in this episode of mind pump. So in this episode, um, this is our quaw episode. This is where we answer, uh, fan questions. These are my favorite episodes. So people go on our Instagram page, mind pump media, and they ask us questions about fitness or whatever else we pick them. And then we answer them on this episode. But what we also do is we open the episode with the intro portion. This is where we talk about current events. We talk about our sponsors and partners. And we talk about awesome articles in science regarding health or technology or other stuff. That's interesting us. Here's what we talked about in the first 34 minutes, which is the intro portion. I started by talking about photo bio modulation. This is like light therapy. So if you've seen those red light, uh, companies that sell the red light therapy, our favorite, by the way, is juve. Um, and they do everything from treat, uh, skin issues, uh, decreased wrinkle wrinkles might even actually speed up. Yeah. Boost testosterone that might actually do that. There's some studies that show that that might actually happen. Improve recovery. Well, this study shows that it might actually treat brain, uh, issues. So shining this on your head, uh, some of it actually gets to the brain. I'm not making this up. The article was published in science daily. It's in the show notes. Uh, by the way, our sponsor juve is in our opinion, the best company to get red light therapy that there is. It's the highest quality. And that's what you want. You want high quality. There's hope for Adam. So here's how you get the maps hookup or excuse me, the mine pump hookup.  This J O O V V. Dot com forward slash mine pump. And if you buy one of their lights and it's $500 or more, you'll get free shipping and a free maps prime program. Then we talked about the show on Netflix unnatural selection. This is a CRISPR technology. I didn't know that people are doing some weird stuff except done really badly in their garages with animals. Kind of weird. Then we talked about, uh, Ky Green talking about how you might go plant based to improve his health because, you know, he's so concerned about it's going to make a massive difference. So we talked about James Cameron, the producer of game changers. And, uh, this might be a, uh, conflict of interest, but apparently he's invested like a hundred something million dollars in a vegan protein powder company. Weird. That's weird. I talked about a stalker in Japan who found his subject by looking at the her pupils in her pictures and looking at the reflections stalker level 5000. That's creepy. Then Adam brought up a fight at McDonald's. Then we started telling all kinds of crazy stories. Uh, Adam knows way too much about strip clubs. You'll find out on that part of this episode. Uh, and we talked about how Cosby, Bill Cosby, the evil person, um, got like three to 10 years in jail. He got off easy. Then we got into the fitness portion of the episode. Here are the fitness questions that we answered. The first question, this person wants to know what we think of daily undulation. So what they're referring to is changing your workouts every day versus sticking to one for like three or four weeks at a time. So what are the benefits or what are the detriments? Next question. This person says, Hey, look, I know I'm supposed to keep my feet and knees straight when I squat, but I like to turn them out a little bit. It feels more comfortable. What's the deal? Is that bad or good? So we turn out for what? We have a nice discussion in that part of the episode. Next question.   What is going to make the biggest impact? Uh, how do you weigh that out? And the final question, this person wants to know what the difference is between having a good relationship to food and having a bad relationship to food. So we talk about that in that part of the episode, my fruit is brewed. Also this month, maps and a Bollock is 50% off. So this is our most popular fitness program. So what comes with this program? Workout videos where we're demonstrating the exercises, blueprints telling you how many reps to do, what the exercises are. It's phased out for you and you do this, uh, over the course of like 12 weeks. So it's a full program. It's our most popular program. It's great for building muscle, boosting metabolism. Um, it's an awesome program. Anyway, it's half off 50% off. Here's how you get the discount. You go to maps red.com MAPS, R E D dot com and use the code read 50 R E D five zero, no space for the discount. Go buy it. I was thinking the other day that, uh, I was like, why is Adam seems smarter to me? You know, like why? Yeah. Like, I was like, I was like, what's, uh, so weird. Yeah. He was saying stuff and I was like, that's pretty sharp. Something different has been happening here. Yeah. And, uh, then I read this study that was published by science earthquake weather. What? I don't know. Why would you say that? Because it's like a random natural disasters on its way. And that makes Adam smarter. Yes. Is that like when you're, like when your old aunt is like, my corns are small. Yeah, exactly. Like you feel like things in nature about to happen. So when Adam's, that's when things don't line up. So when Adam's not smart, shit's safe. Like, oh, yeah, like everything's fine. Nothing happened. Yeah. There's nothing impending, no impending doom. No, I'm like, man, he's getting sharp-ish. What's going on here? I got this article. This, but this was you coming after me from all the nerd stuff that we had. No, no, no, no, no. So then I read this article from science direct. I'll make sure to put it, uh, in the show notes. By the way, all of our podcasts have show notes. So you can go to mind pump podcast.com or use our free mind pump media app. And then you can see all the links and stuff. And, you know, people always ask where you get these studies or whatever. Anyway, so I'm like, rack of mind. Like what's going, he's getting sharper. I got to up my game and then read this article makes all the sense in the world. Science direct published this paper called shining light on the head. Okay. Photo bio modulation for brain disorders. So true. That red light, that red light can be compliment, right? Say he's got brain disorder. I'm not trying to say that, but this, this article says that, okay, so near infrared light, uh, or, uh, near light, I should say, um, like the one you get from the juve light that you've been shining on your head. Cause I do, you've been using it for your hair loss and for psoriasis, which is working for that.  Yes. And so it's absorbed by the cytochrome C oxidase in mitochondria. So that's basically gets the mitochondria to produce more energy. This is why it makes skin heal faster and all that stuff. But they're saying that increases blood flow, energy, neuro protection, less inflammation, brain repair. And according to this paper, I'm not making this up. Please go to the show notes. It sounds crazy, but it's true that it can potentially treat traumatic neurodegenerative and psychiatric diseases. So it's good for the brain and they may be used for treating things like Alzheimer's dementia and that kind of stuff. And they say to shine it on the forehead because there's less hair and it's easier to penetrate, but you're, you're, there's none anywhere. So we were so skeptical about this science dude, right? When we first were talking to you, cause it's like, it literally sounds like like medieval magic. Like I'm going to shine this light on you and you're going to get all these amazing health benefits to it. Okay. It's going to help your skin. So that's kind of like, okay, well, I could see that. It's going to help grow hair grow back. All right.   Fire. Fire. How crazy is that? I knew I was onto something photo mild modulation for the brain.  Have you guys seen the new Netflix series? I had a couple of people DM me and I was like, whenever I get multiple people sending me DMs about a specific series, I'll normally go watch it. And this one, people have been telling me about it's called unnatural selection. Oh, everybody's been posted. Yeah, I watched a little bit of it, but that didn't make it always. It's about CRISPR, right? Yeah.  Dude, it's crazy. It's really crazy. So what is this thing? Well, it's, it's not saying anything new that I don't think that we've ever brought up because I think Justin brought up CRISPR technology a long time ago. Didn't you bring it up? He was talking about, no, he was talking about something else. He was talking about crispy technology for the chips that he was using. No, no, no, no. One of you guys brought up crispy chicken. It was me. It was me. Somebody brought it up and I remember not knowing what it was. And so I didn't know about it. Justin, it was you who brought it up. Yeah. Yeah. So have you seen this at all? Yeah.  Like I saw a little bit of it. I know that they're doing like, you know, at home, like sort of, right. Like alchemy, basically of, of genetic editing and whatnot. Cause you can literally go in and change the gene. There's people that are taking like fireflies and they're breeding them. They're taking the DNA and they're breeding it or whatever. And with the, uh, like a rat and then the rat glows. Yeah.  It's crazy. That's messed up. You know, and this guy's doing it with dogs right now. Well, I know they did that too with, with jellyfish and they, they took that and they, they put it into like certain other kinds of fish to make them glow.  Like you're going in there to buy a goldfish, but it's like a fluorescent goldfish is, is way cooler. What? Am I the only one? Are these people not, did they not read comic books and sci-fi movies? Yeah. We're not supposed to play God. So the scariest part about this to me was, you know, cause originally I think when I heard you talk about a thing like, well, you know, this is in some lab and you're going to be some crazy scientist to be able to figure this out and this technology is beyond the average person. These are like bio hackers at home. Yes. You've got these bio hackers, some of these guys without even degrees or anything that have just researched and learned, figured it out. Terrible idea. And guys sharing kits online and selling them. We got to ask Bishop Baron about this. Dude.    It's, I'm extremely fascinated by it.  And they talk about this fish like this.  You can cut an organ out of it and it'll regenerate it. And the possibilities of that, that we could actually set instead of us having to, no, I'll put it. Regrow limbs. Yeah. Regrow organs.  It's like just because we can do it. I mean, we should do it. Look at this. Yeah, I see that. The glowing rats. Yeah.  These are, okay. So put yourself in the mind of these, these bio hackers. What would you do? What do you think they're going to try and do next? Right now they're just having fun. Glowing rat. It seems safe. Oh, they're going to start doing things that give them like abilities. Like they brought it up on the show, like, you know, having like eagle vision. Yeah, they seem to like X-Men. Right. Right.   Yeah, what's wrong with you, bro? I don't know. Yeah, I don't know. It's, it's, it's crazy to me that we're here. I know. It is getting weird. It's getting really weird.  You know, it's not like, the thing is we, but it's real. It's not alchemy.  This is real.  Well, how long do you think until athletes start doing this to make themselves? Real soon. Real soon. Steroids don't, don't even, there's no chance. They don't have a chance. If you could just alter your, like- Yeah, they were showing the mild statin one. And I think that's going to be the first one, the cheetah legs. I think that's gonna be the- Gorilla arms.  I think that's the first one that's, I mean, if there's, if there's these weird bodybuilder guys, and I don't like to call them, throw them in the bodybuilder category, just weird people that do the- They already do it anyway. They inject themselves with whatever.  The Cynthol. Cynthol.  All fluid.   I get all the attention.  I don't know, man. That's weird.  It's a whole series. I only watched the first episode last night. Katrina fell asleep too. She's like, are you really into this? I'm like, yeah, you're not fascinated by this. There's certain things. There's, you guys have shows that you can watch on, you have to watch. I have to watch them. Actually, that's the reason why I stopped watching it was because like Courtney comes from medical background. So that's like nails on a chalkboard right away. Yeah. So like, I can't watch this. I'm like, ah, all right. I guess I can't watch you either. I'm going to check it out. That sounds really, I'll have to follow up on that. Speaking of bodybuilders, did you see the Kai Green posted the other day? Well, we both fired on there. Yeah. Did you see Jesse Sarkozen? Yeah. Oh, so now he's like talking about trying to become a meatless athlete. He watched, um, game changers, the documentary that, you know, the vegan propaganda, well made, very well made vegan propaganda, uh, documentary. And we've covered that in a previous episode. You could actually search our podcast. We went and watched it when it got released in theaters, but here's a pro bodybuilder, massive. He's on, I don't know how much, how many different drugs or whatever to make himself look like all of them. And then he says, Hey man, I'm thinking about going, you know, meatless, you know, this, this, uh, this documentary's got me all shook up or something like that. He's healthier for me. What? Maybe I'm wrong. He reposted the blood vial one. Remember that part? Yeah. Where they show the blood. That's not science, dude. That doesn't mean anything. What they did there, it doesn't mean that's not science. That's not how it works. That's not a study. The whole documentary was that way. But I find it funny that you got this because then Dennis James chimed in. Remember Dennis James? Yeah, he's thinking about doing it too. For what? For your health? The last protein. Yeah. The last person I would ever, never take health advice from a pro bodybuilder. See, don't be surprised. Okay. Now you brought up the James Cameron thing. I didn't bring it up on the show. Oh yeah. We didn't talk about that. No, I know, but we, we brought it up on the show originally. We said that a hundred percent we think at some point they're going to be selling vegan protein or supplements. Yeah. And sure enough. Yeah. James Cameron is how much? 140 million or something. Yeah. So, okay, if him and Arnold are in on something, it would be a smart strategy to go get a Kai Green, a Dennis James. So who knows if these guys are all setting the table for promoting a vegan protein that's going to. Sponsored athletes. Right. Yeah. So I wouldn't be surprised if- Vega powder.  Sure. I mean, if I was, if I was doing it and trying to make money, that's exactly what I do. And find somebody who has a lot of attention like that. And I would tell him, hey, post like this, just, you know, I'm thinking about doing this over the next couple of weeks or whatever. And then we'll come out in a month and you'll be officially promoting our vegan protein. That would be a smart way to do it versus just come out and be like, Oh, I switched to vegan now and here's my protein powder. It just, it's funny to me. It reminds me of, um, I had like buddies that they would ask me about supplements. They'd be like, Oh, here, you know, creatine, take creatine. It's one of the most effective ones.  What do you mean? You don't know if creatine is like, you know what I mean? Oh, I don't know if I want to take protein powders because I don't know. I'm more into the natural stuff.   I might go to, yeah, I don't know. I might go plant-based. It got me all shook up like, Kai Green, let's take a list of priorities for your health.  No, it's this, this, this is getting to, it's getting, it's reaching a fever pitch with this movement. I have nothing against, if you want to go eat this way, but I have something against the way that they are demonizing meat. It's totally inaccurate. It's not supported by actual research and science.  In fact, what is Rob's Instagram? It's, it's DOS, D-A-S, Wolf, I think. Is it DOS Wolf? You should look that up. DOS Rob Wolf? Is that- Oh, DOS Rob Wolf. Yeah. For those that are listening that, you know, hear what Sal is saying right now and the, you know, these people, right? Show me a study, Sal. Show me a study. Yeah. He, Rob is just dropping them left and right. He's in the middle of actually writing a book to counter this terrible message. So if you want the researcher, you want to hear somebody who's like speaking to this specific topic and you want more information. Rob Wolf is the man. So he's, he's posting on this and speaking out on it. But you're right, it is smart for them to go after the, because that whole documentary showed- Athletes and Boners. Yeah. Like that was their whole agenda. It was. Yeah. We're going to knock this down. And it was all to counter the, you know, that makes you weak or it makes you less manly or whatever. So like, no, it doesn't. Yeah. These guys got better boners. That's in the documentary about that. So it's DOS Rob Wolf. So D-A-S and then Rob is R-O-B-B and then Wolf. So- Wolf. There's an L in there. Sorry, sorry. Wolf, wolf, wolf, wolf. It's silent, fucker. Yeah. It's a silent L, dude. Yeah. Dude, you want to hear something crazy? Yeah, I do. I love crazy. I read the- Crazier than the red light there. Yeah, no, this is really crazy. It's magic. There was this, in Japan, there was this fan that was stalking this Japanese celebrity girl or whatever, and he got arrested on indecency or whatever. So I guess he was, I don't know what he was doing, but anyway, he was a stalker. He found out where she lived by her Instagram and social media pictures by looking at the pupils of her eyes and the reflection off of her eyes. Oh, Jesus.  By watching the reflection off her eyes, he used Google Maps, figured out where she was.  And found her.  Yeah, dude. That's what she could do with technology now. Did you guys ever watch that series? I think Doug watched it, the creepy guy that shows you kind of how stalkers are today. Oh, yeah, we all watched that. Did you watch it, too? Yes. It was really good. Oh, yeah. I forgot the name of that one, too. You. You. Yes. You. I think you're coming out with another season of that. But yeah, that was a great show. I just found that one fascinating because we grew up. Okay, if there was ever a time in your life when you had a little stalker in you, it was probably when you were in high school-ish, right? When you're in love with some girls. Sure. Yeah, right. And what did you do? You don't say you looked at her across the classroom. Maybe you had a yearbook that you looked at. Like you. I mean, that was like. Always trying to get hugs. That was the, yeah, always trying to get hugs. Every excuse you get to get a hug. Hey, Stephanie. She's like, uh, that is the move though, right? I swear to her. I mean, that's the extent. Oh, you feel good.  Let me see your hair. Yeah. And then maybe if you're like a real weird creep, you go stand outside her house, right? And you look through it. Oh, yeah. Then you're, yeah. Oh, my God, yeah. Then you've gone too far. That's extreme, right? But I mean, now. Can't follow them home. Now with like all these social media platforms and everybody putting themselves out there so much and recording videos of themselves, the level of stalker has just gone through the roof. Oh, stalkers are having the best time ever. Easy. It's the best time to be stalker. It's the best time ever. They're like, this is like peak stalker. This is the best time ever. I watched that. I can find anybody. I can stalk like five people at once. I don't even need binoculars anymore.  I watched that show with Jessica, where that guy is like finding where she is and he's doing it through her Facebook and where she commented in the location and her friends and they show on the show like how you can actually do this. And Jessica's like, this isn't, and I'm like shocked. I'm like, oh my God, that's crazy. She's like, this isn't anything new. I'm like, excuse me. She's like, yeah, girls have known this since social media came out.  Whatever they want.  I remember I found out the hard way. I remember when we first got on Instagram, which is because we just got so liking and imposing.  Yeah, exactly. I remember coming home one day and Katrina being like, hey, could you like calm down with the booty pic likes and stuff? And I'm like, what? You know that? Yeah. She's not even on Instagram. Oh man. I'm like, wait a second.    She should have, she played her hand wrong, dude. Katrina, you should have played your hand differently. I would have lied. I would have been like, totally. She ever liked like, but you know, you don't, that's weird. That would never work for me. She'd ask. I'd be like, yes. You know what I'm saying? That's who I am for sure. You know, but she's more like embarrassed about it because she's like, hey, could you stop?  And they, I'm like, they could see that? How do they say it? I have no idea. Oh man. I enjoy the sport of bikini. Mind your own business, everyone. Exactly. My God. It's a, so I wanted to, speaking of crazy like articles, I was reading a complex this morning, it popped up. This lady, two chicks, McDonald's. Okay.  And then she is waiting, she comes in the store, she parks her car, she's got four kids in the car, whatever she comes in. And she's like waiting to like tell them like the order's wrong. And they're like, she's arguing back and forth with the manager. This thing goes on for like 20 minutes. And finally, the lady gets so pissed.  She gets all the food that was in her car. And then she slams on the counter. She's asking for a refund, return, managers fighting with her cross-counter.   Bro, she's, you should have seen the pictures of her face, dude.  That escalated. Yeah, big lawsuit coming for sure.  Yeah, yeah. This lady just threw a french fry at me.   I'll send a video to Doug, so Doug could show you guys because it's like having a hard enough time. Have you guys ever been in a situation like that where you're just at a random restaurant or something? Well, I told you about my incident with my carbine toad. Remember that one? Oh, I got to start for you. I forgot about that. You don't remember my carbine toad situation? No. Okay, so when I used to, my first house had a gated community and you can only have, you had one parking pass and then you had the garage. And I had a three bedroom and I used to rent it out when I first lived there. So I always had two roommates plus me. I've always had like two cars. So my two cars in the garage and then we have two other cars that we're trying to park in our complex. Well, and you only get one parking pass. So it was like the shell game that we were always trying to play and this is something I found out later. I don't know if you guys know this, but gated communities make contract deals with tow truck places. Oh, right. And they basically give them the code. This happened to me. So they can come in and plot cars that don't have passes and it's like a kickback, you know what I'm saying? So they hit me for like 270. So if you get hit and towed, it cost me like 270 to get it out. And then obviously, they get the longer it sits in there.  I'm, you know, mid-20s. I'm making really good money. I'm renting two rooms. So I kind of have this attitude or like it's more of a game at first for me. I'm just like, whatever. I'll try and hide it and, you know, figure out. If I get towed once a year, big deal. Yeah. Well, it's more like once or twice a month. But so it's happening. It's happening quite a bit.  That was just part of the deal. I was just like, okay, I got to pay this. So I'm a good sport about this. Right? I've been towed a bunch of times. In fact, it gets to the point where- Wow, a lot of times. A lot of times. You can't even think of it like park outside over there outside the gate. I lived on East Side San Jose, bro. You don't want to do that.  Right. I was way rather paid 270 than take the risk of parking it in East Side San Jose.   So no, I'm parking it in my getty community. I'm rolling the dice. If I get hit a couple of times in the month, it happens, right? So I kind of have this like attitude like, I think I'm a good sport about it. They, I come down there and there's like the same, the same process, same people too. It's like a family owned tow truck place.  And then this big like, it reminds me of like the member of the master appointment book that we used to have in training. Oh, it's the big appointment book. Yeah, big old appointment book that and everyone's stuff on there. So I'm in there and I'm in there like every other week, right? Dude, it's crazy how much I've come down here. I know everybody there.  She's being rude and stuff. And I'm like, Hey, I come down here and give you guys money all the time.  And she tells me that I need my registration. And like, I was just here last week.  You know, this is me right here. That's my name. That's me right there. I'm, this is, this is my car limit. Nope, you can't have it unless you have your registration.  And I'm getting so mad that I have to go all the way back home to go look for this registration. And of course I go back home and Katrina I think is out of town or something like this at this time. And so she doesn't have, doesn't have, she had it on her and I couldn't get it. And so this lady will not give me my car. So we had to go through this whole process where she comes back, she gets me the registration and we're hours into this.   And I remember going, you know what?   This is what I'm gonna do. So I go to the bank. I get $270 in quarters.  And I come and I slam. Use your money.  Boy was she pissed. And they were fighting with me about that. I couldn't do that. And I said, what do you mean I can't do that?  And you have to. They're like, you gotta count it. I said, I already did count it. I know there's enough there. You gotta count it if you don't believe me.  So, oh man, it took three people to come in. They're spreading the quarters out. And they wouldn't let me have my car until they counted the money. I went next door to McDonald's.  And I just sat there. Who's that, the kid there? Yeah, I just sat there watching them count the quarters. I did that.  My day was already ruined. You know what I'm saying? Day's already shot and ruined.  You know what I'm saying? I'm gonna go have a drink, sit here and watch these guys count my quarters up after they did that. Yeah, no. I had like, so when I was in college, I had a few of these jobs where I was working at restaurants. And this was where I just knew, like this new red robin just opened up. And I'm like, okay, I could get a job there easy. Like, I've worked at a restaurant before, whatever. I'm just gonna do this. In the meantime, get cash because they have a car and they need like spending money around campus and whatnot.   And so I'm in the back and we're really close to the kitchen. And so there's like the cooks are kind of yelling back and forth. He thinks it's about his stuff. And I'm like, no, no, no, this has nothing to do with your order. Like, because he kept pulling on me. He's like, what's going on back there? What's happening?  I'm like, look, there's nothing to do. You guys are fine. Like you'll get your food. Like everything's cool. And so like finally get their food, all that kind of stuff.   And I like put it down like, so you guys, you know, interested in it. And so he grabs it right away. I'm not interested. I've had the worst service here ever. He throws it right at my face. I was like, oh, now you've done it. Grabbed a blender. Now you've done it. Took my apron off. Wait, wait. He actually threw it at your face.  And I was just like, okay, now you've done it. Took my stuff out.  Like let's, let's handle this outside. And so I just like walked right outside and I was away in there. And they ended up escorting him out to his car with his wife and everything.  I'm going to take that, dude.  I could not believe it. I wanted to kill him. Did you get fired? No, I didn't get fired because everybody saw it.  And like I, at first they were like, if you do, they gave me a warning, you know, and they, the corporate like had to like make calls with me. And I had like, I don't know. They almost like had to have like a, I think it was like three or four managers had meetings with me, you know, after that.  You know, they agreed with me.  Oh no. That's biohazard. Bro, I had someone spit in my face one time. Oh my God. What? Yeah. No, that would, we were like, I don't know how it reacts. That's straight, like right to the face. I almost feel like I'd explode or something. So we were, is this is like, oh my God, I'm going to say probably 24-ish or so. And I'm with, I'm with my cousin and I'm with like four other buddies of mine. So there's like five, five or six of us that are all going out. We go down to the Pink Poodle, which is a strip club. It's a very classy one. I know, I've frequented there a few times. That is the worst strip club ever. Well, I know. Here comes Susie. My 18-year-old birthday. Back from maternity leave, you know. Bro, she's got brand new stretch marks to show. So I know one of the dancers there, right? So I know one of the girls there. And so we go and we have a great time. There's nothing like going to a strip club and actually knowing one of the dancers. One of the best ways to go to a strip club. It's like cheers, you know. Yeah, no, no. You get all the love when they know you like that, right? So anyways, so we have an incredible time. Yeah, you have a chair. Yeah. Your seat. Adam, you're back. So it's me, my buddies and everything that's, we have a great night. We stay, we close the place down. It's two in the morning. Now strip clubs have this, a very strict rule that he knows so much. No touching. You're going to break it down. No touching. It's this rule I don't see on your hands. I don't know if you've heard of it before. Everybody knows that rule. It's a different rule. Lookie, no touchy. They have a rule. Okay. If you don't know this, that a stripper cannot get picked up in front of the club. So she cannot be, she cannot like get in a car. Why do you know this rule, Adam? I'm going to tell you why, though. I don't know what this rule is. You want this spinning in your face story or not? You want the story or not? All right, sorry. So it's two o'clock, place is shut down.  So it's more like 2.30 in the morning. And you know, we're out there.    And as I call the cab, like it's like 2.30 or so in the morning. Well, it takes like 30 plus minutes, dude, because it's no, there's no cabs out there at this time. Before Uber. Right. This is before Uber existed. So it's taking forever. It's like after three o'clock in the morning, she's already came out now, we're waiting down. And I see the cab come up.  Oh, they take your. Oh, they take to get in. And so we go fucking, we're running down the block, you know, to get down there. No, no, no, no, that's my cat. That's my cat. We're jamming down there. Cab starts to take off. He flips around heading the other direction and then he pulls over right away. So obviously he's probably PDC-ing together that these are the, it's not them people. So I come running up.  I'm Adam. And the guys in the back like, no, I'm Adam.    Get them. And we're all, you know, all my buddies are active. They're all acting tough because they're inside the van.  And it was like a reflex. I just whack as fast as I could.   But that was the first time that that had ever happened to me. And it was really funny because it wasn't, I didn't even think about it.  Like that's pretty much, yeah, I would assume that would happen. Dude, that's biohazard. I mean, that's, you could catch some weird disease with spit. In fact, that's probably one of the most dangerous weapons you have. And it's also, it's one of the most disres, it's like the top of the top, right? Contempt, it's like pure. Oh, it is. You can say a lot of things to me. It's like a backhand slap. You know what I mean? It's not like a front slap. Is that, you get slapped with the back of a hand? Yeah.  And even that is, is still one step under that. Like spinning in the face, right? It has to be the on, unacceptable on all levels.  I wonder if the pink poodle's still there. I think it is, yeah. Is it really? Yeah. It's not that far from us guys. You know what I mean? It actually is close to it. It is close to it. Was it that bad? Just by memory as well. I mean, it's like one of the better ones in San Jose. I mean, you can't, you don't get a lot of options over here. We're not the strip capital. It's not like Vegas.  Hey, I've been working there for two years. My name's Alexis and I'm offended. Terrible. Anyway, well, I had an article I want to bring up. I don't know if it's, whatever. Let's see you transition to some science there, buddy. I had, it was a cool article, but I don't know. Well, you know what? I have another like weird, not weird, but another transition along these types of topics since we're talking about strippers and sex and all that crazy stuff. Yeah. What's up with Bill Cosby? Oh, jeez. Yeah, that's good. That is a good transfer. He went to jail. You know it was only, he only got sentenced for three to 10 years. That's it? That's it. And they're going to, I think they're in the middle right now of an appeal of that. And even if the, He'll only do like one year. That's all he's going to do. Even if they get, if they don't get the appeal and the appeal gets denied, if it's three to 10 years, he'll serve half the three, right? Isn't that kind of standard? You're going to hold an 81 year old in there for, I mean, he's going to be in good behavior.  I mean, his age probably plays a big role, right? How old is he? He's 81. 81?  You know what I mean? But just what he did, three to 10 years. It's nothing. No, that's nothing, dude. Oh my God. You don't pay four speeding tickets. I think you have the same six, dude. There's people who get ridiculous. There's people in some states who get caught with, you know, 20 hits of asset and go to jail 15 years. Yeah. And didn't do any, they didn't rape anybody. You know what I mean? They just had 10 hits of asset on them. And you got Bill Cosby and he's, that's crazy to me. I know. But I wonder if his age is the biggest. You know what I mean?   Hold on, hold on. Tarnished all his work. Yeah, but hold on a second. Let's think about this. Just think this through. Three to 10 years is life for 81-year-old guy, you know what I'm saying? Well, yeah. Yeah, you can 40 years in prison. He's going to serve three years and then die of old age, maybe.  The rest of your life. Right. You know what I mean? Yeah.  Because you may only live another 10 years, you know what I'm saying? That was such a crazy situation for me to experience because You love Bill Cosby? This is how powerful media is. I don't know Bill Cosby at all.   Completely constructed. Totally fake. Yeah, fabricated.  They're really, really fake sociopaths or whatever. What makes the worst ones the worst is that they're really good at acting in a particular way. Well, the best actors in the world are actors. You know what I mean? They're Hollywood. So those, boy, what a terrible combination. Like you get a famous actor who's like Academy Award winning or just really good at making himself or herself look likable.  It's like you can't, I heard this about Bill Cosby. I swear to God, I was like, there's no way. Yeah. This Bill Cosby.  He was the moral guy. Yeah. Crazy. First question is from Dressfit. What do you guys think about daily undulation? They're referring to probably what, nutrition? No, no, no, no. They're talking about exercise programming. Oh, so changing, so undulation essentially is changing your exercise programming but on a daily basis in terms of the fundamental aspects of your workout. And because when you see studies, the studies will show that this, so they did a really good study where they did this, where you did it daily or you did it in phases or periodized it. Periodized it. Did I say that right? No, I can't say it. Periodized it. Periodized it.  Now I can't say it. Periodized. Periodized. There we go. Doug, are we saying it wrong? I don't even know now. Periodization. Periodized. Yes. Periodized. Right? That would be the short? Yeah. Okay. I said it right the first time. So you got me all over from that intro, bro. I'm all over the map right now. So yeah, they did that. And then they did some on a group that didn't change at all. And they found that daily and then like every three weeks was almost the same and a little bit better in daily. Now this was a shorter study that was only I think a few months long, but at the end of the day we know that it's important that you change your exercise up. Now I used to do it every single day for a long time. Now, why I like the way we do it now, like how most all the maps programs are where we phase it in blocks. Yeah, in blocks of two to four weeks. What I think is better about that and even though the studies have shown that they're pretty much equal, what I think is better about that it's easier for me to gauge and see things and see progress. So if I'm constantly changing exercises and changing rep ranges and changing all those things up on like a daily basis, it's really hard to see like if I'm doing something and I'm consistently doing it if it's making progress or I'm getting better at it or which movement or exercise I'm getting the biggest bang for my buck because there's an individual variance and all that with each of us. So maybe a movement for Sal is just okay, but for me like, wow, I've noticed huge strength gains when I did that or it feels great when I do it or whatever. And when you are constantly changing it every day it's kind of hard to measure those types of things versus following a program or sticking to something for a good solid three to four weeks and then starting to change everything up. I found that was something later on that when we, and this is something that we did together. I know Sal the first maps in a bulk was created this way but all of the maps programs are based off of this foundation and to me I think that's superior. Yeah, so it's really a difference between doing training in the let's say four to six rep range for three weeks versus training four to six reps on Monday and then Tuesday is, you know, 12 to 18 and then Friday is 15 to, you know, 15 reps and then it's one to three reps. So daily, and we're just changing it all the time versus staying within a particular rep range or a particular style for a week or two or three or even four. Now here's the thing, our opinion is based off of our experience training a lot of people and I would surmise that if you took a large enough sample size if you did enough of these studies and took a large enough sample size you would find that the blocking or phasing of, you know, people's workouts rather than daily undulation will be more successful because for one of the reasons which is what Adam said people can learn about their bodies more. They learn what it feels like and how their body responds when they train just in low reps for a little while or just in moderate or high reps or when they train in a particular style for a little while and that is priceless by the way Here's the thing if you want consistent long-term success the most important factor is knowing your body Learning your body is what's going to contribute to long-term success more than anything else and for general population learning your body means you should stick in something for long enough to learn it understand how it feels and how you respond to it The other thing is this and this again you're not going to find this in a couple small sample studies you'll find this in large sample studies or lots of them is that when you're training in a particular phase for three weeks there's a mindset that accompanies it So what I mean by that is when I go to the gym and I approach a workout where I'm training in the four rep range and doing longer rest periods it's a very different mindset than going to the gym and training with supersets for 12 reps and that mindset actually develops and solidifies over the course of one, two, or three weeks versus flipping all the time back and forth Well, it's like sharpening up on your skills versus like, you know just reacting to the day so it I don't know I look at it differently in terms of I want to get better and I want to be able to see progress so I have to have benchmarks to sort of guide the way and there's certain exercises that are you know going to be, you know higher in the priority list and you know a rep range that I want to see you know how my body can improve upon and the only way to do that is to repeat it enough times to really have a proper assessment of that and that so I mean and the argument of like just changing it up all the time I mean it it maybe you could make that for just if I'm constantly I'm in an environment where I have to react to things constantly and I like it I'm you know I have the ability where now I'm creating this this new skill of you know being hyper responsive towards like all these different forces at once but in terms of like me like being able to really assess that I'm getting strong in this direction because of this I have to like extract that out I mean I see value in it if you're somebody that has been training for a really long time and you've already gotten good at all the skills you know your body yeah you know your body really really well you're not this isn't the learning process this isn't your first five years of training you've been training for decades and you know and so you like to switch these things up on a very regular basis I don't see any problem with that I just to Sal's point that our advice comes from training the general population and when I think of that I think I'm I've spent most of my career trying to teach the skill of lifting weights to my clients for a long time and if you were if we looked at this like a skill just like an athlete is learning their sport whether it be basketball or soccer or baseball you what you wouldn't do is constantly be throwing different skills at them every single day or multiple things in a day you would you would focus on one thing and you would stick to that until they got really really good at that and then you would move on to the next thing and stick to that until they got really really good at it because there is the the learning curve of the skill of the movement which I think later on plays a that's why those studies I think are a bit flawed because if you take you know if you take the three of us who've been lifting for a really long time and we do something that's daily you do something that's every three to four weeks and then you don't change it at all I think it would confirm what those studies showed but if you took somebody who you took thousands of people to your point Sal that have that are really still learning how to squat properly into deadlift and do some of these movements and would it be better for them to stick to that squad or stick to that deadlift for four weeks consistently versus squatting one day then doing Bulgarian squats another day then doing lunges another day that for sure right they they they're they're barely figuring each one of those movements are extremely difficult in themselves and to be flipping them all the time I don't think you're you're giving yourself the opportunity to get really good at the skill of it the next question is from Dave Colquitt how do you work on keeping your feet knees straight during a squat or deadlift I've tried keeping them that way but it's a lot more comfortable to point my toes slightly out you don't have to you don't have to point them I remember when I first learned that and I thought oh everybody has to be but when you look at anatomy and the different variances from individual to individual you find that there are there are some things there's a lot of things you can change your your your tissue pliability extensibility how it extends how tight it is how strong it is or stable it is but there's some things you can't change like your bone like you have a hip bone structure and the femur that fits in the hip and your joint and that structure it's not going to change it's it's just it is the way it is and if your joint is a particular way it may actually be better for you to have your feet pointed out a little bit when you squat or when you deadlift and for some people and maybe straight you have to identify if it's a a tissue issue or if it's a bone or joint issue and sometimes it's a bone or joint issue I'm glad you went that direction because I'm going to go the other direction the the reason why I don't completely like that advice even though it's true is that people hear that and then they they just meant it now yeah well not only that but then they justify this extreme external rotation in their feet because they're so tight they can't get their feet straight and I say this because this was me if I were to squat if you looked at my squat just two years ago maybe three now I had I had to have a really wide stance my feet were pretty externally rotated and it was just because I was extremely tight everywhere and when I started to work on the 90 90 it really started to mobilize my hips which the ability to mobilize the hips allowed me to rotate my feet in even more so I think that yeah if you have a slight external rotation that's that's okay but I also want to be careful of telling people that because then what ends up happening is you have these people that stand really wide open and it's not because that's their anatomy that because if you your gait okay it shouldn't be that far off from your gait and if your gait if your gait or your feet are that wide open that you have a problem you're not you're not walking properly if that's what your gait looks like so we should be able to squat in the same position with our feet that we walk in our gait and so and if it's that wide open I guarantee if you looked at that person's gait they would be completely externally rotated or what they call a duck feet out like that and that is something that a lot of people can address and work towards and a lot of that has to do with hip mobility yeah I think unless I mean if you're a competitive power lifter you're going to kind of know that like a bit of external rotation is beneficial to you you have you know a sort of strength and comfort zone within the way that your feet are set up there however I personally tend to lean on the fact that we should be able to squat with multiple angles of your foot position and I should have them in a narrow stance and I should have them wide and I should have the mixed and rotated and rotated and so to limit yourself to you know like just to justifying it based off of limb length or like a lack of you know mobility there I think is is not really doing yourself like the full justice I think that maybe that's just because we need work you know in that direction we need to actually mobilize hips or ankles more effectively so that that would be more of like a revealing thing to me that like you have restriction versus like maybe this is an anatomy issue yeah well it it also I mean it depends on how much they're pointing out he says toes slightly pointed out right I'd like to see what the squat looks like yeah toes slightly pointed out sounds to me pretty normal but he may be thinking it slightly pointed out and he may be doing like what you're saying but I love what what Justin's saying because even to that point like because I do this like I will squat you know with my feet completely straight and fucking four to six inches apart sometimes and then I'll do a big wide sumo sometimes I will let my feet to come out and so I think that's an excellent point why would you why would you limit yourself to a exact unless you're a power lifter because that's different you're just trying to perfect the skill over and over again of the same that's where like when you're a power lifter you don't you don't want to be messing with all these stances you want to take the stance you're going to take every time you get under the bar for the lights to go right like that's what that's what you care about it's performance based totally different for the average person who's listening who cares about just overall health strength and building muscle those people absolutely you should challenge yourself to be able to squat with a narrow stance a wide stance your feet straight your feet externally rotated a little bit and I think that's a fucking incredible point Justin. Yeah but I I guess at the at the end of the day work on practicing your squatting and work on your mobility this is why it's a skill because over time you'll find as you improve on your mobility your squat may start to look a little bit different and it may start to feel more more comfortable over time but ideally I think that's great advice work on different stances unless you're a competitor next question is from Brandon LPZ 26 when over training you talked about lowering the volume should we reduce volume on our big key lifts or smaller lifts? you could reduce volume on either of them but you want to consider the following the big lifts have a larger impact on your body's ability to recover than the small they're doing way more squats doing way more damage than a tricep push down so if you are but it's also sending the loudest muscle building strength building you know adaptation signal as well so if you're like really over trained and you've identified that wow I am I am not good at all and I'm really fatigued then you probably going to want to look at reducing volume in both big lifts and smaller lifts if you're like teetering on the edge you're like you know I was progressing like two weeks ago or three weeks ago and I feel like I'm kind of overdoing it then I would look at the smaller lifts and kind of chip away right there but that's really you got to weigh it out that way really I I feel like I would challenge that too I feel like if someone's over training I doubt it's because of your seated bicep curls your tricep extensions your lateral raises your seated cable row your machine chest press I doubt it's any of those that are and I doubt that that just reducing those exercises at all would actually scale back your over training well it could be like this like you could you could go to the gym and say I'm not doing any small lifts I'm just going to squat because I've been over training so I'm just going to go squat just do the main four and just do the main yeah the main ones or you could do the reverse and go to the gym and be like man I am fried I'm just going to do small lifts I mean in my experience I've never been over training because of all those isolation movements I was saying it's always because I'm doing too much of the big four and I've always got to pull and the only thing that will make my joints feel better allow my progression allow myself to recover better the only thing that does that is is reducing some of those big compound lifts because to me isolation machine cable exercises don't do nowhere I feel like I could do those five times a day every single day non- they're almost like trigger sessions I mean they're yes you can still train some of those movements to be tough but I don't get inflamed joints I don't get like severe fatigue I don't see myself setting back my strength and truly over training unless I'm see I've experienced the opposite how do you yeah and I don't mean that I'm not disagreeing that those that those exercises don't cause a lot of damage to the body I agree with that part what I mean is that I've gotten to the point where I feel like I'm overdoing it so then I go to the gym and I do 30 minutes of you know bench and deadlift and I keep the intensity moderate and I just perfect my form and I leave and I'm doing anything else and now here's what you want to do you want to weigh all these things out because the big lifts they cause the most damage but they also cause the best results the small lifts cause the least amount of damage but they also cause the least amount of results so you want to weigh all that out when you're trying to determine what you want to reduce or cut out of your routine and there's a lot of different ways to do this here's another way to do this go to the gym do the same routine here's my favorite way by the way go to the gym do the same routine lower the intensity that's my favorite way to handle overtraining is I don't even change my workout I just go easy it's the same workout at 50 percent intensity now I'm moving full range of motion I'm feeling the squeeze I'm at half the intensity I normally am I'm still doing all the same stuff try that out for a little while uh-oh still not enough I still need to cut back a little more cool now I'm just going to do same thing low intensity but big major lifts yeah I would assume there's a scale like of yeah reducing like one thing at a time so that makes sense in terms of intensity and then maybe lowering the reps or like one of the other variables and then you know addressing which one causes you the most fatigue and you know to kind of like program accordingly don't you guys feel it's kind of obvious sometimes for you too I feel like like like when I to us when I I know what I'm feeling right like I know I know when I overreached on my dead lifts I know when I overreached on my squat and it's normally those things you'll feel in a certain area oh yeah well I just know I know because today I decided to do eight sets and it was programmed for me to do four five and but I was feeling it or today I was supposed to be training in the eight to 10 rep range but man I was feeling so good I wanted to see what singles look like today you know what I'm saying it's normally those that you do where you probably knew you were overreaching when you did it and then you get the signs of the achy joints the inflammation the not recovering in time like or sitting going backwards and weight like I don't know I feel like if you're truly because I want to be careful there too because like this I don't want to scare people to think that they're all because we did an episode on overtraining so now everybody's like freaking out like oh shit I was tired yesterday mind pumps that I could be overtraining right like okay it's you taking that yeah most people you'll have multiple those signals that we talked about if you're truly overtraining so I I don't want to stress everybody out and think that and you probably have a pretty good indicator that you did because you probably did something out of the ordinary or you weren't following your fucking programming well I think too it's tough for your average person who doesn't have a whole lot of experience yet because they see a plan and they see like a rep count and they see like oh well other people should be doing this and they haven't really learned their body enough to get like pay attention to those signals that like wow well for me personally actually this is probably too much right now and I'm just going to adjust on the fly like that that that comes later with experience but to keep that in mind and in terms of like not being so rigid with whatever plan you have going into the gym like it it could be interrupted next question is from Rachel Beach you always talk about the importance of relationships with food can you give an example of what a good relationship with food looks like versus a poor one well there's take my food to dinner there's a there's an obvious you know there's obvious poor relationships to food you know when it becomes medically pathological you know anorexia bulimia and you know those types of things so that's the obvious ones but let's talk more just kind of general average person I'll say this a good relationship with food is understanding all of foods total values not feeling shameful when you're eating a particular way not using food to medicate or numb yourself when you're when you're in a position where you eat what you truly want I'm going to make a little caveat here what you truly want after you understand the real total values of food most of us only choose when we think of the foods we want we only relate it to the foods that we enjoy the taste of the experience of eating so when I say hey eat the foods that you want people are great donuts pizza french fries whatever okay no that's that's limited you're that's a limited experience that means that you're only picking the foods that you want for the palatability but there are other values that food provides you oftentimes I will really really want to eat vegetables is it for the palatability no I know it doesn't taste as good as other foods but I really want it because my digestion's off and I do this sometimes when we travel we'll travel and I won't eat very many vegetables and I'll come home and be like man I want to eat some vegetables or maybe my energy's a little low or my workouts and I'll so I'll know I'll really want some starches for the carbohydrates or whatnot so a good relationship is it's a relaxed relationship you know it's like any other good relationship it doesn't feel like it's just crazy stress all the time again you're not shameful it's not I can't it's it's that I don't want to or I want to and it takes time and like any relationship so I want to use the word relationship and apply that to everything relationships need constant work so it's like how do you have a good relationship with your spouse you constantly work out it's something that it doesn't just happen and if you don't get one and then just leave it alone and it stays that way it's something that you remind yourself of always I'll say it even more simply than that a good relationship with food is pretty simple what it looks like it looks like somebody who's in good shape and good shape is a a wide spectrum not what Instagram considers good shape good shape off of like health standards which is a relatively decent body fat percentage and you don't have to track or really pay too much attention to your food you're not stressing out yeah you don't think about it you don't stress about it you don't you're not worried that you decided because we're out at the nice dinner tonight the four of us and I decided I want to have a glass of wine with our dinner and have the cream of spinach and I'm not thinking about it I just you know how to navigate around when you make food choices that way and you know how to incorporate your exercise that's a really good relationship with with food is that you don't really have to worry and stress and think about it too much and you're able to keep yourself in good shape if you're way outside of good shape you probably don't have a good relationship with food it's just a matter of how extreme that is you know if you if you're in bad shape you don't have a good relationship with food it's that simple you don't know how to regulate yourself on what your body needs you you have just totally disconnected from those signals and you overeat it's a it's a fact and then another poor relationship with food is somebody who's obsessed with it so much that they're constantly tracking they're constantly thinking about it and that's still a poor relationship with food now I'm the one on the show that I think advocates for tracking the most and I do think that's extremely important to learn and figure out what your body needs but the ultimate goal is to move away from that and to be able to intuitively intuitively eat and intuitively train and that to me is what a really good relationship for with food looks like it's like you it's like imagine this you're at a party and uh they bring out the cookies somebody with a relationship that with food that isn't great would be like oh I can't eat those no no no I can't I can't have a cookie can't eat it someone with a good relationship with food would say I don't want a cookie or I want a cookie now a lot of people listening right now are thinking what do you mean I always want a cookie okay that's and that's my point you don't really always want a cookie when we say I want a cookie or whatever what we're saying is I want the taste and experience of that cookie but when you put the entire thing in context okay when I eat a cookie I don't feel too good maybe I'm a little gluten intolerant it gives me heartburn or maybe you know I'm training for a competition and it's not really gonna benefit it's not really gonna benefit me in terms of my competition or whatever I'm still acknowledging that it tastes good so if they ask me hey Sal do you think this took cookies gonna taste good my answer is always gonna be yes but if they say do you want this cookie when I weigh all those things out and this happens more naturally as you practice this when I weigh all these things out my answer is gonna be either yes I want it or no I don't and when I say no I don't it does not mean that I don't acknowledge that I'm gonna enjoy the taste of that cookie or said food and so this is a practice and it takes time and remember this be empathetic to yourself because the way you have learned to value food if you're listening to this podcast right now and you're in any and you're in a developed country the odds are that you developed your relationship with food based around its palatability and the experience or context or emotion those are all the things that you've developed your relationships with food so you know I'm at the movies and I crave popcorn that's context or I learn to eat these types of food when I get sad because they make me feel better so that's emotion or I think of foods in terms of what's gonna taste the best that's what that's how I value food so that has to do with the the palatability or the experience that what is known as the hedonistic experience but there's so many other things that food provides to us and those are all fine by the way I'm not demonizing all of those those are valid values that food can provide us I'll give you an example if I'm you know like I'm one of my kids birthday parties and you know we make a cake for my kid and I'm celebrating with my kids for the birthday I'm gonna have the cake for the experience the context maybe there's an emotional component I'm happy my my kid turned however old or whatever I'm also gonna enjoy the hedonistic value of that piece of cake I'm also gonna know that it doesn't benefit me physiologically it's not good for my body I'm probably not gonna feel as good digestively or whatever but it's okay it's okay because I know all the context now let's say I'm at home I'm by myself I'm watching TV and I'm like man I want to have a piece of cake I'm craving the taste of a piece of cake and I think well you know there's really no other value besides that the taste of it and I'm at watching TV at home alone I really don't want a piece of cake so that's the difference between the two and be empathetic because it takes time to understand those things I remember years ago I had a client that I've had several clients that didn't like to eat vegetables and one of the ways that I got them to eat vegetables is I would first I'd talk them into having very very small portions but then I would have them journal and track how they felt before during and after and then how they felt the next day and what ended up happening with a lot of them is that over time they started to notice things like my digestion is better I feel better oh I don't like the taste but oh man I start to feel really good my skin is looking better and then over time because they became aware of all these other values besides the the taste of it they actually started to want to eat the vegetables and so this is people talk about balance with nutrition this is what balance looks like so every once in a while or whatever you eat the pizza and the in the cake and whatever and other times you eat all the stuff that's good for you physiologically I think it was Confucius but it was one must go through the cookie to transcend so I don't think he said that bro that's some wisdom yeah that's some wisdom I'll leave you with that and with that go to mindpumpfree.com and download our guides they cost zero we got a ton of them for you it's basically free information to help you with your fitness goals there's a lot of stuff there you can also find all of us on Instagram we have our own personal pages with their own personal flavor Justin's is cheesy you can find him at mind pump Justin's Cheddar like Adams is a little salty you can find it at mind pump Adam and that's all two nuts and mine's a nice balance between the two find me at mind pump sal


So, let us estimate the error term first thing today we had this error term. So, some n going from 1 to infinity. So, here we have a pretty good handle on these values this lambda n is bounded by log n at all ways this x to the c r n to the c these are all very clear this is the slightly problematic one log x by n and the reason is that as n goes from 1 to infinity when n comes close to x then this can become very very small and therefore, the whole thing become very large. So, we have to handle that aspect little carefully otherwise bounding this is pretty straightforward. So, in order to achieve that let us split this sum into 3 components n equal to 1 to x by 2. So, everything reasonably below x plus 2 3 x by 2 n plus n greater than 3 x. So, the first and third are pretty straight forward to estimate just let us see that. So, here since n is always reasonably smaller than x the largest value that log sorry the smallest value that log x by n will take is a constant when n is exactly and x by 2 this will be log 2 which is a constant. So, denominator we would like to give make as small as possible. So, this is certainly order n going from 1 to x by 2 lambda n x to the c divided by r n to the c and this we can just write as x to the c by r summation n going from 1 to x by 2. Well, lambda n we can replace by log n at the most or make it even simpler let us take out log x lambda n always less than equal to log x. So, take that out also 1 by n to the c and this sum is what well c remember is greater than 1. So, when c is greater than 1 this sum actually converges actually n going from 1 to infinity 1 by n to the c converges to some constant value. So, we just this similarly if you look at this. So, here again since n is substantially bigger than x always. So, log x by n would be substantially as n increases log x by n will become a larger and larger negative number but larger and larger and here we are taking with absolute values we need not worry about positive negative. So, again it will attain its minimum when x is 3 by 2 x and again it would be a constant. So, therefore, we can again write this as and now here 1 has to 1 cannot substitute lambda n with log x when you keep on increasing to infinity. This series the convergence will depend on exactly what is the value of c is that right lambda n as most log n. So, when would this converge log n that is what I am saying is smaller than any n to the epsilon. So, this is going to converge anyway in does not matter which way you take as long as c is greater than 1 this is going to be order x to the c by r for any c bigger than 1 that is fine. So, that leaves out only the middle term which is this sum and this we have to handle slightly more carefully. Now, here the log x by n sort of is within a small range of values but the problem is that it can become very close to 0. It may become very close to 0 and then there is that is not so nice. So, to handle that let us do the following replacement. It is again simple trick and will easily you know can write this in a more sensible form. If I just say x by n to be let us see what I want this goes between x by n varies between 3 by 2 to half right. So, what I want here is 1 minus z or let us z I am using elsewhere. So, let us 1 minus s. So, what we get is minus half is less than s less than equal to plus half as x n goes between x by 2 to 3 by. So, that is x by n or x by n varies from 2 by 3 to 2 make n by x. It does not matter actually here just by changing the sign you can replace log x by n but will by log n by x. Now, also for this one this once you do this log x by n is same as log 1 minus s and log what is log 1 minus s that is a standard power series for this which is s plus s square by 2 plus s cube by 3 plus and so on right. And now we use the fact that s is in absolute value at most half. So, this is equal to s plus let us just say this sum the of the rest of it is not going to be very large actually. Let us just write it as s by 2 plus square by 3 plus so on. And since s absolute value at most half this will be at most first term will be at most 1 quarter 1 by 3 1 by 12 1 quarter plus 1 by 12 if you take absolute sum which is going to converge to some constant in fact very small constant also I guess it will be less than maybe one actually not I guess I am sure it is going to be less than 1 because this is actually worse than the geometric series. So, this is less than equal to 2 s oh is that what I want no I want I want the other way I want I want to lower bound on this. So, this would be greater than equal to well if you just take s and plus this much the rest cannot reduce it. So, this would be so take the first two terms fine this is square so it is always positive. So, s plus s square by 2 that is positive and the rest can suppose it is all negative and try to subtract out as much as you can you cannot probably you cannot subtract out more than half s from there because this sum of all of these would be at most half s. So, I am leaving out some details for you to work out whether they are pretty straight forward. So, basically what you get is an s is at most half. So, this would be this s could be 0 also so let us say at most half s at least half s. So, now with this we can write this as order summation take this out x to the c by r summation x by 2 less than n less than equal to 3 by 2 x lambda n by n to the c and log x by n is replaced by half of s which is half goes away. So, s is an s is x minus n divided by x this is not quite what I wanted y is an x coming there but anyway seems I cannot of course that is right. Now, n is at least half x. So, this is n to the c I can replace by x by 2 to the c c is a constant 2 to the c goes away. So, it is x to the c below and c is greater than 1 which cancels out with the x above. So, this is at most and this is log x lambda n since n varies between x by 2 3 x y 2 this is log x by 2 x minus n and as x minus as n varies from x by 2 to 3 by 2 x what happens to x minus n. Yeah let us assume x is half integral this is the it is never 0. So, what happens to what x by x minus n will go from minus x by 2 to plus x by 2 right or so we can fold up. So, and it will take half integral values there which is again we can take it to make the interval a little larger and make it take integral values because you can always multiply out by constants and absorb it in it which means it is something like a sum of 1 plus 1 by 2 plus 1 by 3 up to some 1 over x y 2 over 1 over x and that sum is. So, if you just go back to this it is order x to the c by r the log x here and this sum itself is log x. So, that becomes log square. So, just adding all three what you get is now this error is a bit can be a bit too large if c is c is 2 or 3 then the error actually is much bigger than your main term recall what was the main term psi this is the main term right whatever it is psi x what is and this estimate psi x psi x is this plus the error term psi x would be how much at the most x log x can not be more than x log x in the most silly counting you will give you x log x. So, if your c is a little more than say 1.1 then the error is already going through of course, assuming r is fixed number. So, that we do not want. So, let us fix once and for all the value of c which is going to be very close to 1 I cannot make it 1 unfortunately I cannot make it 0.1 kind make it 0.0001 because any of this will just be too much. So, I have to make it a function of x it has to because it has to be bigger than 1 and yet not be a constant. So, this is what I am going to fix 1 plus 1 over log x and then what you get let us look at this psi x is 1 over 2 pi i now you have integral from c minus i r to c plus i r minus zeta prime z by zeta z x to the z by z d z plus order what is x to the c for this value of c that is pretty simple what is x to the c. Last error term there would not be x to the power c because we have cancelled power c by denominator and the minimum value of n can take is x by 2. So, that will cancel out I got it yes you are right. So, it is quite does not matter I am saying even if you take s to x c because I am going to choose a c to be 1 plus 1 over log x. So, what does this make x to the c equal to is e times x which is just a order x. So, x to the c is order x. So, basically you get x log square x and this is you know a pretty good starting point that I have managed to write psi x which is kind of a cousin of the prime counting function as a complex integral plus an error term which is well which is still too big if r is small of course, no the nice thing is that r is in the denominator. So, if I can always send r to infinity and make this vanish completely and then the error term does not matter also. In fact, then c the choice of c is also does not matter can be any constant, but as we will see later on it is more beneficial to not send r to infinity instead set r to about square root of x, but that we will come to later, but for now let us just keep in mind that we will not be actually sending r to infinity. However, if we do send r to infinity what happens then error vanishes and psi x is. So, this is certainly a corollary of this that psi x equals 1 over 2 pi i is precisely this integral along this line minus infinity. But as I said this is only for fun we will not really be using it because in any case as I said we do not expect to sort of precisely compute a value for psi x a formula for psi x because that is although we will compute a precise formula for psi x later on, but that is not going to be very useful in terms of giving us a good estimate. To get a good estimate we will actually use a finite value of r. So, that is the setup we have psi x we have a complex integral which tells us what psi x. So, the next question is how to be evaluate this complex integral and here I will revert back to C plus i r and C minus i r because this evaluation is what is going to determine the psi x for us. Now, the nice thing is that we know how to evaluate this integral at least we know how to go about evaluating this integral we already done this we already have tools for this. What we need to do is make a contour or make a domain whose boundary one of the boundary is this C plus C minus i r to C plus i r close that make a contour integral around that boundary and count the poles inside that region that is the value of the integral and then estimate the value of the integral on the remaining boundaries and somehow set it up.  So, let us attempt that. So, here is the let me write the integral also C minus i r to C plus i r minus eta prime this is the integral and we have one here and very close to one we have C we already chosen that minus r plus r and we want to integrate from here to here this integrate. So, what is the contour that or what is the domain that we should choose for this? We can extend it this side we can extend it this side we have seen examples of both actually, but if you recall that example that there was in that example x to the z by z was the integrant and depending on the value of x we either went left or we went right and the reason was that if x is less than 1 then you need to go on this side positive side because when you want to bound later on those integrals on the remaining boundaries. If x is greater than or x is less than 1 then on those boundaries you can make z to be very large positive and large. So, x to the z will become very small right and as you you know expand those boundaries that will vanish. On the other hand if x is more than 1 then this strategy will not work because as you go further and further on the positive direction x to the z will keep blowing blow up. So, when x is positive you want to go on the negative side because when and then large then when z takes large negative values again x to the z will become very very small and again you can try to bound it away bound it to 0 or small. So, what this tells us is that we cannot hope to go on the right of this line to integrate this we should try to go to the left of this line. So, the moment you conclude this there is a problem what is the problem. So, let us define this domain. Again possibly the most simple domain that is possible exactly as we did in case of the delta function there is a rectangle going all the way up to minus u here coming down and this is how you integrate along this counter clockwise direction. So, that is the domain D, but as I said we immediately run into the problem what is the problem look at the integrant what is the integrant zeta prime over zeta x to the z by z is the integrant defined on the domain. You want the integrant to be analytic on the domain otherwise nothing works right that is an essential requirement. Now forget the being analytic is it defined on the domain worse. Is that the part of the problem? Yes that is certainly true, but it is even worse x to the z is defined in everywhere 1 by z is defined everywhere except for the pole at z equals 0 that is pretty well understood. What about zeta z? zeta z is this infinite sum where does it converge only for absolute value of z greater than 1 otherwise this infinite sum does not even converge. So, which means this is a line z equals 1 to the left of it including the on the line zeta z is not even defined. So, nearly on there is just this tiny strip to the right of it where the zeta z is defined everywhere else is not defined. So, there is really serious problem there. By the way do you know or do you believe that zeta z actually converges for absolute values at greater than 1? The real z greater than 1 not absolute value greater than 1 real z. So, at least we need to know or we need to ensure that it converges on the right side of this line. So, are you convinced that it does? We did oh then it is perfect. Anyway it is pretty simple just take the absolute value of every term and then you say it comes out to be bounded by a simple geometric integral and then the integral converges. So, zeta z is defined everywhere here, but there are problems on this side. Going further is fine zeta z is defined on the right hand side, but is it analytic on the right hand side on every point? How do you prove it to be analytic? This is an assignment. It is a reasonably straight forward proof just use the tools that you have learned. How do you prove something analytic and apply those tools? So, now we will believe that zeta z is analytic on this half plane which is on the real z greater than 1, but it does not even exist seemingly at least on the other half. So, this approach appears doomed except that we know that an analytic function can be extended beyond its domain. We have seen some examples of this and we know that it can be extended in a very nice way unique way actually. So, maybe zeta has that property there may be it can be extended it is not defined through this infinite series, but maybe there is a way to extend it on the other half. Let us explore this. There is no harm. So, let us write zeta in a slightly different form. This I can write as let us see let us make another attempt. Do it look at a partial sum. This I want to write as integral. So, the key thing is to translate it into integral because that is easier to manipulate and just trying to work out what is should be the form of the integral plug in this integral. So, how far away is this sum from this integral? This integral I can write as equal to sin and regular series is all together. This is equal to summation n going from 1 to x integral y going from n to n plus 1 and n going from 1 to x minus 1 actually. And here you will have n divide by y to the z plus 1 d y and this is equal to what is the integral value? This is dy over y to the z plus 1 is equal to what dy over y to the z plus 1 integral of that is y 1 over z y to the z and a negative sign somewhere and this n to n plus 1. And this is equal to what? Now, what is this sum equal to? We want to expand this. This is equal to n equals 1. What happens? 1 by 2 power z minus plus n equal to 2 2 by 3 power z minus 1 by 2 power z minus 1. Is that right? This does not seem very good. No, these are not very good. Maybe it is not too bad. Now, look at this and this. You get this gives you 1 by 1 power z minus, of course. What does these two give you? What about these two? What about these two? Which is exactly what you want? Except that at the end, the last one, this guy will survive. So, what I get is minus 1 by z and here minus can go away because that will make all the rest of it positive. So, sum n going from 1 to x 1 by n to the z, I think x minus 1 or so. And then the last guy survives which is n which n which is n equals x minus 1. So, that is x minus 1 divided by x y z. So, now, let us get back to this. Good. So, now, we can write using this. What I can say is as equal to x minus 1 over x to the z plus z times this integral y going from 1 to x. And now, sent x to infinity. I should not have chosen x here because x I am using for something else. This is a different x not to be confused with x parameter x of psi. So, send x to infinity here. What happens? The left hand side becomes the zeta function. What happens is the right hand side. The first term, see absolute value of z or real part of z is greater than 1. And because of that, this the first term goes to 0, x as x goes to infinity. So, what I get is zeta function equals this integral. That is nice to know although it does not solve our problem yet, but it is nice to know that there is a not so nice integral that I can write zeta function as because it is not so nice because there is a this floor of y sitting out as integral. But anyway, we are not really integrated interested in getting a value for zeta z. We only want to see if it can be extended beyond its defining domain. And that we can use this integral for. Well, how do we do that? Let us just write this 1 to infinity y minus bracket y, where bracket y is just the fractional part of y. Now, the first integral can be evaluated. That is equal to minus 1 over z minus 1, 1 over y to the z minus 1, y goes taking value 1 to infinity. Now, when y is 1 and infinity, of course, it goes away. The infinity at in pi equals infinity, this is 0 because z is z minus 1 has a real part, which is bigger than 1, bigger than 0. And y equals 1, this becomes of course, all of this is for real z greater than 1. And let me just write zeta z. So, this is also important equation. Now, let us look at the right hand side. Right hand side is an analytic function. Why is it analytic? Well, this is of course, analytic except for z equals 1. And the integrand here is bounded. Well, of course, I have to be careful here. This part, this is analytic for real z greater than 0, z not equal to 1. For z not equal to 1, there is this does not diverge. So, the first term is fine everywhere for real z greater than 0. If you look at this integral, as long as real z is greater than 1, sorry real z greater than 0, y to the z plus 1, the real part of this is bigger than y equally y to the 1 point something. And the fractional y is bounded by 1. So, this integral converges absolutely as long as real z is greater than 0 to some finite value. For every value of z, it converges some finite value. And it is of course, the dependence on z is analytic because the only dependence of z is y to the z plus 1. 1 over y to the z plus 1 is an analytic function in z. And we have seen that when you integrate an analytic function, actually it is both analytic in z as well as in y or for every value of y is analytic. And therefore, its integral is also analytic in z. And it converges for every real z greater than 0. So, the right hand side therefore, is an analytic function which for real z greater than 0 and z not equal to 1. The left hand side is an analytic function for real z greater than 1. And these two coincide on real z greater than 1. Now, just invoke the uniqueness of analytic continuation. This right hand side is the analytic continuation of zeta z. For of course, this is still not fully done. This is 1. And we had zeta z analytic here. And we have the right hand side analytic also in this strip except of course, this a whole at 1. So, it has allowed us to extend the definition of zeta z slightly to the left. Not all the way, but at least slightly to the left. So, that gives us hope that perhaps by doing something more we can extend it all the way to the left everywhere which is what we will do in the next class.


Hi everyone, Joe for jaspyscasebreaks.com coming out to a 2021 Panini prism draft picks football one box random pack break number 20 Big thanks to this group right here Tony Raymond Brandon and Mac for getting in on the action. We're gonna pop open this box first Then we'll label the packs And then we'll randomize your names and the pack numbers and then we'll see Let's see what's inside. All right, so now those are officially labeled now. Let's roll it randomize Names and pack numbers one and a two three times each one two and Three got Mac down to Brendan three times for the pack numbers one two and Three easy One five three four two All right Mac with pack one Tony with five and three Raymond with four and Brennan with two We're gonna go alphabetically on this. So Bren you'll be up first. We're not the labels for these two Really quick standby. We'll place these into the the bags super quick Get this out the door Let's taste the 19th Monday 21 prism draft FB football one box RP random pack number 20 Good luck. All right, Bren you're up first To there's the autograph right there. Let's slide that autograph to the back. There's a Devonta Smith red wave Marquez Stevenson to 249 And behind James Connor is your autograph and it's gonna be Landon Dickerson Alabama's Landon Dickerson A big dude All right, thanks Brendan for giving this a go. All right Mac. You're up next with pack one It was Adam Thielen right wave of Patrick Surtain to 199 Chris Godwin Jamar Chase fashion Shuster Kyler, and we've got a Najee Harris Purple ice autograph Najee Harris nice and that is 30 out of 99 Raymond pack four. Good luck. Got purple ice Cam Newton to 149 Najee Harris all-american silver Justin Fields all-american There's the auto right there. There's Rashad Bateman red wave and your autograph behind Big Ben Rothsberger is gonna be Rashad Weaver Rookie Otto silver and there's the Cam Newton purple ice as well. Thanks Raymond Tony you're up next with the last two packs packs five and three Five and three Good luck. There's Odell Beckham jr. 57 out of 99 Blue ice Jordan love to 199 blue parallel and behind Stefan digs is Ernest Jones Nice blue ice to John Lesser hit a home run 419 feet. That's to 75. All right And your final pack Tony. Good luck. You got Jonathan Taylor red wave Jalen Rhaegor to 299 Big Ben silver and behind Patrick Mahomes is an Austin Watkins jr. Rookie Otto red that is number to 199 Thanks, Tony. Thanks everyone. I'm Joe for jaskies case breaks comm. We've got some more of this prism draft picks In the store. So check it out on jaskies case breaks comm. We'll see you next time for the next one