Hello friends, I am Sanjay Gupta. I welcome you on Sanjay Gupta Tech School. In this video, I am going to explain how you can print the words of a stream. So I will be explaining this concept with the help of two different methods. So before starting of the explanation, I want to share one information. If you go to detail or description of this video, you will find set of links related to playlists. Those are available on my channel and those playlists are related to C programming videos. So you can follow them if you want to watch both programming related videos. So now I'm going to explain this method number one first. So let's say we have one stream and the content is India. So reverse means we need to print this A, I, B, then M and then I. So this should be the output. So I am taking this as STR. So starting the program. So I am declaring a character array. Let's say it's size is 10. Now I'm going to take two variables, I and L, then print F, enter a string, then I'm going to read it through GATES. So STR, right? So this way I have received the input from the user. Now I'm going to first calculate how many characters are available in this stream. Because I need to print A first as output, then I, then B and so on. And in case of a string, we don't know how many characters will be entered by the user. But we know after input, the last character will be having null character after that last character. So we just need to identify where null character is available so that we can identify how many characters are there. So we can do counting in two ways. Either we can iterate a loop or we can use a predefined function that is STR array. So as of now I am going to use that predefined function. So this predefined function is available in string.h, string.h header file. So you need to include this header file, the whole main function. So I think this is very easy to write yourself. So this L will be having a total length. So in this case, we have total five characters. So length will be five. So we have total number of characters available that are five. So now you can see we have value L as five. So we can start loop from four. So here I am writing I equals to L minus one. I need to repeat a loop till zero and I minus minus. And here I can print for string C STR of I. So here you can see length is five. So five minus one will be four. So I will be initialized with four. Four is greater than equals to zero. That is true. So fourth index value will be printed on output screen and see it carefully. I am using percent C here. So percent C always print a single character on output screen. So it will print A then I will be decreased. So its value will be three. Three is greater than equals to zero. So STR of three is having I so it will be printed. Then again I will be decreased. So it will be two. So D will be printed. Then I will be one. So N will be printed. Then I will be zero. Again, this condition is true. So I will be printed and then I will be decreased to minus one. That time this condition will be false and the output will be in print. So this is the first method that I explained you in that case. We received as input a string in STR and then we printed its reverse. So here you can see we are printing that string in reverse order, but actually it is not having the contained in reverse on. So my method number two will tell you how you can reverse the locations of these characters within the string and then how we can print that string without any loop in reverse order. Right. So I'm using these. So for method number two, I will implement a logic to swap characters. So I will be wrapping these two positions. So I will be swapping with A and A will be swapping with I. So it will happen. And I will come through. Right. And this way I will swap these two and the middle one will not be so if we have odd numbers, if we have odd locations, so one swapping will not be done, but if we have even positions like if we have six characters, then all three will be swapped with other three. So this way it will be reversed. So this will be the output. Right. So India is reversed here and we can just print this SDR so that we can have output. So we can print this string without any loop. So now I will tell you how to swap the elements within the string. So I'm writing initial values. So this is the actual string that we need to reverse. So here you can see I have identified the length. So length is five. Now I'm going to start loop from zero. And here I'm using J as L minus one. And I'm declaring one more variable as J. And then I'm putting up 75. So here you can see I'm putting common between. So I started from zero and J is having L minus one. So J is having value as four. Right. And this is the loop condition. Then I plus plus or J minus one. So this is the loop. Then inside this loop, I'm going to apply the logic for stopping the characters. So I'm beginning one more character variable P. So P equals to SDR of I then SDR of I equals to SDR of J and SDR of J equals to T. So this way the characters will be swapped so that we can have reverse of string. So starting from the beginning. So I is below and J is four because of this. L is five, length is five. So five minus one. So J is four. Now check this condition. I less than J. It is two. Now the swapping will be performed. So A will be here. And I will move to here. Right. So in first rotation of this loop two characters are swapped. Right. Now I will be incremented. So I will become one and J will be incremented. So it will be three. So again, check this condition. I less than J. It is again true. So one index and three index will be swapped. So one is having N and P is having IXY will be here and N will move here. So two rotations are completed of this. Then I will be incremented to two and J will be incremented to two. Check this condition. Two less than two. It is false. So no further rotations will be taken place. So this loop will be terminated and you can see string is reversed with this loop. And after completion of this loop. You can print. Reverse equals two percent S and STI. So without any loop, you can directly print that string. So the actual string is being reversed through this logic. And we are printing that one. So the original order of that string is gone and the reverse one is available. So you can use that. So this was method number. So in some scenarios, you just need to print the string in reverse order. So in that case, you can apply method one. In some scenarios, you need to reverse the actual string in memory location. So in that case, you can do method number. So I hope you understood how we reversed or how we printed the words of a string that is provided by the user with these two methods in C language. And I hope you explain. Sorry. I hope you understood whatever I explained in this video. And if you want to watch more programming related videos, you can open my channel and go to playlist and see various variety of videos. Also, you can go to a detailed or description of this video where you will find links of various playlist related to C language on my channel. So thank you for watching this video. Thank you.


Alert. From the SCP Foundation. As of yesterday morning, a significant reality-altering event has occurred, resulting in a massive relocation of matter. The majority of the landmass and population of the United States of America has vanished, as well as parts of Canada. The area marked in red is expected to be affected by severe storms and flooding as air and water rush to fill the vacuum created. The SCP Foundation is working alongside the Manacharitable Foundation to provide disaster relief to the affected areas. The SCP Foundation is also working alongside the Global Occult Coalition to attempt to recover the displaced population. Marshall, Carter and Dark have generously provided resources and paid for the majority's recovery and relief efforts. The SCP Foundation will continue to secure, contain and protect the people of the Americas will be defended and recovered.


A couple months ago, MrBeast's best friend Chris Tyson came out as non-binary. Now the internet proceeded to explode. People were concerned that this would be the downfall of one of YouTube's greatest creators or at least most popular creators because Chris was proving to be such an eyesore in what should have been family-friendly content. People could see a shift in Chris's appearance and it only became more evident when Chris came out as non-binary. Now we covered this on my channel because it had implications to how we understand gender in light of God's design. While Chris has come back again with another interview, what could be seen as a tell-all on Anthony Padilla's YouTube channel. So today we're going to respond and react to this interview, at least important clips of this video because as Christians we need to know how to respond to the zeitgeist of our day. People deciding that they can change their gender all of a sudden based on how they feel. Is this something that we can support as Christians on how do we lovingly and compassionately yet truthfully talk to people in our lives that are struggling with gender dysphoria. These are all important questions but without further ado, let's jump in to the video. Before we do that though, just a huge shout out to everyone on Patreon who supports what I do on this YouTube channel and equipping people to follow Jesus Daily. If that's a mission that you want to get behind, head to the link in my description and sign up today. You get access to our Discord. We do video chats and there's exclusive videos on there for you as well. Now back onto the video. Hello Chris. Hello Anthony. I feel like the earliest videos you guys made really were capturing that friendship. Yeah, it was just us just doing whatever we could think of it because there was no rules back then. There was no like algorithm or anything, it was just whatever you could film put it on YouTube. Yeah, I feel like about how many maybe six months to a year ago I started to see a change in your appearance. Yes. You started to look a little bit less masculine, a little bit less beardy. At the exact same time you look like you were glowing, like you were more excited, like your posture was better. Like you were like excited about life. They're trying to paint this as a very joyous, happy thing. But even as you're watching that, you can see the stark difference in contrast between how he used to look and how he's trying to show himself to be. He talks about how he's on hormone replacement therapy now. We talked about in the previous video that I made about Chris, how he has a son. I believe it's a son. And now his son is basically like literally losing his dad. You think about how important father-son relationships are. I wrote a whole book about it, for goodness sakes. Chris is basically abandoning his child in pursuit of what he sees as his fullest self, his most happy self. But sin, it can lead you into a happy place for a little bit. You feel like you're my most authentic self, but until we are actually stepping into God's design, until we are stepping into who God created us to be, then we'll never truly be, find that joy, find that happiness. Because Chris is finding that right now in his temporal circumstance, in his ability to dress up as a woman and to put makeup on his face. But truly true joy and happiness is found in the unchanging God and knowing that God is for us, that he will never leave us nor forsake us. And having a right relationship with him, it won't be satisfied in these temporal things. And that's why I'm so sad for Chris, because I know this is an empty road for him, but he's choosing to do it out of his selfishness. But we've all been there. We've all been there when we're following our own sinful desires, pursuing that, that which we think will fulfill us ultimately. But sin is an empty road. It is a dead end. God did not make a mistake when he made Chris, and I know that is seen as hate speech nowadays, that is bigotry. But it's the truth. You've had a lot of people speculating on your gender, you know, just you growing out your hair.  I did because I am a woman. Oh, she has never said that before. I've never said that publicly enough, but I've been fully like confident in that decision for over a year now. This is a lie. But it really is.  But it's just like it's the ultimate gaslighting. It really is. And I'm not sure if that's necessarily the correct term, but it's him trying to convince everybody, Hey, I am a woman. I'm a woman. I if I say it, that means it's true. It's this embrace of a subjective morality, a subjective truth that I can create my own reality. I what I say is truth, but it totally disregards the fact that God has oriented the universe in a certain way and that he has the veto on truth. He has the decisive and the definitive word on what is true. Ultimately, what I want for Chris and I want to be very straight up with you, if you're watching this video and maybe you vehemently disagree with what I'm trying to say here, I want you to know that we were all in this place of rebellion against God, whether it's you want to transition your gender or your in lust or pornography or greed or hatred or whatever else it is. We were all caught up in sin, dead in our trespasses and sins.  The death we deserve to die for our sins against God, our sin, our rebellion. It says that the wages of sin is death. That's what we deserve. But Jesus took that for us so that we could have a right relationship with him, but he did not stay dead. He rose again on the third day, defeating sin and death. And what he calls us to is repentance. It's to say, God, I was wrong about this stuff. God, all these things that I sought and I tried to fill me. These were wrong. And I turned from these things and I put my faith in you as my Savior, as my Lord.  He gives you a new heart with new desires. It says that he makes you into a new creation. And when we talk so much about identity here, Chris is trying to find his identity, a fulfilling identity. He talks about, oh, I just want to be myself, my true authentic self. Well, our most truest and authentic selves are when we are stepping into who God created us to me. And we can only do that through Jesus. And he makes us into a new creation where our identity is not in the things of this world, is not in our sexuality, or whatever kind of proclivity or desire that we have. It's not in our sin anymore. It's in him. There was a voice in the back of my head that just hated me. Everything I did hated me. And it wasn't until I started accepting myself and being kind to others and just listening to what my body wanted, that's when that voice started going away, started taking HRT. Haven't heard from that voice in a very long time. Where do you think that part of you has gone? I think it's, I hope it's locked away. And truly the only one who can silence that voice of guilt and shame is Jesus. And when we are not in Christ, when we do not have a relationship with God, that voice of guilt and shame, that's legitimate. Because we are rebellion against God. We are guilty before him. We are shameful before his sight. But yet we still have that image of God in us that we still have dignity and worth.  I've taken all this guilt and shame upon myself. And that you can be free now. You can be free of it. It was kind of weird to see people be like, oh, now Jimmy's put in between a rock and a hard place where he obviously doesn't want to support Chris, but the left would be so mad if he didn't.  I told Jimmy like seven years ago, we were all hanging out and we were having a good time. I had just been really like thinking all that day about like my gender and everything. I just blurted out to them, I don't want to be a man anymore. Just out of nowhere. Out of nowhere in front of all of them. And then started like crying. And the support I got from them in that moment was like, that's when I knew I was like, okay, like I have my friends back, everybody else. You can see how emotionally volatile these situations are. Someone that is struggling with their gender identity, somebody that feels like they should be somebody else, right? I'm a boy, but I think I should be a girl. I'm a woman, but I think I should be a man. And you understand, you can begin to sympathize if you don't experience it with that deep tension going on in their hearts, in their minds. But what you don't do and what you can't do, you can't do this as a Christian, is affirm the lie, affirm that this will make them better or make them happier or make them fulfilled. You're not disregarding of the tension and the pain that they're experiencing, right? Cause that's real. And you can't just say, oh, stop feeling that way, right? They might still feel that way. Gender dysphoria is a real thing. People experience this. But what is not helpful for them is to just say, you know what? If that's how you feel in this moment, you're right, that'll fix everything. That'll give you the identity that you've always wanted. All that tension and that, you know, dis-ease within your soul, that will be cured by this one decision. By putting on makeup, by growing your hair long, by wearing a dress that will make you happy. That's not a friend. A friend is someone who says, hey, I understand that you feel this way and I'm here for you in the midst of it, but I don't think this is gonna be best for you. I don't think this is what God wants for you. And I don't know really how to help you work through some of those emotions at this point or some of those feelings. I'm not really equipped in that, but I'm here to listen to you. I'm here to be there for you, but I can't support you taking these hormones or doing these things to your body that I just don't think are gonna be beneficial to you ultimately.  And if you can understand that, if you really do understand that I don't think this is good for you, then you can understand that as a friend I just can't support you in this. So you've changed the way you present. You've changed the name and the things that you prefer people to call. You've changed your pronouns. Yeah. Have you changed your voice? Because I know that that's something that trans people also work towards. I am working on it. So currently I'm working with the Seattle Voice Labs. It's a Discord server where they offer free lessons, they offer paid lessons, but Claire, she is incredible. She's been teaching me and she really pushed me to do her voice. Like today? Yeah. She thinks I'm ready to do a sentence, to give it a little sneak peek. Hey, Anthony, congratulations on buying Smosh. Oh, thank you. That was good. I can't help but notice Anthony's response to that voice just pure and utter laughter. Why? Because what Chris is doing is unnatural. It's like if I were to make a woman voice, people would laugh at that and I would want them to laugh at that because it's not natural. That's what comedy is. It's doing something that is catching people off guard with something that they didn't expect. When you see Chris, when you see a man and he's speaking as a woman, that's funny. And it should be. But at the same time, it's sad because this is the play. They want you to be like, this is a completely normal thing. It's a normal thing for somebody like Chris to all of a sudden try to change his voice in order to appear more feminine. To me, it's just heartbreaking. It's heartbreaking. And I know people said in the last video too that he's making these choices and he's hurting a lot of people as he's doing this. And I totally see that as well. And I feel great compassion for those people deeply. His former wife and his son, that is awful. And it's so sad. And I also feel a deep compassion for Chris too because this isn't going to satisfy him. This isn't going to lead him down the path that he thinks it will. But he's rebelling against God right now. And I know how that feels. And I want him to know Christ. I want him to see the transformation that can happen and the true identity that's found in God. That's what I want for all of you. Whether you're struggling with gender dysphoria or not for you to find your identity in God and not see these other things, these things of the world or these identities that the world is trying to push at you. They're cheap substitutes for the true and secure identity we have as children of God in his family. I think for the most part that closes the chapter on Chris Tyson and his transition. Thank you for watching this video. Thank you for being kind in the comments down below. I know there'll be people that do not agree with what I'm saying that are watching this video. If you've watched to the end, thank you for giving me this time and space to explain what I believe according to the Bible. I truly appreciate you. I love everybody that watches my channel and thank you for supporting what I'm doing as well if you support on Patreon. Until next time, God bless.


So, last class we started looking at the linear one-dimensional first order wave equation, right, that is where we left off. We were looking at a problem, let me just write the equation dou u dou t plus a dou u dou x equals 0, a is greater than 0, okay. And we saw, we called this, we also called it the advection equation, we also called it the advection equation, advection equation. So, it was basically it captured the idea of just pure propagation. The propagation speed is A, right, the propagation speed is A. And we were looking at a problem, I sort of started talking about the boundary conditions, but I did not quite state what the problem was, so let me just state the problem, okay. So, we could have say a pipe of length L at the, this of course is the x direction, at the left hand side, there is a valve and maybe there is hot water to the left of that, okay. Or there is hot air to the left of that, whatever, fine. One-dimensional problem, this pipe may go on forever, but we are only interested in a length, we are only interested in a length of pipe of length L, okay. The pipe could go on for quite some distance. So, the idea is that t equals 0, we are going to open the valve, okay, right. So, at t equals 0, so what are the boundary conditions that we have and the initial conditions, so at t equals 0 we have an initial condition, at t equals 0, the u throughout this pipe has a certain value, right. You can move the datum to that, you understand. So, we can move our reference temperature or whatever it is property that you are talking about, so at t equals 0, u of x comma 0, that is x comma t is 0, okay. So, we are interested only in this length, wear that in mind. And for all t at x equals 0, right, for all time, for all time t greater than 0, in fact we do not know how long the water has been getting hot, so we do not go there, right. So, for t greater than 0, greater than or equal to 0 if you want, at x equals 0, u of 0 comma t equals, you always scale the problem, equals 1, is that right? So, this is the simple problem that we are talking about. If you were to pose it in the xt plane, in the xt plane, so that is the length L, we have prescribed conditions, we have prescribed conditions from 0 to L on the x axis and we have prescribed conditions for all positive t, that is basically what we have done, okay, is that fine? And we know that our characteristics go out in this direction, so the characteristic that goes out here, I do not care for, right, it is going out. So, we know these characteristics go out in this direction. So, this is the orientation of the characteristic, okay, is that fine? Everyone? So, whatever is the value here on the axis is the value that is going to be propagated along the character at the speed a. So, yesterday I called this point x0, I will call it psi today for my own reason. So, what happens to this, what happens to this function now, so the initial function, if I plot just the function, I plot the function u versus x for the same length L, right, it is a rather simple function just to the left of it, which we are not interested in, apparently the function value is 1, right and then it drops to 0 and there is indeed a discontinuity at that point, okay. And I do not actually care the fact that there is a discontinuity, if it bothers you, you can choose some other, we can choose something else, right, but let us start with this, let us just start with this. So, this valve is open because this is the problem we have been looking at so far, the valve is open and what you would expect is that the water propagates left to right. So, whatever value you have here is going to propagate out, whatever value you have here, I have not drawn the characteristic going from that step, you have a step there, that step is going to move along that characteristic, it is going to propagate along that characteristic, is that fine? So, in time as you go along, this step is going to come out, so at some time t, the step is going to propagate out. So, what you would expect is if this were an animation, you would expect that this step is going to travel, right, left to right, is that fine? Right, and if you think about it, you have a tap, you open the valve and what water is going to travel and you would expect something like this to happen, okay. So, pure propagation, that is all it picks up, no diffusion, nothing else, no other physical phenomenon, pure propagation, there is no decay in the size of the step, there is no smoothening out of the step, nothing, okay, it is just going to propagate left to right, is that fine? Okay, so this property is important, so if you had instead of your initial condition, instead of being a step, if it had been some other function, you can imagine now that you can choose other functions. So, you can choose a function, the initial condition, for example, to go from some value, right, go down, so some value, it can be anything, I mean, you can just pick, right, you can pick any initial condition that you want and that condition will be propagated, it will just basically flow out of the pipe. What I am trying to say is that if in your pipe, whatever it is that you have, right, whatever, there is a temperature distribution, that water is going to flow out, right, so if it turns out that the pipe, because it has been sitting there is warmer, there is a region where it got warm due to conduction through the valve or whatever, then when you open the valve, it is all going to flow out, okay, so this function is just going to propagate left to right, fine? So, you can try out a few different functions. The other thing that you can do is, at, you can try this out, for t greater than or equal to 0, you can replace the function at x equals 0, you can replace the function 0, t, instead of it being constant, which is what I have done so far, pick a function and see what happens. So, you can take that to be say cos t or you can take this along with cos t or sin t, right, or try another one. So, this is one function, try a different function, right, 0, t equals cosine 2t, try various functions and see what happens, okay, try various functions, these are varying in time, see what happens, is that fine, okay, let us get back to this picture, so this characteristic, a typical characteristic intersects the x coordinate, right, at a point psi, yesterday's class I called it x0, today I will call it psi, at a point psi. So, what is the equation of this line, right, remember that this line, you are going to propagate a distance a in unit time, right, the property is being propagated at a speed, a units per second. So, it is going to be, x is going to be psi plus a t, in fact in reality the differential equation, the differential equation that we are actually solving is dx dt equals a at t equals 0, at t equals 0 that tells you x equals psi, that is what we are doing, you understand. And I am integrating that to get essentially x equals psi plus a t, because a is a constant, this is integrated out, but I want to make that statement a little more precise, right, I want to be a little more careful with that, okay, because I will use this fact later, a in this case is a constant everywhere, but what is more interesting to me is, a is a constant along that characteristic, a is a constant, right, but I want to say, I want to put it in a peculiar fashion, a is a constant along that characteristic, that it happens to be constant everywhere else I do not care, right, okay. So, a is a constant along that characteristic and in fact it just turns out that x equals psi plus a t, is that fine, okay, so that is the equation of that line. Now, so what does this equation do, what have we discussed so far, what this equation does is, if on this length L you prescribe some function, this equation will propagate that function, because a is greater than 0, it propagates it left to right, okay. So, I repeat, at t equals 0, if you prescribe u equals f of psi, u of x, 0 equals f of psi, this is going to be propagated how, what is going to happen to this f, so something of the form f of psi, but what is psi, psi in fact from our equation is x minus a t, so given the initial condition, so now we make the jump, given the initial condition f of psi, in fact given any function f, right, which has the appropriate derivatives as far as we are concerned, then f of x minus a t is a solution of this equation, is that fine, okay, right and you can verify that, what is dou f dou x, dou f dou psi dou psi dou x, chain rule, right, what is dou f dou t, dou f dou psi dou psi dou t, substituted into that equation, which is how we would verify whether it is a solution or not, what is dou psi dou x, dou psi times 1, dou psi dou x is 1 and what is dou psi dou t times minus a, is that okay, everyone, so this is indeed a solution, so you can just substitute there and you see that dou f dou t plus a dou f dou x equals 0, so it is satisfied, the equation is satisfied, okay, so any function of this form, any function of this form, equation is satisfied, so you remember where we started this, earlier when we started off I said can we guess the solution, right, so now we are probing, we are looking at, there is a way by which we can construct a solution, there is a way by which we can construct a, geometrical way by which we can construct the solution using characteristics, right, those lines were called characteristics and using those lines we can actually construct the solution, geometrically, okay and from there maybe we can get some, but now what we have done is, we have seen from that just using a little analytic geometry that we are able to say that any function of this form is a solution as long as these derivatives make sense, okay, you can complain that I took a step function what is going on, right, okay, we will see, we will encounter a lot of those situations but as long as these derivatives make sense, right, something of this sort is going to be a solution, everyone, right, so how do we do, where do we go now, I want to still be able to guess the solution, we have a general form, I want something a little more specific, okay, so we will repeat what we did, we will repeat what we did with Laplace's equation, you can of course if you give me a function f of psi, it is possible it is on a finite interval of length l, if you give me a function f of psi then I can use possibly Fourier series, I can use periodic extension, okay and I can use Fourier series to represent this function, okay, why do I use Fourier series, maybe I am getting a little ahead of myself, if I look at this function do I have to do t, so see please remember I am now, I am trying to explain the process of how we go about guessing, see this equation is simple, you can easily, I am pretty sure you can sit down, but the process that we go through is very important, it is more important, okay, so I see first derivatives here, so from my differential equations I am thinking exponential, right, I see first derivatives I am thinking exponential, so okay I am going to get an exponential, but an exponential can go two ways, an exponential can go two ways, in this case the function value did not decrease, in this case the function value did not decrease, right, so I do not want an exponential in the form of e power minus x kind of a thing because the function value will decrease, I want only pure propagation because that is what this equation represents, fine, which means that, so if I have an oscillation it is going to continue oscillating, that is what, that oscillation is going to be propagated, that is all that is going to happen, okay, so the minute you say oscillation not decay, we have something that looks like a Fourier series, is that fine, so what we will do is, we will represent, if you give f of psi, I can write this f of psi in terms of Fourier series, okay, and how does that go, that goes over a summation over n, not going to bother with the limits right now, okay, go from minus infinity to plus infinity because I am going to write it in a, or because I will take it from 0 to l, I meant from minus l to plus l, it depends on the lengths of the interval, you have to be careful, so I will just leave it vague, I will just leave it as n, right, depending on what exactly we pick, the n range will have to be picked appropriately, whether it goes from 0 to infinity or minus infinity to plus infinity, okay, so then we have an exponent i n wave number 2 pi psi by l, is that fine, everyone, is that fine, okay, and then you can do, you can take l is 2 pi, you know the 2 pi's go away, all that kind of stuff, that is fine, so you can have a function of the sort as a solution, so let me swing over here to the other side, so what we have is we will get back to the, so what it basically says is that my u in general, I can write it as summation over n, a n exponent i n 2 pi x minus a t by l, fine, okay, so we started off looking at this equation, right, so and by sequence of investigations trying to find out what it is, we were able, we were lucky, not always possible that we were able to do this, but we were lucky, we were actually able to guess the form of the solution, okay, a general form of the solution, fine, okay, so we can construct it geometrically using characteristics or given the initial conditions, you can try to see whether you can use Fourier series to actually construct the solution, okay, right, are there any questions? This course of course is not about analytic, it is not about analytic solution, we are interested in analytic solutions only because we want to know how the solution to the equation behaves, so that when we do the numerics, we are able to compare the, compare it to the solution, right and where the numerics does not work that I mean, I am sorry, where the, where you do not have an analytic solution, right, generally then we will appeal to some theory of differential equations, so that we can figure out, right, theory of differential equations, for example, when we talked about Laplace's equation having a maxima or a minima on the boundaries, that is the kind of a result that would come out of theory of differential equation, right, then we would appeal to theory of differential equation saying this is the problem that I have, does the mathematical theory tell me anything about the solution before I even solve it, okay, that is what the theory does for you, it will tell you something about the solution before you even solve it, okay, it is very important, right, so the theory part is very important, so then we do the numerics and as a first cut then you can compare it to see whether what the theory has told you is satisfied by the numerics, okay, in this case we have constructed a solution because this is actually a simple model equation for the kinds of equations that we are going to solve at a later point and we will run into some of the difficulties that you would run into with say the full Navier-Stokes equations or whatever, you will run into them with this equation, right, there are certain elements of this equation that so we start with a simpler equation for that reason, okay, so we have a solution but now to the basic point of this course I do not want to solve this analytically, if I did not know the analytic solution how would I solve it numerically, what would be the method that we would use to solve it numerically, fine, okay, the usual progression would be well we use central differences for Laplace's equation that seemed to work, why not use central differences here, that is you use for the equation dou u dou t plus a dou u dou x equals 0, you use a central difference representation for this and a central difference representation for that but I look at the boundary condition and you know at t equals 0 I have a condition but I do not have anything beyond that, I have a concern, okay and for the central difference here on the right hand side there seems to be some issue but I will ignore that, right, but it is clear that I do not have if I am going to, if I am going to march in time I am given conditions at t equals 0 I have nothing beyond behind that, right, so if I were to draw grid lines like I did for Laplace's equation, if I were to draw grid lines I really do not know how to take, I really do not know you know whether I do not know this value, I do not want to guess that value, the future value that would be like guessing the future value, you understand the issue here and I do not want to start here because then I do not know anything behind it, if I represent the equation here I do not know anything behind it, right, I am only giving you a complaint, I am not saying that it is impossible to use central difference, right, I am sure we can think of ways of getting around it but my objective is that is not my objective, right, so I have central differences here, I want to keep life easy, so I will use a forward difference here, is that okay, is that fine, that seems reasonable, that I use a forward difference, I use these two points, I use a forward difference at that point, I use a forward difference at this point and I use a central difference for the spatial derivative, okay, so for any arbitrary grid point, for any arbitrary point okay, so p, q, we can use central differences, so I will zoom in on this, so for any arbitrary point p, q and as we did in Laplace's equation maybe at least in the x direction we will take equivalent rules, right, in the t direction also we can take equivalent rules but we will see what happens here, so p, q, this is p plus 1, q, this is p minus 1, q, this is p, q plus 1, these are the points, right, I have gone with p, q because I have already used i for square root of minus 1, right, up here I have already used i for square root of minus 1, so I do not want any confusion, that is why I have gone to p, q, okay, okay, so this very often in books, textbooks and so on you will see this referred to as a stencil, okay, this is just for you to get the jargon, you will see this referred to as a stencil, right, so this stencil occurs everywhere, so you can take these four points and presumably solve wave equation and what we are doing here, very important, what we propose to do is we propose to represent the wave equation at the point p, q using p, q and these three other points, okay, so dou u, t, dou t, dou u, t, dou t for difference u, p, q plus 1 minus u, p, q divided by delta t plus the truncation error which I am not, so I should actually write an approximate there as a truncation error, dou u, x, dou u, dou x is approximately up plus 1, q minus up minus 1, q divided by 2 delta x, is that fine, so what do we have substituting into our governing equation, so you have up, q plus 1 minus up, q divided by delta t plus a times up plus 1, q minus up minus 1, q divided by 2 delta x should be approximately 0, we set it equal to 0 and this is the objective, this is what I am trying to get, this is the objective, the objective is up, q plus 1 can then be written as I take everything else, you notice this is only one that is a time level q plus 1, all of these are a time level q, so all the ones that are a time level q I am going to shift over to the right hand side, so I have up, q minus a times up plus 1, q up minus 1, q by 2 delta x, I miss something a delta t, a delta t, is that fine, is that fine, everyone, so this is what we have, so in fact this expression a delta t by delta x is going to appear so often, we are going to give it a symbol, we are going to call it sigma, this appears so often that we are going to call it sigma, so that this is going to be up, q plus 1 is up, q minus sigma by 2 up plus 1, q minus up minus 1, q, okay, we have what I would call an automaton, right, so given something a time level q, you can then increment it to the next time level, you understand what I am saying, you can find out what happens at the next time, given at q plus 1 you can go to the next time level, right, so at each time level you can move forward, you have an automaton, so if you are given an initial condition, you are given an initial condition, you can march that initial condition forward in time, right, q plus 1 is given explicitly in terms of, explicitly in terms of expressions that are occurring at level, at time level, time q, time level q, okay, so this scheme is called an explicit scheme, the scheme is called an explicit scheme, simply because you have q plus 1 occurring and this is, these are all at time level q, we are able to solve for it, it does not occur in an implicit fashion, is that fine, it does not occur in an implicit fashion, so this is called an explicit scheme, explicit scheme, sometimes you will hear it being called an Euler explicit scheme, we will call it forward time central space, FTCS, forward time central space, is that fine, okay, everyone, right, so you can actually code this, you can go ahead and code this, right, but this is different, this is a little different from Laplace's equation, so we are going to do a little analysis, right, there are two things that we have got going for us now, one is when we did the stability analysis for Laplace's equation, we substituted exponentials for the error term, we substituted exponentials for the error term and the error check whether the error term decays or not, what happens to the error term, okay, in this case, we have a similar set, they are there, in that case that was, they were if you think about, if you remember back, they were Eigen functions of the, the exponentials were Eigen functions of the Laplace operator, in this case, they actually represent the solution, the equation is a linear equation, the equation is a linear equation, you understand, remember linear equation basically means, equation is linear basically means that, equation is linear basically means that if you have, if l is linear, right, what does it say, l of u plus v is l of u plus l of v, right, l of u plus v is l of u plus l of v, fine, okay, that is basically what, so l again and usual stuff, I mean the l of alpha u plus beta v is l of alpha l of u plus beta l of v, I mean this is, this is the usual description of something being linear, right, which is why we know that A x equals A x, y equals y is linear, right, or the classical, when we talk about a straight line, this is the, this is the usual trap that people fall into, so if you take the equation y equals, and this is an aside, y equals A x plus b, this is not linear, right, linear it necessarily passes through the origin, this is not linear, it is a straight line, it is a straight line, it is not linear, so if you say linear, wait a minute, it is a straight line, does not that mean linear, if that confuses you then think of curvy linear, right, linear, all it means is a line, curvy linear means curved line, do you understand what I am saying, so but in this case linear, this is, this is a, this is a technical term, right, the function is linear, the operator is linear, basically means that l of alpha u plus beta v is alpha l of u plus beta l of v, okay, it is a very precise definition, as a consequence of which y equals A x plus b is not linear, though it is a straight line, okay, unless b equals 0, fine, okay. So that equation is linear, we ask the question again, so if you have dou u dou t or just for, just to keep this, just to keep this clean, let me write a very general equation, so if you have dou v dou t plus A dou v dou x equals some h of x, it could be h of x, t, h of x, keep it as h of x, right, the solution is v, the solution is v, okay, the solution is v, but v is disturbed, a perturbed v, it has an error, right, in Laplace's equation in case I called it e, in this case I choose to call the error u, for obvious reasons that I already have an equation for u. So if the solution, if our candidate solution is v plus u, if our candidate solution is v plus u, okay, and you substitute into that equation, because this is a linear equation, what is going to happen, what is the equation governing u, our original equation, you understand what I am saying, so if I substitute this back in, I get dou v dou t plus dou u dou t plus A dou v dou x plus dou u dou x equals h of x, I am sorry A dou u, oh it is very important A dou u dou x, thank you, right, this combination, this, this and this, the combination of these three satisfy each other, they knock each other out, the combination of the three knock each other out, leaving us dou u dou t plus A dou u dou x equals 0 as the equation that governs the error term, looks the same, suspiciously the same as the original equation, right, and you would expect that because it is a linear equation, even in Laplace's equation case, the equation governing the error was the same as the equation governing our original function, our solution, is that fine, okay, right, so what do we have, we go from here, so I want to substitute now in dou u dou t plus A dou u dou x equals 0, which is now the equation for our error and I want to ask the question, when does this decay for this particular scheme that we have chosen does it decay, okay, for the particular scheme that we have chosen does it decay, I want, what was our u, u is like summation over n A n exponent i n 2 pi by l x minus E t, what do you say, okay, and because the equation is linear, right, also equals summation of u n over n, right, where the individual u n's, where the individual u n's are A n exponent i n 2 pi by l x minus E t, okay, we will play the same game we have played for Laplace's equation, we will say for which wave number, is there a wave number, is there a bad wave number, right, if we can say that for the worst, for the wave number which has the largest gain, as we march in time, see this is what we are doing, we have a perturbation or an error u in our solution, okay, we want to know that if I march from time t equals 0 to t equals delta t, 2 delta 3, 3 delta 3, so on, as I march, is that error going to grow or is that error going to die out, does that make sense, is the error going to grow or is the error going to die out, that is the question that we are asking, so basically what we will do is we will pick one wave number, then ask the question for the wave number which has the largest gain, what happens, right or is there a wave number that has a problem, that is essentially what we are looking for, is there some particular wave number for which we are going to have a problem, okay, so because you know I have already got subscripts p and q, right, so you please allow me to drop this n, we understand now that I am going to take a look at the nth wave number, right, so you just otherwise we will just be carrying along a lot of these subscripts, so let us just assume that we understand that I am dealing with the nth wave number, so what do I have now, my automaton was upq plus 1 is upq minus sigma by 2 up plus 1q minus up minus 1q and as I did earlier what is the relationship between up plus 1q and upq, so if my x is p delta x, right, if my x is p delta x and q could be you know we do not really need it but q could be t could be q delta t, you could keep delta t constant if you want but it does not matter, if my x is p delta x and t is q delta t, okay, if my x is p delta x t is q delta t, what do we have, what do we have up plus 1q equals, where do I get this now, an exponent i n x is p plus 1 delta x minus a t at the time level, I will leave this as t, okay and this is nothing but an, so you can just check, so it is e power exponent i n, I forgot to 2 pi by l, i n 2 pi by l delta x into upq and in a similar fashion because there is a p delta x here which is x, this is basically x plus delta x, right, this is basically x plus delta x and up minus 1q in a similar fashion is exponent minus i n 2 pi by l delta x upq, fine, so I can now substitute into my forward time central space upq plus 1 is upq minus sigma by 2 up plus 1 is this one, this is a mess but these two expressions are the same, I will just redefine something, I will redefine that as theta equals n delta x 2 pi by l, right, I mean I could take l equals 2 pi and that will go away, it does not matter, so this is e power i theta minus e power minus i theta into upq, everybody with me, okay, I can divide through by upq, right, I divide through by upq and I get the gain G as upq plus 1 divided by upq 1 minus sigma by 2, what is e power i theta minus e power minus i theta 2i sine of theta, remember I am using Euler's formula, e power i theta is cos theta plus i sine theta, so this gives me the gain as being 1 minus i sigma sine theta, is that good news or bad news, what do you say, so you want the gain, the magnitude of the gain to be less than 1, right, you want mod G to be less than 1, okay, but what is the mod G, what is the magnitude of G, mod G is or mod G squared to be less than 1, what is mod G squared, 1 plus sine squared theta, sigma squared, 1 plus sigma squared sine squared theta, see the only parameters that we have to choose, right, now this brings this whole thing start, you know, we now face the fact, you cannot change A, the only parameters that you have to choose are delta x and delta t, okay, so sure, I mean theta depends on delta x and so on, but there will be that, we know that there is a highest wave number that we can represent, right, I mean so that all of that is very clear, we have no problem issue with that, so the only parameter that you can change is sigma, right, it is not complex, I mean it is real, so sigma squared we are stuck with it, am I making sense, we are stuck with it, so sigma squared, there is no sigma value for which this is going to be less than 1, so this does not work for us, theta equals 0, yeah it equals 1, right, but theta equals 0 is that dc component essentially, so theta equals, it does not help us, we are not getting anything, right, is that fine, so forward time central space, FTCS is unconditionally unstable for, be very careful now, be very careful, when applied to one dimensional linear first order wave equation, right, yeah, you have a question, why should mod g be less than 1, see when you go from time, when you go from time q to q plus 1, if you go from time q to q plus 1, mod g is greater than 1, right, so you are basically, look at this as a sequence, so you have a, you are generating a sequence, I will leave out the p part, you are generating a sequence which is uq, you are generating a sequence of u's indexed on q and we are asking the question, does this converge and we in fact wanted to go to 0, u in this case is the error, right, v was our original solution, u in this case is the error and we are asking the question, does u go to 0, u is the error in v, you can have a u which is non-zero, this can still satisfy this equation, you can have a u which is non-zero in the Laplace equation case because we have a unique solution for Laplace equation given a boundary condition, u equal to 0 was the e equal to 0 was the only solution, but here you can have a conditional boundary solution which is non-zero for this same equation. u can be a solution which is not, but it does not satisfy our original equation, does it satisfy our original equation including the boundary condition, well if it has homogeneous boundary conditions and homogeneous then you can add it, then you can add it, right and then the question is what happens at the initial condition, so it has to be 0 at the initial condition and somehow it magically came up, you understand, okay, right. What we are basically saying is even from our numerics, typically see where these things come from is even from our, the source of this error, it is not that there is an error that you have injected into the solution and does it decay, there is a source of the error which comes from our, say from our round off, right, from various reasons, there is a source of error that you have, what happens to that source of error, right, it is no different from the whistling that you hear in an amplifier, the whistling that you hear from an amplifier, it is not that there has to be a input, just thermal noise, there is thermal noise in the resistors in the circuitry, that thermal noise is enough, so the issue is the question that you have is it is not that you have, the disturbance is there, you have the thermal noise which is the equivalent here would be I have round off error, right, every step I have these errors, I am making these errors, what I want to know is do those errors, errors grow or those errors do not grow, so in a sense I am not actually answering the global question, what you are talking about is answering the global question, right, so the stability analysis that we are doing, both what we have done here, what we are doing now and what we did in Laplace's equation is a local one, we are only asking at a given grid point what is happening to the, what is happening to the solution if I were to integrate it out, but not for all x, am I making sense, okay, say I see where you are coming from, you are saying it is a homogeneous the equation, this equation is a homogeneous equation, it basically does not disturb, you can add any amount of this homogeneous equation solution that you want, some constant time this homogeneous equation, but normally in your differential equations if you look at it, the way it works is the homogeneous part actually takes care of the boundary conditions, in this case even the boundary conditions are 0, so in a sense this does not, this is truly sort of in the null space of that operator, it does nothing, okay, but potentially if it grows it can blow up, the problem that you have is if I, the difficulty that I have is if this uq, if the gain is greater than 1, the sequence that I get is diverging, so I am going further and further away from my v that satisfies the original equation. No, no, there is no issue of boundary condition here because I am only working at a grid point. No, no, I am not looking at a solution, that is the whole point, no, no, no, no, the solution is v, the only question that we are asking is there a wave number that is unstable in a sense, is there a wave number that is going to grow, that is going to become unbounded, if I disturb that wave number or there is an error in that wave number, is there a wave number that is going to grow, that is all, that is the only thing that is happening, is that fine, right, so that is basically it, so we are generating at this point p, so though I have removed it, I will stick it back there, at this point p, we are generating a sequence indexed on q and we are asking the question what happens now at the point at that x location, what happens as q goes, but it is the analysis that we are doing as a local analysis, it is not a global analysis, maybe at one of the other classes I will do a global analysis where I take the boundary conditions also into account, right, that is really what we should do, okay, here is sort of I beg of saying well this we are engineers, this is, you know, we get this, this gives, and this is already unstable, forget doing the full thing, this is already unstable, right, we are already out of luck, this is already unstable, fine, and you can try to implement this and you will see that it is unstable, try to implement this, so this analysis actually works, okay, this analysis actually works, which is not justification enough, I cannot just say that it works and therefore it is right, but what I am saying is this analysis actually, so this gives you this q's, the sequence of q's that you are going to generate using that automaton which is FTCS, if there is any disturbance it is just going to blow up, is that fine, it is going to diverge, if there is the smallest disturbance that disturbance will grow, that is the key, fine, is that okay, is that okay, so this is where we are, FTCS did not work, okay, so the next thing is, next obvious possibility is we did forward time, why do not we try forward space, now we are groping in the dark, if it did not work, now we are sort of groping around, trying to figure out what to do, so forward time, we use forward time, let us try forward space, is that fine, so Monday's class we will do forward time forward space FTFS, right and let us hope that it works, fine, okay, I will see you then.


Hello everybody and welcome to video number 21 of the online version of the fusion research lecture We are in chapter 4 and in the last video we talked about Particles who are referred to as passing particles. Those are particles who are not trapped We have a strong parallel velocity component and who are traveling around the torus This video we will talk about particles who have a significant perpendicular velocity and this might lead to trapping as we will see So if we have particles whose parallel velocity is small compared to the or smaller than the a perpendicular velocity Then the particles can actually be reflected. So the particles can be reflected when they are traveling in the torus because they can be reflected on their trajectory from the out Board side to the inboard side and What causes this reflection simply the gradient of the magnetic field because you know on the outboard side We have a lower magnetic field as compared to the inboard side So this is basically the same as a magnetic mirror where reflection can happen And this has of course the or this has the result that no full Pull loyal turn Sorry is possible So particles Who are reflected they can no longer make a full pull loyal turn now has a let's have a look at an example So here you see I will start a video now where you can see color coded the magnetic field strength in a typical talk amok And you will see now the simulation of a particle. So this is just the Equation of motion solved and you can see how the particle bounces around if it reaches regions of higher magnetic field now to get lower magnetic field strength higher magnetic field strength Lower magnetic field strength higher magnetic field strength, and you see how the particle bounces around the talk amok so Here even you see the gyro Orbit is included in the solution something which we will usually not do But this is definitely possible and this is a rather old simulation which has been done years ago Okay, this was an example of a Standard talk amok with the particle bounces around. Let's have a look at another video So once again color coded the magnetic field strength blue means low magnetic field strength red higher magnetic field strength You see the particle bouncing around between regions of higher magnetic field strength it is reflected When the field strength is too high Assuming it has a significant perpendicular velocity And now you see a cut Through the experiment To realize the trajectory as it might look from this perspective Okay Now Here you can see snapshots of the two simulations which I have just shown as a video and You might have seen that the duration of The particles or the duration of their stay rather is longest at the reflection points as longest around the reflection points and that is simply because the Parallel velocity decreases when they reach the deflection points more and more energy is Transferred into the perpendicular velocity and since the duration of this day is thus the longest there The displacement is the strongest there. So they have the strongest Displacement at the regions close to the reflection strongest displacement there Okay, so now let's have a look at the explanation for this Deflection or for this displacement. This is basically the same. So this is an equivalent not the same Let's rather say an equivalent explanation as Compared to the passing particles as equivalent like the passing particles Now, what does that mean? So let's make a drawing where we have First we start with a coordinate system So this is X and here we have the Toroidal angle Phi and Here we have R Then let's try to draw a flux surface Maybe something like this So a pole loyal cross section here. We have our not Magnetic field points of course into the board What else is missing? We start with the Positive charge located here. So there's a positive charge here and Since we have a positive charge you remember the Drift velocity for the ions is pointing upwards Look at the video from the last video to to get a reminder on that Now let's start with an iron which moves along the magnetic field line direction then here the Sorry, the theta the pole oil velocity will oops. Sorry will point downwards So this is the pole oil velocity for a particle moving downwards And if we follow that Same explanation as for the passing particles the radio getting smaller due to the drift velocity pointing upwards But then as compared on it different which is now different to the passing particles since we do not have enough Parallel velocity, but a stronger particular velocity the particle will get reflected here then it moves upwards again Like this then it moves upwards again, and then it will again get reflected And due to the drift velocity pointing upwards in the end it will close its orbit, and we have such an orbit which is called due to its shape in the projection here a banana orbit, so this is a banana orbit and Due to the so here we have a poloidal projection, of course the poloidal projection As you have seen in the videos the particle also moves in the toroidal direction But due to the shape of the trajectory in the poloidal projection We call these particles also banana particles So we have banana orbits banana particles oops and This is what is also referred to as a trapped particle. This is a trapped particle and Usually a definition for a trapped particle is that the parallel velocity is 0 somewhere on along its trajectory somewhere along its Trajectory now an important quantity is the width of this orbit, so this From here to here. Well roughly this is referred to the width of the banana orbit data B a for banana and Since this is an important quantity for transport because again this allows the particle to deviate from its original flux surface We will have a closer look at that so important for Transport processes is the width of the banana orbit This one is an important quantity To estimate to get an estimation of this width what one has to do is compare the flux surface radii so compare the flux surface radii at the outboard midplane Where we let our particle start at the outboard midplane and in the reflection point and in the reflection point Inserting this into the equation of motion, which we do not do here and then One ends up with an expression for the banana width, which is approximately Data be a which can be approximated by the Larmor radius which we had introduced yesterday times epsilon or This is maybe more interesting here. We are the Displacement of a particle for passing particle and then times one over epsilon And the square root This is maybe more and I said this may be more interesting because it tells us that the banana orbit is by a factor of One over the square root of epsilon larger then the drift plane displacement The drift plane displacement This is important to realize that the banana orbit is larger by a factor of one over the square root of epsilon Then the drift plane displacement so another Important quantity which we will use later is the reflection time The reflection time sometimes also referred to as bounce time. So it's an expression which you might read somewhere bounce time which is the time it takes between two reflections and That is expressed or can be expressed by usually we use tau and then be a for banana Then this is first we have to pie Because tau is defined as one over omega usually so we have to pie and then the width So the distance over the drift velocity and then Assuming an average angle since the particle is above the mid-plane and below the mid-plane moves around Assuming an average angle then we get something like first inserting the expression for the banana width Now also inserting the expression for the polar llama radius so fully inserting everything then we have MV parallel square root of epsilon over qb theta Then the drift velocity for an average angle as I said then we have qr b Then over MV parallel squared and now having a close look at the equation We can see that this one cancels out This one cancels out Square here goes away. This one cancels out and we end up with an expression where we can insert now what we had used yesterday the transit time for passing particles times one over the square root of epsilon and again You have an expression of the transit time for the banana particles Sorry of the trapping time of the banana particles, which is larger by a factor of one over the square root of epsilon as compared to the passing Time for the trapping particles So let's write that down. So the banana particles take Factor of one over the square root of epsilon longer To complete their orbit to complete their orbit Then the passing particles then the passing particles Again two important factors here, which I have written down here Okay, finally This is all interesting, but maybe a few words about the fraction of trap particles So it is an important effect or not the fraction of trap particles So as I said the mechanism for reflection, this is basically a magnetic mirror and a magnetic mirror In a magnetic mirror particles are trapped if The sign this is something which you had learned a plasma physics one the sign of alpha or the squared of the sign of alpha Which is V parallel over V if this is larger than some critical angle often referred to as alpha not Which is the minimum magnetic field strengths over the maximum magnetic field strengths and Using now the expressions for the minimum magnetic field strengths, which we had introduced the beginning of this chapter So be not R not over R not plus R on the output side the magnetic field is the smallest and the maximum magnetic field strengths Be not R not Over and now this is the magnetic field springs at the input side the high-field side inserting this now years for the sign of alpha not then Be not an R not cancel out so it's R not a minus R R not plus R and this can be written as we can get R not Out of the expression meaning that we have here minus one minus small R over R not We do the same in the denominator one minus R over R not and Well, of course these cancel out right so this can we can get rid of this and Then we can write this as one minus epsilon over one plus epsilon and Now one can integrate one can perform an integration Over a Maxwell-Boltzmann velocity distribution so integrate over a Maxwell-Boltzmann Velocity distribution in spherical coordinates in Spherical coordinates using alpha not as a boundary because this defines the trapping condition and then we get a relative fraction of the banana particles The banana particles and just to be sure that the sticks to your mind banana parties here means trapped particles We get a relative fraction of banana particles with respect to the velocity distribution the respect to the velocity distribution We have Ft the fraction of trapping particles of trapped particles. Let's write it. Yeah, that's right a small t FT trapped particles This can be approximated by two times Epsilon and the square root of it Okay That's it for this video where we talked about trapping particles So those are particles who have which have somewhere the parallel equal to zero along their orbit This means like in a magnetic mirror They can be reflected and since in a toka muck we have low magnetic field springs on the outboard side high magnetic field springs on the Inboard side it simply can get reflection due to that gradient in the magnetic field The shape of the particles in the pool oil projection looks like a banana this is why they are called the banana particles and There the width of this orbit is Larger than the corresponding Displacement of passing particles and also the bouncing time the reflection time is longer by a factor of one over square root of Epsilon then for the passing particles and finally we look at the Estimation for the fraction of trap particles which can be estimated by the square root of two times Epsilon yes, that's it for this video. Hope to see you in the next video


It's not that hard to wonder about plants and their lives. Every time I step onto my balcony, I wonder. I can't even begin to describe the impact that plants have on us. They are literally responsible for keeping us alive. But what about their lives? How do they grow and reproduce? Do they have a life cycle just like ours? Let's find out. A plant's life cycle consists of two prominent multicellular stages or generations. A haploid generation and a diploid generation. And every plant alternates between these two generations. That means a part of their lives is haploid and then diploid. This is why this phenomenon is called the alternation of generations. The haploid stage or generation where the cells have only one set of chromosomes is called the gametophyte. The gametophyte. And the diploid stage or generation with the cells having two sets of chromosomes is called the sporophyte. Sporophyte. The gametophyte is the stage where haploid gametes are formed by mitosis. That means the gametophyte itself is also haploid in nature. The male gametophyte produces male gametes and the female gametophyte produces female gametes. When these gametes fuse together, they undergo something called fertilization to form a diploid zygote. This zygote undergoes mitotic divisions several and mitotic divisions to grow up to become the diploid sporophyte. This sporophyte will eventually produce spores via mitosis. That means these spores are going to be haploid in nature. Each of these spores then give rise to a new gametophyte via mitosis and the cycle continues. Now generally this is how alternation of generation really works. But the specifics changed as the plants evolved. The simpler more primitive plants ended up exhibiting a dominant gametophyte which was more prominent and long-living. But this dominance reduced like crazy as the plants became more and more complex. Now if we check out the bryophytes which are the simplest plants of the kingdom then we'll see that they show a dominant and independent gametophyte. That means the haploid stage occupies a major portion of their lives. The main plant, the one with the leaf-like structures or the root-like structures or the stem-like structures, this main plant belongs to the gametophyte generation which is this portion right over here in the picture. This is a moss plant by the way if you're wondering.  So we have a gametophyte which is haploid in nature and it's going to give us haploid gametes. When these gametes fuse together they undergo fertilization and we end up with a diploid zygote. This zygote will undergo several mitotic divisions and give us a diploid sporophyte. Now this sporophyte is short-lived, barely noticeable and it is heavily dependent on the gametophyte for pretty much everything. Food, shelter, water, everything. And it gets all of these things by staying attached to the gametophyte. So that means this long stock-like thing that you can see, this is the sporophyte. Once the sporophyte matures, it is going to produce spores via meiosis. That means the chromosome numbers are going to get halved and these spores will end up being haploid. Let's change the color so these spores are haploid. These haploid spores are then released into the environment and they will give rise to new gametophytes via mitosis when the conditions are ideal. And this is how the entire cycle continues. However, this is not the case with more complex plants. In teritophytes, the sporophyte is dominant and independent. That means the main plant that you see, the one with the leaves, roots and stems, that belongs to the diploid sporophyte generation. Now when the sporophyte matures, it's going to produce haploid spores via meiosis. Each spore will then mitotically divide to give rise to a new haploid gametophyte, which is so tiny that it is barely one centimeter in size, barely that tiny. But despite its size, this gametophyte is also independent, just like the sporophyte. It can even gather raw materials like water and minerals from the soil. Using these thread-like structures, I believe you can see it right over here. Let's zoom in a little bit. So do you see these thread-like structures over here? These are called the rhizoids. And using these rhizoids, this gametophyte can easily gather raw ingredients like water and minerals from the soil. The gametophyte also bears the sex organs, which are going to produce the male and female gametes. These gametes are going to fuse, undergo fertilization, and give us a diploid zygote. This zygote will develop into an embryo, which is again also diploid. And this embryo will further differentiate into root stems and leaves and we will end up with our beloved diploid sporophyte. Now, this process basically remains the same in gymnosperms and angiosperms too, except those plants also produce seeds. In the seed plants, the sporophyte continues to remain dominant and independent. That means the huge pine tree that you see, or that plant with the pretty flowers, that's the diploid sporophyte. But unlike the pteridophytes, this sporophyte produces seeds, which keep the embryo protected and nourished inside. When a seed gets the right conditions, it will germinate into a new sporophyte. Then, where's the gametophyte? The gametophyte generation in gymnosperms and angiosperms is extremely reduced, so much so that you will find them inside the sporophyte. In gymnosperms, the gametophyte generation takes place inside these hard structures called cones, which are born on the sporophyte itself, the diploid sporophyte. Inside a cone, haploid spores are formed, which develop into haploid gametophytes.  And a female cone has the female spores, which will develop into the female gametophyte with the female gamete.  And that will lead to fertilization. So, the male and the female gametes are haploid, by the way. And they will undergo fertilization and we will end up with a diploid zygote. This zygote will develop into an embryo encased inside of a seed. So, this embryo is also diploid. And you'll find multiple seeds like these with the embryo inside on the scales of the cone. Now, I do have a picture here. So, let's take a closer look. So, do you see the shelf-like places right over here? These are the scales of the cone and inside it you can see several seeds placed. And each of these seeds will germinate into a new sporophyte once the conditions are ideal. And then, the entire cycle will continue all over again. In angiosperms, the gametophyte generation takes place inside the diploid flowers which are born on the diploid sporophyte. So, the flowers are also diploid.  So, this small purple color M means myosis and the capital blue color M means mitosis. So, via myosis, you are going to get the haploid spores which will then develop into haploid gametophytes.  And the female sex organ is going to produce female spores, haploid spores. And that, those haploid spores will develop into the female gametophyte with the female gamete via mitosis or via mitotic divisions. Now, when these gametes come in contact with each other, they will fuse together and they will undergo fertilization and we will end up with a diploid zygote. This diploid zygote, just like how it was happening in the gymnosperms, this diploid zygote will develop into the embryo within a seed via several mitotic divisions. And this seed, when it gets the right conditions or the ideal conditions, it will germinate into a new sporophyte and the cycle will continue. Now, if we had to name all of these life cycles, then it would go something like this. All plants have a haplodiplontic life cycle, haplodiplontic life cycle. Now, in this type of life cycle, myosis doesn't directly produce the haploid gametes. In haplodiplontic life cycle or in plants, what we have seen is that the diploid sporophyte produces haploid spores via myosis. Then these spores undergo several mitotic divisions to give us a haploid multicellular gametophyte. This gametophyte will then produce the male and the female gametes. So, this gametophyte right over here, it is multicellular, multicellular and haploid. On the other hand, the sporophyte is multicellular and diploid. And this happens in the seed plants too. But because of their reduced gametophytes, you'll find some scientists and textbooks saying that gymnosperms and angiosperms have a diplontic life cycle. A diplontic life cycle is characterized by unicellular haploid gametes that are formed directly via myosis. And only the diploid stage is multicellular in nature. That means these male and female gametes that I have on this picture over here, they are haploid, definitely haploid. And they're also unicellular, unicellular in nature. And they are formed directly via myosis. Meanwhile, this diploid organism is the one whose multicellular and diploid. So, it's multicellular. And I've already didn't diploid, but we'll also write 2n here. So, these are the characteristic features of a diplontic life cycle. Actually, we have a diplontic life cycle, like us human beings. Our sex organs produce unicellular haploid gametes directly through myosis. And from the zygote onwards, we are multicellular and diploid. But technically, the gymnosperms and angiosperms aren't like that. They don't have this direct gamete production through myosis. Both of them produce haploid spores via myosis that develop into haploid multicellular gametophytes with the help of mitosis. Now, plants aren't the only ones with such life cycles. Another type of life cycle called the haplontic life cycle also exists. But you will find this one mainly in algae. Here, the haploid stage is always multicellular. And only the haploid stage is multicellular. The main organism is haploid in nature and it produces haploid gametes via mitosis. These gametes fuse together to form a diploid zygote, which immediately undergoes myosis and gives us haploid spores. These haploid spores then undergo several mitotic divisions and develop into the haploid organism that we started with. So the diploid zygote is unicellular because it never gets a chance to become multicellular. And meanwhile, the haploid spores, they undergo so many mitotic divisions that they end up becoming multicellular. So the organism is haploid and multicellular. The zygote, which is the diploid phase, is unicellular. In a way, you can say that the haplontic life cycle is kind of like an opposite of the diplontic life cycle. In the diplontic life cycle, the main organism is always diploid and multicellular. And in haplontic life cycle, the main organism is always haploid and multicellular. And that's all about plant life cycles. Every plant with their unique styles of reproducing. Probably that's what makes this kingdom so fascinating.


This morning as you've heard, we've unveiled a beautiful sculpture and if you have not had a chance to go and take a look at that as well as other aspects of today's celebration I encourage you to do that because it is breathtaking to see what has been created. Facilities and grounds have been an essential part to certainly every day when you see how beautiful the campus is but also as you walk out and say this is so inspiring they've worked together to really bring that together. I thank each of you for spending the time to make sure that today is a success but also for the beauty that will be enduring for many many generations to come as they enjoy what we have created here today. It is truly an honor for the University to provide this historic landmark for Tuscaloosa. Manderson Landing has been a part of the University for many years and now right here on the banks it allows us to look forward about where we will be. The river is a continuous reminder of the life and the vitality of the world that is around us but also the life and vitality of the city of Tuscaloosa. People that are looking from the opposite bank can see it clearly. People that are moving along the river which happens often will be able to see it. Students or people walking along Riverwalk will be able to see it from a distance and experience it and scale as they move closer to it and then past it and around it. These sculptures will remind generations to come how important the University of Alabama is to the city of Tuscaloosa and how important the city of Tuscaloosa is to the University of Alabama.