How to benefit using IKAL? Intelligent computer assisted language learning, IKAL, the subfield of call integrating natural language processing and artificial intelligence, has typically equated language acquisition with the learning of linguistic patterns and grammatical structures. IKAL's system, therefore, has focused on identifying and providing feedback to problems in student performance that are exclusively related to linguistic structures. Basis of two Lyara acquisition. Some of these systems include learner models to adapt explanations, offer advice or decide on how a given student can advance through material. But the view of second language acquisition focusing on absolute linguistic abilities has the development of learner models that take into consideration the ability of students to perform language tasks. The strategies they must master to successfully use language in context and their linguistic abilities relative to the linguistic context and the task. How does it work? The eTutor is an IKAL system that functions as an electronic workbook for German students at Simon Fraser University. eTutor is fully integrated into the language program and several papers have been written about its development and use. The student model of the eTutor keeps track of individual grammar skills a student is acquiring, such as subject verb agreement or the subcategorization in form of arguments. The system collects positive and negative evidence from the student's production and uses it to keep an absolute score of the student's knowledge of each grammar skill. There is no overall classification of a student's grammatical competence. The learner model keeps track of the student's level of performance per grammar skill. To classify a student's knowledge, the numeric performance scores are grouped into three levels, beginner, intermediate and advanced. When the system identifies a specific grammatical error in the student's input, it checks the level of proficiency of that student for the particular grammar skill concerned. The system then decides which feedback message to use. eTutor's learner model is not based on a specific theory of SLA. It models the learner's language acquisition as the acquisition of individual grammar skills, which are modeled independently, without interconnection among them, using absolute scores for each one. IKAL is a system developed to provide writing assistance to a native speaker of American Sign Language Learning English as a second language. It receives short English essays written by the students and provides tutorial feedback on grammatical errors. The goal of student MODEL, Steps of Language Acquisition in a layered organization model, SLALOM, is to capture the status of the grammatical structures of English in terms of acquired, being acquired and unacquired. Do you want to teach English abroad? Take a TEFL course. The knowledge units of the SLALOM architectures are grammatical concepts connected to two sets of grammar rules, English rules and Mal rules. These knowledge units are grouped and hierarchically classified. The hierarchy proposed is based on studies in second language acquisition that showed stereotypical sequences in the acquisition of grammatical structures by learners of the same one layer of community. This hierarchy is used to identify the current state of knowledge of a learner and to predict the next grammatical structures to be acquired. Speak with an ITTT advisor today to put together your personal plan for teaching English abroad. Send us an email or call us toll-free at 1-800-490-0531 to speak with an ITTT advisor today.


Hi everyone, let's look at a couple of examples of how to use the two limit definitions of a derivative in order to answer some questions. The first one we're going to look at is the curve negative 2x squared plus 3x plus 1. We are asked to find the slope of the tangent to the graph, in other words, the derivative. And then we are asked to determine the location of the horizontal tangent to the curve. So since there is no point at which we are asked to find the specific derivative, it's implied that we're just looking for that general expression that will give us the derivative at any point that we were to substitute in. So that requires us here to use the first limit definition of the derivative. That's the one with the h in it, you will remember. So we're trying to find f prime of x going to be the limit as h approaches zero. Now for the numerator, remember the first part is f of x plus h. So into each of these x's here, you need to substitute x plus h. So let's go to the second slide just so we can write that out for ourselves just so you can see what it looks like. So remember our given expression is negative 2x squared plus 3x plus 1. So when we evaluate that for f of x plus h, we're doing negative 2x plus h the quantity squared plus 3 times x plus h plus 1. So it's really just a matter now of having to multiply all this out. So that binomial squared of course is x squared plus 2x h plus h squared. We can distribute our 3, distribute our negative 2, and that should be about all that we can do. That's a minus. So that becomes the first part of our numerator. So let's go back up and copy that in. And I'm going to put it in parentheses just to offset it a little bit. So it was negative 2x squared minus 4x h minus 2h squared plus 3x. This was from distributing the 3 in the linear term plus 3h plus 1. So that whole thing is f of x plus h. Then we have to do minus the original function. Now I'm going to go ahead and just distribute that negative right away as you do minus this f of x expression.  Now if you did your algebra correctly, a lot of terms should cancel out. If they don't, you did something wrong. So the 2x squareds will cancel out here and here. The 3x's will cancel there and there. And then the constants, the ones cancel. So notice all you're left with is a minus 4xh minus 2h squared plus 3h. So let's just rewrite that so you can see pretty clearly what we have. So now you're at the point that you have a limit to evaluate algebraically. Of course, you cannot substitute h0 in for the h because of that denominator. So we are going to have to algebraically manipulate it by factoring out an h from the numerator. And then your h's cancel out. So now you can substitute 0 in place of this h in the numerator. When you do that, you're left with a negative 4x plus 3. And that is our general expression for the derivative of this original function f of x. So what this is giving us is an expression that will help us obtain the derivative, the slope of the tangent to the curve at any x we were to substitute in. Now the follow-up question for this was asking us to determine the location of a horizontal tangent to the curve. We'll think about that. A horizontal line has a slope of 0. This derivative expression here represents the slope of the tangent line to the curve. So if we want the location of a horizontal tangent, hopefully it makes sense that this derivative, this expression that's giving us the slope, has to be equal to 0 because we want it to be a horizontal tangent. So all we need to do is set negative 4x plus 3 equal to 0 and x ends up as three fourths in the end. So it's at three fourths, the x value of three fourths, that we will have a horizontal tangent to the curve. So let's go look at that graphically and we can confirm that is correct. So here I have our f of x function under y equals. So let's take a look at the graph.  So the horizontal tangent to the curve is going to occur at that maximum. That's the only place you will have a horizontal tangent. So let's go ahead and find the maximum and I'll bet you it happens at the x value of three fourths. And we'll just use our calculator features to do that, to find the maximum. So let's see, go a little to the left, a little to the right. And voila, there's your x value of 0.75, three fourths. Let's look at another example. Now this one we are given the specific point at which we want to find the slope of the tangent line. So in this case we can use the second limit definition. So the three is the x value at which we want to find the slope of the tangent. So we're trying to find f prime of three. So remember the way the second limit definition goes. It's the limit as x approaches c. So in this case it's x approaches three of f of x. So that's the original expression minus f of three, f of c. So that's going to be minus a negative five. So of course that becomes plus five over x minus three. Once again you have a limit that you're going to evaluate algebraically. So in the numerator, so we are going to have to factor this. So it factors as x minus three times x minus three. Then that's how the x minus three in the denominator is able to cancel. Of course that's where the quote unquote problem was, because you were not able to substitute three in place of that x in the denominator. So these cancel. So now you can substitute three into the x that remains and notice you get zero. Now think about what that implies. Since we found the derivative to equal zero, think about what kind of line has a slope of zero. We just figured out where it is that this particular parabola has a horizontal tangent line. And it's going to be at that point three comma negative five.


The Cavalcade of America presented by DuPont. Ghost town. City of illusion. Wind echoing in the yawning pitheads of abandoned mines. Wind shrieking among shadows that once were miners' shacks. Millionaires' mansions, saloons, an opera house. A fine gingerbread hotel. This is Virginia City. Ghost of the gutted Comstock load. The biggest bonanza in history. Tonight on the Cavalcade of America, DuPont presents the story of the legendary Comstock and of Eileen Bowers, the woman who played so great a part in its making. Our play is adapted from Vardis Fisher's great new novel, City of Illusion. Our star is Agnes Moorhead of the Cavalcade players portraying Eileen Bowers, the queen of the Comstock. City of Illusion. It is the 15th day of November in the year 1903. A carriage draws to a halt before a crumbling and deserted mansion. And three old ladies dressed in black enter the house. You sure this is the Bowers mansion, Mr. Watson? Yes, and that's their pictures. They're still hanging over the metal. Mrs. Bowers. That's Mrs. Bowers. And the other one must be Mr. Bowers. She must have been real handsome. You ladies didn't know her when she lived here. Oh goodness, no. She was just another lonely old lady living in a hotel like ourselves. Except... Except for her memories. Well, no. No, it was more than that. It was this place. She made us feel it was really important to bring her back here when she'd gone. She's never really been away. None of them have. Yes, they don't call them ghost towns for nothing, ma'am. You mean you think... in this house... She always said she'd come back to live here. And when people say that, they usually do. Even after they're dead. They're gracious. Believing in ghosts at our age, Matty. I'm surprised at you. Let's look at the rest of the house. I'm still here. But not just here in this mansion. I was queen of the Comstock. Look for me over the hill. Sagebrush covers the stones. Pat-brat shelter there. But what you see is illusion. Look behind the veil of time and you will see us as we live down there in Virginia City. There were others there, too. Listen. Howdy, Ollie. Mr. Bowers, where have you been? I've been looking all over town for you. Wells Fargo Post come through. Sagebrush has found close to death up on the pass. Does Ethan Grosh know yet? Found Ethan dead in his cabin. Run a pick into his foot. Game green set him. Mr. Bowers, did the Wells Fargo man say whether they found any papers on his air garage? No, just a letter. He took it on ease. Then we must act quickly, Mr. Bowers. What are you driving at, Ollie? Mr. Bowers, I think the riches of Sun Mountain are about to be revealed. We're going to be rich. We must make plans. Rich? What are they? Here comes old Comstock with his ears wide open. No need to shuck sand in, Miss Cowan. Old Pancake knows all about it. Mr. Comstock, you'll state your business and be on your way. Mr. Bowers and I are discussing our gold mine. Gold? Then it's gold. Mr. Comstock, have you jumped the Grosh Brothers' claims? Madam Comstock Mountain, previously known as Sun Mountain, has always belonged to me. It was I who mapped it. I who drew the chart. You mean you stole Ethan Grosh's drawings? Miss Cowan. Now, ain't you ashamed talking that way to an old man?  Miss Cowan, I wish only to be loved and remembered. I'll give Comstock Mountain to whoever wants it, but this place will always be known as the Comstock. Now, you mark my words. Mr. Bowers and I want only our 20 feet that's already staked out. You're welcome to, Miss Cowan. Thank you, Mr. Comstock, and good day. Oh, yeah, a good day, Miss Cowan. So long, Pancake. Really? How did you come by all this knowledge? I have it in me to know. It lies southeast by south, and then I'll never mind that. We have more important things to discuss, Mr. Bowers. To begin with, you and I are to be married. Married? Wiley. Since we're to be rich, I think it'll be more suitable after we're married for you to call me Mrs. Bowers. Oh, yes, Wiley. Now then, tell me, what did you find up the gulch today? Well, it was a muck up the gulch, Wiley. Quite a piece. A muck? And the Grosh brothers weren't telling the truth. It's not gold after all. You said it was... It's silver, Mr. Bowers. A coin of the realm. I could use a little coin, I reckon. Yes, yes, that's true. Now go and take a bath, Mr. Bowers, and we'll discuss our wedding. And still, Miss Cowan. Now brace yourself. There are again that table, so as I can get a purchase on it. That's it. Now I try once more.  That's the best I can do, Louisa. Then I'm afraid there's no use, ma'am. I'll never be able to get you into these courses. Nonsense, Louisa. I've been married twice before in these courses. Filled over this wedding as well. Well, hold me a tight, ma'am. Here we go. Huh? Mr. Bowers, why aren't you dressed for our wedding? Oh, it's these hip-pointed shoes, Eiley. Care to wear my boots, maybe? Not to your wedding, Mr. Bowers. Here, I'll help you get into them. All right, Eiley. Mr. Bowers. Yes, ma'am. Did you use the hot water I sent you to take a bath? Oh, sure. Sure, I did, Eiley. Mr.  Matrimony is a holy estate and not likely to be entered into but in fear and trembling and abiding faith. The union of man and woman in the bonds of matrimony is indeed an event worthy of the designation of holy matrimony. For in the words of the Apostle Paul... Mr. Reverend. Yes, Miss Collin? That's a very pretty speech and all that. But Mr. Bowers' feet are hurting him. We just want to be married. Oh, yes. Yeah, I see. Very well, then.  And so the curtain falls on the first act of City of Illusion brought to you by the State of America by Dupont, maker of better things for better living through chemistry and starring Agnes Moorhead as Eiley Bowers, the Queen of the Comstock. City of Illusion, the fabulous 60s. It is Eiley's wedding day. Gosh, Eiley. You and me married after all these years. I'll try to make you a good wife, Sandra. I guess I sure am lucky, Eiley, being hitched to a smart lady like you. Do you love me, Mr. Bowers? Oh, of course I do, Eiley. Then you may kiss me. Oh, Eiley. Eiley, hope so. I hearted, Jen. I heard you folks got hitched. Congratulations. Well, see you later on. Oh, what's your hurry, Jen? Gotta get up to Gold Hill. Got my claim. Got your claim? How come? Look out over the mesa there. See you on the hill, Sandy. I don't wonder what got into him. Mr. Bowers. What is it? Look out over the mesa there, like he said. Well, look, darling. It looks like a dust storm coming up. That's no dust storm. Look again, Mr. Bowers. Eiley, but don't seem possible. It's like I told you, Mr. Bowers. They've heard we struck it rich up here, and they're coming up to steal what they can. Like a swarm of locusts. Far as you can see, wagons and horses and mules. And some of footnets. Well, there must be thousands of them. Easily, Mr. Bowers. And this is only the beginning. Sure, Julie. Hey, Joe. Now, Sam, look at what you're going and done. Killed another man. What'd you have to do that for? I didn't like the necktie he's wearing. You keep this up, Sam Brown, and you'll be wearing a necktie made out of rope. Ha! This is the... This ain't Boston. This is Virginia City. Oh, yeah? It ain't enough these poor fools getting killed in the mines every day because of them rotten timbers. You gotta come round shooting them in the time off. Well, you ain't doing it in no place where I work, Sam Brown.  All right, sister. If the black crook cellone don't like my silver, I'll take it somewhere else. Rattle snake.  Oh, well, Julie, you see, Arleigh, well, she thinks you got too many gentlemen, friends, Julie. Ha! That's a good one. Hey, what's going on over here? A convention? Howdy, Julia. John Mackie. I heard you stuck it rich. That's right. I aim to own every mine in the Comstock before I'm through. Oh, that's the kind of talk I like to hear. Oh, boys, I'll see you later. Cellone Sandy. Cellone, Julie. Oh, Mr. Bowers. Yeah? Tell your wife I read her statement in the enterprise about holding on to a mine. Tell her I wish her luck, but I aim to have it. The wizard that delayed me, Arleigh, my cough got bad and stopped for a little brandy. And there wouldn't any brandy do with the brandy and the black crook cellone. Well, who was there, Mr. Bowers? Sam Brown. Who did he kill? What such? Good riddance. And Julia Bullitt. Did you talk to her as well as the town murderers? Now, Arleigh, she does a lot of good. Dr. and the men that gets hurt in the mines, feeding the men that's down their luck. That's what she gets from them warrants. However... Well, Arleigh, you ain't yourself lately. What's got into you, honey? You didn't use to judge so harsh of folks. It's a time of judgment. I feel it in my bones. Well... 10 o'clock, Mr. Bowers. Time I was in bed. Wake up, Mr. Bowers. Come on. Oh, it sounds like thunder. In the midst of a blizzard, sound sense, Mr. Bowers. Getting louder, Arleigh. I'll go and... No, Mr. Bowers. Don't open the door. Look out the window. Arleigh, the snow's like a regular avalanche. The whole far side of the valley's covered. It's still coming. Close the storm shutters over there, Mr. Bowers. Please. Yes, Arleigh. Now then, kneel down, Mr. Bowers. We're going to pray. Our father, my daughter, Howard, because of my name, I can't... Tabin is buried, Mr. Bowers. But the snow is soft. We'll have time to dig ourselves out. All right. I reckon I'd better get the shovel and start in. No, wait. Just a moment, Sandy. In case one of us don't get out, I want to tell you a few things. First of all, I've laid in some provisions. Flower and beans are buried under the floor. You know this was going to happen? Sandy were to have a child. So I used the money for this instead of starting on her mind. Maybe it can't eat silver. Now it's for the others. Oh, gosh. I bet they never stored a thing. They stored plenty of whiskey. You can bet on that. But they're not to know about our hiding place. You want to stand, Sandy? Yes, I mean. Supper's ready, Mr. Bowers? No, I... I can't eat nothing, Ali. Think it of them people. What happened?  Get the flour all right out of the bags, Ali. That's how starved they were. Don't shout, Mr. Bowers. You'll wake our baby. Poor old fella. How is he, Ali? Don't seem to gain much. It's a hard time to be born in, Mr. Bowers. Ali, do you think we've done right? I mean, keeping our food store a secret. We had our own to think of, Mr. Bowers. Hard woman, Ali. I had to be to keep it alive. Now listen, Sandy. When the snow falls, we're going to begin digging our mind. Then we'll build our mansion. The silver doorknobs, like you said? Silver is a good metal. It has a good shine. You'll have to wear different clothes when we're rich, Mr. Bowers. We'll give a grand party when we open our mansion, Mr. Bowers. But Julia Bollett will not be there, nor Sam Brown, nor any of the other thieves and murderers and fancy ladies. Who do you get to come? Only the best people. Someone has to set an example for Virginia City. That'll be us. Such a lovely party, dear. And you mention. It's amazing. That silver all came out of my own mind. It was worked up by artisans in Paris, France. Imagine. And those portraits of you and Mr. Bowers. Who was the artist? Well, Ali and me never could pronounce them Italian names, but he was a real art artist, all right? Well, I'm sure he was. Mr. Mankey? Yes, Mrs. Gellar. Have you seen these exquisite portraits of Mr. and Mrs. Bowers? Well, not bad, Ali. What are you being not bad, Mr. Mankey? Them are real hand-painted pictures. Mr. Bowers, stop pulling at your car that way. Yes, as I was saying, Mr. Mankey. Mr. Bowers and I intend to establish a real society here like they have in Paris, France. Only the best element of Virginia City will be in violence. Yes, I was wondering where everybody was. Well, shall we dance, Mr. Gellar? With pleasure, Mr. Mankey. Jury of Allet. Oh, Ali, I can't leave. Got you all tricked out, too, Sandy. I must say you look better in your work clothes. Jury of Allet, you wasn't invited here this evening. I'm not here for a social call, Madam Moneybag. I'm not good enough for your junk shop here. But I do know where your money comes from. It comes from my mind, which Sandy and I staked out legal. Go up there to your mind now. I hope it makes you sick. What are you talking about? There's another case in. They just started digging the men out. Men are dying so you can act like your queen of the Comstock. And it's needed. Hey, love, this man's dead. Get him out of here. That makes 20. We're dying this rat hole where Ali Bowers gallivants around Europe. What's that, you sir-luck McCoy? Oh, it's you, Madam Moneybag. Look at these men. This is your fault, Ali Bowers. Are you crazy, Jury of Allet? What are you doing on my property anyway? You go ahead and say it. It's not as bad a word as murder. You dare say that to me? Yes, I dare. Oh, you aren't the only one. Others are worse, maybe. But they start with the rest of us. The winter of the snow slide.  We all knew you had food hidden away.  I had my child to think of.  Not anymore. He died on the boat. I'm sorry, Mrs. Brown. But I'm not. Not really. Oh, I don't know what I am. Here we sit on top of enough silver to bring on the millennium. And what do we see? Corses and broken bones. Your husband is very ill, Mrs. Bowers. Shall we talk about it alone? Well, don't you leave me. Mr. Bowers can hear what you have to say, doctor. It's his lungs. There's nothing I can do. Let me know what happened. Good night, madam. But, doctor. Harley, something I got to tell you. About the mountain. Better try to rest now, Sandy. Tell me in the morning. No. The morning will be too late, Harley. I made a mess of things. I was drunk, and I signed some papers. You mean the mine is gone? I'm afraid so. We ain't even rich now, Harley. It doesn't matter, Sandy. What'll you do, mom?  Don't fret yourself, Sandy. Always be queen of the Comstock. Always. So once to be rich anyway. City of Illusion. Virginia City. For there was a man even to the Comstock. At the turn of the century, to a ghost of a mansion, three old ladies dressed in black have brought her ashes home. Here in the garden, it's overrun with weeds. This is where she said she wanted to be. Very well, ladies. Let us pray. Unto Almighty God, we commend the soul of this our sister. And we commit her body to the ground. You can't bury me. You brought me home, but you can't bury me. The stables are empty. The mansions of the hill are empty. The train won't run anymore. The tracks are torn up. There's no city now. Just empty building. Just deserted money. But I'll be here always. Virginia City is mine. With tonight's Cavalcade of America, what brings you a story of the most precious thing in the United States today? It is time, hours, minutes, precious seconds. The tools with which we mean to defend democracy are worthless unless they come to our hands in time. The race we have entered is a race against time. In the imperative business of clipping an hour off an assembly line, ten priceless minutes from a process, chemistry is proving itself a friend in need. Production of vital parts for trucks and planes has been speeded up by new DuPont electroplating processes which give protective or wear-resistant coatings in much less time than was previously required. And another example, most of the rubber we use in this country goes into automobile tires. And tires for trucks and planes are rolling off production lines faster today because of DuPont rubber chemicals that hurry along the vulcanizing process. And something you might not connect with defense is the bleaching of cloth. But of the millions upon millions of yards of cloth required by the Army and Navy, cloth that goes into everything from uniforms to dodgers for the bridge of a destroyer, virtually every yard is bleached. In the past, fabrics were bleached one batch at a time. A batch of cloth was put into a tank and boiled for hours to remove the impurities. Then lifted out and put into another tank where it was bleached for many more hours. After bleaching, it was rinsed carefully and then removed. Then another batch was put through the same cycle. Was there any way this time wasting process could be speeded up? DuPont chemical engineers said there must be. And out of their research has come the greatest advance in bleaching in almost 100 years, the continuous peroxide bleach. Suppose a manufacturer gets an army order for 200,000 towels. He weaves them into strips with hundreds of towels in each strip. At the bleachery, the strips are twisted into long, loose ropes, and the ends are fastened together, with endless belts that can travel over wheels or rollers like any other belt. A round and a round goes this belt in tanks of a bleaching solution perfected by DuPont, never stopping. Bleaching that used to take days now takes hours. Another time saver is dynamite. Far down in the earth, along the glistening veins of coal mines, special DuPont dynamites loosen tons of coal in an instant. Other dynamites break enough iron ore flash to keep the mouths of giant power shovels working for hours. And special DuPont dynamite used in seismic prospecting save time in locating new oil wells. Thus, in scores of ways, minutes and hours of the precious ingredient known as time, are conserved by the chemist who brings you better things for better living through chemistry. Next week, the Cavalcade of America presents the distinguished American poet Carl Sandberg in the first of two new contemporary programs called Native Land, starring Burgess Meredith of stage, screen and radio. Native Land is America today and the effect of our great traditions of the past on this modern democracy of 1941. We hope you'll join us at the same time next Monday night when DuPont presents Native Land on the Cavalcade of America. At this time, may we remind you that you can join our nation's great defense effort by buying defense bonds at your local bank or post office. In support of Agnes Morehead as Eileen Bowers on tonight's program with the Cavalcade players with Frank Reddick as Sandy Bowers, Carl Swenson as Comstock, Betty Gard as Julia Bollett and Kenneth Delmar as the narrator. Tonight's play was based on the novel City of Illusion by Vardis Fisher published by Harper and Brothers and the original musical score for Under the Direction of Don Worries. On the Cavalcade of America your announcer is Clayton Collier sending best wishes from DuPont. This is the red network of the national broadcasting company.


Ya, this is why 254 and this is why in the morning it is interesting that you are here with us and we promised you we are here for the conversation of the day, conversation of the day being what men look for in women, what men look for in soul mates. And my name is Senior Comrade Oguda Bradley Sankara Kaisu or you can summarize it as Sankara Kaisu. It's at Sankara Kaisu in our social media platform and we are live from broadcasting house Nairobi. So you are welcome, you are right in time. You are going to tell us through our Facebook pages that is the way we are going to converse with you. You are going to tell us, you are going to comment and tell us what you think men look for in soul mates. And with me to help demystify this are two interesting people and we are starting with the lady. It's lady and gentleman and lady is Natali. Natali, how are you? I'm fine, thank you. What's your other name? My name is Natali Kakai. Natali Kakai? Yes. Everything about you, what do you do, your social media platform and everything? I'm a journalist in training. I go to Masenno University. My social handles are, I am Nat Kakai. Oh, in every social media platform? Yes. Nat Kakai. Which level are you now? Which class? How do you say? Which year are you? I'm in third year. Oh, you're third year. You're almost there. Almost there. Salute, I said something in the morning about Masenno. I'll tell it to you again. And we have a gentleman in a colored tie. I told you the first time we met that you resemble somebody. Do you remember? Tell us everything about you. I go by the name Muchi Sambulmuchi aka Mr. Elmuchi. I'm a simply Muchi Muchi. Producer, director, businessman, entrepreneur. The ladies man. But most important of all, the son of the most I manje. Blessings kwa ke. How do you call yourself ladies man? The ladies love me and I love them too. So it's a mutual relationship. How do you know that ladies love you? Love, so let's move on. Actions too. Oh, they act in a manner to show that they love you. And in fact, that brings us to the topic of the day. Let us start with you. Oh, what do you in turn also love those ladies? So much. What do you love about them? What informs what you choose to love and do not to love? For you, give me your top five list. My top five? For me, number one, as a self-sufficient woman, by that I mean, mon am keniza jismamiya. You can do one, two, three for yourself.  And there is the aspect of understanding that in growth and in doing what you're doing, you need each other with somebody else. Number two, Number three, the face. Niki amka subuyu kukandu hangu na deitu kumbi amungu. Thank you Lordy. Bara katwi ya ni manze. Ata na ring katwi ya ni mbeli ambogi na wapi wapi. Number four, the love for God, yote ni nafa kwa number one. Lago penda mungu tu bili, mipi amina penda mungu tu bili, siyoko wezu wezu hake. Number five, number five, ambitious like, poverty is not our portion. So, akuna vision, dreams, na goals, ya komo kana to know a way of how to do it. Kabisa. Did you realize that your number two and number three are all about looks? There was attractiveness, beauty, usafi. They are all about physical being. Very, very true. So it's like you are too much in this look. I think it's cool really. Ya, I love, I mean it was a soft life manze. I love it's nice, I love it's set to ya ni na. It's a tunnel for me, niki kuja kwa kuna unanuka. Mini meji take time, asubu ni mihoga, ni majipa kapaamba, ni ni ni ni ni. Nia, fubu, ya kama msupa. Una fanya iso zote, yaura manze. Kabisa. I thought men are not supposed to be. Man, manana fakwa kuwaka once a week or twice a week. Ya, we go zote. Una waka kilo siku na di. Ya siku. Mbu kiangu adiunita msupa, manze, jua tvil na penda maji yolo over the place. Okay, that is it's trusty. Don't let us come back to Natalia. Natalia, you've heard what Muti is saying, We are really unreasonable and realistic We need toiter give up People they have very unrealistic expectations They want a career woman who's earning a lot Career women, I do have a job They are busy people At the same time one in the last two One said, I just want house help So, I think for what he said, it's very reasonable and realistic. What was his number one reminder? Number one was understanding. Understanding. Do you think ladies go here in understanding? Pardon? Do you think ladies go in understanding? Are ladies understanding enough so that they meet the expectation of men in their understanding? Can you understand me one name?  Oh, that can make you leave me. Yeah, women can be understanding. Inna depend na villa nakupenda.  So, na then ya you can understand him, but in real sense, kumampia opportunity. So sometimes you make demands because you want, unata kujikata. So you're looking for a reason. But that's so unfair na tell you what you think. That is so unfair. Also think women in a depend na mtu. There's some more understanding and there's some more not understanding. Need to be lame to ako. Okay, Muchi, I'll come back to you. I'm coming to you next.  I know you saw it, it belongs to same group. There's somebody who advertised for a vacancy that he wants, finally wants to marry. So these are the qualifications he wrote. I want to read everything like I have finally decided I'm gonna get married. Applications with the following requirements should apply. One, should have natural air. Two, no makeups. Three, satisfactory income source. Backed by a bank statement. Four, a great cook with experience of at least five years. Five, no tattoos. Six, should be black without any trace of bleaching or chocolate in complexion. Seven, able to sustain an intellectual conversation as well as post some sense of humor. Eight, attach a written recommendation from at least two former boyfriends. And then nine, extensive knowledge in public relations. Ten, fluent in at least one local language that is vernacular. Then Kiswaili, Sanifu and English language. Then 11, well trained in home management and hospitality. 12, must have a great character of a wife material. 13, no previous record of relationship cheating. And you remember you have to attach two boyfriends' records there. Then recommendation letters from former boyfriends. Then number 13, no previous record in relation to cheating that I've said, the most ability to make a man successful. Number 15, must not practice sequinism. Number 16, must be slim, coca cola bottle shaped and tall and averagely tall, chabi and not shapelessly fat. Number 17, almost through. I think that is the last. Must be a career of God's presence and then lastly must possess angelic voice. So he said those were the end of qualifications. All applicants must be advised to send the applications. Before the end of August, that is by August 28, 2021, attach the following birth certificate, health certificate, passport photo with a white background. You know this is really, it was true. All educational certificate and only selected candidates will be contacted. NB, this is an equal marriage opportunity. No corruption. Any faking will lead to automatic disqualification. Good luck for more qualifications in box me. So he posted it somewhere and he wanted people to inbox him. You are opinion on this application. First of all, I feel everybody has a right to make a list of what he or she wants in his life. But there is the aspect of we treat marriage and love like a job. This is your job.  So yes, it's good, bas buatate sekasa na juku kikosa kupatao om tu vileo umiliiista upo. What happens? Kuchu, kwa kiviye? One woman without all these qualifications. One with all. Do you think she exists? I'm open minded to say kona posibility hana exist, but the possibility of now you finding her is not the challenge. Because hana kwa yuko. Kwa kiviye kwa patoa na naya we are 7 billion people in this one. Labda ko yuko yuru koe Koki jijifula ni tuapu metulia. So na wu kukenya kanaeru mitu liya tu, so the link up is the challenge, na wu leta kwa wupendo. Ya, the people who are professional linkers, their job is to link people, to link people. Do you think they should come in in such a case? Do you even approve of their job? For sure because I heard a lot of people who ame patana na the marriage is successful through that, but badu kuna ase ame patana na the marriage is unsuccessful. Last but not least, kuna aspecti ya karakta. Karakta hui badili, karakta is who you are when you are alone and what you do on an individual basis. So your character and my character must be able to match. Leave alone the list, leave alone everything. Ame able to accommodate who you are as a person. Niki amo kukupenda na kukupenda niki chikoyo flaws and your good. So niki kabisa. Muchi, thank you for being honest and being plain. You are coming to Natalie. Natalie, do you think this woman exists? And if it exists, if you think she is you, just be frank and tell us. That woman exists and thinks she is me. Tell us. Okay, I am sure that woman exists. I am not interested in being qualified for that. I think kwa yo listi ake, there are some things which is said that it's not black and white. It doesn't have a definite definition. Saka slakunizim. Maybe anaman nisha kutegneza nilis, alafu, for someone else, slakunizim ni nitu anahangauti na wa baba. I don't mind having a list but if you are having a list, are you okay with me also having a list? It's a cool thing. Because it's an equal marriage. It's an equal marriage opportunity. So, kwa yo listi ake na demandi so and so this and this in you. I also want to be able to have my list and demand this and this from you. But the way such a woman should also choose a man should marry her because she has a lot, she has just a lot. It is not easy to have good figure, good brain, good everything, good voice. And then somebody puts application and maybe this person doesn't have these qualifications. And you want to get such a virtuous woman. Have you read it's called which Bible verse? Which Bible verse talks about a virtuous woman? Proverbs 31. Yeah, Proverbs 31. So, do we think we should use Proverbs 31 as a parameter? Man should use Proverbs 31, we start with you as a parameter for the person who should get married to. It depends. Do you walk in the jungle, do you deserve a Proverbs 31 woman?   You see, we have two characters in the money. I say I have good character and all that. I love when I do kujapu and maybe I will want to do penda vinta. You can change me. No, thank you. I'm not in that business. That's another aspect here. Aspect that you can change me. There are a lot of men out here looking for the second mother. Yes, for the second mother. We live in a community of here and we have a community. We didn't end up in a kutumio or a kivibaya. We're not clear. We live here but are calm in the name of it. So, deep down we have to claim these other sides. That's a challenge which we will be addressed. Nicole, kwa kwa kwa kwa hika kwa mizi niisil yun amu. Sava ya haza batya hani hana. Nakuna na kawaya niisil kwa hika. Nisha, na kwa kwa mizi niisil ala kwa hana. Bili ya kwa kwa kye hani? Sava ya hania kwa mizi niisil kwa. Lama na maizipu eno na yun mili kwa kwa kwa hana. Niisil, nisha, kwa kwa kwa hani? Igo, na kwa mizi niko yun mili kwa. Nisha, niisil niko. are being emotional for each other being the mental inspiration a yamka subunapatia crizaw na kona kusaka a kenda kupigia hota wakai see w last matia ifa nya kuna wamii, kuna options kumise kuja kuhaya kuhilipigi ye kwa kwa opi nunapata kwa kwa hensa ni wantatu kwa hensa ni kwa kwa ni and i agree with him yes ok women kutu nuna kukilipa but also the guy should be able to do so Una pacha maybe the wife is heavily pregnant na maybe bad one end a job Uo me kwa kwenyumba the whole day Anakuja home of Yombon Chafu on Japi Kani and ya already she's she's exhausted Sisa tutaka kwenyumba Chafu ju wa wezi clean Awezi nirumi alek nime choka you see and that brings me to the point of sacrifice kwa relationship lezima kukuna sacrifice you see even in that list so you can find maybe he finds that woman but kitu too much maybe maybe when a li kwa abadi a li ya kata too and then akafka point aksama akat change katlifika changing but this a tata too bad do you kwa but she's the perfect woman sacrifice so we should not judge like by only physical but you you brought up an interesting point of maybe helping in the house man African man African man ya do you call that toxic masculinity you can do everything in the house like I've been raised mainly by my mom kila mtu is equal in the house so kama ni siza na usha viyombo lewa nimi na pika kama ni ya na pika nimi na usha viyombo kama ni viyombo kuna nyuomba there's this whole aspect of the responsibilities of the house so akutu train aki sema kuna kazi ya wana wake na kuna kazi ya wana ume kazi ni kazi na zote zineza fanya it is better good than said than done do you think you will do that to your wife for me I feel am at a position with God at an elevate so lazman nifani yo job nita kuna finances ya zini cleaning nita jika kufanya cleaning lady na kama kufanya that oh you'll have so many workers that will handle everything my wife aspect ya kwa pregnant she needs to be taken care of the most so anything that I can to ease of the burden ya ya aki wana the pregnancy I should undertake that and do that so how do we achieve that kujua kuna kazi fanya fanya so ease of kazi ni gani 1 2 3 4 mimi niza fanya yes but almost occupied in other things na visions na nini so I delegated to someone who can handle that I see that is but you know there's also always a starting point you can't start but kama pia mimi ni maskini na sina yo do si maindu kusha viwambo si maindu kupika si maindu kummasaj miku you know it's different when it is your daily duty and when you do it out of passion maybe kuhela Natalia says that it should be your duty you know I'm cooking I should be cooking when African men think cooking is for women we're talking about African men squishy with the west who affect kachetu and all that and also just the growth and there is the equality and all that you'll find that even African men squishy they are open to helping the house but kuna wengine too bad but I'll go back to that point I understand ya mimi mine zakuwa ni miyamua si tawai yosha viwambo si tawai pika si tawai fanya 1 2 3 so the moment na approach msupo aangu before to aane mi na maambia miku na kazi siyezi fanya no si expect ni fanya time marriage me anza be clear be open with it do the moment ni mi fika asia nziku create certain expectations which you can't fulfill I know situation where somebody used to do such jobs no and to an extent that waifana tokainya na zema wei baduma tuji kali siya pona wana jwa akuja pikuwa ayon kiburi sa ayon kiburi na ifai love love it's a structure ya kuna respect ya na ifanya jio upendo so kama ni jio upendo sphe kuja na arrogance ya wei beb kuna viwambo apa uta osha mani uta osha okay back to that was by the way back to what we are discussing today there is a small chance here for men that is on the men's priority list of choice for men looks come on top looks men mostly selected that is on that research where 68,000 people interviewed men mostly selected attractive bodies that is 70% and attractive face at 68% the most important features of question partners they say nice eyes and all that so there are other things that came that is confidence confidence at 44% kindness at 38% and 36% came intelligence I want you take you take on this research this is science now I want you take on this research Natalie it's reasonable because when you see someone for the first time you don't see their intelligence until you interact with them so it makes sense that the first thing that they will look at is the looks I'm not saying that your priority is looks or anything like that but the first attraction when you see someone is how they look how they present themselves how they carry themselves that's why it will be at a higher percentage some people think Natalie that looks and brains are inverse that if you have good looks that if you have good looks you can't have good brains if you have good brains you don't have good looks it's simple because you know how to do it you know how to do it that's a false notion it's a mentality that was brought by movies when you watch movies the beautiful blonde is always there it's a mentality when you watch movies but it's possible that somebody can be cute and at the same time brain and you are an example here the perfect example when it's true you are doing that also I love it but you are not saying that you have a good look because you are a professional you are very intelligent you have a good look but you are not saying that you have a good look so to add on what she was saying there is an aspect of human nature and for us human nature is we are programmed to be biological beings by that I mean we are reproducing beings we are reproducing this earth so in the aspect of reproduction when you are looking for someone who is healthy you can provide security food praise person in your life so when you are in health like you are physically fit to a certain extent it's an automatic turn of when you are in health 100 men are in a room but 100 women are attracted to one 100 women are attracted to one because it's the physical part first and then you will get to know the person Muchi before you we get of you the other factors like good organization adventure and then cooking and the clean house able to converse try new hobbies, want friends is it is cooking a factor for you ability to cook it's not a must kwangu jumi kwangu like sike to finely Mr. Teki waifu na kufanya jabo me aq waifu na pende a ni dunze a ni kufil naka king a ni pati e the feeling that I'm supposed to get from a woman but kiki taki kwaifu na kufili kwaifu na aq za job mtm ni bata to kwaifu upo na. Muchi, our time is much spent If we want your final remarks on this, tell men on what they should have in their priority list. Ni wukwem janja in this life, desire one will ever, we are changing people. So in this world in the aspect of change, wujue kila siku we choose to feed something, either physically, mentally, ama spiritually. So whatever we decide to feed is whatever we are. Meku su mechaguwa na feed the same thing piawana feed, na akona the same ambition na the same goals piawana da ikuachiv. Wukwia iwa ato wacha nina listi ama nina mingi, karakta tu yeh nyu uta accommodate witu mbaya wicha the flows, nita penda zile zi niziko pohazaidi na zahidi nizia ribuku grow from that. Na akuna maisha perfect watu cha tichiana, watu upendaana, na in the end of it, kamuna penda msi, uta stick to na yad the end of it.  No man, I'm solomoni. Most men are this way. When they talk, they talk a lot of sense, but when it come to practicality, it's another thing, another thing altogether. I have too much love, too much love. Give us your final remarks, Natalie. Ukiwa na expectations, it's okay to have expectations, but you should also be okay with the woman having expectations if you're having expectations. And also sa zingina expectations are what destroy a relationship. So you should know and decide whether you're willing to sacrifice and you should know whether you're willing to sacrifice your expectations sa zingina kwa relationship or just let the relationship go. Nice. I had wisdom on set, wisdom. I had brains, somebody who said brains and looks are inversely professional. I've proved you otherwise because Natalie's brain is here and very attractive. Muchi has also said that yes, people are very brainy


So, I think that is the only fastest way that you can use to file a case again what you call your employer in case you need to have grievance and in case you have grievance in any way for you to get off. Most of us will go to this labor court or file cases against our employer. You have to wait until the next question.


Great. Hi, everyone. Welcome back now for the last part of this short lecture series. So in part two, we just finished proving this beautiful theorem by I and Les Feltz-Smith, Hoxson-Hunikey, Mashweed, and Muayama in the setting of character CP. And the main point to prove this theorem was to, well, to prove it in character CP, as Hoxson-Hunikey did, the main point was to cleverly use tight-closure techniques and to use the pigeonhole principle. So we use tight-closure because it's easier to prove that something is in the tight-closure of the power than it is to prove that it is on this power. And then our ring was right there, so ideals were tightly closed. The pigeonhole principle really gave us this containment when I take not an N, but when I take a power. And remember, the idea was that you reduce to the local setting and realize that you really just want to prove something about an ideal generated by each element. And you want to see which Frobenius power does the HQ power live in. And we noted that we could do this sort of slightly better containment by sort of using the full power of the pigeonhole principle. Now, I want to point out that if you follow the exact same proof we did, we sort of applied the pigeonhole principle K times. So I take some K, that's at least zero. So when K is zero, this is exactly the same that I just showed you, right? The first symbolic power is I. So there's a kind of souped up version you can do by kind of repeatedly applying the pigeonhole principle to kind of keep getting things in the Frobenius power. So you're kind of in the local setting, this thing will turn into a power. And, you know, it's essentially the same proof. If you want to do it carefully, I'd recommend first go over what we did today and try to do it on your own first, right? Reduce to the local setting, see what it is they want to prove, try to apply the pigeonhole principle. And then you can look at my notes where there is a complete proof of the statement. But I want to write it out because this is going to be important in this second half. So I want to talk about how do, how do we, one can improve this theorem. And there are many ways to sort of go, right? One could think about non-rightly remains. And there's lots I have planned to comment on about that, that we might not have time for. If we do have time, I'll do it at the end of this lecture. If not, you can take a look at the notes. Or remember, there's this much, much longer version, 200 pages in my website that you can look at. But I want to stay in this setting a regular range instead. And I want to talk about whether we can try to write tighter containment. So remember, my goal was, give an I, I want to find and a power, I want to find the best possible symbol of the power you can put in it. And we prove a positive statement. We prove a certain containment holds. We did not say that was best possible. So let's look at some examples. On Monday, we thought about this square free monomial ideal and three variables. So here k is a few. Well, there's theorem. Well, what is the big height here? Remember, we wrote on a primary decomposition for this black light. All the minimal primes, all this is your primes of height two. So the big height is two. So the theorem says the two n symbolic powers can be in the n power for all n. So for example, it says the fourth symbolic power is contained in the square. Okay, can we do better? Last time? Well, okay, two times to go on Monday and lecture one. We saw that the second symbolic power is different. And that's not contained in the square. So if you could do better, the only other possibility would be considered the third symbolic power, right? I already know four words and two dozen. Three is what's left. And you can check this. This is an easy exercise that the third symbolic power is contained. So sometimes you can do better. You can even do better for primary. So for primes, this big height is just the height. And so for example, if you look at the prime p that defines the curve, t to the fourth, t to the fifth, three, four, five. So I really mean like the kernel of the map. So it's the prime and three variables that defines this curve, right? So if you like, it's the kernel of the map to kt, that's anything to be valuable to one of these powers. That's that's what I mean by this. Well, it's a really cool exercise to prove that the second symbolic power is p is not the square. And I wish I had time to do it. It's one of my favorite examples to do. Never seen it. Think about it or look at look at my my my longer notes. We'll see we'll see this and carefully. But anyway, this is a prime of height two, right? It defines a curve in three dimensional space. And so the theorem again says two n symbolic power is contained in the n power for n. And so in particular, again, the fourth symbolic power is contained in the square. It's a cool exercise to prove the second symbolic power is not contained in the square. It's different. But once again, we can improve this a little bit. And write three instead of four. So I'm going to keep first heard. So before he and Mel Hoxer had done that poof, we just covered when he heard of iron lasso spells and sniff words. This is a very first question he asked. Well, first he was surprised. It was a truly wonderful, surprising result. And then he said, Well, you know what? I cannot find any prime of height two in a regular local ring for which I can't get the third symbolic power. Not only couldn't we find it. No, yes, found such a prime in the 22 years since he asked this question. And despite this what I'm going to say next, spoiler alert. And there are good reasons to think about this. So let's see, this is a situation where our big height is two. And there are some things we have, we have seen that are better than two n contained. Right, we proved in frankly, you can do that. So in fact, Harbourn asked a few years later, a very reasonable question. Now I'm back to the setting of any radically ill. Okay, I should be careful here. His original question was for genius ideas in in in phone on the wings. So I'm kind of restating it slightly. But I think it makes sense to consider this this generality.  Can I take the statement from that nice theorem? And could I possibly always improve it? Like subtracting h minus one. So remember, Oxford and he told us, this is true. If you take a power of p, in characteristic p, right, like the pigeonhole principle told us that it would sort of it's, it's, it's a very fundamental elementary fact. So it seems extremely reasonable that you should be able to do this problem. So in particular, notice that when you're in primary care to see two, you get this containment of the third symbolic harm in the square for Bay high two automatically, right? That's that's what this says when you take people to do an n equal to the thing that is strange. And sometimes mathematics is very bizarre. The thing that is truly shocking is that there are radical ideals that don't satisfy this conjecture problems. So the first kind of example was signed by doing each key number. And so there's three algebraic diameters from full ends. These are experts on on points configuration. So what they did was they found a very explicit configuration of points of 12 points into that field. I'm very bad at drawing. And I forgot to prepare drawing beforehand. So if you want to see a picture of the special configuration of 12 points into two, please look into it. If you're more out to break, maybe I can write down what it is. Because it looks very simple. But I'm actually going to write, not just your original code example, but I'm going to write a generalization of it. My hard one in Switzerland. The first current example was published in 2015. And this is a paper from 2015. And I'm going to be over some fields. Okay, I just did not to. Because it's going to be a kind of example to put some bulk on the square, which is true in characteristics. We're going to fix the number and to be at least three. Oh, maybe shouldn't be called in, but I'm really bad at letters, C, maybe. And it's x y to the C minus x to the C, y to the C minus x to the C, z, x to the C minus z. When it takes C to be three, this gives you 12 points. This is in general some special configuration of points. So this is a radical ideal. It defines, as I just said, some points in P2. So that has the height, too. And it fails the third symbolic value. So somehow the magical thing in here is that somehow, even though we have this containment, so by the way, it is originally a receipt but then however in Switzerland, you can do this over in the field, you've got to be careful about what you pick. So notice that, you know, somehow, even though for, for powers of your characteristic, you're going to get this containment, that's somehow not sufficient to get the containment for all the other ends. It isn't forcing to be true for all the other ends. If you go back, and you will, you look at our proof, you know, our tight closure proof in the Hoxer-Kineke theorem, we didn't use this more powerful version of the vision hope but if you try to put it in there, you'll see that there's a true obstacle. Like you can't just, you know, it shouldn't work, right? Because otherwise, this thing would be kind of example, but it really, it really doesn't work. It's sort of a sad, sad way. But in my opinion, it just kind of makes the story more exciting. Now, I will say that despite the fact of now there are other families of kind of examples, all of them are so very special, you kind of have to take some special configuration of points or, or hyperplanes or whatever in some really, really special way. And in some sense, I want to convince you that actually, many ideals do satisfy this, this context. And we're going to prove that a large class of ideals and characteristics need to satisfy this context for all and not just for powers of people. So maybe let me start by saying that hard ones can get your goals for various things. Let me list some and then we'll, we'll prove we'll think about a large class. It's satisfied by minimal ideal. I added the word square feet that is not necessary. But I added in because for us, all ideals are radical. So I should think quickly. It's satisfied by generic sets of points in P2 and P3. So P2 is a result of boxing and hard one. P3 it's a result of the niche. It's likely to MPI. There's no proof. These proofs sort of rely on geometric things that are especially in these dimensions. It's true for some special configuration, like things like metroid configuration, star configuration. So sort of for certain special nice things is all And today we're going to prove what I think of as a generalization of this monomial case. So this is sort of a philosophy that I learned from the recent aspect of core. Whenever you see a theorem that is true for monomial ideals, ask yourself, hmm, is this true for ideals defining efferents? Because I want to say most of the time, now this is me talking most of the time, it ends up being the version of that theorem that makes sense for all ideas defining efferents as a large class of ideals in character CP. And it contains the class of spur-fee monomials. Now I believe you've seen if your if your range a little bit in these lectures, but again, it's kind of the end. We're all tired. So I'm going to just give us a quick reminder, but I'm going to cheat. I'm going to first tell us what is an F split ring? Well, a radius of character CP is F splits. If for being a splits, right, that's what the name says. So, but for being a map, you can think about it as a map of our models. I don't know what notation he's been using before, but I'm writing this to mean the R module that has underlying billing group R, but our module structure is given by the Frobenius map. So this is Frobenius. And for Frobenius to split, that means that there's a map of our modules data that gives a splitting for this. So beta F should be the identity. So when R is F finite, so I guess maybe let me put it like this. So today, so that I don't accidentally say something lucky, all rings are F finite for the rest of my lectures. So remember that means Frobenius makes R into a finite derivative model that's F will restart R as finite. So that's adding F split is the same thing as F pure. You've probably seen the proper definition of the pure A means is Frobenius map is pure. I'm not going to write it carefully, because I think it's easier if you're not comfortable with these things, it's easier to think about the splitting. But I mentioned our pointings of definition at all, because I'm going to think about this condition in terms of a criterion by Fetter. So theorem I think you've seen several times that I want to remind you because it's so important. Fetter's criterion was proved by Richard Fetter as part of the species. This is Sunamal Hoxter and this is in his thesis in 1983.  We take some wrap with you. The quotient is F pure. So F splits. If and only if when you look at your ideal I, and you ask, who sends you into a Frobenius power. You want this colon ideal, so the things that send I into the Frobenius power to be pretty big. So you want it to not be contained in the Frobenius power of M. And you can take this for all or some. This is a typical thing to do. So you can rewrite this to just P if you like. And now you have a very computable statement, like you could put this in Macaulay, for example, compute the colon ideal and decide, oops, whether you're not sorry, whether you are not in the Frobenius power of M. So I think again, I think of this statement as a measure of largeness, right? This is saying this colon is really big. I will say, you know, this also works. If you take a polynomial ring of our fields, you know, you grade it, and you take a homogeneous ideal instead, this also applies, right? So I said, my ring was regular local, but you could also take this graded setting. So in particular, wink, wink, it is okay to use this in the graded setting in the in the problems that you're working on. So in unrelated news, one thing you will prove is that if i is a square free monomial ideal, then our mod i is guess what? So that's, that's an example of such a thing. These guys are contained in ideals that define a few rings by which I mean what I could be quotient by them, I get it.  So for example, the renaissance rings, well, renaissance rings, the polynomial rings are renaissance rings of other afterings, they're all afterings. If you take the generic, the terminant ring, these are all afterings. More generally, if you take sort of a nice swing of the variance, you also get an aftering. So this is to say, this is a nice, interesting class of singularities. These rings that are pure, they sort of have mild enough singularities that you have some control over them, but it's also a really large class that includes lots of classical rings that you want to be interested in. So what we're going to prove is that we're going to prove, so we'll show is that our mod i is pure. Okay, I might as well write the theorem now and then I'll just move it to the next stage. Theorem. This is the theorem of myself with pregnancy, published officially in 2019. I guess I should say that ours regular, it has characteristic p. It should be a finite. i is radical of big height h. And if our mod i has nice singularities, if it is at pure, then it satisfies carbons convention. Well, let me copy page. But before I write a proof for the theorem, and when I give you an idea of what it is that we're going to do, so I want to motivate that the proof. Because I think this is a sort of technique that is helpful in obviously, what we want to do is we want to prove some containment. So we talked a little bit about proving containment. We said that proving a containment is a local statement. So in particular, that's going to help, right? Because this this condition of authority is something that localizes well, so we can easily reduce the problem. Actually, let me let me add that here already, prove kind of the first step is reduced to the local statement. How do you do that? This localizes proving containment is something that's sufficient to do after you localized. So you can you can start by doing that. So I mean, it's sufficient to prove the setting. But a different idea, if you ever want to prove an ideal is contained above, you can sort of measure the failure of that containment by studying the colon ideal that corresponds to it. So this is the same as showing, is it? So proving that an ideal is contained another is the same as showing that the things that send i into j contain one. So it's the same as showing that this quality ideal, let's always contain it, but it's the same as showing that this colon is R. What does it mean for an ideal to be R? Well, it means it's very, very big. It's as large as an ideal can possibly be. How do you show? So if R is local, so maybe let me say Rn, when the way you would have guessed that's what I meant. I take a local ring and I want to show some ideal does not R. This is equivalent to showing that M is not contained. Sorry, but this guy is not contained in M. You were so big that you're not contained in the largest possible ideal. So to prove a containment, it's sufficient to prove that this colon ideal is not contained in M. And in fact, you know, in character Cp, you can even take for convenience powers. For convenience powers should preserve containment. It's sufficient to prove that the, the forbenious power of your colon ideal is not contained in the forbenious power of the maximum. You know, whatever this aim they are, this is a strategy that would always look to prove a containment. What do I have at my disposal? I want to prove a theorem or I'm assuming that I have an ideal that defines that theorem. This fetish criterion gives me a fantastic ideal to look at that somehow, you know, is too big. That does not contain an M to the forbenious power. So what you could do is, you know, if I guess now this I is not the same eye, but whatever, if arm or eye, they're pure. Okay, maybe I need a different letter. Let me call this I everywhere. A suppose that I know that arm or eye is a pure. All I need to show is that that quote, that special corn, I for being his cute lie is contained in this special colon I study. Why? Because this guy is not contained by Fetter, Fetter tells me the special colon is not in the forbenious power of the maximum ideal. But if I have shown that this guy is contained in here, you know, that forces this bigger thing not to be contained in the forbenious power. Because if this thing was contained in a forbenious power, so would the other thing do. So in general, you want to prove some containment. And you have access to some special ideal that you know, defines an F theory. If you somehow prove that this special colon is contained in the forbenious power of the colon that has that comes from the container you want to prove. Then that's what you do. Then you're done. So in my special case, I want to prove this containment. I really want to prove this implication. This is the one thing we couldn't do if we're in person. So to prove this, you show this, what you're going to show is that kind of independently of his security condition, you're going to show that the special contained colon ideal is always contained in the colon. Well, the forbenious power of the colon of the thing. And then that will say, so we'll say this always for large q. And if I prove this, and by this always, I mean, no conditions on I, right, give me a radical ideal I, big I H, I'm going to prove such a containment always holds. And this is going to force that when our money isn't pure, this guy is too made to be convenient. That's the thing is part of the maximum ideal. So this guy is not in the previous part of the maximum ideal. And then you trace back to this, and you're going to find this colon is all of R. And so the containment you want to prove is true. Does the strategy make sense? Are there questions about the strategy? Okay, now I'm going to do something evil. I'm going to prove not this. The thing I just described is what we prove in our paper. I'm going to give you a better version of the theory, which is what we should have written in our paper, but it's not always written there. Because I'm going to prove something was wrong. But it's the exact same technique with that same idea. So I will show coffee. So So now I have our am local. And I will show the following. Well, in principle, I would show this that I'm going to change the total want to change it. I'm going to prove it slightly different contain. So I'm going to do, okay, whatever this is. This implies the following. So suppose I show this containment that I'm claiming, right? Maybe let me call this claim. This is what I'm going to show. This imply that if our mind that pure, this is going to give me the containment and plus H containing an I am. Remember the strategy when I'm an eyes of pure, this thing is not convenient, the convenience power the maximum ideal. So this thing cannot contain the convenience power the maximum ideal. So the containment corresponding to this point. So when I'm an eyes of pure, this is the containment I will show. And somehow this from this containment, we will obtain the big theorem. So I should make a decision now what we're going to do first. Okay, let me first tell you let's leave the claim. Okay, so the claim gives this containment. Let's prove that theorem from this campaign. Right? So I have. So if I n plus H is contained in an eye, I am. I want to show you that hard ones conjecture is true. So how do I do this? And what does this even mean? This looks strange, right? I think of the statement of saying the problem. Each time you take your symbolic power, and you raise the order a little bit, you raise the order by H. That kind of gains you an extra power. So it's more the more copies of H you have in your the order of your symbolic power, the higher power you're going to win. This is kind of what I love the most made of oxygen, we can actually build a black hole that's right. But this is stronger. Because now if you want to consider this is the the symbolic power involved in harboring, right? The one that somehow should end up in either the end of the end of the story. If I go from here, you know, you can even think of this as kind of like H appears n minus one many times. Each time I subtract an H. So for example, I could write H times n minus two. So I subtract an H, right? I went from this guy to this guy, and I gain a power of up. Okay, so I'm literally applying this statement. It's bad that I'm using the letters and the both right, but I'm applying this statement with like, I don't know this and here is actually what I'm calling here. A gen minus two, plus one. But the idea is each time you lower this guy by H, which is what I did from here to there, I lower the power by H, I gain a power of up. And now I'm going to repeat this. And you keep doing this, you keep subtracting for as long as this makes sense, you can keep subtracting H from it. So notice that throughout the process, the sum of this number, and this number are going to remain constant today n minus one, right? Here I have zero, there's no power of I appearing. And I have an n minus one, then I have n minus two, one, which also sum to n minus one. And then you keep doing this, right? Like the next term would be hn minus three, plus one, I squared, and then you keep doing this. It's kind of an induction. It's an induction that stops. Where does it stop? Well, when you get to the end, you've subtracted H as many times as you could, you get this one left, right?  And then, so I guess this is like h times zero. And that plus whatever I write here should add up to n minus one. But now the first symbolic power is I. And so this is this idea, which is where I wanted to learn. That was already written before. So kind of the point is that from here, this sort of gives you a simple induction that goes all the way from this guy, the idea at the end of the one. And this is actually a stronger statement than the one than just proving hard ones. This is actually more powerful. And I may or may not have time to tell you why. But let me actually prove the claim, right? This all only makes sense if I do this claim. So let me prove. So I will prove this awful looking thing, but I promise it's actually not awful. So to prove this, I'm going to do the problem. I'm going to take an element F in here. And then I'll prove that it lives in here. Okay, you know, you are going to prove in your problem sets, the Frobenius powers commute with columns. So this is so I can I can move the Frobenius to inside the phone. So on this side of the phone, and I have I and plus eight. On this side, since Frobenius power commute commutes with products. This is I for being is key. Like to be anything. Oh, and here is the thing. Right. So what I did was I applied for being is on both sides, instead of having the Frobenius outside of the column. This is it. This is only true because firing is right. Right. This is what you're going to prove in your problem set. So hints to the regularities is necessary. I think I added a problem to prove that to prove that the regular is missing. Okay, so somehow I want to start with an answer in this column and land in this God awful, horrible looking thing. Wow. What does it mean to live in this horrible quote? Well, to live in a corner and heal means that you take the thing on the right inside of the thing on the left. Right. So I really want to show that as times I n plus H. Q is contained in that mess on the other side. That's my goal. That's what I'm going to do. And all I know about it is that it sends I into I for being. So let's start. I start off, I'm going to write a string of in a plot of containment that lead to where I want to start here. Along the way, I'm going to make a lot of steps that seem wasteful. This is the most wasteful of all the steps. I'm going to replace the convenience power by the ordinary. So the convenience power is contained in the truth power. And then I'm going to take that ordinary power. Okay, I'll write it out. This is the ordinary power first. So now that means I'm going to break that ordinary power into two pieces. One of them will get q minus one. And the other one will get just one power of the symbolic power. But now I'm going to do another really wasteful thing. I'm going to remember that a symbolic power is contained in I itself. So this seems like the most wasteful step of all times. But somehow I claimed that at the end of the day, even though we were very wasteful, we're going to end up with the best possible statement. So it's kind of kind of magical. Okay, so so far, I just replaced a convenience power by a ordinary power. Took one power out. And then remember the symbolic powers continue the ordinary in the ideal itself. Okay, what did I know about it? Yes. Remember, is something that sends I into the Frobenius town. So when I do as times I, I land inside the Frobenius town. So so far, I started on the left. And I haven't gotten here yet. But this is where I'm at. And then we compare where you want to go with what you have. And I'm halfway there. I want to land in a product of two things, and I got one of them down. So I'm going to keep that put and going to focus on the other. So I actually want to show that what's left here, and plus H, the q minus one, I'm going to show that that is contained in. I want to show that it's convenient. So I'm going to continue and I to the end. That's where I'm going. That's my goal. And remember, I'm really trying to show this horrible looking thing. So sorry, I've only done it very silly steps, right? I took Frobenius power, replace to the power, reorganize stuff, use the definition of where s lives. So now I have nothing to do with that s, I have nothing to do with that complicated stuff. I just want to do this new, still horrible looking thing, but maybe not as horrible looking now, right? It's a little slightly less confusing. So first step. So now again, I want strength containment. I'm going to do something I've done today, which is remember, though, that when you take a product of symbolic powers, it lands in the symbolic power of the sum of the orders, right? So this guy lives inside q minus one, and that's where this lands on. And now I want to somehow go land in here into the symbolic power. In the beginning of the final part, I wrote down a limer that I claim what's going to be helpful. And this is when it's going to be when I get it back. Okay, I'm not going to get all of it. I'm going to get this. So remember, I said this is just the general principle again, and just use in a fancier looking way. So recall, I have this. So let me even put it in a different color or something. Color. This is a recall from earlier. I'm going to apply this thing in gray. What am I going to do? I somehow want to land in here. That's my goal. And this is a previous power of the symbolic power. So it looks like this, right? So let me apply this statement, right on the other side, I just have a symbolic power, which is awesome. So I'm going to apply this apply, apply with k equals what where, well, I want this side to be this side, right? So I want to keep this one to the end. So I want k to be n minus one. So I'm going to copy what's in here and put an m minus one i h q plus n minus one q minus h plus one. My God. So that's theorem from before guarantees, maybe let me put it in a box or something. Okay, the theorem from before somehow that was just a given whole principle guarantees this containment hole. I want to prove this funky thing lives in the symbolic power. I've already shown the left hand side lives in this other symbolic power. If I somehow prove that this symbolic power lives in this symbolic power, then I can complete the dots, right? I can go here, here, here, here, and then I'm done. Like that's why I want. So it's sufficient to show. Somehow I want to show that inside of here lives that other symbolic power I have that lives the q minus one h minus one symbol. If I prove that I connect all the dots and I go from here to there and we've already reduced the problem to this thing. But now look at the amazing thing. I have a containment of two symbolic powers. Note that should go ahead. Yes. There shouldn't be in plus a two n plus gauge. Where should there be? Oh, down here. I see it. I see it. Thank you so much. You mean I miscopied this thing over here. It's q minus one 10 then plus h. Thank you so much. Wonderful. Any more comments or questions? Good, we're almost there. So when is the containment of symbolic powers true? This is when it's easy, right? It's just they're both symbolic power. It's like taking two pounds. This is saying a is bigger equal than b. What's that? What do I want to do? I want to prove the symbolic powers in another symbolic power.  It looks awful, but I've reduced the problem to an inequality of integers. It can't get any easier than this. I have an inequality of integers. I just want to prove it. But remember, I don't need this for all q. I made a claim earlier. I actually only need this for q large. Because what I'm going to apply remember is fetish criterion and fetish criterion just needs large enough q. So I'm actually going to show not that this inequality always holds because it doesn't always holds. But that it holds for all q very well. And once you have an inequality involving integers, where all that matters is that a particular integer be really big. It's actually sufficient to focus on the coefficients on that very big thing. So we're going to look on both sides of inequality and read the coefficient of q. On this side, I have n plus h. On the other side, I have eight and n minus one. I need the coefficient on the left to be bigger than the coefficient on the left. Because the point is, I'm going to make q to be really big, right? So when q is really big, the terms involving q are going to win against the rest because the rest is not going to change. It's constant. Or if you like another way to think about this is you can take it in a problem and divide by q.  Now I divided by q, q is going to infinity. So that those pieces become zero. So only the things that had a coefficient of q are going to matter when you send q to infinity. And so you can see here very explicitly that this is extremely high. You can't do this any better. Because this was literally as good as you could have been, right? I mean, you know, the only thing you can do better is would be if they would be the same. And then you would have to worry. So actually, not saying it's right, I really want this to be bigger, because if not, then I actually, you know, if they were the same coefficient, then I'd actually have to worry about all of those. So this is both great timing and terrible timing. It's great because we finished this proof. It's not great because I have one minute I had so many more things I wanted to tell you. But let me in one minute summarize some of the other things that I wish I had time to do. So first, can you do better than this? And I told you where you'd know, there are examples of things that are pure for which you really cannot do any better. This is the harbour to conjecture is the best possible scenario. If you want to prove something for all ideals that defy a theorem. But if you restrict the class of a pure to strongly afraid, you're going to exclude things like square free monominal ideals, but you're still going to keep interesting things like vergenesis and two minantal rings. And you can do better. What the crag and I showed is that when our my strong way of regular, let me at least try that out. And I think you've seen the definition of strongly afraid, so I'm not going to write it. This isn't the same paper. It proves that if our more I is strongly at regular, let me write in a serious statement can replace age by age minus one. So I satisfy this harbourn's conjecture, but it's going to behave as if its big height is one less than what it actually is. So you're going to get I like to the age minus one and minus eight minus one plus one, which looks awful that there are ways of rewriting this that don't look awful. I'm just like literally writing harbourn's conjecture and putting in age minus one with age is three. But no, this is a corollary. What happens when each is two? This is a really great question to ask the audience to end, but it's very dangerous to do this in a zoom call. So tell me, what happens when you do is you plug in one where these ages are. And you end up proving that the age, the n symbolic power is contained in the n power. So you prove equality. And there are plenty of ideal of defining after range of big height two that are not generated by regular sequences. So this equality doesn't follow for many elementary facts about symbolic problems. It's giving you new things, where you have this equality. You could also have talked a little bit about what happens when it brings up. But I really am out of time. So I think I'm going to have to stop there. So thank you for your attention. Alright, thank you so much for the beautiful talks. Are there any questions? Hi, Eluisa, I had a question. Yeah. Um, so most of the containment problem they've all around ideals in a regular look for some ideas. Why? Most of the containment problems revolved around radical ideas in a regular way, which either define FPO or strongly of regular singularities. Do the does the containment problem become completely intractable if we move away from regular rings to security, say, taking ideas in a war in team ring or re-defining a good singularity? Yeah, very good question. So can we study things like carbon's conjecture of the containment problem when a ring's not regular? There are many obstacles that one faces, right? We use regularity in so we can see much of waves in various places. One big question that is open, and I think would be really beautiful to answer, is the problem. And maybe that explains how hard the problem is, more generally. So when I told you this beautiful theorem, I made a comment, the big theorem that we started from. I made a comment that says in particular that you can take this constant instead of eight, you can take the dimension of R, like you lose best possible mess for all ideals, but now you have a statement that works for every ideal, and it does not depend on. So one problem that is open is if you take a ring that is not regular, say a complete local domain. And so you focus just on prime ideals. Is there a constant D for which this happens for all prime ideals and for all N? The constant doesn't depend on the prime you consider. That's an open question in general. It's proved that such a D exists in some special classes, especially, but I work with various people, but especially by what the community cats and about last year and unique in the past, they have various results of that. Now, some of the results, they're not even constructive, right? They're not finding this one's in constant. They're saying that one exists that is independent of five. So a different kind of question is, can you find something constant explicitly? So one thing I didn't have time to discuss, but it is possible to get in the notes, is that it's a different proof of this theorem using test ideals and characteristics P and multiplier ideals and characteristics zero. And if you follow along with that proof, it depends on essential properties of test ideals or multiplier. And there's some work in particular by Daniel Smokin and Javier Tavarra Juárez, where they sort of examine that proof and examine the one fact that really, really needs a regular way. The rest of it works a corresponding way of regularity. That's right. We've defined something called diagonally a regular range, where they can do a sort of a different proof that still works. And they actually end up proving this exact same result. So for primes, they prove you can take this once in constant to be that high over these diagonally So, for example, that includes the hyper surface defined by the maximal minus of a two by two matrix, for example. And that might sound like a simple ring, but that was open for a long time, finding this constant for primes in that ring. So it gets really tricky. And another thing you can do is you can also study things like hardware instances, right? You can say, okay, maybe can I, first of all, can I do better than like, is this sort of hardware and like saving the best thing to do? And at the same time, you can try to find sort of good, you know, a good, good swanson constant that works. So in particular, this theorem that we proved in the second half is theorem of myself and Freikiniki. This required them to be right here. And the main point was that we were using, okay, we were using some of these components that use regularity. But on top of that, we were using federal criteria. Federal criterion gives you a criterion for an ideal to define an F theory, but the ambient ring should be right. So if you want to prove a version of this theorem overrings that are not regular, you need a federal like criterion, right? Overrings are not regular. So with Lin-Schulman and Carl T. We have a paper where we prove a theorem like this, where our rings are born sign, and we prove a sort of a feather like criterion for born sign wings. But there's the cats. We still need our ideals to have finite projected dimensions. And it is not just so the finite projection dimension comes in several ways. But in particular, our feather criterion really requires finite projected dimensions. But in particular, I think that's a criterion that that possibly has other applications, you know, besides the lower symbolic values. So this is a very long answer to your question. This is to say, like, there is something one can do. The problem can still be attacked outside of regular rings, but it's just much more difficult. And it's still exciting. And I think, in particular, there's more than characteristics techniques can do for the problem. But I think we're just starting to distraction. Any other questions? If not, let's send Luisa again for the beautiful talks. And now on Monday, Baiba was going to tell us about the problems and the problems that are already posted. Yes, I have a question. Yeah, so why we are not considering about the something similar questions like symbolic for a brilliant symbolic power of I, there exists the M such that that is content inside I to the power normal power I to the power of Mn. So, yeah. So, let's see if I if I understand your question, is your question about when was there? No, yes. So, so here we are considering about only suppose for every end. So, large in the symbolic power of N, when it can for this end, can you say that there exists a Mn such that the symbolic power of N is content inside the normal part of Mn? Oh, you mean backwards? Backwards, yes. So, when you do this backwards, it becomes really easy because the end power is always containing the end symbolic power. This is always true. And then, of course, you can now make this guy bigger or make this guy smaller and keep the complete. No, I am saying that symbolic power of I that content inside that normal power of Mn. So, for every end, yes, yes, yes, for every end, is it is there is there any question like this? Oh, you mean you want to fix the end instead? Yes. And you are asking does there exist an end? Now I am sorry. So, see, a silly thing we could do is you could take this Mn to be one, right? So, you always have that the end symbolic power is contained in high. So, that's that's always true. And so, the thing that is not clear is when you start changing this power here, can you still find the symbolic power? Does that make sense? Like once you fix this guy, there's always something to put in here, but it's not very exciting because you can always put the one at the end of the day. You can ask, you know, what is the best thing you can put in here? But that's the same thing that we're asking to continue. But asking for existence of something on this side, asking for existence of an Mn, then that's easy. The answer is yes, when always exists. No, but it's I'm asking about the best thing which can. That's right. So, that's exactly the containment problem asks. The containment problem asks for the best possible thing you want to put. So, asking, you know, given and what is the best possible Mn such that this holds, we're asking, given A, what is the best possible, I don't know, FA to put in here? These are equivalent questions, right? So, I'm asking this for, for like, ah, ah, OK, but maybe now, now maybe I understand your question a little bit. Maybe you're saying, OK, but we're putting like some funky functions here, but what if this guy is not a funky function? What am I going to put in here? But in the end, it's sort of still equivalent, right? Because, for example, when I put statements like this, this is kind of saying that if you want something, maybe I should be using a different way. Maybe you want to use this is kind of saying, let's give me this containment. But in fact, this is equivalent of this, where is the naive big theorem we started from? Kind of equivalent of this. And this somehow should make the problem seem even sillier, right? Because it seems like this should be a very short difference between these two, right? Taking a ceiling or taking a floor seems almost the same. But it really, oh, this is not how I want to write it. I'm writing this back today. Sorry about that. Ha, taking a ceiling or taking a floor feels really similar. But the truth is that there is a massive difference between these two when you think about the problem from this perspective. So yes, the containment problem is also asking the same thing that you're asking. But then you have to sort of switch around how you write with convenience. So if you play with these, you'll see what I'm saying. With these two arguments. Good question. I mean, we can also stop the recording. And if somebody wants to ask more questions, I can stay here and answer more questions.


what's going on guys it's your boy dpj here today with another destiny video and in today's video I'll be showing you guys how you can get this secret Kings 4 raid shader which is called bone marrow now if you have not taken part in the raid or if you have but haven't received any armors if you go to your shader vault this shader does not appear it only appears once you have obtained at least one piece of Kings 4 raid armor now that is a little suspect it really is it makes you think what else is hiding inside of these vaults that ain't unlocked until we get a certain clue makes you think it really does now taking a look at the requirements needed to unlock this shader you can see it states you have to acquire gauntlets, helmet, chest and leg armors to unlock it also notice it states normal mode as well confirming that all pieces of armor do drop in that normal mode so I'm sitting here wondering what reward you get for heroic mode weapons and armor wise are they going to look different who knows now if you have taken part in the Kings 4 raid or if you've completed it let me know down below in that comment section what you got loot wise has anybody got a helmet or is anybody seeing anybody with a helmet let me know I am super curious to find out now I ran through it once and got gauntlets and leg armors and no one in my team got a helmet at all I haven't even seen one of these things so I don't even know if they're obtainable in the raid yet I'm thinking you've probably got to get all of the pieces of armor first then you'll have a chance to get in the helmet but who knows I mean taking a look at the shader it does look pretty badass matches the color of the weapons you obtain from the raid that white and purplish reddish brown looking badass the shader actually looks pretty good in my opinion as well and you can see me showcasing a few armors I have including full water glass and full cross-end raid armor but guys I hope you enjoyed the video do drop a like I do appreciate the support subscribe for daily destiny and I'll catch you on the next one peace out until next time


We have now 11 confirmed cases and 7 cases are from the coast, the total now is 281 cases confirmed here in Kenya, and of course we have 2 recoveries and one of them is a doctor, so the number now stands at 67 of the recoveries made so far and the death remains at 14. Over that team tests have been done up to this and over that 10,000 tests have been done to this end and following the defiance of some Kenyans, 455 of them have been quarantined at their own cost. This is mandatory quarantine because of the unbecoming behavior we have been asked time and again to observe social distancing and we have seen them even trying to party. A few others were arrested just yesterday evening when they were in a wedding, a number of them now coming to 455. The CS has also made a special appeal to the following counties which is Machakos, Muranga, Kajadu and Kiambo to protect Nairobi since they were put or they were considered to be part of the metropolitan. They should also ensure that they are protecting their members and as well as the people in Nairobi from spreading the coronavirus and a few of them that have areas that have been mentioned are Thiriva in Machakos, Kiamboi have Zendegwa and the dry has been mentioned as well and of course you will understand these are some of the areas that are highly populated and the CS has called on the people from these areas and the authorities to ensure they protect each other. Now today being at the Kenyatta National Hospital assorted protective gears have been distributed totaling to 4,759 and the CS says 33 centres are active up to now and of course Dr. Amod has said local manufacturers are being engaged to ensure or to see the production of protective gears to our frontline doctors when they are dealing with the COVID-19 and just to emphasise the CS has called on Kenyans to ensure social distancing. Remember it has been a call since day one because it is believed and it has been tested that the spread of coronavirus is through contact and therefore we have been asked time and again to observe social distancing and of course to have a high level of hygiene by washing your hands with running water and soap and of course using sanitizers and if you are having symptoms like flow like symptoms that maybe you would suspect them to be of COVID-19 then you need to seek medical advice. We have a total number to call that is 719. You can have other information you want to know about the COVID-19 from that. So those are some of the things that have been mentioned this day and of course as Y254 will be keeping you abreast with the latest as far as COVID-19 is of concern. So the number now stands at 281 and globally the cases that have been confirmed is over 2.4 million and the deaths are standing at over 165,000. Kenya has 281 confirmed cases. Now 69 recoveries. This has been due to the two recoveries and one of them is a doctor who was infected by COVID-19 but the death number remains at 14 which is a good thing and we are hoping we will have more recoveries other than deaths and of course as we try to flatten the curve of spreading coronavirus we need to observe the directives of the government have been given. Regarding the 455 people who have been mandatory quarantined it's because of the behavior and just yesterday the CEC Dr. Mwangangi said those people who will be arrested or found bleaching the curfew will be quarantined mandatory and of course now we are seeing it has taken effect. It's a call to everyone observe the rules that have been put in place ensure you do not spray the coronavirus because you won't tell whether you have it until you have the symptoms or until you have been tested. Mass testing has begun in Mombasa and we are expecting it to roll out to other places in the country even as the Moai teaching and referral hospital coming up to help the Kenyatta National Hospital and Bagadi in testing so we are hoping as a nation we will have a mass testing rolled out throughout the country and to get to know how to mitigate these measures. My name is Dereva Hillary I will be leaving you with that and okay here we have the numbers now in new cases in Kenya stands at 11 active cases are 198 we have 281 total cases 14 fatalities those are the deaths that have been recorded and 69 recoveries one of them of the two who have recovered today is a doctor and 13,872 tests have been done so far so this is how the situation is in Kenya and of course Nairobi remains to be leading in numbers followed by Mombasa therefore Kenyans we need to be cautious and try to control the spread of coronavirus. See you again my name is Dereva Hillary good afternoon.