit you. It may go from like 120 blood sugar down to 50, like so quick that you may not really realize what's going on. When I say so quick, it may go within a couple of hours, say three, four hours. But if you are really checking your blood sugar every two to three hours, at least you will see a trend. If your blood sugar is dropping from like 180, now 150, 120, 190, you're like, wait a minute, maybe you should cut that fast.  So if you are really not able to do that, just don't do it.  Important fact, this is very important. Now this DEXCOM and Freestyle Liberate, those glucose monitoring systems, they are not perfect by any means, but they are great to understand your trends.   These are all estimations. These are devices are created around the estimations, but they are good at understanding the trend analysis. And then all alerting you like Liberate 2 system, for example, alerts you to DEXCOM, alerts you. So if your insurance covers or if you have the ability to pay self-pay, I think there are very good investments for any diabetic.  So, but then, you know, let me talk about the repegulonide, or the repegulonide as other solutions, compared to cell phone ureas, if you have to do some sort of cell phone urea because they're cheap, et cetera, but if you're an insulin, that's a whole different ball game. Because if you're an insulin, the question is, are you really taking insulin correctly? How controlled your diabetes is? We don't really know all of this. So you may be taking insulin like 300 units, and your blood sugar may still be 300. That doesn't mean that you're in control. You know, you're some somebody else may be taking only 20 units of insulin, and your blood sugars may be perfect. So there is a difference there.  I wouldn't really experiment myself because a lot of people go with either all or nothing type of thing. And like they will say, oh, I'm fasting, I don't need insulin. If you do that, if you say, especially if you're type one diabetic, we always kind of try to focus on the type two. But if you're type one diabetes, you have to have long acting insulin in your system. Now, when you're fasting, you may need less long acting insulin, and you may need very little to none short acting insulin just maybe for corrections and so forth. But it's going to be a very delicate balance if you have been diabetic, like type one diabetic for 15, 20 years, because you really don't have the defense mechanism that the type two diabetics have. Like your body doesn't really make enough gulcogon, you're not really making enough epinephrine and so forth. Type one diabetes is very tricky. I would not recommend playing with your insulin at all, especially if you're not controlled when you're fasting. You have to consult with an endocrinologist, make sure you know what's going on, and stay in touch with them. Type two diabetes a little bit easier, but when you are on multiple shots of insulin a day, I'll give you an example. I mean, if you are controlled, let's say you're taking 30 units of lontas and 10 units of hemolog, and your blood sugars are really good, you don't have any problem with it. If you were my patient, I would probably tell you, okay, well, since you're not eating, when you don't eat, you don't really take the hemolog or novelog. When you eat, you take the insulin, you take the short acting insulin. Of course, when you're fasting, at some point, you eat, right? So at that time, if you're going to have a bigger meal than usual, I may say, hey, okay, so you normally take 10 units per meal, but if you're going to have a bigger meal, and you're going to have only one meal a day, I would say take 15 units, still have a reasonable meal, not too much, not too little.  Now, of course, if the long acting insulin, you still need long acting insulin, but if your overall calories are going down, more than likely, you will need less long acting insulin, unless you stop exercising, you know, that also another contributing factor. If somebody, you know, stops exercising when they were normally exercising every day, but then fasting, that kind of balance it out each other, right? So you don't have to really change your insulin too much for the long acting. But if you are not really a, you know, person who exercises and you decide to take on this whole Ramadan month, and you want to fast every day, well, in that case, you may need less, you're more than likely will need less. If you're my patient, I will say, you know, if you're taking long to 30 units, go down to 20 units, and then monitor your blood sugar very closely. But again, that's in a perfect world and a perfect patient, you know, the problem with diabetes, it is such, it's a disease that is so individualized, like when God forbid, when you have a cancer, they have protocols. Well, if you have stage three disease, this is the chemotherapy you get, this is the radiotherapy you get, you know, this is very kind of standard and whatever. Sometimes they put you on experimental treatments, but diabetes is not like that. Everybody is so unique. I tell my patients, sometimes they come into the door and they say, I want to be on this medication. I'm like, no, you're not. No, you know, I'm trying to be on this medication. You know, and I explained, I'm like, the person just left before you got on that medication, but you're not going on that medication because of this. You know, we have so many medications and we have so many comorbidities, so many conditions that I have to choose the right, the best medication for you. So just because you want this medication doesn't mean that you're going on that medication, right? So otherwise, everybody can read about medications and can be their own doctor. But diabetes is a very unique disease where everybody's so different. Some people will control their diabetes with just, well, just eating, stop eating carbs or eating less carbs or just walking a mile or two every day. Boom, their diabetes is gone. There's some people, no matter what they do, their blood sugars will be in 200, 300. Or some patients may need five medications to control their diabetes. Some patients may need only one single medication. So it's a disease where you really have to understand what's going on with your doctor, constantly with your doctor. But as I said, if you're on pills, not insulin, it's a little bit easier. Like we discussed before, I would say, you know, you can make your own mind what to do, but always confirm with your physician. And they'll appreciate that you're thinking about it, definitely. You can tell them that I heard from a young, handsome YouTube doctor. That's okay. They want mine. Guys, we'll see you next time. And guess what? I want you to watch this video now.


Welcome to our InterGEO TV studio here in Essen. Looking back on InterGEO 2022 means many important topics, many main issues for this trade fair in our studio. I just welcome now Christiane Zalbach. She is the representative of the organizer of the TVW and she's also general manager of the TVW GmbH. Christiane, how do you feel about these topics, about the InterGEO 2022 in Essen? I don't want to pick up any individual topics, but the InterGEO conference has set the right topics this year. The innovation in the geospatial industry continues and I am convinced that this year's InterGEO will accelerate this development. Geo-information will be in the future. The following characters will be autonomous, they will be smart and they will be connected and also geo-information will be focused around the world and very strongly interdisciplined. And very, very important, they have always the people in the center. In summary, this means that they are simply out of our lives to think away, because important decisions will not come out in the future without reliable modeling. And for the TVW I am particularly proud of the TVW as a event, it has succeeded. A lot of young people are looking forward to bringing this year's InterGEO here to Essen. For example, here were the pitches of the students, but also the InterGEO School Day, which is called, where over 70 students have visited the InterGEO. Shortly after, it has felt like a real InterGEO, the InterGEO is back, the conference and also the media visitors are excited and finally the demand for the Geo-Community to be served by direct communication and immediate exchange. The InterGEO here in Essen has succeeded. Christiane Saalbach, InterGEO 2022 means a big, big success. Now we're looking forward to Berlin 2023. I think you feel the same. Thank you.


Coming up at the end of September, I'm gonna be the emcee for the east coast rep rap festival And I'm actually making a project out of the out-of-marbles that is actually not this size But world record breaking my name is Jim and this is the edge of tech That's right coming up on September 30th and October 1st is the east coast rep rap festival or earth They invited me out to be the emcee of the event. They're giving me a microphone.  We're gonna be doing death racers We're gonna be doing dragsters all of the cool stuff, but something bigger is happening This is an out-of-marbles Vortex if you haven't seen the out-of-marbles towers they kind of look like this and then you put Obstacles in here you put a marble in the top and it goes all the way down I first saw these when Ellie of Coco press created hers, and I fell in love The thing is it is actually normally this size, which is very manageable Gotta get it back together there. It's close enough. I thought I wanted to go bigger I reached out to the Guinness Book of World Records. I submitted an idea and they accepted that's right I'm gonna be creating the tallest 3d printed marble run And if everything goes right we are gonna be attempting to create a world record at the East Coast rep rap festival the thing is they had some criteria on the marble size and Their marble size will not fit on the original scaled marbles I probably should have built this so you could actually see it in action now that I think about it So I had to scale it up and as you can see here. It is much much bigger I'm talking. This is my prototype size. This is 180% of the actual size this thing is gonna be enormous And I cannot wait to bring it out to the East Coast rep rap festival Set this thing up and try to create a world record now All of this is gonna be a ton of work and to be fair. I literally just started printing two days ago That's right. I have one month to get everything together Tested and ready to go and then get it out to earth Before the world record is attempted. So let me know in the comments right now Are you gonna be at the East Coast rep rap festival? I need to know because I would love to have you in the crowd while we drop that marble And the other thing is if you haven't been to the East Coast rep rap festival It is in Bel Air, Maryland All you have to do is fly in to Baltimore and it's about a 45 minute drive from there Super easy flight super easy travel and there's hotels all over Bel Air the event itself is amazing And it's held in a college. It is is super cool to be at I've been there a couple times now and if all the normal suspects show up It looks to be another amazing rep rap festival So I got a bunch of boxes of polymaker in front of me and this is four of like 35 kilograms Uh, I don't know how much of this I'm going to use. They sent me this before we knew we had to scale it up And as you can tell this is is quite the scale. I mean, this is so much bigger If I look in the monitor, I can see it, but this is so much bigger I'm going to use so much filament for this project and I could not do it without my friends over at polymaker Because they are amazing. Nick, you are amazing Hopefully we'll be close to you guys at IRF so you guys can see the monstrosity of the tower we're going to build I'm not going to release how tall it's going to be Officially, but I can tell you this it's going to be at least 20 feet tall So take that for what you will I don't know how tall that building is in there I'm just hoping we can reach it and I'm hoping I can reach it to put the marble in Because that's going to be absolutely crazy. Also my friends at 3d gloop Those guys the andrew's Phenomenal people I I reached out they jumped on this and they sent me a couple of bottles of gloop So I can start pre-assembling before I get to IRF It's going to be important that I do that because not only do I have to do this But I have those mc priorities and I get a talk and announce all the cool things at IRF And I just I'm going to have to have time for everything So gloop, thank you guys so much for your gloop here I cannot wait to start using this and get this thing built as soon as I have enough parts to start assembling it The printers are working around the clock. I I'm hoping to have this printed within Two weeks total, which is a crazy amount of time for how big this thing is going to be So cross your fingers and wish me luck I it's it's going to be an adventure no matter what we do and speaking about adventures and projects I just wanted to shout out to pcb way another huge sponsor of the channel Thank you so much and thank you for sponsoring today's video as well If you guys are looking to do any projects, whether it's a 3d printing cnc Lasers i'm talking metal 3d printing if you want pieces reach out to pcb way throw your model in the website It is super easy to use the staff over there is great if you have any questions The shipping is fast and the turnaround is super fast as well PCb way they will hook you up They will get the parts you need to make your project happen And if you need a bunch or just one if you need a custom pcb or maybe just a part machined They can help you out Thank you so much to pcb way for sponsoring this video and also Sponsoring us and the death racers at earth because that's going to be another Story for the books because there's some battles brewing in the background again Thank you so much pcb way for sponsoring the channel sponsoring this video and sponsoring us at earth as well So now that the cat is out of the bag about the world record that we're going to attempt at earth September 30th and october 1st in bel-air, maryland I also have to say thank you so much to fabrico for shooting me over a rat rig 500 kit Um, it is almost finished It is almost finished and then I can start printing out the really big pieces that I need Like these right here, and I really appreciate you hector and fabrico and the crew over there for all the help You're giving me on the rat rig 500 too. I have not filmed any of that I spent a fun build doing it myself and it's it's been a lot of fun So thank you so much So I guess the bottom line is that I hope I see you september 29th and october 1st at the east coast rep rap festival Bel-Air, maryland. I hope I see you there I want you in the crowd when we drop that world record marble from the top of however tall this thing ends up being Uh, I would love it if the camera could pan around and see your face and everybody else out there I will put the link in the description below for earth so you can jump on there Check out all the accommodations and tickets and all that stuff. Let's get this world record set I cannot wait It's been a dream of mine since I was a little kid Reading the actual Guinness a book of world records to get my name in there and this could be it We can do it with all that being said I'm going to be posting updates here and all across the social medias Going forward for the next month I have one month to get this done to pull it all together and make it happen The bamboo lab printers are just whirling I have seven bamboo lab printers printing around the clock to help make this happen So thank you again to all the sponsors polymaker gloop Fabrico pcb way everybody that's going to make this happen And I hope to see you guys there at the east coast rep rap festival coming up in about one month Don't forget to check that one out too. That's a good one right there. This one. Yeah, good one Bye. See you


This is stock number AT171A. We are here at Summit Automotive in Fond du Lac, Wisconsin. You're new and used RAM headquarters. Remember we are the guys with the trucks on 41 today. We are taking a look at this Super Clean 2018 RAM 1500 Crew Cab 6 foot 4 inch short box. This truck has the big horn package on it and it also has the 5.7 liter V8 motor in it aka the Hemi. This one also has the blackout addition on it so you might have noticed the blacked out headlamps bezels, the painted front bumpers, front sonar on there, the painted 20 inch alloy rims. We shoot all of our videos in 1080p so if you have HD capabilities on your computer tablet or smartphone device turn them on right now because it is like you're right here looking at the truck with me. Factory chrome step bars, the 20 inch rims like I said and they have the Goodyear Wrangler SRA 27560 R20 tires on them and they have just about all the tread left. Lots and lots of tread left on those tires. Very low miles on this truck. Brilliant black crystal pearl is the color on here. You get the blacked out front grille with the blacked out RAM emblem as well part of the blackout package. Just a really good looking truck all the way around. Passing the side rim is a nice conditional scuffs or scrapes on that. You get the blacked out Hemi badge, the blacked out RAM 1500 badge. Down this side you can see just how mirror like and reflective that paint is. We take these HD videos because we sell a lot of these trucks out of state and we want to make sure that our out-of-state customers get the most accurate representation possible so that they can have confidence when they make the trip down that they're getting a very high quality vehicle. Once again the factory chrome step bars, back rim, no scuffs, no scrapes on that. Very nice condition. This one does have the plastic wheel well guards which keeps your frame and underbody looking good and the back tires have just as much tread as the front tires. As we come around to the back, rear bumper is absolutely perfect. No dents or dings on that. It does have a full towing package which includes receiver hitch, four pin and seven pin wiring, the dual rear exhaust backup sensors and you can really see that metal flake on the tailgate there. You also get the blacked out RAM emblem, the blacked out 4x4 with the blackout package. This one does have a backup camera. It has a spray and bed liner in it as well and it does have the LED bed lighting and in 2018 you can get those so they have the button back here so you don't have to use the cargo lamp but pretty cool. Tailgate shuts nice and solidly. No denser dings on that and as we go around this side of the truck for the final time, very nice and clean. This back rim is absolutely perfect. No dents, no dings, no scuffs. Very very nice. Inside the big horn package gives you the gray cloth interior. There are no rips or tears on these seats. This truck has never been smoked in. Seats are very nice and clean. Power driver seat. We do have factory all-weather floor mats throughout on this truck. Power windows, power locks, power folding mirrors. Those mirrors are heated and they also have built-in directional signals. Auto headlamps and we will hop inside here. As you can see this truck only has 3,195 miles. You have the 7-inch TFT display. Nice instrument cluster. The heated leather wrap steering wheel. Also the seats are heated as well. Bluetooth audio controls, cruise controls on that side. You get the 8-speed turn dial automatic transmission. Four-wheel drive auto, four-wheel drive lock. This one comes with the factory brake control. You got your heated seats and heated steering wheel button right there. Stability control, dual-climate control. This one has the 8.4 C radio. So this one has the Android Auto and Apple CarPlay capabilities. So you can put your Google Maps or whatever you'd like right on the screen there. And you can also control your heated seats and everything up here. You can also do your backup camera without actually putting the truck in reverse. Dual-climate control, 115 volt, 150 watt plug-in. You have a cell phone holder right there and a lot of nice storage space and two cup holders as well. Dual glove box, passenger seat is very clean as well. No rips or tears on that. Side current and airbags. And that headliner is in excellent condition. Speaking of up here we got the power sliding rear window and the home link system for your garage door security systems and lighting systems. Finally, assistant and SOS buttons, those are always active with the vehicle. You never have to pay a subscription for them. We will take a quick look at the back seats. The back seats are just as clean as the front seats. No rips or tears back here. Power sliding rear window. Latch child safety system. It does have this load floor back here. Which you put that down like so. You can put your tools or muddy boots or if you got pets you can put it down and they have a nice flat place to lay. And they're not going to ruin your seats. Storage underneath both sides. Factory all-weather floor mat back here. And you get two in-floor storage containers. One on this side, one on the passenger side with removable bins for easy cleaning. So you could put drinks, ice, whatever you'd like in there and take it out and clean it real easy. We'll take a quick look under the hood but you can see just how clean that paint is. Really nice truck. Highly recommended from a condition and quality standpoint. Purchased to your brand new. Driven locally and traded in back to us. So very nice truck. Engine bay is very clean runs very smooth. This truck has been fully safety and inspected by our service shop. Has a fresh oil and filter change. All the fluids have been checked and topped off. And this truck is 100% to go. To see more pictures of this truck or one of our other 400 new and used cars, trucks, SUVs, minivans, wranglers, half tons, three quarter tons, one tons. You name it we got to go to our website www.SummitAuto.com. Full pictures and descriptions of every single vehicle on our lot. Videos of every single used vehicle that we have. All at SummitAuto.com if you want to check out just our used inventory. You can go to truckson41.com that's truckson41.com. Go check us out there and if you want to make this super clean 2018 Ram 1500 crew cab six foot four inch short box years. Gives a call right now<PHONE_NUMBER>. Ask for one of our sales associates to make this truck yours today. Once again that number is<PHONE_NUMBER>. Thanks again for checking out the video. If you'd like to check out more you can go to youtube.com slash summit auto. Remember to like subscribe and share on this video and all the videos that you see there. In fact in a second you will see a link to subscribe to our YouTube channel on your left. A link to more half ton truck videos like this one on your right. If you have not been to our website on the bottom link to this vehicle on our website click those check us out and we really look forward to helping with this super clean 2018 Ram 1500 crew cab six foot four inch short box big horn blackout addition. Thanks again.