Now that we know how to identify keywords from our research questions, let's learn how to combine them together so our resources, such as the library's online catalog, or a library database, knows how to retrieve the information we want. We call this constructing a search string. A search string is the combination of keywords and search tips that make up the search being conducted. Constructing a search string is what allows us to enter our keywords in a library resource, such as a database, to find the information we want. This video will show you how to develop a basic search string using three search tips. Bullion search operators, quotation marks, and the asterisks or star symbol. These tools can be used across different library resources and will help make sure you are retrieving the most relevant results for your research. Let's start with bullion search operators, which involves using the words and or and not in our search. When writing them in our search string, these words are always written in capital letters. The and bullion search operator combines our terms together and finds information that includes all of those terms. For example, a scholarly article that talks about both city farming and food security. The or bullion search operator allows for searching by synonyms or related terms. It allows you to broaden your search by looking for information that has any of the terms used in your or instruction. For example, searching using the synonyms city farming or city gardening or urban agriculture will return articles about this method using any of these terms that we have included in our search. And finally, the bullion search operator not removes search results you may not need by filtering out specific words. For example, if you are interested in information only about food security in the United States but kept getting articles about food security in Africa, you could write not Africa and it will exclude that information from your results. Next, we have quotation marks. By putting quotation marks around a specific term or search phrase, you'll be able to keep the words together in your search. For example, we might do this to our keyword food security. Truncation involves the use of the asterisk or star symbol and comes at the end of your root word. It's helpful if you are searching with a keyword that might have different endings. For example, using the star symbol at the end of the word teen. Can you think of some variations of the ending of this word that this tip would search for? We would get teen or teens or teenager or teenagers without having to type in each word using the or search operator. So what might our search string look like incorporating these tips? We would use the bullion search operators to connect our keywords together. So first, we'll ask our library resource to search for city farming and then its synonyms, city gardening and urban agriculture. We'll then add our second term food security using the and bullion search operator so that we only get information that is about both city farming or one of its related terms and food security. We'll wrap food security in quotes to search for that specific term and we'll take the root of our keywords farming and gardening and add our star symbol so we can get any variations of the ending of these words. Remember that there's no such thing as a perfect search, but doing the pre-research step of identifying your keywords and applying these search strategies can help you find better results. Research is a strategic process that involves trying different search options to see what returns the best results. We'll continue to see these search strategies and tips at work in Highway 301, finding information.


Biomass-derived carbon materials, BCMs, have become increasingly popular due to their abundance, low cost, and high carbon content. This review summarizes the progress of BCMS-related research, focusing on the types of biomass used to prepare BCMs, variable structures of BCMs, representative synthesis strategies, and the influence of synthetic conditions on the structure of as-prepared carbon products. The review also discusses recent progress in versatile applications of BCMs based on their morphology, and physicochemical properties, and highlights remaining challenges. This article was authored by Hong Zhihui, Ruoqiun Zhang, Pengcheng Zhang, and others.


So, I'm checking my mic and then I'm Jeff Richardu niyambekilam tuken yume niyambia uja niyambia visuri. I'm busy checking my mic and then, nice to nice, imefka past me and judgeee. See, I'm singing, it's also checking my microphone. Ah, ni mi ambi wata unwa. A chana tro mwenye song a imbe. That was not nice. It's not. Hai, Zuru. At Wifi4 Facebook, Y254 channel on Twitter, Wifi4 underscore channel on the ground. Hashtag is Thursday Vibes. It's over in the morning when I talk about what color me well. And technically, I am in studio with Stefani Ayeta. But technically also, I am with other guests who we are going to say hello to right now. Hello. Hello. Hey, manjibu fiti. Mekula. Mekula. I like that. But me naskianja. Lafuhu yu guys ameni eptokakuniyambia ni chef na mi naskianja. Tuneza jesoti isyuma. After this, after this, most definitely lasiyuma. Ni na don't play about food. What's your favorite food by the? I don't think I have a favorite. But kuna food zingine zyu slapka ugali mepi kwanahi unga kisiagia. Baaz. Ni kapta skizala. Bana is like a hug from the inside. Anyway. Hai guys. We are not talking about food. No. They are artists. Today they are here as artists. So please introduce. Let me start with you. Because one manjia isyuma na chakula. I go by the name Osarendaige. Or Jeff. Miha kwa manjia mbe yu hengine. But nisa hawa. Now you are flexing. No, no flexing. Osarendaige. I'm an artist. I'm in a crew which is a duo. It's called the All Stars Icons. Yeah. With the Z. Okay. Okay. That's not with the All Stars. Why All Stars? Is it the shoes All Stars? No. No, no, no. When you read the Bible. We're taking it way back. Way back. When you read the Bible, stars actually point out a direction which people follow. So out of that and because we are people of faith, we decided to pick the All Stars Icons as her name for us. Because initially we were dancers. No dance choreographers actually. Choreographers were dancers. We eat of you. Okay. Okay. We were dancers. So we had independent crews. He was the leader of a crew called Faluja. And I was the leader of a crew known as Solid Sea. We sent it to MADS and formed the All Stars Icons. Yeah. Way? So chef, dancer, artist, all around cool guy, businessman. How do you balance all these things? I can say, pay in favor of God. We cannot take it for granted because it's not easier to balance all these things because kuna challenges zaki. That's why you had to break for some years. Actually started way back. 2011. Way? And you asked back? Yeah. So took a fanya-fanya. Some projects, awards, nominations and all that. And then commitments came in. Now you've grown. Olioa, o kanzakusga rasta. Yes. Yes. Life goes on. With life you come. And then commitments come in. And then now this dance is not paying off. Napaata. Now you have to choose between what you love and what you have to do. Way? So kachikwa you break for like seven years now. Sasando to make amaz artists. Wow. I'm tired of dancing. We still dance. We still dance. But as choreographers not as active as we used to do. Because you can't do the same thing for years and years and years. Kusakata season one. That was way back in 2009. Wow. Season one. Yeah. With all that. The years going. You turn to see. Kishi denu ni gran. Yes. They're not showing. Ni grace. Ni grace. Nani. Nani. Baden to na shkuru. We are here. And we are here with a serious motive. Right. We are not here to play. I can see that. Yes. In the years that you guys have been doing what you're doing. I can hear that basically we're doing entertainment. Whether it's dance or either one form or the other. Umone magroa je. From Masguimbe now we have dances on Tiktok. People are making money on apps. The thing I can say is that before Kupata Airplay Platforms shows was really limited. That's kind of tricky. Kuskulu kwa na atuwa kwa pwa na. They were like controlling the whole scene and all that. But right now internet is there. You could our platforms they may come up. So say, when there's a sale stuff here kwa anyway. Right? And up and down because you see like dancers right now on Pata Platform Kati Kutukwa but there's a quarter brand and they sell those products and then they can turn it directly. You know what I get? You may remember our time Kasi show you dance in Msani. Msani na kusakuli pa. Msani na kutukwa. There's nothing you can do about it. You get Kusdans itself finer structures. That's why you can go to Trogliza. Kuna danza millionare. Wow. No, it's just a fukutout. Kume sha. Mama Elon Musk hands it hands. Exactly. So coming to that Kirizon in those lines change was needed and I'm so happy that change is happening. Okay. Recently we've had a couple of artists complaining. Okay.  Yeah, yeah. Sasa nikita kering buna na pewa 1200 15 shillings. Yeah, those are the things we are we are also talking about. You don't feel some type of way coming back in the still doing. It's really painful. You know why? Because do sayze as an artist do figure that level you have to be very professional. A professional studio Aikotinia 20 Gs. A professional videographer Aikotinia 50 Gs. And that yume baginsano. So pick up budget 100 K. But you see this return in Aikotinia 1000. How will that make you feel? You get. With COVID-19 to the other day pesos as jakuwa So this is the kind of challenges we are trying to say. I think it will affect. If at all we set up really good structures that can zineza gabanizi revenues that zineza fiki yamsani zineza kwa poa. But sa easy atakuna wasani wajuhu where to go and register for any music society copyright. A lot of artists go to the studio and get money. Una patal kuna mutu juyaki namsuindo So those are the things that we make over and over but to not na shkuru ko squeeze zime punguwa. The internet may change a lot of things and then new artists are being appreciated and na fraia ju pia kuna artists wama kujana new sound things are changing the music is growing professionalism. So yeah. We are going to cut your craft series. Say here, here. Sjus mamia kieti sjume sema umesoma ehe sjiuwa ni musani sjume sema kuna mungu Sjus mamia kieti niku pige kura shua wakati womungu wakati womungu It's notDIOCALLING here. I don't know wana kakus mamia wawana eza maowana eitu waki njuwa nivile tuwa waelana eitu wana eitu zime mamia kieti. master. Here na same waka tiwomungu Bona unati kakuni patiya mese mona umu nita kiki patiya details Udu though, is it something you're interested in and how you feel about it? Leadership? Ya, ya, ya, ya, ya, for sure I like that I like that For sure, for sure Unu juki angali yamta wa watu una wu mii ya, for real And it takes someone with a soul, someone with a heart to identify the problems that people go through And maybe come up with a solution We try that with dance Then it went the way it went Then we focused on other things Now we've taken our skills All the way as a first priority So we always hope for the best and we know Taimi amungu ita kuja So concerning the leadership when the time is right I will step out Onanzaku planti sit sayi Ya, nishaanzaku planti Ya, bana sayo trwa fukiria 2020 Anyway, to skiri eto It's good to drink Hashtag Thursday vibes What's the most intimate song you guys have done together and why? Napenda, oh, that's nijongelile Ya, nijongelile Napenda ngo mani trwa inapis Inapis, inapis Inapis ili kuwa gomatul yandika It's very emotional by the way Kwa yo EPA to latest It talks about our transition That it's like a prayer To nambia god niajitupati inapis Kus inapis is the new success Anpata There's no way you can do something And you're not at peace Despite you loving it Takusumbua So kutuna ili kuka prayer Itupati apis In whatever we're doing Ata sisi tuwezikui penda mo Kus wherever kuna yo pesu Uta penda yo kraft mo Kaya yo pisa kuna Ata we mo nyuha uta fokus You won't be at peace at that period So this song Basically nangia about inapis Tutulu yuza ni fanya mamboa Ata stressiku aji utlaks Fanya mamboa Weveli na kuskia Now get back to you right here What's your most intimate track? Mine For me it's tutashinda No no no Tutajua Tutajua because When you listen to that track It tries to explain the challenges That we went through as dancers And I remember For some time I was so angry Because of the experience And when you're going through something You might forget To count your blessings in the process So when we wrote that song It helped me Like went out again Let all the anger out And continue with the craft that you want to do So it was a It was a letter for me and I I really like the track If you are a dancer Is it very big for you Because you keep referring to Dance is part of us Dance is part of us We are the guys behind Sari sari dance We worked with We grew up Meza figa selfi set 4 times 4 times Dance was Dance was everything to us Get in the photo please Ah yes I met famous people Dance was part of us Dance was Dance was What can I say It was our heaven That's how Special it meant to us But The vision was so big On our side I don't think people saw what you saw That's what I can say I think there may be They are actually seeing it right now But the dream was so big Because we wanted to have this dance structure It works the same way Music society in New York We wanted to have this structure That A dance group Or a dancer can go and perform And actually be paid With a body that governs dance You get We wanted our own dance awards We wanted our own dance awards We wanted our own dance agencies You get Whereby this corporate companies can come And they don't have to look for dancers You go to this company You get this dancers With your prize and with your budget And actually this dancers are supposed To earn a living out of this Because as a dancer that's all you do You have to go daily for practice You get It takes a lot to have a 5 minute performance You can practice like 3 months So you can imagine how many people Lost their dreams The dreams got shut Because the people who actually love This craft This is what they want to do And this is all they have to do But then their dreams just How can you have such a beautiful vision And deniers The vision is still there And they are going to do it That's what I promise Thank you It's like a girl then everything I sound so beautiful We dance We even had a dance school We even had a dance school You've played part in many dance Dance school careers I wish you Stronger Stronger I want to take a selfie Because the weight I love you I love you I love you The time is not timing with me But let me ask you what we are asking on socials And that is Ni nini ili kwa highlight yako Uki took a high school Kenda holiday Now what was your highlight Highlights as in I didn't get that question Uki kwa highlight kwa high school Hati Continue Ask again I'm sorry The question is Ni nini ili kwa I've heard about the highlights What was your highlight Uki kwa high school What you were looking for Yes Holidays Holidays Shows Shows I can tell you from churches To events To tours So Shows events I think life here is going to school events Since primordial I don't know what to say I don't know I don't know I don't know I don't know I don't know I don't know what to say Maybe We are also humans It's life Maybe Your last words of you What would you like to tell your people Everyone at home just keep it real man We need a thousand More real people in life We really need We really need Real people not fake people And you can tell the difference How How It's in your facial structure Your body One day you can hide forever You can't hide forever The real is real I used to hide from myself Never hide from it That's very difficult So say your name Last words Be real Stay true to yourself And always be very loving All right Remember it's at Y-5-4 on Facebook Y-2-5-4 on Twitter Y-5-4 on Instagram Hashtag is Thursday 5 Kwa na salamu Kwa na salamu Kwa na salamu Kwa na salamu Kwa na salamu Kwa na salamu Kwa na salamu Omae producer Wetu angubu sanan Nitu omae Nitu help along Lafu kuno asepia Bili kwa before Kina omae kina reachibeth Kina lemba TDB Huelsa seote Kina lahu Desh Vini Huelsa seote Mahirishweli unable to fanya Kwa seote miko Awa loving family Kila msuetu Lafu piyamu to follow On all social media platforms At Allstars icons everywhere Allstars with the Z Icons with the Z Facebook, Instagram Kila maali Twitter YouTube And like, share, subscribe Na piwa fuluda Y-2-5-4 Campaigns Nene theaters Mamimko connected Sorry Zana pata Pata! Ah находится Okay, now I know. Guys, we're going to play their song and then we're going to come back and wrap it up with Stephanie. Alright? Hashtag does device.


My training stuff, we're doing a lot of the Kresi programs, something we did last year and did this year, so it was good to get two years of that under our belt. But the yoga, less lifting, I do a little bit of yoga every off season, something I kind of do typically. Modifying the workouts, I really worked hand in hand with Eric, trying to, I like these certain drills, I like doing these med ball stuff, these movement stuff, so I think he kind of meant, we just kind of modified the workouts a little bit. I don't think I really lifted, lifted less, I just, you got to stay in shape, it's a long season. So we just made some modifications, stuff that I liked and first couple times working with him, so we got a great relationship and it was, he's good at what he does, he's a great strength guy, so it was a good working of him, this off season. Eric, I understand that it's freaking news to have and you've had a couple, but how confident are you that it can be a, that these modifications can lead to more gains on the field field? I think it's going to make big adjustments, you know, especially a lot of the movement stuff that he has us doing, a lot of stuff with med balls, you know, just getting those explosive movements, I feel like that's where a lot of the things happen, or with those quick, you know, kind of explosive movements, so just training a little bit more with that and, you know, a little less straight bench press and squat, you know, I think is going to help out. We can go next to Eric Boland. Eric, please unmute.  He's a winner, you know, helps lead us into the postseason every single year.   I was everything moving, and, you know, I think I can speak for the whole clubhouse and this whole organization that we're happy to have him back, and, you know, back in Pinschraps where it belongs.  You see, is that important to, you know, not only me, but, you know, the young guys we have on this team and, you know, all my other teammates. He just, you know, even if he's not in the field, you know, he finds a way to impact the game, you know, and he's just a very special individual, you know, in that way. We go next to Tina Cervacio from Fox 5. Tina, go ahead. Tina, go ahead. Yeah, Tina, another bad taste, you know, in our mouths.   You know, that's something that we didn't do. The Rays came out and they were, you know, a great ball club.  I think that's something, you know, first and foremost that we need to do.  And, you know, we're doing the same thing, but I think it starts in the regular season getting, you know, keeping guys on the field, you know, playing hard and winning our division.  But I think it all starts, you know, now in spring training, now in the regular season. That's, this is where the, you know, the work starts and you need to have some fun in the postseason. We can go next to Sweeney Murdy. Sweeney, please unmute. Question to what you're talking about is the work with Eric Cressy and how you, what you hope to benefit from that on. Has anybody talked to you or you thought about how this style of play also leads to, maybe, a certain amount of wear and tear, you know, pulling that big body around the field sometimes. Is there any adjustment in that area, or is it all about the maintenance and getting ready for the postseason? Yeah, good to see you Sweeney.   You try to, you know, try to have a better, you know, if I can get a better first, first read, a better first step, you know, that might help me, you know, not have to crash land into the wall at the same time, you know. So it's little things I've been working on with Reggie Woolitz, you know, we've kind of talked a little bit in the off-season, a little bit now, since we got into the camp, just working on just certain things with footwork, you know, just a quicker first step.  It does a lot of wear and tear, but, you know, I think it comes back to just playing a little smarter. We go next to Marley Rivera, ESPN. Marley, go ahead. Marley, go ahead. Hey Marley, good to see you.     That's probably going to get back to Guardi. No. 150. Sorry. Let's go next to Dan Martin. Dan, please unmute. Dan, you know, one thing I'm excited about actually is, you know, a lot of these new pitchers we signed. You know, I'm glad I don't have to face Corey Clover. Jay Moe, I'm excited to have him in pinstripes.  You know, I'm excited about that.   We've got Jay Moe, we've got Kluber, I think we're going to be, you know, stacking up our rotation to be, you know, something special. You know, we've got a lot of great guys over there and I think just finishing. That's what it comes down to is like, you know, just being ball players in the box, you know, grinding out at bats in the postseason and just getting the job done.   You know, it's moving runners over, you know, playing some small ball, doing what we need to do to put ourselves in the right position. Go next to Pete Caldera. Pete, go ahead and unmute. What do you think of his evolution as player, first as a Yankee and what you might expect out of this? Yeah, I was excited to hear that news.    I'd rather work on something I'm good at and just work on that, you know, with Clint's, you know, he's the opposite. He's a guy that has looked for his weaknesses, you know, if it's stuff with his hitting, stuff with base running stuff, fielding, and he's, you know, attacked it head-on, you know, so watching him progress over the years and, you know, when I'm usually out there in that right field with him, watching him throw, watching him work and it's been a treat, you know, watching him, you know, turn into what he is and, you know, he goes from one year where he's getting a lot of criticism on his outfield to the next year he's a go-go candidate. You know, that just speaks so highly of what type of, you know, player and person Clint is. We're going to let Ron Blum skip in line because it's my understanding it's his 60th birthday today, AJ. Ron, are you out there? I'm here. Yes, man. Happy birthday, buddy. First off, happy birthday. I'm glad you're staring at it with us here.  How do you get it out of your system and how long does it take? Yeah, I usually don't talk to too many people, you know, a couple of weeks after the season's over, where that's where I kind of decompress, get away from things, don't even think about baseball, you know, I don't even want to see it on TV, but after that, you know, next couple of weeks, kind of getting into November to December, is when I start breaking down film, looking at past games, you know, checking it, you know, just different numbers I got, different things I look for to see, like, where can I improve, what can I do to help this team? You know, if I feel like I can do my job, you know, I trust all my teammates to do their job, you know, it's all about accountability, step up and do my job, so then I spend a couple of weeks just looking where I can make adjustments, and then from there, it's time to move forward, you know, last year is last year, and time to start looking forward to the new year and new team and new things to do. Thank you.  Everybody, essentially reported now.  It's been, this is a special group we got here, you know, a lot of these guys I've been with, you know, the past couple of years, and we've built such a great relationship.


Okay, let's go to the store Yeah Yeah, let's go somewhere else. So here's what we're gonna do. We are going to go shopping go buy some things and we're gonna cook for Thanksgiving, okay? Okay, oh, there's some music in the background Ken lifeless house, are you serious? I just moved. Let's get the car. Wait pick up Nanks Hold on wait, what did what did I see see off a beta gamma delta epsilon Z I'm not wearing makeup, and I'm in my sweats You're at CVS. Just buy makeup. I just had a fight with the pharmacist my shoes My suit says a massage person You got the wrong lane and getting the wrong that's illegal. No, it's not legal. No, I'm not he's doing illegal Oh Yeah, no, no, this isn't a train or extreme. Okay Wait To touch you I'm high on oxy free oxy. I'm selling oxy selling oxy going once or twice Doesn't exist Okay, so here's what we need we have fresh turkey Christmas my Czech turkey Brine it tonight. No, I can't brine it. I don't even know what brine means If you keep going straight you're gonna hit the meat market and in the big tubs that they have in there It's gonna be like the second or third Non-frozen one.  Okay on the top from the front of the chicken. It's gonna be right there. Oh, yeah All right, cool. Thank you I have to put the turkey with a chicken Here it is chat Fresh young turkey with giblets Fresh all-match old Butterball is the good of brand right like I don't need to brine this brine right brine. I read it on the internet No, no, do I need to find it or that round there? I don't need to I mean, it's not I don't worry about it. I'll just I'll just I'll just go the instructions I read online. Okay. All right. Thank you Okay You're basting Okay, how many how many pounds do I get? Okay, there's little butterball. Do I get a little one or a big one candy no candy? Stopped on a fit This is 14 laps You get to 14 pounds you get two small ones or else I won't cook this is This is 10 laps get two 10s tickets that's nice man. So two 10s. I feel like getting one big one. It's better Man, but I want a No, I'm saying like I think one big one Okay, what do you think? Probably I don't I've never seen a spider like that I'm trying to chat what kind of spider is that what kind of spider is that oh No It's shot at the way I don't know but I've never seen a spider like that Don't look at it me Just kill it. No leave it be. It's so annoying whenever people don't kill bugs It should slay them all So I got two turkeys, but I don't know I don't know what else I should get No, not frozen this is fresh Wait, but if they look frozen they feel frozen they look frozen it says fresh but It says fresh Okay, let's go back actually. Oh, he's already cooked. This is slow roasted hickory smoked What do you guys think? Yeah, we were at the chicken wall earlier We have we have to have turkey.  I thought it was fresh it says fresh Thanks, what'd you do? Oh, you're putting it back. So it's not fresh, huh? Well, I don't know It looks fresher Okay, let's go to the store. We're kicked out. Yeah, we're kicked out. Yeah, let's go somewhere else Okay, let's leave we're going on much nicer establishment like Walmart. Oh frick that guy went through that Why can't I record? I don't care. I'm recording what they keep going to do to me I'm leaving so I can record.  I'm already walking out here But no, I'm not tricked. It's a word back into HGP. You can't stop me from streaming Wait, they consume me. Okay Chopper We're already kicked out. This is freedom. I don't have to wear a mask They can't do I don't know where to go. There's that fun. Look at them walking with They're gonna give out to me again. They're all staring. He wouldn't got the turkey sub-bottom anyway Are you gambling again? No, I wasn't gambling. I was That feels good to me class chat Frick, how do we-


Next question is from Teenie Tangy. Are there any lessons you have had to learn twice?  Personal, working out. Oh, working out, I learned every lesson 15 times. Yeah, and I feel like that's too easy. I feel like that's too easy if we go the working out. Yeah, like working out too hard. I think every lesson, and I think that's why I think we have so much patience and empathy and I think why we repeat things over and over on this podcast is because I remember and I'm sure you all remember a lot of the lessons that we teach, they had to be taught to myself more than once, you know? Like I had to like change to do it and then see the results from it and then still go back to old behaviors or habits and then again, so yeah, you know, let me think of a personal one that comes to mind that I've had to learn. Yeah, the workout ones are easy. Yeah, working out too hard. Yeah, I've learned that one, I don't know, 1,000 times, you know, not doing proper mobility work. I still am learning that lesson diet stuff.   That happens a lot and I think, you know, I think the lessons I learned once were the ones that caused a huge consequence. You know what I mean? Like, you know, if like texting in the car, like if I did that and then I hit a car and caused someone an injury, I could see myself learning that lesson because the consequence was so terrible, right? But if it's not that, I don't know very many lessons I haven't had to be exposed to several times in a row. Yeah, I'm trying to think of a good, a personal one. Like, you know, I always talk about how your greatest strength is your greatest weakness, right?  And like an example of that is like I'm assertive and a lot of times that serves me, serves me in business to take charge, lead, do these things. But then if my social awareness is lacking that day, many times I could really turn somebody off and it could kill a business deal, it could rub somebody the wrong way. And so this is a lesson that I constantly learn. And I don't think I ever, or I should say I'm constantly learning. I don't think I've perfected it. You know, how do you know always like this is the time I lean in hard to the assertiveness or this is the time where I back off, like always trying to refine that. So I'm constantly learning that lesson. That's got to be a tough one too because I could see how after you had a situation where maybe you were too assertive, it's easy to defend yourself to yourself. Well, whatever, screw it. That was their bad. For many years, that's how I was. For many years, it was like I'm an acquired taste. You either, it is what it is. If you're too weak to stay hanged, I had that attitude for sure. And as I got older and wiser, I think, and more socially aware, I think that I've learned to know when to throttle it down and when to pull it back. But I'm still, again, it's not a perfected skill. It's something that I'm always trying to harness. I have a similar one to that, but it's more geared towards, like I will jump into anything that I feel super passionate about and like an idea or without like completely sitting down and doing the research, the market research, the, you know, like playing, like drawing it all out in terms of like, you know, the mind map and the business plan and all this kind of stuff, which to me I felt has been a superpower because it's been able to allow me to make connections with people I wouldn't have before and learn these lessons that I can, you know, build off of and, but the problem is that mentality persists, which then I get myself back in a hole because I jumped in before really, you know, doing my due diligence of research and really like understanding what I was in for. So it's really, to me, it's more of a calculated risk because I like learning that lesson like over and over, instead of just jumping in with the risk because I'm willing to take the risk is to be a little bit more measured in that, but still, you know, find that balance of like, I can still figure this out like as I'm in there, but just do a better job of actually like looking into it first. That just reminded me of something, a lesson that I've learned. It was choosing partners. I've done a lot of business ventures and many of them I've had partners and failed more than once, right? So I made the wrong decision more than once enough to where I probably thought, oh, I should just never have partners ever again, yet here I am in a four way partnership, right? Never, never even had. Sexy. Yeah, right? But, you know, I think that in the past, I tended to lean towards just either one, a friend or somebody I felt really comfortable with or somebody who I think had similar characteristics as I did. And I really think that what makes this work is that the things that we have in common are core values, which I think that's our foundation and that's important. But then we couldn't be more different as far as our strengths and business and that's why it works so well. And I think I didn't learn that. I wouldn't have learned that had I not had multiple partnerships fail and not work in the past to come to this place to recognize that, you know, understand that everybody in this partnership plays a significant role. And I could never do Justin. I could never do Salah. I could never do Doug. And that's a good thing. And because we have similar values that we always fall back on, that's the core foundation and the partnership. A lot of that stuff wasn't taken into consideration in my 20s when I was partnering up with people. That was a mistake. I think I had to learn more than once. I got one that I learned over and over again. That was recent. So, you know, you talk about greatest strength is your greatest weakness. I like to call it the shadow side of an attribute, right? So, you know, I have a lot of self-belief, you know, self-confidence. I have quite a bit of that, but the shadow side of that is arrogance, right? You can get so much self-belief that it makes it hard for you to listen and hear other people. And I'm aware of this. It doesn't mean I'm good at it. It just means I'm aware of it. So, I'm at the phase before I get good at it, I guess, or whatever you want to call it. But I had this happen with my kids and with Jessica.  And so, you know, we were with the kids and we were talking about, you know, things that we do. And my daughter goes, yeah, she goes, sometimes you'll start a sentence and you won't finish it. It happens all the time. It's super annoying. And I'm like, what?  I'm like, what do you mean?  And then you'll look at your phone or trail off. And she goes, you need to work on that. My daughter says that to me. Now, Jessica's been saying that to me for a long time.  I don't always know me. And so it made me like, okay, I got to listen more and consider that I may not be aware of something that someone else may be aware of. And I got to be able to consider that. But it was my kids. My kids were like, my daughter was like, yeah, that's a problem. She literally said to me, you need to work on that. Oh, yeah. But they're brutally honest. Yeah.  There's another one. We were talking with the kids and I'm like, who's got the worst temper? I'm thinking like, you know, I'm not going to have the worst temper for sure.  I'm like, holy shit, am I that not aware of that? That I lose my temper? Dad yells. Yeah. So learning to listen and consider people's either, you know, criticisms or complaints of me, especially people around me, that's a lesson that I think I got to keep working on learning.


Good morning everyone. It's a real pleasure to be here and I would first like to express my gratitude and appreciation to the whole UNU wider team, but in particular to Yongfu for enabling me to be here and to share some thoughts with you today. Let me just first preface my remarks today by saying that this presentation in Norway is meant to be sort of a comprehensive presentation of all of the work both analytical investment that the World Bank being a large institution does in the area of green cities. I did though of course want to give you a flavor of some of the work that we do do more on the analytical and the normative side and also to but also just to sort of the open up the space for discussion so to present some thoughts that my team and I working on cities and climate change and on sustainable cities in general have been thinking about in recent years and to engage with you on a discussion on some of these issues. So just in terms of background and setting the context especially since I'm the first presenter the just I think as all of you probably well know our planet is becoming increasingly urbanized we've heard many times in recent years that for the first time in human history more than half of us more than half of humanity now lives in urban areas it's never been like this before and just an example from China from the Pearl River Delta you've got the area of Shenzhen where 30 years ago it was not a city and today it's built up and in fact is part of a very very large urban conurbation. Globally if you then think about what you know so that the trends and the projections for the next few decades Africa is really the next frontier for urbanization. Latin America and increasingly many parts of Asia are quite urbanized. Africa has relatively low urbanization rates currently but that's the next wave and what you see here are just some of the analysis we've done in house for a number of African cities this one I'm just showing here is Nairobi. I apologize if you're not able to see very clearly the bottom half of the slide but that's actually just the sort of the GIS analysis based on some of the satellite images from the year 2000 then 2010 and then the projection to 2030 of the built-up area of Nairobi. One of the key messages though which we do communicate particularly to those of us who don't actually work in urban is really to understand that cities are drivers of economic growth and prosperity. For a long time you know there's been a lot of so a lot of the conventional thinking was about how urban poverty is driven by you know increasing cities that rural urban migration is a bad thing but I think the World Development Report 2009 and other recent work actually has tried to sort of shift the paradigm and to say that well there are many many benefits from the ablomeration economies that cities provide and so this slide here shows us one of the figures from WDR just showing this sort of correlation if you will between urbanization and income. One of the more recent reports by McKinsey just out recently also gives some numbers which I just added on to the bottom of the slide on the the percentage of national GDP which is generated in urban areas and I think the estimate is something like globally about 60% but for some of the largest I think five or six hundred cities it's as high as 80% of GDP which is generated in these other areas. Now coming to sort of the core of the discussion for this conference and I think again this message I think I feel sometimes I'm preaching to the choir or to the converted about how important cities are for climate change and particularly for mitigating climate change. You know just based on the IEA sort of calculations from a couple of years ago we do know fairly confidently that at least two-thirds of global energy consumption are attributable to cities. If you then include the embodied emissions food, water, so on and so forth of all of the activity which goes on in cities and which has teleconnections to areas outside of cities we think that as much as 80% of global emissions are ultimately attributable to the residents of cities. So if you compare this or you put this together with the sort of the broader context of urbanization trends globally and just the sort of the pace of development and urbanization and investment actually in infrastructure in buildings etc in cities this becomes actually a very very urgent message and to put it very pithily in a sort of a more dramatic phase the battle against climate change will largely be won or lost in the cities. In other words what we do in our cities in the coming decades will really determine to a large extent what we're able to do for global climate change. Having said that when you're working with cities when you're a mayor or a city administrator or a city level decision maker or practitioner you're faced with very very practical issues and challenges of the day. When we do speak with city officials they'll say oftentimes especially in developing countries especially in sort of smaller and medium-sized cities they'll say don't come and talk to me about climate change you know that's probably number 23 on my list because I have 22 other urgent things which I need to attend to today and I think that speaks on very well to some of what we heard in the opening plenary this morning about the trade-offs about the relative priorities and about how we can work with local government sub-national governments on you know very sort of practical and pragmatic solutions. So now I wanted to show you a number of slides of basically numbers or data which is again just to give you a sense of what some of the numbers are looking at. This slide actually sort of vertical axis is CO2 emissions per capita over time and then the horizontal axis is urban population as a percentage of the national population and the bubble size is the total carbon dioxide emissions of the of the country and what we did was we plotted this over time and so you see for a number of countries so the US for example you know has always been a high emitter both in terms of total national emissions the bubble size but also per capita you see up there but then you'll see some other countries and it's quite interesting Sweden is actually the little one further to the right which has always been well it's a relatively small country and has relatively low per capita emissions but I think what's interesting to see is the developing countries in the large developing countries Brazil, China, India for example where you see this this growth in the per capita emissions and I think we know that quite well you see the growth in the total emissions as well and what's interesting you see a country like Brazil or the Republic of Korea South Korea which is those smaller bubbles and kind of goes there towards the the end of the slide that's actually sort of how this you know the countries have urbanized over time this graph has sort of had a long life it's existed in various forms actually I should say for at least more than a decade now it's actually sort of digging through the data and going back and and and seeing because you'll see different versions of this but I think it's it's it's quite robust actually just in terms of the the the relationship you do see between urban form and and density and the the energy emission energy consumption and therefore the the emissions what you do have on the vertical axis here though are the individual emissions from from transport not the overall emissions and the message of this slide and again this this diagram in this analysis from Alain Bartot and others has been around for for quite a while but the point of this really is that sort of you know when you invest in cities you lock in urban form for decades or even a century scale and I think this is a particularly important message that the investment decisions we take today you know will will influence you know will influence the emissions pathways for very long decades into the future and particularly in places like China which are urbanizing really fast this has very serious implications if you look back though at cities in what I could call I guess the global north this actually just sort of gives more evidence that you know your decisions that you might be able to take today are actually limited or constrained by the decisions in the past just how you've built up so American many American cities in this case you see here Atlanta really it's difficult actually to change the pattern of sprawl to change how people live just because of how things have been built up you know in Los Angeles even Washington DC etc. I like that and by contrast you just look at the built-up area of Barcelona which is actually more or less the same number of people this is fresh off the press so I should say it's actually forthcoming because we haven't published it yet but it's coming out shortly on this new paper that we're calling Towards a Partnership for Sustainable Cities and here what we've done is we've plotted on the horizontal axis GDP per capita for cities right and then the vertical axis GHG emissions per capita for cities and what you see there where we put it in bold and in capital letters are the numbers for which we have the peer-reviewed metropolitan level data and I'll get to this again on in a subsequent slide about just how we measure GHG emissions from cities because I don't want to get into it too much for it now but because there are many different methodologies by which cities are actually measuring and reporting their emissions so the ones in the sort of non-bold or lower case are not capitalized are the ones where they're not actually fully peer-reviewed but what you see here then are you know you can sort of divide into four quadrants and what we've done is you know the green line there is oh I only have two minute slides okay I'm gonna accelerate the green line there is sort of the high income classification according to the World Bank which we usually apply to countries and then 550 parts per million is there again sort of the you know what do our global per capita emissions need to be to keep us below 550 and this is not very good news actually because you see you know this this slope this way and many many of our cities in the upper right hand quadrant and if you want to look at the bottom right hand quadrant which is where we would like to be there are actually very very few cities who are sort of in that quadrant with you know relatively high income levels and low emissions so I need to accelerate now show you again some of the sectoral analysis of GHG emissions from selected cities the point of this slide actually or the point of this analysis is to say that if you're able to analyze the city's greenhouse gas emissions you're able to look at sort of the sectoral breakdown that helps you to prioritize actually you know your efforts for mitigation this is sort of one of the latest sort of advances some internationally in terms of international standard for city greenhouse gas emissions we've been working in this space for a few years now with a number of partners UNEP UN Habitat World Resources Institute C40 and IKLE because of the what we would call the sort of proliferation of sort of different methodologies for calculating city greenhouse gas emissions so the latest advance in this space was the GPC which was actually announced and launched more recently at Rio Plus 20 and I just got to paste it there on the left sort of the the reporting table actually which is City of Rio de Janeiro has according to this new protocol but I'd say the important thing before I move on and I realize I actually only have one minute left is that the it's really important that we do actually have this owned and rolled out by organizations and networks of cities like IKLE and C40 I'm gonna skip this slide I'm not going to talk to it too much except to say that then if you're looking at mitigation efforts within the city there are obviously the number of sectors and there are obviously all sorts of things you can do here and you know this is just sort of an analytic diagram about and where these efforts might be and what they might be I'm going to skip this slide I'm not going to talk to you too much except to say that we've also been working in and on the question of carbon finance for cities again I think you're well aware that sort of the the current problem with the carbon markets actually is just sort of you know the limited demand and the low prices so on and so forth but if you look at registered CDM projects so far one of the challenges for cities has just been the high transaction costs and this approach which is working across multiple sectors has been trying to overcome that problem this is an example of a tool that one of the tools that we have with the with SMAPs the energy sector management and assistance program now I think it's rather unfortunate that because of the constraints of time I'm not able to go in sort of into detail on the adaptation side of things so let me skip a couple of these slides again which you well know this one I do want to talk to for a little bit two years ago the World Bank did a study on the economics of adaptation to climate change it came up with some big sort of headline numbers it's quite a lot of talk about sort of the approach that which was taken and how these numbers were arrived at but the point was this I'd like to make from with respect to sort of what we're talking about in terms of cities and climate change is this if you look at sort of this sector will break down from this table many of the costs actually of adaptation if you know according to to the results of the study will actually fall in cities I'm out of time these last slides which you can see were actually of some recent work we've done on adaptation issues in cities with particular reference to urban poverty my last two slides and if I could just if I may just take another 30 seconds actually will then be to talk about financing issues because then people will always say well where's the money you're the World Bank so what can we do in just in terms of the money we need and the investments we need to make the message of this slide is there are a number of things or a number of sources of funding and so I've given some examples here of some of these sources that the World Bank does work with but unfortunately there is less available than is commonly thought and there are sort of each of these sources for example you know have their own processes and procedures and so on and so forth and some of these are morally fully earmarked and fully allocated so the last this last slide actually is just to say that we we did try and address at least some even some sort of the information gaps if you will in terms of information availability on what's there so there was again from a couple years ago a joint UNDP World Bank effort and you'll see the URL there to provide through the web sort of some information on what climate finance options there might be and there's actually a dedicated section on which is relevant or which is for cities so thanks very much again for your attention and be glad to to pursue some of these discussions and questions with you after this.


This is like the time of year, many of us will be looking back and reviewing our performance. Something 2022 taught me personally is the importance of being consistent. And when you think about it, how can you be a consistently profitable trader if you yourself are not acting in a consistent way? And I kind of put it into two camps. The stuff covered by your trading plan and you talked about your playbook for small caps, for example, the sticking to certain instruments set up at times of day. This is kind of relatively easy to define and follow, but the bigger challenge in my mind is the mindset. It requires you to kind of be extremely self-aware, recognize when you're straying out of the zone, you know, overconfidence, underconfidence, tiredness. Even having too much coffee can change the way you execute. So my question to you, apologies for the rather long preamble, how do you maintain like a grounded, consistent mindset when you're trading? So big question while Pala is just pushing over here. Yeah, I saw that. So now it's halted, but the plan was good. So that's the reason Trillion right here works. So in order to become consistent, you need for me three main things. One is routine and I will talk about these as 3R. One is retraiting, one is report card. So when a trader wants to be mentored by me, the first thing is not looking into his trading, but is looking into his life. Expectation goals, I would say, fear, stress, everything that matters is important to trading. And trust me, traders, if you wanna make a seven figure per year or more, or you're a small trader, you wanna get to that, you know, maybe three or four figure per day, everything starts from a plan. So the first thing is you have to plan ahead what you want to achieve and you have to have a routine. Routine that consists in lowering your stress, having a stress relief, preparing in the morning, or if you trade like me in the afternoon, evening for your trading day and be repetitive. Routine means it has to be boring, repetitive, so you don't have to have like stress, you don't have to have like anxiety when you trade, you have to be simple like on your zone, in your zone. So when you have a perfect environment, when you know exactly what you're going to do when everything is planned, then you can start preparing your pre-market plan or your watch list. This is the first thing. The second thing is you have to make experience, okay? You wanna be consistent. Consistent means that you have to trade a strategy that first of all works, so you have to back test it and then you have to become good in that strategy. For example, the strategy that I teach, but it's not that I teach my strategy, I'm consistent, traders are consistent and the new trader that jumps in will be right away consistent. No, it takes time. It's like if I tell you, play in that basketball team, everybody is great, but you just started, even if they tell me they teach you how you have to play, you still need a lot of training in order to become good. And this training becomes like experience, practice and retraining, actually really taking even book map, you retrade, in order to become better in recognizing pattern, faster analyzing the structure, and reading order of the flow. And the third thing is having a report card, so analyze what you're doing every single day and then every single week, every single month, see if you're actually evolving as a trader, why do you have to improve, why do you have to overcome in order to get to the next level, why do you have to fix? And this is a valuation that has to be totally honest with yourself. It's you analyzing yourself and then work with a mentor. These are, I say, the pillars for becoming a great trader and to become consistent. That's awesome, I love it. And the first thing you said there was, be consistent in other areas of your life as well. And I totally agree with that. And it looks like you as an individual are extremely consistent in other areas of your, obviously you have a fitness program and you think this kind of helps you, like things outside of trading helping you within the trading environment. The fact is that a trader, generally, wants to always control everything. So the more control, the more planned is your life. The less, I would say, space forever, anxiety and stress you will have. That's the reason why when you trade, I prefer you to trade a strategy with set rules, set risk, set number of shares or contract, whatever that is. Because everything has to be planned. If you don't trade with the correct plan, or the correct strategy, then you're gonna fail. Yep, absolutely. I also believe it's the skill needed to follow your plan, to sit on your hands, to be patient. It's the same skill that you need to keep grinding in the gym. It's discipline, right? And that's, it all ties together. Exactly, exactly, exactly. It's everything that's based on your goals, your motivation, your plan, dedication.  And this is the point of trading, like it never stops.


Well, welcome everyone to day five of the USS. This is Paul Melbourne. I'm one of the steering committee members and you're in a Swanson as you know is the other one. I just want to make a brief announcement. You then and I have prepared a survey to discuss your experience your engagement your concerns and so forth from the first week of this summer school. So that is not yet online but it will go out to you it's a survey monkey. At the end of the day, we would very much like your responses by Saturday evening if possible if not then certainly by Sunday noon, and we will try to act on those. So that's all I have to say, and welcome and kill, take it away. Thank you. So, yeah, let me set up the screen chair. So, yeah, so I guess, right so so last time we introduced introduces construction of the, the pediatric numbers. So last time. So I introduced the field QP for prime number P. And so let's just recall that this was. So this was a complete. So, this is a complete non Archimedean field so sorry so I mean that QP is equipped with some sort of it's a it's equipped with the pediatric absolute value so an extension of the pediatric absolute value on the rational numbers. So this is a complete non Archimedean field. So with an absolute value. And right so it was, it was defined as the completion of the rational numbers with respect to this absolute value sort of analogous sort of defined as somehow analogous to the real numbers. The way the real numbers are defined as a completion with respect to the usual Archimedean absolute value. And right so we also saw last time. So you can sort of represent an element of QP you can you can represent it by a by a pediatric expansion. So expand elements of QP via pediatric expansion. You can write an element of the form, the sum of a a sub I times P to the I, as I ranges over all integers but it sort of starts at some minus and for some, and large enough, and goes to infinity. And the AIs are between zero and P minus one. Yeah so QP is is constructed as a means constructed as a completion of of the rational numbers. And it's supposed to be kind of like kind of analogous to the construction of the real numbers as completion of the rational numbers. And in general whenever you do sort of when whenever you have a completion process it usually becomes much easier to solve solve equations in a completion, because how there's more more room to maneuver. So for example it's much easier to solve polynomial equation or to show that there is a solution of a polynomial equation over the real numbers, then over the rational numbers. So over the real numbers, if you're given a polynomial. So given a polynomial f of x, in R of x, then you can detect roots well let's say let's say your polynomial just has simple roots. And you can detect at least simple roots by looking for changes of science. So you can detect existence of roots by looking at sign changes. And when we're over the periodic numbers there's no longer going to be an ordering so I say in fact there's there's no way of ordering QP because minus one is going to be a sum of squares. Similarly, but well somehow analogously. There are ways that you can you can look for. You can look for solutions of polynomial equations over the over the periodic numbers, and these come from from from looking at Congress's. So that's sort of a general tool for, for, for producing solutions of polynomial equations over the periodic numbers. And so this tool is called Hensel's lemma. I'm sorry. So there's some drilling that just started in the background and so if it becomes noticeable please let me know and I will try to switch devices in such a way that may or may not help with that. Okay. Right, so Hensel's lemma is the following statement. So let f of x, the polynomial in ZP brackets x, and Hensel's lemma is going to produce is going to give a criterion for, for the existence of a solution of f of x. So let alpha be an element of ZP. With the following properties.  But the derivative is not too small. So the value of the function is absolute value less than one, but the value of the derivative. So f prime of alpha has absolute value equal to one. So the definition is there exists a unique element beta in ZP such that alpha minus beta is less than one. And f of beta is equal to zero. It's saying that if you have you have sort of an approximate root of of the function, function f, and its derivative is not too small, then you can refine that uniquely if you're within a disk of a one open disk radius one to an actual solution of your equation. Okay. So, another formulation of this.  So this polynomial has a simple route, which I'll call alpha bar. So alpha bar is the reduction of alpha on p so so saying that f of alpha is less than one is saying that f bar of x has a has a root of alpha bar, and the second condition is saying that the root is simple. Then this route lifts uniquely to a root. So let's call it beta in zp of the actual polynomial that we started with f. So if you have a solution if you have if you have a solution of your equation mod p. And it's a simple, simple route so the derivative doesn't vanish at that at that point, then you can, you can upgrade that to a solution in in the patic integers, and it's unique within its congruence class. So this is Hensselstam. In fact, there are many different formulations of Hensselstam and this is one, you know, one formulation or one, one case of it. Okay, so I want to give sort of a proof sketch of Hensselstam and the proof sketch is basically use Newton's method. So use Newton's method to refine alpha to the root beta to alpha sort of an approximate solution. And then you're going to sort of iterate the this process. And then it's going to converge in the limit to to the solution beta of the equation. So explicitly what what this means is that you define a sequence. So define a sequence of patic and address alpha zero alpha one to dot dot dot. Alpha zero equals alpha.  And the amount of the amount you're supposed to change the solution or the approximate solution is is given by linearly approximating your function at that point. So the first formula is alpha n plus one equals alpha n minus f of alpha and over f prime of alpha. And so notice here that we're doing a division. And so we want it. So in order for this to work well we want to be dividing by something which is not just small, which is exactly this condition that you know we're starting with an element alpha such that f prime of alpha has a value one it's a patic unit so we can sort of divide by it without causing too many problems. And so then, what's going to happen is that. So this is exactly right so this is your new method. And then what you find is that the patic absolute values of FN are going to converge to zero as n goes to infinity. And in fact alpha n is a is a Cauchy sequence, which is going to converge to the limit beta. And that's going to be the root of your polynomial. So, so essentially the, you know, so the way the way that you prove this is that right so I mean, the way that you prove this is essentially you prove that f of alpha and n plus one is is going to be less than the absolute value of f of alpha and I think even you can put a square here, maybe less than or equal to. This way. So it's going to converge very rapidly and the reason is that you're going to use the Taylor approximation of F near.  So big O of f of alpha and divided by F prime of alpha and squared. So, I think that means we get to we get to say this in our inequality. And this is this is this is I mean this is the Taylor expansion so so there's no issue with the expansion because you're just working polynomials. And I mean you have to check a little bit that the Taylor expansion is okay that that you have this and the denominators and the Taylor expansion sort of cause problems. But essentially what you're going to use is that if you have a polynomial minus minus the linear approximation at a point you're going to, you know it's going to be divisible by like x minus alpha, alpha n square point. Right, so then you observe that these two terms, the first two terms are just going to cancel. So this is big O of F of alpha and divided by F prime of alpha and squared. And so then that's going to that's going to show that since you started with something where the value of F was less than one, it's actually a converge very rapidly the F of alpha and are going to converge very rapidly to zero. Okay, so this is really I mean this is really again this is really the construction of Newton's method. Probably seen over the real numbers. But in fact that can, you know all these questions involving the convergence and so forth are, are a lot easier in the setting because one is that you're working with polynomials but I mean, more silently is that you're working in a non or comedian setting. So instead of the triangle inequality for for bounding absolute values you have an honor comedian property. Okay, so, yeah, so maybe that's a leave it at that I think. So on the problem set you'll explore this I mean there are many there are actually many ways you can formulate the proof of Newton's method and you explore this a little bit more on the, on the homework. But I mean just just sort of to give an idea of how this is going. Essentially, what what you want to do is that this is a this is approved by successive approximation. And that's sort of made very clear in this in this procedure via this method, but you could also sort of formulated in a, maybe slightly more. Well, in a slightly more direct but maybe less efficient fashion, which is that what you're trying to do is you want, you want to beta, such that f of beta is equal to zero. And so beta is given by some some patic expansion so so beta has a patic expansion. And, well, that you're trying to find beta such that the first patic digit or the zeros patic, I guess it's the same as the spirit of patic digital alpha. And after that, well after that you have to find the rest of the patic digits. And, right, so the observation is that the first and patic digits of f of beta depend on the first and patic digits of beta. And then you sort of inductively solve for each of the patic digits. And then you sort of inductively solve for the patic digits of beta. So again this is not unlike how you might, you know try to solve such an equation over over over the real numbers you sort of, you know you first work out that first end digits of the decimal expansion and then you work out the next one and so forth and you sort of do the same thing in this in this patic context, except that the expansion is going in the opposite direction. So it really is very analogous to how you, you would try to solve some sort of equation over the real numbers. Okay, but so let me give some examples of this. So first let me give an example which, which came up on the exercises yesterday. So let me assume that actually let me assume that p is greater than two. And let's consider the polynomial. It's one minus one. Right so so this is this is a polynomial and in zp brackets x. And right so when you reduce mod p, it's, it takes zero at all the classes of fp. So if you have any element of fp, it's, it's p minus first power, or sorry, if you have any non zero element of fp it's p minus first power is equal to one. It's possible that you to the p minus one is equal to one. If you use any element of fp cross, exactly crosses a cyclic group of order p minus one. So, any element x with which belongs to zp minus p times the piece of any element which is a patic unit has satisfies that the absolute value of x to the p minus one minus one is less than one. However, the derivative is not zero. So but the derivative, but the derivative is p minus one times x to the p minus two. So this is a patic unit. And so then what Hansel's online saying that you have this sort of approximate root of the polynomial extra p minus one minus one is in fact any element any patic unit is a sort of an approximate root of the polynomial extra p minus one minus one. Oh, sorry, so this means a set theoretic difference. So this means all x and zp such that the patic absolute value of x is equal to equal to one. Okay. Yeah. So a conclusion is that in each congruence class of of zp, except the zero congruence class in each congruence class mod p, except zero. So there is a p minus first root of unity. So, I mean, I guess a way of saying this in terms of in terms of groups is that you can take the patic integers and take the group of units. And this maps to the units and FP cross, which is the mod p minus one. And in fact, this produces a well canonical section of that map. So that's saying that you can, you know, that that's saying that for, for this element, you know, for each, each congruence class might be except zero, you're producing a p minus first root of unity. And in the pre-medge. And in fact that's unique by the uniqueness part of Hansel's online. Right, so I in fact I didn't, I didn't actually prove the uniqueness part of Hansel's online I think the existence proof is more less going to imply that but in fact you're going to do that on to see that on the exercises so. Okay, so conclusion is that QP contains the peanut P minus one P minus one, a primitive P minus first root of unity. And you see that by looking at this polynomial X to the P minus one minus one that has that has a solution that has solutions want to be in fact every every element not zero of FP is a solution and it's a simple route. So you can lift that to roots in ZP. So I think on the problem set. Yesterday there's there's sort of another way of producing it explicitly. So, on the problem set, there's an explicit way of producing this which is that you start with any element x which again is in ZP, ZP cross. And then consider the sequence x, x to the P, x to the P squared dot dot dot. And this converges periodically to a P minus first root of unity, which is congruent mod P. So in this case you can really mean you can really produce this. You can produce this via an explicit limiting process by sort of raising to lots of P powers but Hansel's Lambo works sort of very generally. So what's another example of this. So let P be a prime number such that he is congruent to one module of four. So, then the square root of minus one. That belongs to QP. So I guess yeah so technically this is a this is a special case of the previous. Previous example. So, so this is a special case of, but right so we can also do this very explicitly so we can consider the equation x squared plus one in FP. And this equation has a root. Because P is congruent to one month four, it's it's necessarily a simple route, because the derivative is 2x, and then you apply Hansel's Lambo. So, right. So in fact, sort of thinking more about this and thinking more about Hansel's Lambo, it lets you completely determine which elements of the periodic numbers are squares. So, Hansel's Lambo gives a complete determination of which elements of QP cross. So which non-zero periodic numbers are squares. And in particular determines the structure of the group QP cross modulo QP cross squared. So this is this is a group we want to sort of have access to if we're going to think about quadratic forms over QP. Okay, so let's, let's start to do this. So let me do this when P is greater than two. So, so this this determination is going to be a little bit more complicated when P is equal to two. Because then when you take the derivative of something like x squared minus a you pick up a factor of two which is not a not a two attic unit. But when P is greater than two, we can do this pretty explicitly as as follows. So if we're given an element you in ZP cross, then Hansel's Lama implies that you admit the square root. If and only if the reduction mod mod P, which lives in FP cross admits a square root. Similarly you apply and so summer to the polynomial x squared minus you. Okay. Right. So, right. In general, if we're given some element of QP cross. So if we're given some element in some non zero periodic number, we can always. We can always write it as some power of P. So let's say P to the I times a pediatric unit you cross so where I is an integer, and you isn't CP cross. And right so if this is going to be a square then necessarily I is even because necessarily if you have a square, then the pediatric valuation has to be even. So what we've seen is that sort of is a necessary and sufficient condition in order for this element to be a square which is that I should be even and you should be a unit, you should be a square mod P. So this is a square in QP cross. If and only if I is even. And let's say you bar in FP cross is a square. And so the conclusion is that we've determined the structure of the group QP cross mod QP cross squared. Namely, it's Z mod two cross Z mod two. And where do these factors come from well on the one hand you have P. So PS, P is not a square, because P at evaluation is one. And over here you can take any not quadratic non residue. Let's say some you in ZP cross, such that you bar is not a square. So you bar in FP cross is not a square. And so you've got the structure of QP cross one QP cross squared as the mod two cross Z mod two in this case. Right, so I think there might be a question. Oh, yes, please. Um, so for this reduction of a periodic and number to the field FD, how does that work. Right so sorry so you mean this construction you bar. Yes, exactly. Right so for this. Okay so if you if you consider the ring ZP, we can take the quotient so so ZP is is is going to be the the sub ring of QP, consisting of elements of Norma most one. And inside ZP we have the multiples of P so these are equivalently the elements that have patic up to the value less than one so P times ZP. And this quotient is the same as FP. So if you think about patic expansion so right so explicitly if you think about patic expansions and any patic number. It has a patic expansion and we're just remembering what happens in level zero. Sweet. And so this, like we avoid the possibility of dividing by P, since we're not working with QP as a whole but just with the subring as EP. So exactly so much. So any patic integers of the form, some of a I times p to the I where I goes from zero to infinity. And then we're mapping this to FP by sending this to the class of AI, sorry, a zero. Right. And so this is, I mean this is sort of, I guess, very fun to work out explicitly because, like if you want to find if you want to actually find a square root. You can, you know, you can try to, like, however you might try to calculate a square root, the decimal expansion of a square root you can you can do that in this case as well and sort of inductively work out the Okay, so Hensel's Lama completely determines the structure of this group QP cross mod QP cross squared. It's a little trickier for p equals two. So what you can show so it's not necessarily true that it's not so easy to see, or it takes a little bit more work to see when a two added unit is a square. So Hensel's Lama is going to run into problems now because you have squares and so when you take the derivative it's it's zero mod two. But what you can show is that I think you'll see this on the problem set that any element you and Z two cross such that you minus one is divisible by eight is necessarily a square. So you can see this either using sort of a slightly more general version of Hensel's Lama than the one that I that I stated just earlier where where you don't necessarily have a simple root module of P but instead like the value of your function is a lot smaller than the value of the derivative. So this, well, it's sort of equivalent by sort of changing variables in a clever way. So, the, right, so the conclusion, in this case, is that Q to cross mod Q to cross squared is isomorphic to Z mod to cross the mod to cross the mod to, and as generators well, you have to so because the two added numbers necessarily even. And here you can take minus one and five to be to be generators to be sort of basis elements for this F two vector space Q to cross my Q to cross square. So, right, so it's a little bit more complicated at at the prime to when you're talking about squares in this way. But you can still, you can still work it out pretty explicitly and in particular something that's kind of nice about the periodic numbers is that if you look at units modular squares you got a finite group. So that's one way in which the periodic numbers are I mean that's also true the real numbers. I mean that's one way in which the, you know, the two added numbers are a lot simpler than something like the rational numbers, because whether or not something is a square sort of a finite condition. Okay. Sorry maybe I should pause for more questions here. Sorry. So yeah, so now let's apply this to to quadratic forms, quadratic forms over the field QP. So let's try to understand what are quadratic forms over QP when can classify them, you know when are the isotropic and so forth. And so the first step in somehow doing that was to think about units modular modular squares in QP. So, right, so let's fix a quadratic form. Let's call this brackets a one through a and over QP. The AIs are living in QP cross. And let's imagine we want to try to sort of classify these. So the first thing that we can do is we can, well, if we wanted to classify something like this we can we can multiply all the AIs by square classes. And right so. So in particular we can multiply everything by even powers of P. So, by rescaling. We can write it in the form. Say, you one through you are direct some PV one through PVS, where the UIs and the VIs live in ZP cross. So by multiplying by some even powers of P everywhere we can assume that the AIs are all either patic units or like P times patic units so we can sort of break it up into into two pieces like this. Right, and so it turns out that if P is greater than two this sort of breaking that breaking it up in this way and using the classification of quadratic forms over over FP is going to give a complete classification of quadratic forms over QP. So P is greater than two, which again is is the assumption that makes everything much simpler. When you're talking about quadratic forms in this context. We can use this to completely classify over QP. So, namely, the idea is that if you have a quadratic form over QP. Well, you can break it up in this way, and then you can reduce each piece mod P. And so you can reduce the UIs mod P and you can reduce the VIs mod P to give you quadratic forms over FP. And if P is greater than two, then this essentially gives you a sort of more or less canonical decomposition of of quadratic forms over QP. And since we've been saying things in terms of, I mean, right, so since we introduced this this bit ring, I want to, I want to try to formulate things in terms of the bit ring. Or in terms of this. So, what's going to happen is that the bit ring of QP, at least as an ambient group is isomorphic to two copies of the bit group of FP. Okay. So, right, so I want to make the following construction. To see this fact, what I what I'm essentially going to do is to reverse the above construction. So let you one bar through you are bar be a quadratic form over FB. So let's say I have a quadratic form living over over the field FB. And then what I'm going to do is I'm going to produce a quadratic form over QP. So you can produce quadratic form over QP by lifting UI bar to a periodic unit UI.  So note that, right, so there's there's some non uniqueness. There's some non uniqueness in the choice of flips. So, there is non uniqueness in the choice of the UI, but all the unique all the non uniqueness is up to scaling by square factors. Right, because if you have, if you have, if you have two different choices of you one, then their, their, their quotient is congruent to one month P and therefore it's a square since he's greater than two. So, so yeah so this is. So the choice of the UI is not relevant. And in fact, this gives a well defined, well defined map from, isomorphism classes of quadratic forms over FP to ISO classes of forms over QP. So, so this is not this is not sort of immediately obvious because in order to define this map I've sort of chosen a diagonalization I said let's, let's take a quadratic form over FP. Let's, let's put it in this diagonal form you one bar through you are bar. And let's lift it to this diagonal form you one through you are. But in fact, one has to argue that this is actually independent of this is actually well defined up to isomorphism it only depends on the isomorphism class of quadratic forms over over FP. In fact, this is something that now we have the tools to do using the chain equivalence theorem. So I guess in this case you could, you probably just check it directly because isomorphism classes of quadratic forms over. I think in this case you could, you could sort of check it directly. But in fact, you can see this using vets chain equivalence theorem. I mean if if you were to change the, the sequence you one bar through you are bar in such a way that you get the same quadratic format P, then the only way that can happen is that if you do this particular sequence of moves on the you one bar through you are bar. You know this particular sequence of moves that we saw like you can multiply something by a square you can interchange to consecutive elements, or you can do this one. You can do an extra, extra construction that comes from a suitable change of basis. And the observation is that if you if you do any of those moves on the you one bar through you are bar, you can also do those, you can do the exact same moves on the other side, you one through you are. And so if you think about this it's it's saying that the moves that you're allowed to do on the, you know, on the, on the diagonal form over FP, you can, you can do them on the on the right hand side as well. All right, so there's a question, right so how do you lift. How do you lift UI bar to UI on so any, any, any lift is okay. So, right so I mean, the lift is coming from ZP mod P times ZP is equal to FP. And so if we have an element in FP cross, we're saying let's let's let's pick some element of the pre image so some element of ZP cross, such that not P it's it's that element of FP. So that's not unique. So that's that's not going to be unique but if you have any two such choices they're going to differ by a spread their quotient it's going to be a square. So for the purposes of writing down the associated quadratic form, it's not going to affect the isomorphism class. Right so this produces a map of right so this is going to produce a map in fact it's a it's a ring so it's going to produce a map of commutative rings, which goes from the bit ring of FP to the bit ring of QP. And again P it's greater than two for this. Because in order for this to be all fine we need we need this, we need to take P, P odd. And so this this map is sending. Sorry, I guess I should say GW here because we said isomorphism classes so this is sending brackets you one bar through you are bar to brackets you one through you are. Right, so this this produces a map of commutative rings which is some sort of lifting, lifting construction. So let's call this map fee. We can also produce another map. P times fee, or let's say yeah P times. I guess brackets P times the I should say, which sends brackets you one through you are to brackets P you one. So it's exactly, it's exactly the previous construction but you've you've rescaled everything by P. So it's very well defined because it's so so it's it's literally brackets P times times the math fee. And so one obtains a map from GW of FP direct some GW of FP into GW of QP. Sorry so there's a question. Just to confirm so the map fee is sort of a section of that restriction down to the finite field, but the map brackets P fee is not a section right. Right so it's not. I guess I'm not sure I would say it that way because there's no map so this is kind of a funny map because there's no. There's no map in either direction between QP and, and FP. I mean there's a map from ZP to FP. And I guess we did not talk about we did not try to define growing the rings for something which is not a field. So you could, you could indeed phrase it as saying it's, I guess I think you could phrase it this way that it's, it's something going from the growing the equipment ring of FP to the growing the equipment ring of the ZP appropriately defined and then you're composing that with the map to the growing that ring of QP. But I think explicitly I'm just sort of thinking about an ordinance like you have a you have a quadratic form over FP, and then you lift that to like a diagonal form you lift that to a quadratic form over ZP, and then consider that as over QP. And it's not completely obvious that that's well defined and you know independent of the various choices that you made along the way but in fact that is the case and that's using that piece greater than two. Okay, so, right so you obtain a map like this and this is the map fee and P times fee. And so this map is so so this map as we've seen is basically so this.  So that's going to tell you that this map is surjective. It's not quite injective. And it's not injective because brackets one minus one is the same thing as P times that so it's the same thing as brackets P minus P these are both the hyperbolic plane. But somehow that's the only redundancy. It turns out that to the theorem of Springer, and right so in fact GW of QP is the quotient of GW of FP direct some GW of FP by the, well, I guess by the hyperbolic plane. So minus hyperbolic plane, or the VIT group of QP is isomorphic to the VIT group of FP direct some the VIT group of FP. So, I mean essentially what this is saying is that yeah so this is saying that a quadratic form over QP so if you like an anisotropic quadratic form over QP, and an isotropic quadratic form. So what this gives you to give an anisotropic quadratic form over QP is the same as giving a pair of anisotropic quadratic forms over FP.  And that's essentially the sort of classification of quadratic forms over QP when P is greater than two. It's in terms of essentially a pair of quadratic forms over FP modular this this redundancy involving the hyperbolic plane. And in particular there's following. I guess it's sort of a corollary or you can prove it directly. Which is that any quadratic form over QP and at least five variables is isotropic. There is in fact an anisotropic quadratic form of, of dimension four. So the you invariant of QP is exactly four. Right, so, you know, so why, why is something like, so, so let me, maybe I should answer maybe let me end by at least explaining the proof of this. So you can prove this by sort of examining the above proof if you have a quadratic form in at least five variables, well then you break it up into like sort of a quadratic form involving petic units, and p times a quadratic form involving petic units. And if you have at least five variables and one of those two pieces must have at least three variables. So if you have a quadratic form in three variables over FP, we know that's isotropic. And so in particular contains a hyperbolic plane. And that, well, that's going to mean that for QP it also contains hyperbolic play. So, this really follows from the previous arguments. But I also want to explain I mean this is sort of equivalent but I also just want to explain this kind of directly. So explicitly. The quadratic form say in five variables. So let's say your quadratic form. I mean again you, it's a diagonal form a one three five, and by sort of rescaling by even powers of P you can assume that it's at the form let's say brackets you want you to brackets PV one PV two PV three, where the you guys and the v is our, our petic units. And what we want to do is we want to see that this is want to see that this is isotropic. And to see that this is isotropic. Well in fact, just this last part this PV one PV two PV three is going to be isomorphic isotropic. So in fact, and that in fact well that's equivalent to saying that v one v two v three is isotropic. Because you're solving a homogeneous equation. And to see this in turn, what you need is a solution of v one x one squared plus v two x two square plus v three x three square equals zero. So this equation has a solution over FB by Hensel's I'm up. So this has a solution let's say x one bar x two bar x three bar over FB by sorry not by Hensel's I'm up because because quadratic forms in three variables over FB or isotropic. And this is a right so this is not a not this is a non trivial solution so so one of the outside bars. And then you can lift this to a solution over over ZP or FB using Hensel's I'm up. Right, because so so one. So, well Hensel's I'm up was formulated as a one variable statement, but essentially so let's suppose x one is not zero. So let's let's suppose x one bar is in this solution p is not zero. And then we're going to consider this equation, we'll just pick any x to an x three that lift x two bar next three bar and consider it as a function of x one bar. And so it's a one variable, you know, then it's really just a one variable polynomial, and it has a root mod p it's necessarily a simple root because, you know, because it's a quadratic polynomial.  So just fix two of the variables. Okay, so this is this is really. So this is really a fundamental feature of QP, which is the quadratic forms and at least five variables are isotropic. So I guess I didn't quite get to prove that there is an isotropic quadratic form of dimension force and maybe I'll pick that up on Monday. I also write so so so far I've been working for p greater than two which makes things a lot easier. But in fact this result is also true when p is equal to two. So that's this quarrel areas also true when p equals two. But you have to work a little bit harder to prove it. You have to work a little bit harder to prove it, because for example being a square is a more complicated condition, and because applying Hensel's I'm is a little bit trickier, because you're going to have repeated roots month two. So you're going to have to apply for example some sort of refined, you know some sort of more general version of Hensel's summer, which you will explore on the exercises. But in fact this, this corollary is is still true over over Q2 with a little bit more more work. So I guess I will, I will stop here for today, and we'll continue I guess I think we'll go into sort of more details. I'll try to put some more details of these arguments and what you need for this on on the problem set, because a little bit rushed at the end. And otherwise I'm going to, yeah, going to continue on continue on Monday so. Yeah, so I stick around a little bit for questions. So there's a question. Right so, so in this, the proof of this corollary I considered a quadratic form, the form you want you to PV one PV two PV three, but the same argument would also work if you have you want you to you three and then PV one PV two, I guess, because you would say that the first three is isotropic. So this works. Similarly for you want you to you three and then PV one PV two. Namely, you would show that this particular, you would show that the first, first piece of the quadratic form involving the three patic units. This part itself is already isotropic. It's isotropic mon P. And, well, it's isotropic mon P and then you can lift that using counsel so much to show that's actually isotropic in QP. Okay. Next question was what why does where does the argument fail for just three variables over QP so. Okay, that's a good question I'm going to, I'm definitely going to put more details on this on the problem set. But if you just had three variables, right so let me give the example or even a four variables of something which is not isotropic. So, so maybe I should, yeah, should be more. So, so let's say one comma you one comma you bar is an isotropic over over FP. Right so, ie minus you bar is not a square. So let's let's pick some anisotropic form of dimension two over FP. And then let's lift you bar to you in ZP cross so again doesn't matter which left because it's all going to, you know it's all going to vary up to a square. And then the claim is that the form.  And again, assuming P is greater than two. Right and so, so I'm going to put this on the, I'm going to put this on the problem set but I mean so let's let's suppose.  So let's suppose you had a solution of this equation over QP, and we want to get a contradiction. So, so without loss of generality can assume that the Xi's actually live in ZP right so you can so by rescaling your solution you can assume that it's a vector that lives in in ZP. So you can assume that not all the Xi's are divisible by P, right so you have x one x two x three and x four, they're they're all in ZP but not all of them are divisible by P. And now you sort of have to split into two cases. So suppose, for example, that x one is a pediatric unit. And you just, you just reduce the thing on P. So if you reduce mod P. You get that x one bar squared plus you times x two bar squared equals zero. And this is a contradiction, because, because this doesn't have any non trivial roots. So if either x one or x two is in ZP cross, then already you get a contradiction because you can look at what happens my P, and you find that this quadratic form reduce spot P has a has this has a non trivial solution which we assume does not happen. Conversely, suppose x one and x two are divisible by P. One of x three and x four lives in ZP cross.  Well, then you do the same thing but you first divide everything by P. So, then if you call this star, then the left hand side of star is divisible by P. If you divide by P, it's not zero. So it's, well, but it's not zero mod P square. So if x three or x four is not zero mod P, then you observe that when you consider the sum of four terms the first two terms are divisible by P squared but the second, the sum of the second two terms is cannot be zero mod P square because again, it's not isotropic mod P. Okay, sorry. So I think I'm going to put this, I will put this on the problem set so can work through it in a little bit more detail but yeah so this is how you construct an anisotropic format for variables that are QP. Yeah, okay, so there's a little bit more discussion in the chat, which I think is good. More questions. And I can ask you a question please. Yes, please. Sorry, I don't always see the hands so it's just. Oh, sorry. So basically I wanted to. Whoops. I still hear. Yes. Okay, hope everything's fine. So, I just wanted to double check basically whether I, I understood the, your, your proof about why the U invariant of QP is less than or equal to four. So the idea is that if I have a form of dimension five a quadratic form of dimension five over QP, then, well, we have this that we have this orthogonal direct sum where you know I have a part which reduces to you to F P star and a part that reduces to, well, zero modular FP. So I have this you, you, the UIs and the PV eyes, and one of those and one of those two parts must have dimension at least three. So if I reduce it to FP, then because you you invariant of a piece to, there's a hyperbolic plane that split. And then if I lift that I still have a hyper hyperbolic plane that split, therefore I have isotropy in my original form, right. That's exactly right. Yeah. Yeah, so you can also say it without using the words hyperbolic plane which is, I mean so you can also say it using hence the summer which is that you have a, I mean you have a solution when you've reduced that, like, you have a solution, let's say the first piece has dimension at least three, and then that has a solution on P and then you lift that to a solution using some you can also phrase it in terms of hyperbolic. Yeah, I see, I see that's great. Thank you. So, so for example, this is going to tell you that if you have a this this arguments maybe I'll put this on a home site and so if you if you have a, if you have a field, if you have a field of characteristic not to then the you invariant of the power series of the Laurent series field over that field is going to be twice the you invariant of the field that you started with. We can see how this could relate to the case of why you have FP is to and you have QP is is for because we can think of that as some kind of Laurent series in the case of QP for with a pediatric expansion right. Yes. I see. That's great. Thank you. Yeah, so I guess the more generous you would be like if you, I mean so so FP Laurent series T and QP, and what's sort of analogous about them is that they both have complete discreet valuations. So discreet, I mean evaluation and we saw like the pediatric valuation, you can also do a tea attic valuation for FP Laurent series T and being discreet means that the value the valuations are all integers. So it can really formulate it as a statement about complete discreetly valued fields. So this would be the devaluation from the exercise set from yesterday. Is that the, that's great. Yeah. Thank you. More questions. Thank you. Thanks. And thanks everybody for wearing PC my t shirts for our special day. Has everybody gotten one. Yeah, some of you are wearing it. Yes. I did not get mine because you were sent to a wrong address. All right, well, I'll see you in my office hours. See you to all either in office hours or Monday. Have a nice weekend.


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