Hi everyone and welcome back to another episode of our road to rank roulette series My name is Leigh also known as Osiris and we are kicking off with this team that is on the screen right in front of you right now We can activate those bonus buttons come today, but because we've had two Amazing games this week already I really want to stick with the team and the other two games that we've lost were two and two at the minute We're very close and I feel like we could have maybe won those if maybe one or two things It just went our way in those games if you've missed those you'd like to check those out I would definitely recommend checking out the last two days Episodes because the hype is crazy in the game So you've got one hit kills all over the shop not to spoil anything But king and haxer is having a competition between themselves in this team and it's just been incredible so far So we'll give this team one more run out today We'll probably make some changes activate a few bonus buttons tomorrow if you've got a preference on which bonus button You'd like to see activated remember We've got a switch up randomize that legend maker and a patreon button and do let me know down in the comments section And we will make sure to do something with those tomorrow So without further ado, let's get into it today The music is on will select a cool track as always if you do enjoy this sort of content Please remember to drop a like on the video do subscribe to the channel for more pokey content and Leave your comments down below Let me know what your thoughts are on these episodes will kick off with the aether foundation music today Because it's a good old classic in this ultra series and And we'll wait for our next opponent. Hopefully it doesn't take too long, but the team so far I'm just loving it that I I don't know like the restricted pairing could probably be a bit stronger We are very weak to fairy types But we do have a good means of getting rid of fairies pretty quickly with that one hit kill option And we've been doing not too bad so far With it, which has been incredible. So I couldn't have really asked or wished for anything Better with how things have been going on that side of things. I'm super hyped about it It's been great, and I really hope you've been enjoying the team so far this week That is the main thing and remember as well this week We're doing a little trial on the channel where I will be putting up a more serious competitive content on an evening and bringing back our school of hard knocks Series which will be 7 p.m. UK time. So Greenwich Mean 10 plus one every night Monday to Friday this week The team we've got over there is a lot of fun. Hopefully you're enjoying it and do let me know What your thoughts are on having the double content? I'm myself probably gonna drop it next week because I need to just assess what it is like from The content perspective. I still have a full-time job that I do I still need to spend time with looking after the year Tying out the tash and doing other things outside of the content So it might be a little too much at the moment for me But I will definitely let you know how I've been coping with it But remember we will be streaming Thursday this week. We're gonna be doing some battle spot Over on twitch that'll be kicking off 8 p.m. Thursday So do come and hang out on Thursday over on twitch the links down in the description Or just follow me on Twitter. I always tweet out when I go live. That's Osiris VGC over on Twitter So you can follow me there and I'm getting a bit better active wise I do tweet and then to tweet sometimes I have busy weeks at work It means I can't really find the time to to tweet out as much as I would like to But like a lot of the time I do especially when the stuff going on I like to and I try to be more active on there going forward as well Because I'm it's it's difficult. It's difficult and I'll always say it's difficult It's difficult finding time to be really active on everywhere and and trying and stretch yourself And sometimes you do stretch yourself a bit too thin and yeah, that's it. So concentrating on different areas But yes, follow me over onto it basically and I'm just jabbering now Now it's taken a bit longer to find our first opponent and I would have liked to So what we're gonna do is cut this now and come back straight away when we find our first opponent of the episode And we got our first opponent. It is ZG and they are running a team of we'll get into team preview Very cool-looking team consisting of Greninja, Incineroar, Mewtwo, Tapulele, Gothetel and Kyogre So we've got that trap option here with the Gothetel You've got the psychic spam option here with the Probably mega Mewtwo and the scarf tapulele then supporting the Kyogre Which is the other restrictor of the team They've got Incineroar the standard on most VGC teams and then Greninja And it's interesting pick here that offers that dark type support And probably helps out a bit more against threats like Lunala And that this team might struggle against in things like Duskman and the Crosmo with its dark typing there Also benefits from the rain with its Water typing as well Right, what are we gonna do? I think Trick Room is a nice option for us to go down here. Dalg has a decent lead for sure We could go Haxorus again to try and get some one hit Kyogre action I do expect my opponent to lead off with The tabulele they may go down Gothetel Incineroar but I think the psychic spam route is a bit More optimal for my opponent to go down And we will bring Palkia and do we bring Shiftry could be decent Kingler also could be very good I'm gonna bring Kingler actually because if we can get a trick room up Kingler can do some nice work Especially with the rain up on the field. So I'm quite happy with these four. Let's get into it So Here we go against Pikachu trainer. Is this Lu in disguise using a Japanese card?  I know she does tend to I don't know if she watches as much these days, but if you are I'm it's just a perfect trainer for you, isn't it? I'm U2 Incineroar coming out interesting I'm not gonna see the Lele making appearance just yet. That makes a bit of sense really Um, I think we've got trick room and we got guillotine. What are we gonna guillotine now? The Incineroar or the Mewtwo could cause more problem to us probably the Mewtwo. Let's trick room. Let's go guillotine Mewtwo Come on Huxerous Kingler put you to shame in our last episode Let's let's show Kingler who the real one hit KO King is Mewtwo mega evolving Mewtwo Y will probably get faked out into one of our slots It's more likely to be the dialogue or I think to prevent the trick room going up. Um, I think that's where you would expect it But maybe into the Huxerous there we go.  It's just Um, let's bring in Palkia Okay Now what do we do? Oh, do you see? um I think we protect Dialga And we'll go trick room with Palkia because it feel like Aurora Sphere is probably something that you have on Mewtwo Um, and I think the double end to Dialga is more appealing at this point to go Aurora Sphere and Flair Blitz into the Dialga slot than it is into the Palkia I'm sad that yeah, there's the Aurora Sphere. I'm sad that Huxerous has went down without getting it and one hit kill We're gonna see knockoff So much damage. All right, but we do get the trick room up, which is nice. Um Do we double the Mewtwo? Or do we double the Incinerole? Uh, I don't think we double We could double the Mewtwo. It probably protects here though. That's the problem. The Incinerole is going to be probably slower um We could bring in Kingler. The knockoff's a bit annoying really. Um Although Palkia will take another knockoff, I think um Let's cover the Mewtwo. Let's go earth power into Incinerole and go for a spatial round into the Mewtwo I think the Mewtwo will protect here, but I'm just going to cover in case it doesn't Okay, it's switching out Uh Not so ideal. Not so great. Hmm At least we'll get some damage onto Incinerole here Darker's Lariat Huh Palkia takes it. It's not really going to do very much though. Uh, we get the earth power into the Incinerole Um Let's go flash cannon into the Lele. We'll not be able to pick up the knockout. I don't think we're not powerful enough You never know though. Um, and then we'll bring Kingler in on the Palkia slot It's just otherwise.  I think Kingler is strong enough as well to come in on A darkest Lariat from that slot anyway, and then we got some more opportunities to go one hit here style Style points, but the Kyogre's lurking in the back in the problem is now the Palkia Um, yeah, we would have been better. They're going for the double into the Incinerole for sure Uh, the Palkia's it's such a low health now It's gonna struggle to to really come back against like Kyogre and stuff like that There's a darkest Lariat coming in. I'd imagine a moon blast into the Dialga here Um, but we should be able to take that Kingler taking this pretty comfortably Um, yeah, I'm picking up the knockout quite Oh, it's a z move twinkle tackle twinkle tackle Lele What?  Um, but I never really Got around to properly using it Dialga takes it pretty comfortably. We're just out of berry range, which is a little bit annoying for us Um, we'll go for that and then we'll just Crab Hammer the Incinerole We should be able to pick up the knockout onto Incinerole with Crab Hammer I mean, hopefully we underspeed it as well. I don't know if we're gonna be though. They know that's the thing I think we probably Don't underspeed the Incinerole and that's the biggest problem for us here Uh, darkest Lariat is this into Dialga? I don't know, protect him maybe there Yeah, and we're gonna get yeah, we'll get the Incinerole But we will lose Kingler and then we're just down to to Palkia Crab Hammer at least again a knockout here with Kingler Hmm Side shock. Yeah, uh, just too much for us Too much If Dialga's buried activated, it would have been a little bit better. Uh, but Palkia just not in any position now to do anything Uh, the switch into Lele from u2 as well if we hydro pumped I think doubling into the Incinerole that turn like we kind of Touched upon would have been the better way to go Uh, but these psychic teams are always difficult and Less and less people I feel are running scarf Lele now and it's to that assumption that everyone runs scarf Lele Um, it's not really much we can do special rend into the Dialga, but we're gonna get knocked out either way Uh, we just Not got the time not got the resources not got the support to deal with with this combination Uh tap Lele and I think it's highlighting areas in our team that where we are weak, you know We are weak against uh these fairies and we've got very little Um to be able to to Really come back against them. We've got to rely heavily on on Dialga a lot for our trick room Um and uh part steel typing Um, we're just not protecting Palkia well enough. I mean the Incinerole there having the random knockoff as well Taking palkia down way more damage than what we expected it to do. So I mean, it's fine Good game to my opponent. I just think a few things that we need to look at Going forward with the team and I think tomorrow is probably a day where we make some We activate some buttons. So I think we'll finish up today Hopefully we can have a really good finish like we have been doing We're on a bit of a streak at the minute where we're having kind of a mediocre game that could have definitely been winnable and then going into The second game where it's it's a lot better So I'm hoping that trend continues league title defense. Oh Now let's go on the cross my version two the epic the cross my version two music To end with an epic finish to this team because I think this is going to be the last time we actually have this Specific builder team before we activate some buttons. So it'll be nice to Give it a good send off. I guess before we go on to some different stuff Tomorrow and Friday, but the team so far this week been incredible. I've been really enjoying it. I hope you have that's the main thing Um And that'll be interesting I'm just trying to think now if we go for the switch up which pokemon it maybe take out to switch up Um, and if we go for the legend maker, I think I probably Right now Do we get rid of palkia or dalga probably I probably would get rid of Palkia, I think of the two restricteds to replace it with something else um Although I do really like palkia. That's the other thing but it's just not It's defensively bulky as I like it to be so We'll make some decisions though and that'll be in tomorrow's episode Doesn't look like we're gonna find it upon anytime soon. So I'll cut it here We'll come straight back my friends when we find our next opponent of the episode And we got our next opponent of the episode. I'm so sorry. I'm not paying attention We got oh a very similar team to what we just played. So Maybe you can make some different plays and see if we can adjust a little bit better going into this one So our next opponent is running incinero, me too, kaio ga, tapulele, crowbat and the katana. So Hmm trick room again gonna be very good. My opponent really relies heavily on tailwind from Like the crowbat or the katana and then the fast mode from the lele and the me too now I would say the team probably does have Scarf on lele in this team, you know, we've seen less and less players are running it but at the same time I think It is probably something that my opponent is going to be running here. Let's go Do we go double trip? We could go Hmm Just a taunt is definitely a thing um On the crowbat. It's just if we had a way to to nuke it early on would be nice However, we're not going to be weak to The fairy types. I mean, we could go shiftry dalga might be a nice option there Kupaxorus for the late game And I think palke we definitely need as well But salazzle could be good here just for the katana although um If the rains up it makes it very difficult for us to hit it Especially with the overheat. So maybe Or better not better off not bringing it to this match and maybe preserve it for a later episode this week so Um Katana could be tricky The crowbat is very tricky It makes me feel like mental hurt would be better on one of our trick room setters So at least we've got a way around it, especially when we see something like crowbat come out Um, and we are going to see crowbat kaiaga hit the field Now I think If you're crowbat, you definitely taunt dalga here One of the things we could potentially do is bring in parkia Uh, just so we've got a bit more offensive presence in front of the kaiaga Um at the moment, we haven't really got too much and at least with shiftry Um I mean we can worry seed it But i'm gonna fake out I know yeah, I'm gonna bring in parkia I think we can take the water type attacks better from the kaiaga as well Um, and I totally expect the the crowbats to go trick the taunt to prevent the potential trick room and then tail in the next turn So there's a fake out Uh, oh just Really we could have just went for the trick room. Are you kidding? Are you actually kidding? Okay Man that is disappointing, isn't it? That is really disappointing. Um It's got leaf blade And I don't know whether it's worth going for the trick room now I think we'll just attack because I think the taunt will come out from the crop Really really really really no taunt crowbat doesn't have taunt. Why are we thinking about overthinking this so much, aren't we? So taking down to our sash It's a bit of recoil damage And there's a nice beam. Where you going into shift tree? Really wants to get rid of that shift tree Really concerned about it special rend does hit. Let's see what the damage is like on to kaiaga It's decent. So we'll pick up a knockout next turn for sure. Uh, we're going into hiaga And i'm gonna go for the trick room I feel like we get brave birded from I'm gonna go water spout into the Hydro pump into the crobat water spout park here. Can you imagine there's another brave bird? Where are you going this time into park here? going to die Not taunt and another ice beam Okay We do connect on to the crobat And we'll be able to set up a trick room which we could have done from turn one But we're playing around expecting our opponent to have taunt which most croats do Whether that crobat had taunt or not, I do not know but kind of The victim of our own thoughts in this match, but we're sitting pretty nice at the moment me too going to come in So we're not in the worst position at all We can pick up the knockout on to kaiaga Or we can double in to the me too, which I think more inclined to do Here go for the hydro pump and the flash cannon into me to pick up the knockout there if we can The kaiaga can ice beam away all at once, but It's not picking up the knockout on to anything right now So both our dragons sitting in a pretty nice position It's just a me too that could cause us a few issues kaiaga is going to protect so I mean too good to protect. Okay, so I mean It's next turn. I'm definitely just going to go well do I because I really want to get rid of the Why should we just go after the me too? I think we just chase the me too down Their tailwind pit is out We'll chase the me too down again because I think that's the the one thing that can cause us a few issues later in this game Park here sitting so healthy at the minute doesn't really care about the kaiaga We don't need to concentrate down on that just yet. Um And if we get the knockout here with hydro pump, which we actually just miss out So it's good that we doubled in there just to make sure that the me too goes down Uh, we're probably seeing another ice beam from the kaiaga Yeah Ice beam or where we're going this time dialga We activate the berry We get froze. How many times have we been frozen this week twice? I think twice three times. I don't know Uh, dialga does freeze which is really Not ideal and if it's incineral then it's fine if it's tapulele makes it a bit more difficult cartona Uh, we could really do with We could really do with dialga not being frozen right now Uh, the earth power would be extremely useful there again. I'm going to double into the cartona hydro pump and earth power here I don't think the hydro pump will be able to get the cartona in this range Uh, it does just protect them with a kaiaga. I'm gonna actually attack I think here and could have went for that with the spatial rend You need dialga to thaw out. Come on dialga thaw out ice beam again He's not freeze this time. Um Yeah, so we've got what we got. We've got a haxaris in the back Um, I'm hoping that we can thaw out with dialga here hydro pump And earth power the cartona for the victory. Come on Let's do this Please kaiaga protecting. Okay Still not this last turn It's a hydro pump. What's the damage going to be like on this? Wow, take him down to the sash. Come on dialga thaw out. Please dialga. Come on Yes, dialga you heard our cries for help and you did it and we are going to take a victory tying up I score for the rest of the week The hype train continues through all adversity And everything else dialga pulls through thawing out Getting the knockout onto that sash cartona and making sure that we can pick up a nice Plassy win here today with our cup pocket and dialga core. So that is really good for us there's the earth part from the dialga going into the kaiaga and um Spatial rend will be enough to wrap this one up Take this primal color down and pick up a nice victory first end the episode today Which is incredible. Um, I'm gonna end today actually because we've got Now we will we'll call it a day there because I want to come back and before we have any more matches Want to change up want to activate those bonus buttons tomorrow. So we'll we'll knock it on the head now We'll be back tomorrow we will activate some bonus buttons tomorrow and Make some changes to the team and see if we can continue this run on for the rest of the week The two pokemon that I don't want to see going on their king lair and the hax race I feel like they're doing an incredible job this week although the dragons here did did really well So fair play to them. Thank you so much for tuning in though guys. I hope you've enjoyed today's episode Have a great rest of your day. Make sure you take them care yourselves and I'll see you for the next one So till then take care and bye. Bye


Nigeria has recorded 41 new cases of the virus bringing the total number of persons who have tested positive in the country to 165,702. The Nigeria Center for Disease Control announced the severe atoficial tweeter handle. According to the tweet, there 41 new cases were from seven states. A Lagos state reported the highest number, with 16 new cases, Ube or second with 14, while Ado state had the lowest with one new case. The NCDC also treated that us of 15 to of May 165,702 cases have been confirmed while 156,412 cases treated and discharged for 2,066 deaths were recorded. And don't forget to hit the notification button so you get notified about fresh news updates.


Libertarians are usually on the defensive, because their principles seem so extreme and some of their views so radical. This is true whether the opponent or critic of libertarianism is a liberal, a conservative, a democrat, a republican, a centrist, a populist, a socialist, a moderate or a progressive. All are united in their attacks on libertarians in many areas. But it is these groups who ought to be on the defensive. They all believe that it is appropriate for government at some level, one, to arrest, fine, imprison or otherwise punish people for engaging in entirely private, peaceful, voluntary and consensual actions that do not aggress against the person or property of others. Two, to fund education, research, cultural and construction projects. Three, to regulate, oversee and sometimes prohibit commercial activity between willing buyers and willing sellers. And four, to take people's resources against their will by force if necessary and transfer or redistribute them to other citizens or foreigners as the government sees fit. Libertarianism is a philosophy of liberty, property and peace. It is the philosophy which says that people should be free from individual, societal or government interference. To live their lives any way they desire, pursue their own happiness, accumulate as much wealth as they can, assess their own risk, make their own choices, engage in commerce with anyone who is willing to reciprocate, participate in any economic activity for their profit and spend the fruits of their labour as they see fit. As long as their actions are peaceful, their associations are voluntary, their interactions are consensual and they don't violate the personal or property rights of others. Libertarianism respects personal privacy, financial privacy, free thought, individual responsibility, freedom of conscience, free exchange, free markets and private property. Libertarianism celebrates individual liberty, personal freedom, peaceful activity, voluntary interaction, laissez-faire, free enterprise, free assembly, free association, free speech and free expression. The creed of Libertarianism is the non-aggression axiom. Everyone should be free from aggression against his person and property as long as he respects the person and property of others. As long as people don't infringe upon the rights of others by committing or threatening to commit acts of fraud, theft, aggression or violence against their person or property, the government should just leave them alone. But because the government is the greatest violator of personal and property rights, Libertarians oppose and seek to limit by peaceful means the intervention, regulation and control of the government. Instead of being on the defensive and making excuses for or backing away from Libertarian principles like the recent Libertarian Party presidential ticket, Libertarians should boldly say, of course I support this and that, of course I oppose those and so, of course I am a Libertarian. For example, of course I support medical marijuana. It is not the business of government to decide what medical treatments are appropriate. The government should not have anything to do with the practice of medicine. Of course I oppose gun control laws. If guns are outlawed, only outlaws will have guns. Criminals who want to commit armed robbery or murder are not deterred by gun control laws, there should be a free market in guns, just like any other product. Of course I oppose public education. It is not the job of the government to educate anyone's children or force anyone to pay for the education of anyone else's children. All schools, vouchers, grants and loans should be private. Of course I support the legalization of all drugs. Vices are not crimes. Every crime needs a tangible victim with measurable damages. It is not the job of the government to prohibit or control what a man wants to smoke, snort, swallow, inject or otherwise ingest into his body. Of course I oppose Medicare and Medicaid. It is not the job of the government to provide or pay for healthcare or health insurance. And it is immoral to force some Americans to pay for the healthcare of other Americans. Of course I oppose all foreign aid. No country should receive foreign aid for any reason. What the government gives in foreign aid must first be taken from Americans. No American is ever asked if he wants a portion of his taxes to go to some particular country. If individual Americans want a particular country to be supported then they should write a check. Of course I support all tax deductions and credits. Reducing tax deductions and credits increases government revenue just like raising tax rates. It is always better when Americans are allowed to keep more money in their pockets and out of the hands of Uncle Sam. Of course I oppose refundable tax credits like the earned income credit. Refundable tax credits are a form of welfare. No one is entitled to receive a tax refund of money that someone else paid in. Of course I oppose the US military providing disaster relief. It is the sole purpose of the military to defend the country. It is never the purpose of the military to provide disaster relief even in the United States. Of course I support the right of people to sell their organs. If there is anything that a man owns free and clear it is his own body. Everyone has the natural right to sell his organs to the highest bidder while he is alive or after he is dead. Of course I oppose food stamps. It is not the job of government to feed anyone. All food aid should be undertaken by private organisations. Of course I oppose all minimum wage laws. It is not the job of the government to set a minimum price for labour any more than it is the job of the government to set a minimum price for bananas. Of course I support free trade. Government managed trade is nothing but Soviet style central planning. It is not the job of the government to protect industries at the expense of American consumers or calculate the trade deficit. Individuals and businesses have the natural right to export goods to or import goods from any country they choose. Of course I oppose government unemployment compensation. It is not the job of the government to pay people to not work. Unemployment insurance should be provided by the private sector just like auto, life and fire insurance. Of course I oppose all anti-discrimination laws. These laws are an attack on private property, freedom of association, freedom of contract and freedom of thought. In a free society any individual, business, employer or organisation would have the right to discriminate against anyone for any reason. Of course I support ticket scalping. Ticket scalpers perform a valuable service for event hosts and patrons. What could possibly be wrong with an exchange of tickets for cash between a willing buyer and a willing seller as long as their activity does not violate the property rights of the owner of the ground where they make their exchange. Of course I oppose welfare payments. It is immoral to take money from those who work and give it to those who don't even when the government does it. All charity should be private and voluntary. Of course I oppose occupational licensing. Why should anyone have to get permission from the government to open a business, engage in commerce, work in certain occupations, have a particular vocation or provide a service to willing customers? Why does anyone need permission from the government to work? All occupational certification could and should be done by the private sector. Of course I support these things. Of course I oppose these things. Of course I'm a libertarian. Libertarians, as economist Robert Higgs has said, should never concede the moral high ground to those who insist on coercively interfering with freedom.


Here is a recipe for cilantro lime rice. We have some long grain rice, onion powder, garlic powder, chicken bouillon.  Half an onion, two cloves of garlic, half of a lemon, excuse me, lime, some cilantro and some jalapeno. So taking the onion with the garlic, sauteing it with some olive oil and some red pepper flakes. Now I'm going to put in my rice. I'm going to saute this rice, stirring it frequently to make sure it doesn't stick to the skillet. Going to saute it for about four to five minutes. Here's my chicken broth here. The key to making great rice, and I'm pretty sure most of you know this by now, is to pour the liquid over your rice. Just barely over it, okay? Turn it on low or actually turn it off. Put a lid on and leave it like that for 15 to 20 minutes. Do not touch it because you basically want it to steam. All right. So here is my rice bubbling away and I'm going to go ahead and turn it off. And this is what it looks like all done. Yummy, yummy, yummy. I've been eating this rice a little too much. So putting in there, half a square of the lime. I'm going to put in about, I love cilantro. I know a lot of people don't, but I'm going to put in about half a cup of the chopped up cilantro.  So I hope you guys enjoyed this recipe. Take care. Bye.


Okay. Then I want again to welcome to all of you and let me thank my co-organizer, Professor Verma and Chung for the beautiful idea in my opinion to provide to young researchers a training in the foundations of commutative algebra and algebraic geometry in prime characteristics. We can say that we start in the best way. It is a pleasure and honor for me to introduce Professor Craig Eunice of the University of Virginia. Craig Eunice and Melvin Oxter developed first the theory of the tight closure in 1988. They published one of the first papers on the subject in the new series of bulletin of my American mathematical society. And let me recall that in the true introduction, they wrote with the fine, the tight closure of an ideal in characteristic P and use it to give a new much simpler proofs of many theorems in a great strangeness form. This is the beginning of a fundamental part of international mathematics. Then we can start the first lecture by Professor Craig Eunice and the title of the talk is where does the tight closure come from? Please Craig. Thank you very much. This is a can, first of all, I should ask, can everyone hear me all right. It's good. I'm going to share my screen and hope it works. Everyone see that. Okay, I will start so. Everyone can read that. I can't see the chat while I do this, so I want to encourage anyone that has questions to ask them at any time. And if somebody could check the chat for me and relay them thanks Kyle. That would be great. I also want to thank ITCP for hosting this event. I love going there, the times I've been there. Everyone has a chance to participate in another program in person there sometime. The organizing committee has been fantastic. And I want to thank them. Also, it's a huge amount of work to organize something like this. The list of tutors who are going to be giving the lectures is incredible. Many, actually all of them have contributed greatly to this theory in one way or another over the years. And finally, I want to hope the participants have a really good time and have as much fun learning this as I did, helping to start it. So with that, let me let me go ahead and start I don't have a lot prepared I have a few remarks I also wanted to leave time for questions about not so much the actual details of the theory but maybe questions about to expand on where it came from or how we originally did it. So I'm going to give three answers to start with, I guess I should say, in terms of the years, literally, the, the start was a talk I gave in the University of Illinois in 1986 at a conference there I believe in November. And I spoke on something involving the integral closure of ideals which I'll actually talk about in a little bit. So the talk Mel Hoxter came up to me we'd been talking about various things over the years, and he said, basically he said, what's in answer three down there he said, maybe we should think about the crucial idea in the proof and try to conceptualize. So that sounded great, and we arranged that I would visit the University of Michigan and as soon as possible in the next year which was sometime I think in February of 1987. And we spent a very intense week, working this out. This magical week of my own mathematical career, it was, it literally was almost like magic, we had this idea and we started thinking about how to apply it, and suddenly all these results just sort of came for free almost. And we were very excited at that time. And since that time of course it's been developed by a huge number of people in many different directions which will be hearing about. So let me go to my first answer here which you probably have already read where did it come from. Well, through the late. Well through the decade of the 70s basically characteristic P methods started coming into community value. And it already existed in number theory for a long time, for instance, the proof of irreducibility of cyclotomic polynomials and things like that that those are characteristic P proves. But really in community of algebra, the fundamental paper of Pesquina and Spiro which proved some of the homological conjectures by reducing them to characteristic key was a crucial step. So Joel Roberts proof that rings of invariance or Colin McAulay which you'll hear about in these lectures. Keichi Watanabe go toe and others started taking up the idea of F purity which you'll also hear about Paul Roberts use characteristic P methods to also study homological methods. And then Lipman and Reese are very important in terms of their work on integral closures, even though they didn't directly use characteristic P methods. So that's my first answer it came from a long history of characteristic P methods and problems to which you could apply. Another answer was really to some extent although Keichi Watanabe and others were also very important here. And that was the desire to find algebraic proofs of results which were purely algebraic but at that time had only been proved by analytic methods, or in some cases algebra geometric methods. The professor gave a very important series of talks at George Mason University, where Neil Epstein is right now. In 1919, I forgot the year 1980, maybe, where he talked about analytic methods, proving algebraic results, and he threw down the gauntlet to try to find algebraic proofs of these. And finally, the third answer is one I learned from Mel Hoxter also, and I already mentioned it, whenever you have a germ of an idea that proves something, a method, a lima, whatever it is, conceptualize. And what that means is make a definition if you must, but take that nugget of an idea and make a concept of it and start studying that concept. And it's amazing how far that will go sometimes. So, let me start not with tight closure, but with closure operations in general, because tight closure is a closure operation. So here's a question for all the students. My rings are going to be in the theory and it's not always necessary but I, it's much better to stay in that case. I is going to be an ideal inside are I hope you can read this. Here's a question to define a new ideal which contains I by the following property. Say an element is in J this bigger ideal, if and only if for every homomorphism from our to a field. The image of X is always in the image of I in that field. Now of course, it may not look like you're saying much because, you know, fields have no proper ideas so the image is either going to be zero or the whole field. Well, I'll give you a second to think about it. Especially the students. Anybody have a guess they want to give in the chat students only. Anybody brave enough by Bob suggests the radical of I exactly very good. The nil radical. So J is actually the nil radical of I, which is the set of all X and R. Such that there exist in X to the innocent. In fact, let me give a theorem here. Which says the following are equivalent. First of all, the first property is what I just said that for all maps to fields the image of X is in. The second is that X is in the intersection of all prime ideals, which contain I. The third is that X is in all the minimal primes containing I, and they're only finitely many of those because it's a material ring. And the last is the property that usually defines the nil radical. It's not hard to prove one implies two implies three. Three implies four is a in a team McDonald or common technique involving localization and four implies one is more or less trivial. Do it for an exercise if you want. But the thing I want to emphasize here is that this property right here. The fourth property is is an equation. And it's very useful as you all know. But the first one has to do with maps to arbitrary fields if they're very different. And it's by combining both of these that you often find a huge amount of strength in applications. All right, let's do another closure operation. Second question. This one. I don't think the students would necessarily know because they may have never seen it before. So the same setup. I have a ring are a netherian I have an ideal, and I wanted to find another ideal. I wanted to define it by the property that for every homomorphism from our into a one dimensional regular local ring, or also known as a discrete valuation, rank one discrete valuation. The same property I talked about earlier is that the image of X, which is fee of X is in the extension of it to be. By the way, I maybe I should say it just for commonality. The field of course, right here this is just a zero dimensional regular local funny way of putting it. But that's all the no radical is and now I'm just sort of popping the dimension up by one and saying well what if you do the same thing but you only look for maps to discrete valuation. So the integral closure bar of iron, not our, and it's usually denoted with a bar. And I believe this understanding the integral closure was at least for me, a very important part of the development of tight closure. Now let me give you the theorem about integral closures which parallels the theorem. I gave earlier about the new rack. So basically I want to assume right here. Maybe I should mark this in red right here. That I is not in the union of the minimal primes of our so for instance if ours reduced or domain. I just say ours a domain I just be saying I is non zero. And I have an element X and R then. This means the following are equivalent. So first it has the property I just talked about right here. That for every map to a DVR I should have said that's a DVR. This is a DVR. So one dimensional regular local. The images in IV. The second one is a very useful one. If you know more about the integral closure of rings which is what typically you study first. The fastest the integral closure of the restring. RIT. So this is the subring of the polynomial ring, which is generated by the elements little I times T. So just take the integral closure, then inside our tea, then the property one is equivalent to saying that X times T is an S third one is an equational. So this is the equation characterization. It says that something very much like the nil radical. It says there's a monic polynomial satisfied by X, such that the coefficients get into higher and higher powers of by a one is an I, a two is an I squared, etc. Asa been is an IDV. So this obviously by the way implies that X is in the nil radical. And since everything here is in I already. So this certainly says X to the n is. And the fourth is maybe the most important for tight closure. It says that you don't actually have to check one equation. You don't have to check maps to rings, but you can check. There are infinitely many equations in some sense that there exists a C which is not in the minimal primes of R the union, such that C X to the n is I the end and I should have quantified in here so this is for all in sufficient. So all these are equivalent characterizations of the integral closure. I see a chat here. I can see the chat didn't realize I could. Good. Yes, they're in the George Mason. That was that George Mason conference was, I believe, the very first time I met Kichi Watanabe, which played a huge role in all this. It was a real pleasure to meet him. I believe you can correct me if I'm wrong. So I'm just going to repeat. As I said, the set of all X satisfying these equivalent conditions called the integral closure. And it's an ideal containing I and contained in the mill radical. And it's extremely important ideal for many reasons. But there are two directly related to tight closure. And I want to mention one of them and give you a small proof about the other. So, I think I said right at the start that several things had been proved using analytic methods that there was no known algebraic proof for. And this is what Hoxter based many of his lectures on, and one of them was something called the Briance and Skoda theory. So I'm going to give you the Briance and Skoda theory in the statement of it. I hope you can see that. You have are in this case is a power series ring and in variables and an arbitrary ideal. Then the Briance and Skoda theorem says that the integral closure of the nth power of I is contained in I. The end here. Notice these ends actually match up. So this in and this in are the same in that was very critical. This was inspired by a question of Mather actually originally. What do I have on the next slide. Oh, good. Maybe I should mention that. Mather's question was based on the following so are here is a. R is again a power series ring over the complex numbers and F is in R. It was known that F is always integral over its part. This is, to me, one of the fundamental relations about integral closure, which is always fascinating. And Mather asked. Now remember the integral closure is always inside the nil radical of the same idea. There exists some in such that F to the end is in the partial. This is not obvious. Try to prove it directly sometime. You'll find how hard it is. And Mather's question was that. Well, I think his original question is can in be chosen. And the Branson's code of theorem gives this answer and I'll say why it's because. So let me go back up to what it says. It says that the nth power of any the integral closure of the nth power of any ideal is in the idea. And all I really need to do is note here that if you take the integral closure. to the nth power, it's always inside this engine. This is very easy to prove using the equivalent characterizations again. So if you go down here. If you combine this statement. With the theorem of Branson scota. You see that in fact, F to the end is. The Branson scota theorem answered in the affirmative. The question of Mather, and it did so by giving actually. It actually told you what the uniform number was. By the way, while I mentioned it on this. You might ask what it means if in is one. It's a very famous theorem of Saito, which says that if, if F equals zero is an isolated singularity, then in is one if and only if F is so called quasi homogeneous that there's some waiting that can make it homogeneous. So this was the focus of one of the focuses of Hoxters lectures at the George Mason meeting. And then satay gave an algebraic proof soon after that. But I was a colleague Ellipman at that time, and I was working hard on the integral closures of ideals David Reese visited Purdue at that time. And I started thinking about more about powers and integral closures of ideas, and I want to show you an argument. This was a time that was, I wrote up in paper and I think 86 and it was what I was talking on at this Illinois conference, because this shows you a tight closure argument without ever mentioning tight closure. So I want to end with this basically because to some extent this was the beginning for me of this journey. Let X one through XD be a regular sequence. By the way, I guess I should have said there is a very easy tight closure proof of brands discovered, which you'll probably see in the lectures. I'm not going to give it now, of course. So, remember for the students here, you've just started studying some of this this means X one is a non zero divisor and Xi is a non zero divisor on our mod the first I minus one. And the yoga is that regular sequences. I'll use this a little bit behave. As if they are variables as much as they can, at least when they interact with themselves. You can make this precise using the Cohen structure theorem flatness but I don't think I want to do that in this talk right now. Here's a theorem. It may not look like much, but it actually has a lot of consequences. And this was, I should give credit this was done by Eto. In all cases. And by myself, if it contains a field. And I'm basically there's a more general statement but I'm going to prove the simplest case of this right now because I want to show you how characteristic P can come in to help you. So the theorem is very simple. But always an Ethereum ring. As you have a regular sequence. And I is the integral closure of the ideal. If you want a specific example you can think of the case where F is a hyper surface with an isolated singularity. The partials would be a regular sequence in this case. And we just saw that F is always in the integral closure for instance so that would be an example of such a phenomenon. Then, if you look at this intersect I squared. Even I squared bar. Maybe I should put bar here. It's just the X's times the ideal. So what does this mean. So that's the statement. If the summation of our X side is in the integral closure. Then, the R is can be chosen and the integral closure. Now in fact, because all the scissor G's on a regular sequence are are generated by the casual scissor G's. You can see that in fact all choices of our I will have to be an I bar, but literally it doesn't say that start. I just want to stress something in my own thinking about tight closure that one of the things in community of algebra that I've always been very curious about is, when you have some relation like this. I mean, it could be a summation or a Xi is in some other idea. What can you say about the coefficients. These are the hardest things to pick out. You have some general relation, but how can you more easily determine the coefficients and this is where I think analytic methods were incredibly powerful in problems like this. And what tight closure is done to some extent is allow us to replace analytic methods by these characteristic key methods in almost all cases. A question will quicken the chat. So, in your question there, are the R s is supposed to be or the R is supposed to be in itself or an I bar. R eyes. Oh, sorry. Thank you. I bar is I because this is I. Yes, thank you. But what I really meant to write is, is this right. But when you take the interval closure of an ideal if you do it twice you, nothing new is added. Thank you. So let's prove this, give a sketch. So proof. So we have this sum. I'm just going to do it with ice squared. Well, I guess it doesn't. And let me go back to the I'm going to have to go backwards I know in zoom that's a real pain to see these screens flashing by but I'm going to go backwards for a second or two anyway. So I'm going to use this fourth property of, of integral closures that there's a C, not in any minimal prime above in our such that C x to the n is if if x is integral. So that has the following consequence there exists a C not in the union of the minimal primes of our such that C times I squared bar to the end is in either the two. You can do it at an element in time but because the ring is the theory and they'll be a finite set of generators, you can do it for each one and take the product of the seas that work for each one and that will So I want to now assume we're in characteristic P. This is prime characteristic and I want to take the element. Why, which is this element. And I want to use that. See why did the end is in. I did the two and actually sorry. I did I the same mistake again. I better just erase. It is the statement was correct but I want to use the stronger statement that it's in this idea. And that's simply because I squared is integral over x one through x d square. And I want to do the same thing here. And I want to use this with n equals P to the. The nice thing working in characteristic P. As I'm sure all of you already know, is that when you, it's a endomorphism so called Frobenius so it has this property. Simply because all the binomial coefficients, if you raise it to the p power all the binomial coefficients are zero characters, when they operate on the grid. So when we do this to why we get the file. But now, if we apply this boxed property here. The fun, we get the much nicer expression like this. And now what have we achieved by this. So what we've achieved is we've now got an equation. It involves essentially the x, the integral closures disappear. They are asking if you can kindly show the previous page. One more. It's okay. Thanks. Yeah. It's all right. Yeah, thanks. You probably want to specify that this is for all. Here. Yeah. That would help. Thanks. Actually just large in is enough but in fact you can do it. Like this. Thank you. So, I want to make a side comment about these coefficients and why characteristic P is so useful, which is a more of a heuristic philosophical. You see, if, if you're in characteristic zero. You take an expression like this, and you expand it out. Of course, there's all these terms. Multinomials in the X's with mixed, you know, mixed terms with the RIs. And it's extremely hard to usefully analyze that maybe there's a new method out there at some point that will discover that allows you to do it. Sometimes, at least I don't know how to do it truly usefully. And that's where things like harmonic analysis and stuff really helped in my own estimation. But what characteristic P does is it removes all the static all the noise and allows you just to look at the P to the powers themselves and this is really incredibly useful. So, finally, we need a little exercise now to finish off. So the exercise, whoa, what did I just do. I don't know what I just did, but I didn't want to do that. I'll write it in red. So the exercise is to prove if effects one through XD is a regular sequence. And you have a sum that looks like this for any. The SI are in the nth power. So I know you're doing a lot of exercises here. If they were variables, this would be quite easy. They always behave like variables so you can do this any number of ways but try it and see, see how you do. So I want to apply this exercise up here now back to the proof. And the conclusion from this exercise and this equation is that hence for all P to the CRI to the P to the is in X1 through XD to the P to the. And if you recall the criterion for integral closure that was number four on my list that I used in this. This exactly implies that our eye is actually in the integral closure. This is for all. And that's the end of the proof. Now, there's no tight closure here. But this was the argument. That in some sense. It's not the only argument. There were arguments like this done by Hoxter and his amiable system of parameters paper, where he talks about big coin Macaulay algebras. There are arguments, somewhat like this in the Hoxter Roberts paper about rings of invariance or coin Macaulay. But maybe the first time it had been so explicitly used with with this particular type of multiplier here. So, I guess I should give you the definition of tight closure before, but I mean you'll see it in a second anyway. And then I want to share a screen. I'm just showing you some of our first thoughts on tight closure so the characteristic of ours. By the way, I should have said here too. This is a characteristic P proof but by reduction to characteristic P which I hope you'll learn a little bit about this, these lectures. This immediately proves the same theorem, as long as the ring contains a field. I do want to point out it to did it by other methods without any restriction at all. Create a professor sponsored window in the chat but upstairs when you were working with a closure. It should have been infinitely many and or equivalence with. Yes. Did I not write for all large and or something like that. I think you're a for all and Yeah, yeah for okay I won't go back to correct it right now but yes, that's absolutely course. So characteristic P.  We say X is in the tight closure of I, if and only if there exists to see. Minimal primes of our such that for all, he sufficiently large. C x p to the E. Now if I just wrote, I to the PD that would be integral closure. So this is tighter than that. And it says, so that's the original definition of tight closure. One of the most important aspects of tight closure which I think you learn about in the second series of lectures is that the seas here can actually be chosen independent of X and I, this is enormously important called, we called such things test elements. So I want to stop this share now and share one other screen but let me before I. Before there may be some more questions. Yeah, there's one more question here. I'm going to chat. After proving the resulting characteristics, he, how do you conclude it in terms of zero. So that's a whole process.  So this is a general process if you want to if you have some equation you've shown or has to happen. And you want to show it. So you start in characteristic p. So you start in characteristic zero containing the field.  If you're not local already localized to make it local. Then you complete. And then you have to use a very big theorem called the art and approximation theorem. You assume you have a counter example basically, and you're going to carry that counter example from characteristic zero to characteristic p. Given a counter example, use art and approximation to get a counter example in a finitely generated over a field over the some base field. Then you use something called generic flatness. Sorry, we get a lot of spam calls. Stop in a second. generic flatness and collect coefficients to get a counter example in a finitely generated algebra over a finitely generated Z algebra, or the coefficients are, which is inside the F.  Let me call this a of a to get a counter example in characteristics. So it's a very complicated process. But it was worked out by Hoxter. So Pesky and Spiro did a similar thing, starting here in their paper, using characteristic p methods, and Hoxter realized you could use art and approximation to do things in general and his previous work. So this whole machine as it were had already been set up where you can almost automatically conclude in characteristic zero, equi-characteristic zero theorems you've proved in characteristic p. I don't know if this actually will be covered in these courses or not, but it's sort of a black box if you, you can take it. Any other questions before I stop. Craig, there is another question if this process for this machine is often called reduction to characteristic p. Yes, exactly. That's correct. So when somebody says reduction to characteristic p, they mean essentially you, you've gone through this, this whole process and just anything else. Thank you. Can I stop this here? Okay. I just want to show you, I've been going through my old notebooks. I, by the way, I, I've always kept notebooks with my, just my calculations or work or, I mean, most of it's junk, truthfully. But I would highly recommend you do this yourself for anybody starting out. You will be glad you did. Often you think to yourself as you get older, oh, I think I did something about that 20 years ago and you can't remember. Well, I can find it in my notebooks. So I've been going through these and putting them in PDFs actually. And luckily I was doing one just before this. And I want to show you what I found. This was in the first notebook in tight closure. I started working on it with Mel. So can you see this? So this was a list of problems and questions we had almost after that first week of magical week. So to start with, we called it sharp closed action. So the sharp you see there is really tight closure. You can see the first thing we noticed is that if every ideal is tightly closed, we could prove Cohen Macaulay. We wanted to give a definition in characteristic zero, which we eventually did. We wanted to understand what class of rings had the property that every ideal was tightly closed. That's one of the F singularities you'll learn about. We wanted to give a criterion for quasi homogeneous hyper surfaces for when they had this property. We knew it had a relationship to rational singularities but at that time we didn't know what it was. Property five is to give a good name. It didn't really have a name for a while. And we eventually chose tight because it. As you'll see it's it's a very tight fit to your original idea. We wanted to know about what happened when you completed or if you localized course localization became a big open problem for a long time until the counter example my runner and monski and there's still open questions about that. Well, we wanted to sharpen results in our original notes so we wanted to apply it to direct summons of regular rates you'll hear about that and some of these. We wanted to know what it meant in low dimension. We wanted to give criteria other criteria for what it meant to be in the tight closure, what it means. And then at the very bottom you'll see that some notes about every ideal and a regular ring colons were in the shark closure. These are all things you'll, you'll be seeing about in this. But I think right away we knew pretty well that this was going to be related to all these things and working this out has taken a lot of people over a lot of years to do. Many, many people have made substantial contributions to this. As you'll hear about. So that's really all I wanted to say. It's been a great fun for me working on this and I hope you all enjoy it. Maybe not as much but at least someone.


October is Famous Monsters Month at 50th Street Studios. Join me and my co-hosts at our Twitch channel for classic monster double features every Saturday evening this month. See you there! Hello everybody and welcome. I'm Keith at 30 Seconds High Five. This week I'm going to tell you about Sennelin Ascends. And I don't know how to tell you about it. Where do I even begin? Josiah Bancroft, the author, has created a really compelling, complex and complete world here with the Tower of Babel. It is an infinitely tall tower in the middle of the desert with radically different societies living on each level. Nobody knows how tall the tower actually is, who built it, where it came from. A heartless society has built up around the tower. A society of people scrambling to make a living, to improve their lives, to find meaning, whatever, and all getting caught up in the awful world of the inside of the tower. Sennelin is a schoolteacher from far away small town and his young bride, who have just arrived at the tower, they're just sightseeing. He's a historian. He thinks he knows about the tower. His young wife disappears almost immediately and he is doomed to spend the rest of his life struggling through the levels of the tower, trying to find his wife as a desperate tragedy about loss and also about personal development. By the end of the book, I sure as hell wanted to know what was going to happen in the next level. And I was surprised over and over again throughout the book about how the various levels and the different societies living on those levels related to each other. Bancroft has really created something special here. I recommend you look it up. Thank you for watching this week. Please come back next week for another book review and in the meantime, please become a sponsor. Follow the link on screen and in the description where you can make a monthly pledge or make a one-time donation to help me make more videos.


Hello and welcome to this presentation of the STM32 SDMMC controller module. It covers the main features of the controller which is used to connect the CPU to an SD card, MMC card, or an SDIO device. The SDMMC controller provides a communication interface for the microcontroller to communicate with multimedia cards, SD memory cards, and SDIO devices. This interface is fully configurable, allowing the easy connection of external memories thereby extending mass storage capability when more memory is needed. Applications benefit from the reduced pin count required to interface with memory cards. With the SDMMC interface, applications can easily manage high speed read and write operations in external flash memories. The SDMMC controller supports data bus widths of 1-bit mode, default 4-bit mode, and 8-bit mode for enhanced data throughput. The SDMMC interface interconnects with the DMA to offload the CPU during data read or write transfer periods. The SDMMC clock generator can generate signals up to 400 kHz for the initialization phase and up to 50 MHz for cards supporting high speed mode. To enhance power consumption, the SDMMC clock can be disabled when the command and data busses are idle. The controller can interface with SDIO modules with advanced features like read-weight, suspend or resume operations, and standard operations like multi-byte transfer and interrupt signaling in 1- and 4-bit modes. The SDMMC controller is an SDMMC bus master that provides all SD, SDIO and MMC functions needed to interface with cards. It consists of an SDMMC adapter and an APB interface. The SDMMC adapter provides functions such as clock generation, command and data transfer, while the APB interface manages the control and status registers, FIFO buffers, as well as DMA and interrupt requests. Two clocks are available for the SDMMC controller, the APB clock or PCLK for the APB interface, and the SDMMC clock or SDMMC CLK for the SDMMC adapter. The SDMMC adapter includes a control unit that contains a power management module and a clock control module with a clock divider for the card clock, SDMMCCK. The clock control module provides an 8-bit pre-scaler for SDMMCCK clock generation, which allows it to generate a clock equal to one-half SDMMC CLK. It also provides a bypass mode for communications up to 50 MHz. The control unit can disable SDMMCCK generation when the bus is idle. The command path circuit is used to program a command response sequence. When enabled, the command path shifts out the command index and argument on the SDMMC CMD pin. After the last payload bit is sent, a CRC7 is computed and sent on the bus before generating the end bit. When a response is expected, the command path is configured to SDMMC CMD in and waits for the device response. The transmission and reception of commands is controlled by the command path state machine or CPSM. When no command or response is in progress, the command path is in idle state. When the CPSM is enabled to send a command, the command path moves to send state until the last bit of the command is sent. Then depending on whether a response is expected or not, the CPSM can return to idle state when no response is expected or move to a wait state and wait for a start bit on a command pin. Start of the response transmission. When a response start bit is detected within the allocated time period, the CPSM moves to receive state. After receiving the last bit of the response, the CPSM verifies the response's integrity using the received CRC and then returns to idle state. The CPSM returns to idle state after a timeout if a response start is not detected. The CPSM can be configured to send a command synchronized with the end of data transfer. When this feature is enabled, the CPSM moves to pending state and waits for the end of the MMC stream transfer. When the last data signal is triggered by the data path, the CPSM moves to send state. The SDMMC controller offers high flexibility for configuring the command indexes and arguments. With a flexible 32-bit register for configuring arguments and an independent 6-bit field for the command index, this architecture ensures that the firmware can address any type of card. The command path state machine is able to generate all command tokens with no restrictions on command index or argument. In addition, the start bit, transmitter bit, CRC and end bit fields are automatically generated and sent on the bus. Our response is a token that is sent from the card as an answer to the previous command. There are two types of responses, short and long. With four 32-bit response registers and no response constraints, the SDMMC interface supports both long and short responses to correctly initialize the card and communicate with it. Short responses have a total length of 48 bits and are composed of a mirrored command index, 32-bit command status, start bit, stop bit and CRC7 checksum. When a short response is received, the command status is saved in the SDMMC RESP1 register and the mirrored command index, when available, is copied to the SDMMC RESP CMD register. Long responses have a total length of 136 bits and are composed of the 120-bit CID CSD register content with the start bit, stop bit and CRC7 checksum. When received, the CID CSD card register is copied to one of the four SDMMC RESPX registers. The SDMMC interface also features the automatic detection of a start bit, command index extraction, 32 or 128-bit response extraction and automatic CRC7 verification. Once the SDMMC ARG and SDMMC CMD registers are programmed with CMD index, weight RESP equals 01 or 11 and CPS men equals 1, the CPSM moves from idle to send state and the host starts driving the SDMMC CMD line to send the command to the card. If the CPSM is programmed to wait for a response, weight RESP equals 01 or 11, it enters wait state and the command timer starts running. If the card doesn't respond within the maximum NCR time, the timeout flag is set and the CPSM returns to idle state. Once a start bit is driven by a device, it is detected on the command line and the CPSM moves to receive state. When the response is fully received, the received CRC code and the internally generated checksum code are compared and the appropriate status flags are set in the SDMMC interface status register. Note that for responses without a CRC, for example in the R3 response format, the SDMMC controller generates a CCRC fail flag, which means that the command response was received but the CRC check failed. After a complete command with a response is received, the CPSM remains in idle state for at least 8 SDMMCCK clock periods to meet command to command timing or NCC and response to command timing or NRC constraints. The datapath transfers data both to and from the SD-SDIO or MMC card. On each SDMMCCK clock cycle, the datapath can send 1, 4 or 8 bits depending on the bus width configuration. Transfer logic is clocked by the SDMMC CLK clock. It is divided into two subunits, one for data sent and one for data received with a dedicated control bit and status flags. The databuffer is not part of the datapath. Transmit and receive FIFO logic are mapped in the APB domain. All signals from the different subunits are resynchronized. The CRC calculator guarantees data integrity between the card and host. At the end of the datapacket, the CRC is calculated automatically. Depending on the configured data bus width, the datapath sends data blocks over 1 SDMMCD0 for SDMMCD0 to SDMMCD3 or 8 pins, SDMMCD0 to SDMMCD7. First, a start bit is generated on the bus followed by the datapacket with the first to last bytes of the sequence, the fourth byte in our example. Then the CRC16 and end bit are appended to the datapacket on the bus line. In a 4-bit data width configuration, each line has its own start bit, end bit, and CRC16 checksum. In this example, the four bytes are sent over the SDMMC bus in 8-bit mode. For each SDMMCCK clock cycle, a byte is shifted out with a start bit, end bit, and CRC16 checksum on each data line. The datapath state machine or DPSM controls the transmission and reception of all data. When the DPSM is in idle state, the first transition is triggered when the DPSM-enabled bit and transfer direction are set. For data transmission, when enabled, the DPSM moves from idle to weight S and then to send state. While in the weight S state, the DPSM waits until the data-fifo-empty flag is de-asserted. When data is available in the FIFO buffer, the DPSM moves to the send state. In send state, the DPSM starts sending data to a card according to the bus width set in the control register. At the end of the data packet, the DPSM sends an internally generated CRC code and end bit and moves to the busy state. In busy state, the DPSM waits for the CRC status flag. If it receives a positive CRC status, it moves to weight S state if the SDMMC D0 pin is not low, meaning that the card is not busy. From weight S state, a new packet transmission can start or the DPSM can return to idle state when all the data is transmitted. A negative CRC status from the card or a FIFO underrun error can force the DPSM to return to idle state. For data reception, the DPSM moves from idle to weight R state. When a start bit is detected on the bus, the DPSM moves to receive state where it remains until a full packet is received. As long as the end of data transfer flag and errors are not detected, the DPSM will keep switching between weight R and receive states. If an error or the end of data transfer flag is detected, the DPSM will return to idle state. A read weight state is an SDIO specific operation to stall the transfer in order to execute other commands or internal operations. It can be reached from receive state while the transmission is ongoing or from idle state. When the firmware requests a read weight stop operation, the DPSM moves to weight R state and waits for a start bit from the SDIO device. The FIFO is a 32-bit wide 32-word deep data buffer mapped on the APB domain. A data FIFO packet is the data source for the data path transmit and receive packets. Depending on the DPSM status, the data path FIFO can be disabled, transmit enabled or receive enabled. Dedicated receive and transmit FIFO status flags are available to ease firmware implementation. When the data path is disabled, all FIFO flags are D asserted. The hardware flow control function is used to avoid FIFO underrun when DPSM is in TX mode and overrun when DPSM is in RX mode errors. The hardware flow control logic stops the SDMMCCK PIN signals and freezes the DPSM when a risk of underrun overrun is detected. In send state, the SDMMCCK PIN clock signal is stretched and the DPSM is frozen when the FIFO threshold of two data words is reached. In receive state, the SDMMCCK clock is stretched and the DPSM is frozen in receive state while the FIFO is full. Threshold is 30 words. The clock and DPSM are restarted when the FIFO full flag is D asserted. The read weight operation is an SDIO specific operation that allows the host to temporarily stall the data transfer while emptying its data buffer or sending commands to other functions of the SDIO device. The SDMMCC controller supports two read weight modes, either by stopping the SDMMCCK or using SDMMCCD2 signaling. The advantage of SDMMCCD2 signaling allows communication with the card while in read weight mode. With multifunction cards, there are multiple devices that share the access to the SD bus. When the function supports suspend-resume, the host can temporarily halt data transfers to perform other internal operations or to communicate with other functions and then resume the suspended transaction. If a card supports the suspend-resume feature, the host can temporarily halt a data transfer operation to one function or memory in order to free the bus for a higher priority transfer to a different function or memory. When the SDMMCCD bit is set to 11, the CPSM knows that the current command is a suspend command. If a suspend request is accepted, the DPSM will wait as the function sends a complete packet and the application empties the reception FIFO before going to the idle state. Only then can the firmware start communication with a higher priority portion of the card. In order to restore a suspended transaction, the firmware needs to reconfigure the DPSM to read the remaining data before requesting a function resume. The interrupt concept is used to inform the host of changes in the card status using the SDMMCCD1 IRQ pin in 1-bit or in 4-bit data bus mode. SDIO interrupts are sent from the card to the SDMMCC host when the card detects an external event. The SDMMCC host detects interrupts sent on the SDMMCCD1 pin once the SDIOEN configuration bit in the data control register is enabled. While the DPSM remains in idle state, all low levels on the SDMMCCD1 pin are detected as interrupts from the card to the host. Here is an overview of interrupt events. This slide shows events related to the command path state machine. This is the list of flags for the data path state machine with events related to the transfer direction and transfer status. Here is the list of flags available for FIFO management in interrupt and polling modes. DMA requests are internally generated when triggered by FIFO threshold events. Here is an overview of the peripheral status at specific low power configuration modes. The device is not able to perform any communication in stop mode and lower. It is important to ensure that all transmissions are completed before the SDMMCC controller is disabled or the system is switched down to stop. Performance depends mainly on the SDMMCC bus width and clock configuration. The SDMMCC interface can generate clock signals up to 50 MHz but real speed can be decreased by the application and depends on several factors. The SDMMCC bus capacitance has to be considered as PCB track and card input capacity can play a significant role. GPIO settings also have an effect. Fast GPIO mode should be applied on command, data and clock signals. Lower power supply voltages and extreme ambient temperatures slow down the edges. And in some cases the application can't always manage fast data flows, especially due to overly frequent exception servicing or long time spent in interrupt handlers. The SDMMCC interface can be used in a wide range of applications where a low pin count is needed to interface with removable or permanent mass storage data memories. The SDMMCC controller can be used to extend device connectivity when using external SDIO devices, for example Bluetooth SDIO modules. Here is a list of peripherals related to the SDM32 SDMMCC interface. Users should be familiar with all the relationships between these peripherals to correctly configure and use the SDMMCC controller.