For more videos on people's struggles, please subscribe to our YouTube channel. Hello and welcome to People's Dispatch and our Mayday special series of interviews where we talk to representatives of trade unions from across the world. This Mayday is especially significant as the workers and trade unions across the world are facing a series of heightened challenges. So how are these organizations planning to confront them? What are the modes of organizing that they're thinking of? Today, we are joined by Edgar Desiree of the United Food and Allied Workers Union of Zimbabwe. Thank you so much Edgar for joining us and could you first start out by talking about what is the situation of workers under COVID-19 in the country, especially workers of your sector, the food and allied workers who are considered an essential service. These workers have to go out during the time of the pandemic. So do they have enough protective equipment? Do they have enough other other amenities that are being provided by the government and their employees? We have the greatest challenge is the provision of personal protective equipment in those sectors that have been declared by this government to be essential sectors that are operating within this lockdown, which were within our sector, the food industry. The service that we have recently carried out as a trade union showed that there is no compliance to the provision of personal protective equipment, especially masks, the sanitizers and so forth. It's not there. That's the first challenge. Then we have the other challenge of the suspension of the rights of the workers to the extent that the working hours which have been, which we recently picked in terms of legislation at nine hours on average, the workers are now working six to six low hours in an effort to cover up for the challenges arising as a result of the of the pandemic. We also have the other challenge, which is the the psychosocial fear in the workers, which is which we are getting, where the workers are there in apprehension as to what will become of them after or if the lockdown is to continue. And as a result of this pandemic, what will become of their jobs? Are they going to be safe? Are they going to be laid off those fears and the fear of how this they are going to support their families is the greatest fear? The other challenge is that the legislation is sort of being suspended in that the employers, they are forcing the workers to take their leave days. These days are a value. They are money. And they are just being forced to take their leave days as compensation for the loss that the employer is making during the lockdown period. The other challenge is that the working hours they have been shifted instead of working for nine hours a day, the workers are now having to wait for six hours for from six to six, which is 12 hours in a bid to contain the social distance. What has happened is the numbers they've reduced in the plans. And by reducing the numbers in the plans, the work cycle has been changed to accommodate social distancing. And by trying to accommodate social distancing, they have they have actually increased the number of waiting hours to 12 hours instead of nine hours. And the number of shifts have also been increased. Those are the greatest challenges that we are facing. Right. Has the government taken any step at all to ensure that the workers' rights are being protected or that they're not being exploited? Because we are getting reports of employers actually exploiting the workers. You mentioned of course the extension in time. So what steps are being taken by the government if at all? Currently, in Zimbabwe, there are no measures that have been taken to protect the workers. Instead, we have taken this special offer. It has been on the ground trying to ascertain the situation. This is where we are seeing the violation, the fundamental violation of this right by the employers, taking advantage of this situation. We are planning that if it is extended, we will then approach the government to say, where are the inspectors to inspect what is going on in the industry because the workers themselves are under threat from both the disease and the violation of their own rights during the COVID-19 period. Absolutely. And the other key question of course is that Zimbabwe over the past year especially has seen a number of protests. There was the strike by the doctors towards the end of last year, which went on for quite some time. And even before that, after the elections, there were a series of protests. So in this political context, what is the credibility of the government? Is it able to implement measures to tackle the disease properly? We did hear that there was a lot of suspicion about the government's testing efforts because they were not enough. On the ground practically, you may want to believe that there is a lockdown in theory but practically what is happening on the ground is it seems to be taking a political twist where a certain political party is now taking the COVID-19 as a campaign platform where the issue of looking at these people are having together and receive donations and so forth. That's the greatest challenge that we have seen. And the other key question in many countries is that how does the labor movement actually organize at a time like this? So the possibilities of strikes, the possibilities of rallies are no longer there right now and even in the coming months, it might be difficult to actually bring people in large numbers. And this is not even counting government suppression. So how is the union engaging with its members right now? What is the kind of communication that's going on? What are the strategies that are being used? Currently, as we were discussing recently on how to mitigate this in terms of organizing, we have devised the individual visits to companies to get information on what exactly is going on the ground. Small groups, we meet small groups of workers, four or five in a day and talk to them. We have also created WhatsApp platforms so that we are alerted to whatever is going on in the plans if we are not able to get there because the mass gatherings are no longer wanted. And in some companies, we are not allowed to get in simply because they are saying keep out because of this coronavirus while they violate the rights of the workers inside the plan. So that's why we use this WhatsApp platform to say WhatsApp so that we get to know whatever is going on inside there. And what are the key demands that the union is putting forward to the government right now? The greatest demand that we are putting forward is that there has to be a mitigation of the loss of jobs in the event of any in the form of a social cushion so that if those workers are to be laid off as a labor response to the COVID-19, the government should then be in a position to provide a social fund for those people to begin a new life after the layoff. Right. And what about regarding in the next coming months? Because also there is a question of governmental support in terms of food, in terms of other resources as well. In the coming months, we are actually planning and we can foresee this going up to as far as September. However, in terms of our demand, we are saying the government must come out clear as to its position with regard to the workers because currently here in the bubble, it's only the government is only meeting business to say how do we move forward without involving labor. And so we are demanding that even if it is to meet business, labor has to be involved to some extent and we are demanding that what is the current position, can you come out clean in terms of the money that you are saying you are going to give to industry that those that have been affected, how about the worker? How are they going also to sustain this worker, given that there is threat of cutting of wedges? How they are they going to get this cut of wedged by the by the by the driver? Absolutely. The way the workers in the food industry will sustain this country with the workers in the food industry will continue to fight the system until the Thank you so much for talking to us. Thank you very much for your idea. That's all we have time for today. Keep watching People's Dispatch.


Good afternoon. It is Thursday, May 13th, 2021, and it's 12 o'clock noon. I'm calling the meeting to order Governor Baker's March 12 order suspending certain provisions of the open meeting law allows us to hold this virtual Town Council meeting. Given that we have a quorum with the Council President I'm calling the May 13 2021 special meeting of the Amherstown Council to order at 12 o'clock to PM. I will call upon each Council by name at that time they should unmute their mic and say present will indicate that they can hear me and we can hear them. Please remember to mute your mic after you say present. I will start with Shawnee Balmain. Alyssa Brewer. Present. Pat DeAngelis. Present. Darcy Dumont. Present. Lynn Griezmer is present. Mandy Johannike. Present. Dorothy Pam. Present. Evan Ross. Present. George Ryan. Present. Kathy Shane. Here. Steve Schreiber is absent because he is at graduate graduation at UMass. Steve Andy Steinberg. Present. And Sarah Schwartz. Present. Okay. We have 12 of us present. This meeting includes audio video and is available live on Amherst media. It is also being recorded. There is no chat room. If you have technical difficulties, please let Athena and I know we are going into executive session in just a moment. And I will make a motion for that, but I want to mention that we will not reconvene an open session. Following the executive session. We have a special town council meeting. And therefore there is no public comment. Counselors will log off of the present session and log onto the link provided for the executive session. And I need a second is to convene in executive session under mass general law. Chapter 30 a paragraph 21 a three to discuss strategy with respect to litigation. Chapter 21 a zero C as in Carol V as in victory zero, zero, zero or three. Carol, comma, gray Kara, comma, at Al versus Susan Odette in the capacity acting town clerk. It is clear that an open meeting may have a detrimental effect on the litigating position of the body. The town council will not reconvene an open session. Following the executive session. Is there a second seconds Ross. Thank you. I now need to ask. Ask each counselor to log off. And we need to vote. We will not return. We need to vote. I'm sorry, thank you. We have to vote. Okay, all those and they are there any, is there any question or discussion. We have to vote individually roll call vote like normal for executive session. Yes, I do. Okay. Then I'm going to start with Shalini. Yes. Alyssa. Hi. Happy Angeles. Hi. Darcy demand. Yes. Thanks so much. An eye. Dorothy Pam. Hi. Evan Ross. Hi. George Ryan. This. Kathy Shane. Yes. goat's Schreiber. Andy Steinberg. Hi. There are Schwartz. link. Thank you.


Our next speaker, so Arina Swanson from Purdue, tell us about the Uniform Martin Reese. Okay. Thank you so much for having me here and I especially thank Juga Verma for organizing everything and putting so much effort into organizing this and just having the big vision for everything that we're doing and this is not the only course that we're doing in our tight closure. So I want to, what about, how do I get back to the, somehow it scrolls way down. So I'm talking about Uniform Martin Reese Lama. And in blue there is something that wouldn't that be nice to have.  And is it true that there exists a positive integer K such as for all ideals I and all integers and bigger than equal to K. We either have this strong version of the Arina Reese Lama I to the end in times M intersect with N is equal to I to the N minus K times the sum the module I to the K M intersect with N. So maybe we have this, or maybe we only have the the weaker version. And so we only have this inclusion here. Oh, that's why it's not right. Maybe we only have this inclusion. Well, the way I wrote this right now, neither of these two versions is true. And what happened here. Okay, so what is true is this uniform Martin Reese Lama is true if we impose certain conditions. And so Craig Hewnecker wrote a paper and published a paper in 1992. And he proved that this is true if I was essentially a finite type over in the theory local ring. If I was a finite of characteristic P or if I was essentially a finite type over Z. And I cannot possibly go through all three of these in the three lectures that I have, especially I cannot go through part three because that requires reduction of characteristic P. But these three cases are not the only places where we have some version of uniform Martin Reese Lama. And there are some open parts you can possibly prove another one. But it is not true in the big generality. And so let's first talk about our first talk about that but before I get to that. Let's talk about the classical. Art and Reese I think we have to start with that. This was published by David Reese in 1956. And it was a lemma inside a paper that he had for the proof of the crop. And it was lemma one and ever since then this has been called a lemma, but it's a very important result so we are usually told that if something is a big result we should call it a theorem. Yet here we have a lemma, and it's staying a lemma I think maybe that's for affectionate not just historical reasons, and you really like something then you call it cute and little even if it is giant and big importance. So it has the name art in there, apparently according to David Eisenbach's book, commutative algebra with a view towards algebraic geometry, he has a footnote that Emile Arton lectured on this in the greater generality so art David Reese proved it for rings. And so there exists he proved that there exists K, depending on N and M. Well, this is the general statement that will have such that for all ideals I and for all integers and bigger than equal to K, I to the end. No, no, no, no. This also depends on I. I to the end M intersecting M is I to the end minus K, I to the K and just say the end. So David Reese only proved this in case M is the whole ring and N is an ideal, and then Arton lectured in this in generality and Matsumura writes in the book that this was proved independently by Arton and Reese. Okay, and we have all seen many uses of Arton Reese Lemma so when you're doing completions, Hilbert functions are for the while you could can be used to prove the current crew principle ideal theorem co intersection we compare iodic topologies with this. It has also been used for existence of superficial elements and sequences the prove Reese is there for comparing valuations to determine final generation of ideal transform rings this was in some lecture notes by Peter Schenzel. So there are many notes and exercise zero which will not appear in the final version of this notes but if you can give me other examples of uses of the classical Arton Reese lemma then I'll add them here and move exercise zero from the back. Well, so this classical Arton Reese lemma is already powerful. But why would we want to have this uniform version where K doesn't depend on I. The first time this appeared in print but only as a question was in a paper of Eisenbad and hawk hawkster and hawkster. It does the question, but they only want said when I varies over maximal ideal. And they in their paper, they were proving the. One of the risk is main lemma so the risky has several main lemmas and in fact, there was a main lemma he had them homomorphic functions. We don't need to worry about what that is. But the risky had a main lemma inside there and then, again, another lemma perhaps it should be a theorem and hawk Eisenbad and hawkster then gave another proof in a greater generality but it said well. Another possible proof would be if we had a uniform Arton Reese of this former the ideals could vary over maximal ideals. And then that version was proved. The weak form, which is all that was needed week form was given by O'Carroll him O'Carroll. And that was in 1987. And, and in that paper, he used arbitrary Eminem, but he used to prime filtration and we'll get to infiltrations in the proof of the uniform version of Unicus as well. And then Duncan and O'Carroll had a paper. Two years later. They proved it in a strong form. And then there was I want to mention one more thing. O'Carroll proved another strong version. And that was later I can't remember the year, and he proved it when I varies over principal idea. And for this he needed some primary compositions of what it was doing. So, so this is what I'll say about the background and now we will go towards proving the this uniform Arton Reese lemma well, we will only be proving the week version. And, and, but I don't want to start with these restrictions here, I want to start with an arbiter in the theory and ring and do as many reductions as I can, until I'm forced to make some extra assumptions. So, that's how we'll proceed. Okay, so what we will do in the step one step one, we will do a reduction to the case where M is the whole ring, and N is a prime ideal. Perhaps it's not clear why we would want to do that but in general working with some the whole ring and prime ideals or some ideals in general is much easier than working with arbitrary modules. Okay, so. All right. So the first thing to do is one part a is we do and is a sub module of M and let's call this L zero and now form a prime filtration. And maybe we finished it else of us. Okay. And so we have a prime filtration that means that for all I, the relevant li mod li minus one is isomorphic to our mod PI for some prime ideal piece of our. So, I trust that you have seen this before, so I will not go through the proof. And then one part so still reduction to M the whole ring. I suppose that if there exists K sub by such that for all ideals, and I, and for all integers and bigger than equal to K sub by L sub I plus one, I to the. Maybe I should call them because ideals are I so indices should be J ideal I to the N times L sub J plus one intersect with Elsa by. So if we know that this is containing I to the N minus case of I times Elsa by. And this is for all of we're doing J's here. And this is true for all J for all J. Then it's an easy exercise that for all and bigger than or equal to the sum of these case of J's I to the end. And for all ideals I, I to the end and intersected with and intersected with and is containing I to the N minus the sum of case of J's when he didn't know that's it. So we have a prime filtration. And then if we can prove the result for each one of the adjacent modules, then you know it also for the big chain. Right. And then the final reduction is. M mod n. So we reduce to the case with M mod n is isomorphic to armot P and P is a prime ideal. And, and if there exists K such as for all integers and bigger than equal to K. And for all ideals I, I to the end. And then intersect with P is containing I to the end minus K P. Then we conclude that I to the end and intersected with N is containing I to the end minus K P. That is supposed to be. So, this is, I'll actually prove this. So, go up here. This is our reductions. Okay, so proof of one C. One C. Okay. So, what we have is, since M mod n is isomorphic to armot P.  You know, X in M, such that the set of elements, the ideal of elements that multiplies X into N equals P. So that, that, thus we have I to the end and intersect with N. Well, I to the end and M is just N plus R X intersect with N. And now we use the following fact, we have a plus B intersected with C. And if a is a subset of C, then this is just a plus B intersected with C. So we will use that up here. So this is just I to the end N plus I to the end X intersect with M. And now we have to work with this part. So, let Y, okay. Let Y be in I to the end X intersect with N. So we can write Y is ZX for some Z in I to the end. Okay. And then, but Z also multiplies X, Z is an element of R, multiplies X into N. So, Z is therefore in I to the end intersected with P. So we have an assumption. We're assumption here this is in I to the end minus KP. So that's where Z is. So why, which is the X is therefore in. This is in I to the end minus KP and then we still have X. But P times X, that's in N. Okay. So we proved this with one part C. So what we just accomplished, what we just do. Why are you deleting things? We just reduced, I changed the red color didn't work. It only works in one screen. All right. So we may manage to reduce to this from now and we don't have to deal with arbitrary modules. We only have to deal with the whole ring and a primary deal inside. Okay. Any questions about this? Okay. Then I guess I should scroll. Now, so now let's go step two. Red color. Okay. Step two. What are we doing step two. Oh, I skipped what is in my notes. Step one was a reduction to faithfully flat extensions. Okay, so I'll write here. Okay. I'll write step zero or is we may, we may always pass to a faithfully flat extension. So if we can prove the uniform married and we slow mine faithfully flat extensions, then it's true wherever we started. So now I'll call the new step to agree with the notes on Moodle. I'll call this not step two, but step three. Step three is a reduction to only using I being just a primary ideals, ideals that are primary ideals that are primary to maximum ideas. And maybe I should say something more about step zero step zero is trivial. I said a lot more.  So, if we can work with M primary ideals then you have a little bit of a power about understanding of primary decompositions and when we lift things from various parts. Things will be, we can force things to be contained in those high dimensional high height ideals. So this is a pretty an important step as well. So, first, let I be arbitrary, I'd be an arbitrary radio. Then, if we need to understand this. Well, take an arbitrary maximal ideal. Actually, let's take an arbitrary maximal ideal, raise it to the elf power L could be large, and then raise this to the end power. We don't need m's and m's anymore. Well, I could just work with RSNPs, but it's fine if you have modules. Okay, and this is true for all positive integers L and for all maximal ideals of the ring. And what is true. If we know the weak uniform art and re SLEMA for M primary ideals for zero dimensional primary ideals, then there exists a K. There exists K that makes this intersection here. It's contained still over all L and all M that's containing I to them L and minus K times M. But then we can be generous. This again L M. This power here is containing I to the M minus K plus M to the L. We're very generous, but it's okay. Because now, when we vary over all L and all N, all maximum ideals M and all exponents L, this is just I to the M minus K. Right, so that, that at least we have only N primary maximum ideals so accomplished. We accomplished this part here. Any questions about this. And so we're here. And so in particular, in the class of generality. We only take I is primary to a maximum ideal. And we need to show there exists. So, without last reality, we have to show that for all prime ideals P. It's okay, such that for all primary ideals that are primary to maximum ideals, I to the N, such that for all and this is containing I to the M minus K P. Now, we may be tempted to say well why don't we just localize what is true that if I is M primary. Then, I to the N intersect with P is I to the N minus K P, if and only if it is true locally. After localization at M. Well, globally, then it's true locally. If it's true locally at this M and I is M primary. Well, globally, when we take this thing and this thing here. We don't need to localize it any other prime ideal, any other maximal ideal, if they don't contain I, then the left side is just P localized that maximum ideal and the right and so it's the right side so of course, this inclusion here words that works after localization anywhere else. So, well, this looks pretty promising. So is it true that it's suffices to prove this. It's suffices to prove this reduction to using a weak art and reslam uniform art and reslam only after all the localizations. And that's not quite true, because if you first localize it all these maximal ideals, then you get K that depends on this localization. And we want a global K that works before localization. So what is really happening. So, what we want is global K, but inclusion can be proved locally. And sometimes locally we have quite a bit more leverage for proving this. So we need it. Yes. Professor I guess there is a correction here pointed out by somebody in the chat in the if and only if statement when you're localizing. Yeah, is it. Oh, right. Thank you. Yeah. So the intersection should be with prime ideal. Yes, this should be localization that and was that the question. It says intersect with our localization. No, this we need. Oh, here. Yes. Yes. Thank you. Yes. Thank you. Yes. Thank you very much. Thank you, Alessandra. Okay. Now it might be true. Yeah, it's true. Okay. So the important thing is, we need a global K, we can just first localize find a K, we need a global K, and then we can localize to prove this inclusion here. And then, and the consequence of this. Oh, you can just prove things. Other consequence. Okay, here I can show more. Well, it's no good spot.  Another consequence is, if you have to prove inclusion locally, then we may in addition assume that locally. We find a global K, then we localize at a maximum ideal we need to prove the inclusion there locally. And then now we use the, that's step zero that I forgot to say, first, that that locally we can pass to a faithfully flat extension. Because to get infinite residue field. So, I won't say much about this but if you start with an Ethereum local ring, then you, you join a variable X. That's not a local ring anymore. But if you invert all elements that are not in the ideal extended, the prime ideal extended from M, then now localize. This is a faithfully flat extension. And this ring here has infinite residue field. So, maybe we want to prove something for all ideals in here, if we could prove that there exists that this K, this global K key here works for the uniform art and reslam after localization first in our localize the M and then after passing to this, then we will know it also in the original. Okay. So what's the advantage of infinite residue field. Advantage reductions exist. Minimal reductions exist. Well, with minimal reductions. That's why do we need minimal reductions, we're trying to prove the art and reslamma. There are, there's nothing about integral closure at all. Minimal reductions have to do with integral closure. However, at least in Unicus proof. And actually in his paper, he shows that this uniform art and reslamma and uniform, integral closure inclusions are very tightly related. And we'll talk more about that. So, for now, let's just keep it at this and we will use it eventually. Any questions about this first. Okay, so let's go for step four. Well, step four says reduction to minimal reductions. Okay, so this is two different meanings of the word reduction. Except this is not quite true. Not completely we won't just replace. Said, all right, we only need maximal ideals that are primary to maximum ideals. We won't say now we only need ideals that are minimal reductions of primary deals to maximum ideal so no, it won't be so it won't be a complete reduction. However, it will be. It will be good enough for whatever we want to do. So, here's a lemma in my notes it's lemma 3.8. And in Unicus notes it's lemma 3.1 in Unicus papers lemma 3.1. So, let I J. PB ideals in the ring R. So this is global ring. I is contained in J, and P is the prime ideal. And let H and K be positive integers. Or at least non negative such that for all integers and I will stop writing and bigger than equal to K or and bigger than equal to H because I to a negative far as a whole ring so it all works. So for all integers and we have two parts one is J to then intersect with P is in J to the n minus K T. So here you may want to think about the J is probably I read. Oops, I said it the wrong way. J is J. J is a sub ideal of I. And J is, to some extent, a reduction of I is not quite. But here I'm not stating that it's a reduction but in the future maybe we want to think of J as a reduction. So here, supposedly now we may have a uniform martin Reese for this reduction ideal maybe for all reductions. So suppose that we know uniform martin Reese for J. And now we need to know something about I suppose that I to the end intersected with key is containing J to the n minus H plus key. Okay, so here, maybe we're getting a hint that J is not quite a reduction of I but maybe J is a reduction of I mod P. So this is true for all. This part two is true for all integers, large integers and that means that mod P. Well, no, it doesn't quite say that. But at least the mod P something good is happening. All right. And so, if we have these assumptions, so this is and then what's the conclusion. I to the end intersected with key so now we get the uniform martin Reese for ideal I which might be arbitrary. So this is in I to the end minus K minus H minus one P for M. This is true for all me. Okay. So if we know it for these, maybe today's a reduction of some sort, then, and we know some good relation between I and J so these are assumptions one and two. And we also know the Artemis, a uniform argument is for arbitrary ideals. Okay, so proof. First to claim that for all and I to the end is containing J to the end minus H plus I to the end minus K minus H minus one P. And this, the first end that we have to worry about is when any escape plus H plus one. So then this is just I to the zero. And I to the zero is just the whole ring. So this is just. This is the assumption to. Right, so my assumption to this whole so now we have a base case. If any slower than that then it's still base case. So, if we know the claim for some end. It's bigger than or equal to K plus H plus one. Then, well, what about the next and I to the end plus one. So since we know it. So, so we if we know it for some end, then we have this J to the end minus H plus I to the end minus K minus H minus one P. I didn't mean to you have extra intersection. You wrote an extra intersection in two intersection says no so I'm a suit. I'm assuming this. Where's intersection. This is lost here in hypothesis to you don't have intersection. You don't need into six right there. This is not an intersection. But you wrote I to the end intersect with me. I mean the left hand son. Yeah, which part are you talking about. Yes. Condition to this and. Yeah, there's no intersection here. Here there's. Oh, thank you thank you. There's no intersection here. Thank you. Yeah, there's no intersection there. Thank you. Who's that. Thank you. Yeah. Okay, thank you. Yeah, because otherwise this wouldn't be to over here. Right. No, this wouldn't be to here. All right. All right. So if now by assumption we do have the base case because of the correction that we just saw. And now suppose that we, we have induction step here. And then let's multiply through by I. And then we get this. And we also know by, by part two. This is contained in J to the N minus H plus B. So, oh, I see if I can. I have more space here. So we have a now this again. This part here is a sub ideal of the left side so I can put this in here. So I'm going to pass the intersection of the rest. Interceptual J to the N minus H plus B. Oh, no, no, no, no. This, because I have N plus one here I also have to have N plus one. Plus one. Except that I don't need that N plus one now I will be a little generous. This is contained. I to the end some, I keep copying this. And then we're here, this is generously containing J to the N minus H intersected with J to the N minus H. I guess plus one plus B. Now, this here is a sub ideal of that so we get I to the N minus. This is a sub ideal of the left side. And this is a sub ideal of the right side. Oh, right. I'm still just proving the claim. Okay. N minus K. Plus H. P. And this is J. N plus one. Minus H. Plus the intersection of the rest. Intercept. And this here is by our assumption one. This is now contained in I to the N minus K plus HP plus J to the N plus one minus H plus J to the N minus H minus K. I approve the claim. I to the N plus some going back here. I to the N plus one should be contained in J to the N plus one minus H plus I to the N plus one minus K minus HP. And that's, well, it's, I need just one more line here. So today is a sub ideal of I. So this is a sub ideal of I to the N minus K plus H. So that proves the claim. Let me just go back to blue color. So now let's go back to we really want to prove this, this inclusion then. Right. So then what do we have. I to the N intercept was P. Well, we use the assumption to this is containing J to the N minus H plus P intersected with P. Actually, no, we will use the claim. What does the claims say. I to the N is containing J to the N minus H plus I to the N minus K minus H minus one P and we intersect that with P. And this again, this is a sub ideal of peace so we can pull it out. Plus the intersection of the rest. And now we use assumption one. This is contained in. So first a couple of this. J to the end minus H minus K. And J is sub ideal of I. And anyway, this is contained in there. So we crude what we want it. So this proves the lemma. So what this lemma will enable us to do is that well, there are sometimes when so called reductions or almost reductions will be the workhorse and if you can prove it for those almost reductions which we have to make more precise then things will be pretty good. Now, this is, I only have two more minutes so let me finish. So let's look at this assumption. I to the N is containing J to the N minus H plus P. So if we knew that so this means that J, which is a subset of I. This is a reduction mod P. So, if this is true for all in sufficiently big. This means that we have a reduction mod P. So we do have a reduction. Well, so well, why don't we always use reduction. The problem is that here age, perhaps depends on the reduction, or at least on the ideal I. And then that would not satisfy uniformity. So what I will make here is like a wish and definition. And I think this really shows the power of how unique it thinks. And sometimes I met he has these definitions, and maybe they're true. And, and then he works with them and well when are those wish definitions to, and that's why the, the uniform mind we see Lema doesn't work for all rings because these wish definitions are not always true. So, for every positive integer for every ring R and every positive integer K, define T sub K of R to be the intersection. You vary over all the ideals. So for all integers, and you take the integral closure of I to the end, and see what elements in our multiplied into the ordinary. And then you take lower power of the ideal I so that's T sub K, and then let's define T of R to be the union of all of these T sub case. So, one big thing that can unique at the proven we'll get to it next on Wednesday is these T of ours are not always zero. In fact, they're not always continuing minimal prime there, they can be pretty big. So this gives us like a uniform. So, analytically unremifed. So, these rings have to be analytically unremifed for these to exist. So, uniform analytically unremifed, unremifed elements, elements or something. I don't know they he he did not name them. So, so this is where integral closure does come in well that's still not convincing that we will need it but we'll get to it next time that it is indeed needed. Okay, I'll stop here. Are there any questions. There are no questions. Thank you so much arena. Thank you. Thank you. See you later. See you all in sick.


Hello, thank you for joining me. I'm out for a walk in the London Borough of Harrow just walking up Cannon Lane. We are going to a miniature railway, but right now we're going up over a slope over the Metropolitan lines, Uxbridge branch also carries Piccadilly line trains, so we're coming to this rather cage-like separate pedestrian bridge over the railway. Food of trees over there is Roxbourne Park, which is where we're going to, but if you can just see through the mesh, here's the railway line. So up there is Rainer's Lane station. It's the junction where the Metropolitan line trains turn left to join the Metropolitan main line at Harrow and Hill, and the Piccadilly line trains turn right, head down through Alporton to join the rest of the Piccadilly line network. Rainer's Lane, there was nothing there, it's just when they opened the railway they named it after a local farmer, Mr Rainer. So this is back onto Cannon Lane, which only really goes to a school and Roxbourne Park. So the park's down there. Now in Roxbourne Park is the Harrow and Wembley Model Engineering Society's miniature railway. The society's been around since 1936. The railway's been at its current location since 1976. So we're going to go and have a ride on that railway, but first we've got a bit of a walk through the park, so I'm going to do it in a moment. I'm going to be on the other side of that fence. We'll walk along beside the railway and we'll find our way from Standard Gauge Metropolitan line to the miniature railway. So we just saw a free car Piccadilly line train heading for Roxbourne. It must have been on railhead treatment, which is going to nearly always six cars. To get to the railway, you've got that avenue of London Plains, but what we're going to do is we're going to follow this path along here beside the railway for a short distance, because there's something down here I would like to show you. As I said, Rainer's Lane Station is in that direction, but this park is called Roxbourne Park and there is a river called the River Roxbourne and down in that corner over there is a river. So you'd naturally think it must be the River Roxbourne, but it's not. It's the Eddingbrook. The River Roxbourne is about half a mile that way, so I'm not quite sure why they called it after the river that doesn't flow through the park, but that's what they did. So as the railway goes off that way, there'd be a bridge or a culvert just down there under the river. I'm going to cut off the grass over here. I'm going to go right here to Brook itself because it seems we might as well feature the river that flows through the park and then I'm going to walk off in that direction over towards the railway. So right now I'm in the London borough of Harrow, but the railway itself is in the London borough of Hillingdon. So this park is actually in two boroughs, the fast borough jobship department is in Harrow, which is that corner where the railway is, which is in Hillingdon. So as we come down here, I'm at John's Park, and that's where the entrance is. A few words come here from our train from Rainer's Lane. It's trust me, in Rainer's Lane, there was nothing there before the railway, it was just the junction of where the metropolitan frequency lines joined or split, whichever way you're travelling, and they named it after a local farmer, Mr Rain. So what we're going to do, you can hear a train, let's watch that train. It was a 49S stop train. I always thought it was something nice, no matter where you are, you know, when you're in a park and a train gets passed, even more when you're in a city like London, and you know, you've got trains passing through, it's this sort of green area in the middle of an urban area. So we come down here, this is the bridge over the river, the river Rocks, well, not River Rocksworn, the Yedding Brook, as I've said, the River Rocksworn does actually flow through Rocksworn Park, and then that way goes to Rainer's Lane station. So if you come by train, that's the way I would suggest coming to the park. There's the river down there. What I'm going to do now is follow this rather pleasant part down here. So it says Rocksworn Park. I'm going to follow this rather pleasant path down beside the river, and that will take us to where the railway is. So I'm just going to walk through the awesome leaves and beside the river up to the railway. So this woodland path comes to an end. I'm still following the Yedding Brook down there, come out into another one of the many open spaces of Rocksworn Park. I still need to get to the other side of the river. So somewhere up here, there must be a bridge, and then, like I say, across the bridge. And we should go to the railway, which I'm quite excited about. I have been to this railway a couple of times over the years. I remember coming here as a child, not long after it had been featured on the BBC Children's Programme play days, where they used to go to different places. I remember first seeing it on there, and we looked at the end and said, how am I going to get to the railway? So I was like, okay, perhaps we should go there. So my parents did take me there. You might just be able to hear the wind's not blowing it out. I can hear whistles of steam engines. So that's exciting. We know we're going to get some steam, which is always good at Minnets Railways. So I think it can't be too far now. Admittedly, I have been here before. I've not been this way to the railway before. It's an explore as I make the video. So it's got to be this way. If this is the Yedding Brook below us, which it looks, it's quite a built-up bridge, but yeah, there. That's the Yedding Brook. So as I keep saying, you can see the water nicely down there. I keep saying the Yedding Brook flows through the Rocksbourne Park and there's a river of Rocksbourne. So why is it not called the Yedding Brookside Park? I don't know. Okay, right. I think we're nearly there now. It's a very pleasant quiet park for Great London. I bet you ask a lot of people from London, have you heard of Rocksbourne Park? They'll probably say no. It's a shame because it's a nice place. Right, where is this rail? I'm going to keep going until we find it. I'm now just crossing this final piece of grassland area of the park before we come to the Yedding Brook somewhere down there in the trees. The Yedding Brook is just over here. So let's go and have our autumnal trip around it. I can actually hear a train already. So what are we going to hear that? Diesel locomotive of some kind. What have we got out here? So the track is literally the other side of this fence.  Just all train going in front of me. It's a nice little circuit they've got here. So basically it covers this end of the park. So we'll wait and see what comes along and then we'll go for a ride ourselves. So if you have a look here, you can see there's like an inner bit track and an outer bit track. I believe you go around the inner one first. You kind of do a big loop around. So basically the train just follows it around. Can I hear a train coming or am I? There's going to have to be a train coming soon. Let's just walk a little bit. I'm sure we'll see a train go past before we get there. So the station is at the other end of the site and if you want to come by car, you can park at this end of the site. There is a car park. So let's see where we end up. I can clearly hear a train. So the railway, you can see there's two tracks here. They go right around there and there's like a tree-like tunnel through the trees over there. So if we look somewhere over there we should see a train emerge. There we go. Class 20. Funny because class 20s have been known to work on the Met in preservation. Well here we have a Met. So what should happen is that class 20 will go around and coming along on the track closest to us. So the track is the widest to seven and a quarter inch and you've got five inch and three and a half inch. So the class 20 expects is seven and a quarter inch. Here another horn of another diesel. So let's just watch class 20 go by. So the class 20 has gone by. We know they're running steam because we've heard some horns. So what's always exciting about miniature railways is you never quite know what you're going to find. Whether it's going to be steam, diesel, electric, whether it's, you know, what locos, you just don't know. It's really a bit of a whatever turns up. As like a heritage railways, as nice as they are, you tend to go to a heritage railway knowing what locos are. So if you're going to travel behind that miniature railways it's really, yeah, just wait and see what you get when you get there. So there's a complex bit of track there. See the steaming bays? Oh look at this, this looks exciting. There's an American steam locomotive. So what do you mean about the, what you don't, you just don't know what you're going to get. Look at this. That's pretty cool. We're just going to have a ride behind that right. So the station's over there. Let's go and have a ride on the Roxbourne Park Miniature Railway. Oh, I see. I've had a great afternoon here at Roxbourne Park. I've really enjoyed having a few trips around the railway. The railway very kindly allowed me to go line-side. So I was able to get some more shots of the trains there. I did also have a trip behind the Class 20. I'm just coming now. I'm just telling you about this because I think this is quite cool. This is a miniature narrow gauge loco. I know we've had confusing discussions in the past about what's miniature and what's narrow gauge. This is a narrow gauge loco on the Wildtime Railway. It's called Knaven Castle. And this is a miniature version of it. So the real loco is one foot 11 and a half inches. But this is scaled down for a miniature scale. So it's just coming into the station now. And then when this train finishes its journey, it'll be the end of my visit. I'm going to go home. As I said, I've had a great time. Just watch this for a while. A little steam loco. Quite like that as well. So yeah, I've had a great time here. Thank you very much to the Her and Wembley Model Engineering Society for being so friendly and allowing me to line-side and allowing me to visit. I'm going to go home. So if you're out this way, do come and visit them. They're very easy to get to on the Metropolitan Lines, Uxbridge Branch or the Piccadilly Line for that matter. Just get a train to Rainers Lane. And it's a very pleasant walk beside the Eddingbrook through Roxbourne Park to the Miniature Railway. Here comes that class 20. So we'll see that class 20 and then it'll be the end of the video. So thank you very much for watching. Please do feel free to like, subscribe and comment from the Roxbourne Park Miniature Railway. Good bye.


Hi everyone, in this tutorial I'm gonna be showing you how to use slicing in Python to reverse a list. In this example I'm gonna create a list, let's call it myList. This list will contain the numbers from 1 through 5. Now I'm gonna use this myList, I want to slice it to reverse the order of the list and I want to store it in a new variable. For example I'm gonna call this new variable reversedList and finally let's print this reversedList and let's run it to see the results. And here we're having reversedList of our initial list. That's all in this tutorial, don't forget to subscribe to our channel, see you in the next tutorial, happy coding!


Hey Lena, this is Megan with Oxmore Toyota. I just wanted to send this quick video greeting just to personally introduce myself. It was a pleasure speaking with you on the phone a few moments ago. In this email you'll actually find a link to all our new and pre-owned RAF4s and I will give you a call back in April. If by chance you'd like to come out sooner you can either reply to this email or contact me at<PHONE_NUMBER> and I'll talk to you soon. Bye!