This video is going to talk about linear inequalities. So a linear inequality looks like ax plus b is greater than c. Again, it looks like a lot of variables, but we really just have this one variable of x and a, b, and c are going to be real numbers. And again, we can't let a be equal to zero like in equations because if it were, then we wouldn't have a linear equation.  So we're looking at graphing, and we want to graph x is less than or equal to one. Well when we graph, we can use, some of you might have learned using closed and open circles, but we're also going to talk about interval notation, which deals with parentheses and brackets. And we might as well just talk about that now. If you have parentheses, it means that your endpoint is not included. And if you have brackets, it means your endpoint is included. So I'm looking at this, and it can be equal to one. So when I come here to one, it's going to be a bracket. And it's going to be less than one, so it's going to be, I'm going to open my bracket to the left because that's where my numbers are less than one. And I would put my bracket in here, and then I'm just going to put my arrow here. And that means that it goes on and on and on and on and on. Now, if I want to write the interval notation, again, we have to think about the endpoints. And it also, we've talked about the endpoints, but come down here and it says the smallest interval and then the largest interval. That's how we write them. Well, down here, this is negative infinity at the end of that arrow because it just goes forever. That's our smallest value. The farthest thing to the left is our smallest value. So it's a negative infinity. It's a sideways eight. And if you notice here, positive and negative infinity always have a parentheses with them. Okay? So I'm not going to include the lowest point, which is negative infinity. And I'm going to go all the way up to one. And in this case, I am going to include it so it has a bracket. X is greater than negative six. Okay? This means not included, and this means include. So X is greater than negative six. It's going to go to the right of six because it's bigger than negative six, but it's not going to include it. So this time, I have a parentheses here at negative six, and it's going to go this way forever up to positive infinity. So the interval notation, we start at negative six, and we don't include it. So it has a parentheses, comma, the largest value, which is going all the way up to infinity, and you never include infinity, so it'll be a parentheses. You can't include infinity because you can't ever get there to say, this is exactly where I'm stopping because there's always one more. All right. What happens, though, if we have a compound? This is what we call compound inequality. What are we going to do here? Well, we're going to really think about as negative seven is less than X, or if you want to, you could write the X first, and since it's open to the X, I rewrite it open to the X in negative seven, and then I also have X is less than four. It doesn't include either one of them. So down here at negative seven, X is greater than that, so it's going to go this direction, and it's going to be a parentheses because I'm not going to include it, and it's going to go this way forever, and when I have X is less than four, it's going to come here at four, not include it because it's a parentheses, or less than, and it's going to go this way forever. And if you notice, in between here, in between my negative seven and my four, there's both a green and a purple line, so I'm going to shade that in because that's really where my answer is. My answer is really from here to negative seven, and up here to four, and everything in between. Over here, those numbers are less than four, but they're not greater than negative seven. Over here, all these values up here, they're greater than negative seven, but they're not less than four. The only thing that satisfies both of those inequalities is what's in between. So how do you write that interval notation? Well, what's the smallest value that's in black here? It's a negative seven. And do we include it? No. And then we go all the way up to four, that's the biggest value in our interval, and do we include it? Now, the inequality says we don't, so it would be a parentheses. So we need to learn some rules before we start working with inequalities and trying to solve for x. So if I add two to both sides, we're starting with four is less than six. If I add two to both sides, four plus two is six, and six plus two is eight. And six less than eight is still true. If I subtract two, that's going to be four minus two, and we want to know if it's less than six minus two. Well, four minus two is two, and is that less than six minus two, which is four? So two is less than four, which is true. So we want to multiply both sides. So four times two is going to be eight. And we want to know if that's less than six times two, which is 12. And yes, that's one's also true. What if I divide both sides by two? So four divided by two is going to be two. And is that less than six divided by two, which is three? Yes, two is less than three. Let's multiply both sides by negative two. Four times negative two is negative two. And is that less than six times negative two, which is negative 12? Negative two is not less than negative 12, so it's false. What if I divide by negative two? Four divided by a negative. I'm going to write it below here, is a negative two. And is that less than six divided by negative two, which would be less than negative three? No, negative three is farther to the left, so it's smaller. So again, we have a false here. There's a rule that says if I have, multiply or divide by a negative, I have to switch the inequalities to make it true. I have a y, and I want to do the multiplication and addition properties just like I do with equations. So I'm going to subtract three from both sides. And I find that y is less than negative five. Well, if I'm going to put that on a number line, I would do something like this, negative five and zero. Just two reference points, that's good for me. And y is less than negative five, so it's a parentheses opening to the left because I've got to go less than, and it's going to go in this direction. And for an interval notation, it's going to go to negative infinity. And it's going to go all the way up to negative five, but not include either one of them. Now, here's the tricky one. I have to divide by negative one. But I'm dividing by a negative and that was one of those rules where we had to switch the inequality. So negative x divided by negative one is x. And four divided by negative one is negative four, okay? So let's make sure that we did this right. Let's get it graphed and then we'll see. So we say negative four and zero. And x is less than or equal to that, so it's a bracket opening to the left because it's got to be less than. So let's pick a number that's in there. Let's say negative five. So the opposite of negative five has got to be greater than or equal to four. The opposite of negative five is five and that is greater than or equal to four. If I tried something over here like zero, the opposite of zero has got to be greater than or equal to four. Well, the opposite of zero is just zero and that is not greater than or equal to four. So we know we did the right thing here. And the interval notation would be negative infinity, because it went all the way down there to the biggest point of negative four. And it includes negative four because of my inequality set and equal to. But never includes an infinity. Finally, I've got a fraction here. I need to fix it up. I need to multiply everything by three. And when I multiply it by three here, it gives me three times negative one-third is going to be negative one times x. Plus three times five, and that's going to give me plus 15. Greater than and then three times four-thirds, which the threes will cancel each other out and we're just left with four. And if I subtract 14 from both sides, I'm going to have to fix my negative x. It's going to be greater than, still carry my inequality. And I have to switch the inequality because I'm dividing my negative one. So that means that x is less than 11. Finally, we have to look at the kind where we have a compound inequality. Now what you do to the center, you do to either side. So this is just three x in the middle, I'm trying to get to x here. So in order to get to x by itself, I'm going to divide that by three. But that means I have to divide the left-hand side and the right-hand side. That gives me negative four less than or equal to x, which is less than three. And on my number line, negative four is the smallest value and three is the largest value. I can include four and it's going to go this direction. I can't include three, but it's going to go this direction. So you can see again that it's anything between the two values. And it's going to be a bracket negative four to three with a parenthesis. One more. What you do to the middle, you do to the outsides. So I need to subtract three from the middle. So I'm going to subtract three from the left-hand side and subtract three from the right-hand side. 15 minus three is 12, less than negative 2y. Less than 25 minus 3 would be 22. Now I have to divide by negative 2. But since I'm dividing, I've got to change my inequalities. I'm dividing by a negative. Since I'm divided by a negative, I've got to switch those inequalities. And divide by negative 2. And that means that 12 divided by negative 2 is negative 6. And this just gives me y in the middle because the negative 2's cancel. And 22 divided by negative 2 is going to be negative 11. And normally, we write our inequalities that are facing to the right. So I really need to write negative 11, because it's the smaller value, which is less than y, because it points to the negative 11. And then it points to the y and the negative 6. So we have negative 11 here, and it's greater than. So it's going to go this direction. And we have y is less than negative 6. So we have negative 6 down here. And it's less than that, it's going this direction. And we have anything in between. If we want to go back up here, we could look at this and just think about this. You could just disregard this if you wanted to. And you could say y is less than negative 6. So that's what we did in red. And y is greater than negative 11. And that's what we did in green. And then it's everything in between. And that means that our smallest value is negative 11, not included, in parenthesis, comma, negative 6 is the largest value on that interval. And it also has a parenthesis.


But as the Jesuits came in, they did not want to change the Indian culture, and this I think is one of the brilliant facts of Catholicism. As they encountered the Indian culture, they decided that that culture was fine in and of itself, but like the human person, when the human person encounters the salvation of Christ, encounters the redemption of Christ and the sanctification of Christ, we physically do not change, our bodies do not change, but the moment the turning of the soul makes us radically new people. And the Jesuits applied the same theory to a culture. A culture should not become European. That is not what God intended that culture to be, but it should be Catholic. And so the Jesuits did an excellent job of taking an Indian culture and sanctifying that culture with a soul of Christianity, without rechanging that culture and making the stamp of Europe upon it. And this is the story of the Jesuits throughout Canada, throughout much of North America, and then later into the 19th century West.


Hello everyone, this is financial author Amit Dan of AmitDan.com on this channel You'll find financial travel and more videos just like the one you're watching right now If you like what you see, please give it a thumbs up and consider subscribing Hello everyone, this is AmitDan of AmitDan.com today I'll have another episode of unboxing and brief review of a travel charger slash travel adapter or travel converter this is Let me go travel adapter universal travel charger So the reason I like this this one should have USB output So whenever you travel These days usually all the electronic Accessories or gadgets are USB powered. So for that reason you'd It's better to have a USB travel adapter rather than having a regular adapter I do have a regular travel adapter, which I'll show you at the end And let's go through some of the specs of this product and before that I'd like to mention that this is not a paid or affiliated review I purchased this with my own money. So the opinions are totally honest and unbiased So now the some of the specs are that this should work in 150 countries around the globe This is all-in-one all-in-one travel charger do dual USB ports adapter and you can charge simultaneously More than one device and it has a safety protection. So it has a built-in safety shatters so which supposed to protect your Devices and the travel adapter performs very well under 100 to 240 fold So which is that you know anywhere you go most of the countries you're covered and this is a very lightweight portable and So let's go ahead with unboxing and see what's in here and the total video shouldn't be more than 10 minutes Okay, so I'll open up here So here is our Adapter and looks like the box is empty here We even have a small card which is user user manual Okay, so that's a good thing that they touched something here It's a very Small it's like a flip card. So I'll just put it aside here for now and let's Get going at the real part Okay, so here is what I have so it looks like this way and I'll just try to Open it up and then Okay, actually looks like I don't have to even open it up. So if you want the pin to come out so you have to push it this way and Then so if you do it like this, then here is The standard two pin this should work in USA, Japan, Australia and Thailand so you can retract it and then if you have If you want this one to come out. This is mostly in Asia so just do it this way and Then you can just plug it in here and then Here I can see another switch. Let's take see what happens when you push it. Okay, so this one is it says here that UK and I Cannot read the other country could be great Britain Okay, so this is another it should go this way so which is good and Here are the two USB ports, so this is USB one and USB two and I believe these are USB 2.1 Output so which is fast, but not too fast, which should be okay. It doesn't have to be really fast and then Once you lit Put it in there should be light somewhere which I'll test every shortly I think I don't think you can actually open this up That's what I was expecting, but which is fine now. I'd like to show you something else. I had purchased a adapter in the past which is like this and You can just you know open up the whole thing. So for example you can It comes out this way and then it comes out this way. So everything is like a removable And Then you can put it back This way and then everything goes inside this way now the problem with this one that it doesn't have USB output. So for that reason I bought the a new one this type of Travel adapter actually cost almost half of this one So if you are buying something just make sure that it has USB output because USB output is the one You'll be using most of the time This is only good for like, you know Product or electronic Equipments without USB input, but nowadays everything has USB input for example Camera cell phone or anything possibly can think of so I think this is just a waste of money So for that reason you should get something which has USB output. So you don't have to buy more than one So now what I'll do I'll put it Into the power outlet then I'll I'll show you How it looks like if there's a light or if you can see something is lit and if it works So just give me a second here So you have a standard 110 120 output here and I'll just take out the Pin for the Canadian or US outlet. So it should be This way Okay, so I'll just push it in here Okay, so as you can see I see that there's a red light, but it's gone. Oh, actually yes It came back. So I believe as long as the light is lit It means that it's connected and now What I can do if I push if I put my phone in here so I have my one plus one phone here and I'll try to charge this using the USB output and see if it works Okay, so this is the USB Cable so I'll just put it in here and Okay, so I put it there and I don't know why it's sometimes losing the power so it could be That it's not just tight enough Okay, so Which is right here. So this light is lit and it's red. So that means that it's working and it's charging So this is this works. However, sometimes looks like it's losing the connection. So I don't know that if this is my My the problem with my outlet or this is the problem with this product so that I'm not sure but I can what I can do. I'll just put it in here and Okay So it's on right now and it's charging here So The good thing is having a light is that you can see that, you know, if it's on or not So if it's not on that means that it's not working then you have to just, you know, push it a little bit To make it work However, looks like I'm having a little bit difficulty with this particular outlet So at this point, I'm not sure that this is the problem with this product or this is the problem with the outlet However, as you can see once it move a little bit, then it just works fine. So it's a good this way Okay, so In terms of the price and everything this is not this is a very well-priced product and This is about below $20 Canadian you can find easily on Amazon or probably other at other retailers It's called. Let me go travel adapter, but there are similar products which you can find with the different brand name and Different manufacturers So whenever you're buying something just make sure that you have the USB outlet and it has it covers most of the countries And it has the surge protection Which is very important to have Because that way your device will be protected once you travel or once you you're in a foreign country Because some foreign countries the power Supply is not so stable and it fluctuates It's not like in Canada or USA that you're getting steady output or steady current power all the time I've seen I have had the situations in the past where I was traveling to Asia or in the Caribbean and my devices got Broken or got messed up because of the power problem. However, this is a very good feature to have the surge protection So that will wrap up my Review of this product. If you like what you see, please give it a thumbs up and consider subscribing And there will be more videos coming up like this. Thank you for watching and I'll see you shortly


As World War II had spread around the earth until it became all-out global war, so the Cold War in the late 1950s and early 60s became a danger period of ever-increasing tensions that were worldwide. In West Germany, United States Army troops patrolled the border, the iron curtain that divided the communist world from the free west. Khrushchev threatened that communist East Germany would soon take control of the Allies' supply lines into West Berlin, the same kind of action which once before had threatened death to that free city during the Berlin blockade. Refugees from East Germany fled to West Berlin by the thousands, half a world away in Korea. American soldiers continued to stand guard at the demilitarized zone on the 38th parallel, where an uneasy truce dragged on intermittently. The Cold War had its hot spots. From the mainland, Red China shelled the offshore island of Kamui, held by nationalist China. A revolt against Chinese communist control was crushed by force. Refugees fled to India to escape the Red Terror. In Southeast Asia, communist guerrilla forces set out to conquer the new nations of Laos and South Vietnam. In our own hemisphere, a Cuban revolutionary betrayed his own revolution to become part of the international communist threat to world peace. Thousands of freedom-loving Cubans fled their homeland. Provocation and without warning, Red China attacked neutralist and militarily weak India in an aggressive drive for territory. Late 1950s and early 1960s, making very clear the purpose and designs of a new generation of aggressors. And early 60s, the combat readiness of the United States Army had to be one of great versatility. Along with the rest of our military establishment, the Navy, Marine Corps and Air Force, the Army served as a potent part of a massive deterring force to prevent the big Cold War from suddenly becoming a big hot nuclear war. The Army had to have the capability of fighting conventional wars, as well as little unconventional wars created by communist-inspired subversion, insurgency and guerrilla activities. So-called nasty little wars of long duration, such as were occurring in the jungles, swamps and rice paddies of Southeast Asia. The Army's responsibilities included not only the immediate defense of the United States, but its readiness to meet our obligations as a member of the United Nations, NATO, the Southeast Asia Treaty Organization and other alliances, if called upon to take action along with Allied forces to maintain or restore the peace, to meet overt acts of aggression as we had been called upon to do in Korea. In the 40 or more years since World War I, we had learned a hard lesson that one trouble spot anywhere in the world, like the proverbial rotten apple in a barrel, could infect the rest of the world and us. We had paid with our lives, our blood, our energy and our resources to learn that lesson. The important thing is that we had learned it. It had taken time for the aggressor to learn his lesson too, that a new strong and ready armed force had come into being, not for conquest, but as a potent force to help deter aggression, however uneasy the peace, and to work for the freedom of men who would be free everywhere. The United States Army. Ready military strength. This was something the aggressors understood and respected. You are witnessing the review of an entire United States Army armored division, a part of our ready forces in West Germany, only a small part of an active army of 870,000, 40% of whom were serving overseas. In all, a combat ready fighting force of 16 divisions, with a reserve backup of 29 combat divisions, a formidable arm of defense, a deterring force to discourage any aggressor from rash military adventures. Complimenting these forces in West Germany was a growing inventory of Army missiles designed to give flexibility of firepower to Western Europe's defense. The Hawk. Nike Ajax. The Sergeant. The Nike Hercules. At missile sites throughout Western Europe, a 24-hour operation is maintained, with frequent day and night alerts to ensure operational readiness. The Army's arsenal of missiles emphasizes how far it has come since the last weapon was fired in World War II. Here is one more reason why an aggressor would pause before unleashing conventional war. But a military organization designed solely to fight so-called conventional wars is not geared to fight the kind of unconventional warfare that had broken out in the jungles, swamps and rice paddies of Southeast Asia and could erupt elsewhere at any moment. We are faced with a clear and one-sided threat of a change in the internationally agreed position of Laos. This threat runs counter to the will of the Laotian people, who wish only to be independent and neutral. It is posed rather by the military operations of internal dissident elements directed from outside the country. This is what must end if peace is to be achieved in Southeast Asia. The situation was no better in neighboring Vietnam. Communist-controlled North Vietnam, in violation of its 1954 Geneva Agreement, was using the same guerrilla tactics against South Vietnam. The United States not only increased its flow of weapons and equipment to Southeast Asia, but sent military advisors as well. South Vietnam were men of the Army's special forces, specialists in counterinsurgency and guerrilla warfare, to aid in increasing the combat effectiveness of the pro-West forces in their struggle against the communists. Tensions continued unremittingly throughout the world. Cuba had become a rigidly-policed totalitarian state. Execution was the bloody answer to opposition. To Cuba, now came Russian technicians who arrived to make sure their newest satellite would rapidly become a Soviet-dominated stronghold only 90 miles from the shores of the United States. The Chinese were there, too, to disseminate their own particular variety of communist doctrine. Cuba was now a base for international communism in the Western Hemisphere. So, freedom-loving Cubans who had fled the Castro police state vowed to return and reclaim their homeland from the communist spoilers. Many gave their lives in an heroic but disastrous attempt to do so. More than a thousand of the invasion forces were taken prisoner. They were submitted to long lectures by a triumphant Fidel Castro. In a three-and-a-half-hour May Day speech, Castro gloated over his victory and told the world that Cuba was now a socialist nation and would hold no more elections.  Crisis followed crisis. Khrushchev threatened that Russia would sign a separate peace treaty with East Germany, a Soviet creation which the West refused to recognize. Refugees from East Germany poured into West Berlin, fearing that the few remaining escape exits would soon be blocked. Those would be justified. In West Berlin, our army units were ready and alert to the dangers that a separate Russian treaty with East Germany might bring. World attention again focused on Berlin. Our troops there were training in mock riot control exercises. American soldiers, playing the part of would-be agitators, were confronted by specially trained troops employing riot control techniques. It was also employed in the civil disturbance test. The situation was still tense. Representatives of 14 nations met, seeking a solution to the Laos problem. The conference ended in an east-west deadlock. Tensions were increased, not lessened. Being the critical state of affairs throughout the world, President John F. Kennedy in May 1961 asked Congress for additional funds for worldwide defense commitments. Khrushchev, along with other communist strategists, continued the bellicose threats. Refugees from East Germany increased to more than 2,000 a day. Since 1945, nearly 4 million East Germans had fled from communism to a haven in the West. But on the 13th of August 1961, virtually all escape was suddenly cut off. Here was a new symbol of communist ruthlessness. Overnight, East Berlin became a concentration camp, sealing in the inhabitants of that sector. Barbed wire and cement were the answers to the refugee problem. It was a sudden and forceful intrusion into the lives of people caught on both sides of this grotesque barrier. There were desperate attempts to escape. A few East Germans made it. This was communism's ruthless answer to those who sought freedom. Khrushchev demonstrated our resolution. We strengthened our forces in the Ligert West Berlin. Souls between East and West for a Berlin solution continued. But the stalemate and the tension persisted. United States and Russian arm are gunpoint to gunpoint across the East-West dividing line in Berlin in a dangerous confrontation which needed only a spark to ignite the powder keg of open warfare. Explosive situation, the American soldier gave one of his finest demonstrations of discipline and cool courage. This was his business, and he was conducting it and himself extremely well. He was there to keep the peace, but he was prepared to fight if he must. In crisis reaching a boiling point, the combined armed forces of the North Atlantic Treaty Organization were brought to a high state of readiness. The fighting team of the NATO countries, ground, sea and air, symbolized the collective strength of Western Europe's defense. The Allied stand in Berlin and the might of these NATO forces throughout Europe once again gave the Russians cause for second thought. 1961, Russia's Khrushchev withdrew his end of the year deadline for signing a separate peace treaty with East Germany. But it was only a slight pause in the communist's game of strategic irritation. In Southeast Asia, communist-inspired subversion was unrelenting. South Vietnam looked to others for assistance in stemming North Vietnamese aggression. Military and technical assistance was offered to South Vietnam in a determined effort to thwart this communist's drive for conquest. In South Vietnam, flushing out the communist Viet Cong meant moving fast, locating the area of enemy operation and taking action to deny him any foothold. South Vietnamese soldiers were trained by our military advisors in the latest techniques of helicopter movement. The South Vietnamese were resolved to resist communist aggression. To do less would have meant complete communist takeover. For Little Laos, the future was uncertain, as was the future of big but militarily weak India. Missed China, having overpowered peaceful Tibet, now attacked India along the Tibetan border without warning. Neutral India was rudely awakened. She too was vulnerable to communist aggression. The people of India were quick to respond. Her young man answered the call to the colors. The men assisted in the emergency by helping to finance the country's defense effort. Red Chinese forces ceased fire, but remained belligerently poised on India's doorstep. Still making territorial demands of this peace-loving neighbor was the state of the world in the early 1960s. Threats of big war, little wars raging. The complacent, they were in for a shock. The communist world was demonstrating its collective image of belligerence, power, and solidarity. The world was moving toward a new crisis. The secretary of defense spoke to the nation about the grim details. Russia's men in Havana, Soviet military technicians, had done their work well. Our reconnaissance planes over Cuba had photographed Russian missiles pointed our way from nearly completed bases only 90 miles off the Florida coast. Missiles that would be fully operational within a few days. Time could be running out. October 22, 1962. Good evening, my fellow citizens. This government, as promised, has maintained the closest surveillance of the Soviet military buildup on the island of Cuba. Within the past week, unmistakable evidence has established the fact that a series of offensive missile sites is now in preparation on that imprisoned island. I have directed that the following initial steps be taken immediately. To halt this offensive buildup, a strict quarantine on all offensive military equipment under shipment to Cuba is being initiated. All ships of any kind bound for Cuba, from whatever nation or port, will be found to contain cargoes of offensive weapons be turned back. This quarantine will be extended if needed to other types of cargo and carriers. The Cuban missile crisis tested the combat readiness of our entire defense establishment. The army stood poised, ready to respond. Air Force tracked the progress of the more than 20 Soviet ships heading toward Cuba. If Russia chose to send these ships carrying weapons of offensive war through the quarantine, our response would be automatic. They would be sunk. That was the clear and concise ultimatum given Russia by the President of the United States. The most dangerous crisis in the Cold War years had focused the entire world's attention upon our military strength. The leadership of our army, Air Force and Navy had been given the awesome responsibility of meeting the aggressor's bold move within our own hemisphere. That move was checked by our resolute action and by the fact that we now knew well through past experience that weakness is a certain invitation for the aggressor to attack. This time we were prepared. We know now that the Soviet ships carrying weapons of offensive war turned back and that the Russian missiles were removed by the Soviets when the President of the United States made it clear that if they did not remove the missiles we would take action to do so. The Cuban situation itself was not resolved but an hour of great peril had passed without nuclear holocaust. Again two things had become apparent. The aggressor would go as far as he could to the very brink of war if he was allowed to get away with his aggressive acts. And second, that only the will to fight backed up by a strong ready modern fighting force could deter him from his designs for conquest.


Hello everyone, I'm Randy, I'm a surgeon and I like or love to cut people up and sew them back together and I also teach young doctors and I like to introduce to you here's a conductor. Hi all, I'm here to tell you about my skills as a conductor. I've recently defended my PhD pieces about transferring my skills, the conductors, to Randy's profession. Randy has been my supervisor and the results have been developed in a close cooperation between the two of us. And we are here today to share the idea and our project with you. But first I would like to invite you to take part in a small experiment. My experiment I never before had the opportunity to conduct the chef's choir. So please help me on this one. Please help me on this one. All you have to do is to sing. Simple. I'm sure you know the song and we have the lyrics. So please follow me carefully. Please Michael, please follow. Okay I have to say this twice. Please follow me from the top. And please sing. Chefs came to Ding Deng Dong. Right? Ding Deng Dong. First time. Please follow. How do you manage to do it? No. I've been watching your cuisines and well okay. Come on. Ding Deng Dong. Twice. Last time. Ding Deng Dong. Bravo. I'm told today is two very brave chefs who like me have never been conducting the chef's choir before. In fact they have never been conducting before. Mida and Dave. Where are you? Thank you. Welcome. Welcome. You have to learn how to conduct what we call a basic four beat pattern. It looks like this. One, two, three, four. Simple. I teach you. We go one, one. Come on. And one, two, right? Three, four. Again. One, two, three, four. Now you should be able to do. Will you please start? Once again. No. Yeah. You go. You go. Okay. Once again from the top. Only one hand. Ready? And please tell them what you think of the musicianship. Once again. You aren't paying attention. Good. From the top. Beautiful. And we are going to talk about today's a little about this. You'll see more of this. So we invited here to share our product with you. And I think now I have to turn my paper that René realized that chefs face daily the challenge of leadership. And he understands the parallels between leadership in different intents and stressful working conditions. And here the chef working as chef. And here I am working as team leader in an emerging situation. And this is a training situation, a cardiac arrest. We have about 10 minutes to rescue the patient, give him medicine and a lot more. And the pressure is high and the outcome might be fatal. Like chef, surgeons are not formally trained for skills in leadership. But one day when I was watching a conductor, I got the obviously idea that asking him for help because he represents a profession where leadership is a core skill. When working with the leadership training in challenging and stressful situations, we found that it was necessary to address anxiety in the situation as well as fear of the situation itself. Evidence tells us that anxiety blocks for learning and anxiety blocks for performance. You may be familiar with the phrase freeze, fight or flight. The biological explanation for this is that the reptile brain, the one that treats the basic rights for food, sex, fight or flight takes over our rational thoughts during moments of stress and danger. When this happens, we are hit by the mcnulla hijack. If the conductor is afraid of the orchestra, she doesn't have a chance. He will be paralyzed, unable to overlook the situation when standing in front of the orchestra. Unable to think a rational thought. And certainly not able to gain respect and demonstrate authority or power. The mcnulla will have taken over his or her body and thoughts and the conductor will just want to get away far away. An anxiety can cause strong physical reaction.  The list of reactions to fear is so much longer and the reactions are very personal and therefore very different. The body signals fear and uncertainty are clearly visible and read instantly. We feel this ourselves in the situation and then a self reinforcing negative spiral starts. There are many reasons for anxiety in different situations. Worry about personal recognition, the desire to please and be loved and be popular. Concern about the professional impact. In some cases the reason is a little alive for death. If I'm not a good team leader in the clinic, a patient might die but even lesser anxieties can be paralyzing. If I'm not a good conductor, the music won't sound right and you would get fired. If I'm not a good chef, I will get a bad review, guests will stop coming and the rest might go bankrupt and the people who are still having a job under my will lose their job. So what did we do? We designed a course for young doctors to train in team leadership in emergency situations but not in a clinical setting. To be able to train them we had to create anxiety but in safe surroundings. So two are designed musical exercises and we use them over the course. First the doctors learn how to conduct a basic four beat pattern. Just like you did. You saw how I did it. It looks quite simple but as you saw as it turns out, it's not. An important purpose was to throw the course participants into a completely unfamiliar situation far from their comfort zone. This creates uncertainty and fear of performing but as exercises were about conducting something as friendly and simple as far as yuck and playing with ghetto blasters and not professional skills for the doctors. The frame was most safe. As you will see, there's a lot of fun on the course which is very important. What did you do? I didn't do anything. You go back. There's some technical stuff here. A little further please. A little further down. I have to read this text. There it is. By teaching simple conductors exercise we give the students an unreasonably or even ridiculously difficult obstacle. You just saw it. Which they do not have any strategy to handle. Everyone starts at zero which creates cohesion and safety in the classroom but also a fear of performing. Conducting is a surprising element that helps participants concentrate and focus. Here from the course four ladies learning to conduct a forebeat pattern. It is not simple. Look at her. Oops. Okay. Crossing the legs, scratching the throat, twisting the hands, throwing the hair back. Okay. The conductor is specifically trained to lead. Traditionally over the course of 100 years this training takes place through apprenticeship. And the very first thing the conductor, the aspiring conductor learns is that his fear of the situation is the first thing he must overcome. He does this by getting control of his body language. He learns to stand up straight and erect, raises his chest and raises his chin in order to radiate calmness and authority. This confidence will be instantly read by everyone on the team. The conductor feels this and now so does the orchestra. The self reinforcing spiral becomes persistent. Basically anxiety is related to the perception of yourself in the situation. It is a mental battle between you and you or me and myself. Here is a student from the course whose fear is making her struggle with herself. Before we can address fear we have to decide to act as a leader. It is absolutely necessary to have taken the mental decision to take on leadership before wearing this model of authority. This can't be done halfway. This can't be done hesitantly. As a professional leader of a group you assume all the responsibility for the outcome of the entire group's work, whether it's good or bad. This decision will help you to focus right. As a leader you are no longer a part of the team. You are not friends with your team. You are simply in front of the team and thus responsible for the professional outcome. But you can of course choose to be friendly. Focus is no longer about being popular, well-liked or loved. The focus is to raise the team's professional potential to a higher level than the sum of the present skills. The development doesn't take 10 years. It takes three hours as this video shows. At first she is not really comfortable, but after a short instruction she improves, straightens her back, starts to use her eyes, but it's not really there still. But now you will see how it looks when she makes the decision to assume the leadership. It is about clear communication without interfering elements. Leadership is basically communicating a message, whether it is communication to the whole team or to individuals. The communication must be absolutely clear. No noise on the line. But noise to communication happens often without our knowledge, especially when we are stressed. My own noise, my own reaction in a stressful situation as a conductor was surprisingly my lift thump. Because I've been playing trombone for 20 years standing like that before I took up conducting. So I got used to stretching my lift thump when creating music. So when I started conducting it looked like this. I didn't notice until a colleague conductor pointed it out for me. Wow. For sure I would like to be known as a good conductor and not just the guy with the strange thump. A rubber band around the finger solves the problem. I'm good. But this is only to say we all have a version of my lift thump when challenged in a stressful situation. And now for very short before after we do it. You have to look carefully because they are very short. So please. This is another René and this René is thrown out of his comfort zone a bit. But now look at his back, look at his chest and now he raises his chin and he gains authority. It works. You can respond to anxiety in many ways. Fooling around is another valid response. We got him straightened up in control of his back. And just and she is very serious I must say. We just have to add a few comments to the next video. This is a young woman with a huge problem. Her smile. It is a wonderful smile. But whenever she feels anxiety the smile blocks her face and takes her authority away. And in acute emergency situation in the hospital this smile is totally inappropriate. It is a huge problem for him. It is real. But indeed it is a wonderful smile. And for the first time there she is able to show power in front of other people. And this is another smile. A very proud smile saying I did it. So if you are able to calm yourself and know how to control your body and if you have fully understood the responsibility as being a leader you are on your way. And then you are able to deal with the next elements of the world work as leader in front of the team. It is most important to establish confidence between leader and team. Confidence that the leader is competent, confident that the leader is able to pursue the project and focus on the work. Confidence that the leader can identify the human and technical resources available and that he or she is able to respond to them and make it all play together. To establish this confidence the leader must appear credible and it is actually possible to consciously work to establish credibility. Does the team sense that the leader is professionally focused, that he knows what he wants, that he has a global view of the whole picture? Does he act with the calmness the team needs to feel confident and safe? Does he have presence? That is does he demonstrate the ability to actually see every individual on the team and respond to their work? If so, he achieves a high level of credibility. This is the end point, the ghetto blaster exercise. We show a little about it later. The purpose of the course is this authority you see here.  It is not something they simulate, it is not something they imitate, it is something they are feeling. If the leader is able to show that he or she is committed to the project, that the project is important, she or she gains even more credibility. But it requires the courage to show vulnerability. It may sound like a paradox in a situation requiring authority, control and competence and power, but it is met by the team's respect and thus gives high credibility. This is the last before and after meeting. It is interesting because this guy is trained in the military and has therefore awareness of his body language. He really does a fine job and imitates what he has seen too or do, but takes a closer look and he doesn't seem so confident. The slips are gone, he hits big holes in the air and he is not present and even listening at all. There is a slight panic present in his eyes. After a brief instruction he begins to listen to the team and ends up providing a small but proud smile. A very important tool is the ability to improvise. Improvisation is not a card you choose to play in absence of a better one. It is actually the strongest card you can play, but it requires that you are well equipped professionally. Improvisation is at this level enables you to get above normal standards, procedures and check marks and actually find the best solution in a given situation. Improvisation shows a high level of presence and provides very high credibility. Improvisation includes the courage to turn off the PowerPoint presentation and talk without a script. Improvisation is the courage to choose another tempo at the concert. Improvisation is the courage to try out new ways if they can improve the moment. It is the courage to come up in the middle of a service with a new scallop dish because you realize the oysters aren't good enough. Improvisation enables you to embrace the unexpected. So please listen carefully, I repeat. Improvisation implies that you rely on your own assessments, intuition and judgment in the situation. Improvisation requires extensive knowledge and experience in your field. We designed another exercise, the Ghetto Bass exercise, you saw part of it before. The intense feeling of leadership is experienced exactly as when conducting an orchestra. The focus is the sensation of leadership, the feeling of dealing with a complex situation, including many people performing different tasks. Many leadership parameters are at stake in this exercise. It is all about presence, engagement, the courage to show vulnerability. It's about taking decisions under pressure and finally about improvisation. The feedback is given at our course, it's very personal, each individual has his or her own challenges. I allow myself to be subjected based on my own knowledge and experience for what is needed. My feedback is intuitive and swift. I interpret and respond to what I see. The students later tell me that they are happy about what they call straight-on feedback and as it can be seen, I'm allowed to come very close to each individual. As someone said, no matter how much training you have or how many books you have read, nothing can fully prepare you for the challenge. In this respect, leadership isn't easy, it's difficult, necessary easy. You have to practice leadership to learn it. Tourist research resulted in the five Cs, command, with means, authority, professionalism and power, credibility, your trustworthy, competent, the ability to handle in this situation, calm, create calmness for the team and communicate clear. Three days ago we recorded the next video from service at NOMA and here we'll see an incredible lot of Cs. So, we'd like to say thank you. Thank you.


Welcome. So last session of the day, I guess. And let us start that session with a shockingly high number. So apparently, 75% of container images that run in production have high or critical vulnerabilities. Furthermore, 85% of these container images have a vulnerability for which a patch or fix actually exists, or at least this is what the numbers from a system report show linked to the report on the slides. So the report is a bit of a year old, but I kind of assume the numbers haven't changed dramatically at the instant. And now you ask yourself, these numbers seem high. What can we do about it? Is there something we can do against this? It's certainly not the lack of great tools. There are a lot of great tools to scan container images for vulnerabilities and other things. So it's maybe also something that we just have to run them. Now, we think we actually have one idea where we can actually make it a little bit easier to run off those tools for you automatically so that you don't have to think about them too much. So today we would like to talk about how you can nicely integrate tools like security, vulnerability, compliance tools into your build process if you happen to have a ship right around. Now, this won't obviously fix the numbers that I just showed you in production, but at least our goal should be we shouldn't increase those numbers. Like, if you build something new, let's make sure that it doesn't have any vulnerabilities right from the start, right? So let's not make this number even bigger. So this is Karen. Hi, Matthias. We work at IBM. We work on a project that's called Code Engine. It's a serverless platform as a service, which has build capabilities. You give us the source code, we will build it. And in this capacity, we work on the CDF project ship right and in cluster source to container image build framework. Now, we are happy to share some of the latest changes we did in ship right that go around how we can integrate more tooling into the build process to help you with things like the scanning or other compliance requirements you might have. One example could be, for instance, software built from materials, S-bombs, how can you create them as part of your build flow automatically without having to do anything about it? Well, the idea is to make your life easier and that you don't have to think about those things, unless, of course, the tools find something in your image. And then, of course, you have to think about it and have to fix it. Now, for those of you who have not heard or used ship right before, like I said, it's an in cluster source to container image build framework. With other words, if you have to happen a Kubernetes cluster laying around and you use that for your project, ship right is one option you can use to put it into your Kubernetes cluster and do the builds for you in that cluster. For instance, application builds or container images for run to completion workloads. Now, ship right is an incubation project in CDF since 2021. So fairly new, but we've been around. We have community meetings every Monday. Everybody's happy, can join. We're happy to have new contributors. Details are on shipright.io. Links are also in the slides. So please join us. Let's talk about ideas. Now, we are currently on our slow but steady march towards better API status. It shouldn't scare you that we are from alpha to better. Actually, ship right runs in IBM Code Engine for three years now. So it's doing its job diligently and perfectly fine since three years with thousands of builds per day. It's just that the API, we still think we want to keep the door open for things to change the API. Because the API is, we really wanted to keep it simple, not too opinionated, but also powerful enough. And as you might guess, that's not that easy. So we wanted to keep the door open for iterations on the API. But in terms of its core capability, it actually really works wonderful for us, which actually brings us to today's talk. So we want to talk about one of the things we changed recently and what we think could be quite useful for everybody. And before we go into that, we actually want to speak about some of the inner workings because it makes it easier later in the demos to know the terms and what does what. So with that, let's head into some of the inner workings. The microphone. So in ship right, there are a few core concepts which we would like to explain first. So these are builds, build strategies, and build run. So build is what you want us to build for you. So here you can specify where we can get the source code from. Or you can also provide the source code from your local machine. The build strategy is how you want us to build it. So here you can build strategies somewhat comparable to GitHub action definitions or some other CI tool where you define the steps. And in our build strategies, there are steps to steps to which you can define how you want to build your image. And then there is a build run. So build run is the actual build that runs. Or you can say that it is the actual execution of those steps in the cluster. So with regards to the build strategy, with regards to the build strategies, with regards to the build strategy, these are usually maintained and rolled out in the cluster by the strategy creators. And we do provide some sample build strategies in the sample section in our shipwright repo. So for example, we have sample build strategy which shows you how you can integrate buildkit to build Dockerfile-based applications. And then there is we can also integrate buildpacks with shipwright where ship, if you don't know buildpacks, buildpacks is basically it detects the programming language from your source code. And it builds it and it pushes to the target registry. And there are other tools such as Kaniko, KO, Builder. So the point is you can integrate a variety of different tools with shipwright. So under the cover of shipwright, there is tacked on. So we use it to automatically create the execution plan of steps, both system-defined boilerplate steps and the steps defined by the strategy creator. You must know that some common elements are provided by the shipwright itself. For example, the strategy creator does not have to deal with how he will obtain the source code from Git or from other means. So shipwright does that for you. So yeah, our main goal is shipwright is convenience, abstraction of complexity, and usage of standards, and proven tools, and processes. So the problem is the way we use buildpack for now, what it does is it eventually pushes the image to the target registry. But it sounds very nice in the beginning, but the problem is it makes it very difficult or somewhat impossible to extend the build process. So for example, if we want to run the vulnerability scan on an image, and we eventually, it would be really nice if there is a possibility that we want to abort the build altogether and not to push the image to the container registry if it has critical vulnerabilities. So there are other use cases also where this default behavior for now is counter predictive. So these use cases might include image signing, vulnerability scanning, metadata post-processing, validation checks, et cetera. So this led to the initiative we mainly want to talk about today. We want to break the build strategy into two flavors. So the left one here, here you can see the orange boxes basically represents those steps which are introduced by the shipwright. So these kind of will be maintained and handled by the shipwright. And the blue boxes are those steps which are from the strategy itself. And these are usually created by the strategy creator. And on the left-hand side, this is our current behavior. So here, we can integrate different tools, as I told you. We can integrate buildkit, buildpacks, canico. So here you can integrate. And then your strategy will eventually push the image for you. So that is why we call it as strategy managed push, very creative. So on the other hand, so there is now. We are working on it, and we have done a few examples for this. So on the other hand side, there is a possibility to tell the strategy that shipwright is supposed to do the push. So on the left side, it was strategy creator. He has to define who will do the push. But here, shipwright will actually do the push for you. He will be able to do the push for you. So here, on the right side, we do inject the steps which will get the source code. And it will be managed by shipwright. And then these are the steps which we will get from the strategy itself. And then shipwright will do the push for you. So this enable us two features and two areas of features and functionality. So there is possibility for strategy writers to add more steps into the strategy definition. That happened prior to the push. So eventually, Tecton will take care. Everything happens in the due order. And at the end, shipwright will add the final step for the pushing. And there is also possibility for us in shipwright to put additional logic into this push step at the very end. And we have two examples. Like we will show you two examples today. So we'll talk about two things, software build-off materials, that is S-bombs, and vulnerability scanning. So yeah, Matthias will cover the demos for those two examples. All right, so we will do just two demos that kind of show what we think would make sense if we now have the capability kind of to break up what you just saw. Like coming back to the LEGO pieces, right? Like Buildkit is such a LEGO piece for us. Or Paquito Buildpacks or KO Builder, you name it. And sometimes it's easy to put them together. Sometimes it's missing. So we're kind of hoping to solve this one here or make it a little bit easier for some of the use cases. So the two demos are S-bombs and vulnerability scans. Now it's just two demos for the sake of having two demonstrations so that we don't mix too much. We kind of think that in the real life, people would just choose opt in and opt out what feature they want. So I guess most of the time it kind of makes sense to have the full package and actually do vulnerability scan and also the creation of the S-bombs. But you can use either of them or none of them depending on your use cases. With each of these demonstrations, we will create a build. So again, what you want to build. And then just trigger a build run so that we have one example. Now what you see here is just on the bottom, you're not supposed to read it. It's just the K9S output of the cluster just to give you an indication. Something is actually happening. Tecton is doing its magic, spinning up a pod, putting in everything. Images are being loaded. On the top left, we are using the shipwright CLI to create a build and a build run. And then on the top right, just peek look into the container registry to see there's actually nothing there yet. And if everything goes well, then we hopefully have an image in the container registry at the end. Now what we want here is, again, breaking up the build. We just don't want to, in this case, it's a sample go application with a Docker file. And up until now, how shipwright worked with the build kit, build strategy is you give it a source code and a Docker file. It will figure it out. It will build it. It will push it ready. But now we kind of modified the build strategy in a way to say, build kit, you do your magic with the Docker file and the build. But just don't push it. Create a nice OCI raw directory on the file system and store the raw data there, because we actually want to do some post-processing on it. Now this is happening as I speak. And eventually, you see on the top right that the image will show up. So what's happening is once the build step is done, we go into our post-processing and shipwright. We call just whatever tool makes sense in this demonstration, just an implementation detail here. We use SIFT to work on that directory, let it create an S-bomb in whatever form it you want. And this output is then captured, put into an OCI file attached to the registry to the image and then pushed into the OCI registry. And then eventually, just that you can see that it actually happened, I use Crane to just as is, pull it out of the registry to give an indication where the build run actually created here. Now, again, the idea is, and especially me coming from a bit of a Cloud Foundry background, you always have to be careful with opinionated systems. So we want to keep it flexible enough so that it makes sense for you. SIFT is just an example here. We know there are build tools that actually create the S-bombs as part of the build process anyway. KO is such a great example that creates the S-bomb automatically. So we also envision that depending on, again, what Lego pieces you put together, some will end up creating the S-bomb themselves and we will just post-process them. If you happen to have a build strategy that uses a tool that doesn't create an S-bomb, then at least you have the possibility to configure, please create one automatically for us so that we can use it more non-at-all if you just don't need that for your build. And again, the core idea we have here is since we do the build anyway and we have all the pieces of information in the pod anyway, best is to actually do the post-processing right there and then just push the image already around. So that would be basically to automatically rate S-bombs with whatever strategy you want. The second demonstration, again, very, very similar. We create a build and then later a build run. Again, just for the sake of the demonstration, shipwright has the capability to not have these two steps. You can also do it in one step, but I didn't want to overload the demonstration too much here. Now we fake the build a little bit because we actually need an image for the demo that has vulnerabilities. And it took us actually a bit of time to find an image that works all the time with vulnerabilities. But we found one in the Python namespace. So the build is just taking the image, again, kind of a build. The output is, again, not a container image that you can then push, but again, just raw OCI on the build.parts file system. So at some point, you will see that the build step is done and then gives back control to us in the POST processing. And in the POST processing, we use, in this case, Trevi, again, implementation-specific. You can, technically speaking, configure whatever tool that makes sense, or you use, or you're familiar with, to do vulnerability scanning. We said there are a lot of tools out there. So what happens here is Trevi has executed on the build.parts file system with the OCI. And what is important, or what we find important, is the image is not pushed. So it's still only in your build system. We find a couple of vulnerabilities. And what we now do is the build run, as we said, is the execution of a build. And the nice thing about the build run is once it's done, it will actually be enriched with information from the actual build. One example could be, for instance, the digest that was created, or sizes, and so on. And these status fields are now enriched with the vulnerabilities that were found in the respective image. Now, again, multiple use cases.  Another idea is you just leave it with the details you have of the build run. So the build run actually, at the end, contains the status. It's completed in three minutes. And by the way, these are the vulnerabilities I found in the image that I just created for you. And then it's about the rounding framework or tooling you have, whether you want to read those information and process them further. Or what we kind of think makes the most sense is that you configure your build in a way to say just the boarded. Like, if you find a vulnerability during the build, just a boarded right there. And the great thing is the image isn't pushed to the OCI registry already. Because, I mean, it doesn't make sense to push something if you actually know there's something in there that's not supposed to be in there. And that gives you feedback to see, OK, maybe I have to look into, actually, what's the problem? CVE number, you will find out the package and then can iterate on your source code and say, OK, let's do it again. Now, these are just two examples that we picked because we figured makes the most sense. And these are the ones that we are currently working on. There are other ideas we have. For instance, BuildKit has, no, Buildpacks. I always confuse them. Buildpacks has this feature for reproducible builds that it actually manipulates the creation timestamp of the image. The idea is the same input should produce the same output, independent of the time you build the image. A lot of container image tooling actually uses the current timestamp of the build for the container image. So another thing we now have the opportunity with separating the things is actually that we can do post-processing on whatever fields in the container image we want, especially the creation timestamps that we could just set a neutral timestamp like 0 or the commit timestamp. So whatever makes sense, we can do post-processing before we actually release the image into the container registry. Other use cases might be removing annotations labels that are actually in some base image. People actually sometimes get annoyed if they have some base image that already comes with labels and they just inherit that and they don't really make sense anymore. So we think there are more possibilities and we would like to hear feedback. So what are other tools that you know and love and use for container images where you think, OK, that kind of makes sense to integrate them into a build automatically so that you don't have to think about them. And of course, questions. So I would open the floor for questions. Exactly. So the question is whether you have to write Tecton Yama. So in ShipWrite, we use Tecton and this is our tool. But we kind of decided to say, OK, we abstract that away from the end user. So our interface is really what we just said, the build, which is just a very simple Yama file with this is the source. These are the credentials to obtain the source. This is the strategy we think makes sense. This is where we want to put it. This is the simplest contract you can have. You can actually extend it with more things, but to keep it simple, this is the simplest contract. And such a build definition is really just 10 lines of Yama. And the idea is that we provide a convenience layer and do basically all the Tecton stuff under the cover for you, including obtaining source code and setting up the task run that the steps are in the right order and everything. So the question is whether it makes sense to have ShipWrite as part of a Tecton pipeline. And I just noted to my colleague, yes, there are such ideas. We want such integration also to be possible. Andrika, do you want to? You have to think about our goal is with Code Engine, which is a serverless platform as a service. So we have compute resources for people to use to serve or have jobs. So the idea is, of course, would be great to use the same resources actually also to create the build. So whenever that makes sense, and there are a lot of use cases where we also have Tecton pipelines. And of course, it would kind of make sense to put that together. All right. So then thank you. So in the name of all ShipWrite contributors, thank you very much for your time and interest. Check us out on ShipWrite.io or GitHub.com slash ShipWrite.io. So we have a couple of projects there. It's not just the build that we just showed you. There's also the CLI, which is again like a convenience layer where you just call the CLI with, this is my Git repository, or also this is my local source code. So we also support that you kind of say, ShipWrite, please take this local source directory, push it up, create a container image for me. And we also have other projects. So take it out. It's all in this area. We have some actually great getting started guides. Like if you happen to have a kind cluster laying around, it's just like two commands you can copy, paste, and set up ShipWrite in less than five minutes. With the samples we have, just to get an idea. And with that, thank you very much and have an awesome day.


Hi everyone, it's MJ the fellow actuary and in this video. I want to talk about the Gumbel Copula Now Gumbel was a German statistician who was very much anti Hitler And he kind of got kicked out of Germany because he was very much against the Nazi party Anyway history aside You'll see Gumbel comes up quite a lot with extreme value theory and other statistic things But in this video we want to talk about His copula because Gumbel's copula is part of this Archimedean copula Which we discussed in the previous video Essentially we have these Generator functions where we add the marginal distributions together and then we take the inverse of the sum of them Essentially the big idea is taking space space zero to one to zero to infinity Adding there and then bringing it back to zero to one so that we have a probability Gumbel's generator function is given by the following Negative lin f of x to the power of alpha where candle's tau is equal to one minus one divided by alpha So we're seeing that this alpha is the correlation parameter Alpha lives between the values of one and infinity We see that when it is equal to one we get the independent copula We see that if we had to take it all the way to infinity we'll actually get the minimum copula and What this essentially does or how this Gumbel copula flexes dependency is through the following It'll have a upper tail dependency But it won't have a lower tail dependency and this is something that we can use when it comes to say Credit portfolios specifically with credit losses We'll see that their dependency increases during times of recession. So this is an application of the Gumbel's copula and What we can do? I mean, I'm just explaining that in the bivariate case You can take it all the way to the n-dimensional multivariate case and that is essentially your formula It's not that scary when you kind of break it down and you look at the components Anyway, let's look at some pictures of the Gumbel copula So what I'm gonna be doing is I'm gonna be playing with the parameter alpha and I drew these in a statistical program known as R So you can see here when we have it as one it is the Independent copula and it's kind of just this big block. Okay, nothing too too special But as soon as we start adding some sort of correlation You can see how the graph starts to kind of take this the shape It almost looks a little bit like a stingray the big idea or something that I want you to focus on is The upper tail so the tail of the stingray over there You can see one of the ways to to read These these type of graphs is to also pay attention to the axis. So you can see here is from one to one Here we're going from two to one here's five to one fifteen to one sixty to one And that's just showing how much more weight is getting put in the upper tail So we call this the stingray's tail Well, it looks like a stingray's got its little wings little head there I don't think anybody else causes the stingray Don't refer to it as the stingray in the exam It's just how I kind of see this but the big idea here as you can see that as we're Changing this parameter and you can see the minimum copula was full positive Dependency so as we make alpha bigger, we're gonna see more Dependency between our marginal distributions and what the Gumbel does is it allocates a lot of the weight in the tail And that is something that we're seeing when we start increasing The parameter you can see sixty that is a lot of weight over there Compared to to these other ones and the reason why I say focus on these axis is here is because otherwise the picture looks Like very much the same but when you realize that they're being scaled you can actually see the the effect of it Anyway, this is very much a quick introduction to Gumbels copula