What's going on in the family? Welcome back to another video. It's your girl Janice. If you guys are new to the channel, make sure you hit that subscribe button and join the family. So guys, as you know, Isaiah just got a brand new BMW M3. This car is his dream car. He doesn't want anyone touching on it, anyone breathing on it, anyone even sitting in it. But obviously he has to let me sit in it sometimes. But guys, this car is super expensive and he is just so freaking OCD about this car. It's insane. So today I am going to be convincing Isaiah to let me drive his brand new expensive BMW M3. Now guys, you know that I don't drive often and I actually asked him to let me drive on the trip down here and he did not let me do it. But I have a trick up my sleeve to let him tell me that I can drive his car. So guys, if you are ready for today's video, don't forget to smash that like button. Commenting Janice in the comment section below if you think that I'm going to get my way today. And let's go ahead and ask him the big question. Hello? What are you doing? Nothing. What? What do you want? I just have a question for you. No, the answer is no. The answer is going to be yes. No, it depends. What? Why do you have the camera? You're scaring me. What? I'm going to drive your car today. No, you're not. Yes, I am. No, you're not. Yes, I am. Over my dead body. No, I'm driving it. Please. No, why? Why do you want to drive my car? You have your own car. Because I just want to take a spin around the block. Nothing crazy. I just want to take a spin around the block and your bends. No, I want to drive your car. Don't touch my car. No, I like your interior.  What do you mean? You don't need to know how it feels to be a big boy. Yes, I do. Big girl, go to your bends. No, I want to drive your car, not mine. No. Well, guess what? Get a bike. You're going to let me drive it. No, I'm not. Yes, sir. You're going to drive my car for a while. Why do you want to drive my car for a while? Because I want to. For a while. If you love me, you'll let me drive it. Bro, you know you can't be doing that. Yup, if you love me, you'll let me drive it. You're killing me. I'm shitting myself. I'm literally shitting myself. This interior is going to be brown in a second. All right, so how do I turn the car on? Just hit the start button.  Press on the brake, step them. Oh. I'm so scared, guys. I'm so scared. All right, let's go. No, no, no, no. You got to let it warm up. But don't matter if it's hot outside. You got to let it warm up. You see this? Oh, my goodness. All right. How long do we got to warm it up for? Give me a good, like, at least five minutes, bro. All right, we'll see you guys in five minutes when I'm a grandma. Let the oil circulate. All right, dang. I'm nervous. God help us, bro. God, please. I look good in the driver's seat of the BMW M3. I think you need a regular 3 Series, not a M3. What do you mean, a regular 3 Series? What does that mean? Like a 330 or something like that. 340 or? Isn't that what you had? I had an M340. Oh, M. There's an M3 now, OK? Oh, I'm scared. Do you know how to put it in a drive? Because it's not a regular trip. No. OK, step on the brake. I'm so scared. Oh, my God. Oh, my God, bro. I'm telling you right now.  You mess up the car any way we're selling the house. We can't sell that. We're going back to my mom's attic. No, we're not. You could go back to my mom's attic. I'm going to need the money for the repairs. I can't really see nothing around me. That doesn't, don't say that. That doesn't help. I can't. I'm being honest. I can't see anything. All right, bro. Step on the brake. OK. You're going to shift it to the right. All right? It's in drive. All right, like this? It's in drive. You're scaring me. You're really scaring me. Go ahead. Let off the brake. It's a regular car. Now just don't overthink it, bro. Don't overthink it. You're making me nervous. It's a regular car. All right. It's a regular car. Can you watch that side? Because I can't see. There's nothing. We're in our driveway. There's nothing. OK. I just don't want to. Am I far enough from it? You're good. You're good. OK, just go straight. OK. Turn right. Go that way. We're going to go right. Signal first, baby. There's no one coming. OK, OK. It's so scary. OK. OK. Woo! OK. This thing got a kick. You're going to give my heart a kick is what you're going to do. Ooh, I like this very hard. OK. Take it easy. I want to get an M3. Take it easy. No, you're not. All right. Turn a little more. There you go. OK. All right.  All right. Why are you speeding up? Relax. I want to see. Get a feel for the car, bro. Get a feel for the car first. I want to see how fast it goes. No, it goes fast. All right. You're talking about 500 horsepower. Oh, this is nice. Babe, do not speed. Why? Because you're going to give me a heart attack. Don't speed. Stop like that. There's a stop sign. OK. Look, soft brake, see? Yeah, soft brake. There you go. All right. You want to go? Now, soft gas. Which way you want to go? That's fine. Go straight. Go straight. Go straight. Soft gas. There you go. Damn, this car better driving than mine. I like it. You guys think I'm probably being extra right now? I'm not, OK? My dream car we're talking about here, and she's over here trying to play games with it. I'm not nervous no more. I want to go full speed. No, that's what we're not doing. All right, stay away from the trash cans, please. I'm not even close to that. Stay away from the trash cans, please. All right. You're getting a little too close for my comfort. All right, which way? Left or right? Left, because the garbage can's over there. The garbage truck. Got it. Oh, God, OK. All right, go ahead. I don't want to go this way. You're good. All right, easy, easy. I'm fine, though. All right, yeah, fine, yeah. I'm pulling over to the side, you know? Not too close to the curb. There is no curb. Not too close to the curb. There's a curb right there. Oh my God. It's raised. Don't you're going to scratch my rim. It's raised. All right, go right. Go right. All right, I don't see. Oh, right? Yeah, right. Why are you turning left? I'm sorry, I'm sorry. You don't know you're right from left. All right, come here a little more. Cut the wheel a little more.  Don't, bro, don't be doing that. You're turning a little too wide. This isn't a truck. How am I turning too wide? All right, go all the way to the end of the street. This is just a straightaway. It should save my heart a little bit. I'm just trying to avoid. There's a person all the way up there. Don't hit him. Oh my God, should I turn right here? No, go straight. But there's a dead end. I have to do your turn. Oh my God, all right. We already passed the turn. Shit. All right, go. No, just go. Just go, just go, just go, just go. See, see, I don't know. You got me driving in the middle of the street.  Oh, there's a car coming.  I'm scared, I'm scared. Oh my God. All right, pull over to your right a little bit. Not too much because the curb is there.  You're good. You're good. You're good.  For real. That was a choice. Turn right. Right? Yeah, right. All right. Now watch the curve. Get a little close. Okay. Curve, yep, you're good. Cut it, cut it, cut it. Okay, cool. Curve. Woo. Do you want me to pull over? No, no, we're good. Don't pull over. Okay, but there was a tractor trailer. No, it's not a tractor trailer. It's a truck. It's a pickup truck. He needs to get out the way. He's in the middle of the street. No, no, he's good. He's good. Go ahead. He's waiting for you. Thanks, guy. He's scaring me. He's scaring me, too. This is a curb. So, like, what if a car was coming over me? It's all right. It's all right. Ooh, look at that Miata. I can't look. I'm driving. Yeah, you're right. Don't look. God, that's actually really nice. Wow. I like Miatas. Remember I wanted one? Yeah, I do. But let's focus on the task on hand, okay? Okay. Stop doing it. Don't do that.  I'm going to cut this right shirt, bro. All right. Go left. Left? Go left. What if I don't want to go there? I'm not going to even lie to you. I'm lost. I don't even know where we're at. Me either. Living in a new neighborhood. This car need a move-over.  I am. I'm trying to move over. You're in a little too close. You're in a little too close. Yeah, he likes it. You see that? He flicked the lights. Oh, really? Which way? He flicked the lights. He likes the car. We're about to leave the freaking area. Okay, well, turn right. There you go. Right? Okay. Perfect. Oh, my gosh. That's the best turn I've seen you do all video. Yeah, because you're not.  Is this a circle? It's a loop. Oh, my gosh. No, it's a loop. Okay. Okay. It's a loop so then I can go all the way around. If you're real, you'll drift it. I'll do it right now. No, don't. Stop. It was a joke. Well, I'm taking this curve. All right. Yeah, you're going a little too fast around the curve. Babe, I see where I'm going. I am good. Okay. Stop sign there. Yeah. I see the stop sign. You might have to teach our kids how to drive in the future because all right, hold on. Let this car pass.  I'm not. I see him coming. All right. Yeah, go ahead. Follow him. Keep your distance though. I'm not going to keep my distance. Keep your distance. No. This car has moved over a lot for me. Yeah. Respect.  Garbage cans. Oh, sorry. I was too busy talking to the car. Don't talk to the car. It's not going to talk back. All right. Well, where do I go? Straight? Keep going straight, bro. Oh, this car is stopping. Why is he stopping? I don't know. Don't worry about him. Worry about you. If you hit Janice, you're going to hit the car. You're too close to the curve, Janice. I'm moving all the way because this car doesn't look like they know what they're doing. Go ahead. Go ahead. You see? I'm letting them pass. You see? You're good. All right. Easy on the gas. Gosh. I know how to drive. All right. Well, maybe you should drive your car. I don't know why we're on mine. Because I wanted to just take it for a spin. Take it for a spin. You have your own car. Why don't you take that first? Oh, look. It looks like we're driving into the sky, no? No, it doesn't. It doesn't. No, it doesn't. I promise you it doesn't. Yes, it does. It looks like I'm driving into my freaking, you know what? Into the sky. Let me not say that. It's a G. Go left. You better be taking me the right way. I don't know where I'm taking you, but we're trying to figure it out as we go, right? Take me home.  All right. Well, I'm dead. First of all, it was your idea to drive. It was, but now I have to go here.  I'm not coming at your car crazy.  Yeah, yeah. I'm putting the GPS on to get us home because now Yeah, put the GPS on because we're lost. No, we're not. We're lost in this neighborhood. Garbage cans, Janice. Garbage cans. Wait, I'm trying to see. Are we on our street? No, don't worry about it. Just drive straight. I'll worry about getting us home. Oh my gosh. All right. It says we're two minutes away. All right, so take me home. Okay, all right.  All right. You want me to drive like two miles per hour? I would prefer it, yeah. All right. I'm driving two miles per hour. Watch the bumps. Watch the bumps.  Suspension. All right. Okay, Janice. All right. Easy on the gas. What? What's the rush? There's no rush to get home. Janice. Janice. Janice. Stop. You're going to miss the turn. Right here. Turn. No, I'm not turning. Turn.  But there's a freaking thing there. No, you're good. Turn. Just don't hit the trailer. All right. There you go. You're good. All right. You're going to be here. It's going to be you backing in. I'm not pretty. To the driver. Start with me. Yes, you are. You're going to put it back the way it was. No, no, no. I'm not doing that. Yeah, you are. No, I'm not. Okay, so. All right. Now I'm scared. What? Move up. Move up. Go ahead. Go ahead. Turn this way. Go ahead. Uh-huh. Oh. Go ahead. Go ahead. I can't see. You're good. Stop. Stop. All right. You're in reverse. Cut the wheel to the left. There you go. Make sure you don't run over my lawn. I've been working on that a lot. You're good. You're good. Guys, I kid you not. My heart's bracing. Keep going. Keep going. You're good. Go. Back it up. Go. Go. Go. Go. Go. Go. Go. Go. Go. Go. Go. You got a little bit more. Go. Go. Go. All right. Look. Now you're going to put it down to the right and you hit P for Park. It's Park. That's it. We're done. You can let go of the brake. I have to poop. Get me out. This was your idea. This was your idea. You wanted to do this. I have to throw up. We're never doing this again. You did a little better than I was expecting.  You really do. I'm about to throw up. I'm so nervous. All right. We'll open the window and get out if you're going to do that. Because my origin's here. You don't need to be green. Bye. All right. Geez, man. All right, guys. So Janice came back in the car. She walked around the driveway. She got some air. You're good? Yeah, I'm good now. I was getting a little nauseous, you know. I still am. I still am. Okay. So go ahead. Close out your video, bro. We're never doing this again. I'm letting you know that right now. Well, today was fun. Never going to do this again.  Wow. Wow. I'm just kidding. I'm just kidding. I'm just kidding. This car is beautiful. This car is amazing. The garbage truck needed to change its brakes. All that squeaking. It's a eeee.  Anyway, go ahead. Close out your video, bro. So I can go ahead and drink some milk inside. Calm myself down. My nerves. Calm your nerves. Milk is going to calm your nerves. It's going to do something, but go ahead.  But anyway, it is time for today's post notification shout out. Today's post notification shout out goes out to Antonio Mark. Thank you so much for your love and support. Izan, I love you.  Turn on those post notification bells so you're notified whenever we post a new video. And if you want to see a part two. No. To this video. I guess. There's no need for it. I guess. I will do that for you and for the fam. No. And we will catch you in the next video.


So last time we met, we were talking about lists and I showed you a few examples of the ways that you can manipulate lists or query them for certain of their elements. I'm going to show a couple of more little list manipulation commands here at the start and then we'll talk about matrices or if you like two dimensional lists and start to see some nice things that you can do with them. Okay, so I'm going to first show you how you can join two lists together. So I'm going to define the first one as fruits. So I'll say apples, comma, oranges and then I'll define vegetables or veggies equals cucumbers and squash and now if I want to put those two together I can use the command join so I can for example define a new list all equals join and then I list the lists that I want to join sorry that should be a square bracket so it would be fruits and veggies. Okay, so now you see I've got a list where veggies, these two elements have been concatenated to the end of the fruits. All right, so that's nice if you have a couple of sets of data and you want to put them together into a single list. All right, now sometimes it's nice to be able to sort lists, okay, so you may have a list of numbers and you might like to order them from high to low or low to high or if you have a list of names such as this list here you may want to alphabetize. So let's see how those things work. All right, so first of all let's just try sort all, all right. So now you see when you have list of things that begin with letters the sort command will order them in alphabetical order, right, A, C, O, and S. How about if you have numbers? So let's remind ourselves how the table command works so I'm going to define squares is going to be equal to the list of the squares of the first 10 integers, so we did this last time. This is a handy way of making that list, all right, and now what I can do is actually let's make this in decreasing order, so I put the increment minus 1 and then start from 10 and go back to 1 and now you see it's arranged from high to low. If I want to reorganize that and make it go from low to high I can say sort squares. Oops, I have to spell it correctly. And now you see it orders from low to high, all right, so let me make this equal to something else. I'll call it new. Now what if I had a list like this that went from low to high but I actually wanted to sort it from high to low? Well, there's an option to the sort command so let's define newer equals sort new and if you say greater with a capital G that will sort from high to low. So now you see this list new has been reordered from high to low in newer by including this option greater. Okay, so these manipulations can be useful. All right, now so that's about all I'm going to tell you about manipulating simple lists, okay? There's obviously a lot more that you can presumably do but these are some of the more useful ones and many of which we will actually use at some point in the course. All right, so the next thing I want to do is introduce you to matrices, okay? So a matrix is a two-dimensional array of numbers, all right? And the way that they are defined in Mathematica is as a list of lists, all right? So what I'm going to do now is we'll type one in and then I'll tell you a little bit about the format and how to interpret it, okay? So I'm going to define a matrix now which is called data, okay? And it's going to have three rows and two columns, okay? So in Mathematica format you enter a row as a list and a matrix as a list of rows, all right? So I'm going to show you that now and then you'll see how it works. Okay, so a list of lists means I should have a curly bracket and then within that a curly bracket for the first list or first row and in this case it's going to be 0, 10, okay? So that's the first row in my matrix, then a comma and then the second one, 1, 25 and then the third row is going to be 2, 50, okay? If I enter that all it does is regurgitate to me what I entered, okay? Now there's a couple of different ways that you can re-display a matrix so that it actually looks more like what you intended when you input the data. So for example, if I type data and then use the post fix, so that's double slash and then type table form, what this does is it will re-display this matrix which doesn't look so pretty in mathematical format in a more appealing form. And now you can see that in fact this corresponds to an array, two-dimensional array of numbers with three rows, two columns and the first row, 0, 10, second, 1, 25, third, 2, 50, okay? Now there's another way to display a matrix that looks even more like the mathematical version of a matrix and we'll learn later on in the course how to actually do linear algebra using matrices in Mathematica. That command is matrix form, okay? So what that does is it puts parentheses around the table form and now you can see if any of you have had experience with matrices in Math classes that that looks like what you write when you define a matrix in Math class, okay? Now there are many, many ways to manipulate and do calculations with matrices and we'll save that for later on in the course but there's one thing that can already be useful early on, there's one operation and that's what's known as the transpose. So I want to show you how that works. So if you type transpose and then the name of your matrix, what happened there? Well it took a matrix that had three rows and two columns and reorganized it, the elements within it such that it now has two rows and three columns, you could see that if you put the post fix here, table form, okay? So essentially what it does is it moves elements across the diagonal of the matrix, okay? Maybe that's a little easier to see if we were to just make a 2 by 2 here. You can, whoops, I typed in the wrong spot here. If you just nuke this last row, okay, and then enter this command again, you can see that the diagonal is the same but one and ten have been exchanged across the diagonal, okay? It's just a way to reorganize the elements in the matrix, okay? So that's the transpose. Now, the next thing that I want to introduce you to is the idea of using a matrix as a way to store and manipulate data. So for example, it could be some kind of experimental data, all right? And also to display it in a nice fashion. So what we're going to do is we're going to type in a matrix and this matrix is meant to correspond to data that you might measure in a chemical kinetics experiment where you're trying to figure out the rate law and the rate constant for a chemical reaction by measuring the concentration of one of the species as a function of time. So for example, with some kind of spectrometer, all right? Okay, so what I'm going to do is clear data because I like to use that name. And now, I apologize for this but it's worth going through the pain once in a while to appreciate what it's like to enter data in by hand and we'll see more efficient ways of doing it shortly. I'm going to define a matrix so you can think of us as having been in the lab earlier today, made the measurements, we wrote them down in our lab notebook and now we're going to copy those values from the lab notebook into Mathematica and then later on, we're going to play with those values and do something useful, okay? So what this is, as I said, it's going to be concentration versus time. So basically, what I'm going to have is you can think of this as being time in seconds and then there's going to be the concentration of some product in molar, okay? And then I'm going to have the values, all right? So I can think of this as being a matrix that consists of two columns and the number of rows is going to be dictated by the number of data points that I have, okay? So that's what we're going to enter in now. All right, so data equals, first point is zero seconds and 0.01 molar and then we wait 1,000 seconds and we take another measurement and we get 0.00625, so our reactant concentration is going down and then we take another point at 1,800 seconds and the value there is 0.00476 and then a little later, at 2,800 seconds, we measure 0.0037 and then at 1 hour, 3,600 seconds, we measure 0.00313 and then 4,400 and 0.00270 and then two more points, 5,200 that value is 0.00241 and then finally the last point at 6,200 seconds, we measured 0.00208, all right? So there is our matrix and it looks like I have a little problem here. I need a bracket here, thank you, all right? Okay, all right, so I can enter that and now what I want to do is I want to make a nice looking table. In general, maybe it's not worth doing but I just want to show you that you can actually do things like format a table and put headings on the table, all right? So I'm going to type in a command here and then I'll explain to you how it works. Okay, so I'm going to say table form data and then I'm going to put in an option that allows me to make some headings on my table. So table headings, arrow, okay, and now I can put in, first of all, a set of headings for the rows but in this case I don't want any because the time is basically a good indicator of the rows. So for the rows I'm going to say none and then I'm going to put in a heading for each of my columns, okay? So for the time, I'll put in time in seconds and then comma and then concentration of A in molar, okay? Now if we enter this, we get a nice looking table, all right? So if we would have just said table form data, we would have a table without any headings so all we did was make it look nice. We didn't put any rows, labels on the rows, not necessary but we have these nice things that help us to see what we actually entered into our matrix, okay? Now next thing I want to do is plot these points, okay? So we have a new plot command and the difference between this new command and the one we already learned about is that the plot command we learned about previously plots a function that's evaluated over a range that we specify. Here we don't have a function, we just have a series of points, data points, in other words a list of data points and so we have a different plotting function, it's called list plot and it has many of the same options as the plot. The only distinction being what's the nature of the thing that's being plotted, okay? So when you have discrete data points you use the command list plot, okay? So here's how it works, list plot, bracket, you give the name of your matrix and if we want we can put some labels so I'll put axis label arrow and the x-axis can be time in seconds and the y-axis concentration in molar, okay? And now if you plot that, you see that it plots discrete points. Those are the ones that we entered in, okay? Now one thing that I like to do is to make these dots a little bigger. So Mathematica has a default and I remember what it is, it's probably 1% of the plotting width or something and as you can see those are pretty small dots so I like to increase it and we can do that by adding another option which is plot style arrow and then point size and I believe, whoops, that should be a capital S if I put in 0.02 so the fraction of the plot frame that'll basically double the size of those dots so if you enter that you see that your dots are much more visible, okay? All right, now suppose, okay so just to recapitulate here, so this is some data that you may have measured something very similar in G-chem lab in an experiment in general chemistry, all right? Now suppose I have a good idea that this reaction is either first order or second order in this reactant A. Let's see if anybody can remember and tell the rest of us how it is that I can determine the reaction order by looking at the data, yes. What he said is you look at the basically the functional form if it's exponential it's first order, what about if it's second order? Right, so remember we have from chemical kinetics we learn about integrated rate laws which tells us how the concentration of a reactant or a product depends on time and later on we'll remember how it is we derive those but for now let me just write down the formulas, okay? So first order, the concentration of A at time t is equal to the log of that actually is equal to the log of the concentration at time 0 minus the rate constant times time, is that what's familiar, okay? What about for second order? Well in that case 1 over the concentration at time t is equal to 1 over the initial concentration plus the rate constant, the second order rate constant times time. Now what these formulas tell us is in both cases we can make plots to tell whether or not the reaction is first or second order because here if I define y as the log of concentration at time t and log of concentration at time 0 is equal to the intercept, the y intercept then this tells me that I should have, so if I define x as t, m now is minus the rate constant, the slope is minus the rate constant, okay? So if I plot log of concentration versus time I should get a straight line whose intercept is log of the initial concentration and whose slope is equal to minus the rate constant. For the second order case I plot 1 over the concentration and then I should get a line whose slope is equal to the rate constant and whose y intercept is equal to 1 over the initial concentration, okay? So what I want to do next then is manipulate those data either by taking the log of the concentration or the reciprocal of the concentration and we'll plot them and we'll see if one of them gives us a line in which case we can conclude what the order of the reaction is and then probably maybe, yeah, next time we'll see how we can determine the rate constant by fitting the data, all right? So this next exercise is to learn how to take a matrix of data that we've entered and then manipulate it. In this case we're going to either take the log of the concentration or the reciprocal. All right, now the way we're going to do this, so now we have the matrix data, all right? Here it is. Concentration is a function of time. What I'm going to do is I'm going to copy that matrix into a new matrix and then in the second column I'm going to put the log of the concentration in place of the concentration itself, okay? So now I'm going to make a new matrix called new data and I'm just going to call that data, all right? That's just going to duplicate it. Now I'm going to type in something and then I'll explain to you what it does, okay? So what I'm going to do is say new data, bracket, bracket, all comma 2, bracket, bracket equals log, bracket, data, bracket, bracket, all comma 2. Now I'm going to go ahead and put a semicolon and then we'll just look at new data in table form, okay? We're missing a bracket here, all right? Now what happened? So this thing inside the double brackets, this is a way to refer to all rows second column, all right? So what I'm basically saying is take all rows in the second column of data, take the log of each one of those and put them into the corresponding all rows second column of new data, all right? So now what I have here, it's not the concentration anymore, it's the log of the concentration. So this is kind of a nifty way to do an operation on, in this case, the second row, I mean second column, that's why the comma 2, all in one shot. And now I can simply plot new data and see if it's a straight line. If it's not then I would conclude this reaction is not first order, okay? So let's do that. So we can say list plot, new data. And you can kind of tell by eye whether or not that's a straight line. Does anybody in here think that looks like a nice straight line? Not really, all right? So what do we conclude? That these data that we entered are not consistent with the first order integrated rate law. All right, so now let's check second order. So what do we need to do for second order? We need to create instead of the log of the concentration of the reciprocal. And if we plot that versus time and it's a straight line, then we conclude that the second order rate law is consistent with these data, all right? And we can do it more or less the way we did here. So I'm just going to mouse all this in, put it down here. And now instead of taking log, all I have to do is say 1 over. I don't need these brackets because I don't have a function here, okay? So now we've replaced the concentration by the reciprocal of the concentration, okay? And now let's go ahead and plot that, okay? Now, does that look like a straight line? It looks like a pretty good straight line to me. All right, so then we would conclude, in fact, that these data are consistent with the second order integrated rate law. And like I say, shortly, not yet, but pretty soon, we'll figure out how to actually fit the data to a straight line and get out the rate constant, which is going to be equal to the slope, okay? So any questions on entering matrices, doing simple arithmetic operations on them? And plotting them using list plot, okay, good. All right, now the next topic is going to be fitting, okay? And so I'm going to explain to you what it is that we're going to do. How many people have ever done least squares fitting? Anyone? No? Okay, good. Then you'll learn something useful. All right, okay, so what I'm going to suppose that I have is a series of points, okay? So each of these is just a set of points in the XY plane and I could be anywhere from 1 to the length of my data, all right? And the objective of a fit would be to see if I can find a function, an analytical function that traces these data, in other words, that goes through the points very closely, hopefully, okay? And so this, all right, this is the data and the fitting function I'm going to call, I'll just call it half of X, okay? And so the idea behind least squares fitting is that it optimizes the parameters of the function F of X, so it's going to have some parameters in it, I'll explain that in a second, such that it fits the data as well as possible. And so now I'm going to tell you what is as well as possible, okay? So there's a quantity, which we denote chi squared and what this is is going to be a sum over all the data points of the absolute value of a quantity known as the residual, okay? And the residual is just the difference between our function F of X evaluated at XI minus the actual value of Y sub I, okay? So this is a measure for each point of how far my fitted function deviates from the actual data, okay? And this is the sum of that quantity squared over all the data points. And the idea behind least squares fitting, as the name suggests, we minimize chi squared with respect to the parameters, okay? Now there are different types of least squares fitting. The simplest one is called linear least squares fitting. And the nice thing about linear least squares fitting is that you are guaranteed to get the best possible answer. It's an analytically solvable problem, okay? So you choose a function F of X. You perform a linear least squares fit and you're guaranteed that the parameters that come out are the best possible parameters for that function that you chose. There's another version which we'll talk about later in the course, which is called nonlinear least squares fitting, and that one you're not necessarily guaranteed to get the best answer, so you have to be a little more careful with that. But I want to give you an introduction to fitting by showing you the simplest version, which is already very, very useful, as I hope you'll see, okay? So what do I mean when I say linear? This means linear with respect to the parameters, okay? So for example, if I have F of X equals MX plus B, so that's just our formula for a straight line, this function is both linear and X, and it's also linear in the parameter M and in the parameter B. If I had F of X equals AX plus BX squared, is that linear with respect to the parameters? Yes. It's not linear with respect to X, okay? It's quadratic with respect to X, but with respect to A and B, the parameters A and B, it is linear. So this is linear and X and A and B. So I can use both of these functions as trial functions to do a least squares fit. What about log or A log BX? Can I do a linear least squares fit with that function if the parameters are A and B? The answer is no, okay? So this is linear in A, but not in B. B is inside the log as written, but you can actually expand the log and make it into a linear function of parameters. You see how to do that? So I can define this as some other parameter C, and now I have a linear function, okay? Okay, one more example. Suppose F of X is equal to A cosine BX, same story. Linear in A, but not in B. And that one, I can't use the same trick of rearranging it to turn it into a linear function of parameters, okay? So the next thing we'll do then is to see a simple example of how to do linear least squares fitting in Mathematica and then we'll do a couple of more examples next time. Any questions on this? I know from experience that what is a linear function in parameters is not always obvious to you when you first see this, yeah. We're doing this because this is something that is done routinely in experimental science. So for example, for this problem, suppose now I want the rate constant. What I'm going to do is I'm going to fit these data, we won't do this right now, we'll do it next time, fit these data to a line, and then the rate constant's the slope, okay? But I want the best possible line that goes through those data if there's a little bit of noise going point by point it's going to give me a different answer at each point. I want to consider all the data at once, fit it to a line and extract out a meaningful parameter, in this case the rate constant, okay? And this is done all the time. And we'll see other examples of what you can do with it, okay? So then let's go ahead and see how it works, all right. So the first thing I'm going to do is I'm going to define a function that is a formula for a straight line, okay? So I'm going to define y of x underscore colon equals 2 times x plus 3, all right? And now I'm going to make a table of values of this function, okay? So I'm going to say my data equals table and in the first column I'm going to put x and in the second column I'm going to put y of x and I'm going to evaluate this at a series of discrete points, x goes from minus 1 to 5, okay? All right, so if I do that I get a matrix of data, okay? Now I want to fit this and so now I'm going to explain to you, sorry I keep going back and forth, how it is that we specify the functional form in Mathematica for the linear least squares fit. So I want to fit this function to the formula, a generic formula for a straight line, all right? So I'm going to have y of x is equal to b plus mx, okay? So in order to specify this function in Mathematica I need to specify for each term in my function what's the dependence on x, okay? The first term doesn't depend on x, all right? So it's proportional to x to the zero power or 1 and the second term is proportional to x, okay? So that term I'm going to specify by just putting in an x and if I had a third term that was like, you know, plus a x cubed I would add an additional term x cubed here but we don't have that for this one. We'll see that a little bit later, okay? So we have to specify these two terms and I'll show you how to do that now. So the command to do a simple linear least squares fit in Mathematica is just called fit, okay? So what I'm going to say is fit and you say, whoops, it's a bracket, excuse me, say what you want to fit so it's my data and now I put in the functional form and I do that by listing in curly braces the terms in my function so I have a term proportional to 1 and a term proportional to x, all right? And then I have to say what is the variable and here it's x, okay? So this here is how you specify the formula for a simple straight line in Mathematica and if there were additional terms that had other dependencies on x I would just list those here. Yeah? Mm-hmm. And we're looking at the equation above. Mm-hmm. It doesn't matter if you do that. The equation doesn't matter. It won't matter, it doesn't matter, no, no. It doesn't, he's asking if I wrote the function like this do I have to put the terms in the same order here and the answer is no. Mathematica is smart and can keep track of the terms and match them up appropriately. But in this case this is an artificial example so all I want to show you with this example is I've created exactly data that corresponds to this formula so when I do the fit I should get exactly this back, all right? We'll add some noise to it in just a second. This is just to show you that it works, okay? All right, so if we enter we get a formula and this formula is something we can actually use to do calculations now if I set this equal to something, all right? So this is a trivial example where I got back the answer I put in because my function corresponded perfectly to the 2x plus 3 form and so I get exactly that back, all right? So now I'm going to finish up just the last couple of minutes by adding some noise to this data and then we'll refit it, okay? And you'll see we won't get exactly the same answer back. We will get the best possible answer but it won't be exact anymore because there will be uncertainty in the data. Okay, so now what I'm going to do is I'm going to take the same function and I'm going to add some random numbers to it, okay? So here's how I do that. I'm going to say plus outside the bracket plus random, real and then bracket and curl is minus 1 to 1, okay? Now what does this do? What this is going to do is it's going to generate random numbers on the interval minus 1 to 1 and it's just going to add them to my function y, okay? So there you have it and now if you just do a list plot of my data what you see is it looks more or less like a straight line but notice it's got some jiggles in it and that's because I added random numbers in, okay? So this is the same line as before except now it has some random noise in it, okay? And now what we're going to do is we're going to fit that with exactly the same command and now you see I don't get exactly the same answer as this, okay? So my y intercept is actually 2.45 now and my slope is deviating slightly from the slope of 2, okay? But this is the equation for the best line that goes through those data. And now I'm going to show you quickly and we'll do it again next time because this is something that I ask you to do in the homework and I'll tell you how to do that next time. I'm going to show you how we can plot both of these, both the actual data and the line on the same plot so you can see how good or bad is to fit, okay? So first thing I'm going to do is I'm going to set this equal to a variable, data plot, okay? Reenter it. So data plot now contains the plot of my data and now what I'm going to do is I'm going to create another plot called fit plot. Well, first I have to store my fit, okay?  So fit data here is just this formula and it's a function or you know it's an assignment that I actually use with the plot command because it's not a series of data, it's a continuous function, all right? And now I'll plot it over the same range as my data which is x going from minus 1 to 5, okay? And if I enter that I get a line and now I can show you how to put them both on the same plot, there's a command called show and now I can say simultaneously plot on the same plot, my data plot and my fit plot and if you do that now you see your best fit line plotted on top of your data and you can see that in fact this line generated by a linear least squares fit goes through the data points rather well and later in the course we'll learn how to actually get and interpret some diagnostics to tell you just how good the plot is but for now we'll consider eyeballing it as a good judge of the goodness of fit, okay? So that's it for today, we'll continue with linear least squares fitting next time. So see you tomorrow.


I like I'm gonna try to do this every show and have a positive story. This is kind of a fun story So in in South Carolina, I think this is a South Carolina Farmer a farm on a plot of land is just planted the first The first GMO trees. These are trees that both grow faster produce better wood for lumber in a much shorter period of time because they grow faster and Suck in more co2 than normal trees This is an amazing product, you know, assuming it works It's this is kind of a first example in the wild of these trees being planted developed in Silicon Valley by a Silicon Valley bio tech company These are GMO trees in, you know, they're being planted I guess You know, it's a they're planning 5,000 of these trees. They're modified papalars P O P L A R S papalars And and they they they're keeping track of them. They come from two different lines of cloned and modified Trees of the purposes of research there. They have different names different each each clone gets, you know series of clones gets a different name And and part of the research is to is to follow these but This is pretty amazing the goal of the company is both to provide farmers with Better trees and and quicker ability to quicker get lumber off of the trees So a profit motive and at the same time Dramatically increase the suction of co2 out of the atmosphere This is the kind of examples that I love of trying to find ways where we can reduce co2 emissions And at the same time make money Reduce, you know, not even reduce co2 emissions Not reduce co2 emissions, but just increase the amount of co2 we suck out and Taking the sucking out of the co2 and turning it into something that we can actually make money off of in these popular trees One way in which this can be done and it's super exciting It turns out and I don't know the exact mechanism by which this work But it turns out that these trees made it from lab to being planted in the ground in record time They managed to found a loophole in the law in the regulatory environment that allowed them to go very very quickly Because generally genetically modified plants take forever to get approval for example There's a team of scientists who has genetically engineered a blight resistant chestnut tree And I believe that John Allison is involved in this project He's in in returning chestnut trees to the southeast Chestnut trees used to be the dominant tree in the southeast and I think John Allison is involved in this effort Anyway, scientists have genetically engineered a blight resistant chestnut tree They died out died out because of some parasite. Well, this is resistant to that parasite And it's a very similar method to these these other trees, but They are being held up by the regulators and they've been waiting for decisions in 2020 since 2020 Another engineered apple a bioengineered apple Grown on a small scale in Washington state Now has finally grown took several years to be approved These popular trees found I don't know some kind of Some kind of loophole in the regulations and managed somehow to to get approved very very quickly I think it's these kind of things that are super exciting. It's these kind of things that make me Optimistic about the future Science or in technology applying science and and solving human problems, which is what technology? innovation does in ingenuity does Keeps chugging ahead. It keeps moving ahead That the you know, I think we're gonna see over the next 20 years super exciting things and in Biology of plants and fruits and food and all of that We're gonna see super exciting things in human longevity in cancer treatment and heart disease and all these things that will allow us to live longer The science is moving forward and one could only really imagine If we didn't have all the regulatory burden of regulators weren't holding all this up And of course he is an example where the science is not just producing better trees But also solving or if there it is a problem at least addressing what people believe is a problem of too much CO2 Okay, just plant more of these trees and we solve that problem this related to Alex's thing There is a fascinating article about this and I know many of you will resist reading it because it's where it published But the reality is this publication publishes a lot of good stuff Now I'm gonna get a bunch of people unsubscribing and that's the New York Times So it's called living carbon is the name of the company and it was published. I think published today Yo yesterday in the New York Times And it's it lays out the environmentalists attacking this that lays out the story about One of the things they're gonna have to do with this grove of trees is protect it Because it's likely that some environmentalist organizations are gonna come in and try to uproot the trees or burn them down or destroy them Because they're so manical about anti-gmo So they're actually gonna have to protect these trees and the New York Times gives a very objective. I think a very objective presentation of the story So, yeah And and they make the point of all their hang-ups in terms of regulation, you know They also point out who you know GMO can be dangerous and so on but that's okay. You can read it You can come to your own Own conclusion They say while outright destruction of genetically engineered trees have dwindled Thanks in part to tougher enforcement of laws against acts of eco-terrorism The trees still prompt unease in the forestry and environmental worlds major organizations that certify sustainable forests ban engineered to trees from forests to get their approval Some also prohibited member companies from planting engineered trees anyway to date The only country where large numbers of genetically engineered trees are known to be implanted is Guess which country allows for massive numbers of genetically engineered trees to be planted And this is why this is why they might they might in spite of their political system overtake us one day in terms of you know economic growth and that is China China doesn't have all these hang-ups doesn't have it China doesn't have all these you know crazy Insane irrational Environmental regulations so in China they're building new cup pop plants. They're building hydroelectric dams like crazy and the planting genetically engineered trees We should be too Anyway, I thought that was a fascinating story really really cool And I hope you did too Thank you for listening or watching the Iran book show if you'd like to support the show We make it as easy as possible for you to trade with me. You get value from listening. You get value from watching Show your appreciation. You can do that by going to Iran book show comm slash support I go to patreon subscribe star locals and just making a appropriate contribution on any one of those any one of those channels also if you'd like to see the Iran book show grow Please consider sharing our content and of course subscribe Press that little bell button right down there on YouTube so that you get announcement when we go live and for you those of you Who are ready subscribers and those of you who ready supporters of the show? Thank you. I very much appreciate it


The article presents key principles and actions for developing interventions to improve health, including involving stakeholders, reviewing research evidence, drawing on theories, collecting primary data, understanding context, considering future implementation, and using iterative cycles of development with stakeholder input throughout. Researchers should consider each action's relevance to a specific intervention in a specific context. This article was authored by Ramana Nulans, Heather Cassie, Ali M Duncan, and others.


Stories and content in Weird Darkness can be disturbing for some listeners and is intended for mature audiences only. Parental discretion is strongly advised. Pandemics and Plagues They've come and gone ever since humanity rolled out of bed and placed its feet onto the earth for the first time. Fortunately, thanks to a greater knowledge of disease, scientists and medical professionals have urged our world leaders to initiate changes to help cut the crisis short. Don't leave your home unless you absolutely have to. Wash your hands frequently. Only gather with others online, which would have been hard to do during the Black Plague when the internet hadn't been invented yet. Keep events like weddings as small as possible. In past moments of public health crisis, though, we weren't always so knowledgeable. Perhaps the most unusual response was a ritual folk remedy in which the living were married on a field of the dead in front of their entire village. Black weddings were the joining of two people previously unwed but done in a graveyard. Nowadays this might be done because the couple live and love the goth lifestyle, or they admire the beauty of old cemeteries, or they want to make sure as few relatives come to their wedding as possible and this is a surefire way to do it. But the betrothed for Black weddings were often poor, orphaned, disabled or some combination of the three. In fact, sometimes the bride and the groom didn't even know each other before arriving at the cemetery to take their vows. The hope was that the communal gathering and assembly of all their kindness and love combined with these blind date graveside nuptials would somehow stop the diseases cold. It was believed joining a couple in the presence of the dead allowed for a more direct appeal to be made to God, hoping for him to intervene. I'm Darren Marlar and this is Weird Darkness. Welcome weirdos, this is Weird Darkness. Here you'll find stories of the paranormal, supernatural, legends, lore, crime, conspiracy, mysterious, macabre, unsolved and unexplained. While you're listening you might want to check out the Weird Darkness website. At WeirdDarkness.com you can find paranormal and horror audiobooks I've narrated, streaming video of horror hosts and classic horror movies. Plus you can visit the Hope in the Darkness page if you're struggling with depression, anxiety or thoughts of suicide. You can find all of that and more at WeirdDarkness.com. Coming up in this episode of Weird Darkness. Imagine a young mother coming home late one night after a trip to the ER for a minor illness. The next day she's missing. That is just the beginning of the strange and unsettling case of Charlotte Polis. There was a time when people would get married in cemeteries. Not because they were goth or because they loved the macabre surroundings, but because they felt it would keep them from getting sick. The five-pointed star is ubiquitous. We learned to draw it in grade school. Teachers draw it on our test papers to show we've done a good job. But turn the star upside down and suddenly it becomes mysterious and strange. It becomes a pentagram. Why does the pentagram hold so much power over some people? And what are the secrets it holds? It's a supernatural creature that appears at night to travelers. If you see a white one, it is there to protect you from harm and help you along your journey.  We'll look at the Central American cryptid, known as the Cadejo.  In some ways, the wedding of Harry Fleckman and Dora Wiseman in November of 1918 was traditional. The elaborate ceremony in Winnipeg, Canada had been a month in the making. It featured music, scripture readings, and two rabbis as officians. But despite the familiar customs, it would have been hard for guests to forget why they were there. The solemn grave markers, the sounds of a nearby funeral, and the ever-present specter of Spanish influenza were all reminders that the ceremony was not a typical wedding. The ritual was part of a decades-long tradition that was more about preventing illness than celebrating a holy union. Various religions throughout history have responded to pandemics by praying to or trying to appease a higher power. During the Black Plague, the Christian Brotherhood of the Flagellants marched through Europe, whipping themselves with scourges to earn God's mercy. Muslims reacted to the same pandemic by giving greater importance to communal forms of prayer, like processions and mass funerals. In some Eastern European Jewish communities, one plague-fighting ritual that took root was the Graveside Wedding, which came to be known as the Plague Wedding. Plague weddings, also called Black Weddings, likely originated during the cholera outbreaks that ravaged Europe throughout the 19th century. The thinking behind a plague wedding was that holding a sacred ceremony among the dead would make the participants and witnesses more likely candidates for divine intervention, as, in the Jewish tradition, weddings bring people closer to God. Even accessories associated with the ceremony were believed to hold spiritual properties.  For plague weddings, the bride and groom exchanged vows in a cemetery because being surrounded by death was thought to make the holy ritual even more appealing to God. There is no textual basis for this obscure practice, though, so it was likely interpreted many ways. An alternative explanation is that seeing what should have been a joyful ceremony in such a dreadful setting would provoke pity from God, who would then show mercy by ending the pandemic. Plague weddings were also notable for who was getting wed. According to Itzik Goetzman, a folklorist at the University of Texas at Austin, the community arranged marriages between people who were difficult to marry off, which usually meant they were poor, orphaned or disabled. The organizers may have viewed this as an act of charity, thus boosting their favor with God, but such matches, which were often between two total strangers, tended to be dehumanizing. These marginalized people were typically viewed as property of the community, and thus didn't have much say in whether they wanted to be props in the ritual. Though it was then spoken of as an ancient practice, the Black Wedding was a relatively modern invention that never expanded beyond the fringes of Jewish society. When they were practiced during the cholera outbreaks of the 1860s, Jewish leaders in Eastern Europe condemned the practice and tried to suppress it. But with cholera claiming millions of lives in Russia alone throughout the 1800s, any source of security, even if it was symbolic, was hard to stamp out. The tradition could be applied to any new plague Jewish people faced. During World War I, at least one Black Wedding was held in Warsaw, Poland, to fend off typhus. There is even evidence of the ceremonies being performed to combat locust swarms in the Middle East. It wasn't until the 20th century that plague weddings landed in North America. When Jewish immigrants came to the continent, they found a new pathogenic menace in the form of Spanish influenza. The Spanish flu was one of the deadliest pandemics ever to sweep the globe. Between 1918 and 1920, a third of the world's population was infected, and 50 million people died. Many public spaces, including synagogues, closed in response to the threat. Meanwhile, some immigrant communities took the new scourge as an opportunity to revive an old superstition from Europe. The 1918 wedding between Harry Fleckman and Dora Wiseman in Winnipeg was one of a handful of Black weddings recorded in North America during this period. A report of the event in the Winnipeg Evening Tribune described the scene. The ancient Jewish Song of Life was played. On the west side of the cemetery at the same time, Jews were chanting the Whale of Death as a body was committed to the grave. That same year, two strangers were wed in Mount Hebron Cemetery in New York City. Another such wedding took place in Philadelphia around this time, when Fanny Jacobs and Harold Rosenberg were married under a chupa installed at the first line of graves in a cemetery near Cobbs Creek, Philadelphia. More than a thousand guests were in attendance. As had been the case in Europe, the Black weddings of North America continued to sow discord in Jewish communities. Following that October 1918 ceremony, the newspaper The Jewish Exponent published an editorial criticizing the practice. The wedding held in a Jewish cemetery last Sunday for the purpose of staying the ravages of the epidemic was a most deplorable exhibition of benighted superstition it read. Unfortunately, the publicity given to the occurrence will convey to many people that this is a custom sanctioned and encouraged by the Jewish religion. The people who do such things do not know what Judaism means. Plague weddings did nothing to staunch waves of disease. In fact, it is possible they helped to spread them. In some cases, all it takes is one carrier to infect a large group of people, as Typhoid Mary Mallon demonstrated when she caused a Typhoid fever outbreak at the summer house where she cooked in 1906. There are no reports connecting plague weddings to outbreaks, but similar events contributed to the Spanish flu pandemic. A 1918 Liberty Loan Parade led to thousands of Spanish flu infections in Philadelphia, the same city where a plague wedding was documented the same year. Large gatherings like weddings were known to be vectors for the virus, which prompted some cities to ban them completely. Fortunately, like a virus unable to find a host, the tradition of plague weddings appears to have faded away. Up next, imagine a young mother coming home late one night after a trip to the ER for a minor illness. The next day, she is missing. That is just the beginning of the strange and unsettling case of Charlotte Polis when Weird Darkness returns. Remember staying up late on a Friday or Saturday night, either at home or at a friend's house, and watching your local TV stations' horror host presenting a terrible B movie with aliens, monsters, ghosts, alien monster ghosts, vampires, werewolves, and all other kinds of crazy creepy characters. Those were fun nights, weren't they? That's what the Weirdo Watch Party page at WeirdDarkness.com has to offer, all day, every day. Thanks to our friends at the Monster Channel, you can visit WeirdDarkness.com slash Watch Party right after listening to this episode and immediately be entertained by a horror host and horrible movie. Or should I say horror-ribble movie? And not only can you watch the B movies and horror hosts streaming there 24-7, but once a month, we all gather together to watch a movie and talk about it in the chat room on that same page. Get your frights and funnies on the Weirdo Watch Party page at WeirdDarkness.com. Imagine a young mother coming home late one night after a trip to the ER for a minor illness. The next day, she is missing. That's just the beginning of the strange and unsettling things that would happen, in this case, over the next several years. Sadly, 26 years later, there is still no resolution. Charlotte Polis was born on July 4, 1965. She grew up in a large, very close family and was the oldest of six children. Charlotte was a very loving and caring person with a huge heart who loved to bake and do crafts. She met Paul Polis through a friend around 1985. They would eventually marry and have a daughter Layla and a son, Aiden. Although Charlotte's family was not thrilled with Paul as he seemed to be the controlling type, they had no reason to think anything was really wrong. That is until March of 1994. Charlotte was feeling ill on March 11. She'd previously made plans to go out that Saturday night and celebrate her brother Ali's birthday. However, she phoned Ali's wife earlier in the day and told her that she wasn't feeling well and would reschedule. Charlotte continued to feel poorly throughout the day and was suffering from some dizziness as well. The decision was made that night to take her to the ER. Paul arranged for his parents to watch the children as he drove Charlotte to the ER. She was diagnosed with an inner ear infection, given some medicine and sent home. She was also given a prescription to be filled. After arriving home late and speaking to her mother by phone around 1.30 am, Charlotte went to bed. This was the last time her mother would ever speak to her. At around 8.30 the next morning, Charlotte's mother tried to phone her daughter to check on her. They were normally in constant contact and were very close as Charlotte was a big help to her parents as well as they would help her with her young children. Paul answered the phone and said that Charlotte was still asleep.  But that phone call would never come. After hearing nothing from her daughter, later that morning Charlotte's mother called the house repeatedly. There was no answer.  There was still no answer which meant Paul wasn't home either. Charlotte's aunt even went to the Polis' house that day to try to contact Charlotte and knocked on the door. But the house appeared empty. Then around 7.30 pm, Paul called the Nagy family. He said he was concerned because Charlotte was gone. Paul claims that when he woke up on March 12, 1994, Charlotte was in bed. He said he spoke to her and told her to go back to sleep and he had some errands to run and would take the children with him. He claims he was gone from about 11 am to 4 pm with the children. However, Charlotte's mother said she had phoned the house before 11 am when Paul claims he would have still been home and there was no answer. He stated he made many stops, a scrap yard, laundromat, pharmacy, fast food restaurant and drove around town looking at houses to buy. When he arrived home, he said he noticed Charlotte was gone. The couple only had one vehicle and Paul had been using it all day but her person medicine was still there. The only thing that was missing was Charlotte. He did eventually phone Charlotte's family that night to see where she was. Ollie recalls the message from Paul seemed odd and he sounded under the influence of something. He found it strange that Paul would be so concerned with Charlotte only being missing a few hours. They were all immediately concerned and went to the Paulus' house. When they arrived Paul's parents were already there. The neighbors even told Charlotte's parents that Paul's parents car had been there all night prior. The Nagy family noticed that there was some extreme cleaning being done at the home that night which was concerning to them. Paul's father who was at the time the deputy coroner of Trumbull County by Ollie's recollection seemed to be instructing his son what to say and what not to say. As the family looked around the property searching for clues, Charlotte's sister noticed a little shed on the Paulus' property. There had been a small amount of snow on the ground and she noticed two sets of footprints that appeared to be dress shoes and boots leading to the shed. Charlotte's sister also said the doors were bulging on the shed and it was locked with a padlock. With the Nagy family asked Paul for a key to see what was inside, he refused. Ollie said the next day his brother returned and chopped the padlock off with an axe and it appeared everything had inexplicably been moved to one side and the doors were no longer bulging. The police were called and Paul's father seemed to be in charge of things by what the Nagy's saw. As it started to sink in that their loved one was actually missing, the Nagy family started to get very suspicious towards Paul. Ollie recalls that about a month before Charlotte went missing, his sister phoned him one night and asked them to bring Dairy Queen over. Ollie and his wife arrived and Charlotte was there with her children, but Paul was upstairs asleep. As the three were visiting, Charlotte said something unsettling to Ollie. She asked him if something happened to her, would he raise her kids? She also wanted him to give the children her boxes of Princess House Crystal or sell them and give them the money and say the money was from their mother if she was gone. As you can imagine, Ollie was very concerned at this question. He asked Charlotte if something was wrong or if she was being abused by Paul and she downplayed it and assured him that everything was fine. Another thing concerning was Paul's list of alibis for the day that Charlotte was discovered missing. Paul listed several places where he was with his children and everyone interviewed either didn't see Paul at all or didn't see the children with him. He did appear to drop off Charlotte's prescription at the pharmacy but never picked it up. The scrapyard Paul said he went to was actually closed that day. Also, Paul went to the laundromat that morning after Charlotte went missing. The person working at the laundromat was a relative of Ollie's wife and she said the comforter Paul brought in appeared to have a large dark red stain on it. When he left the comforter in the washing machine Paul was adamant that no one be allowed near the machine. Paul left and as the worker looked out the window she said the children were not with him. The idea of Paul looking at possible houses to buy didn't make sense either. Neither Paul nor Charlotte worked and lived on welfare so the idea of them purchasing a house seemed very unlikely. A neighbor of the couples said she saw Paul the day he was supposed to be running errands.  The car was filled with boxes and bags and the children were not inside the vehicle. Also, according to Ollie, soon after the disappearance, Paul Polis had a yard sale and sold all of Charlotte's belongings. As the police were investigating the disappearance they wanted Paul to come in for a polygraph exam. He used the excuse he didn't have a vehicle so one of Charlotte's brothers offered to drive him there and Paul accepted. However, when the day arrived Paul goes missing. He left behind a note talking about how difficult it was being suspected and the way he was being treated. Looking back now Ollie did read the note and said it appeared to almost be a confession note of someone who had committed a crime. Charlotte's brother Ollie worked as a Trumbull County reserve deputy and then when his sister went missing transitioned over, volunteering his time for the next 26 years with the Gerard police department. He worked hand in hand on his sister's case with the detectives to try to get justice for her. He's still working the case even today despite health issues. No one knows for a fact where or what Paul was doing those three months he disappeared. Ollie's opinion is he was being kept at his attorney's house the entire time out of sight. When Paul arrived back he was arrested and charged with obstructing official business. However the judge dismissed the charges saying that he had a Fifth Amendment right to remain silent. He also said Paul had cooperated by speaking with them and initially allowing a search of his home. Weeks after the disappearance Charlotte's family was alerted by a relative late at night that happened to be driving by that the police were at the police's house. The Gerard County PD was also contacted and the family discovered that the Ohio BCI Bureau of Criminal Investigation were there spraying luminol looking for any presence of blood in a house in the car. There was a spawn of blood found in the trunk of the couple's vehicle. It was discovered to not be animal but before the sample could be further tested it was accidentally destroyed. There was also blood found in the hallway of the police's house. Paul's mother insisted it was menstrual so it wasn't tested. He learned that Paul's father was telling the police where they could and couldn't spray. Why was Paul's family allowed to be there during the investigation while it seems Charlotte's family was not even made aware of the search? Ollie believes that Charlotte was possibly being poisoned in the weeks before her disappearance. He said that Paul would have Charlotte drink milk at bedtime. Charlotte was never much of a milk drinker in the weeks before her disappearance. She did start complaining of feeling ill and tired all the time. Ollie said she was to see the doctor the week after she disappeared to try to get some answers. Of course she never made it to that appointment. Ollie also recalls that a few weeks after Charlotte disappeared there was a 9 foot by 13 foot area of concrete floor dug up at Paul Polis' parents' garage. He said they refused to explain why the area was there, citing it was their own personal business and soon after had the area filled in with more concrete. To his knowledge this area has never been searched. There were many concerning things that happened as far as the police investigation into Charlotte's disappearance. Ollie said after Charlotte went missing Paul had scratches from his elbow to his wrist. Paul claimed he got them installing a hot water tank, but Ollie said there was no recent installation of one at the house. Paul then said he got them installing drywall at the house, but Ollie said that wasn't true either because he had been the one to help Paul install the drywall. Police took photos of the scratches on Paul's arms, but not long after the photos disappeared. About a month later when the Nagi family went back to the Polis' house to grab some things for Leila who was staying with them, Ollie noticed some disturbing things and took photos of them. He saw what happened to be blood splatter on the kitchen ceiling, master bedroom, pillow on the couch, the curtains on the back door and on a blanket in their hutch. Ollie gave a copy of the pictures to the police for their investigation. The copy of the police's pictures go missing soon after. Being focused on his sister's disappearance, Ollie mistakenly then gives the original photos to the Gerard police. Those photos have now gone missing as well. Another very surprising thing they discovered at the Polis' home after Charlotte went missing was found in the bedside table of the couple's bed. Divorce papers that Charlotte had drawn up. This seemed to even surprise most of the people in Charlotte's family. Apparently the situation in the Polis' house was more serious than anyone had realized. The family then wondered if Paul had seen the papers. In the years after, there seemed to be no real advancement or leads in the case. When Charlotte went missing, her daughter stayed with the Nagis while the Polis' raised the couple's son. The daughter appeared to have witnessed something unsettling around the time her mother went missing.  She told the Nagis as well as the police and many others that grandpa helped daddy hurt mommy and put her in trash bags. In 1995 it was reported in the media that Leila Polis had went missing. She even had a Charlie project missing persons page. Ollie did explain that he got his parents and Leila out of the area to protect her. Charlotte's father was determined not to let what happened to his daughter happen to his granddaughter. Paul was trying to get custody of Leila and the Nagis had custody so before they could be served with those papers, they fled the area. They made the sacrifice to hide out to protect their grandchild and returned several years later. Paul Polis' parents had been raising the couple's son Aiden since Charlotte's disappearance. In Ollie's quest to find out what happened to his sister, he tried everything he could think of to get Charlotte's case more attention. He contacted all the major talk shows and news programs in hopes that they would air the case, but no luck. Then he decided to contact his congressman, Jim Trafficant, and ask for help on the matter. Representative Trafficant was so amazing in helping with the case. He dedicated his time and resources to it and wrote a letter to the TV show Unsolved Mysteries, asking them to air Charlotte's case, which they did. Although Representative Trafficant has since passed away, Ollie is still grateful for all the help he gave to bring attention to his sister's case. Then around 1999 Ollie got a really strange call. It was from an anonymous person who phoned him about 10pm one night, directing him to check the dumpster near the Trumbull County Police Department before midnight when the trash would be picked up. Ollie brought a friend with him and hurried over and was completely shocked at what he found. Charlotte Polis' entire police file was thrown in the trash along with numerous others. Ollie was livid as him and his friend retrieved all the files from the trash. He also immediately contacted the media to let them know the outrageous thing that had been done. He was eventually told a story that it was just a mistake made by a janitor working in the building, which Ollie found pretty unbelievable. Angered at Ollie bringing the media into the matter, the PD also threatened Ollie with theft if he didn't return his sister's files, so he did comply. In the years since his wife's disappearance, Paul Polis has certainly not led a quiet life. He has been in trouble with the law many times from felony drug possession to pulling a gun on a chief of police to embezzlement of over a million dollars along with his then girlfriend. There were also a few people that Paul was connected to who died under strange circumstances and there is speculation as to what if any connection Paul may have even had to their deaths. I'm sure you can imagine what my theory is on this case. Paul Polis killed his wife and with the help of his family, disposed of her body. Paul was known to have affairs, so it's unclear whether he just wanted to get out of his marriage without giving up his children or killed her in a violent rage. He has a well documented history of substance abuse. Perhaps Charlotte even threatened to leave or he came across the divorce papers. It seems like the most unlikely scenario on the planet that Charlotte would be abducted or leave the house in her pajamas, feeling very poorly with no purse or vehicle. Sadly, this case appears to be like so many ice cold. Ollie Nagy is still determined to find answers and find his sister, but the leads seem to be few and far between at this point. He has no doubt what happened to his sister and Paul Polis, even with a very lengthy criminal history, is still free. After so many years and so few leads, sometimes the public's interest can wane. As Ollie was quoted in an article, it's like no one really cares, but we care. There's always hope in any case that the family will get answers and that Charlotte can be restored to her family. Until that day happens in this frustrating case, Paul Polis appears to have made a clean getaway. When Weird Darkness returns, the upside-down five-pointed star, why does the pentagram hold so much power over some people? Plus, a supernatural creature appears at night to travelers in Central America.  These stories are up next. Every month in the Find Bigfoot calendar by Timothy Wayne Williams. Each month, you'll be captivated by an original Timothy Wayne Williams painting. Beautiful and captivating, but within each painting hides a monster. Bigfoot is hiding somewhere in each painting. Search for Bigfoot and invite others to do so as well with the new Find Bigfoot calendar, available now at WeirdDarkness.com slash Bigfoot. That's WeirdDarkness.com slash Bigfoot. TC Lethbridge once said, What is magic today will be science tomorrow. He was an English archaeologist, parapsychologist and explorer, and also a controversial figure in British archaeology. His significant research is related to a certain magical pentagram, a five-pointed star and its secret use. When he died in a nursing home in 1971, his name was generally unknown. Yet, even more strange, is that the last thing he was thinking about before he died was, again, a mysterious pentagram known and used by cultures throughout the world for thousands of years as a protective symbol with the power to banish evil spirits. Today, those who admire Tom Lethbridge and his contribution to paranormal research know that he is the most prominent name in the history of psychical research covering subjects like life after death, dowsing, poltergeists, ghosts, second sight, the nature of time, and precognition phenomenon. His ideas were described in a series of books and published toward the end of his life. Curious, but Lethbridge has never been particularly interested in psychic phenomena until he came to the crucial point at a later time in his life and began to take a serious interest in the subject. Disappointed with the hostile reception of one of his archaeological books and his job as the keeper of Anglo-Saxon antiquities, Lethbridge left Cambridge and retired to Holhouse, a Tudor mansion on the south coast of Devon in southern Britain.  He came in contact with his neighbor, an old white-haired witch who lived next door and possessed a few extraordinary powers. One day, the old witch explained how she managed to put off unwanted visitors by drawing a pentagram in her head and then visualize it across the path of the unwelcome visitor or on the front gate, for example. Her secret use of the magical pentagram introduced psychical researcher Tom Lethbridge to the world of the paranormal. In the beginning, Lethbridge was very skeptical until something extraordinary happened that finally convinced him for the rest of his life.  A few nights later, Mona woke up with a strange feeling that there was someone else in the room standing at the foot of the bed, but she could only distinguish a faint glow of light which slowly faded away, leaving the bedroom in the darkness again. The next day, they both met their neighbor who asked if someone had been putting protection on them. She explained that she came to their bedroom on another night and couldn't get near the bed because there were triangles of fire around it. Three years later, the old lady died in rather peculiar circumstances, and her death resulted indirectly, of course, in one of Tom Lethbridge's most significant insights in the realm of the occult. One day, passing the cottage of the witch, he experienced a horrid feeling of suffocating depression, and his scientific curiosity pushed him further to investigate this strange, nasty feeling. He walked around the cottage, and all of a sudden, he discovered that he could step right into the depression and then out of it again, just as if it was some kind of invisible wall. This disturbing and inexplicable incident made Lethbridge convince that he must look for other clues. Another strange incident occurred about a year after the death of the old witch. On a wet January afternoon, Tom and his wife, Mina, drove down to the beach to gather seaweed. Suddenly, a blanket of fog descended upon them. It was a ledrom bay, Devon, that Lethbridge experienced the blanket of fear and gloom. The next day, he mentioned what had happened to Mina's brother, and from him, he heard about a similar incident that took place in a field near Avebury in Wiltshire. A week later, Lethbridge and his wife set out for Ladrom Bay once again. They stepped onto the beach, and both walked into the same bank of depression or ghoul, as Lethbridge called it. The feeling was intense, unpleasant, and made them both dizzy. They found the place frightening and sinister and not only for them. Nine years later, a man committed suicide there, and Tom Lethbridge was wondering what could make people feel so bad in this particular place. What was this intense, bad feeling that imprinted itself in the area? Have feelings of despair or perhaps even those evil ones been recorded there? 1. Tom Lethbridge was convinced that the key to the puzzle lay in the water. He knew that underground water produces changes in the earth's magnetic field. Suppose the magnetic field of running water can record strong emotions, which as we know are electrical activities in the human brain and body, and such fields could well be most energetic in damp areas and during foggy weather. 2. Lethbridge was a keen and accomplished dowser, and the pendulum was the key to his interest in the unknown. He had known for years that a pendulum could be used for divining and its accuracy could convey a lot of extremely complex information. He was confident that a pendulum responds to the mind, not only to some vibration. Human beings possess powers we are not even aware of, because there are powers of an unconscious mind which go far beyond that what we understand. In his book The Power of the Pendulum, which I have linked to in the show notes, Tom Lethbridge wrote about the unknown realm of our mind, the super-conscious. It, the super-conscious, knows far more than we do because it does not have to use the brain to filter out everything. It lives in a timeless zone. All of which may be true and probably is, but is also incomprehensible to us. What Lethbridge tries to say to us is that everyone has experienced moods of unusual vitality, sudden ecstasy of excitement. Having memories of such moments, as well as our power to recreate them, we are equipped enough to research the unknown realms of our mind. The year was 2012. In the Zona Rosa section of San Salvador, the capital city of the Central American nation of El Salvador, an urban revival of sorts was taking place. In a place known for its museums, cafes and garden city feel, new businesses were moving in and living spaces were being created out of old buildings to accommodate the artists, young urban hipsters, and those Central American forward thinkers who wanted to experience something new. Amid the fusion sushi places and new cosmopolitan discotecs, a brew pub emerged, inspired by the relatively new craft beer movement in the United States. The logo of the new brewery featured a red-eyed, snarling dog-like creature. Anyone who came to the new business knew why the brewery chose that image. The name of the brew pub and the name of the animal were the same. El Cadejo This Central American business may be the first microbrewery in the world to have itself named after a cryptid or legendary creature. The stories of the Cadejo range from the country of Dicaragua through El Salvador, Honduras, and Guatemala, all the way to the state of Chiapas in southern Mexico. The specifics of this creature and the stories about it are varied across the many countries and terrains of its supposed habitat. Most cryptozoologists or those who seek to discover and describe previously unknown animals dismiss this creature and believe that the Cadejo is merely the stuff of legend. Many Mexicans and Central Americans who have actually seen the Cadejo would argue this point. The Cadejo has been described in several ways. It is usually a big dog-like creature with hooves of a deer and abundant fur. It runs more like a deer than a dog. It has been spotted in both rural areas and highly urbanized settings. There is a black version and a white version. The black version can be broken down into three different types which often cause the cryptozoologists to dismiss the Cadejo out of hand. The first kind of black Cadejo is the most supernatural of the three darker ones. It is often described as a pure manifestation of evil and may even be the devil himself, taking it earthly form to conduct his nefarious business on earth.  It is often described as having a thick, glowing metal chain around its neck, which some language scholars believe gives the creature its name. In Spanish the word chain is cadena and from this we get the word Cadejo. This type of Cadejo never attacks a person but appears to terrorize the victim by its mere appearance or may present the person with visions of hell or other scenes of death and destruction. In one account of this type of creature, a teenage boy named Santiago encountered a Cadejo that showed him a beautiful tree being consumed by hell fire. Santiago believed that the vision was a sign to help him to stop drinking and to serve as a warning for him to stop disobeying his mom. This first type of creature while not threatening physically is usually a bad omen and may predict a horrible event in the person's life. Usually the witness is left with feelings of post-traumatic stress and they are reluctant to speak of their experiences. Sometimes prayer or the showing of religious objects such as a cross may cause the creature to slowly back off and disappear. Another precaution to take when coming across this type of Cadejo is to stand with your feet together so that the creature doesn't run under your legs to whisk you off. The second version of the black type of the Cadejo is the most troublesome of all versions. It is more like a wild dog or wolf than anything else. When a person comes upon this manifestation, it may mean a fight to the death. The creature may appear first in the shadows, making noises to alert the victim of its presence for reasons of pure terror and to generate a feeling of demoralization in the victim. When the Cadejo senses an intense level of fear, it will go in for the kill.  The only thing that can save a victim from certain death is the intervention of the white type of Cadejo to be discussed later. The third type of black Cadejo is the least powerful of all the manifestations. It is said to be a combination of the second lethal version and a real dog, because it is a mortal hybrid it can be killed, although it is difficult to do so. This type of creature surprisingly does not bite its victim, rather it kicks and pecks at the person with its snout. While causing little physical harm, this harassing interaction with the Cadejo can make the human go mad. If this third version is killed, it is said that its body rots very quickly and disappears within minutes. On the ground where it dies, according to legend, nothing will ever grow as it has left a stain of evil. Religious objects or prayer can ward off this creature. One who encounters this third type can also try another trick. The potential victim can spit in his own hand and offer it up to the Cadejo. If the creature licks the palm of the potential victim, it is assigned that everything will be okay and the human does not need to fear the Cadejo. In addition to the three kinds of black Cadejos, there exists a white one. The white one is benevolent and is described as having fluffy, downy-like fur and light blue eyes. This big canine is more of a protector and appears when needed, usually at night, often to guide a person out of trouble. It is also the only thing that can protect a person from the second and most dangerous type of black Cadejo. It is a human's only real defense against the most malevolent black form of this type of creature.  Instead, it eats the small bell-like flowers that grow in the mountains of southern Mexico and throughout Central America. Some believe that the white Cadejo is not an animal in the strictest sense, but more of a benevolent spirit that shows up to help in times of distress. In a popular story about the white Cadejo, a man named Juan repeatedly returned home at midnight from a long day at work to see a huge white dog hanging around his house. Juan had a wife and small children and sometimes was concerned for the well-being of his family because he spent so much time away. Whenever Juan would see the dog and try to get close to it, it would shake, wander off a bit and then disappear. One day Juan tried following the dog and when he got closer to it, he touched its paw and the big white dog opened its eyes and began talking. Juan was frightened. The dog said, I am leaving. You don't need my help anymore. Juan asked, What help? And the dog replied, I was sent from heaven to protect you and your family, but you have showed me that you no longer need my protection. The white dog then just closed his eyes and died, at which point Juan buried him. To date, no bones or other physical evidence has come to light to prove the existence of the Cadejo as a real, living, breathing being. So investigators of strange creatures usually dismiss the Cadejo as just a legend. Legends are based on something, as is often repeated, so what would explain this phenomenon? Critics often cite the many instances of the Cadejo being seen while a person is intoxicated and dismiss the whole idea of the creature as coming from an impaired mental state, combined with stories previously heard by the witness or experiencer. Investigators are still left with the stories previously heard. Where did those stories come from? It appears as if the whole Cadejo legend is a blend of the pre-Columbian and the Hispanic European belief systems. The indigenous of the area believed that people were connected to a spirit animal and that the animal was also a protector. In the white Cadejo we see this, along with the Catholic concept of the guardian angel. This creature is thus a blend and serves to bridge two cultures. The Black Cadejo counterpart also may have its pre-Spanish origins in the Nagwall, a starling, upright dog-like creature present in nearly all of the Mesoamerican cultures, from the Maya to the Zapotecs to the Aztecs. The combination of the Nagwall with European Christian devil imagery is very apparent. The malevolent Black Cadejo has hooves, its smells of sulfur, and may threaten its victims with images of a very Catholic hell. Like most stories about the devil coming to earth, the stories surrounding the Cadejo are cautionary tales.  Don't stay out too late. Don't wander away too far from home. And don't disobey your parents. From this perspective, the Cadejo is a very interesting study in social control and serves to keep certain members of the believing population in line. If you made it this far, welcome to the Weirdo Family. Please share a link to this episode in your social media to help spread the word about the podcast. And if you could, please recommend Weird Darkness to your friends, family, and co-workers who love the paranormal, horror stories, or true crime. Maybe they'll become a Weirdo Family member too. Do you have a dark tale to tell of your own? Fact or fiction, click on Tell Your Story at WeirdDarkness.com and I might use it in a future episode. All stories in Weird Darkness are purported to be true and you can find source links or links to the authors in the show notes. The disappearance of Charlotte Polis is from Lost and Found blogs. Plague weddings is by Michelle Debsak for Mental Floss. Secrets of the pentagram is by A Sutherland for Message to Eagle. And the Cadejo is by Robert Bito from Mexico Unexplained. Weird Darkness theme by AlibiMusic. And now that we're coming out of the dark, I'll leave you with a little light.  And a final thought from Paul Bose. Forgiveness does not change the past, but it does enlarge the future. I'm Darren Marlar. Thanks for joining me in the Weird Darkness. The Thinking Behind a Blague Wedding. We're just making stuff up now today. The Thinking Behind a Plague Wedding. But the leads seem to be few and far between at this point. He has no doubt that he has no doubt what happened to his sister. The third type of black Cadejo says back Cadejo. There's also a front Cadejo, and a right side, and a left side Cadejo. And there's one behind you and above you. This episode of Weird Darkness is brought to you by the number one and the letter K.