If you sense the call of God on your life, then you are going to want to hear this series. I'm talking about the seven trials of the call. These are seven ways God will test you before he can use you. Seven ways he will process you before he can really anoint you for the next level of ministry. Now all of us are being used by God where we are right now, so long as we're filled with the Holy Spirit. But in order to get to that next place in ministry that God has for you, in order to grow into a new season, in order to grow into a new place of influence and authority, you have to pass your divine test. Now here's the thing about tests. Because once you fail a test, you have to take it over again. And perhaps in your ministry, in your life, in your walk with the Lord, you feel as though you're going in circles. Well it could be that you're not passing your test. I'm going to show you how to do it in this series. First, Stephen Moctezuma is here with me. He's going to lead you in worship and then we're going to get right into this lesson. It's going to challenge you. It's going to bless you. It's going to inspire you. It's going to equip you to fulfill the call of God on your life here is Stephen Moctezuma. So several years ago, before we were actually on television, the Lord had put it on my heart that I would be reaching people through media. Now this is something that I knew when I was 11 years old. But if you fast forward from the time I was 11 years old to probably maybe five, six, maybe seven years ago, there was a season in my life where I was very frustrated in the ministry. And during this series, I'm going to be very open with you. I'm going to be very honest with you.  But while we're going through my journey, we're also going to go through the journey of Joseph from his dream that God gave him to the fulfillment of that dream. So as I was saying, there was this year where I was really discouraged and this had been the third or fourth attempt on my part in response to what the Lord had spoken to me to raise finances to take the ministry onto television. Now at this point, I didn't have any resources or special connections outside of, you know, the resources or special connections that anyone in ministry would have. I wasn't connected to anyone in TV. I didn't personally know anyone who you might consider a big name. I had no political connections. There was no real big financier that I knew personally at the time. And so everything I had done at that point was really all by faith. And to this day, I tried to challenge myself to stay in the place where what I do requires faith because I like to say if where you are doesn't require faith, then you're not in the will of God. And so I remember I had this conversation with the Lord concerning this frustration that I was experiencing because I told him, Lord, you're the one who told me to do this. You're the one who put it on my heart to go into media ministry. You're the one who showed me the vision of reaching the masses through television. And here I am obeying what you've told me to do, yet I'm not seeing any fruit from this. I'm not seeing the vision come to pass.  I was frustrated. I was tired. I felt like nothing was going anywhere. And I know many of you have been there in your life before. You haven't been there. You'll be there at some point because you have to go through this. God will process you this way. So in my rebellious way and my immature way, I spoke to the Lord out of emotion.  It's not a good idea. And so I remember out of emotion, I just said to the Lord, I said, Lord, if you don't come through for me in this season, if you don't give me the breakthrough you promised, I'm going to quit the ministry. And immediately I sensed the conviction of the Holy Spirit. And the Holy Spirit very firmly but very clearly spoke to my heart. I said, if you don't respond, I'm going to quit the ministry. And the Holy Spirit spoke with perfect clarity, perfect love, but perfect authority. He said, then quit and I'll anoint somebody else. And I remember that shook me to my very core. I said, Lord, I don't want, I don't want to quit. I'll keep going. And you know, if the Lord talks to you that way, it'll shape you up rather quickly. And so then another year goes by and I repeated the same thing. Went through the fundraising. It looked like it was going well for a while. Then we hit a wall. There's frustration. And all of a sudden I find myself in the same position praying the same ignorant prayer. I said, Lord, if you don't come through, I'm going to quit the ministry. This time the Lord was more firm with me, reminding me of the way I acted a year earlier. And this time what he said shook me even more. The Lord said to me at that time, you just bought yourself another year. What I didn't realize was that the Lord was testing how I was responding to the frustration. And until I was able to respond to the frustration with faith, I wasn't the person that could have received the fulfillment of what God wanted to bless me with. And so it's important to pass your tests. And so as I said, I was 11 years old when the Lord saved me. I got a vision for ministry from 11 to 13 years old. There's this very key time in my walk with the Lord. I'm spending time with him in prayer. And then I want to now correlate my story with Joseph because this is a very similar season I found myself in. And I want you to consider the scriptures in Genesis chapter 37 verses 12 to 14. Now Genesis chapter 37 verses 12 to 14, this is after Joseph had shared the vision with his family. Keep in mind, Joseph's brothers were very jealous of him because they saw the favor of their father upon him and they didn't like that. So Joseph carries the favor of his father has against him the jealousy of his brothers. And so they decide to never speak to him. The scripture says they couldn't even say a word to him.  Now imagine your family dislikes you.   Well, this stirred the embers of their jealousy deep within their souls.  But instead they decide to do the merciful thing and sell him into slavery as we'll find in the coming lessons. But what I want to focus on here right now is the fact that Joseph received instruction from his father from whom he had favor. Joseph receives the instructions from his father to do something very menial. And the scripture tells us in Genesis chapter 37 verses 12 to 14. Mind you, this is after God gave Joseph the dreams about him being a ruler or a person of authority, of a person in power. Genesis chapter 37 verses 12 to 14. The scripture says, soon after this Joseph's brothers went to pastor their father's flock at Shechem. When they had been gone for some time, Jacob said to Joseph, your brothers are pastoring the sheep at Shechem.  I'm ready to go, Joseph replied. Go and see how your brothers and the flocks are getting along, Jacob said. Then come back and bring me a report. So Jacob sent him on his way and Joseph traveled to Shechem from their home in the valley of Hebron. Now what's interesting to me here is this took place again. I'm emphasizing after Joseph had a dream about his family bowing to him, about him being a person in authority and in power. Now the scripture says in Matthew chapter 20 verse 26, Not so with you. Instead, whoever wants to become great among you must be your servant. Joseph passes his first test here by obeying the instructions of his father even though he already knew there was greatness in himself. Joseph passes the test of number one, service. God will look for a man.  The one who is devoting themselves to prayer and the word and worship, not just in public, but in private. God looks to anoint servants. The scripture says God resists the proud, but he gives grace to the humble. God resists those who want to exalt themselves. In fact, he will humble those who try to exalt themselves, but he will exalt those who humble themselves. And Joseph, even though he sensed greatness within himself, even though he knew he was anointed, even though he knew that he would one day be a man of great power and authority through the vision, through the dream that God had given to him, he obeyed the instructions of his father and did the practical, did the daily, did those things that don't seem like they matter. He passes the test of service. Now, when I was in the ministry, in television ministry, when I started in the television ministry, I should say, excuse me, people said, wow, this guy came from nowhere. In their minds, in their view, I just popped up on their TV screen. But they didn't see the years that had led up to that moment. And please don't misunderstand what I'm saying. I'm not saying that television ministry is the end all for everybody or that being someone who is seen in the public eye is the goal. That's not the goal at all. I'm simply saying that for me, the vision that God had placed in my heart was to use media to reach the masses. And so the beginning of that came and people started saying this guy came from nowhere. But the truth is, I didn't come from nowhere. I was processed. I was serving. I didn't step into the ministry the day I went on camera. There were years that led up to that moment. In fact, I can remember how I started in ministry. I didn't start in ministry on television. I didn't start in ministry on a pulpit. I didn't even start in ministry by teaching the word. I started in ministry the day I was at church. And my youth pastor at the time asked me to run something as simple as a projector. Now, when I was doing the projector in the church, we were using not the modern-day digital projectors. We used what was called an overhead projector.  And on that glass, you would lay a transparent sheet of paper that had writing on it. And that light would go through up into the magnifying glass and it would be projected overhead onto a screen. So I had to literally change the slide by hand. There was no button I pushed that changed a digital slide. I would have to flip the songs. They called them the song flipper. That's where the name came from if you've ever heard it. And so, when I devoted to that ministry, I didn't do it half-heartedly. I didn't say, well, I believe that I'm going to be someone who's used of God greatly one day. I'm going to be in the healing ministry. I'm going to be in the television ministry. I'm going to be an evangelist. Therefore, this is beneath me. No, that service, that projector was the gateway to the pulpit. I did not enter ministry the day that I walked in front of a television camera. I entered ministry the day I stepped in front of that projector. And God saw my heart. I remember saying to the Lord, every song I flipped, I said, Lord, this is for you. Lord, this is for you. I wasn't looking to be recognized. I wasn't looking to be celebrated. I was just looking to please my Heavenly Father. This is the thing about the act of service. It can feel frustrating because it can feel to some like it's a waste of time. Well, it's not. Service is never a waste of time because service never goes unnoticed by the Lord. But too few people want to serve. They want to enter right into what they call major ministry. They want to enter right into the dream. You will never walk through great doors until you can first walk through the door of service. This is the first test. And again, we see Joseph passes this test. We see that even though God had given him these very clear dreams, even though Joseph knew deep within his heart that he was to be a man of greatness, he still obeyed the instructions of his father to do the little things. Are you doing the little things? If God has called you, that's wonderful. If God has anointed you, that's wonderful. If God has given you a gift of healing or a gift of prophecy, those are great gifts. But are you serving? Are you doing those things that are small? Jesus said, if you can be trusted with the little, you can be trusted with much. If you could be trusted with the matters of the material and the natural, you could be trusted with the matters of the supernatural. So number one is the test of service. And next week I'll talk about the next test in our series on the seven trials of the called. I want to pray with you now. And I want to believe that the Lord is going to anoint you and to use you. Let's focus now our hearts on saying, Lord, we want to be servants. We want to be used by you. We want you to anoint our lives. So Father, in Jesus' name, I pray for that one watching now. And I ask, Lord, by the power of your Holy Spirit, that you would reveal to that one watching the areas where they can serve. Father, I pray that as they serve, the anointing of the Holy Spirit would come upon them mightily. In Jesus' name. I want to pray for you now. Just keep your hands stretched toward mine. I sense the anointing of the Holy Ghost. There's someone watching. You were convicted as I was talking. And in your heart, you know God's given you a gift of prophecy, gift of the prophetic. And you've been asking, Lord, when is this going to be recognized? When are people going to recognize this? When am I going to be able to minister? The word of the Lord to you and to everyone watching is simply this. Do what God places before you. And promotion comes from the Lord. Leave the promotion in God's hand. Father, in Jesus' name, I thank you, Lord, for not only the anointing flowing, but your healing power flowing. Someone's just been set free from depression. I thank you, Lord. There's a heart condition being healed right now in Jesus' name. Someone with arthritis and diabetes is being healed.  You have arthritis and diabetes.  Lord, I thank you for the anointing of the Holy Spirit. Thank you, Father. There's someone else watching me when I said arthritis and diabetes. You said, well, I have arthritis, but I don't have diabetes and I want to receive a healing from the Lord. You can claim that, too. You can claim that, too. I thank you, Father, in Jesus' name. Lord, I thank you. And I pray, Father, we would commit ourselves to serve your ministry for your glory, Lord. We'll do anything. In Jesus' name, we pray. And I want you to say, if you agree, say amen. Well, I want to welcome now the new members of Spirit Church. There you are up on the screen. We love you. We are praying for you. I always say that because I always mean it. If you'd like information on how you can join the Spirit family, then go ahead and use the information at the bottom of the screen to join for free Spirit Church. Now, we do encourage Spirit Church members if Spirit Church is their primary church to give their tithes and offering, but it is a 100% free ministry. Now, if you sign up, you're going to get a weekly email from me. You'll be able to reply to that email for prayer support from our staff, and we'll all be one family. You'll be joining a family. There's over 2,000 Spirit Church members from all around the world. I want to read now your comments, and these are your comments from a lesson I did a while back. If you'll remember, I was at a conference, so we had to rearrange the schedule a little bit, and I didn't get a chance to read some of the comments on some of the videos. So I want to read now for you some of your comments from the video, seven keys to answered prayer, which we had a great response from that video. But here are some of the comments now. Christmas Abigail writes, Thank you, Lord. Thank you, Pastor David. Sending love from Nepal. Jay Jerica writes, Thank you, Brother David, for the Spirit-filled teaching. I have noted so many hindrances in my prayer life, but now I can stand. Many thanks to you. May God bless you. May God bless you. Thanks for the lyrics to Our God is Able. I thank your team writing from Zimbabwe. Tatiana Z writes, Honestly, I am really thankful for this message in my life. I faced so many unanswered prayers, I was even thinking that God does not hear me. Thanks for this hopeful message. Thank you, Lord. Thank you, man of God.  Clarissa Luna writes, Wow, this message truly couldn't have come at a better time. I love these Spirit Church teachings because through you, I know God is expanding my knowledge, softening my heart to his message, and building me to accomplish all I was made for. Thank you, David and the team. God bless your ministry. And the final commenter writes, This message came exactly at the right time. God is using you to speak breakthrough in my spiritual life. Thank you, Brother. I'm believing for a big miracle through your ministry. The best thing about you is you remind us that we have to win souls. You have a true heart for the Lord. God bless you abundantly. God is great. Glory to Jesus. And you're absolutely right. My heart is to win souls. And this is where I need your help. We want to keep this content free. We don't want to charge for this because some people, they can't afford it. And we should not charge for the word going out. So you'll notice I never charge for my podcasts. I never charge for my sermons.  So help us keep this channel free and the teachings free. Also help us with our fundraising campaign. Ultimately, the goal is to win more souls. But you know that we needed 1,000 new $30 a month partners to expand to the next level of ministry. And here we are on that campaign. Look at where we are. Look at how close we are to our goal. We need your help. Sign up to become a $30 a month partner today. I'll tell you where that support is going to go. That monthly support is going to go to two things. It's going to go to the monthly support, the monthly costs for our new television studio. Now in the new television studio, we'll be able to make more content, higher quality content. We'll be able to bring in a studio audience, have regular meetings in that building. We'll be able to go live from studio. We're going to have a 24 seven prayer room and we're going to be able to reach more souls than ever before. The second place that monthly support is going is to our ministry events. I want to do more miracle services all around the world because I want to win souls. So help me do it today. Become a $30 a month partner. We're almost to our goal that monthly support is what's going to help us expand the ministry. If you become a $30 a month partner today, I'll send you either carriers of the glory or 25 truths about demons and spiritual warfare. I will sign it as my initiation gift to you in thanks for being my partner. Now remember this support is going toward the facility and toward the events and it's going to help us to win more souls. Ultimately, that's what it comes down to. We want to win souls. So if this ministry has blessed you in any way, if you feel connected with me in any way, let me a helping hand. Let's do this together. We are on the verge of a huge breakthrough as a ministry and you're going to see us reaching more than ever before. God is expanding this ministry. Big, big, big, big things are on the way. So if you would like to give, wait until the end of this video. If you're watching on YouTube at the end of the video, a button will appear that you can click and you can sign it to become a $30 month partner today. If you're watching this anywhere else, like on the app, then use the information at the bottom of the screen. Actually, on the app, when this video is done, you'll see pop-up partner with David. You go ahead and click that. Other than that, use the information at the bottom of the screen. Well, that is it for this edition of Spirit Church. Until next time, remember, nothing is impossible with God. Also, help me win souls by spreading the gospel through events and media. Make a one-time donation or become a monthly supporter by clicking on the donate link now.


This paper describes the design and implementation of a digital research recruitment registry to optimize awareness and participant enrollment for COVID-19-related research in Baltimore. The registry was designed to connect interested participants with recruiting study teams. The initial pilot effort was successful in connecting interested participants with recruiting study teams. This article was authored by Ana Pila, Haley Miller, Alubon Mio Gombe, and others.


So the store and monkey are both gonna be the team captains and it's the ox battle team versus team. For every one round, the other team loses.  I'm gonna go into y'all rooms one by one back and forth and I'm gonna ask you, who are you drafting? So monkey, I'm giving you the first draft pick. Who would you like? It's the first pick in the biscuit ox battle draft. I will select TJ. Let's go. Hey TJ, you the first draft pick, bro. You should be proud of yourself, my brother. All right, distort. So who would you like to pick? With the second overall pick in the 2023 biscuit draft, I'm taking ZCD 27Z. Come on now. Bro, so you taking the Brit? W Brits in the chat, you know what I'm saying? They not gangsters, but they know they juice world. All right monkey, so who do you got? I'm gonna go with Black Hawk. I ain't gonna lie, that's a sleeper chat. I didn't expect that. All right, so distort. Let me go for Kwot, cause she knows her juice world too. That's a horrible pick. No, it's not. Three more picks.  I'm gonna go with plants with my next pick. That's a valid pick. Let me get basketball. I think I'm gonna have to go with Joker.  I'm gonna have to go with Rube, bruh. That's so disrespectful, bro. He chose somebody without a mic before he chose the rest of y'all. I'm gonna lie. I think I got a steal here. I'm gonna go with spooky. One more pick. But right now I'm gonna have to pick factors. All right, boys. The first category, Tiz B. I ain't gonna lie, that's it. Hold on. Drugs. All right, chat. So look, they're choosing drugs. Team one. So what song are y'all going with? Dirty Soda. I got dirty soda on no coast. She do coke, no coke, no coke. I'm gon' flow. I'm gon' brag. I'm gon' boze. Keep the tags on my clothes. You gon' put them scary. Bitch, I'm a dog. Juice world, Terry. I'm gonna rock, rock, rock, rock. If it's the category, but let's see what the other team got. Cause maybe there's fits better. We got purple substance.  I think I need it. Okay, so the chorus is hella drug. Purple substance, I think I need it. I might have to. What do y'all think? Which one, how y'all feeling about drug talk? Which one y'all feel like, you know what I'm saying? I would say purple substance. Yeah, I agree. Damn, that's a wipe. Yeah. Hey, you know what that means, chat?  So who's on here? I'm out, bro. Man, joke me. Get that point out. Bro, you just been a... So you think you're the weakest link? Why are you taking your stuff out? I just trust in these guys. Like, if they win, I win. You two, if you guys were playing, would you have voted yourself out for your team? Cause I ain't allowed to meet personally.  You got a broad mirror, so y'all better kick me out, bro, tie my hands behind my back. Bro, nothing, I'm not, like, no. And it's lips on. Okay. All right, chat. We gonna play their song first. Wake up. We get along almost every way. I write my wrongs every single day. I would die if she ever tried to run away. Got a smile on my face cause I get to wait. The past ain't relevant. I'm right now. I've been through the toxic, toxic women. So toxic that I cannot just be friends. Let go to your team and see what they chose. We win confide and we're gonna stick with that. Yeah, confide. I need to confide. True love's hard to find. Whoa, whoa. If I'm edging for the love, bro, I have to pick up Wake Up. So I think it is also very even. I think Wake Up is more love. Hey, I bet they chose, they chose chat. So they gotta give it to somebody.    We're setting it off spooky. Goodbye, Spooky. So when the other team, like, stopped being so annoying in chat. After Spooky. He's just spamming the chat.  She's a rascal. Man, pull up in real life. We'll see. We'll see who wins, dawg. Oh, god.  Whoa. All right, let's see what the next category we got, chat. Let's see what we get, chat. All right, chat. It's upbeat. So, monkey, what is your song? Uh-huh. Chop on me, go, brah, brah, brah. I made the cash appear and talk, brah. Badger's a shrimp with a dollar sign.  Yes, I'm in a day.  Chop on me, go, brah, brah, brah. Spaceship life so far. I made the cash appear and talk, brah. Chop on me, go, brah, brah, brah. No loaves, baby, I'm too hot. All right, with our pick, we got run it up. I'll make a bet. For random. Run it up, chat.  She gave me top and a back seat.  Drink away my flaws. Sipping money till I drown it all. Jalai drink, I'm just saying you can do better. I got you at it. I got my handle barra- Oh! BVS on the drone. Out, this is not high, bro. This is not high.   My hand on my toe, here, just by the ceiling.  So, which one do y'all think is the better upbeat song or the better song? It's not even, bro, it's Ha Ha, bro. It's not, bro. Yeah, I think it's Ha Ha.  But they're both, it's just 2B1, so. OK, I don't want to kick anybody off the team. Who do you want to leave the boat with?   Why? Because I want to leave.   What do you keep doing? You just call her a board hopper? Oh, man. I'm God, I'm not getting the drama I want.  Y'all niggas, all y'all niggas, I'll leave, God.  I'm trying to hear niggas argue, bro.  I didn't do this for y'all to be happy. OK, chat. So hear me out, listen. The more unique beat. Let me go ask for their song. Let's ask Team 1 for their song.  Uh-huh. What? Um, man. So we had to pivot, but we're going with horrible. I'm surprised nobody played Purple Devil. That's what I was expecting. I was really expecting a Purple Devil. Love that song. All right. Let's go to the next team. We came in with a banger. We got fireflies. All right, chat. Fireflies. We're shooting the more unique beat. Not the better song. I mean, we're probably going to both agree on this anyway. You going with fireflies? Hell yeah. Hell yeah. Yeah, yeah. Oh, God. Let's see who they feel they get rid of. Man, I told y'all we should have just put it in. Bird.  Well, yeah. Comeback season. Hey, yeah. Comeback season. Yeah. Them niggas not coming back, chat. I'm sorry. It's over. It's over, bro. They got three people left. Five, bro. It's over. They don't even got a starting team anymore, bro. That's been the will. Let's see what we get. And it is going to be. Man, I'm trying to feel depression.  You know what I'm saying? Hey, hey, hey. Make me depressed, chat. We're going with maybe. Maybe what? No. Maybe the name of the song. I knew that. I knew that. You know? All right. Let's go to the other song. Let's see what they got. So what y'all so, bro? Our song is my fault. All right. Yeah. It is your fault. All right, chat. Let's listen to my fault.  I can't. I can't keep listening.   W song, too. Do we even got a consult about this one, boys? No, no, no, no. No, no, no. It's over. It's over. I'm not even the juice fan, but it's all my fault. My fault is like top is top tier. My fault is top tier. All right, team one.   Who are you? Who are you going to be?  Bro, he's pretty.  I don't know. I want to hear. Personally, I don't think discrimination is allowed. Hold on.  Are you going to get rid of the British or the racist? All right. While I was gone, it looked like y'all, you chose who y'all got rid of. Whoa, wait, wait, wait a second. You got rid of the Brit. Shout out, TJ. No hate, TJ. I was named. You got rid of the Brit. Yeah, no hate. And you kissed him. I hate Brits though. I didn't think about that, but I knew hate though. No, he didn't say, no, no, no. He said he hates Grits. Grits like the food. That's what he said, Chad. He hates Grits, bro. Like, what are y'all talking about? He didn't say it. No, Chad. He said Grits. I didn't go out.  It's 2v5 right now. Jesus Christ. It's not going to get any better, Chad. I didn't go out. They're going to have to, if they come back, I'll be so genuinely shocked right now. I'll be really shocked if they came back from this. Because I didn't, you know, because I didn't, it didn't see it.  All right, Chad. We need a feature. So we need a song with at least two minimum, two artists on it. You know what I'm saying?  Like, don't play one of them songs, bro. Please don't. Hey, Mr. Biskett. Hello. Why you sound zesty when you said that? Yeah, I can't afford to lose again. So who y'all got? What y'all got? Oh, we went with Hey, Mr. All right. Hey, Mr. Chad. That's a banger. Hey, Mr. Mr. That's a banger song. I don't know what they can do to beat that, but let's go here. Casio. Which one of y'all niggas just called me old? I didn't say it. No, don't be quiet now. Which one of y'all niggas just called me old? Oh, so we quiet now. That's crazy. Niggas is one of each leg.  Okay. Man, you want to go first? I can go first. I'm unbiased because I'm not a juice fan, you know what I'm saying? But even as not a juice fan, I've heard both of these songs somehow. The first one's a little bit overplayed. The second one has it better be. And G Herbo, they went crazy. And G Herbo is not that good of a rapper, but he rapped here, which is crazy. And Juice went insane too. I like the second song way more. This is going to be an uncommon take, right? But I think that G Herbo and Juice are a better duo than Ski and Juice. Oh my God! Hold on, hold on, hold on. I like Ski more as an artist than G Herbo.  No, bro. Yes, bro. I know. So which song are you choosing, Clue? Cash out. What would you pick, Miska? I would have definitely chose Hey, Mister. But I heard that like 20 times. I can't do that. I can't do that against the chat. Look, they voted Hey, Hey. I voted for Hey, Mister, but they chose Cash Out, you know? Because Cash Out is still good, so I'm not even really tripping over it. You know what I'm saying?  That is crazy. I didn't realize there were people that existed that actually liked G Herbo's music. I did not see those. Do you regret the people you chose for your team? No. I put it all on the judges.  So you think the judges, so you blame the judges? I'm not going to lie. There's at least two rounds that got stolen from us. So I have five packs on my table. Tell me which one I should smoke first. Oh my God. Hey, how you going? I'm broke.  Bro, the story is getting too comfortable in all the... He's in B World. Shit.  Yo, yo. Who lost? Who lost? Listen, listen. Go back to the corner store.  Go back to the corner store. Go back. Okay, okay. Bye, YouTube. Hey, subscribe, like, and comment. Y'all need to come in here because these battles are lit. I ain't going to lie.  Trust me, you don't want to miss out. Y'all should be part of these.


Hello. Yeah, so good morning, everyone. So welcome to the morning session of our workshop program. And in this morning, we have three talks. And the first talk is given by Eris Berg. And he's telling us about the basement of France law in strange metal. So let's welcome. Okay, right. Thanks very much. Thanks to the organizers for the opportunity to speak here. So I'm going to talk about the minimum France law and it's an application to strange metals. And these are the collaborators on the works that I'm going to mention and I'll highlight them as I go along some of them are here. So, yeah, so this is my outline so I'm going to start from basically the most conventional strange metal confined. That's a that's a metal tune to the vicinity of a van Hof singularity just as a test case. And it turns out that just just from that you can actually learn something interesting about the minimum France law. It actually is predicted to break down in an interesting way close to a van Hof singularity. And then I'll move on to a to more exotic cases, a strange metals like a marginal family liquids, and I'll discuss what the tickets recording. And I'll discuss what the minimum France law can actually tell us about the mechanism of of a strange metals a little bit more, more broadly. All right, so I'll start with this example of tuning a metal to the vicinity of a van Hof singularity that's a relatively conventional case but beautifully it was actually realized recently in a search to one for very, very clean metal so it's a very good test case. And what's been done is actually applying uniaxial strain to the system so that that can tune a one of the three bands in the system to the van Hof singularity this is a very quasi 2d metal so we can really think about the 2d band structure. But this, this band is a function of strain actually reaches touches the, the zone edge. Okay, and this is a plot of the logarithmic derivative of the resistivity as a function of temperature, and as a function of strain. Okay, so here's zero strain. And this is where the van Hof singularity is that's actually been confirmed by our first measurements as well. And this is temperature. Okay, so you see this beautiful fan shaped a behavior. So it really looks like a quantum critical point today is a quantum critical point but presumably a very trivial one it just it just the van Hof singularity. But you see here very clearly. Okay, so if you look at the exponent of the resistivity as a function of temperature. It's, it's close to T squared on either side of the van Hof singularity, but close to the van Hof singularity and in this quantum critical region. It actually deviates and it looks like somewhere close to 1.5. Okay, and this is the actual resistivity as a function of temperature. Okay, the black one is is just shifted. Okay, this one is the actual data. Right at the van Hof strain. So this is as a function of temperature at the van Hof strain. And in this paper, they actually try to fit it either to T to the power one half, or to T squared log T. And over this range of temperature. Okay from from DC which is, which is enhancing this close to the van Hof singularity three Kelvin up to 40 Kelvin. It actually fits more or less equally well both both forms. And there's a more recent experiment actually by the same group by by a Clifford Hickson company, where they suppress superconductivity by a field, and they actually find that T squared lock T actually fits the data visibly more than than a T to the power of 1.5. Okay, so where is this actually coming from. And if you think about a little bit this is actually somewhat surprising so think about it in the following way. Okay so the only change from zero strain to the van Hof strain is that there's this small area of the Fermi surface and one of the bands that's going through the van Hof singularity so in that part of the van Hof singularity the Fermi velocity is going to zero, but other parts of the Fermi surface are actually not affected but much, at least in terms of the band structure. So it's somewhat surprising that the transport, which presumably would be short circuited by the parts of the Fermi surface that undergo the small scattering rate is actually affected at all by this by this minute change in the Okay so the question is why isn't there no short circuiting of the van Hof singularity by all the parts of the Fermi surface that are that are far away. Yeah. So, so you know I mean it's a, a, a, I think that it is elastic but for this purpose yeah I mean there's no plastic effects the residual resistivity is actually not affected much. Okay. All right. So, yeah, so the question is, you can imagine it in this way, the parts of the Fermi surface that are undergoing the van Hof singularity are somehow becoming becoming hot. Okay there's a very high density of states there, but why. Okay, so maybe there's an enhanced scattering there also, but why aren't they short circuited by all the all the rest of the Fermi surface that actually, which is actually not undergoing any dramatic change. Okay and to answer that you have to think about different scattering processes. So, here's the system where one of the bands is tuned into the one of singularity. And we can, it's useful actually to classify the different scattering processes into these different groups. Okay, so we'll distinguish the cold electrons which are which live far away from the van Hof singularity and the hot, the two electrons which are living in the vicinity of the window of singularity. This would be a cold cold cold cold, a scattering process process of two, a, of two electrons. Okay this would be classified as a cold cold to cold hot scattering process okay so we start from two electrons, which are far away from the van Hof singularity. One of them is scattered somewhere over here, and the other one is scattered into the van Hof singularity Okay, so that's another type of scattering process. And thirdly we can have a cold hot to cold hot so we have we start from an electron far away and an electron at the van Hof singularity, and they scatter by small momentum, such that the hot one remains hot and the cold one remains cold. Okay now we can ask about the contribution of these different scattering processes to transport. Okay so the cold cold to cold cold. Doesn't know about the van Hof singularity so we expect this to have the usual Fermi liquid type scattering, scattering rate of T squared. Okay this cold cold to cold hot process picks up the logarithmic singularity of the density of states from the vicinity of the van Hof singularity. And two other important points to note this process involves a large momentum transfer. Okay, actually might as well be a process so it does show up in the electrical resistivity. However, it turns out that in the presence of multiple bands this process can actually occur anywhere on the premise surface it's not limited to any particular portion of the premise surface so it's not short, short circuited the scattering rate from this process if it's large would actually appear anywhere on the firm surface so it will appear in the resistivity. And finally there's this cold hot to cold hot process, which can occur anywhere on the Fermi surface but always involves a very small momentum transfer because the hot electron has to remain hot. Okay so this would show up a very strongly in the single particle scattering rate but would not contribute much to the electrical transport because it involves a very small momentum transfer. Okay so from the cold cold to cold hot is scattering rate you actually get this T squared log T behavior. So in this paper with Sean and Connie from a couple years ago, we actually applied the same mechanism to a slightly more exotic in a behavior okay in the three to seven the bilayer structure mutate material. Apparently there's a stronger van Hof singularity than this this is just the ordinary logarithmic van Hof singularity. So we invoked that to get to the behavior which is observed there which is actually leader in T resistivity in a specific heat diverges like T log T, okay and possibly that can explain that material. Okay but I think that here things are much clearer. And we believe that this is actually the right mechanism. Yeah. Yeah, so, you know, one, one thing you can do is scatter right from here to here. Okay so that's a very particular momentum. So that that won't won't allow you to scatter cold electrons from anywhere to anywhere on the Fermi surface. Okay, so that that again was not expected to show up in travel. Okay, any other. What's that. Yeah, so. Yeah, so, so in the quote marginal Fermi liquid applies to the three, three to seven not to this material. Okay so there it looks like. Okay, the, the, the resistivity there in a special value of the magnetic field scales like T, and the specific heat is T log T. So that's, that's a behavior which we associate with a marginal Fermi liquid, where the self energy goes like omega log omega. But but this is not here. I'll actually discuss that later, but in this situation what you get is not the marginal Fermi liquid. Okay, it's from the usual from adequate behavior but it's not the marginal. Yeah, any other questions, comments. Okay. So, yeah, so so now the question is, well, there is this, if these cold hot to cold hot processes that they're actually much more singular than the cold cold to cold hot processes and the question is where do they show up so this is small momentum transfer to show up in the electrical resistivity, but these are inelastic scattering processes you might expect them to show up in the electric in the thermal connectivity. Okay, and that's that's what we actually decided to look at. So, just as a brief reminder, the, a widemann France law is is is a relation between the electrical and the thermal connectivity that was actually observed empirically very long ago. That appears in metals. And it's basically the following statement. Okay, so you define the Lawrence ratio, which is the ratio of the thermal connectivity over temperature times the electrical connectivity. And in a very, that in a very broad in variety of metals at at low temperature, this ratio actually approaches a universal value. Okay, so it's, it's pi square over three times a k Boltzmann over e to the power to. Okay, and here are some examples. Okay, so the open squares here are nickel. This is a, a quasi one dimensional metal. The ratio divided by L not as a function of temperature and you see this nice approach to one. Okay, this is a, this is silver. Okay, so so again at the lowest temperatures approaches one. And here's even a heavy Fermion compound. And this is very typical behavior of it starts from one at the lowest temperatures. It deviates down and it actually recovers back to one at higher temperatures, but here I mostly focus on the, on the, on the low temperature bit. Okay, and what's kind of striking is that in these different materials is this ratio really approaches one within a few percent so it's really a quantitative, a quantitative prediction. So what's the origin of this of the of the Whitman-Franc's law. It's basically elastic scattering. Okay, so it comes very naturally out of the of the Boltzmann equation when the scattering is elastic, and the argument is very very simple. Okay, so the idea is that at low temperature both the thermal connectivity and the electrical connectivity are both basically carried by electronic quasi particles. So to get a resistivity of any kind you need to relax either the electrical or the thermal current. Okay, so here's the expression for the electrical current in terms of the quasi particles. And here's the thermal current. Now, if the scattering is elastic, it cannot modify the energy of the quasi particle it can only modify the velocity. So if you see that to relax the electrical or the thermal current you have to relax V of k. Okay, you cannot relax the energy. And therefore it's exactly the same scattering processes that contribute in the same way to both electrical and in thermal resistivity. So that's the origin of this law. Okay, so the explanation of this behavior. Okay, of the of the Lawrence ratio is that at the lowest temperatures, the scattering is elastic, okay it's dominated by purity scattering, then at higher temperatures phonons kick in, and they cause inelastic scattering this is why deviates down. As you as you raise the temperature, as soon as the temperature exceeds some typical if the frequency of the phonons, the scattering from phonons becomes quasi elastic again. Okay, so the phonons can only absorb a energy of order their frequency. So when the energy of the of the electrons becomes higher when the temperature is higher than that, then the scattering from phonons is essentially elastic. And this is why the Lawrence, the Widman-Franc's law actually recovers at high temperatures again. Okay, this is a very, very difficult behavior in many metals. Okay, so coming back. Yes. Yeah. No, so yeah, so what what happens here is that this is a this is a heavy fermion compound. Okay, and at high temperature you have quasi elastic scattering from from the local magnetic moments, and this is the reason for this recovery. Okay, so it's actually quite different physics, but yeah, it's not phonons here. All right. So, coming back to the example of a metal tuned close to a venous singularity. Okay, so what's expected to happen there. So we saw that there are these ch to ch scattering processes that are that involve a very small momentum transfer and don't show up in the electrical resistivity. But these are in elastic scattering. Okay, so they can relax the quasi particle energy. And therefore we expect them to show up actually very strongly in the thermal resistivity. And this might lead to a very strong violation of the Widman-Franc's law. So we calculate the, the cold electron cross section to scatter off hot electrons. Okay, which is described diagram. And that it turns out to scale like the like t to the power three halves. Okay, so very, very different from the usual Fermi liquid behavior, which is, which is the squared. If you, if you calculate the thermal connectivity, kappa over t, that's that actually scales like one over t to the power three halves. Okay, so it is actually dominated by these processes. And you compare that to the electrical connectivity that goes like one over t squared log t. And the ratio, which is the Lawrence ratio actually scales like square root of t times log t. Okay, so as a function of temperature, this is what happens the red curve here is when our system is exactly at the one of singularity. Okay, so you, there's this, this very, very dramatic drop of the Lawrence ratio, basically to zero. And then if we tune away from the one of singularity, this is what we expect. Okay, so, by the way, this is a perfectly clean. So the calculation is for a perfectly clean system will discuss what happens in the presence of impurities shortly. But in the clean limit, this is what we expect to find this with this very, very dramatic violation of the vitamin front slot. Okay, and a similar effects were actually predicted in a variety of situations where there's a source of very strong inelastic scattering that involves a small momentum transfer. And some other examples are just an ordinary to the firm liquid is also expected to be a violation of the demand France law but only a logarithmic. Okay. And, yeah, and some other examples are a hydrodynamic metals, or metals close to specific quantum critical points, these all are expected to have similar violations of the vitamin front slot. Yeah. Yeah, yeah. So it comes from this. Okay, so, so, if you just calculate the spectrum of chart fluctuations close to a but no singularity. Okay, so, so that turns out to have a stronger singularity than log.  And over a small range close to the vinyl singularity. This is where this, this, this singular behaviors is coming from. Yeah. Given the high density of states of singularity why don't we also consider the hot hot and cold cold scattering process. Right so so in the question is why, why not consider a hot hot. Why not consider a hot hot to hot hot scattering process. That's because most of the electrical and the thermal current is carried by the cold electrons. Okay, so this would occur this is what what you might measure if you measure the the single particle lifetime of the hot electrons, but we're interested here in the transport properties. Okay, so it's really mostly the cold electrons. Okay, so. Okay, so, so we see that the Lawrence ratio actually carries some interesting a information about the scattering processes in the system the inelastic scattering processes in particular. And now we'd like to apply that to understand the mechanisms of strange metals more more broadly. Okay, and here's just one example of that. So, this is a recent example from Andy Mackenzie's group. So this is a palladium chromate material, and that over some range of temperatures shows a very nice linear and T resistivity, and from comparing that to a sister compound to a palladium cobaltate. They argued that this linear and T may not be due to phonons that's usual mechanism to get linear and T resistivity at high temperature. And the reason is that a palladium chromate cobaltate actually has a identical structure. Very similar, very, very similar a phonon spectrum. And the only difference between these two materials is that the chromate actually has low localized magnetic moments on the chromium ions and the cobaltate doesn't. Okay, and you see the big difference in the resistivity and also the resistivity and the cobaltate is not quite linear. It turns out that if you look at the Lorentz ratio in the chromate material in the region where the resistivity is linear that actually approaches one within a few percent. Okay, so this is a very strong hint that the source of this linear and T resistivity is coming from elastic scattering processes from some slow degree of freedom presumably the phonons, never the list. Okay, so why, why the big difference between the cobaltate to the chromate materials. Also, the slope of the scattering rates as a function of temperature here turns out to be close to the so called Planckian limit. So it's h bar over KBT. And the question is why is that a okay so these puzzles, I think are still open. So this is just an example of how, how you can learn about the mechanism of the strange battle. Okay, from from the Lawrence ratio. If it's one, it's a strong hint that it's all elastic scattering. Okay, but now we get to the main puzzle. Okay, so we'll, yeah. Yes. Yeah, so in in right so the Lawrence ratio is is Kappa over T sigma. So if Kappa for some reason is much bigger. It can go above one for instance in an insulator it will go infinitely above one because sigma goes to zero when Kappa is carried by phonons. Okay. Yeah, so in in metals usually what happens is that even at quite elevated temperatures the electrons are dominating both the both Kappa and sigma. And that's because the Fermi velocity is so much bigger than the phonon velocity, the speed of sound. Okay, and then typically what you what you find is that it's either below one or one. What. Yeah. Yeah, yes, yeah, right, right. So yeah, so there are various reasons for that. The electrons are not so fast. And they're probably strongly coupled with the phonons, but yeah. Okay, but now let's let's get to the corporates. Okay, so now we get to the maybe the strangest of the strange of the metals. So here's a particular corporate. This is a new demium doped LSEO. And a close to the so called critical doping. The resistivity is actually linear on one hand down to extremely low temperature and on the other hand up to quite high temperature. This is a clearly a violation of what we expect from a Fermi liquid. So we're over at this value of the doping the specific heat actually has a logarithmic is a singularity it goes like T log T. And the question is what can we learn from the weight of our front ratio on this on this behavior. It turns out that at at the lowest temperatures measured. Surprisingly the weight of our front slot is actually obeyed within a few percent in this system. And most strikingly, even in this magic doping this critical doping when the resistance is linear down to the lowest temperature, the real one France is actually obeyed. Okay, it's also obeyed for the transfers, transport a coefficient sigma xy and cap xy. Here's a different corporate is this is a thallium corporate also that shows linear resistivity down to the lowest temperatures over a range of doping actually, where super connectivity is suppressed by high field. So the lowest ratio is a obeyed up to up to one percent. Okay, so yeah. What's the temperature here. Here. Yeah, so so so this is trying to extrapolate to equals zero. Okay, so they measure they measure the lowest ratio. So, so super connectivity is suppressed by high field. Okay, and, and, and this is going. This is trying to go to the lowest temperature, because because really the, the way them on front slot should only be obeyed, even in a normal metal only at zero temperature. Right. Yes. Right. Yeah, so that's the question. Okay, so presumably it is an impurity dominated regime. Okay, but the question is if you have a some kind of nonfiber liquid, which this presumably is. Okay, so should the we run front slot be obeyed even at zero temperature. Okay, the argument for a normal model was all relying on the existence of essentially free quasi particles. Okay, so what what does this tell us does this tell us that there are quasi particles or just that it's obeyed because of disorder or what is it said. Yeah, so as far as I remember it's always obeyed within our bars. Yeah, yeah, they never saw a clear large reproducible violation anywhere basically at low temperature. Okay, so what what does that tell us. Okay, so that's that's going to be the question for the reminder of the stock. So maybe it tells us that actually underlying this, there is an ordinary Fermi liquid, and just the scatterers are quasi elastic but for some reason the scattering rate is linear in temperature. Okay, there's some low energy degree of freedom that we didn't know about that scales linearly with temperature. Okay, and in particular what we'd like to know is, for instance, what's the lowest ratio of the marginal firm liquid. Okay, so suppose we have a system where perma liquid is just marginally violated. And that's a that's a phenomenological model that was proposed a while back to explain the behavior of the of the corporates. Okay, so this would be a system where the single particle self energy scales in this way. Okay, the real part has a logarithmic correction to the usual Fermi liquid behavior. Lambda here is some coupling constant, and the imaginary parts a scales like the maximum of T and omega. Okay, and this implies in particular that the single particle scattering rate at zero frequency scales like temperature. In some situations, if this scattering of single particles is a large angle scattering, that also implies that the resistivity actually scales linearly with temperature. Okay, and the specific heat would get this logarithmic behavior both of these behaviors actually seen at this magic doping in the in the corporates. Okay, now in a couple of comments here. Okay, so a in the marginal firm liquid is really a phenomenological theory. Okay, to, to make predictions we really need some model that we can solve that gives this behavior and allows us to calculate any other quantity, for instance, we can write models that give this single particle scattering scattering rate or self energy, but the resistivity would not scale linearly in temperature because the scattering all comes from small angle scattering. Okay, so this is actually what happens in the model that Avi mentioned yesterday. If you have a 3D Fermi surface coupled to critical boson, this is this is precisely what happens. Second, it's kind of interesting to note that in the marginal firm liquid has a nice property if you look at the single particle scattering rate that's the imaginary part of the self energy divided by the derivative of real part z factor. Okay, and that's a from this the coupling constant lambda, if lambda is big, it actually drops out. And this is T over log T. So it's, if you like this plankton behavior, it's bounded by a KBT over H bar. It never exceeds that value. So, in that's worth, worth noting a property. Okay, so, right then the question is what would be the Lawrence ratio of this type of system. And for that will actually need some concrete model that we can solve and gives us this behavior.  Okay, and just describe the strategy. Okay, so in general, we're interested in a some kind of lattice model of electronic degrees of freedom. So suppose we have an n band hybrid model, and on every site we have some number of orbitals, and the type of Hamiltonian we'd like to solve is a each each orbital has some hopping into some interactions between the orbitals on different sites and so on. And, of course, we're interested in, in these strange metal phases that are difficult to get perturbatively we know that if we just do perturbation theory and the coupling will typically get a firm liquid phase. Okay, and in addition to that we'd like to include disorder lattice defects phonons and so on. Okay, so these are of course, of course very very hard to solve. So that's the interactions. Okay, the strategy I'll take here is to imagine that the number of orbitals on every site is actually very large. Okay, so that provides us with a small parameter one over N, which is not the interaction strength so that can potentially give us some access to these strange metal phases. Okay. So that comes of course with the price. Okay, we have to introduce a symmetry, at least statistical symmetry between these different orbitals in order to be able to solve it. Okay, so there's the worry that this large number of degrees of freedom on every site provides a bath for dissipation. That's not something that we that exists in the real system. Okay, but this can give us access to these non-quasi particle regimes that are hard to access otherwise. Okay, so this could give us at least some window into these behaviors. Okay, let me just give you one example that we worked on and kind of relates to something that Nikolai talked about yesterday. Okay, so we wanted to study a scattering of electrons due to phonons close to the material for Reagan limit where the mean free path of electrons becomes a photo one. Okay, and that can be done using this approach. Okay, so we introduce a model which has N electronic orbitals or bands on every site, and has n squared flavors or modes of phonons on every site. Okay, so this is our Hamiltonian this is the electronic part. This is the phonon part and this is the interaction. And as Nikolai described, we considered two types of interactions one of them is on site. Okay, and one of them the phonons are really coupled to the electron density on the same site. And the other one is the SSH type where the phonons are coupled to the bond density. Okay, we can solve this model when the electron phonon coupling is of order one or even exceeds one as long as n is big enough and we're interested in this context in the in the in the regime where the temperature is much bigger than the by frequency, but much smaller than the permi energy, permi energy. Okay, and it turns out that these two models actually give a very different behavior in this case. Okay, and the hosting type model, the, the resistivity grows linearly with temperature. And then when you reach the material for Reagan limit a mean free path of the electrons becomes order one. There's a kink in this slope, but it just continues linearly with a different slope. It's not a crossover, but there isn't the so called resistivity saturation which occurs in many, many real metals when, when a KFL approaches one. Okay, on the other hand in the SSH type, a variant of the model. It starts linearly as well. That's the usual block a runeism behavior, but then when KFL becomes of order one time some function of the density. And this is the resistivity saturation to a value of the order of H over E squared. Okay, this is a 2d model. Okay, so this is the so called resistivity saturation is a possible explanation of that behavior which is actually seen in many metals. And in this particular case we can test a how bad is the approximation of setting end to infinity. Okay, so in this temperature regime where the temperature is much bigger than the phone and frequency we can treat the phone as relatively static. Okay, and we can solve the problem in numerically and calculate the resistivity using Monte Carlo. Okay, we can calculate the resistivity for finite end so here's n equals the same model n equals two, four, six, eight, and here's infinity. So at least qualitatively, the end the large end limit actually captures the behavior of this model, even for in a small as four or four. There are other other predictions I won't get into that. Okay, but now what we want is basically a controlled theory of a non firm liquid at low temperature. Okay, where we can ask about the Lawrence ratio. Okay, there is a model that does that's just that that's the such the vehicle type model. So that's basically a model of quantum dot a zero dimensional system that has an electronic orbitals, and they're interacting all to all. Okay, so there's only interaction in this model, no hopping. Every four of these orbitals are interacting via this, this type of term, and the interaction matrix element is random, we take it to all these interaction constants to be independent random variables, such that their average is zero and their variance is some value use Okay, and it turns out that this problem can actually be solved in the larger limit. It gives a very interesting non firm adequate behavior, pay the single particle self self energy scales like like one over square root of Omega. So this model is argued to be self averaging meaning that in what we can compute is only the, the configuration average self energy over all realizations of this interaction matrix, but the claim is that even a typical realization would actually show the same behavior and that's been checked numerically. And that's the maximum chaotic but I, which I won't talk about. Okay, so the idea is to try to use this model as sort of a window into non quasi particle transport. Yes. A green function. Sorry. Yeah. Okay. To study transport we actually need to generalize this model into a higher higher spatial dimensions. Okay, so here's a lattice lattice generalization of this model again on every every unit cell we have now these n orbitals. Okay, and there's hopping between the sites and there's a random all to all interaction on site. There's a single particle dispersion and there's a local interaction in real space that has this s a s y k form. Okay, so all the orbitals within a site are interacting in this way. And we also want impurities. Okay, the solar scattering. So for every realization of this random interaction. If we didn't have this last term of the Hamiltonian the problem would be exactly translational invariant. Okay, so a this w equals zero that's that's like a clean metal in every realization okay it's random we only know how to compute things averaging over these years, but for every realization there's actually exact translational symmetry. So we also want to introduce controlled disorder so this is, this is a single particle potential on site which is actually random inside dependent so this breaks translational symmetry. Okay, so this kind of model one can actually solve in the larger end of it shows a crossover from a firm liquid at low temperature to a non firm liquid at a high temperature. It's not quite what we want. Okay, so this is the translational symmetry. It's not what we actually want is something that shows a non firm liquid behavior even at even at very low temperature and for that you need to generalize this model a little bit. So there's a variant of this model which is analogous to condo lattice problem. Okay, there are two types of electrons the sea electrons, whose Hamiltonian is shown here, and they're much heavier so called f electrons that are dispression less. They also have their own local view, but they don't hop at all. And there's this interaction between them which is like a density density on site interaction between a between the sea and the f electrons that also has this is like a platform. Okay, and yeah. Okay, yeah, I mean I'm almost done. So it turns out that this model this condo lattice type model actually shows a marginal firm liquid behavior down to the lowest temperatures. Okay, you can you can solve this model exactly in the larger limit that's equivalent to solving basically these self consistent equations for the self energy and then you can compute connectivity. This model actually realizes exactly the type of marginal firm liquid proposed by by Varma et al. Okay, so it has a electrical resistivity that scales linearly with the temperature down to the lowest temperatures, an optical connectivity that go like along squared, specifically to just T log T and so on. Okay, it even has interesting one hope singularity. Interesting, the husband up in oscillations that a display the Fermi surface of the sea electrons but don't have the difference cost of which behavior. Okay, so what about the Lawrence ratio in this model. Okay, so we can compute the electrical resistivity we can compute the thermal kind of a resistivity. So we start from the clean limit that is without the single particle disorder term. So that gives us that the electrical resistivity is just linear and T. Okay, and it turns out that in the clean case the Lawrence ratio as far as temperature independent, and it has a value which actually deviates from the firm from the conventional value. So it's a some number this number was computed in this paper, but this number is actually not universal it depends on on details. Okay, but now we want to introduce impurities. So in the disorder case. Okay, so a somewhat to our surprise it turns out that at zero temperature the Lawrence ratio actually approaches one, even though this isn't. A marginal firm liquid. In fact, there's a slight generalization of this model which is really a non firm liquid and nevertheless the Lawrence ratio approaches one in the limit of T going to zero. But what my students, if it downloaded is that if you look at the first correction as a function of temperature to the Lawrence ratio, it's actually linear in temperature. So this might actually serve as a sharp, a sharp signature for marginal firm liquid. If you do the same calculation and an ordinary firm liquid with impurities and with electron electron scattering. The first correction would be T squared. The electrons would give you even even higher power at low temperature. Yeah. Okay, so so this is our main point. Okay, this is what the Lawrence ratio of off of a marginal firm liquid, at least within this model should show okay the Lawrence ratio to the approach one at T goes to zero. So it's a non firm liquid. But if you deviate from zero temperature the first correction is actually linear, and the scale of the variation of the Lawrence ratio is off the scale of the elastic scattering rate. Okay, and when when the temperature is much bigger than the elastic scattering rate that reaches some non non universal constant value. This is what happens in this month. Yeah, yeah. Yeah, yes. Yeah. Yeah, this this model actually satisfied Madison rule. Yeah, yeah. Square root of temperature. A good question. I need to think about that I don't think so but Yeah, so that's a good question. Yeah, let's let's chat about that later. Yeah, yeah so I'm basically done. Okay so so this is this is what looks like in silver. Okay so much more and more flat so this is really the, the prediction here. Okay, that's that's my, that's my summary slide. Okay so the winner my front line is violation can tell us something interesting about the mechanism of ordinary and straight metals. And this is what it done. It's doing it a clean metal near a no singularity. And this is a disordered marginal family equity. Okay. Thanks very much. Yes. Okay, thank you. Can you explain a little bit. How the disorder happened is between the site is among sites. Yeah, so, so the model we studied is a model where the disorders, the disorder is on site. It's just, if you like it's a it's a random hybridization between these different orbitals. I don't expect it to be different if we added a little bit of randomness and the hopping between the sites. Okay, but that's not what we did. I'm wondering in detail why there's this minus alpha T correction in the disorder. Yeah, yeah, yes, so, so right so I can give you an intuitive my intuitive understanding of that. That is based on sort of a Fermi liquid picture so I don't know what extent that's really correct but the idea is very very simple so you say in the scattering in this model of electrons on near the Fermi surface essentially momentum independent. And that's coming in both to the thermal and to the electrical resistivity, but it's always in elastic. So these scattering processes are weighted differently for the electrical and the thermal resistivity, unlike the elastic case where they're weighted equally. Okay, so if you go to zero temperature it's all elastic, and then the Lawrence ratio is obeyed. If you look at the first temperature correction, you get a different correction to linear but different for the sigma and the kappa. And that's why if you look at the ratio you get the linear. Yeah, about the model, you had the interaction on sites to be same for each of the sites. Yeah, so yeah, but wouldn't that lead like after this order averaging to some non local interaction between different sites. Yeah, it does. Yeah. So, right, if you disorder average it looks like a non local interaction. Yeah, but you can still solve it turns out. No, there's an online question. Yes. Right. Yeah, if all the scattering in the system is elastic. Is the way we learn front slow violated. I have not found an example of that. Yeah, I mean, I think that's, that's an interesting question. If there is, if we can construct a model with that is the case. I don't know if an example. What is a, what is the experimental property of a firm liquid and non firm liquid. Okay, so, right, I mean a firm liquid we know a lot of properties right so they're well defined quasi particles and that has all sorts of experimental consequences. So scattering the resistivity is D squared in a specific heat is constant and so forth a non firm liquid is anything that's not that okay a metal that's not that it's not, it's not just one thing. And I presume that the Fermi surface of your marginal firm liquid is small, not, not necessarily, I mean it's, it could be large. It's not essential I think to anything that I discussed. Okay, so, so I understand that you have okay one minus alpha t and then presumably, you know, minus beta t squared minus gamma t cubed, etc. Yeah, but then the question is about the pre factors, because I'm guessing that alpha goes as one over one over w or one over w squared. The pre factor itself of the ratio between one over one over w squared and your interactions right. So the question really would be, what would with the next order is it, which come from phonons actually completely dominate over this. Yeah, so, so in a clean metal, you want to look at the relatively clean marginal firm liquid and then the slope is very large. But it would also have weak electron phonon or then it's weak electron phonon would also help, which may or may not be the case. Okay, so I think you need to move on. Thanks. So the next speaker.


The Columbia Broadcasting System presents yours truly, Johnny Duller. The next half hour has its baggage packed to take a trip with America's fabulous freelance insurance investigator, Johnny Duller. At insurance investigation, he is just an expert.  Expense account submitted by special investigator, Johnny Duller, to Home Office, Oriental West Cargo Bonding Company, San Francisco. The following is an accounting of my expenditures during my investigation of delayed cargo aboard the SS Shanghai Wayfarer, or the case of the slow boat from China. Expense account, item one, $181.52. Plain fare from Hartford to San Francisco and answer to your urgent call. Expense account, item two, $3. Lunch on Fisherman's Wharf and answer to my stomach's urgent call. Item three, $1.20, cab fare to your office. Dollar, my name is Fundy. I haven't had the pleasure of meeting you before. And may I say it's a pleasure meeting you. It's a rough trip. I'm glad it's over. Over? All right, it's just begun. Here, dollar, this is your plane ticket to Singapore. Singapore, hmm? You know, Fundy, I had a choice. Really? To come to San Francisco to see you or to take a case in Boston. A nice old lady on Beacon Hill clubbed her husband with an early American bed warmer. But no, rather than New England broil lobster, I'd rather have San Francisco crack crab. Now all of a sudden, Singapore. May I ask why? Yes, we've bonded against a delay, a $120,000 cargo over or tin aboard the Shanghai Wayfarer. The ship was due to sail from Singapore three weeks ago. Still out there, tied up in the Tanjong Pagardoc. What for delay? Muteness, mechanical or just plain mysterious? I'm afraid it's a little of each. We flew an expediter out there 10 days ago to see what he could do. All the satisfaction we've had from this man, Harrison, is a report that since his arrival, the Wayfarer's main shaft has burned out, the freshwater pumps have fouled up, and her steering machinery has gone on the fridge. You don't need an insurance investigator, you need a good plumber. Ha ha ha, well, maybe you're right. But anyhow, you'll find our man, Harrison, William Harrison, at the Crown Colony Hotel. He'll fill in the details. Dollar, you have only a matter of hours after you hit Singapore to get the Shanghai Wayfarer started on its way. I must impress upon you the fact that any delay after that will cost this company $2,500 a day. Well, all I can promise is the old college try. Times like this, I wish I'd gone to college. Well, anyway, I'm in the right sound to make my last night in the States a good one. A few drinks with the right gal at the top of the mark. A few rare steaks at Alfred's. A few dances to Freddie Martin's music at the St. Francis. A few moments alone in the arms. $2, that sounds mighty good. But your plane leaves in two hours. Two hours? Well, I guess I'll have to do without the drinks for dinner and the dancing. Spent a count on item four, $120. Lost in the course of teaching fellow passenger how to play poker. My mother warned me not to never to play cards with strangers on trains or steamships. I wish she'd included airplanes. You'd implied, Fundy, that the situation smelled. Well, you should have caught a whiff of the city, especially the native sections, through which I had to pass on my way to the Crown Colony Hotel. I found it on Anson Road. I found myself a room. I also found William Harrison's room. Didn't find Harrison. All I found was a calling card from my old friend Trouble. Whatever Harrison was, he didn't want to be. And he left a trail of broken furniture and blood to prove it. I searched the dresser. Shirt size, 14. Socks, nine. That meant Harrison was a small man. I went through the bathroom, shaving brush and toothbrush, still wet, indicating that he'd been there not too many hours before I arrived. Then I tried the wastebasket. In addition to one large glob of used chewing gum and empty cigarette package and some old fling eggs, I found a swizzle stick with a name on it. The Collier Key Bar. All that meant was that Harrison had a head cold and been trying to cure it with Singapore slings. But at least I knew where he'd been drinking. The Collier Key Bar looked out on the harbor and was dark enough inside to give a man a good excuse for drinking night caps at noon. Your pleasure, sir. Say, how are you on mixed drinks? Mixed drinks. Governor, if I don't know how to make them, I look them up in the book. If they ain't in the book, I fake them. Now, what about me? Straight bourbon. Right, your sir. Oh, hey, bartender. Yes, sir. Are you by any chance acquainted with an American named Harrison? Harrison, sir? Yeah. He arrived in Singapore about 10 days ago. Small man with a coldness head. Oh, Harrison, sure on him, right enough. He's been coming in every night with a chief engineer from one of the ships in Port. Oh, yeah? What ship is that? Well, the Shanghai Wayfarer, I think. Well, the Shanghai Wayfarer. What's this engineer's name? Eh, now, old arm, I ain't getting him into any trouble, am I? He's a nice chap, he is. A handsome tipper. This handsome? My governor, $20 American dollars. Why, compared to you, sir, Mr. Frank Moore is downright tight-fisted. Well, now that's it, ah, I've done it. Well, let Mr. Frank Moore's name slip right now. My Mrs. is right. For a little man, I've got a ruddy large man. Expense account, item five. Rich short fare to the Tanjung Pagar docks. 10 cents. Tip to Pony Boy, $1. The ships moored four and a half to the Shanghai Wayfarer, were busy stuffing the pungent treasures of the east into their deep steel pockets. The only sign of life aboard the Shanghai Wayfarer was the right hand of the burly gangway watch. It was holding a knife with a six-inch blade and slicing thin slivers off a plug that looked more like tar than tobacco. As a gangway watch, he might have been fine. But as a reception committee, he was no Elsa Maxwell. That's far enough, mate. There's nobody aboard, and there's nobody coming aboard. It's all right with me. All I want is a little information. Where can I find your chief engineer, Frank Moore? You've come to the wrong place. Try the icebox over at the Singapore Police.  Oh? Any idea who did it? The whole in some dame he's been playing around with. No, I don't know her name. Have they got anything else? Listen, mate, my job is to guard the ship, not answer questions. OK, OK. Have it your way. A watch out for pirates. The British chief inspector, Singapore Police, gave me everything except an invitation to tea. But unfortunately, he never even heard of Harrison. He took me into the morgue, and a look at Frank Moore's body told me nothing I didn't already know. Aidman stabbed all right.  As for his personal effects, his maritime union card confirmed the fact that he was indeed the chief engineer of the Shanghai Wayfarer.  And finally, a photograph that made me admire the late Mr. Moore's taste in women. Whoever it was that said, never the twain shall meet should have met her. She was half-cast and all woman. Her picture was inscribed of Frank Moore, yours forever, Chandra. From the inspector, I learned two more things. One, the fact that the police had already questioned and released her. And two, her business address, the Wardlow Bar on Melee Street. Say, young, you like to read nice sing-song girls? No. The only girl I want to hear sing songs is Dinah Shore.   Yes, do I? Where's Chandra? Oh, she go across to Penang tonight. You buy any drink, we stay. We sit right over there. Come on, job, the gunner.  Oh, come on, darling. I put you away home to say hi one night in your own coffin. Complete with stab wounds, no doubt. Why, you say that. Why, you ask for Chandra. I'm a stranger in town. I can't find the local chapter of the Lonely Hearts Club. So shall we find a quiet table? I don't know you. No, but you knew Frank Moore. That gives us something in common. Over there is one. Oh, is that it? Yeah. Come on. This doesn't sound like a very quiet table to me. In Singapore, you will learn whispers stand out in the quiet. They disappear in the noise. I'll bow to the wisdom of the native guide. But who said I had any secrets? You talk about Frank Moore, shall I know if you do not have secrets to give, there must be secrets you like to learn. But I tell the police everything I know which is nothing. No, you are disappointed in me. No, no, none of all. You make good scenery. And I'll bet there's quite a story that goes with you. You'll find me interesting. I'm a man. Why do you come to me? Well, there were two places I could go for what I'm after. And you're much prettier than the SS Shanghai Wayfarer. I'm looking for a lead and a man named Harris.  You are wrong. I do not know him. I do not even know you. Oh, well, that's soon fixed. My name is Johnny Duller. Your name is nice. Especially the, uh, stalloport, huh? You are very droll, but I see when you make this joke there is no smile on your face. You are worried about your friend Mr. Harrison? Yeah, that's right. Maybe he was lonely tonight. Maybe he does not want you to find him. Ah, you certainly make me feel much better. How about a drink? I never drink before midnight. All right, then I'll wait. We'll have one then. All right, Johnny. But we don't have it here. We go to my house. There it is cool on the river, and there it is quiet. So we do not have to whisper. Midnight must have been invented for Singapore, and her house must have been invented for midnight. Only one thing looked out of place. Up on the wall was a souvenir of Chandra's war effort, a real American baseball bat, a Louisville slugger. An honor was written, remember the U.S. Marines. Everything else in the place was soft, the lights, cushions, Chandra. It is nicer to drink here, no. Yeah, may I say it's a might intoxicating without a drink. I wish the boys back in my high school senior class could see me now. What do you mean? In the graduation annual, they predicted I'd be a bookkeeper. Oh, I do not understand you. And neither did the boys in my senior class. Johnny, please say things I can understand. I want to know you better. Maybe if I stop talking all together, you'll get to know me better. I stopped talking, but I didn't stop thinking. When I'd mentioned Harris and Chandra earlier, she said maybe he was lonely tonight. If she didn't know him or anything about him, I wondered how she knew that he was missing tonight and not for a couple of days, or maybe even longer. Besides, the boyfriends of women like her don't keep secrets. I still assume that if Frank Moore had known Harrison, Chandra had known Harrison. I also assumed that she'd spidered me into her parlor for purposes other than social. And that notion was seconded soon after I had it, when somebody kicked the door open. Yes, it's your journey, darling. The two boys in the door were not from Western Union. And ugly as they were, Chandra left my side to join them, which made me think that maybe my senior class had been right. Looking at that trio's six eyes and two guns glaring at me, I wished I was a bookkeeper. In just a moment, we'll return to the second act of yours truly, Johnny Duller. But first, here it is almost the end of February. All over the country, people are thinking about their new car. All but one man. And he remains quite content with his old automobile and wearing apparel. An ancient Maxwell and a well-worn toupee. For these reasons and for several others, named Mary, Dennis, Don, Phil, and Rochester, he now has the number one comedy show in America. All over the country, people think about him too, every Sunday night. Here, the Jack Benny show with Claude Reigns as Jack's special guest next Sunday on all these same CBS network stations. And now, back to yours truly, Johnny Duller. Described from left to right, we're a fat man with three chins and a bald dome. And with him, a punk with a sneer and arms that were too long for the rest of him. A gun muscled me into a chair and started making anything with sense. Well, well, son of my dear. We had at last face to face with the mysterious stranger, Johnny Duller.  He knows why. He came to the world low bar. He knew about Frank Moore and he was looking for the other one Harrison. That is why I phoned you. Well, it would seem then that this unfortunate chain of events is nearing the final link. Yeah, this guy uses his head better than Harrison did. Well, Duller? I'm using my head right now. Blend it, blend it. So doing, you may well prevent Harrison's death as well as your own. Oh, well, that's better than nothing. But is that all you can offer? Skip the bargaining rust line. It takes too much time. Quiet, Corgi. At times when money is cheaper than the results of your kind of blind violence. Well, Duller, you do have a price. Take a tip from my last name. Don't start bidding.  You aren't sure he knows where it is. He must know. He was looking for Harrison. They both know. You be quiet, both of you. 500 pounds English dollar. Where is it?  Surely, Duller, since you've entered the situation at such a late date, that is profit enough. Oh, I'm a man of expensive taste. I've always aspired to such things as $200 cigarette lighters. Go ahead. Keep spitting out that wise talk that you'll be spitting out teeth. How'd you like to go swimming with your hands and feet tied? I could bite my tongue. Oh, not to a chat, Corgi, my boy. This man is worthless, dead. Perhaps, Duller, we can induce you to talk in much the same way as we could prepare a parrot by... slitting the tongue.  Why, you... Either this guy is nuts or he doesn't know anything. What I know would fill a police blotter. Corgi, you know nothing of psychology, my boy. What this man is attempting to pass off as a show of bravery is based purely on the knowledge that he is, momentarily at least of some considerable value to us alive. Now, Duller, be careful.  Now, sir, what shall it be? I was a squirrel. A squirrel said to the little girl when she asked him what he wanted for Christmas. Nuts. They will know that. Corgi. Thanks. He finally came to, in the dark, trust up like a turkey and lay there trying to figure it out. Obviously, the two rude dudes thought I knew something I didn't know.  Also, that I had accomplished exactly nothing towards speeding the SS Shanghai Wayfare over the bounding main. While I was comforting myself by repeating over and over that old insurance company, Soother, never say die, I discovered I wasn't alone. Hello. You, who are you? You were here first, you tell me. Well, my name is Harrison. Harrison? Yes, who are you? I'm Johnny Duller. I was sent out here by the Oriental West cargo bonding company. Oriental West? Yes, I was supposed to do what you couldn't get done. And look at me now. Getting hit over the head and dumped in here must be par for the course. How long you been here and why? I've been driving myself crazy trying to figure that out. Well, this little guest house, wherever we are, must only have one set of proprietors. I can tell you who they are, at least when their names are using tonight, Rosalyn and Corky. They offer me 750 English pounds to tell them where something called it was. What is it? What's a package? What's in it? I don't know. It belonged to the chief engineer of the Shanghai Wayfare, Frank Moore. He was helping me try to get the ship on its way and I owed him a favor. He asked me to drop this package at a bar, the. The Warlow bar, yeah, go ahead. That's right. I was supposed to give it to a girl named Chandra. She wasn't there, so I got her address and went out to her place. You mean that package is at Chandra's house? Yes. When I got out there, the Chinese made let me in. I waited as long as I could, and then rather than leave what might be a valuable package just lying around loose, I put it into the bottom drawer of a dresser and left. Oh, great. For such things, I go around laying down my life. It's obvious that these men will stop at nothing to get their hands on that package. Well, when they asked you where it was, why didn't you tell them? Then neither one of us would be here. What's more, I'm beginning to think the sooner they get the package, the sooner our ship sails. Frank Moore has been a good friend to me. He wanted Chandra to have it, and I couldn't just turn it over to those two. Well, I've got some news for you. And this should make you really unhappy. Those two happened to be in business with Chandra. They're on the same team. She's one of them.  Well, here we are all roped up. You know, for a pair of guys who came out here to speed a shipload of raw tin on its way, we're doing just dandy. We're lucky if we gather this thing alive. Offhand, I'd say our host probably murdered Frank Moore trying to get that package. Maybe we're next. Oh, maybe right now. A beam from a powerful flashlight stabbed us in the eyes. The sudden change from too much dark to too much light kept us blinded. Well, look who's here. At least the voice behind the glare wasn't Rosalinds and it wasn't Corgis. But it was a familiar voice, one I'd heard and heard lately. He walked in on us, the flash in one hand and in the other, a knife with a six inch blade. At first I wondered whether it was the one that had been buried in Frank Moore's bank. And then I remembered when I'd seen it before. The man bending over us was the burly gangway watch from the Shanghai Waffer. And you told me to watch out for pirates. Well, this situation is getting a little overtrouted. I didn't think there was room for any more. What do you want? You know what I want, darling. The same thing Rosalind and Corgis are ripping your hotel room apart for right now. Now don't tell me you're looking for it, too. Two things I know about that package, Mr. The name is Roark. OK, Roark. One thing I know is that it's dangerous company. The other is I want no part of it. The only thing I'm interested in is getting the Shanghai Waffer out of port. That won't be hard once I get that package. Where is it, darling? I'll trade the answer to that question for a little freedom. OK, hold still. Oh, sir. Thanks. Harrison's next. I want him with us in case he's lying. All right. OK, Harrison, roll over. Hey, you! When Roark bent over Harrison, I dropped a flashlight out of his hand, ran across the darkened room, threw the open door, and kept on running. Sometimes the long way around is the shortest way home, so I headed for Chandra's house. I not only had some getting even to do, but I had some curiosity to satisfy. Somehow the Shanghai Waffer's failure to sail on schedule was tied up with a mysterious package. But how? Why? I decided I'd earn the right to see what was in that package. She'll be lonely. I heard your playmates are over making themselves a home in my room. So I thought you and I could have a little chat. Maybe I've got a surprise for you. What, Johnny? I think I know where that package is. Johnny, you gave that package. We both don't worry for the rest of our lives. But we must hurry before Rosalind and Corgi come back. We go now. OK, where's your bedroom? Johnny, what do you mean? Oh, come on, where is it? Come, I'll show you. There's another one now. It's been here all the time. And now while I open this thing, you can go and have yourself a nervous breakdown. This is more fun than unwrapping Christmas presents. And now I take off the cover. Wow! Now I know how the winter feels on hit the jackpot. The package was paper all the way through, brown wrapping on the outside and green spending on the inside. Big bundles of fresh, clean American 20s. Thousands of the same kind of bills that the Singapore police had found in the late Frank Moore's wallet. It would have taken half a day to count it, and I'd wasted too much time already. There'll be no good to you without me, Johnny. You have to know how to get rid of them. Oh, color fit, huh? Yes, they are made in China. Frank Moore brought them from Shanghai to Roslin to take to the States, but Roslin was not here in Singapore. He was late, so Frank had to make some accident happen to his ship to keep it from sailing. But then he changed his mind. He decided he would give the money himself, but Roslin caught up with him. Oh, I see. He was sending them to you by way of Harrison just before he was knifed by Roslin, huh? Who talked him into that? You, by any chance? You and I could be very rich, Johnny. You never give up, do you? It's $500,000 there. Mm-hmm. Yeah, that should buy about 50 years in jail. I'm taking this down to customs and you're with it. No, I do not think you do. Uh-huh. Time to play another visiting team. Come on, beautiful. I know what you went away. Is it worth me asking? I grabbed her, lashed her wrists with a cord from the package, and since she liked money so much, I stuffed her mouth with a fistful of those troublesome $20 bills. I locked her and the rest of the loot into a closet, dashed into the other room looking for a weapon. And then I remembered that Louisville slugger from the US Marines. I was glad they'd landed. I grabbed it off the wall, got a toehold in the carpet on the left side of that door, wrapped my fingers around the bat, swung it in the back of my shoulder, and waited. Under my tail, we just came out. Two outs and one to go. Three outs and the side is retired.  First, I take her guns, and now we sit and wait for you to wake up. I'll take over from here on in dollars. Oh, I don't know about that, Roark. I happen to be the guy who has the gun. Oh, well, here. Take a look at this. What's in your wallet that I want to look at? More hot 20s?  OK, I'll turn around with my hands up, and then you can look at it. OK, fair enough. But if you so much as move, I'll start shooting. That's the deal. It's a fine time to learn this. Are you satisfied? John Joseph Roark. US Treasury Department. Come on in. I'm sorry I couldn't come out into the open before, darling, but I was too close to the payoff of this case to take any chances. Well, you know, I'm beginning to think that just being in this town is taking chances. That counterfeit's been funneling through this port on its way from China for months. We had Moore staked out for a long time, but this is the first shot we had at the top. That's him lying there on the floor, Rosalind. Now I've got him. Oh, your pal Harrison told me where I can find the only other thing I need, that package of hot money in the dresser drawer. Oh, it's now moved into the bedroom closet, along with a package of hot woman. Well, then, darling, it looks like my job out here is just about done. Yeah, I guess so. Hey, what a man. You're from the Treasury Department. Well, and after you get all these birds into their cages, how about helping me make out my income tax? Expense account, item six. Hotel bill, one night in Singapore, $5. Item seven, one new outfit, replacing mine, which was ruined in course of taking midnight dip in Singapore River, $200. Item eight, $20. Bar checks for cheering up one William Harrison, your expediter, whose innocence had him running errands for the man who was holding up the departure of your ship. Item nine, $375. Spent while killing time until the departure of my plane back to the States, after the Shanghai Wayfarer finally sailed. You see, this time I had four hours on my hands instead of the two you allowed me in San Francisco. Expense account total, $1,407. Signed, yours truly, Johnny Dollar. In just a moment, we'll tell you about next week's Johnny Dollar adventure. But first, for more exciting drama in the mystery and adventure line, remember CBS two thrill-packed Saturday night shows, the adventures of Philip Marlowe and gangbusters. Be sure to hear Philip Marlowe and gangbusters tomorrow night on most of these same CBS network stations. Next week, CBS will take you adventuring with Johnny Dollar, hitting the hotspots in Palm Beach and the Orleans, with the star of Hades Diamond on a trip all points south. Charles Russell plays the role of Johnny. Our music is composed and conducted by Mark Warner. Yours truly, Johnny Dollar, is written by Paul Dudley and Gil Dowd, and is produced and directed by Richard Sanville for CBS, the Columbia Broadcasting System.


Hi and how are you all today? The question says evaluate i is equal to integral 0 to pi x sin square x upon 1 plus cos square x dx. Now let's proceed with the solution. We have the given integral as 0 to pi x sin square x upon 1 plus cos square x dx. Now here on subtracting sorry we have we know that sin pi minus x is equal to sin x right. So we have i equal to integral 0 to pi pi minus x sin pi minus x upon 1 plus cos square pi minus x dx. So as we know this we have or i equal to 0 to pi x minus sorry pi minus x sin pi minus x is sin x and cos pi minus x is also cos x. So we have 1 plus cos x over here. Now this be first equation and let this be the second equation. So on adding the first equation and the second equation we get 2i is equal to integral 0 to pi pi sin x minus x sin x plus x sin x upon 1 plus cos square x dx. Now here these 2 will get cancelled out. We have 0 to pi pi which is a constant which will come out of the integral sin sin x upon 1 plus cos square x dx. Now if we put cos x equal to t then minus 2 sin x dx will be equal to dt and our limits will also change from 0 to pi to become minus 1 to 1 dt over 1 plus t square. So we have 2i equal to pi tan inverse t back it close minus 1 to 1 on applying the limits we have pi tan inverse 1 is pi by 4 minus tan inverse of minus 1 is minus pi by 4. So we have i equal to pi into pi by 4 plus pi by 4 is 2 pi by 4. So we have i equal to pi square by 2 and this is the required answer to the given question. Hope you understood it well and enjoy it too. Have a nice day.