I'm here with Anna to such a joy always to be together, friend. And we have an incredible show coming up, talking about that very thing about people feeling lonely. Yes, we certainly do. And welcome. We're so glad that you're with us. And we get to welcome Justin Whitmore early to talk with us about the power of friendship and how to overcome loneliness. This book made for people comes at just the right time as our country admits to being in an epidemic of loneliness. If you long for community and strong relationships, this conversation is for you. And so, Sydney, I just, I wanted to ask you, like, what has it been like for you building friendships as an adult? Cool. Let's talk about this all day. And I think like, I think there is, it goes from a time when you're growing up from, you know, high school and then to college and then people fall off and then you're changing and you're evolving. And I was actually having a conversation with one of my dearest friends just about this very thing of like how you evolve, how you heal, how you change. And it's sometimes people fall off or you're in different seasons of life. And it can feel very lonely.  And in the season and time of just, you know, even though we have social media, even though we have all these ways to connect and engage with people, there's a lot of loneliness that can happen. What about you, Anna? Absolutely. Yeah. You know, I, the first time that I remember like really hearing a lot about loneliness was whenever I worked with the mops group, the mothers of preschoolers and you know, that season of life in particular being a stay at home mom where you really are with your kids so much and isolated in a lot of ways really starts to plunge women into that loneliness. And we really found that those, those groups to like intentionally put ourselves into where we could connect with other women in the same season. It just helped to build those strong relationships. And it's really beautiful to see all these years later how the friendships that we made there. And stepping out of our comfort zone to talk to a new person and be able to share from our hearts and how those friendships have continued to grow.  And I think a lot of times shame or we try to act like everything is okay. But when you're able to open up to people, then you're like, oh, you're walking through the same thing too. Cause we know that God didn't call us to walk alone. I'm so glad he made other people.  We are made for a relationship or we're going to get into that conversation here in just a minute, but I want to say you'll have to stay tuned also for after the conversation, because we have a God story that we're excited to show you about a wonderful family that is in our local area that through their business, they're doing incredible things for Christ. They're called sacred gate farms. So that will be coming up later in the show, but for today, our guests, Justin Whitmoe early joins us to talk about the profound impact of friendship. In his book, made for people, Justin offers transformative strategies to combat loneliness. If you're struggling with feelings of loneliness or have struggled to build strong friendships, know that you aren't alone. Justin, we're excited to get this conversation going. So welcome to hope today. Thank you. I'm so happy to be here with you all. So Justin, tell us a little bit about yourself and why you felt this was such a critical time to put this book about friendship into the world. I'd be happy to. So my name is Justin early. I am a business lawyer here in my office this morning, but I'm also a writer and author and a father of four boys who are now between five and 11. So there's a right answer to why I wanted to write this book and there's a real answer. The right answer is that if you look at the statistics, America and the West is in an epidemic of loneliness. It is actually shortening our lifespan. We are dying younger. We're experiencing all kinds of mental health impacts. And we desperately need to talk about how we do relationships. It's an art that we've lost and I care about that. But the real answer is not really the out there answer. It's the in here answer. I am somebody who has experienced the difficulty, the devastating difficulty of loneliness. So this book starts from my experience in high school when I was moving to a new town, knew a lot of people around school, but nobody really knew me. And I remember everything about life seemed hard. Everything was a cause for anxiety until this magic moment where I found my first friend. And I remember from that point in my life on everything began to change, not the circumstances, but the way I faced the circumstances were different. And ever since I've had this feeling that I was made for friendship. I was made for people. And I see that exact same thing happening around me every day. Difficulty of circumstances for fathers, for mothers, for the stage of parenthood for your job of mental health. And I just want to tell everybody that those circumstances are difficult, but they radically change when you face them alongside a friend because you are made for friendship. So right. And in your book, you talked about how miracles can come out of friendship. But before we delve into that, you also talk about loneliness with God. That is a real thing. Can we start there and then begin to unpack the rest? It's a great place to start because we're looking out on the world at this epidemic of loneliness and seeing the mental health and the physical decline that's causing. And there is a reason for that. If you look at Genesis, Genesis, too, has an incredible and very interesting moment where God says to Adam, it is not good that you are alone. Now, remember, God has just created the world. He's called everything good, the water good, the light good, making Adam good. And then he looks at him and says, it's not good that you're alone, which is just so interesting because of course, remember, God is there. So it's an incredibly important theological moment where we realize that we can be lonely next to God because we can't experience God the way that we were made to until we experience him alongside others. And that is an ode to marriage, but it's also an ode to friendship. When God creates Eve, Adam is now in the kind of communal relationship with God. Of course, we're made in the image of the Trinity, right? He's in the kind of communal relationship with God where now he could walk alongside God the way he was made to. And that's true for us today. Walking with Jesus is essential, but it's also essential that we do it alongside others. So I want to push back on this Just Me and Jesus narrative and say that we need to walk with Jesus alongside other deep relationships. Oh, that's so fascinating. And so out of that, talking about Adam and Eve and how we can parallel that with marriage, but also friendship because it's not everybody is married and you talk about covenant friendships. What does that mean? How does it differ from a regular friendship? Well, friendship is a word that has suffered recently. In the past century, a friend has gone from being something that was just held in the utmost high regard to now something you can do with a click on Facebook. So it's a tough word to use. So I use the phrase covenant friendship as a way to recover, not to invent, but to recover the glory of friendship as the way Jesus described it when he said, I have not called you servants. I've called you friends. And he says the greatest act of love is to lay down your life for a friend. So covenant friendship is an idea of spiritual friendship. It's where you dive into relationship, not just into connections and conversations alone, but where you're actually vulnerable and you're committed. I like to put it like this. A covenant friend is someone who knows you completely and sticks around anyway, because this is exactly what Jesus did for us and his active friendship for us on the cross. Jesus said, I know you're sent fully. I know all your deepest secrets, but I am here to commit to love, to sacrifice anyway. And when we know our friends fully through repeated conversations and being with them in presence, when we know them fully and they know us fully and yet we love each other anyway, we're acting out a gospel sort of relationship to each other, a reflection of what Jesus did for us. And I call that covenant friendship. So beautiful. And you say that men struggle with friendship. It's like in a pretty dire state. Why is that? Well, it's important to note that we are all struggling with this, but it's very important to note that the male outcomes are extremely dire right now. I think there are a couple reasons for it, but one of the reasons is just the sense of modern manhood is usually not to be somebody who is vulnerable, who tells their secrets. It's usually to be somebody who is quieter and carries the world on your shoulders. And I really want to speak to men particularly and say that one of the most manly things that you can do, one of the most masculine acts there is, is to live without secrets, is to be known amongst your friends. I think that everybody, but particularly men, should have one or two other people who know them completely and love them anyway. That's a covenant friend. And I do this with my friends Matt and Steve on a regular basis. They know everything about me. I don't have any secrets with them, but I know that despite my flaws, stick around anyway. And that makes me feel not just strong, but strong in the Lord. It makes me feel like the man that Jesus was. And I think that's really important for men to hear in our modern moment. And you say that there are strategies that we can implement to build strong connections. You talk about a couple of your favorite strategies. Yeah. If you read the book made for people, I talk about friendship as an art and as a habit. And it's important because relationships are totally habits. They're not things you can put on autopilot, relationships are somewhat an art. But there are really significant things that you can do to put yourself in the way of friendship. One of the most important that I say is one hour a week spent with friends. One hour a week. This is not really hard. You know, for everything else important in life that we need for survival and thriving, food, sleep, work, parenting, we have to do all the time. But friendship, if you set aside one hour a week, maybe that Wednesday morning Bible study or that Thursday morning coffee group or that Saturday running club, whatever it is, if you set aside one hour a week and commit to that rhythm, your life will begin to drastically change. I see this all over the place. It can be a small group. It can be accountability group. It can be a workout group. But as long as you stick with it and actually talk to these people and make yourself known, you're doing that incredible act of vulnerability with commitment. Remember being known and sticking around anyway. And that is the rhythm of covenant friendship. So I would recommend anybody just start there. It's a radical thing just to devote one hour a week to friendship. Yeah, it's all about being intentional, right? Because we live such busy lives, our schedules. We compact them with so many things. And we have to schedule in that intentional time with friends to be able to build those strong connections. You also talk about the different habits that are so important. And one that stuck out to me was forgiveness. I can't say enough about this because anybody who tries a deep relationship, like covenant friendship, will very quickly realize that people are flawed, right? Every friendship has two ingredients, a center and a center. So it's no surprise that we hurt each other. It's no surprise that we let each other down. And I think for men and women, this is particularly important. We have to be ready for our friends to hurt us. We should expect for relationships to be hard, not easy. And that really, when we change that expectation for our friends, it really helps because we're not surprised when our friend doesn't completely rise to the level we hope they will be. We're not surprised when we found out that we've hurt somebody, but rather we realize this is an opportunity for forgiveness. And if you don't practice forgiveness, then any conflict, any argument, or any moment of disappointment is kind of the end of the road for friendship. You need to discard it and find a new one, someone who will do better. But when you realize that forgiveness is the heart of the gospel and the heart of human life and relationships, then difficulties become the doorway to deeper relationship, not the end of relationship. And I'm not saying this is easy. Nothing important is easy, but this is crucial because if we don't know how to do forgiveness, if we don't practice forgiveness, then we can't do friendship. I'm thinking back to conversations that I had with the mops' moms, the moms who were full-time at home with their kids and their struggles with loneliness and found that a lot of insecurities would get in the way of stepping out into new friendship. Can you talk a little bit about being brave to step out? Absolutely. I want to note that it is precisely those hard times of life. Let's say infants, new stages of motherhood, a move, new jobs. Those pivot points, that was for me in high school when I moved to a new place where everything was changing, everything was hard, and I didn't have a friend yet. And it's at those times that we become scared to admit that life is messy, life is difficult, or failing in lots of ways. But here's my encouragement. What you think is your flaw. When you share it with someone else, sounds like bravery to them. So when you act vulnerably and say, I am so struggling in my mental health at home with the children, you feel embarrassed because you don't want other people to know that, but the person on the other side of the conversation hears that and sees courage. They see bravery because they're saying somebody else who is brave enough to speak the truth about what was going inside their heart and their head. And so I am with you. This is difficult. But when we're in those phases of life and we are vulnerable anyway, we, one, do an incredible thing because we invite help. And that is exactly what you need. And two, we inspire others to do the same. Vulnerability is contagious.  And that is a beautiful and magic moment. So I just encourage everybody towards that courage to be vulnerable. Yeah. Great miracles come out of strong friendships and communities. So Justin, thank you so much for being with us today. Again, your book is called Made for People, Why We Drift into Loneliness and How to Fight for a Life of Friendship. Thank you again for your time and for your heart that you have shared. So wonderful to be with you all. You're very welcome. All right. Well, we're going to be unpacking more of that conversation in just a few minutes. But as promised, we want to share with you the God story from Sacred Gate Farms. I tell you what, I learned about this wonderful couple, Lorraine and Martin. And I visited their blueberry farm and we picked blueberries and we went home and made a delicious blueberry dessert. But I so fell in love with their story of how God used them in such a mighty way to be a blessing in the community and show people the love of Christ. So here is their story. We moved to the area in 2010. Back then we really didn't have any farming knowledge or any farming background. And we always knew it that we wanted to be out of the city. So whenever we move out of the city, we purchased that property and there was 20 acres here. We were thinking like, how can we engage our kids? Because planting 20 acres of corn would be only so engaging to them. It was the same time when we're discovering who God is. We can't contribute this to anything else than divide intervention and just putting it on our hearts that that's what we are going to do in the future. And sure enough, 10 years later, here we are. Sacred Gate Farm is a Christian faith-driven farm located in Apollo, Pennsylvania. Martin and Lorraine are the husband-wife team behind this family-owned business. They, along with their seven children, operate the farm daily. We heard great things about this unique place, so we reached out to hear their story. One thing that was really significant for both of us, all through my journey through trying to discover Christ and trying to listen to him, I was always kind of big on like, Lord, please just send me a sign, just send me a sign. And although my husband was a little bit more skeptical of the signs thing, when we, this whole field was tilled and we had to add a lot of fertilizers and a lot of nutrients to the soil in order to prepare it to plant the blueberries. So if you can imagine our whole about four or five acres of this area was just brown and black dirt, just completely bald. And my husband said one day, walk up to the middle of the field there. And I'm like, why? And he's just go up there.  And of course, you know, I'm starting to get ulterior eyed and I start praying. I was like, Lord, is this really that sign that I was looking for, you know, just really trying to... I am looking at it and I don't believe it. So I... It was hard to believe. There is a two different sides to the story, but it was literally there for three days, walking back and forth in that field. Never seen it again. Never came back again, yes. But it was definitely very revealing and confirming for us. So it was amazing to, you know, to feel just God's presence there because I wouldn't take that for anything else. When we came here and we got everything set and we're ready to plant our blueberries, we had 2,400 bushes to plant. So my husband planted the first couple and he looked out in this field and he's like, oh my goodness, how are we going to do this? It was literally two plants and four bear acres. And when I turned around, I was like, I'm not sure if I can do that, you know. Yeah. So we were kind of praying. We're like, Lord, please, you know, should we reach out to other people to help us? Should we, you know, please send somebody and everything? And I remember our children at the time, our two older boys were now teens. Six and seven, seven and eight came out and said, hey, can we help you guys? Can we dig holes? And we looked at each other. We're like, oh, you're going to be in the way. I just discredited a little bit. I hate to say it. Yeah, because it was a lot of work, physical work. And so we said, all right, come out with us. And sure enough, it took us what, maybe two months to plant the whole field. Yeah, it was about two and a half months. They were here for every single bush that we planted and through the process, like after planting all of them, mulching them, like every one of our children helped us in planting this farm and growing this farm. So they have their feet deep into it. And I just pray that not only that they have it in the farm, but that they're rooted in Christ through that, even though it's hard to see sometimes. It was a spectacular experience for me as a father too, because I discredited that I'm like, oh, six, seven year old boys. I'm like, I don't think that they can handle it.  You just chose not to use it. So they literally dug 3,000 holes here that I put the plant in backfill. But they were measured and everything, it was great. And sure enough, they like to push the limits. And like we learned parenting as we go. I believe that God is working in their hearts. We all went through the difficult times. I hope that we are just walking along and pouring into them and just being available when the troubles come. But I believe we have spectacular children and obviously biased opinion. But they are often on their knees. And I think that they're heading in the right direction. So I think that that farm became more of a mission to where we can invite people and talk about God and spread the kingdom of heaven here on earth. It's not easy, as you know, probably from your experiences. Not everybody will be willing to be exposed to that truth through our behavior and our large family. We're just trying to at least make people think, like, oh, why is that like that? You know, like, why don't have a, why they are so nice? Why there is no set price per pound? Why I can take what I want and leave without paying for it if I choose to or leave all the money that I can as well? I think that we're just trying to make that very accessible to everybody. We have never been disappointed yet. We never wanted to take this away from, you know, for example, single mothers that, you know, putting up a certain amount per pound would be like, I can't do that. And there are people that will leave more than what they need to. And, you know, we are always grateful for generosity and how God is moving in our lives and to this farm to advance His kingdom, you know. So we're just praying that we are aligning ourselves with His will. So we had an older couple that we were just recently met at the time, Tom Gamden and his wife, Evelyn. She has recently passed, but she was diagnosed with cancer. And he brought her up here to pick some berries one afternoon. And we came up to say hi and, you know, just kind of spend some time with her. And she just grabbed my hand and she just said that, you know, for this one hour while I was here picking berries, I forgot that I had cancer. I was just in her peace.  And, you know, one thing that I always, you know, when we pray in the evening as a family during our you pick season, I always say, Lord, please let every berry picked here on this farm, be a prayer for those people who need it, wherever, however they need it, let every berry just be a prayer for them. What an incredible story and testimony. I want to say a big thank you to Sacred Gate Farms. I'm just sharing their story and, you know, and I did a phenomenal job with that and putting it together and just even thinking about the beauty of when we come together. I know we earlier, we're talking with Justin about loneliness, but do you see the way that God designed it with the human family that we come together, we use our gifts and we're able to bless the community? I just think it is phenomenal to see that in action. And all those blueberries, I wish I could just make them and eat them and put them in my mouth. They looked delicious, right? Something that stood out to me that is applicable to today's conversation is how they have set up their family and their life to be outward focused. You see, they understand that they were created for so much more beyond themselves that this world, this life is not about them. It is about what God has called them to. And so as a family, they set about digging thousands of holes and planting those bushes and praying over each one for the people that would come out of faith. They had to trust that the people would come and that as they picked berries, I love that. Silurian said it's one of her favorites that every berry picked is a prayer. And to see the love of that family, the happiness and the joy of that family, even in all the grueling hard work that it takes to do what they do. But the Lord is using them in such a powerful way and friend you at home as you step out, whether it's into friendships, into being there for other people as a family to be a blessing in your community. Know that your life is not about you. It is about what is God is doing in you to be a part of his kingdom on this earth, what he is doing in this earth.  And you will find blessing and miracles and happiness as you step out and be vulnerable and see what God will do.  Am I forgetting one? I can't remember. There's like eight of them, but you all know what I'm talking about with the fruit of the spirit. But I was just thinking about when Anna was just speaking and sharing, is that isn't it amazing that God is calling us to have fruit, that it's not just for us to hoard in ourselves. And that's what the world teaches, is just have the success, have this fame. It's all about you and hoarding it.  I have created you.  The hope of glory is in you. And as you seek me and as you look to the Father, he's going to bear fruit in you, that you can give out to others, to those in your community, to those in your neighborhood, to those in your family. That's what's important. That's what God wants us to do. You know, we're talking a lot. I just think about even the family and how they are leading in their community in a way. But at the leadership, I think we have taught a lot of ways that, you know, it's how many people are under me. No, it's how many people do you serve? That's what God is calling us. Jesus was the servant of all. So how much more should we be servants? How much more should we just look to the heart of Jesus and see a need and fulfill it? And that's what I love. We saw with Sacred Gate Farms is that they saw a need and they're fulfilling it in their community. And we want to leave you with this today. What can you do for that one person in your family, in your community, wherever God is calling you to go? Because that is the greatest that we have is to give that love out to others through the power of Jesus so that they will know us by our love. We are so glad you joined us on Hope Today. Have a great day. On Tomorrow's Hope Today, discover God's character by exploring the book of Revelation. Author Jeff Kinley examines 13 characteristics of God that Revelation uniquely and vividly illustrates through its foretelling of Earth's last days. Don't miss Tomorrow's Hope Today. We wish us to thank all our faithful viewers whose consistent prayers and financial support have made this program possible.


Hello. Today I'm here to talk about understanding strategy and deception using game theory. A little bit about me, my name is Juno. I work as a red team consultant and also do security research. I moved to Texas in 2018 from Alaska and since then have become really involved with the local hacker community. I currently run DC 214 Dallas's DEF CON group and I've been very involved in the Dallas Hackers Association, which is where I actually got my start speaking about game theory. Outside of the hacker community, I also volunteer and teach through Girls Who Code, Black Girls Code, and it's in the local schools. I've got my bachelor's degree in computer science and economics and I'm getting my master's degree in cybersecurity from NYU. And my contact information is up on the screen if you have any questions or comments or just want to know more. First, a little bit of a roadmap. First, we're going to start by talking about what is game theory and how has it been used previously. And from there, we're going to look at the prisoner's dilemma, which was one of the most popular game theory examples. We're going to look at how to find a simple equilibria and then look at a repeated game where the players repeat the same interaction more than once. Next, we'll look at brinksmanship, how drastic action can force your opponent's strategy to change, and a signaling game where players have the opportunity to trade information that may or may not be true using a very simple attack-defense scenario and we'll look at finding an equilibria there as well. And there are a couple of definitions to keep in mind, the first of which game is the term I'm going to use to refer to the interaction that we're modeling, and players is the term we'll use for the ones doing the interaction, whether or not they're human. And rationality is defined by economists is a little bit different. It's defined as doing what is best for oneself 100% of the time. Now, that's quite a different definition than I'd say most of us have, but we need this as a rule in order to explain why players choose the highest payoffs in our models moving forward. So we're actually going to start with one of the most common game theory examples, the prisoner's dilemma. You're probably familiar with it. It's pretty popular either in the pilot episode of half a dozen cop shows or intro to psych and philosophy textbooks. But if you're not familiar, it's really where two individuals are arrested after committing a crime. They're separated and offered the same plea deal. That gives them two choices. The first choice we'll call defect, confess and you will go free, only your partner will be charged. Or cooperate, stay silent, don't say anything. However, if both players choose to defect, both will be charged. If both cooperate without communicating, both will get a lighter sentence. Now, obviously that's the best option, but is that what really happens? So here I have a payoff matrix for our two prisoners, Alice and Bob. And Alice and Bob both have the option to cooperate or defect. And the numerical values I have assigned as the payoff just represent the amount of good to each individual player. For example, if Alice chooses to defect while Bob chooses to cooperate, she gets that highest payoff of two, the best possible outcome for Alice getting off free. While Bob gets that payoff of negative one, the worst possible outcome for him, which is getting the maximum sentence. Now, because Alice and Bob are separated, they're essentially making their decisions independent of each other. This is a simultaneous game. So all each player has to go off of is what they believe their opponent will choose to do. So if Alice believes that Bob will choose to cooperate, she'll choose to defect for that higher payoff. Similarly, if she believes Bob will choose to defect, she'll choose to defect as well again for the higher payoff. And because Bob's in the exact same situation and has the exact same payoff, he'll make the same choices choosing to defect if he believes Alice is going to choose to cooperate or choosing and choosing to defect if he believes she'll choose to defect. This leaves us with a logical outcome of the game or the Nash equilibrium where both players defect. Now, obviously, this isn't the best outcome of the game for either player. If both players choose to cooperate, we have what's called a Pareto optimal outcome, an outcome that's better for both players. However, in real life, this doesn't just happen in a vacuum. Nothing in my previous model addressed any of the potential consequences of these decisions outside of the direct interaction. So we can start to make the model more realistic by looking at a repeating game where Alice and Bob end up in the same situation over and over and end up having to choose whether to cooperate or defect each and every time again and again. And when the same interaction happens repeatedly, players can begin to use a punishment strategy to motivate each other to make the choices that they would want. And there are many types of punishment strategy, but we're going to look at what's called the Grim Trigger punishment strategy. Cooperate until betrayed and then choose to defect forever after that. Now, obviously, this leaves the player open to betrayal at the beginning, but there's retribution if the player is betrayed. One thing we have to keep in mind when we're looking at the math for this, though, is the discount factor, which is a measure of patience. Essentially, future payoff is worth less than present payoff. $10 now is worth $10, $10 two days in the future is worth 10 times D, a number from zero to one that represents the decrease in payoff. So if you're in this scenario and you believe the opponent is using a Grim Trigger punishment strategy, you know that as long as you haven't defected, your opponent will cooperate with you. So we can find the payoff to cooperate is going to be equal to one minus the discount factor multiplied by one. The best possible payoff you can get in the scenario if you're sorry multiplied by one, the payoff you get for cooperating plus one, the payoff you get for cooperating multiplied by the discount factor on out and repeating out into the future. And this actually just simplifies down to one. And then if you choose to defect, your payoff is going to be equal to one minus the discount factor multiplied by two, that payoff you get for defecting right now plus zero, the best zero, which is the best possible payoff you can get after choosing to defect. Now that you know your opponent is choosing to defect every single time, multiplied by the discount factor on out into the future, and this simplifies down to two minus one D. And so when we, and when we put these equations together, we find that the payoff to cooperate is greater than the payoff to defect, as long as the discount factor is greater than one half. So for cooperation to have a higher payoff than defecting, the future payoff must be worth at least half of the current payoff. However, if you do choose to betray your opponent, you know that they will play defect indefinitely. So then your payoff to cooperate is going to be equal to one minus the discount factor, multiplied by negative one. The payoff you get for cooperating when you know your opponent is going to defect, plus zero, the best possible payoff you'll get the next round with your opponent defecting, multiplied by the discount factor on out into the future, which simplifies down to D minus one. And the payoff to defect is going to be one minus D, multiplied by zero, the payoff you get for defecting along with your opponent, plus zero, the best possible payoff you can get multiplied by the discount factor again on out into the future. And so the payoff to cooperate in this scenario is only greater to the payoff to defect when D is greater than one, which isn't possible because it's a percentage. So if you believe your opponent is using a grim trigger strategy, once you have defected, it is better to continue to defect. So next we're going to talk about brinksmanship, which is defined as the practice of trying to achieve an advantageous outcome by pushing dangerous events to the brink of active conflict. So here we have another scenario between Alice and Bob, where Alice and Bob are playing a game of chicken. And each player has the option to swerve or to stay. And if both players swerve, they tie. If one player swerves while the other one stays, the one that stays wins the game of chicken. However, if both players stay and neither of them swerve, they crash. Again, like in our last model, we can translate this into a payoff matrix, essentially ranking the outcomes for each player. So like our previous game, this is a simultaneous game, and all our players can base their decisions off of is what they believe their opponent will do. So if Alice believes that Bob will choose to swerve, she'll choose to stay to win the game for that highest payoff of two. And if she believes that Bob will choose to stay, she'll choose to swerve to avoid the crash and for that higher payoff of zero. And Bob is in the same situation and has the same payoffs, choosing to stay if he believes Alice will swerve and swerve if she stays. So our two Nash Equilibrams are where Alice stays and Bob swerves, and Alice swerves and Bob stays. Now, what if they had different motivations? What if instead of Bob being Alice's opponent, Alice's opponent was say a stuntman in a stunt car who could take a crash without as much damage as she could? In this scenario, our stuntman would choose to stay whether or not he believed Alice choose to swerve or stay. So this makes our Nash Equilibrium an outcome where Alice chooses to swerve and the stuntman would choose to stay instead of what we saw with two equally matched opponents like Alice and Bob. So going back to the Alice and Bob example, here we have a decision tree that models the same payoff matrix we showed for Alice and Bob. Alice has the choice to stay or swerve, and then next Bob has the choice to stay or swerve. Now, because this is a simultaneous game, Bob doesn't actually know whether Alice has chosen to stay or swerve. Swerve, he doesn't know which node on our decision tree he's at, and so this dotted line between our two Bob nodes denotes that lack of information. So, and brinksmanship is essentially the act of removing one of these options. So let's say, as Alice is driving, she calls out, hey, I cut my brakes. I can't stop. What Alice essentially does in that scenario is she removes half of the tree. She says, I am not going. I cannot swerve. I'm going to stay. You have to make your choice based on that. And so then Bob knows exactly where he is. He has to choose to stay or swerve based on his knowledge that Alice will choose to stay. And that's brinksmanship is, is essentially removing half of the decision tree using a dangerous action to force your opponent into making a decision based on that. So next, we're going to look at a very simplified down attack-defense scenario between two players, an attacker and a defender. And essentially the defender has the choice to choose to monitor their stuff or not. And the attacker has the choice to attack a system they come across or to ignore it and move on to the next thing. And so, and if an attacker believes that a defender is going to be monitoring a system, they're not, they're going to pass on attacking it. However, if they believe the system is not monitored, they're going to choose to attack. Likewise, if a defender believes their systems will be attacked, they'll choose to monitor, but if they believe they won't, they won't choose to monitor. So unlike in our previous games, we don't have a Nash equilibrium or a logical outcome to the game. So to start to find an equilibrium, we're going to look at P, the likelihood the defender will choose to monitor their systems in the first place. And so for an attacker, the expected value of attacking something they come across is equal to P, the likelihood the system is monitored, multiplied by the payoff the system they attack, they get if the system they attack is monitored, that payoff of negative 10, plus 1 minus P, or the likelihood the system is not monitored, plus 10, which is the payoff of attacking a system that's not monitored. So the expected value of attacking is going to be equal to negative 10 P, plus 10, multiplied by 1 minus P. And the expected value of passing is always going to be equal to 0. So when we set those equal to each other, we get P is equal to 1 half. This means that P is equal to 1 half, or if it's 50-50, whether the system is monitored, the attacker is forced into indifference about whether or not they attack. And we can see this with this graph that I have here. Essentially the expected value to the attacker goes down as the likelihood that the system is being monitored increases. And so by changing P, defenders can actually force the attackers into indifference or into making a different decision. Now, what happens when one player can lie? We saw how information can change an outcome of a game, but how do we actually change that information? Well, using a signaling game, we can start to look at what happens when a player starts to signal with information that may or may not be false in order to force their opponent into a different outcome. So for our signaling game, we're going to look at the same game with two players, attackers and defenders. But this time our defenders are either high or low quality, and attackers want to attack low quality defenders. Now, any defender can signal that it's a high quality defender and its systems are secure, but signaling comes at a cost, and this cost is higher for low quality defenders. And I should talk for a minute about what a signal actually looks like because there is a big difference between a credible signal and one that's not. For example, if a company tweets on Twitter, we're unhackable, we're going to see them go down in probably the next three hours. But a lot of signals we see are things like buying overpriced security systems that never actually get properly implemented, stuff like that. And this is a bit of an abstract example, or this model is a bit abstract, but it's good to keep those real world examples in mind. Now, one thing to keep in mind is the attacker does not know what capabilities the defender it comes across has. This makes this game a game with incomplete information. So here I have another decision tree for our signaling game. And the first move is just a move by nature, determining whether any given defender is high or low quality. And we're going to say high quality defenders appear at a percentage of one third or 33% of the time, and low quality defenders who can't monitor their stuff appear at a percentage of 66% of the time. Next is the choice by the individual defender to signal that they're a high quality defender or not. And of course that signal comes at a cost.  And the dotted line between these attacker nodes here signals that the attacker does not know which part of the tree that it's on. The attacker knows only that they've seen a signal, whether or not the defender is high or low quality. So first we're going to look for a separating equilibrium where different players do different things. So we're going to say if a defender is high quality they'll choose to signal and if a defender is low quality they'll choose not to signal. In this scenario if an attacker sees a signal they'll choose to pass knowing that high quality defenders are signaling 100% of the time. And if they see no signal they'll choose to attack knowing that low quality defenders are not signaling 100% of the time. Now this equilibrium actually fails because if we see our low quality defenders are choosing not to signal and are getting attacked for that payoff of negative 10. However if attackers are passing on 100% of signals it's worth it for low quality defenders to choose to pay to signal for that payoff of 10 instead. And so this separating equilibrium doesn't work. So next we're going to look for a pooling equilibrium where all defenders choose to signal. If all defenders are signaling attackers are going to attack X% of the time based on the probability the signal is coming from a high or low quality defender. And the probability of the signal coming from a high quality defender is one third, the rate at which they appear based on nature. And the probability of signals coming from a low quality defender is going to be equal to two thirds based on the rate they appear in nature. So if attackers attack X% and the payoff for attacking a high quality defender is negative 10, the payoff for attacking a low quality defender is 10. So the expected value to an attacker of signal or of attacking given that they've seen a signal is going to be equal to the probability the signal is coming from a high quality defender multiplied by that payoff of attacking a high quality defender that negative 10 plus the probability the signal is coming from a low quality defender multiplied by the payoff of attacking a low quality defender. And this ends up being equal to three and a third. The expected value of passing is always going to be equal to zero. So in this scenario 100% of the time the expected value of attacking is greater than the expected value of passing. So attackers will attack 100% of the time. So next we're going to look for a semi-separating equilibrium. And that's where if a defender is high quality they'll choose to signal and if the defender is low quality they'll choose to signal X% of the time. And so attackers, given that they've seen a signal, will choose to attack Y% of the time and given that they've seen no signal will attack 100% of the time. So the probability, and here I have the equations for the probability that the signal comes from a high quality defender and the probability that comes from a low quality defender, which we'll need later. But I won't get into how I found that just right now. So first we're going to find Y. The probability the attacker will attack a defender given that they've seen a signal. And to find that we're going to look at the expected value to a low quality defender. The expected value to a low quality defender of signaling is going to be equal to Y. The probability they get attacked multiplied by the payoff of sending a signal and getting attacked, that negative 13. Plus 1 minus Y, the likelihood the player is not attacked, multiplied by the payoff of sending a signal and getting passed on, which is that payoff of 7. And we'll compare that to the expected value of not sending a signal, which is the payoff of getting attacked, or negative 10. And when we set those equal to each other, Y is going to be equal to 17 over 20. Meaning that attackers will attack if they see a signal 17 times out of 20. And so what this ends up looking like is the expected value of signaling decreases as Y increases. And when Y is equal to 17 over 20, low quality defenders are indifferent about whether or not they signal. And so from there, we're going to do the same thing to find X. The probability a low quality defender will signal their high quality. And that's going to be based on the expected value to an attacker of attacking, which is going to be equal to the probability the signal is coming from a high quality defender, multiplied by the payoff of attacking a high quality defender, plus the probability the signal is coming from a low quality defender, multiplied by the payoff of attacking a low quality defender. And the expected value of passing is always equal to 0. So when we set those equal and solve for X, X is going to be equal to 1 fourth. So in this scenario, a quarter of low quality defenders will signal that they can monitor their systems. And so again, we can graph the expected value to an attacker. As X, the likelihood that a low quality defender is choosing to signal increases, the expected value of attacking increases along with it. And when X is equal to 1 fourth, attackers are forced into indifference. So when we run through this game with these numbers, we find a semi-separating equilibrium where a quarter of low quality defenders will signal that they're high quality, and attackers will attack given that they've seen a signal 17 times out of 20. So where do we go from here? I hope to have given you a good demonstration of how we can break down conflict into steps. And by going through those steps and determining the payoffs, each step of the way we can start to model the outcomes using game theory. We can also watch how strategies interact and can change each other. And I want you to think about where do you see some examples of some of these similar strategies being used on a regular basis? What do you think you can start to apply this to? Now obviously there are some limitations. I'd say one of the biggest limitations of game theory is determining payoffs and motivation in a realistic fashion. The numbers I use in these examples were just that. Examples to make the math come out relatively nicely. However, when determining payoffs, I often try to look at it as essentially ranking the outcomes for each of the players and from that doing the amount of net positive or negative, but there are a lot of different ways. And it's often hard with some of these more complex games when there are motivations you potentially may not have been able to account for. And as these games get more and more complex, of course they're harder to model, especially without the use of a computer. But we've been using game theory to model conflict throughout centuries and I believe when applied to cybersecurity we can create complex and robust models of threat behavior. Anyways, my name is Juno. Thank you so much. I have my contact information below. Please feel free to reach out with questions and I have some of my sources and further reading up there. Thank you so much.


All right, so welcome back to the second half of the seminar with Changlong Zhang telling us about elliptic losses by the periodic module and the 3d meter symmetry. So please, please go ahead. Thank you. Yeah, so the next part is about this, the materialistic operator in elliptic case. So I use the same notation for, I mean, both of them are denoted by T, but since they are really in different context, I think there should be no confusion. So, yeah, we have just constructed this t-state group algebra and also this periodic module, which are vector bundles over A cross H check. Okay. Again, let me remind you that A is equivalent to a module point and the H check is equivalent to a module G of a point where the torus have to be has to be used replaced by this T check. Torus for the London's dual. Okay. So you really have to consider things over A cross H checks in the same time. So then we can define this the materialistic operators for elliptic case. But for that one first, we have to define the coordinate function just like when you write on this k-series the materialistic operators you have to use this e alpha notation. And so they are basically coordinate functions. So for the elliptic case, the coordinate function are defined as follows. So the ability a is t low star tensor e. We know this is basically e to the n spa and the coordinates are defined as follows. So I give you any alpha in t alpha star, any root or any character, you can define a map going from this ability to e by sending this tensor to this pairing between alpha and mu check. So mu check is a core character. So you take the pairing with alpha and alpha check, which gives you an integer and then you can multiply with t. T is an element in capital E. Okay, so this linear map extends. And so if you take an element C from A, you denote its image to be so the c alpha is denote to be chi alpha of C. Okay. So that's like the coordinate of C at the alpha direction. You can do the same thing for the other ability variety. So if you have Linda in mu cross t, mu tensor t, you can again. So after you fix alpha check, which is a core character, then you can consider the pairing of our check with mu, and then which gives you an integer and then multiply with t. So that's the coordinate of Linda at this alpha check direction. And that's chi alpha check of Linda. Okay, so that's the meaning of the notation C alpha and Linda check. So keep in mind that both of them belongs to the elliptic curve. Okay, so with that, then, first, let's recall this data function, Jacobi's data function, which is the product of so for you in C star invertible complex number you can define you are one half minus you to the negative one half, and then take the product, this infinite product with this one minus q to the s u multiply with one minus q to the s u to the negative one. And then there's an actual piece here. Okay, this has nothing this is does not depend on the variable. So here q is the parameter that you choose to define this elliptic curve. So that's basically the multiplicative version of the Jacobi's data function and we can, that's a homomorphic function over some double cover of this C star because of this one half exponent. And the first we really like to work with the additive version so we can replace this theta by the other theta. So this is the curly theta that's the normal theta. Okay, so in this case, this x belongs to C. Okay, so many other way, but they usually use this theta but we like to use the other one. That's more convenient for us. Okay, with that, then we can define this term as realistic operators. She alpha. So it's basically again, it contains two parts. So here I can. He's the casey reversion. So it has two fractions. So similarly here we also have two fractions. So all the fractions involve the theta function. Additive version of the theta function. And then the variables H, which belongs to E that we fixed at the beginning. See alpha belongs to E, then alpha check belongs to E. So this is difference of two elements in capital E. And so that's why we have that's why this theta makes sense. So we consider this fraction. Okay, and we have this fraction. And also notice that for the denominators, the denominator, denominators are the same which is similar like the case theory case. Denominators are the same. And also if you look at the zeros of the pose of C. You have a zero, you have a pole at zero, you have a pole at zero, now you have a pole at E alpha equal to one, a pole at E alpha equal to one. And the other thing is like, so for this part, you also have H minus C alpha, this corresponds to this one minus Q E alpha. Okay, also I made a mistake in the notation that is this Q, in the case theory case, corresponds to the C star action on the flag right. So this Q has nothing to do with this Q. It's just, I shouldn't have used Q here, I guess. This is more standard. Okay, the two Q have no relation. And also for the case theory case, you have this one, the one group action S alpha. For the elliptic case, you also have this S alpha. Okay, the actual piece is this S alpha D, which appears in both of the two fractions. Remember that S alpha, X on A, S alpha D, X on A check. Okay. So this is the formula obtained by Emmanuel Weiber in their work in the geometric construction of the elliptic classes. They use Poisson resolution and they studied the recursive properties of the elliptic classes, they obtained this formula. One can also obtain this formula if you studied the work of Feather, when he was talking about this, studying this dynamical quantum Young-Baxter equation. He also come up with some solution and the solution involves some fractions, which will be just the two. I believe the his work is for type A, I'm not familiar with other types, but for type A, if you look at his solution, he will have some fractions, the fractions can be put in here to obtain this T alpha, this operator. Okay. And one can also define the non-linear version. So we call it T alpha D. They are very similar with each other. Okay. There's a symmetric property here, which I may skip for now because of the time, but they look very similar with each other. So for instance, you see the numerator that's C minus lambda, this is lambda minus C, and that's H minus Z, that's H plus lambda. Okay. So there's some similarity between the two. So the elliptic version of the modularity calculator and now they satisfy T alpha square equal to one and T alpha D square also equals to one, and also they satisfy the graduation. So that's the elliptic version. So with all the construction of this, the modularity calculators and this periodic module, we can start to mention the main result. The first part is that we prove that T alpha and T alpha D, both of their original sections of S. Let me remind you, S is this drug sum of W inverse alpha star, V inverse D star L, inverse tensor with L. Okay, that's the twisted group algebra. So this T alpha and T alpha D, both of their original sections of this S. Okay. So to prove this, one has to study the zeros and posts of this T, and also by using the property that over elliptic curve, if you have a rational section of some limando, the zeros and posts of the function uniquely determine the limando. Okay. So the other property is that we can define the other result is that these limani webers. Elliptic classes can be obtained by applying this TW inverse applied on C. Now what is C? C is just some rational section of L that you fix at the beginning. Okay, after you fix this, you can let TW inverse act on this periodic module. You will get elliptic classes and then one can prove that this elliptic classes corresponds to the limani webers. Elliptic classes and also so that means they are also equal to agonistic or concoves. Elliptic stable envelope. Okay. I don't know that this C is always there because in their definition in agonistic or concoves definition, they have to fix a polarization. This polarization corresponds to the C. So in other words, if you choose suitable polarization, you will get the result that you can identify this with their work. Okay. And you can do the London's dual version. Okay, so that's the second main result. And now the last one is more related to the 3D middle symmetry that we mentioned before. That is, you can take the sum of all the demodulistic operators. You can also take the sum of the demodulistic operators for the London's dual version. And both of them can be realized as maps between periodic modules. The first one can be think of as maps from M to MD. The other one is a map from MD to M. Okay. Also note that in the definition of the T, there are a lot of the denominators. So they are not, so the T here are not global sections, they are only regional sections. Okay, because they contain holes. So the result we have is the following. That is the T and TD, they are dual, they are inverse to each other. Okay. So with that one can recover, so then recover 3D middle symmetry result of Riemannian Weiber. Okay. So that's the main result. So any questions? We're good. Okay, so I have 12 minutes. So first let me mention some remarks and then if there's still some time I can talk about the idea of the proof or maybe some ideas. So the first one is that in Riemannian Weiber's work, they also have another recursion. So if you have this Bohr-Samson recursion, they have another one, they call it our matrix recursion. We believe that that recursion can also be obtained in this way. That is, it's more like if you have TW, inverse, ectome, something. Now, if you apply a T on the left, you get some recursion. That's the Bohr-Samson recursion. If you apply T in the middle of the two objects, you get the other recursion. Okay. So this is not verified, but I believe this is true. So one just have to write down the calculation and then to verify with their calculation. Okay. And the other one is this Bohr-Samson, sorry, this free Makai transform. So one can define HD to be the elliptic HEC algebra with dynamical parameters.  Okay. Some algebra object over this A cross A check. And then there's also the definition of the Skinsburg-Caplanov-Vasserot elliptic HEC algebra they defined in 1996. That's the reconstruction of HEC algebra. They define some elliptic version of the HEC algebra. There's no demodularistic operator because they do not satisfy the operator ratio, but the algebra is well defined. So you can look at the HEC modules and they are, this gives you a subcategory of the coherent shifts over A. Similarly, you can do the Laland-Stewell version, which gives you a coherent shift, a subcategory of the category of coherent shifts over A check. Now in the middle, you have this coherent shift over A cross A check. And then this competition is called the free Makai transform. That is, you take F, you transfer with F, transfer with the concrete lambda. Sorry, pull back. So if you have a shift over A, you pull back to the product, and then you transfer with the P, the concrete lambda, and then you push forward. Q, so Q low size, push forward. Sorry, that's another Q. Okay. So the competition of the two gives you an equivalence of categories between the two categories, sorry, the derived version, you have to derive them. Okay. It does not give you an equivalence of categories on the obedient categories. You have to use the derived category. That's the so-called free Makai transform. What you can prove is that one can start from the hack, the GKB hack module, and then you can pull back to get a module over A cross A check, and then transfer with L, and then you push forward to this category. Okay. And then this competition will prove that this is the equivalence of categories. And in this case, you don't need to take the derived category. Just the current category is good enough. Okay. That's some work of myself. And this should appear soon. So that's the second remark. And the last one is that about properties about like models. So we know that like models over E are uniquely determined by their sex, their sections. And the zeros of status zeros and the pose of data functions are uniquely determined by quasi periodicity of the data functions. So all the four parts uniquely determine each other. Okay, so that's why if we can determine the zeros and posts and some elliptic classes, you can recover the line bundles. Conversely, if we know the line bundles over E, we can recover the zeros and posts of the expression section. So they're almost, so I mean, one can use the information from one side to get information about the other side. Okay. So I have eight more minutes. So any questions? Very good. Good. Okay. So maybe I can say a bit more about the idea of the proof. So this is the construction in this tensor triangle. Sorry, this tensor category. So remember we define this twisted group algebra to be this one. So SWV is basically the, the piece at the grading WV. And the tensor structure is defined as follows. You just define to be so the tensor of this part, this WV piece with this XY piece to be this one. And by definition, this tensor is seismic to this, this WX, VY piece. Okay, now if you write down the rational section of this, I mean, for people in commentaries, they like to work with actual elements. So what you can do is you can look at the rational section, say a, sorry, a rational section here. But if you want to indicate the degree, WV, you can use constant Kuhlman notation. Similarly, if you want to take a section of SXY, you can take a rational section B, but if you want to indicate X and Y, you have these delta X and delta Y. And of course things are confusing here because this V are really the dynamical part. So that's why in this case we have to put a D here to distinguish between the two. Okay. And the product, if you write on the product, it will be equal to the constant Kuhlman twisted product. That is, it will be equal to A, WVD of D, delta WX, delta VY. And there's a D here. Of course, it's easy to see that the two Y group actions should commute with each other because they act on different pieces. So that's why it doesn't matter how you write WVD or VDW, it doesn't matter, it will be the same. So this product is really the constant Kuhlman twisted product. The only difference is that here you have two Y group actions. Okay, but they commute with each other. So that's why this is still well defined. That's the meaning of this tensor product. Okay. The other point I want to make is that, remember we defined the T and the modulostics for London dual as regular sections of S. But if you can, you can change the S a little bit. TWD will be still there and they can be think as functions from maps from this to here naturally because remember we defined SWV to be this tensor product. Now if you apply W up style on both sides, you're going to get this identity. W up style SWV equal to this line model tends with this line model, but that's the inverse so you can automatically write it as maps from the first line model to the second one. So this model is saying that you can think of it as a map from M to MD. And similarly, if you apply VD style on this identity, you are going to get this tensor which will be giving you this one. So that's why you can think of it as sort of like maps from MD to M. So that's the reason why this TW and TWD if you take the sum of them, they will give you maps from M to MD and the other one maps from MD to M. And then as I said, they are inverse to each other. Okay, so that's the construction. Okay, so, so that's why this T and this TD they are inverse to each other. I'm sorry. I mean, they are maps going opposite direction and then one has to prove that they are really inverse of each other and this proof does not mean it's not technical because one can just use some functionality, sorry, canonical properties and a little bit calculation to prove that they are inverse to each other. Okay, there's no technology involved. Okay, so I think three more minutes. There's also the temporary pairing. So I think I don't have time to go into detail here so I will be very brief here. That is, so we know that this Twisted Group Algebra act on this periodic module. And also, there's another version that's not the London dual just the same version. We are still in the same context of the new bundle of GMP. There's a variant form of this as we call it S prime. And you can also change the M into M prime. So then temporary pairing is really tearing between this M and this M prime. So it's not over the same M is M tensor with M prime. Okay, the tensor of the two will be asymptomatic to this line bundle. This is a line bundle over A cross A check. And this line bundle is defined as follows. This O tilde is defined to be the following. So we know A cross A check you can project to the first component A and then we know we have this coordinate map chi row. So then this O of 2H, this is a line bundle degree zero line bundle over E. The corresponding real divisor is 2H minus zero. 2H is a element in capital E.  So the tensor of the two will be asymptomatic to this degree zero line bundle. So that's why if you fix a rational section of this degree zero line bundle, which is this fraction. One can use this to define a pairing going from M plus M check, M prime, sorry, to O of A cross A check. Of course, we know A cross A check is a billion varieties. So this is basically constants. Okay, so that's the meaning of this Pancré line bundle. So why is to use the Pancré line bundle to, sorry, Pancré pairing to transform between elliptic classes here. And there's another version we call EW prime. So then we have that. So if this is EV, this will be equal to delta WV. So this is the Pancré pairing in this elliptic case. Okay. So basically the construction and the proof of the main result uses the two things. One is this Pancré pairing. And the other one is this construction of this crystal group algebra. Okay.  So I will stop here. Thank you. Thank you very much for a very interesting talk.


Hello and welcome to Chanakya IS Academy. I'm Priya Kumari and today daily current affairs topic is Solidarity and stability deal. So points of discussion for this topic is why this topic is in news, then background, the deal, rationality, then Gulf cooperation, then GCC and India. So let's see one by one. So why this topic is in news? Recently, Gulf leaders have signed a solidarity and stability agreement towards ending the diplomatic rift with Qatar at a summit in Saudi Arabia. The announcement at the Gulf Arab leaders meeting in the Saudi Arabia with the annual summit taking place amid a breakthrough in the dispute between a Saudi-led bloc and Qatar that started in June 2017. So what is the background behind this whole issue? In 2017, Saudi Arabia led an Arab transport blockade against Qatar ostensibly to punish Doha for its ties with radical Islamist groups. The bigger motivation probably was to pressure Qatar into reducing diplomatic and economic relationships with Iran, Saudi Arabia's great rival in the region. Coalition countries put 13 demands as conditions to resume relations, including shutting news organizations such as Al-Jazeera, closing a Turkish military base in the country and downgrading ties with Tehran. So what was Qatar's role? Qatar has gone from being one of the poorest Gulf states to one of the wealthiest, the presence of large gas reserves contributing significantly in helping it to become an influential player in the region's politics. Qatar called the whole embargo a violation of international laws and instead strengthened ties with Iran and Turkey. Qatar also denied backing Al-Qaeda and Islamic state, however. What is the Solidarity and Stability Deal? Members of GCC signed a deal in Al-Ula, Saudi Arabia to remove all the sanctions over Qatar and reopen their land, sea and air borders to Qatar. Bahrain, Kuwait, Oman, Qatar, Saudi Arabia and UAE are the members of GCC. We will see next what is this GCC. So leaders of six members, Gulf Cooperation Council or GCC, which I am saying, signed the Al-Ula Declaration which is named after the Saudi city where the summit is being held and a final communique. For the rationality behind this deal, so the rationality is to unite efforts to promote the Gulf region and to confront challenges that surround them or Gulf region, especially the threats posed by the Iranian regime's nuclear and ballistic missile program and its plan for sabotage and destruction. This rationality is given by the countries who have signed the deal. There is a desperate need today to unite Gulf efforts to promote region and to confront challenges that surround and to control Iranian regime's nuclear and ballistic missile program. So this rationality was given during the signing of the deal. So how this the whole deal will going to affect? So the summit is seen as positive. Yes, it is positive. It is a bignik of a process to heal the damage caused over the past three and a half years which started from 2017. It would open its borders to Qatar despite lingering issues between the neighbors. A positive summit unifying ranks and establishing fraternity. The current changes and challenges surrounding entire world require genuine strength, cohesion and cooperation as well as depth and stability. So what is this Gulf Cooperation Council or GCC? This is a political, economic, social and regional organization which was established by an agreement concluded in 1981 among Bahrain, Kuwait, Oman, Qatar, Saudi Arabia and UAE in view of their spatial relations, geographic proximity, similar political systems based on Islamic beliefs, joint destiny and common objectives. The secretariat is located in Riyadh, Saudi Arabia. What was India's role in this whole situation? So India did avoid involvement in local or regional disputes in the region since Indian interests do not entail power projection but necessitate peace and regional stability. The Gulf is among India's top trading partners. The deepening energy interdependence is marked by growing volumes of energy imports into India. The number of Indian migrant workers in the region stands at more than 7 million. The expansion of the political engagement has been matched by the growing security cooperation especially on counter terrorism. India and its Gulf partners are also taking tentative steps towards defense cooperation. So that's all for today's topic. We will see you in the next topic. Till then, thank you and goodbye. Don't forget to like, share and subscribe to our channel and press the bell icon to never miss an update.


So, again, they can give you mixed radicals, adding subtracting mixed radicals. So you can have something like, let's just stick with 2 square root 8. 2 square root 8. And then let's go minus 5 square root 30. Okay? Now 2 square root 8, we just did this, but it would be 2, 2, 2. Paramount, they come out as singles, there's a 2 waiting for, so it multiplies the 2, becomes 4 square root 2. Minus 32 is 5 2s multiplied together. It's 4 times 8, 4 breaks down to 2, 2, and 8 breaks down to 8, 2, 2, 2. So it becomes 2, 2, 2, 2, 2, 2. Square root means bring out those 2s as a single 2, bring out those 2s as a single 2. So 2 times 2 is 4, it comes out as 4, because those 2 come out as a 2, those 2 come out as a 2. What's waiting for them is a 5, so 4 times 5 is 20, square root 2. Now, because these things are exactly the same, you can combine it like terms. So 4 minus 20, 4 root 2 minus 20 root 2 is just 4 minus 20, 4 of this minus 20, that's the same thing. So it's just become negative 16 square root 2. And that just means it's a negative number. This is not a negative in the exponent where you flip the whole thing, this is just a negative number. Don't get confused, I'm going to say it again, don't get confused between what's in the base and what's in the exponent. This is just a negative number, and that would be the answer to that, because when you get radicals like that, you have to break them down so they're identical radicals for you to be able to combine them, otherwise you can't combine them.