potentially two marathons. So one in the spring, one in the fall. You guys know which one in the fall is. If you hear screaming in the background, don't worry. Michael's rejecting his nap. Michael's rejecting his nap. But we're good. Okay. Heart to heart time. Cheers. Cheers to a great 2019. Yes. Epic 2019. Epic 2019. And an incredible 2020 coming up. Okay. So it's time everyone. A new decade. Yeah. So cool. It's time to start prepping for the new decade, the new year, new racing, new filming. Okay. And to talk about everything we think we can pull off together. Yeah, a lot. Boom. And it's crazy. And we're having a baby. Oh, and we're having a baby. That was a major point we got to talk about in case you didn't know. Well, if you're new, welcome. Baby is arriving. Can I show? When? Okay. February 22nd to due date. But I think Michael came two weeks early, basically any minute. Basically. We're prepping right now for this baby. Yeah. Michael came early. So that is a big point to consider. And what does that make? How many people on a cross country team? Oh, four. Oh, boom. That's all we're doing here. Cross country team. We're doing our bit. Yeah, we're building the cross country team here. Megan, yeah. So two weeks ago on a vlog, I asked everyone to chime in on what they would like to see on the channel in 2020. Like a stop to cover. Yeah, different ideas. And one, some people said don't fix it if it ain't broke. Keep doing what you're doing. Amen. Right? He's got the wisdom. But, but everybody, basically some of the most popular vlogs from 2019 were definitely the racing vlogs. Okay. Were you racing or where? Both. Wow. Yeah. So whether it's Pikes Peak, whether it's Western States. Oh, yeah. So I'm listening to your comments from that vlog two weeks ago. If you haven't seen it and you want to chime in, what I'm going to do in 2020 is go back to that vlog over and over again and look and see what people want me to film. Well, that's fun. And talk about the people speak. So that's up a right hand corner. Okay. But now, true love and I, we're going to have a heart to heart. Okay. About the schedule, keeping in mind, there's a baby on the way. There's a baby on the way. Also keeping in mind, it's our fourth baby and we know what we're doing. Yeah, we got, babies are the easiest. Yeah. It's, you know, your five and six and seven year olds where you're forming virtue. That's where you're the rubber meets the road. Babies, they sleep. They sleep. Here we go, Han. Travel schedule 2020. It's on your screen. And basically, we're actually going to the first one on the screen. I'm actually not going to, it's a, it's a secret. And so we'll say that for, we'll say that, but we got to talk about it at some point. So because of Houston marathon out, Olympic trials out. For now. For now. Yeah. So you went 2024. We, doors have been opened in a good way, meaning I will be healthy in the, I'll give you an update on the, on the knee soon, but I will be healthy, hopefully in two to three weeks, maybe four weeks. I don't know, which means I would love to run potentially two marathons. So one in the spring, one in the fall. You guys know which one the fall is. Yeah. New York, baby. New York, November 2020. The whole fam's going to be there. November 1st. November 1st. But I'd love to talk to you about April. So there's the Boston marathon, April 20th out in Massachusetts. So for that one, everyone, I would not be running. That would be filming. Okay. Boston marathon, April 20th, April 26th, the London marathon. Whoa, the next week. So this is where we got rubber hits the road. So that would be, so here's the deal. I probably cannot get into the London marathon for me personally. It would be more me going to film. Okay. So I'm putting a big asterisk next to the London marathon. But if you look the next week, the first weekend in May. Oh my gosh. Perfection. May 3rd in the Czech Republic. It's an IAAF marathon. It's a fast course, very competitive. Okay. And so kind of like, and this is where we got to talk, but kind of like Vienna. Yeah, and then Riggs. Vienna and Amsterdam. Amsterdam. It's an option. Yeah. As a spring marathon combo. I think that's a good thing for our buck. One, all the time and energy and money's invested in it, but you're only gone once. And you're, we're only out of our routine one time. And as long as you have enough time to film and catch what you want and rest your legs before you race, I think that's brilliant. Plus, listen, we're not big. I'm traveling. You don't need to try across the ocean once. Yeah. Yeah. And ton of people in the UK. Shout out to the Irish. True loves Irish. I'm drinking Irish breakfast tea right now. Thank you very much. So shout out to the Irish, the Scottish, the Welsh and the Brits. Okay. So we're all, so I would go to London. We would do a group run. Okay. Film Kipchogi. Wow. And then that's his next thing. That's his next marathon. Yeah. No, I think London. So and then zip over to the Czech Republic. Okay. Race and then come home. Yeah. So that's an idea. I like that. I like back to back. Is it exhausting to film, taper and vlog? It's exhaust. It's always exhausting. Yeah. But it would be good. And you've done it before. I've done it before. Yeah. Yeah. So anyway, that's just an idea. I like that idea. Now, so with Boston. Yeah. That's like the weekend before. Well, and I'm just, it's just an idea. So this is where like with the new baby, put you on the spot. What do you think? He's two months old. Yeah. Yeah. He's sleeping well. Yeah. So for boss, I think my mom's coming out somewhere. Okay. So I'm not worried about the baby two months old. Okay. He's sleeping. If I come to Boston everybody, we're talking 36 hour trip. Yeah. I would go America. I would show up Sunday. Yeah. Fly home Monday night. See ya. And we do a group run in Boston. Perfect. Film the Boston marathon. Okay. Zip back. Okay. So who's doing Boston? Anybody like Chogie? No, big names, big names, big names. It's a big, it's a big marathon, but not. Yeah, big names. Plus if ever you did run it later, it'd be good to kind of see, do they change the course every year? Yeah. Kind of check it out. No, it's the same. Okay. So, okay. Yeah. I'm not worried about April at all. Moving on to the summer. Veil, perhaps. Yeah, veil games.  Yeah, I'm putting that in right now. That's my favorite. Now, moving on to Western states and the US Olympic trials. Okay. So if I'm gone that much in April and early May for London and the Czech Republic, I don't know if I'd like to be gone for Eugene, Oregon and Western states. Yeah. It's summertime. The kids are off school. That's true. So I'm, this H there on the screen is kind of a, I would say is a big, big maybe for me. Okay. But anyway, but I do think everyone loved Western states this past year. Yeah. Pretty epic filming. Yeah. Definitely helped the channel grow a lot. Yeah, that was a big name for us. Found a lot of new people. Yeah. And no, I'll just say, Walmsley is not running this year. Okay. He's running down in South Africa. Okay. But there will still be a lot of big names. Okay. So I'm thinking most likely passing on the Olympic trials. Okay. Because then I would be gone, let's say the 22nd to the 28th. That's six days. Right. I mean, one thing to note is my mom will be in town. That's true. So her mom is the best. She is the best. You will see her this summer on the blog. She's the greatest woman. She is a teacher and just watching her be with my kids has taught me how to parents. She's amazing. Yeah. So now I would say anything in the summer works for me because my mom will be here. Yeah. Okay. And Kate at some point. Yeah. Okay. So that's June. Okay. Moving on to July hard rock 100. It was snowed out last year. Too much snow here in Colorado. And so they canceled it last year. Everyone was bumped to this year. So you're automatically in. So you're automatically in. So Schlaub and Browning, I think. Dylan Bowman. No, I'm not in. No, no, no, no. The hard rock 100. Yeah. Yeah. It's a hundred mile race. We're not doing 100 mile races for a long time. For a long time. 40s. So I think I would want to do that. And it's in Colorado. Now, this is where it gets tricky. So hard rock 100. We're coming for you. Okay. For filming, not running. Yeah. He's August. So Pike's Peak marathon. So I'm going to register January 1st. Now, are you for sure doing the marathon? Would it be smart to do both? And then we have the option? Or it's like, no, we're registered for both. Yeah. Possibly. Yeah. I mean, no, I mean, although I bet if we register for the marathon and at the end for whatever reason we wanted to switch, they would probably let you. I don't think so. No, they're pretty strict about that. They're kind of, they're setting their ways. So I think that I want to race the marathon because I want to race, even though like Joe Gray is great, I want to race the European guys. And the marathon is part of the Golden Trail series this year. Yeah.  And they did great. And they ran fast. And Kylian. Yeah. Kylian Joannette. And so I don't know if Kylian's going to come back. But this is where it gets tough is L on your screen in August, August 23rd. If Bekele and Kipchogi race each other in the Olympic marathon in Tokyo, which is a long ways away. It's two weeks before Pike's Peak. So I'm in a little bit of a pickle because obviously I want to do well at the Pike's Peak marathon. Right. I think I don't know how long it takes to fly to Tokyo. I'm assuming I would go to Seattle in Seattle to Tokyo. So it's actually not a ton of flights. It's just like a long ways across the Pacific Ocean. But if it's Bekele versus Kipchogi in the U, in, I would, it's like a, it's literally once in a lifetime because they're both. So this year Bekele ran two seconds slower than Kipchogi's world record. In different, but they didn't race each other. Right. And so everyone's wondering like, because Bekele I think is two or three years older and this is, he's coming to the end. Like he doesn't have, he doesn't have much left. So I'm, that sounds like a challenge to me. So I'm in a little bit of a pickle, but I don't know what you think. Well, could we, we definitely should register for Pike's Peak and then see how it all plays out. What if one of them gets an injury? Yeah. And we can even decide what, 12 weeks, eight to 10 weeks before August 9th of how we want to do that. But no, you're right. That would, if you do decide to go there, I mean, I just, I love the world, but America is the best. And I just, I'm not a big traveler. We're homebodies, but I'm going casting out. Yeah, bless you. So, huh, that sounds good. I think wait, I think pause and just wait and see you're right about injuries. If one of them doesn't race, then it's not as epic. Okay. Yeah. No, that's really all I, I'd love to see them race against each other. I bet that guy, who was only two seconds behind him, they're racing in real life. Kipchogi, or I would say the Kalei has more brute speed. Okay. Like he just is faster turnover. Okay. But I think Kipchogi has the more, he can hold a harder pace for longer. But if they were neck and neck at the end of the race, I would say they might get him. Yeah. But he's older. Yeah. Yeah. He's got old man strength. Okay. Everyone moving on, rounding out the heart to heart here is New York City. Yeah, that's the for sure. So November 1st, New York City. I'm in. I haven't registered yet, but I'm going to get in and we'll definitely be doing a group run there. So basically after Pikes Peak hunt, it's like September, October, clear. Yeah. Really no focus in there. No filming, races, no travel. Okay. And then December, nothing either. So hunt. And then we get home from Marath, New York, and we go to Vail for my birthday. Boom. Hello. Come on. Zooming out a little bit. Okay. Is this too much travel, especially with the baby in late April and May? I do not think so at all. First of all, the hardest for me is the first like three weeks. And we didn't talk about the boulder boulder because it's 45 minutes away. Yeah. Amen. So no, I'm not worried at all. April 20th, January, February to March, March, April will be two months or more because I think he's coming early. Yeah. No, I'm definitely back in regular life by then. Yeah. Okay. No, I'm not worried at all. Okay. I'm feeling real good about this. I think you have a good plan, a good map out. It's honing. This is what it takes is just plugging it in, everybody. This is what I do. I'd be curious to hear, this is not the question of the day, but I'd be curious to hear how you lay out your year of racing. And you just got to plug them in and research. There's so many options. How do you even, yeah, choose? You have to go to a million different websites. And one of your brilliant vlog fans should create one spot. There should be one spot. Yeah. Where everything is listed. Every single race in the world. That would be amazing. That would be amazing. But hon, onward and upward. I think it's a great plan. To 2020, I'm just, okay, I'm not saying that the Czech Republic is going to be, and I also threw out Krakow. But Krakow is the same day as London, just so everybody knows. What if you get into London? I know. Would you want to? If somebody can get me into the London Marathon, I would very strongly consider it. But it's hard. It's like New York City as far as popularity. So it's like, it's one of the six major marathons of Tokyo, London, Berlin, New York, Chicago, Boston. So it's really hard to get into. But if somebody has a connection, I've actually tossed this out to other people. Who knows? I don't know. But I get it. Like they have literally hundreds of people wanting to get in. So in Czech Republic, yeah. Sorry, I should be saying Prague. It's in Prague. Prague is beautiful. I've been there before. It's an amazing city. So it's like you've got the gold standard IAAF races, and then you've got like the next tier down. And that's what Prague is. So it's a very big race. Wow. Yeah, it's not a po-dunk thing. Yeah. So basically the spring marathon will be the, so you're really doing three marathons with Pike's Peak. But this is true. This is true. So Pike's Peak, it's different. Yeah. It's different on the body. It's going to be hard on the body, but it is different. You're right. It's not three pavement ones. Plus, third time's a charm. Three, three for a, three's a good number. Right. Yeah, for New York. So honey. I think it's amazing. All right. Well, thank you. I am grateful that you work so hard and that you're willing to get on the plane and go to all these places because God bless America. We love Denver, Colorado. Yeah. We're going to buy a new suitcase. Oh yeah. We are. We are. For 2020 because the travels continue. Why don't you get like stickers in all the places? Yep. Start putting them on the suitcase. Perfect. Okay. Perfect. All right, honey. Okay, success. 2020. Cheers to 2020. Good heart to heart.  It's a good plan. And then once you have the races, then you can start building the training around the races, but it really helps to find your races and say I'm committing and then work backwards. But I'm on bread. Work backwards. Yeah. Reverse engineering. Yep. Otherwise, how would you even process? I mean, you literally need to know what you're doing next December. Yeah. Right now. Yeah. So that you can plan out everything. That's very smart. And what's fun is that I get to mix in the filming races as well, like Hard Rock, Western States. I can't film Leadville just so Boston.  And it'll be nice to because actually Boston 2021. Boom. Boom. Boom. All right. Somebody on the blog just recently qualified for the Boston Marathon. And congratulations to him and to her against Goodman Woman. And to everybody who has qualified and is heading there. Boom. That's a big deal. So you see BQ being qualified in the comments. That's what they're talking about. BQ. So congrats again. And this is going to be fun. And oh, all right. We can keep talking. Good times. Good times. Okay. Yeah. And there you have it. All right. It's about eight hours later. That was fun. That last clip filmed right there on the at the kitchen table. I've never done that before on the channel where I set up two cameras and we just talked and hashed it out. True love and I coming up with a game plan for 2020. It's going to be an amazing year. And yes, question of the day. How is your 2020 racing schedule rounding into form? And what are your top one or top three races that you are anticipating the most in 2020? Okay. For me, well, it would be the first marathon, the Pikes Peak Marathon, and then the New York City Marathon. So whatever that first marathon ends up being, I just don't, we gotta, gotta put out the feelers there and figure out the game plan for that. But at least you are now in the loop as to what's going through my brain for racing and filming for all of you in the new year. Oh, it's happening. It's happening. I can't believe we're, we're almost there tomorrow, tomorrow, tomorrow. So all right, everyone, we're going to toss it back to the vlog from exactly one year ago, December 31st, 2018. That'll be on the right and on the left. We'll toss it back to the Western States 100 on the left since that was such a big blog from 2019. See beauty, work hard and love each other. See you tomorrow.


It's not time for Python on hardware all right There is so much going on in the world of Python on hardware that we had to make a newsletter in this newsletter We're almost like 9,000 subscribers. I think it's the most subscribed to Python on hardware newsletter So this week on the newsletter We have a few things. Yeah The first thing we have is we were celebrating how many guides we had yeah, so it's 2,500 That's a big deal good up to 2,500 lots of guys And then in our newsletter we also point to our other newsletter which we do which is the internet of things monthly that comes out every month We started to cover some things that I think will be interesting in the Python world But it'll probably be interesting in the legal world soon GitHub co-pilot it's an AI coding assistant and The controversy this is funny as hell I mean like everyone's angry, but I just think it's hilarious I think we're actually at a point where we're starting to laugh at lots of stuff versus just get angry So I was just like oh cool like this will be interesting. Yeah, and so the controversy is that this GitHub co-pilot thing was trained on all this GPL code and It then inserts code when you want it you still have to do stuff afterwards like compile it and everything But the debate is should GitHub Microsoft be able to use all that open-source code as a training set to produce code for people and the answer is Donna, I mean what's what did licensing doesn't specifically say whether or not you can train data on it a lot of code that people look a Lot of code like this example it comes from the API documentate, you know, it's like it's a little bit It reminds me a little bit of the The API discussion that just came up in the court in the Supreme Court case between Oracle and Google about Java and everyone the judge said well, it's like if you have a function that determines You know the max like the you know max int int and it returns an inch Whichever is larger gives you the larger the two ints There's really only like one way to write it like at some point. There's when your API is high level enough There's no real other way to write it. So Something that just copy and paste basically is it is a copy and paste smart snippet writer Is there a de minimis of code That is not licenseable because it's it's such a small amount of code And there's no no other way of writing it because the API only works one way Yeah, I mean like like yes, it's like it's like if it's trained it's not going to write the entire program for you It's like it's only going to make sense for small chunks of code because the longer your your markoff chain is right the less sense It makes so it works for small chunks and the question is what is the smallest number of lines of code? That is non licenseable because it's so short Yeah, so I don't have an opinion on this yet because I'm still looking at all the arguments There's arguments that are like well if we were to stop we is a society in laws If we were to stop GitHub Microsoft from doing this a music company could use AI to come up with every song that hasn't been even Every note of every song that hasn't been made yet and just every time a new song comes out Just like well, we already made that you can't do that anymore because we've already did that So there's a lot of like interesting thought experiments with this But we'll see because I think it's going to be really interesting and I think it's going to help people with their code I just know that all these lawyers are so excited. Yeah, they're like Is it like this is a new flavor? Another approach could be where you use this in like why don't you tell me if my code is definitely broken mode? Yeah, I mean there's linters. I think an AI linter could be interesting too. Yeah, okay Next up the HP prime g2 calculator supports two types of Python. That is right more calculators Python so it has a couple different things Next up Oshpark has six layer boards for folks that are really in Oshpark do check it out They have pretty fast shipping and then we went over a lot of the things that you can expect and our newsletter But the highlight this week and the thing I'm going to talk about is Mark calendar eight six twenty twenty one. It's circuit Python day. It's happening. It's Friday the six It's a sneaky day. It looks like an eight looks like a six. These are all snake letters steak snake numbers So that'll be the day that we do Circuit Python day this year. So let us know what you want in the chat. We have a good idea We're gonna have interviews with people in the community projects kind of a whole smattering of things throughout the day Other people will do stuff around the world pretty much up to them We like to have a hands-off We have a central hub and things and discord and stuff where we do things But everyone is free to do stuff and have a good sucker Python day and that is the Python hardware news for this week


All right, so learning, we do it every day, but today I'm gonna share with you what was probably one of the best, if not the best learning experience in my entire life. It all revolves around this image over here. This is a Newton fractal, or a Newton-Raphson method fractal, and if you know anything about fractals, you probably know where I'm going with this, you know where this comes from, and you can probably skip this entire next part of the video. But long story short, it's this function, X minus P of X divided by P dash of X. P of X is any polynomial of any order, and X itself is just a random guess on the complex plane, which is what this fractal is drawn on, like most fractals. Basically, P of X has four roots in the complex plane. They're around these four locations. And what we do for every single pixel is we run the Newton-Raphson equation on it, a certain amount of times, maybe a hundred times or 20 times, until we see which root it lands nearest to. Now we color each root and each pixel according to this, and when we pull all those pixels back, we get this image, which is very fancy look. Now if you wanna know more about this method, you wanna know more about where this comes from and why this is true, you should watch this video by Three Blue and Brown, where he runs through how this actually came to be and how Newton's method is very, very useful for finding the roots of polynomials. However, I just wanna focus on the programming aspect and the logistics of actually drawing an image like this, because that's what really had me caught up for like a month, trying to figure out how to do something like this. I would like to structure this video in the form of tips of general pointers that I give to people who are looking into mathematics or science or programming for fun. The first big tip I give is to set an appropriate scope, because too many people tend to either set too much work for themselves or set too little work for themselves and end up unsatisfied with the end result. In my case, it was the latter. Remember that p dash of x from the Newton-Raphson equation? Well, I wanted a function in my program that could genuinely take the data from p of x and then figure out the real derivative function instead of just estimating it to an appropriate degree. Doing this estimation of the derivative by using a sufficiently small change in x and calculating the change in y isn't only computationally less intensive, it's less rough on the computer, but it also takes less time to program. And if I actually went through and tried to make a complicated system to calculate the real proper derivative, then I would have probably wasted way too much time on that and not on making this very pretty picture. So once that got thrown out the window, I just decided to use a traditional equation that you can have in a function. And as far as the Newton-Raphson part of the program was concerned, it was just smooth sailing from then on. The next big tip I have for people doing science or math stuff or especially programming is to try to be both efficient and creative with your solutions. One of my big problems when I first approached this issue was how exactly do you create an image? What do I do? Do I need to plot all the points? Do I need to make some kind of image file? How am I going to format that? How is that going to look? How am I gonna assign the colors? That's surprisingly difficult to come up with and there's so many options, many of them being, quote unquote, right answers. And the trouble was picking one that was the most appropriate for this problem. Originally, I used Pi plot lib, which if you don't know is a library for Python that lets you plot stuff on a graph, which probably seems very useful in this situation. However, this really did improve to be true. Not only did I have to calculate the position of every single item in the array, which was unnecessary overhead in the program, but Pi plot lib itself is super slow when you're dealing with lots of points. So clearly the far more efficient and better method to go with here was to create an image. More specifically, this led me down the road to discovering all the various creative image formats that exist and ended up settling for something called PGM. Now, if you don't know what a PGM is, essentially you can declare the size and the width and the grayscale of an image and then just have zero, one, two, three, four, whatever, define the level of lightness for the pixels on your image. This is a very simple way of having an image file format that you can write to and create grayscale images. This was exactly what I was looking for and I instantly got to work making a system that could write to these images. So in summary, most of the tools that you probably need, the tools that are best for the job tend not to be in front of you. You have to go out there and research and be creative until you find exactly what you're looking for. All right, so now that I have the origins of our fractal, the Newton-Raphson equation and the endpoint, which is the image file format, I need to figure out how to glue these things together in Python. So it all starts with considering each pixel being a complex number.  You then take this number and put it into the Newton-Raphson method and repeat after a few times and then see which route it lands closest to after a limited amount of permutations. After doing this over and over again with every single possible pixel on the grid, you then compare its value with one of the roots, which either you know or you can figure out by looking at your results. Depending on which route it lands closest to and the order in which you put the roots, which is arbitrary, you can calculate the color of the pixel by simply picking a number between zero and the order of the polynomial minus one. The color can be equal to the index of the root that it's closest to if the roots aren't a specific array. Remember the radius we used before? Well, we can use that in the calculation to figure out the resolution of our picture and we can also output the color depth by using the order minus one. And once all this data, including the actual pixel data for each individual point on the graph is outputted to an image, we finally get our sought after Newton's fractal. But the thing is, is that I'm nowhere near done with this. That's the last lesson I wanna give to you about learning is that you're probably never gonna be finished with what you're learning. There's always more to do. There's always more to figure out. And there's always more things to break for you to fix later. This program already has a few problems. Like for example, if you increase the radius past 100 or so, it starts taking a long time for these images to render. Now that might just be up to Python being very slow, but it's also just poor programming practice in general. I mean, a lot of the pixels here are the exact same color and that can probably be made far more efficient. So long story short, the real point I wanna get across to this video is that when you learn for yourself, you tend to not only develop your knowledge, but also your skills to learn in the future. I now know how to do a lot more stuff in Python. I know a lot more stuff about math, just because I spent a few months figuring out this Newton-Raphson thing. It was fun to do. It was educational, but most importantly, we got some really pretty pictures out of it. So yeah, I've been Denshi. That was learning, programming and fractals. Goodbye.


Today I have the distinct pleasure of hanging out with Frank Bassa from Granada Gold. How are you today? Very good Tracy. Thanks for having me here Tracy. Always a pleasure. Okay. Everybody out there been watching PDAC. What's happening kind of the canary in the market, right? And so what's happening in the gold market, Frank? Well, the gold market got hit a little bit last week, but I think it's that short term. I think it's going to recover. Long term gold. I think it's going to really go. Right time riding the wave, so I think we're in a good place. And for those of you out there going, okay, yes, we see the tidal wave for gold. It's finally going to hit the juniors. That's what the experts are telling us. Frank, you have words that people should love to hear. High grade. Close to the surface. Talk to us a little bit about this. The deposit we have actually was actually mined as a high grade mine historically underground. It was about nine and a half, 9.7, 10 grams a ton. But we also found people in the past who got open, open pitable material, weighing five grams a ton. So the previous operator, 94, took out about 14 and a half thousand ounces of five grams. So what we did was we decided to work on that extension of that going east. So we have about half a kilometer of that material now, but the grades that we drill near surface are above that. So we get grades like 11 grams, 12 grams, 13 grams per ton over quite a wide width. So we're going to carry on with a near surface drill program, extend that. Basically, the structure is about five and a half kilometers long. We only looked at about 500 meters of it. So we'll see how far this thing goes.  We have all our permits. Basically, we ship the rocket anytime, 20 mil in Quebec, Ontario. Okay.  Extensions. What are you talking about? Well, the thing is basically historically what they did was the old timers, they mined only three structures, which was vein one, vein two. And somebody in 1994 decided to take an open pit sample out of vein one, which of a shaft one was, and they took out 14 and a half thousand ounces at five grams. Nobody was there until they took it. So we didn't look at that either. We did a bit of drilling and all of a sudden we found out, sure enough, it could be five grams. And so far, we found about 500 meters of that. So we have a lot of new investors to gold. We're trying to bring in new millennials and say, gold, treasure hunting. We've been treasure hunting for quite a while. So can you give our new investors some tips on what are good results? What should they be looking for? Well, I think, you know, most people look at an open pit at one gram. We're kind of returning grades of over one gram open pitable. You know, we're looking at about five gram average and it's been done in the past. You know, historically, when somebody takes out 14 and a half thousand ounces at five grams, it's a significant sample. So I think we found that structure. We're following it. And it's not normal to have an open pit running at five grams. Most people do one gram or less. So we're on something fairly large here. And like I said, it's five and a half kilometers long. We only look at, you know, 500 meters of it. And your last drilling results, the market responded incredible to it. Yeah. Well, we got a pretty good turnout. You know, we drilled about 33 meters of over 10 grams a ton near surface. When I say near surface, that means right from surface down. We have no overburden. So basically kind of very rare in this industry. So basically, you know, you break the rock, you ship it. And that's really rare, you know. So what should we as shareholders, hearing all of that, we got gold shovel ready, right? Key terms, high grade, yeah, high results. So what should we anticipate, say this next quarter or two? Well, I think we'll carry on with the drill program, we'll try to extend the past of 500 meters.  We find another structure next to it, a little south from there. We haven't touched it yet. And then of course, they can go to our website and take a look at what historically some of the other people pulled out, somewhere in the 30s, 40s, they took us stuff like seven ounces a ton, which is really high.  You know, it's not unheard of that we have these kind of grades on our property. Well, Frank, thank you so much for joining us today. And for those of you out there that do not know we're not a gold, I'm sure they're going to take a look now. Yeah, simple as GGM on the venture exchange. Thank you. Thanks a lot. Thanks, Tracy. Appreciate it.


In this lecture, we will learn the following things. We will learn what is a muon. We will learn how to use the muon as a laboratory for making predictions with a Lorenz transformation. And finally, we will learn how the muon was the first direct test of the validity of special relativity. Let's begin by reminding ourselves one more time about the Lorenz transformation. If we make observations in a frame S' that we consider to be moving, and we want to convert them into observations in a frame S, we consider to be at rest, then the equations in the top left of the slide will do just fine. If on the other hand we have observations that are made in the rest frame S, and we want to convert them into observations in the moving frame, then all we have to do is change x to x' t to t' and v to negative v in the upper left equations and we get the necessary equations in the upper right. This function, gamma, that appears in all of these equations is a function that depends on the relative speed between the two frames, v, and it has the form of 1 over the square root of 1 minus v squared over c squared, where c is the speed of light. This can be expanded into a series representation of this function using the binomial expansion we've looked at this before, and so we get an expansion that looks like 1 plus a half v squared over c squared plus terms that are higher order in v, like v to the fourth, v to the sixth, and so forth. And for sufficiently low velocities, less than a few percent the speed of light, we can usually neglect some or all of these higher order terms in the series expansion and get a simpler representation of the gamma function at low velocity. Now we've looked at some consequences of the Lorentz transformation in special relativity, we've looked at length measurements and we've seen how they depend on frame of reference, we've looked at time measurements and seen how they depend on the frame of reference, and we've looked at the simultaneity of events in one frame and see that they are not necessarily guaranteed to be simultaneous in any other frame that's moving with respect to the one in which they are simultaneous. But is the relativity of time a real thing? Ideally what we would want to do is take two twins, put one in a spaceship that can accelerate very quickly up to speeds well in excess of half the speed of light. That's where the gamma function begins to take on values that are very much in excess of one. And then we would let the twin travel out on a journey of maybe 10 or 20 years and then bring them back, that's going to take another 10 or 20 years. And when the twin gets back from the journey at high speed, we would compare the two twins and see how they have aged from the perspective of observers on Earth. That would be a great experiment, except that it is really not easy to construct a vessel, not only that can hold humans for long duration space flights, but also that can accelerate up to speeds that are appreciably close to that of light. This is an engineering challenge that we as human beings have never really mastered. Instead, what we need to do to test the claims of special relativity is to identify a laboratory where such speeds can be readily achieved, but also one where there's a natural clock of some kind, a regular sequence of events whose time can be well predicted, so that we can compare those things when they're at rest to when they are in motion. Now tiny particles would be a great potential laboratory. Tiny particles have very small masses that are very easy to accelerate, and so it's possible that something like the electron, with a mass of 9.11 x 10 to the minus 31 kilograms, could be ideal for investigating fast moving objects and maybe even the relativity of time. For example, the ground state energy of the hydrogen atom, the lowest energy state that an electron can have when orbiting a single proton, is 13.6 electron volts. You can look back at the conversion factor for the electron volt, look it up on the web, whatever you want to do, but you'll find that this comes out to be about 2.19 x 10 to the minus 18 joules of energy. Now if you translate that into a corresponding kinetic energy for the electron in that state of hydrogen, what you'll find is that you can calculate the speed of the electron that is represented if you pretend that the electron is orbiting like a little planet around the proton under the influence of the Coulomb force. So let's do that. Let's imagine that the electron is orbiting the proton at the center of the hydrogen atom and let's use the energy of this state to estimate the kinetic energy and therefore the velocity of this electron. And if we do this using the classical kinetic energy, one half mv squared, and rearranging it to solve for the speed of the electron, we find that it should come out to be about 2.2 x 10 to the 6th meters per second or thereabouts. That's already pretty fast on its own. That's about 1% of the speed of light without having to do anything exotic except maybe study the electron in a hydrogen atom. Now of course we'd like to get the electron up to faster speeds than that, but if that's how fast it's already going in a hydrogen atom, then you can imagine it's probably not too hard to get it going faster. In fact, JJ Thompson, who's credited with the discovery of the electron, isolated them by ripping them off of parent metal atoms using the Coulomb force, using a strong electric field and a large electric potential. This has the effect of accelerating the electrons up to rather high kinetic energies for the tabletop experiments of his day, representing tens of thousands of electron volts of energy in the electrons. And that would equate to speeds roughly of the scale of 10 to the 8th meters per second. Perfect! Those are the speeds we want to investigate phenomena at. So a particle like the electron would be easy to accelerate, but there's a problem. The electron doesn't do anything. It's an extremely stable particle. In fact, left on its own, an electron will simply be for the rest of the history of the universe so far as we know. So it doesn't have any regular characterizable phenomena associated with it once you've isolated an electron. It lacks a kind of clock that it carries along with it that we could use to see whether or not the passage of time is affected by the motion of the electron. Well, are there any such clock-like phenomena in nature that are associated with very small particles? The answer is yes. Radioactive decay of atomic nuclei is exactly an example of a natural clock that ticks all the time in nature whether we're there to observe it or not and if we do observe it we can use it to measure the passage of time in a system. So for example, among her many discoveries, two-time Nobel Prize winner Marie Curie isolated the element polonium. It is highly unstable and the natural isotope of polonium, polonium 210, transforms spontaneously into a stable lead atom, lead 206, after emitting energy in the form of radiation. Specifically what it does is it spontaneously ejects two protons and two neutrons from the polonium nucleus. These two protons and two neutrons are bonded into the nucleus of a helium atom and this thing is known as an alpha particle. We'll return to alpha particles later in the course. The bottom line is there's some spontaneous phenomenon that happens with regular time intervals that we can use to actually track time in nature. Now polonium 210 has what is known as a half-life of 138 days but what does that phrase half-life mean? It means that if I were to isolate 100 atoms of polonium 210 in a sealed container and have some way of looking at those atoms and counting them every hour of every day, if I were to wait 138 days from the time I seal the container and then look in the container, on average I will find that after 138 days about 50 atoms of polonium 210 will remain in the container. The container will also now be home to 50 lead 206 atoms. They resulted from the spontaneous decay of the missing polonium 210. Now if I further wait another 138 days from that moment and look in the container again, on average I will find that I now have 25 polonium 210 atoms left. Half of the sample I had 138 days ago and correspondingly I'll find 75 lead 206 atoms in the container. A half-life is a regular interval of time and if you had some kind of equipment that could be used to establish the amount of a certain isotope present in a sample, you will find that after one half-life, after every half-life passes, you'll lose half of what was there the last time you looked. So unstable radioactive elements have a reliable built-in clock, a regular process that occurs at the same place that is the nucleus at regular time intervals. However, there's a problem and in the historical context what I'm talking about here is a problem in the early 1900s. Polonium and other radioactive elements were pretty hard to come by in the days when they were discovered and even in the decades after that. And even if you could isolate an appreciable sample of them, how would you know precisely how to count the numbers of those things whether they're at rest or whether they're in motion? And not only that, you got to put them in motion which means you need to accelerate them. And there are thousands or tens of thousands or hundreds of thousands of times heavier than electrons. Particle accelerators that are capable of bringing ions up to speeds approaching that of light of any decent quality and control were decades away in the early 1900s. They wouldn't emerge until the 1930s, 40s, and 50s. So it's nice that we have these regular clocks like radioactive isotopes, but you can't actually do practical experiments of the variety we're thinking about trying to do. That is attempting to see whether their clocks slow as they are put into a state of motion relative to the observing frame. If only we had a tiny particle that combined the lightness, abundance, and ease of acceleration of the electron with the regular instability of radioactive atoms. And it's into this part of our story enters the mu meson, or for short the muon. Now the mathematical description of unstable nuclear behavior and of the strong binding of things in the nucleus took decades to work out. But around the 1930s, with some experience now with other forces in nature, like electromagnetism, it was hypothesized that the forces inside the nucleus that both bind it together so tightly but also occasionally allow it to catastrophically break apart. That these forces were maybe of two different kinds and that they had particles like the particles of light that transmit electromagnetism that acted as intermediaries in the nucleus and transmitted these forces within the nucleus. And so these intermediaries were given a generic class of name, mesons, from the Greek word mesos, meaning intermediary. And by the 1930s, or certainly the 1940s, the hunt was on to find them. Now shown at the left here on this slide is an image that was taken by two physicists, Anderson and Netermeyer, and published in 1936. A previously unobserved electrically charged particle punches through the slab of lead that runs through the center of the photograph. These are two different views taken from different angles of the same particle interactions at the same moment in time. And the interactions are taking place in a lead target that runs through the center of the picture, roughly here in the picture. Now as this previously unidentified particle passes through the lead, it knocks apart nuclei, but in this process it barely loses any energy. This was a really strange beast in its day. It would come to be dubbed the mu meson, or muon for short, as the physicists of that day mistakenly thought that this has to be one of the sought-after nuclear force intermediaries. I mean what else could it be? This turned out to be a bit of a lack of imagination and experience on the part of physicists with the broader picture of nature, a good lesson for all of us of course, and this assumption turned out to be wrong. The particle was real, but its role in nature was not as originally assigned, and that wouldn't be fully understood until the 1940s, 1950s. Its electric charge, however, was pretty well determined from careful experimentation, and it was found to be identical to that of the electron, negative 1.609 times 10 to the minus 19 coulombs, so it carries with it the same elementary charge that the electron possesses. Its mass, however, was very unlike the electron. It weighed in at 207 times the mass of the electron, too light to be a proton, too heavy to be an electron, and crucially, unlike the proton and unlike the electron, it is also unstable. If you trap a muon nearly at rest, and there are some fairly straightforward ways to do this, on average you will find that it only lives about 2.2 microseconds, or 2.2 times 10 to the minus 6 seconds. Now, let me make an important aside while we're on the topic of unstable particles, about unstable particles, half-lives, and the characteristic lifetime of an unstable particle. The mathematics of unstable particles was developed in response to the discovery of radioactive decay of atomic nuclei, and it's a fairly straightforward application of algebra and calculus, and I find it's instructive to run through it here. Consider the kinds of systems we've been talking about so far. So, for instance, let's imagine you have a system of n-zero unstable objects, like 100 nuclei or particles, like the muon, and you observe them to have that number at time t. And then you wait a little bit. We consider some change in time, t plus delta t, at which point we then discover that the number of objects has decreased by negative dn. Now, here, dt and dn are differential units of time and number, respectively. And you find that the number of objects remaining after a time has passed, dt, since the original time measurement, is the original number, n-zero minus dn. But if you double the number of objects, so if you start with 200 unstable objects, for instance, and wait the same amount of time, you don't find that the dn is the same size as it was before it gets bigger. And if you triple the objects to 300 or quadruple them to 400, you again find that dn after the same dt is proportionally larger, and it's larger in proportion to the size of the starting sample. There is some proportionality between the change of number of objects, the change in time, and the original number of objects. And so we can express this observational relationship in a simple equation. Negative dn, the change or decrease in the number of objects, is equal to a constant, which we have yet to determine. We'll denote that with the Greek letter lambda, lowercase lambda, times the original number of objects, n times dt, the time that passed during which the time the number changed. Now you'll notice that this is set up to look like an equation of differentials, and so one could actually integrate both sides of this equation. You can put all the n's and dn's on one side, and all the constants in dt on the other side, and then you could integrate the side with the number stuff on it, from the original number, n0, to the final number, n, after waiting a time t. Now you'll notice that on the left-hand side we have the integral of negative 1 over n dn, or 1 over x dx, if that sounds more familiar to you. And so you should know from some experience with second semester calculus that the natural log of the argument n in this case will wind up being the answer to this integral. On the right-hand side we have a much simpler integral. We're just integrating a constant times dt from time 0 to time t whenever we observe the system later. And that's a very simple integral. You just wind up getting the time t back times lambda, and then you just evaluate it at the endpoints. So if you do that you should find that you get the following equation. The natural log of the original number of objects minus the natural log of the final number of objects at time t is equal to lambda times t minus 0. 0 is the original time at which you observe the system and see that it has objects n0 in it. Well if you rearrange now and try to isolate the number of objects n at the later time t on the left side, you wind up with this equation moving the logs and constants and signs around. And so finally you can solve for n as a function of time. And you find out that it's exponential in nature. If you start with a number of objects n0, the number of objects left after a time t is given by e to the minus lambda t times the original number of objects n0. Now let's talk about this constant of proportionality which we've been calling lambda. So in order to satisfy the requirement that the total argument of the exponential function be dimensionless, it must be true that lambda has units of inverse time, one over time or one over seconds per second hurts in the units of oscillatory phenomena. It's convenient to therefore define lambda as one over some characteristic time which I'll denote with the lowercase Greek letter tau where tau is known as the time constant of the phenomenon. Well what does it actually mean for t to reach tau, the time constant? Well if you allow enough time to pass that one time constant's worth of time goes by, you find that 63.2% of the original number of objects are gone. For unstable particles, this characteristic time is what is known as the lifetime of the particle, and you can actually show using some math we'll develop later in the course that mathematically tau is also equal to the average time that an unstable particle exists. So it has two meanings. One, it's the time after which 63.2% of the original n-zero objects have disappeared from the system, and two, it's the average time that any randomly picked unstable particle will exist. Now where does the half-life come into all this? Well you can show that the half-life of an unstable particle which we could denote as t with a subscript one-half is directly related to the time constant tau by the following simple equation. The half-life is equal to the time constant times the natural log of two. So when we say, quote, the muon has a lifetime of 2.2 microseconds, unquote, we're referring to the time at which there is a 63.2% chance that any single muon has decayed, vanished, gone away from the original sample of muons. Now let's talk about muons and observing them and their origins in the world around us. Muons are not naturally occurring in the same sense that atoms are naturally occurring. Atoms are generally speaking stable, they stick around for a long time, and they form large structures because they have a chance to bind to each other through chemical means, which is just electromagnetism in action. Muons on the other hand are a bit stranger, you have to make them, and because they don't live very long, you have a very limited opportunity to study them once they come into existence. Now thankfully, nature does make them all the time, and it does so because the Earth is constantly bombarded by particles from outer space that are smashing into the atmosphere at very high speed, very high kinetic energy, and these things are known as cosmic rays. And when cosmic rays slam into the Earth's atmosphere, they result in a whole bunch of particle interactions that ultimately spray muons down onto the Earth, among other things. So they do this by smashing into nitrogen or oxygen nuclei, having all kinds of nuclear reactions in the process that produce a whole bunch of other particles, and I'm not going to worry about what those are right now, but ultimately, muons can result from this, and the symbol mu with a minus sign next to it denotes the muon with its natural negative electric charge. There are also positively charged muons, and that's a subject we'll come to later in the course. Now Anderson and Nettermeier, who originally discovered the muon, did so using showers of particles from cosmic rays, so-called cosmic ray showers, and they did so by putting detectors at different altitudes in the Earth's atmosphere. So for instance, they did a bunch of experiments with a detector located on top of Pike's Peak, which is 4.3 kilometers above sea level, and they did a bunch of experiments at home base at Caltech in Pasadena, California, which is roughly at sea level. And it turns out in the decades we've been studying cosmic ray interactions and muon production, we've learned that most of the muons that are produced by cosmic rays are made roughly at a height of 15 kilometers above the Earth's surface. That's not the top of the atmosphere, but it does correspond to the place where the density of the atmosphere gets big enough that these interactions of cosmic rays and nitrogen and oxygen molecules or nuclei get very high in probability. And so we get a lot of muons that get produced at that part of the atmosphere. Now, based on the known instability of the muon, one might expect that if one counts a certain number of muons at a high altitude, say counting a number N1, then by the math of unstable particle decay and using the known lifetime of the muon when it's nearly at rest, that is, tau for the mu is roughly 2.2 microseconds, one could accurately predict the number of muons you should expect to see at a lower altitude, N2. Now, at that lower altitude, because particle decays had a chance to happen, we expect fewer total muons to be found. If we make 100 muons or 1,000 muons at 15 kilometers above the Earth's atmosphere and we go down a bunch of kilometers, we don't expect to find the same number of muons. We expect to find typically fewer. All right, now, this is very interesting here. Let me show you this. So here in the basement of Flandren Science, the physics department has a small experiment set up that allows us to capture muons created in cosmic ray showers above the Earth, trap them by trapping them in atoms and a material in this device over here on the left. And then after trapping them, we can wait and see how long they stick around until they decay. So all of this equipment is designed to establish the time at which a muon is trapped and then the time at which it then subsequently decays because it doesn't live forever. And if we take a look at the data here, what we find is that when we trap these muons and hold them nearly at rest in our reference frame, indeed we see an exponential falloff in the number of muons that survive after a certain amount of time, as predicted by the theory of particle decay. And we can see that after about 2.2 microseconds that there's a roughly 60 to 70 percent chance that any single muon will have already decayed exactly as previous experiments have determined. So this is our own little way of caging muon using atoms to trap them, then waiting to see them decay and measuring the time between those two events in a frame that's essentially at rest with respect to the muon. And indeed this is how we figure out, for instance, that the muon lifetime is about 2.2 microseconds. This experiment alone here in the time it's been operating, which looks to be about 2,300 hours or so, has trapped and observed the decay of about 1.6 million muon. So just think about the sheer number of muons that must be raining down on the surface of the Earth all the time. We're capturing just a tiny slice of all of those in their fantastic laboratory for looking at the little clocks that fundamental particles carry around with them so that we can study time using the tiniest building blocks of the universe. Now the muon's short lifetime should radically cut down its numbers as we go lower and lower into the atmosphere. And in fact, the effect is quite stunning. So let's imagine we give the muon the best possible chance of making it to a low atmospheric height. So close to the surface of the Earth. Now to do that, we're going to crank its velocity up to the fastest that anything that we know of can travel and that's the speed of light. So we're going to set the speed of a muon that's just been produced at 15 kilometers above the surface of the Earth. We're going to set its velocity aimed straight at the surface of the Earth to the maximum it can be 2.998 times 10 to the 8th meters per second. So if you crunch the numbers, you'll find out that in one lifetime a muon can travel just 0.66 kilometers, not even a kilometer. It doesn't even go 1.15th of the way down closer to the surface of the Earth. But at this point, it's already had a 63.2% chance of decay. There's a 63.2% chance that that muon won't exist anymore by this point. But let's imagine it survives and it makes it two lifetimes. After two lifetimes at the speed of light, it could have gone 1.3 kilometers, doubling the distance it's traveled into the Earth's atmosphere and now having made it a little more than 1.15th of the way into the Earth's atmosphere. But by this point, it pays the immense penalty of having a probability of 86.4% of having already decayed. 10 lifetimes will only bring a muon 6.6 kilometers into the atmosphere. That still leaves it about 8 kilometers above the surface of the Earth. But by then, it has had a 99.995% chance of decaying. There's really very little chance that muon really makes it this far. And if you take it twice as far, 20 lifetimes, the probability is even smaller. So the bottom line is that we don't really expect if we produce 1,000 muons at 15 kilometers to find really any of them down at sea level. So what actually is observed? Well, shown at the right is some data. It's real data taken from an experiment that really can count muons at different altitudes. And the graph shows the number of counts per minute versus the altitude where the measurements were taken. And these measurements were taken by high school teachers who were involved in a program called QuarkNet. This program engages teachers in K through 12, typically high school teachers in real physics research environments. And this data is actually taken from an experiment they did that was reported in the article that's listed in the footnote on this slide. Now, what they found was that if the experiment was run 3.5 kilometers above the surface of the earth, above sea level, they found about 300 counts per minute of muons at that height. Now, let's use the Galilean and Newtonian assumption of time, that time passes at the same rate for all observers. That is, whether the muon is moving or not, its clock and clock's on the ground tick at the same rate. Now, that's a total violation of the assumptions of special relativity and, of course, the conclusions that one would then draw from the Lorenz transformation, but we can make a prediction using the Newtonian or Galilean idea. And so we can basically estimate how many counts per minute we expect at half a kilometer, which is roughly the lowest height where the teachers took data. Now, what you observe is that at 3.5 kilometers, the number of counts is about 300 per minute, 300 muons per minute passing through the detector. And if we give the best case chance of all those muons making it down to half a kilometer above the surface of the earth, we find out that we should expect the yield to go as e to the minus y over c times tau, where c is the speed of light and tau is the lifetime. And so after a height change of just 3 kilometers going from 3.5 kilometers to half a kilometer here, we expect to find at most about 3.2 counts per minute from muons that are produced at this altitude of 3.5 kilometers. Is that what we actually observe? And the answer is, heck no. In fact, the teachers observed not three counts per minute, but 100 counts per minute at both sea level and about half a kilometer above the surface of the earth. It's pretty hard to tell the difference between those two sets of data. So why would that be? Why would it be that the Galilean Newtonian prediction, or at least its assumption that time is the same for all observers regardless of the state of motion, would not get this experiment right? It seems so simple. We know the lifetime of the muon when it's at rest. You know the height difference between where you make the first and second measurements. You just do some counting. It should be easy, right? And you don't even get close to the right answer. So why would that be? Well, I think we already know the answer. The answer is time dilation. Special relativity with its Lorenz transformation that's supposed to be valid for all speeds up to that of light will help us to understand this. So let's relate what's going on in the muon's reference frame, which we'll call S prime, and what's going on in the earth's reference frame, which we'll call S. So we choose the earth and the atmosphere to be at rest. We choose the muon to be moving. So it's viewed as a moving object with respect to the earth. And so we can call that the moving frame. Now in the reference frame of the muon, where it thinks it's at rest, its lifetime is 2.2 microseconds. Recall that this is the lifetime as observed when the muon is nearly or exactly at rest, and that would be its proper lifetime when it's exactly at rest. The proper lifetime is measured in the frame where all events happen at the same location in space. For the muon coming into existence and going out of existence, all happen at the same place itself. And so that's the frame in which proper lifetime is defined. That's also how we measure the lifetime of the muon as we stop it and we let it decay and we see how long that takes, typically. And so that's the 2.2 microseconds associated with how long the muon lives. Now the Lorenz transformation would predict that the time measured by an observer on the earth, the time that's passing in the muon's frame of reference, will be different from a person who would measure the time but ride along with the muon, thinking the muon is the thing at rest the whole time. So we can take observations in the frame of the earth, observations of, say, the clock ticks in the muon's frame at T2 and T1 and take the difference, the delta T between those ticks. And we can relate those to the spatial coordinates where all the events happen and the time measurements as observed in the frame of the muon, the s' frame. We're just using the Lorenz transformation one more time here. Now, all the events in the muon's frame of reference being created in the upper atmosphere, decaying later at a time T2 where the earth is closer to the muon, they all happen in the same place in the muon's reference frame. In other words, x2 prime is equal to x1 prime. Therefore, this equation simplifies and the time difference in the earth rest frame is relatable to the time difference in the muon's frame of reference by a factor of gamma times the time difference in the muon's frame of reference. Now, the lifetime of the muon and its frame of reference is 2.2 microseconds. So delta T prime, or T2 prime minus T1 prime, is 2.2 microseconds. So special relativity would predict that from the perspective of an observer on the ground, the muon would appear to live longer than would be expected if it were at rest as well. This is completely in accordance with the observational evidence. More muons, many more muons, are observed to survive to a lower altitude than would be expected from classical physics and its assumption of the absolute passage of time for all observers. So the data told us that of the, say, 300 muons per minute observed at 3.5 kilometers, roughly speaking, 100 per minute of those survive around half a kilometer above the earth's surface. In the reference frame of the earth, we can relate these numbers to the observed decay time of the muons in their rest frame, that is, tau, the proper time and also the lifetime of the particle, and the distance that they travel from the perspective of the earth and atmosphere rest frame y, as well as the typical speed of muons. So what we find is that taking the decay equation, n equals n0 times e to the minus T over gamma tau, then tells us that we can solve for the velocity and the gamma factor of the muons using the data. We know n and we know n0 from the data. We know why, because we know the height difference that the teachers made the measurements at. We can solve for this quantity gamma v, which is related, of course, ultimately to the speed of the muons in the atmosphere relative to the earth. So if we do the algebra here and solve for gamma v, we get the following equation. Now, I'm going to leave it as an exercise to the viewer. We want to solve ultimately for either gamma or v, but since they're each a function of the other, we have to do some algebra to isolate one or the other. And to help you along with this, recall that the gamma factor is defined as the one over the square root of one minus v squared over c squared. And that means that the velocity, if you solve for that, is equal to c times the square root of one minus one over gamma squared. And from this, you can take gamma times v and you can get a nice expression for that. So gamma v can actually be written entirely in terms of gamma, which is interesting. And if you use that trick, you can get to isolated expressions for either gamma or v from this equation here on the right-hand side. So go ahead and try that yourself as an exercise, but you should find the following things. You should find that the gamma factor for these muons, assuming that all the 300 that are created at or appear at an altitude of three and a half kilometers then could be counted or not at 0.5 kilometers above the Earth's surface. And you then, based on that assumption, estimate that the gamma factor is around 4.3. And if you solve for the velocity of these muons, they are radically close to the speed of light. They are 2.91 times 10 to the 8th meters per second in speed relative to the Earth and the atmosphere. Now, from the person on the ground's perspective, that journey from three and a half kilometers to half a kilometer above the surface of the Earth takes about 10.3 microseconds, which is way longer than one lifetime of a muon. So it's no wonder muons make a fantastic and early laboratory for tests of special relativity. Nature is readily making lots of them per second in the upper atmosphere. They can be measured using technology that was available in the early 1900s, at least the first half of the 1900s. They can be observed and to see when they decay and how often they decay and so forth, and all of that together can be used to assess the validity or not of special relativity. And of course, what we find is that special relativity wins the day. It is the correct description of space and time for inertial reference frames. And it's remarkable how well it actually works. Now, of course, the atmosphere is complicated. The production of muons and the atmosphere is complicated. If you really wanted to do a super thorough job of this, you would have to do a detailed simulation of the interactions of cosmic rays in the Earth's atmosphere, producing muons at various heights, and then see how many you count at various heights with and without special relativity. If you do that, we find that with special relativity, we can exactly produce the atmospheric data without special relativity. We utterly fail to reproduce the atmospheric data. It really is true that special relativity is the correct description of space and time and motion. Now, I find it's instructive to quickly take a look at this same situation, but from the muon's perspective. That is, if you could ride along with the muon at its ridiculous speed, what would you observe? Of course, in the muon's frame of reference, it's at rest, and the Earth and the atmosphere are rushing toward it, or in the case of the atmosphere, past it. So you and the muon come into existence very suddenly, three kilometers above the final Earth observation place. Now, that's in the perspective, that three kilometer statement, that's made in the frame of an observer on the Earth. We'll get to the distances in a second in the muon frame. What you can say for sure is you come into existence, the Earth is far from you, it's racing toward you at a speed of V, and it's getting closer to you all the time, and at some point, you'll go out of existence. And the question is, how far is the Earth and atmosphere going to move in the time between those two events coming into existence and going out of existence? So the perspective of the Earth observers on the left in this cartoon, and the perspective of the muon observer is shown on the right in this cartoon. We don't know the distance between the surface of the Earth and the muon in this picture. We only know it from the original experiment in the Earth rest frame. But here we are confident that the Earth is rushing toward us at the opposite velocity that's measured in the Earth frame for the muon heading toward the Earth. Now, from the muon's perspective, of course, it's standing still. In all events coming into existence and decaying, they happen at the same location in its frame of reference. Therefore, the time it typically is going to stick around is going to be 2.2 microseconds in its frame of reference. It sees the Earth below it when it comes into existence, and it sees that surface of the Earth rushing toward it at a velocity of negative v. So how far does the muon have to go to make it to its destination? From its perspective and its reference frame. Time is ticking away at whatever rate it goes at for the muon, and ultimately it can measure time using its own lifetime, which is about 2.2 microseconds. Nothing funny with time in its rest frame. But of course, the distance where the muon was created above the surface of the Earth is three kilometers in the rest frame of the Earth. That's the frame where the Earth and its atmosphere appear to be at rest, and that makes that distance three kilometers the proper length or proper height above the surface of the Earth. That is the longest distance that any reference frame would measure between where the muon is created and say the surface of the Earth. The muon, on the other hand, will see the Earth-atmosphere system as moving, and therefore distances in that system contracted along the direction of flight. And the length or height above the surface of the Earth that it will measure will be the proper length, three kilometers, divided by gamma. And this comes out to be about 23.3% of the proper length, or 0.699 kilometers, 699 meters. That's the distance that the muon perceives between where it comes into existence and that final measuring point, which was three kilometers away in the frame of the Earth and the atmosphere. So from the muon's perspective, we conclude that it observes that the distance it will travel is contracted compared to what observers on the Earth are seeing, and that contraction factor is one over gamma. From the muon's perspective, the distance between the place and the atmosphere where it came into existence and where it ultimately decays is greatly shortened, requiring only a time of delta T prime of about 2.4 microseconds to make the trip because the Earth atmosphere system is contracted and moving relative to it. And so, you know, given that it lives about 2.2 microseconds in its reference frame, it's absolutely plausible that it could make it that full distance that people on Earth said it went. It's just that the people on Earth are confused because the distance is shorter than they claim from the muon's perspective. So while observers on Earth and an observer moving with the muon would disagree on the reason for the muon reaching the lower measurement point, they all agree that it's very likely to happen. The Earth observer argues that the reason it makes it is because time in the muon's frame is passing more slowly than they claim because the muon is moving. And so that takes longer to decay as a result of that. It's able to cross the 3 kilometer gap, even though it should have only lived 2.2 microseconds because time has slowed down for it while it's in motion. The muon observer says, no, our clocks are working just fine. What's going on is that because the Earth and atmosphere are rushing toward us, they're in motion, and so they seem contracted along the direction of motion. And as a result of that, we don't have to go that far to make it to, say, the surface of the Earth. And we're definitely going to make it in about 2.2 microseconds or so. That's why we made it so far. Now, they're both right. Even if they have different reasons for what happens, they both observe the same outcome, the muon makes it to the surface of the Earth, but they disagree on the space and time reasons for that. And that's okay because the Lorenz transformation allows them to relate their perspectives and put their measurements into the other person's frame to see what's going on and resolves the paradox in that sense. So to review, in this lecture we have learned, first of all, what is a muon? It's a subatomic particle. It's about 200 times heavier than the electron. It's about five times lighter than the proton. And it has the same elementary charge as the electron. So it's its own thing, and it would take decades after it was originally discovered to finally fit it into the sort of final picture of nature that we've reached today. The muon, regardless of what it really is, is an outstanding laboratory for testing predictions that are made using the Lorenz transformation. Specifically, about whether or not muons, given their very short lifetimes, should be able to travel the vast distances from where they're created in the Earth's atmosphere to where they can be measured down on the surface of the Earth. And in fact we find that muons in vast numbers make it from where they're produced in the upper atmosphere to the ground. But they're not supposed to if time passes at the same rate for all observers in all frames of reference. So it may seem weird that time doesn't pass at the same rate when you're moving, but it's the truth. We have direct tests of this, not only with muons, but with many other systems as well. And in many ways, the muon wound up being the very first direct test of the validity of special relativity, and it held up against that test beautifully to live another day and make more predictions, which is what makes it such a spectacular theory of space and time.


Hi, this is Sean Brotherson, Family Life Specialist with NDSU Extension at North Dakota State University. Today I'd like to visit with you about your health and responses to rural stress. Stress conditions in agriculture have been severe in recent times in the upper Midwest, so I'd like to share a little bit about the importance of understanding signs of rural stress and finding resources to manage it effectively. We'll begin with a short 30-second video from the North Dakota Bankers Association about responding to rural stress. This is Al Gustin. As farmers we know the importance of taking care of our equipment, but with all the demands of our daily operations we sometimes forget to take care of ourselves. Concerns about production, prices, and policy can weigh heavily on us. If you feel isolated and overwhelmed, talk to someone, family, friends, or professionals. If you feel you just can't keep going, call First Link anytime at 211 for listening and support. This message is brought to you by the members of the North Dakota Bankers Association and Beck Sports. Farming and ranching consistently have been ranked among the top 10 most stressful occupations in the United States. For example, the National Institute for Occupational Safety and Health examined 130 occupations and found that those who were working in farm and ranch settings had the highest rate of death due to stress-related health conditions, including heart disease, hypertension, ulcers, or nervous difficulties. All of us need to learn how to manage our stress levels and reduce the effects of unwanted stress in our lives. Too much stress can make a person more accident prone and it can negatively affect your health. For optimal health, the following things are really important. First, identify some common stressors. Second, recognize symptoms of being affected by stress. And third, manage your stress through conscious choices and healthy living practices and social support. By doing these particular things, you'll make the workplace, including your farm or ranch operation, a safer place. When you think about your farm or ranch operation and the key farming resources or assets that you depend on for productivity, typically you can make a list of particular assets on your operation. These might include land, equipment, seed, fertilizer, labor, capital, or other things. However, I want you to think about another asset that we often don't talk about. After thinking about your key farming resources or assets, ask yourself, have you ever listed health as one of the most important assets in your farm or ranch operation? Health is actually the most important asset for any farmer, rancher, or agricultural worker. Among all the assets on your farm or ranch operation, the health of the individuals involved in the operation and the quality of relationships are the most important assets to consider. Although they typically are not listed as assets. Now, if health and the quality of relationships are the most important assets on a farm or ranch operation, then also the health and safety of individuals need to be the most important priorities in managing any farm or ranch operation. You need to develop a good plan to manage and improve the health of all the individuals who work in an operation. Good stress management and pursuing health is good farm and ranch management. Maintaining the equipment that is used on a farm or ranch operation is important to an operation's productivity. Farm and ranch operations rely on their equipment to be in good working condition. During key times such as planning and harvest, your equipment is not maintained in good working condition, then an operation has a hard time being safe and productive at key times. Similar to the equipment that is used in agriculture, an individual's health is what a person relies on to be in good working condition in times of stress. You rely on your health to function every day, to make decisions, and do the tasks that are needed to be productive on your farm or ranch operation. Your health is critically important as a key resource in times of stress, so giving attention to maintaining one's health is even more important than maintaining the equipment on an operation. For those who are working in agriculture, too often prioritizing the maintenance of farm equipment is common, but farm and ranch workers don't always think about or work on their own physical and mental health. Signs of stress are like warning signals, similar to the flashing red light on the dashboard of a car. They're warning you to pay attention, to slow down, and to take steps to check for difficulty. These warning signs can occur in the physical, mental, emotional, behavioral, and relationship dimensions of a person's life. Let's talk a little bit about what some of those symptoms might look like. First physical and emotional are mental signs of stress. When we talk about physical dimensions of stress, we'll often see things like people having headache, tension in their back, and neck muscles. You might see digestive difficulties with stomach upset. Sometimes people will have difficulty breathing, shallow breathing, labored breathing. Low energy is very common, extreme fatigue or tiredness, and also you might see aggravation of existing health concerns. It's important to recognize that individuals might show signs of stress in different ways. For example, one person's physical signs of stress might be stomach upset or fatigue, while another person may have a severe or persistent headache. When we talk about emotional signs of stress, some of the indicators include being overly irritable about little things or more commonly irritable than is usual for a person, having a sense of frustration or anger about news that you receive or anything that's going on in your operation, feeling impatient, feeling restless, unable to feel settled, a sense of discouragement or hopelessness is quite common when you're dealing with heavy levels of stress. We'll also see withdrawal from others, a tendency to isolate oneself, not engage in some of the typical activities that you usually enjoy or engage in. And then some other indicators include feelings of anxiety or panic or being paralyzed, unable to make a decision or feel like you can get to the day, or similarly difficulty concentrating. So these are particular signs of stress to be attentive to. A person may vary in the emotional signs of stress that they exhibit. One person may withdraw from others or become more isolated while another person may become angry and irritable. So you need to think about what are the particular signs of stress that are manifesting in your own particular life and experience. When we discuss behavioral signs of stress, we're talking about things that we see in the actions of individuals and how they're behaving on a daily basis. Often you'll see difficulties with sleep and that can include either oversleeping or feeling exhausted all the time or having an inability to sleep or to get to sleep. You'll see difficulties with relaxing or concentrating, just an inability to settle down. Sometimes you might find yourself getting angry more easily or having trouble making decisions. For some individuals there will be attempts to escape the stress that they're experiencing by an increased use of alcohol or other drugs. Personality-wise a person might have difficulty being flexible. And then with regard to diet, some people will overeat, we call that eating your stress, while others will have just a limited appetite and not feel like eating at all. So these are some behavioral signs of stress. In terms of people's relationships, often you'll see communication difficulties that start to emerge as people are more impatient or they don't want to listen. Conflict with family members or others that you're working with in a farmer ranch operation may exist. Feeling of not being satisfied, content anymore with the relationships that you're in or the work that you're doing. Sometimes you may see elevated verbal or physical signs of stress outbursts or even abusive behavior. You may just feel strained interactions in relationships with people. You don't want to talk to them. You want to avoid others. And sometimes people will just throw themselves into their work and as a result, they'll have more limited time with a spouse, children or others. Important to realize that one person's behavioral signs of stress might be difficulties with sleep and overeating, while for another individual it might be increased use of alcohol or drugs, aspirin or aids to help sleep or things like that. So be aware of these variations in signs of stress and again, recognize that they're an indicator that you need to slow down, get checked and pay attention to one's health. Now there are some key factors that are helpful in managing one's stress and health. Some of the positive factors that protect somebody from the negative effects of stress or limited difficulty include, first of all, increased family and community support from others. You can also access support from professionals in the community, whether that be for counseling or just to have a listening ear, or to access particular resources that are going to be important to your health. The second thing is to have a regular weekly check with a health care provider, a health partner, a counselor or some other trusted individual to assess how you're doing. Third is focus on healthy living practices. These include things like regular exercise, a good sleep habit, a healthy diet and other health practices that are important to you. And finally, it's important to seek effective medical care or other professional care to get support and treatment for any particular health concerns that you may be dealing with, whether physical health or mental health. NDSU Extension has a publication called 12 Tools for Your Wellness Toolbox in Times of Pharm Stress. You can access that online. It will allow you to look at different practical options for pursuing health and decreasing your stress. What about the experience of others around you? It may not be you as an individual who is experiencing stress and it's related difficulties, but you may see it in the lives of people that you're working with. And so you want to think about what you might be able to do to help others. Explore how you might take some action or approach an individual who is dealing with stress. First of all, be consistent and patient in your expression of concern for someone and how they're doing with regard to health or management of stress without exerting too much pressure on them. Simply express that you're concerned about them. You want to see how they're doing. Second, it's important to be direct in your approach. Treat the individual as an adult. Share any concerns that you have with them openly. A third point is to be empathetic and be genuine. People are more likely to ask for help from someone that they trust and they perceive to be genuine, supportive. And so please seek to communicate that as you're talking with someone. It's important to listen and build rapport. You might give specific examples of behaviors or attitudes that concern you. For example, you might say, I've noticed that you've been exhausted lately. You seem to have a hard time getting through the day. What's going on? How are you doing?  Is something going on? So you might use specific examples of behaviors that are raising a red flag for you about someone's health and stress management. Finally, use specific statements such as, I am concerned about you. I'm concerned about your health. And be willing to share your genuine interest in their well-being. A couple of other things that are important in considering action that you might take to be of assistance to others. First of all, be aware of the resources or organizations that are available in your community, their particular services, and how you might provide a referral to them. Think about if there is help needed that you in particular are unable to provide but you can connect someone with a helpful source of information. You might listen for signs that the person or family needs help in an area where you might link them with a particular resource. That might be financial assistance. It might be legal advice. It might be personal counseling. If you find that is true, then you want to assess what agency or what community resource is going to be most appropriate to address an individual or family's particular concerns. You might discuss a referral with the individual and say, it looks like you are feeding this way. I think that X or Y agency could be of help to you in your situation. Don't worry if you don't say things perfectly. What's important is that you convey your message of concern. You express your willingness to help someone. Also, don't be discouraged if the person isn't ready to accept some assistance. Simply express your support and be patient with the situation. But plan a seed so that the opportunity comes to link them with a particular resource. You can help be that bridge to a resource that can be helpful to them. Finally, encourage people to get proper professional assistance for their health from a primary care doctor, a mental health professional or other trusted professional source. If someone feels a need to access mental health supports or services or if you feel concerned about an individual who's under stress and may have difficulty coping or even it has feelings of suicidality, the resources that you can direct them to may be lifesaving in their instance. I'd encourage you again to be familiar with those resources and how you can direct them to sources of support. In the state of North Dakota, you can call the 2-1-1 24-hour helpline and referral service. They provide camp crisis counseling and also link individuals at any time with particular resources that are needed. You might also refer someone to a local health care provider or mental health professional. And the National Suicide Prevention Lifeline is 1-800-273-TALK or 8255. There are a variety of other educational resources on managing farm stress that can be found online at the NDSU Farm Stress webpage. Simply search for us online at NDSU and Farm Stress. These resources include educational videos, podcasts and other resource publications on a variety of aspects of managing farm stress and improving one's health. We encourage you to access that information. Thank you for listening today. This has been brought to you by NDSU Extension, Extending Knowledge, Changing Lives.


One could say that 2 Timothy is the final word from Paul in Scripture. Sensing his end drawing near, Paul writes to Timothy, his spiritual son, with some final instructions and encouragements. Be strong and grace. Teach others to teach others the good news of Jesus. Pursue justice, faith, love, and peace. Always speak gently and earnestly to those who do not yet know the good news of Jesus. In all reality, this letter is for us. We embody the Kingdom in living these words.