What's going on guys? It's your boy DPJ here today with another Destiny video and in today's Destiny video, I'm bringing you guys some more live mayhem clash. I was just about to say gayhem clash then. Or gayhem mainplay. Mayhem gameplay is what I'm supposed to say.  But yeah, you guys seem to like the gameplay I posted the other day so why not bring you more if that's what you guys like. I'll show you guys my class set up. Basically exactly what I used on my last video. PVP year one where level does not matter. My one, two, three, scissor G, scissor G is what I use. I mean, weapon is an absolute beast so it's what I'm using. Shotgun.  Let's try the sleeper simulator. I mean, I've never used it in PVP. I don't really get a chance to pick up much ammo to be honest.    But we will see. We will motherfucking see. I'm recording you say? I mean dude. What's going on in my motherfucking sky? Oh, my new mic's being delivered. Skype, I'd say Skype for my emails. My new mic's being delivered because the mic I've got now at the minute. Fucking, some of these makes me sound like I'm recording in a church, like a massive hall, I don't know why. I need something with deeper bass because my voice is high pitched enough.  You know what I mean? You know what I mean? So like, I'm getting a new mic.  It's called a razor siren. Recommended to me by my friend. He's in that 1-3-3-7. It's the mic he uses and his audio sounds wicked on his videos. So yeah, why not invest in a new mic? Even though this one set me back quite a lot of money. I'll use this one for when I go away. I'll use it as like my touring mic. You know what I mean?   I don't know. This is the mic I'm thinking about.  This is gonna go so bad. So I did good on my first video. I posted a couple days back. I'm gonna do bad on this video, but I will still post it no matter how bad I do. I will still post it.  They ain't a bad game, plays with me.  I'm not no good at this game.  Play for fun. And rage. And to entertain you guys, that's about it. Let's see what we've got here. No one comes down here no more. That's weird. It's gonna try and flank random if I can. More fuckers here. I know somebody's gonna be hiding around here. Wow, did he just spawn there then? Oh wait, he did.   They're spawning all around me!  Come on. I'll take a free kill. Why not?    Run away. Hammersaw, is that Hammersaw in the ass? Oh did he just get owned in his hammersaw?  Can't see what's going on. Couldn't see what was going on there. I was just blazed off in blue light. I'm just hearing hammers. Hammers everywhere.   You just can't get away from him. You cannot get away from him. You know what? The team wins in mayhem. Whoever's got more hammers. That is it.  That was pretty sick. That was pretty sick to be honest. That was pretty sick. He confused me. He confused that kid. Blade Dancer. Blade Dancer. I found a knife then. Why didn't I find a knife? I'm so dead. Two storm trance. Who range that? He hit him across the map. Almost. I just pulled out my garden and got shadow shot in my face. That guys hit me twice now.   I'm getting more kills here at the minute.   Uh oh. Uh oh. I'm not fighting hammers. Get out there. Sit here.  See me then? Oh no. This ain't going well. I'll put out a rocket launcher. For people. Oh this is horrible. I don't like sleeper. See me then? I don't like it. Just don't work. That just doesn't work for me. Not one shot left. See what we can do. Can I kill anybody with it? Nope. I can't at all. That was a fail. Okay. I'm failing. I'm failing. I hit one. I hit one. That I do. That I do. Oh I'm dead. I'm dead. I'm failing this fucking. I'm third as well. Oh I'm doing horrible. I'm doing horrible.   I'm falling behind. This is a close game. This is a close game. Infinite. Where am I getting shot from?  See if I can get out of here. I don't even get any kills out of here.   I'm lagging. It's a choice just here. Oh really? So you're telling me that in fucking? I'm so dead. I'm so dead. I'm so dead. I'm so dead. I'm going to do six kills behind. This ain't going well at all. This is not going well at all.  I blame the lag to be out. Welcome back heavy weapons. Gadahorn is the heavy weapon. Three rockets. Uh oh. Hammer of soul. Uh oh. Uh oh. That was ridiculous.  But you know what to do. Oh we got an hatch factory. Not bad. Fucking. Why was it twice? I pulled out my golden gun and that game got owned. Ridiculous. Hammer of soul.  You know what I'm going to do? I'm going to carry on. I'm going to carry on. I'm going to play another game. Why not? Might as well make the video about 20 minutes long. First you go round to the end. You're back in legends. I almost raged creating a bit of that game to be honest. I'm not even going to lie to you. Shit wasn't going my way. Wasn't enjoying it.  I just hate that map. Map's horrible.  It really is. Give me a decent map. Give me something like Twilight Gap. Or Firebase Delphi. Or Pantheon.  Anything but the cauldron.     We lost. This game I'm going to do good.  If it finds a get, I'm going to find Steve. There we go. Widow's caught as well. You know what? I actually... I don't mind that map now. The map was Charles. Charles of the Souls. And just how unbalanced the spawns were. I mean, spawning at the top has a clear advantage over people spawning at the bottom. Charles of the Souls, you should mix.  And I probably would have loved that map. I mean, I've played that a lot of it since I last played Charles of the Souls.   But, it's not my favorite map. Charles of the Souls clearly ruined that for me. To be honest, Charles of the Souls ruined cauldron for me as well. Absolutely destroyed that map for me. Everybody just blinking shotgun. To be honest, I haven't played Charles of the Souls since. Blinking shotgun in A and is bad.  And it's gone now. What else? We've got that.  Pulsers.    But yeah. I haven't really got a even set up set up for Charles of the Souls. I know I'm probably going to use my grasp of malloc because it's the only thing I've got. That's a lot like my size of G. I mean my size of G stability is way better. But still this is the grasp of malloc. So a way better roll if you look at that. Head seat and glass half full. Beastly injection mold as well. But standard stability on this thing ain't that great. I mean look at that. It bounces everywhere man. It bounces everywhere but yeah. Ah I said Charles of the Souls is next. Well this Friday is it? I don't even know. I don't even know. I know it's coming. The 30th I think it is. We'll see.   Now to be honest I'm away that weekend. I'm away. So I won't be playing until I come back. I don't know when does it end. Does it end on A? It's a three day event is it? So it will probably be gone Sunday night Monday. I'll probably miss it all together. Oh well. Oh well. But I'm watching. Not bad. Not bad. Not bad. Not bad. Not bad. Not bad. Not bad. Not bad. Not bad. Not bad. Not bad. I don't mind this map. I don't mind this map at all. Reminds me a bit of fire based out for some reason. But yeah.  We're gonna eat my fuckers up now as long as that laggy bus. Is that doing less lobby? I hate moving my phone now if you died about 50 minutes later. Ridiculous. Black screen.  I can just guarantee it. Let me see it. Look at this fucker here.  I don't know why. This is not the map I was thinking it was. This is not the map I was thinking it was but I do like this map as well. I do like this map as well. Okay. I don't know what was I thinking of now. Black something. Something black. That's what I was thinking of. Okay. Let's go. Let's go. You know people are jumping up here. Or not.   My accuracy is disgusting man. I'm so dead.  Change the lead.  Sounds like a heavy machine gun. That's a super cell only. I just pulled this out way too early. No but I got owned.  Okay. Here it is motherfuckers. He's down. He's down. Let's go. Let's motherfucking go.  Ready go. Ready.  Heavy ammo available.  The same guy that I've ever gotten a gun two times in a row. The second time I got up a kill though. So I'm not too bad.  We're just getting trapped in. We're getting trapped in here. Only five minutes left. Oh wow. Set then. I need to dismantle my connect. Just take it apart and break it. Hey one two three.   You guys know that already. More heavy ammo.  Look at both heavies as well. Oh this is gonna go really bad.  I'll get up on my head. Where is it?     Stop running.  Yeah teabag my size.  I don't even know if it was but still.  Come at me bro. Come at me bro. I want somebody to save me. If I can get up there. Don't go. Don't go. Pick it up. Pick it up. Pick it up. There we go.  We can win this game. We can pull his back. I'm scared to pull out of my gun. That kid's gonna get me. I don't even need shotgun ammo to be honest. I'll just grab it. That kid. That kid. He hates me. He hates me. I must have done something to his parents.  Wait I'll give you a herb and evidence.    We're in. I'm confusing me.  That convention he has got.  He's jointed. He's fighting me from there boy. He's not fighting me from there.  You guys see that then? That was insane. Oh that was insane. That was insane. That kid. He hates me.  I was on like one kill off as well. That was ridiculous. That was ridiculous. I'm doing good. I'm doing good. It's just putting me with absolute scrubs. Fair play to Thunder above me. But look at the kids below. One more game.  But yeah. Come on. Find me an enemy. I don't want to play against that kid.  He backed out. He backed out. He backed out.  Every time I pulled out my golden gun. He was waiting for me with his golden gun. He killed me with his golden gun.  I spawned.  I got him as many times as I got me.  Was he the one who teabagged me as well? Get out of here man. Put me on a decent map. I like that map to be honest. But you can get trapped in easy little app. I mean every time we were spawning action straight away.   I still come pretty good. What was it? Can you have a look? Was that 1.54 was it? Yeah 1.54. The Oculate MSC. Yeah to be honest. I'm doing good. I'm just playing with Skrubby teammates.  Perkite balance.  They're fucked. They're fucked.   But I appreciate it if you are.  It's going to be about 35 minutes. I mean like... I can post. In the daytime I don't really have that much time. Because I get up sort of late about 11am. 10am if it's Thursday. I've got to do videos. I've got to prepare videos. I've got to record them. I've got to make them. I've got to put them together. I've only gotten to about half three in the afternoon. Because my daughter comes home with my girlfriend. And they're here in the background here. Because I refuse to be away from them. When I'm recording. Well when I'm actually playing or whatever. Because obviously if you step away. Step away.  If you separate yourself from your family. It's not right. So I refuse to do that. I prefer to do all my work. In the same room where my daughter plays and my girlfriend watches her TV. You see my dattle leg. My leg's a bit bigger. So I refuse to do it. So I haven't got that much time to play. This morning I woke up pretty early. Can't remember what for. Because I was waiting for a package. And black shield. This was the one I was thinking of. Yeah this was the map I was thinking of. Yeah. So I can't really. I don't really have that much time. But because I got up early today. I got a few hours spare. So that's why I'm recording this video.  Then that's what we'll do. That's what I'll do as well. I'll do this obviously with single games as well. If you guys haven't got 30 minutes out of your day to watch a whole video. I'm for battle. But I don't mind. This is all good to me.  I like playing. I mean mayhem clash. I could play it all day. Every damn day to be honest. I just hope it's here next week. Or mayhem control. Mayhem control can be ridiculous. I'd like to see mayhem riff too.  No one's scoring points. You're in the lead.  You're on me already. I know that's gonna happen with death. You know what I need to do. Lost the lead. I need radar on screen when I'm aiming down sights. Because it's just... I just want to see where people are coming around me. I'll just stop the panic. I'm like this. I'm in and out looking at the radar.  I thought that was my dude. I'm in there dude. Let's get killed. Let's get one kill. A single kill. Come on now. This game's gonna be bad. I'm going negative. I ain't even gonna get killed.  Let's get scubby teammates again as well. I'm the scubby teammate here to join us. Got one kill. And I got no health at all. A 10 kills ahead already.  Shit no one's mercy murdered me. Bastards.  Some of them are gonna get the rocket if you don't really want to groove.  Oh he's double killed.  What was that then?  Oh blade dancer versus hammer or soul. Blade dancer got wrecked. Just the way it goes isn't it? No. No. Why are you fucking... Why are you using yourself?   I just don't get it. You should know. Take out a group of enemies with one blow. You should storm. Try and take out 20 enemies with one blow.  Wow I died so quick then. Every one of them must have been in a headshot. Every one of them must have been in a headshot.    I'm doing scrubbish now. I'm doing scrubbish.  Did he die? I'm not even sure. I feel like I'm dead just in case. We are almost 11 kills behind. Heavy ammo available. Hey hey.  No one waits for me. We got the horn shots. Here we go.  His red death is a beast. And again red death is a beast. Oh he got owned out of his heavy then. He got owned out of his heavy. Come on hammers man. You should be getting like 5 or 6 kills every fucking...  There we go. There you go.  That was so close. That was so close and I almost got wrecked. Come on hammers. Just follow the hammers. Hit a kid man. Come on now. Uh oh. I'm lucky with my throwing knife. And nothing happened. There should be one hit killed to everybody.   Heavy ammo is underway. I'm not even gonna move. Try more inside.  Nothing. No boy. I'll do the sniper across there. Oh that was a fail knife. Oh a fail.     Oh thank you.    Look at that. He dies 10 minutes after he gets round the fucking...   You can't fight a hammer or soul.  You see they get the kids. You get one hammer or soul.  I thought that was a guy shooting in the corner then. 10 kills behind. He's not going good at all. Heavy ammo available. No you're watching that thing. 30 seconds and counting. He's telling me that didn't hit me in his face.   I'll do the hovel again.  I got too far hooking up. 31 minutes.  I know I've come negative by about a million.  Absolutely crazy.  But yeah guys I hope you enjoyed the video. If you guys do want to see more live gameplays just let me know down below in that comment section. Also drop a like as well. I do appreciate your support. As always and I'll catch you on the next one. Peace out until next time. Peace.


I want to call your attention to the fact that we have to leave this room because it's too big and bigger classes needed. So effective next lecture, which is on Wednesday, September 1st, we'll be moving the 325 with Hunt. So please don't forget that. All right, last time I was telling you about emission operators. I reviewed your definition of an emission operator, which actually comes in two different forms rather recently equivalent. And I just want to mention a very simple but important theorem that occurs in funny mechanics, which is that if you have two emission operators, then the product of those emission operators is emission only if they can use. Proof is so trivial that I won't go over it in class, but it is something that will occur many times in the course. Now what I'd like to have started dressing today, first of all, is the subject of eigenvectors and eigenvalues, which are eigencets and eigenvalues. Let's suppose we have an operator A, not necessarily emission, acting on some ket space, and we have some non-zero ket U, take it to be non-zero, so I'm simply going to let A act on U, which just multiplies U by a number, lowercase A, which in general is a complex number. Here, capital A is the operator and lowercase A is what we call the eigenvalue, as I guess you all know. And likewise, the ket U here we'll call this the eigenket, or sometimes eigenfunctions if we're talking about way functions or eigenvectors. Anyway, that's the eigenket and the eigenvalue. Now, the eigenvalue as I mentioned in general is a complex number, depends on the operator A. Now the situation regarding, or the theory regarding eigenvalues and eigencets is considerably simpler in a finite dimensional ket space than it is in an infinite dimensional one. So I'm going to begin primarily with, I'm going to begin by telling you about finite dimensional spaces, but I'll make a few comments as we go along about how situation is different in infinite dimensions. Here's the first thing to notice is that if you have a ket U, an eigenket, not supposed to be non-zero, because if it were equal to zero, this equation says zero equals zero, and it's not very interesting. We exclude the zero case. So if U is a non-zero ket, then you can think of it as a vector that sticks off somewhere like this, and the spacing question, here's the vector U, or the ket U. It's obvious from the eigenket eigenvalue equation, however, that if you multiply U by any constant, a complex constant, you get another eigenket. So this equation does not uniquely determine the eigenket, only to within a multiplicative constant. Or to say in another way, if I take this non-zero ket U and I multiply it by any vector I want, it could be a negative vector as well. But since these are complex vector spaces, it could be really complex numbers, including phases and so on. But in any case, the result is what you give is, from a given non-zero ket, you can construct a whole one-dimensional subspace of eigenkets. You could call this an eigenray, if you want. In some sense, this equation really determines an eigenray rather than a single eigenvector. Now, it may happen that there's, in addition to this vector U, this ket U, it may happen that there's another vector, or ket, which is linearly independent of the first one, which is also an eigenket of the same eigenvalue. Let's call this V, let's say. A V is equal to lowercase A V. The point is the A's here are the same, but the V's are different. And let's suppose the V is linearly independent of U, so that it sticks off in some direction like this. Then it becomes apparent that any linear combination of the U and the V is also an eigenket. And so the space of eigenkets you see becomes a two-dimensional vector space like this. In these circumstances, it becomes appropriate to speak of an eigenspace. And that's what we'll do. We'll call this the eigenspace. I'll call this E's of A, script E's of A. I'm generally using the notation script V of the entire ket space, the entire vector space. So with an A subscript on it, it stands for the eigenspace corresponding to eigenvalue A as a given operator A like this. And the result of this is we get a geometrical interpretation of this eigenvalue eigenket problem, which is that if you have an operator A, it corresponds to a certain privileged subspaces of the ket space. In general, we can think of two of them just trying to draw a picture of it. If I've got two spaces, here might be one, and then here might be another one going off like this. I've got one of them being two-dimensional, one of them being one-dimensional. Let's say this is E1 and this is E2. Our two different ket spaces corresponding to eigenvalues A1 and A2. You can imagine it looking something like that. Now, as you probably know, if there is more than one linearly independent eigenket, then we say that there's a degeneracy. And the number of linearly independent eigencats corresponding to a given eigenvalue is something we'll call the order of the degeneracy. So the way I drew it here, the order is two because it spans a two-dimensional subspace. The order of the degeneracy is just the dimension of the eigenspace. If the eigenspace is one-dimensional like I drew here for this E2, then we say it's non-degenerate, which means that the... it doesn't mean the eigenket is unique, but it's unique to within a single multiplicative constant. All right. So the, as we'll say here, the dimensionality of the eigenspace again is what we call the order of the degeneracy. So the order of the degeneracy is always one, but it could be greater than one. All right. Now, the... another thing that comes out of this is that given operator A is associated with a collection of numbers, generally complex numbers, which are the possible eigenvalues of A. That's got a terminology. It's called the spectrum of the operator A. So the spectrum of A is just the set of... it's just a set of eigenvalues A and A. That's all it is. This is the spectrum of our operator A. And it's usually viewed as being a subset of the complex plane. So if I... it's a complex eigenvalue plane, really. So if I have the real A and imaginary part of A as being the... give us the complex plane for the eigenvalue, then in general the spectrum... I mentioned a moment ago that I'm describing the situation in a finite number of dimensions, first verse simplicity. And if we're in a finite number of dimensions, the spectrum consists of a discrete set of points, which can be distributed here and there in a complex plane. That's the spectrum of the operator. All right. That's just a terminology, mostly. Let me give you some examples of the spectrum of operators coming from a lot of mechanics. Let's, first of all, talk about the harmonic oscillator Hamiltonian. The harmonic oscillator Hamiltonian. So let's write hx on u n. It gives us e n u n. And as you all know, the energy eigenvalues are equal to n plus 1 half h bar omega. And so the spectrum... we have to look in a complex energy plane. There's real energy, imaginary energy. The spectrum consists of points first 1 half and 3 halfs, 5 halfs, 7 halfs, and so on. As you see, it's a discrete spectrum and it's on the real axis. Here's another example. Let's take the free particle, h equals p squared of 2 n. And then here's what the spectrum looks like. Real energy, imaginary energy. In this case, there are eigenfunctions for all non-negative values of the energy. So the spectrum consists of the entire real axis like this, right abroad having... excuse me, the entire positive real axis, not negative. This is an example of what's called a continuous spectrum because it's not discrete points. Continuous spectrum only arises in infinite dimensions and I'll say more about it a little later, but right now I'm just giving you a preview of some of the things that can happen. Here's another example. Let's take the hydrogen atom, Hamiltonian. We have to talk about the energy spectrum, so real energy and imaginary energy. In the case of the hydrogen atom, you know the ground state, everyone knows it's minus 13.6 electron volts, so there it is, there's the ground state and negative energy. And then there are other bound states of negative energy and they pile up, they have an accumulation point at e equals zero. There's actually an infinite number of them that accumulate there. And then they don't actually emphasize this enough, but actually for positive energy there's a continuous spectrum. The hydrogen atom possesses a continuous energy spectrum for positive energy, which is similar to the case of the three particles as far as the spectrum is concerned. Anyway, that's the hydrogen atom. This is a mixture of it's creating a continuous spectrum as you can see. And then finally let me just mention the case of the unitary operator. The unitary operator is one whose inverse is equal to its submission conjugate, so this is unitary. And I'll write the real and imaginary parts there for the eigenvalues. It turns out the eigenvalues of unitary operators are always phase factors. And thus they lie on the unit circle in the complex plane. And if we are in a finite dimensional space, there's only a finite number of them, so they're just going to be discrete points which sit on the unit circle and they don't even be spaced or anything. They can be anywhere on the unit circle like this. So these are examples of what's meant by the spectrum of an operator. And you see in general, you may have to be viewed as in the complex plane. You see this in unitary operators. I think what I'm going to do is go over here and enter the next phase to this. So that's the concept of the spectrum. Now, the whole subject of eigenvalues and eigenquets simplifies quite a lot in the case of the special case of permission operators is permissioned.  And fortunately, we don't have to do it very often in quantum mechanics. But in the case of those submissions, there's a number of simplifications which all this here. The first one is, so here we want to think of A acting on U is equal to A U. The first one is that the eigenvalue A is real. You can see this in the case of the Hamiltonians, all of which have a spectrum if I can fit me on the real axis because the Hamiltonian is remissioned. But the unitary operator has eigenvalues in the complex plane. Anyway, for permission operators, they're real. The proof of this is elementary. All we need to do is to take the eigenvalue equation and multiply on the left by the bra U like this. We're going to multiply both sides.  And then we take this equation and we apply the permission conjugate to both sides. The left hand side turns into itself A U A. If they were not remissioned, you get a dagger in the middle. But because of remission, it doesn't. So you get the same thing. That's what's special here.  And so the left-hand sides are equal, so the right-hand sides are equal too. And we're assuming that the eigencat is not zero because that's what we mean by eigencast. So this scalar product U U is non-zero. So the result of this implies that A is equal to A star. You've probably seen this proof before. That's very simple proof. All right. The second thing to say is is that the eigenspaces are orthogonal. And so the picture that I drew, it's still there up at the top of the board right there, it's actually not a very accurate picture. Not to the case of the permission operator. In the case of the permission operator, I had to draw it like this. So here's one eigenstate. Let's call it E 1. And then here's another one. Let's call it E 2 corresponding to eigenvalue A 2. And these are orthogonal, so there's actually a perpendicular to it. And it's like this. So if an ordinary operator, annual operator corresponds to a certain privileged subspaces of the ket space, in the case of permission operator, these are actually orthogonal subspaces. Now, to say these spaces are orthogonal means that any vector in one space, in any vector in one of these spaces is orthogonal to any vector in the other space. I've drawn this so that E 1 is two-dimensional, so it's a two-fold degeneracy, and E 1 is non-degenerative. The same picture applies in any case of any degeneracy. The proof of this is also quite simple. It works like this. Let me take, let's say that A x on U 1, and just the U 1 all over again multiplied by an eigenvalue of A 1, which I'll write up to the right now instead of the left. It doesn't matter which way you're writing. It's multiplying a number times a vector.  But these are two different eigencets and possibly two different eigenvalues. Now, let's take the first equation and multiply the left by the bra U 1. Excuse me, by the bra U 2. On both sides. And then the second equation, multiplied by the bra U 1 on both sides. So we get this. And then let's take the second equation and apply a dagger to it.  I don't dagger A because it's a mission.  But now you see, if we compare the first and the third equations, the left-hand sides are equal. And so the right-hand sides of the first and the third equations have to be equal also. The scalar product is the same.  And so, either the two eigenvalues are the same or all the scalar product is zero. Which is clear on this picture here, because if the two eigenvalues are different, so they belong to different spaces, then the vectors are orthogonal, just like I drew it here. On the other hand, if the two As are equal, then nothing can be said about the scalar product. They're not necessarily orthogonal, and that's clear too, because if I've got two vectors inside space D 1, they certainly don't have to be orthogonal to each other. Of course I can choose them to be orthogonal if I want to. This is related to the fact that a permission operator on a finite dimensional space always possesses an orthonormal eigenbasis, which is something I'll come to in a little while. But we don't have to choose a orthogonal inside one of these eigenspaces if you don't want to. By the way, eigenvectors are not unique. I mentioned that already, because just in the case of the single eigenvector, you can multiply by any non-zero scale factor. Of course, normally quantum mechanics would usually normalize them to make them have a unit norm, but even then, that leaves the phase factor of indeterminate. So even with that convention, you still got an arbitrary phase. If there's a degeneracy, it's worse than that, because it's clear from this picture that you can rotate this any way you want. There's a huge number of ways of choosing what's no eigenvectors inside this two-dimensional space, or any space with a degeneracy of higher than 1. All right. Now, there are further simplifications in the case of permission operators, and let me illustrate them by going back in this picture here for permission operators. In fact, since I'm now talking about permission operators, let me erase some of the rest of the stuff here which applied to other cases that I don't care about anymore. Let's move back. All right, so let's talk about the case of permission operators. We have two spaces here. The way I drew it, you'll probably imagine this is a three-dimensional space, so it's broken up into 2D plus 1D. You can see that here. The question is, what would happen if this were in 4D? I'd have, suppose I had an operator that had the two-fold degeneracy for one eigenvalue or one-fold degeneracy for another eigenvalue. It acted on a four-dimensional space, but there weren't any more eigenvalues. Well, it would be like we weren't enough dimensions here to fill up the space. Okay. Well, this leads me to a concept that I was talking about, the concept of the direct sum of two subspaces. The direct sum, which I'll be known by, in this case for this drawing, we're going to talk about the direct sum of these two subspaces, E1 and E2. What this means is the direct sum of these two subspaces is just a set of all vectors you can get by making linear combinations of vectors that come from either one space or the other space. So, in this picture here, it's a three-dimensional space, you can see. Another way of saying it is that if I choose a basis in E1 and then another basis in E2, and I just throw these basis vectors together, in this case it's giving me three basis vectors, and then consider the span of that. That's what we mean by the direct sum. So, the result of this is that the dimension of the direct sum of these two spaces, E1 plus E2, is going to be equal to the dimension of E1 plus the dimension of E2, dimension to add and then taking the direct sum. This is just the meaning of the direct sum. Now, if we have an operator and we consider this eigenspaces, and then we take the direct sums of all of the eigenspaces, the question is, does that fill up the whole space? Like I said a minute ago, maybe this picture here is in 40 space, so we've got a two-dimensional subspace and a one-dimensional subspace, but it's nothing out of the four. What happens then? So what that happens is that we say that the operator is not complete. So a completeness is a means to say that the eigenspaces fill up the whole space. Well, that was a nice result about our mission operators, which is that they are complete. At least this is true in finite dimensions. They're always complete. This is not true for non-remission operators, and it's one of the great advantages of remission operators. It simplifies a whole lot of things. In finite dimensions, permission operators are always complete. So it wouldn't happen in finite dimensions, namely four, that you'd ever get two plus one equals three of the missing subdimensions. It wouldn't happen, at least not for remission operators. In infinite dimensions, it turns out this is not necessarily true, that there are remission operators that are not complete. By the way, I don't know if I mentioned this last time, but in the mathematical literature, especially if you're able to term self-adjoint, if you're a purist about this, they aren't exactly the same. They depend on domains of definition and stuff like that. I will never use the word self-adjoint, but if you see it, I think it's the first pass of practical applications and quantum mechanics, you can regard them as being equivalent, because they are for most applications. Well, in any case, in infinite dimensions, there's a difference between in infinite dimensions, a remission operator is not necessarily complete. And so what we do is introduce a new definition, which is that of an observable. An observable, an observable is an operator which is remission and complete. It's not necessary to add the second part of here in finite dimensions, the infinite dimensions you should do this. Now the reason this name is given has to do with the physical interpretation of quantum mechanics in which one of the possibilities of quantum mechanics is that if you make a measurement of a physical observable, there's a big possible outcome of the eigenvalues of the corresponding operators. And if the operator were not complete, the operator is supposed to be remission, and if the operator were not complete, then it would mean that if you added up the probabilities of all the possible answers you could get on making a measurement of an ensemble of systems, you could get an answer which is less than one, because it would be some part of some vector that could stick off in these extra dimensions. This is absurd. It violates the physical principles of quantum mechanics. And this is why in quantum mechanics we deal not only with permission operators but actually with observables. These are the only ones we deal with really. One of the consequences of going back to this picture here, one of the consequences which is fairly obvious in a picture like this, is that an observable or any remission operator in a finite dimensional space, let's talk about finite dimensions now, is that any remission operator on a finite dimensional space possesses what we call an orthonormal eigenbasis. An eigenbasis just means, well, a basis means a set of vectors expand in space. And an eigenbasis means that there should be eigenvectors as well of the given operator. Well, it's clear this is true because if there's all you have to do since these different eigenspaces are orthogonal to one another, all you have to do is just choose an orthonormal basis of each of the eigenspaces. And then the collection of all of those becomes an orthonormal eigenbasis for the whole space. Here's some notation for this, talking about orthonormal eigenbasis. Let's suppose that A axon channel called nR brings out an eigenvalue of An times ket nR. Now, the R here is an index that we introduced through what we say resolved degeneracies. As to say nR, in the notation nR, we let R run up from 1 up to 1 up called d sub n. And d sub n is defined as the order of the degeneracy of the eigenvalue An. In other words, it's the dimension of the corresponding eigenspace. And so the idea is that these kets nR, the n indicates the eigenspace and the R is a label of an orthonormal basis chosen inside that space. And as I mentioned, there's a great little barbituriness in how you choose that basis. In any case, let's use this notation for the eigenvalue. Unless the eigenvalue doesn't depend on R, it only depends on n. Well, if you have this, so this is just the notation for the eigencats, the orthonormal eigenbasis, the corresponding to this, we have a resolution of the identity that says that 1 is the sum of n and R of the outer product of nR with nR. This is just the normal thing you get when you write a resolution of identity in terms of the orthonormal basis. Turns out you can also write the operator A in terms of the orthonormal basis. It looks like this. It's the sum of n and R of the eigenvalue An times nR of the outer product of nR. That's just like the line above, except you inserted the eigenvalues under the sum. How do I know these two operators are the same? Well, you see, if you have an operator and you know what it does to a set of basis vectors, then you know what it does to any vector because of linear superposition. So the way you prove that these two sides are equal is you allow them to act well, what basis vectors should you choose? Well, the obvious answer is these basis vectors. You put those in there and you can easily check that both sides give you the same answer. By the way, I should have mentioned something here, which is the orthonormality conditions of these vectors. These are the following that nR scale of product within prime R time is prime of the delta n prime prime of the delta R R time. The two prime of the delta is fair for different reasons. The reason it's a orthogonally independent prime is because of the property over here that eigenspaces are orthogonal or in this picture here that we have long different eigenspaces that are orthogonal. Whereas the orthogonality in R prime is because of our convention of choosing an orthonormal basis inside each of the eigenspaces. In any case, this is the complete orthonormality relation and that's what leads to this resolution of the identity in terms of its eigenvalues and eigen and eigenbasis. Right. Now I defined the direct sum of two sub-spaces up here. I think I hope it's clear what that means. Let me just mention now that the completeness property can be expressed in terms of the direct sum. It says that if you take the direct sum of the eigenspaces, all of them, you will get the entire ket space. So completeness in terms of direct sums. Just a way to say it. As I mentioned, most of the theorems about eigenvalues and eigencets and so on are considerably simpler in finite dimensions. Let me say something about the infinite dimensional case. So you can get an idea of some of the things that can arise there. You can actually see some of it already. You can continue a spectrum in the infinite dimensional case. It's always a finite spectrum. A discrete spectrum, I should say. To illustrate what can happen in infinite dimensions, let's talk about the space of wave functions in one dimension which we know a lot about already. And let's talk about the momentum operator, which I'll denote by p. Of course, minus i h bar d dx in the shorting of representation. And I'm going to put p without a hat in the eigenvalue of the operator. So the eigenvalue eigenfunction problem for the momentum looks like this, is that p acts on some function, let me call it use of p of x, which is supposed to be an eigenfunction of the momentum operator with eigenvalue p. So we let p act on it and it has to just multiply by p. This is an eigenfunction equation. The momentum operator is minus i h bar d dx which becomes d dx and u d of x which is a simple differential equation. And when we solve it, we find that u p of x is equal to e to the i dx over h bar. And that's it. Okay, so within a most indicative constant that's the eigenfunction. Now, this differential equation has a solution of the momentum eigenvalue, not only real ones, but also complex ones. There's a solution for any value of p. Does this mean that the spectrum of the momentum operator is the entire complex plane? Well, the answer is no and part of the reason is that this is in the continuous spectrum and I need to tell you some things about that. But let's just notice that if p had a non-zero imaginary part of real p imaginary p. If you imagine an eigenvalue which is off here in the complex plane with a non-zero imaginary part, then this function would diverge exponentially either for positive x or for negative x and it would rather badly behave. It certainly wouldn't obey the boundary conditions of the used to in quantum mechanics where you want wave functions to die off in infinity. Actually, what you really want is the wave function to be normalizable functions that are physically meaningful because the probability of finding a particle somewhere has to be one and so only normalizable wave functions are physically meaningful. Well, in any case, if p is a non-zero imaginary part it's certainly not normalizable. What happens if p is real sitting on the real axis? Well, then this becomes the familiar sine of e sine of cosine e to cosine x which, however, does not die out as x goes to plus or minus infinity. In fact, there's absolute value of 1 everywhere. So it's still not normalizable and it still doesn't represent a physical state and it still doesn't belong to Hilbert space. And so what we have to say is that if you're looking for normalizable vectors or normalizable states that belong to Hilbert space the momentum operator doesn't have any eigenfunctions at all. Not much less an eigenbasis. So how do we deal with this consideration? Well, I'm just going to describe to you how one normally thinks about this. In the first place we restrict consideration to just real values of the momentum. This of course is the, this is the idea follows the case of finite dimensions in which permissioned operators have any more eigenvalues. If we do that then we can take this eigenfunction u p of x divided by the square of 2 pi h bar. This is normalization constant. But the reason for doing that is if I take the scalar product of u p let's say u p prime I'm, the idea here is to investigate the orthogonality relations of these eigenfunctions. The, let's just abbreviate this by the way my writing is p p prime dropped to u where we just label the eigenfunction by value. Well, this is the same thing as the integral over x of u p of x complex conjugating times u p of u p prime of x. That's equal to the integral over x divided by 2 pi h bar of e to the minus i p minus p prime x over h bar. I ask this out of the way here. And that is the standard representation of the development function p minus p prime. So to summarize this in one spot, we have the scalar and product of p p prime is a direct delta function of p minus p prime. If we divide by this square root of 2 pi h bar is a normalization, that's why I did this to make this delta function come out. So what we see is that these eigenfunctions of the continuous spectrum, that's what we're getting here, are normalized in the delta function sense instead of a chronic delta sense. Moreover, the axes are complete in a certain sense also. They're complete in the sense that an arbitrary wave function psi of x can be represented as a linear combination of these u p's. However, since the eigenvalue p is a continuous variable now, we have to use an integral instead of a discrete sum. And so we write this as an integral over d p. Let's call an expansion coefficients all called pi of p because there exist expansion coefficients such that any psi of x can be represented this way. And the answer is yes, because you write this out more explicitly. It becomes the integral d p over the 2 pi h bar of pi of p. Well, I'll put u p of x first, u to the i p x over h bar under pi p. And so what you recognize is that psi of x is the Fourier's theorem that says an inverse which is integral d x over the square root 2 pi h bar e to the minus pi p x over h bar times psi of x. And so the answer is the expansion coefficients exist. They're nothing but the Fourier transform of psi or as we say the fun mechanics of the momentum space wave function. And in this sense these eigenfunctions continue as a factor marked complete. That's the situation and part of the situation with the momentum operator in a continuous spectrum. I think I want to say one more thing here which is to take this last formula here and let's just reinterpret this. Notice that this becomes the integral over x and if I take the 2 pi h bar under e to the minus i p x bar that's the complex conjugate of our eigenfunction. So let me write this as u p of x that's the complex conjugated multiplied times psi of x and this in the draft notation is the scalar product of the basis function with psi so I'll write it like this as p psi and let me accumulate that in a single box too because that's interesting it says this pi of p is the scalar product of p psi and so the expansion coefficients which we otherwise call the momentum space wave function are just the let's do something similar with the position operator again in one dimensional wave mechanics so let's call x hat is the position operator which is multiplication by x but the hat is the operator and x is the number by the way as in the side here I'll put it over here somewhere there's some terminology that goes back to Dirac which I'll use occasionally in this course Dirac distinguishes a q number and a c number and you'll certainly hear this if you're in a quantum mechanics business very long well a q number is just Dirac's a notation for a terminology for an operator and a c number is a terminology for an ordinary number it has to say an ordinary complex number I remember this because c stands for complex but anyway the distinction here is that x hat is the q number which is the c number here alright and let's let x and 0 be the eigenvalue and let's let f x 0 of x be the eigenfunction so if you want x hat acting on f x 0 you can put x 0 times f x 0 on x like this however since x hat just means multiplication by x this is the same given x times which implies that x minus x 0 times f x 0 of x is equal to 0 which implies that either x equals x 0 or else f x 0 of x is equal to 0 this function must vanish everywhere except that x equals x 0 it can only be nonzero at one point it's what this is saying well the interpretation of this is that we interpret it this way by saying f x 0 of x is the delta function of x minus x 0 the function is not a real function but it can be interpreted as a limit of real functions and in some representations of it you see this looking like something that becomes concentrated at the point yet we take this definition of f x 0 of x as the eigenfunctions of the position operator then we can let's take the scalar product of f x 0 f x 1 this is the same thing I'll just write this for simplicity by writing just down the eigenvalues instead of the full function of x 0 x 0 but this is the same thing this is integral over x of delta of x minus x 0 complex conjugate times delta of x minus x 1 well the delta function is real so the star doesn't do anything and if you do the integral you get yes so comparing to what we have above we have something similar in the extra representation the scalar product of x 0 x 1 is the delta function of x 0 minus x 1 also there's other simple identities that we have if we have psi of x can be written this way it's the integral over I guess I want psi of x 0 it's the integral of dx of delta of x minus x 0 times psi of x is one of the properties of the delta function but this can be also written as the integral over dx of f x 0 of x complex conjugated times psi of x which is therefore the scalar product of x 0 with psi and so we find the wave function psi of x 0 is the scalar product of x 0 with psi and I ask you to compare these two results with the momentum results of the above you see in both cases the wave function is just in fact just the expansion coefficients of the abstract state with respect to the basis which is an eigenbasis of an operator this is one reason for not regarding wave functions and configurations based on the central object rather than the abstract vector is really the central object these are just different choices of components by different choice of basis choice of basis is an arbitrary choice alright now there's finally a couple more relations related to this which I'll just skip over the details and just summarize the results these are resolutions of the identity like both the momentum and the case of position looks like this is as if one is the integral over p of the outer product of pp we've also one is the integral over position of the outer product of x and these become the resolution of the identity relation in the case of the continuous vector here's another couple of remarks about the continuous vector if we have a single value of p let's speak of momentum let's take a single value of p of the momentum axis here single value of p here's the real part p here's the measure of p to some point in the spectrum of p can we talk about the eigen space corresponding to that momentum value we can certainly do that in the cases of discrete spectrum and the answer is no because the eigen vector corresponding to this value doesn't even belong to Hilbert space these resolutions of the identity we have here are work in the function sense but the vectors in question don't lie in Hilbert space this is the difference between the continuous spectrum and the discrete spectrum and there is no such space of the Hilbert space corresponding to a single value of momentum on the other hand if I want to take two interval I to be able to interval p0 and p1 then it does make sense to talk about the subspace corresponding to that interval and later on write down an explicit formula for this but I think you can believe this on the basis of what you know about signal analysis you know about band limited signals or filters in electronics that remove all the frequencies except outside a certain frequency while there is the acceptance of some kind of a filter it accepts wavelengths that lie or momentum that lie between p1 and p2 and reject everything else that's actually a projection operator the projection one to a subspace corresponding to a band in the continuous spectrum so in the continuous spectrum you get subspaces of the Hilbert space corresponding to intervals of the continuous spectrum quite easy as if I just do an example which is hydrogen atom let's let the as usual hydrogen atom language let's let NLM be the wave functions of the outstate wave functions of the hydrogen atom these are orthonormal and send them NLM together product with N prime all prime N prime LL prime N prime the eigenvalue only depends on N NLM X on this because that's E N I'd say LL where I think you know that E N is proportional to minus 1 to 2 N squared but the point is it depends only on N and not on the 200 quantum numbers and this means that the the ion spaces of the hydrogen atom and NLM are degenerate X which is still there I guess it's gone now the R in X I introduced earlier that just indicates any effect of arbitrary choice of orthonormal spaces of each of the energy ion spaces however the hydrogen atom also has positive energy ion states I'll denote those by E L M like this and this is the case where E is greater than equal to zero it's a continuous spectrum now there's no discrete analysis in the following way is that E in E L M scalar product with E prime L prime L prime this is direct delta function in the energies times the product of delta in the L's and the product of delta in the L's and if you do this then so this is the worth of malady relations actually they're part of them here's the worth of malady relations for the bound states here's for the unbound states here's another one too is that the unbound states are zero the bound states and the bound states and the unbound states are orthogonal so three formulas here these are the worth of malady relations now here is the resolution of the identity for the hydrogen atom we have the sum on E L M which runs over all the bound states of the outer product E L M and then we need an integral from zero to infinity over energy and then a sum on L M E L M outer product of E L M this is what it looks like there's also you might call I also mentioned that the operator itself can be written in terms of it's eigen states if we do this for the hydrogen atom and it looks like this all we have to do is put in the energy driving values under the sum of the bound states but also for the unbound states what it looks like for me based on the energy here regarding regarding observables in quantum mechanics is that two observables in theorem I'll write it out observables A and B possess a simultaneous eigen basis only if they commute it's an important theorem for quantum mechanics because we're the question of commuting observables is closely related to the question of the possibility of making simultaneous measurements of two different observables without one measurement interfering with the other it's related to the uncertainty principle and things of that sort so the theorem says that A and B observables A and B possess a simultaneous eigen basis only if they commute now an eigen basis as I mentioned earlier is a basis which is also consists of eigen factors of A or eigen eigen kets of the given operator what I mean a simultaneous eigen basis is I mean a basis that's eigen kets of both operators A and B simultaneously so every eigen vector is an eigen vector of both A and B how do we see that this is true probably the easiest way is to go back to this picture that we're thought of as eigen spaces so let's call this B and here and here a different one called the I'm only going to concentrate on B so here's the idea let's suppose that let's suppose that I have A acting on a ket u gives me a sub n u that means that ket u lies inside this eigen space E n it means with A at son all it does is it multiplies by a complex number so it gets longer it changes its direction a minute but it stays inside alright now what about if we let the operator B act on a vector that lies in this space E n what can we say about that well let me put parentheses around that so in particular B is going to act on this vector what will it do could it possibly lift it out so that it sticks out in this space somehow that's a question so because if I take B u and I let A act on the left hand side this is of course the same thing as A B acting on u but because B and A are assumed to commute that's the same thing as B A acting on u and then the A acts on u and brings out the eigen value of A n which is just a number so this can be written as A n times B acting on u and the result of this vector B u you see is an eigenvector of A with the eigenvalue A n that means it belongs to the space E n and so this picture I drew is wrong instead B u has to be some vector that's also inside this space so this is an example of what we call an invariant subspace we say that the eigen space of the operator A is an invariant subspace of the operator B cuts it into another vector in the same space the space is also an invariant subspace of A because A acts on an eigenvector in this space and just multiplies by a scale factor which certainly keeps it in this space so the eigen space of A is actually an invariant subspace of both operators if they commute now if you have an invariant subspace of an operator you can define what's called a restriction of the operator to the subspace but if the subspace is not invariant of the operator then it doesn't make any sense to talk about a restriction because if I've got a vector in a subspace and I let an operator act on it and it brings it out of the space then it doesn't make any sense to say what is A restricted to the space but anyway we do have an invariant subspace here of both A and B let's call the restrictions of these let's put a borr over this to indicate the restrictions the operator A is actually quite easy it's just the eigenvalue multiplied times the identity and restricted to this space the operator B is not so simple to write but one thing we can say about it is that it's a permission and so the result is that B the intermission on this subspace possesses an orginal basis on this space so there's a B basis here which is more than normal and that B basis is obviously an eigenbasis of A as well therefore you get simultaneous eigenbasis of the two operators okay well that's all I'm going to do for today to remind you again we meet next time in 325 La Hante not here as soon as the game is over


Hey, this is Brett, and this 2018 Dodge Challenger RT Shaker is stock number 13792Z. I am here at Summit Automotive in Fondlec, Wisconsin, you're new and use Dodge Challenger and sports car headquarters. This 2018 Dodge Challenger has the 5.7-liter V8 Hemi engine, puts out 372 horsepower, and it's paired up with the 8-speed automatic transmission. This vehicle has been fully-safetyed and inspected by Elmer Service Shop, has a fresh oil and filter change, all the fluids have been checked and topped off. This car is 100% ready to go. I'm going to go all the way around in this video. Inside, start it up, take a look under the hood, give you the most accurate representation that I can of the car. Destroyer gray is the color, I shoot all my videos in 4K, and if you like the video you can subscribe to my YouTube channel, which is youtube.com slash Summit Auto. Click the bell notifications, you'll get updates on the videos I do each and every day, as well as having access to one of the largest catalogs of vehicle and sports car walk-arounds in particular on YouTube. Comes with the 20-inch painted and polished aluminum wheels, they're in really nice shape. Comes with some general G-Max tires, these are 245.45 ZR20s. I would say they have a really good amount of tread left, probably 60-70% of the tread left on those tires, maybe even more. Front fender is in excellent condition, no dents or dings on there. Has a projected lamp headlamps, the LED running lights, and the projected lamp fog lamps. Front bumper and lower valance are in a very nice condition, I didn't see any dents or cracks on there, looks really good. You get that shaker hood with the true cowl induction, I'll show you how that works in just a minute, you also get the nice stripes that go up over the roof. This is an excellent shape, I didn't see any dents or dings on there, and the passenger side front fender is in fantastic condition as well. Passenger side front wheel, no major scuffs or scrapes, and as you go down this side of this 2018 Dodge Challenger tick note of how clean that body is, how reflective and mirror-like that paint is, I take these HD videos so if you're far away or even if you're close by and you just cannot make the trip down, but you're still interested in purchasing the car, you can see the car, hear the car, and have confidence in the vehicle that you're looking at before you can get here, so when you do get here there's absolutely no surprises. You can make a smart and informed buying decision from wherever you're at. Now if this video helps you make that buying decision or just the decision to make the trip, let your salesman know that you saw the video, that it was helpful and that Brett sent you. Back wheels in excellent shape as well, and the back tires have just as much tread as those front tires can actually see it a little bit better back here than in the front. Coming around to the back of the vehicle, the rear bumpers in fantastic condition, I didn't see any dents or dings on there. You get the chrome-tipped dual rear exhaust, the backup parking sensors, the LED tail lamps, you get those stripes on the trunk lid as well. Trunk lid itself is in nice shape. Back storage area is very nice and clean, and back here you get a Kenwood amplifier, tire inflator kit, and this is where your battery is located. Nice and clean back there, shocks are doing a nice job pulling that trunk lid up, and that shuts nice and solidly. Coming down the driver's side just as clean as that passenger's side, didn't see any dents or dings on the rear quarter, and for full disclosure this back wheel is just as nice as the rest. And down the driver's side door, looks really good, didn't see any dents or dings in there, and it does have heated power mirrors. You get the enter and go system on here, inside, this one comes with some cat skin, tan, or brown leather seats, little rips or tears on the seats, side curtain airbags, they're in really nice shape. Power driver seat with lumbar factory floor mats, auto headlamps, tilt telescopic steering wheel, power windows locks, and mirrors. Back seats are in excellent condition as well, does have the latch child safety system for any child car seats you may have, and factory floor mats back here. We'll hop inside, check out the miles, the radio, and everything that this car has to offer on the interior. Let's see that this one has 23,532 miles on it. You get the seven inch LCD display on there, instrument cluster is very nice and clean, comes with the leather wrapped steering wheel, cruise controls on the right, Bluetooth and information center controls on the left, and audio controls in the back as well as your paddle shifter, comes with the Uconnect seven inch radio, AM, FM, and Sirius XM radio capabilities, you also get all your different apps on here, and you can project your cell phone to the screen via Android Auto or Apple CarPlay. So if you have a nav system on your phone, such as Waze or Google Maps projected to the screen, it's like you have Navra in the screen, we actually did a demo on an 8.4, same system, this is a little bit smaller radio, but you can check that out, that's in the upper right hand part of screen.  This one does have the keyless entry with the remote start on there, passenger side, floor mat and seat, are in fantastic condition, smells very clean inside this car, you do get some, only the strong survive head rests, and the headliner is in excellent shape and this one does have the power sunroof. Up here you get map lights and sunroof controls and your auto dimming mirror. We'll start it up and take a look under the hood. Starts right up, no check engine lights or anything like that. I would personally like to thank you for checking out the video today and hopefully from this HD video you've been able to verify the quality, condition, options, and of course cleanliness of this car all the way around, see those lights are all working very nicely. Under the hood we have the 5.7 liter V8 Hemi engine, engine base very clean, runs very smooth. Once again this car has been fully safe and inspected by our service shop, has a fresh oil and filter change, all the fluids have been checked and tapped off. It is 100% ready to go, you get the Mopar cold air intake there, the shaker air intake as well. Get the shaker sticker on there and that's wide open. There's the emission sticker. And like I said this car is 100% ready to go. I would highly recommend this car from a quality and condition standpoint. I'd ship this one just about anywhere. Whoever's going to get it's going to be very happy with the quality and condition of it. And to see more pictures of this car or one of other 550 new news cars, trucks, SUVs, minivans, Wranglers, challengers, chargers, Mustangs, Corvettes, Camaros, you name it we got to go to that website right there, summitauto.com full pictures and descriptions of every single vehicle all at summitauto.com. And if you'd like to check out more HD videos you can go to youtube.com slash summit auto click the bell notifications. You'll get updates on the videos I do each and every day as well as having access one of the largest catalogs of vehicle and sports car walk arounds on YouTube 8000 videos and counting. In fact in a second you will see a link to subscribe my YouTube channel in the upper left a link to more challenger videos that have done in the upper right link to this vehicle on our website and lower left and a link to one of our latest YouTube videos and lower right click those checks out. We're super excited to help you with this ultra clean 2018 Dodge Challenger RT shaker and destroyer gray clear code. Thanks again for checking out the video. Remember to like, subscribe and share on the YouTube channel. I really appreciate it. Thanks again.


Mae'rmellu gweld geni gwyardzogol yn gyllid ar desbyty y peictr Note one o wirio yn ie uhmie sen. Lyfan symud. … Dieu i mi ar dem ni, Cymru ciumaeth chym graduation yn gyllid sources Aron y cangwyd arcafn oherwydd strikinguryl maen 따�f yn tus plate. Cymru gyntaf i chi eri i dim i ang tenants Bringo fod cael eu rhoi i prototype Yn ein hollad d discretol at seith i weithgoffaith Fyrd Dodd-Dale. I look forward to seeing you all at the reception this evening. Three years ago, almost exactly a month after I was elected, I visited Homestar Glynwathus at their Macedonia base. I will always remember the feeling walking into the community house, one of support and one of empathy, and persuasion, because just a few weeks later I found myself running the Glynwathus 10K in aid of the organisation and the vital work that they do. Homestar Glynwathus was founded 21 years ago, and locally the organisation trains volunteers and experts to support between 60 and 70 families in the area at any one time. Across the UK, 13,500 volunteers support over 27,000 families and 56,000 children. Homestar works across my constituency to support families who are struggling, whether that be because of social isolation, poverty or poor mental health. None of us can predict what life will throw our way, whether that is illness, disability or bereavement. Homestar works to take that pressure off families who are in need. Ahead of tonight's debate, I spoke to Lindsay Brown, our local homestar coordinator. Lindsay told me that I am a great believer in that your postcode cannot dictate what your future holds. We work in Glynwathus to give people opportunities. We have two mums who have just completed their college qualification through a partnership with Fife College, and we have four more interviews next week. I know that that might sound like rose-tinted spectacles, but it really is our job to help them to be the best that they can be. Any of you who know Scotland's best new town, Will, might know of Macedonia. There are real challenges in that part of Glynwathus, but there is also a sense of pride and community that you do not find everywhere. But whilst poverty can be a key driver in terms of who Homestar supports, it is also important to say that supported families can come from all walks of life, and they do. This is also mental health awareness week, and it should be said that poor mental health can play a crucial role in tipping families over the edge. As Lindsay told me, there has been a huge increase recently locally in anxiety and poor mental health, particularly in younger people. That has shifted the focus of Homestar's support with volunteers looking at the importance of teaching young people about their own self-worth, for example. Indeed, the team has recently started a mindfulness class, and it is fair to say that some had their doubts about how those classes would work originally, but now the mindfulness sessions are extremely well attended because people are starting to see the benefits of investing in their own mental health. Support is offered as wide-ranging. Another example is that that is provided by cookery classes to support families, and I am reliably informed that one dad managed to lose over two and a half stone after attending one of Homestar's eight-week cookery programmes in Glynothus. For someone who was living off takeaways and junk food, he became someone who was finally able to cook for himself, his daughters' behaviour improved, and all of that as a result of just being taught the life school to cook. In Glynothus, the Homestar team has also been able to tap into improving nutrition by developing a community allotment, and, as a result, they now have access to two raised beds at the back of Glynothus high school, the produce of which will be used by the whole community. They have also been working in conjunction with the Princess Trust and Fife College to have the community houses' gardens transformed for the benefit of the people of Macedonia. It is fair to say that Homestar is not just a standalone organisation. 33 per cent of all of Homestar's Glynothus's referrals come from Fife Council's social work department. As Lindsay told me, Homestar can get to where other agencies are unable to, because families do not see them as a threat. Homestar offers a tailored approach to each family, giving compassionate, confidential help and expert support without judgement. That is one of the reasons why they have such a strong partnership with social work services and the local authority in Fife, where they recognise the importance of the skills that Homestar brings that partnership. A large part of Homestar's work remains focused though on the more traditional home visits that are provided by volunteers. All volunteers are required to attend a 10-week programme or a preparation course before being allowed access to visit families' homes. Those visits are crucial, particularly in the early days of having a new child, for example. That practical and emotional support provides families to maintain their dignity, even when dealing with their own personal difficulties. One of the key ways in which the Scottish Government has supported this important time for families is through the baby box. I hope that all parties across the chamber can reflect on the policy as having been hugely significant for families in Scotland at a vitally important time. Being a family on what that means has changed over the last 21 years. Indeed, right now we have a Cabinet Secretary on Paternity Leave. I think that that sends a really powerful message about 21st century Scotland. When I reflect on my mum's experience of bringing up three girls in the 1980s, she had no choice but to forfeit her career. It was what was expected. We now have a much healthier view of the roles and responsibilities within a family, and the work of Home Start supports both mums and dads in doing just that. Across the chamber and irrespective of party allegiances, we all want to see the next generation giving the best start that they need in life, not just to grow but to thrive. Home Start has a real focus on early years, as those that make the biggest impact. As they note, children who are raised in a stable, loving family environment are more likely to have a positive and healthy future. We know those formative years that 0 to 3 category are the ones that make the difference, and Home Start makes the interventions to support families at that crucial time. Getting it right for every child starts with organisations like Home Starts. It starts by making tiny changes to support families to teach them how to cook, for example. It starts by giving parents the knowledge to understand what benefits they might be entitled to. It starts with supporting families struggling with poor mental health. Before closing, I say that Home Start Glynothus is always looking for volunteers, and I encourage constituents in the Glynothus area to get in touch if they would like to play their part in supporting the work of the charity. Home Start Glynothus is based at the heart of the community. The work that they do in Glynothus and beyond is vitally important to families in my constituency. I would like to end by paying tribute to Lindsay and her team of amazing staff and volunteers, and I wish them continued success for the next 21 years and beyond. I say to the gallery that I understand why I wish to applaud. It is not permitted from the public area. I call Annabelle Ewing to be followed by Alexander Stewart. At the outset, I would wish to congratulate my colleague Jenny Gilruth on securing this debate this evening. Jenny Gilruth has explained eloquently to the chamber the important role that Home Start Glynothus has played in the lives of countless families over the past 21 years. I would wish to commend all involved, both staff and volunteers alike, and it is good to see them join us here tonight to present an officer for this debate. All I can say is very much echoing Jenny Gilruth here to the next 21 years and beyond, as Jenny said. As the MSP for a Cowdenbeath constituency would come as no surprise, Presiding Officer, the tie, too, we would wish to congratulate Home Start Cowdenbeath for all the work that they have done since they first started up in July 1990, so they are just a few years behind in terms of their celebration of their 21st birthday and we all look forward to that in a couple of years. Home Start Cowdenbeath, which has an office in Cowdenbeath High Street, in fact, just diagonally across from my constituency office, serves the area covering Bellingrey, Lawhore, Crosshill, Glencraig, Lochgelly, Llymphinnans, Kelty, Hill of Beath, Crossgate and, of course, Cowdenbeath itself. Its focus is on enabling parents and carers to be the best that they can be, recognising that there are a lot of pressures on particularly young parents. Home Start Cowdenbeath adopts a very person-centred approach in the family support that they offer, which can include one-to-one support around perinatal mental health problems, increasing breastfeeding through local support groups, getting children into the outdoors, through, for example, promoting gardening activities Home Start Cowdenbeath also seeks to get fathers involved in anti-natal workshops, and in general recognises how important it is that fathers are part of the process from the outset. Home Start also seeks to tackle loneliness and isolation, particularly as far as young mothers are concerned, and groups are run specifically for under 25s to provide a forum where young mothers can chat freely and hopefully gain some confidence. However, it is important to note that home visits remain at the heart of what Home Start does. After an initial meeting with the family to discuss their specific needs, Home Start will match the family to a volunteer. The help given is unique to that family and will depend entirely on the circumstances of the case to see what is needed for that particular family. A key element is building the family's resilience so that they are better able to cope in the future with the challenges that they face. Of course, sadly, there will be many challenges faced by particularly young parents today, in my constituency and in that of Jenny Garrooth and indeed across Scotland, further to the great difficulties that Tory austerity is inflicting on our population. At the same time, Home Start volunteers play a pivotal role in helping families to access local services, such as housing or local mental health services. It is very important, as Jenny Garrooth has said, to access to children's centres, health services and to access to citizens advice, in terms of, for example, helping them to obtain financial advice. The volunteer may even look after each other to allow the parent to attend relevant appointments or the volunteer may help with transport so that the parent can reach the service that they need. Presiding Officer, in conclusion, I would like to pay tribute to the staff of Home Start Coutinbeath with whom I had the pleasure to meet some months ago. I was hugely impressed with their dedication and determination to make a positive impact. Those workers do a very challenging job and it was clear that it is also an emotionally draining job. They do it extremely well and they make such a huge difference to the lives of so many families in my constituency. Finally, I would also wish to pay tribute to their fantastic army of volunteers without whom all that great work would not be possible. Thank you very much, Ms Ewing. I call Alexander Stewart to be followed by Claire Baker. Thank you, Presiding Officer. I am grateful for the opportunity to take part, and I would like to thank Jenny Gilruth for bringing her member's business debate to the chamber this afternoon. It is right and proper that we congratulate the Home Start Coutinbeath on reaching its 21st anniversary year. I am delighted that we have individuals from the organisation in the gallery this evening. I very much welcome the support that Home Start has provided over the years to help families and young children build better lives. My first association with Home Start was back in 1999 when I was elected as a councillor on Perth, Inc and Ross. In my capacity as vice-conviner for housing and health, I had the opportunity to meet staff and volunteers from Home Start Perth. I was a regular attender at many of their meetings and their events, and spoke at a number of their events. I was delighted to visit Home Start Clack-Manager when I was back in 2017, which was celebrating its 30th anniversary in 2018. Home Start is a charity that embraces many of the values that I and others can concur with, and it continues to offer support and encouragement for individuals going forward. I also understand that Home Start relies heavily on volunteers and that the assistance that the charity offers in tailoring each individual family's needs and requirements is vitally important. They also support families and groups and provide them with opportunities for trips, for events, for parties, for access to services. We have already heard about how they can help with mental health, with classes for cooking, ensuring that wellbeing is taking place and also allotments. All of those give individuals the opportunity to expand their potential and support the individuals that they are looking after. As many of us know, birth through for the first few years before they go to school, there are child development and the environment in which that family home can be strongly determined out of the opportunities and the life chances that many children are presented with. It is well known that children who are raised in a stable and loving family environment are much more likely to have a positive and healthy future, and the visits that Home Start do, the home visits are vitally important to engage with individuals who may be having difficulties or may have a crisis to manage, and by doing that they can support them through that. Home Start works extremely well in ensuring that the volunteers are parents themselves, many of them, whom they actually understand that the difficulties that are being suffered and protect and support service users in that issue. They also work alongside parents in their own homes to help them cope with the stresses and strains of the life experiences that they are experiencing. Home Start volunteers make a huge contribution and they help the skill, the confidence and the strength to nurture and support, and that is there to ensure that people are protected. It is very important that Home Start excel and that I would congratulate and commend them for all the work that they do. As so, I follow the highlights of many of the Home Start recent successes and myself, since I have been here member in this Parliament, I have put forward some motions that have helped to highlight the fantastic level of support that has been received by some of the organisations that are there to help and support. I will continue to offer my support and encouragement across Home Start, across my region on Mid Scotland and Fife, and I have already said that in Perth and Cynros and in Clackmannan, but also in Stirling and Fife, there are making a real difference for individuals and families and that should be commended and congratulated. Organisations like Home Start are being needed more and more due to the hectic lifestyles of many individuals, but they are there to support and nurture and make sure that life chances and life opportunities are there to support wellbeing. Each and every day, volunteers and staff go that extra mile to ensure that families have that safe and secure environment, which ensures that they thrive in the lifestyles that they have. I wish Home Start, Gonddorothys, all the very best for its anniversary and its 21st year, and I know that they will continue to support and volunteers back up when. And we have to work together the Scottish Government, the councils, the third sector, they all have a part to play in ensuring that we continue. So I look forward to seeing it go from strength to strength. Thank you, Deputy Presiding Officer. Thank you very much. And I call Claire Baker, to be followed by Rona Mackay. Ms Mackay is the last speaker in the open debate. Ms Baker. Thank you, Presiding Officer. It is a pleasure to speak in this evening's debate and I thank Jenny Gilruth for securing the time in the chamber. As an MSP for Ms Scotland and Fife, I have visited Home Start across my region. It is a few years since I last visited Glynothus Home Start and it was good to hear from Jenny Gilruth about their recent activities and the support that they continue to provide to families in that area. I would like to add my congratulations and very best wishes to Home Start Glynothus on their 21st birthday and thank all the staff and volunteers who make the organisation what it is, a welcoming, inclusive, caring support for families with children. All Home Starts offer support to families in a variety of situations. They can provide valuable support to families who are new to an area and have no family locally, families struggling with poor mental health, families perhaps coping with bereavement and loss and families where there are child protection concerns. Home Start Glynothus is well known in their community and that is reflective of the presence of Home Starts throughout the UK. There are 32 Home Starts across Scotland. Glynothus is part of a network of Home Starts in Fife and I am pleased to have visited North East Byth, Kirkcaldy and Levenmouth Home Starts. As well as the 21st birthday for Glynothus, 2019 marks 25 years for Home Start Kirkcaldy. Like Glynothus, they value their volunteers and I would like to take the opportunity to recognise the commitment of Mary McKewitt, who has volunteered for them for 20 years. Her years of dedication to the organisation is also reflected in the valuable commitment of Fiona Haste and Elizabeth Coop, who have volunteered for 11 years each. All volunteers attend a preparation course and they are matched to the families that they work with, as it is a relationship based on trust and understanding. As well as the one-to-one work, Home Starts offer a variety of other activities and I am pleased to see that BBC children in need are supporting a family support group offered by Home Start Kirkcaldy. I recognise organisations that support the work of Home Start as well as the efforts of their volunteers and supporters on fundraising. As Jenny Gilruth has highlighted, Home Start Glynothus supports between 60 and 70 families at any given time and, as part of their birthday celebrations, Kirkcaldy Home Start is recognising that they have supported almost 1,000 families over the years. The importance of the work of charities such as Home Start to supporting families and children in five cannot be underestimated. We have previously in the chamber highlighted the work of Gingerbread, of the YMCA and on Friday I went to visit Plus in Stirling who support children and young people with a disability. Those are only a few of organisations which provide invaluable support to families and provide a service that invests in families, helps to build confidence and parenting skills, provides opportunities for young people and much more. The work that they do supports the efforts of social work, of the health service, of education and employment services and can help to prevent people from reaching crisis situations where they would need greater statutory support. Although this evening is a celebration of all that they do, we must recognise the financial situation that they operate in. Without them there would be a greater strain on other services and yet too much time is taken up by funding applications that are often complicated, sure to short term and too project focused and having to justify their value to overstretch local authorities who are constantly looking to deliver less for more. That is a common pressure faced by charities providing valuable services in our communities. They are a resilient and determined group of organisations who have to face the uncertainty of funding but they still achieve so much. Third sector organisations like Comestart are able to be flexible, responsive and innovative in the support that they offer. For example, Homestart Levenmouth have just started a pilot perinatal service as they identified the number of young mothers experiencing mental health difficulties and they recognise 80 per cent of their referrals are due to child or parental mental health. Homestarts across the country are often working at the sharp end of family services and they report they are seeing an increase in the complexity of needs faced by families and families facing several challenges. Homestart consistently provide quality support offer guidance and help build healthy happy connected families and I thank them all very much for that this evening. Thank you very much and I'll call Rona Mackay. Thank you, Presiding Officer. I'm grateful to my friend and colleague Jenny Gilruth for bringing this debate to the chamber and I look forward to attending the reception for Homestart Glenrothes that she's hosting tonight. What better way to highlight the great work that they do than by celebrating their 21st birthday? The help that they have given over those 21 years is inestimable to so many families. Presiding Officer, we all want children to have the best start in life. The Scottish Government's groundbreaking baby box and best start grants are testimony to our commitment to make this happen as well as our transformation of early years education. But we know we all know children can be born into circumstances where their parents need just that little bit more help, some practical friendly support in their own homes to get them over the roughest of times.  Homestart's a local community network of trained volunteers and expert support helping families with young children through those challenging times. The organisation was first started in 1973 by Margaret Harrison. The first one in Scotland began work in Perth in 1984. Margaret believed that supporting a family was best done in their home where it can be shaped to the needs of the entire family. She realised that if parents get support and friendship from another parent they will be better equipped to learn to cope with the problems life can bring and will be able to give their children the best possible start. The support parents struggling as we have heard to overcome their own troubled backgrounds in order to do the best they can for their children. Homestart sees first hand the harm that children can suffer when they experience trauma, abuse or neglect at an early stage of their development and in many ways their volunteers were at the forefront of the ACEs movement before it even had a name or the high profile that thankfully it has now. They have a local community network of over 1,000 trained volunteers who support 3,000 families and nearly 6,000 children every year. That's pretty incredible. There are 31 branches in Scotland and the nearest one to my own constituency in eastern Bartonshire is Glasgow North based in Maryhill who do a fantastic job. Some of the issues they deal with are families struggling with postnatal depression, isolation, physical and mental health problems, loneliness, bereavement and many other issues. And they receive the support of a volunteer who will spend around two hours a week in a family's home supporting them in the ways they need. The volunteers are non-judgmental, friendly and professional and support is tailored to the needs of each family. Much thought and care goes into matching a family with the right volunteer. And they too now also welcome dads, grandads and male carers who need support to join the group. So in conclusion Presiding officer I'm going to let the facts speak for themselves on the benefits of home start. A survey of clients said that after they engaged with the service 95 per cent felt their child's emotional and physical well-being had improved. 93 per cent felt felt less isolated, 95 per cent felt healthier and 94 per cent felt more involved with the child's development. I don't think any organisation could boast a better testimony than that. So happy birthday home start Glenrothes and thanks to all home start branches for the amazing work that they do to help make Scotland the best place to grow up. Thank you. Thank you very much. And I now call on Marie Todd to close to the Government. Minister please. Presiding officer Like other members across the chamber I want to thank Jenny Gilruth for tabling this motion and for highlighting the important work of Home Start and Glenrothes and the Home Start Network in Scotland. I'm delighted to welcome some of you to the chamber this evening. It's really great to have you here with us. I've had the pleasure and privilege of visiting many home starts across Scotland and I've seen the powerful work that you all do first hand. It really is a lifeline to those who need your help. I would also like to take this opportunity to send Home Start my best wishes for a successful parliamentary reception this evening. Happy birthday. Which is highlighting the importance of perinatal mental health during mental health awareness week. Maternal mental health is a key priority for the Scottish Government. We know that as many as one in five women are affected by mental health problems in the perinatal period and in Scotland that means 11,000 women are affected every year. Being a parent is one of the hardest and yet one of the most important jobs that anyone can take home. The challenges are even greater for families who are already coping with additional challenges in their lives and Home Start volunteers in Glenrothes and right across Scotland make a really vital contribution to supporting these families to achieve the best possible outcomes for their children. As you know it's the Scottish Government's aspiration to make Scotland the best place in the world for children and young people to grow up and the implementation of the national parenting strategy is key to driving this ambition. By championing the importance of parenting and by strengthening the support and offer to parents and by making it easier for them to access the support. We've made significant progress around commitments on extending the provision of early learning and childcare by increasing the provision of free childcare which will be further increased to 1140 hours for three and four year olds from August 2020. We're improving antenatal and postnatal support widening access to relationship support developing parent club website as a reliable source of information and advice for families and continuing national play talk read and book bug programmes encouraging and supporting families to develop positive nurturing relationships with their children from the very start. By the end of last year we'd expanded the health visitor workforce by at least 440 since March 2014. We're also expanding the family nurse partnership programme by investing around £16 million per year reaching up to 3000 families at any one time. We also recently launched our much-loved and redesigned baby box helping to give every child in Scotland an equal start in life. All of these actions make a real and practical difference for families. Homestart Scotland are one of our key stakeholders and delivery partners in implementing the national strategy. And they're funded by the Scottish Government through the children, young people and families early intervention and adult learning and empowering communities fund. In the four years that this funding's been running we've provided just under £800,000 to Homestart Scotland. We've recognised the important role that third sector organisations like Homestart Scotland play in supporting children and families right across Scotland. In the last year alone the Scottish Government's awarded over £5 million to a wide range of third sector organisations which support children, families and communities across Scotland. In addition, over £2 million was invested in 2018 to continuing funding for national programmes supporting children like the play talk read and book bug. I'm delighted that the Deputy First Minister recently announced the Families and Communities Fund which will provide up to £16 million per year to focus on early intervention and prevention and to improve outcomes for children, young people, families, adult learners and communities right across Scotland. That three year fund will provide more certainty and stability for third sector organisations to plan and deliver services. Earlier I touched on perinatal mental health and because it is mental health awareness week I want to finish by highlighting our investment of £50 million through the programme for government commitments. I hope that the shows that we are determined to improve the recognition and the treatment of perinatal mental health in this country. I would also like to mention the issue of stigma which is something that is still a reality for many women experiencing poor mental health in the perinatal period.  I know that the volunteers at Home Start Glenrothes and Home Start Scotland make a huge difference to families and that includes providing support with mental health issues. I'd like to congratulate them again on their anniversary and wish them all the very best in their vital work supporting families in Scotland. Thank you. Thank you. That concludes the debate and I close this meeting of Parliament.


Good afternoon everyone and welcome to BBBY's The Sox with Show Play of the Day. Interesting looking clothes here in the stock. Beautiful, wonderful, fabulous gap and a great short today in the morning. And had a really big pushback into the afternoon. So let's take a look at it. Great entry this morning. Oops, here we go. All the way back. Love the gap. Beautiful gap. So stock opened. Opened at 56.50 when immediately out of the gate and, you know, grabbed it as soon as I could. Felt all the way down. And actually the stock did break $55, not convincingly, but by a teeny weeny little bit, right before 10 o'clock. So this was the place to get out of some of the positions. Some people get out of the whole position and I said you could do that. But if you wanted to hold it to a bigger target, the thing to do would be to get out of half or three quarters. As it turns out, the stock never went to the larger target on the day. And it really ended up looking great. Get back here. After the rally back here, it looked beautiful to continue lower and based and based and based and based and based until what turned out to be right before 12 o'clock major reversal time. So when the stock had this lift here into this area here right before noon, it wasn't right to continue lower because it shouldn't have done this. And actually it really shouldn't have gone over $56 after it set up here in the five minute chart and it actually set on the 15. Here. So this actually set up here, looked great until this far here, this doesn't look good and it isn't good and then everyone went back down again. So the retracement was too much here into the late morning period to continue lower in the afternoon. But up until this time here, this looked great. Back down again to the next number. I mean that looks perfect. And then here's the problem. So stock could have continued down to $54.53 and eventually the core target on this is $50 and this is definitely going to get there. However, it did not get there on the day. Beautiful, great, fantastic move this morning. I mean really, really nice move. Anytime you can get a dollar, a dollar or more out of something, a dollar, 10, a dollar, 20, $50. Anytime you can get a dollar move in something that quickly in the morning with an aggressive immediate entry. And basically your whole day in, it's a beautiful thing. So nice trade in the BBBY today. I did actually have conviction that the stock would get down last night and it did. And I watched this morning, I rated it. It was a fantastic get. This chart is definitely lower. It had the pushback today in my analysis here I believe because of the fact that the stock did get down quite a bit and fell aggressively in the first half hour of the day and then the market had the rally into the afternoon close and this just gave it up a little bit. However, the stock is still lower. So nice trade in here. BBBY today in the Stock Swiss Show. Live trading room, this was the play of the day. Didn't do anything else. Love it when the top pick I have and the best thing I like goes and works immediately out of the gate. Those are my favorite trades to take. If you are interested in the Golden Gap class I want to learn how to find stocks like BBBY and take these entries and to make that kind of money and the drop down there right away is into the open. You can take the gap class. The next class is July 12th and 13th. And the last day to sign up for the course at the current price is June 30th. So that's Monday. You can pre-register for the July class and pay the current price of the Golden Gap class which is $24.99. Starting July 1st the course is going to be $29.99. So feel free to email me if you'd like to sign up for that ahead of time by June 30th which is Monday. My email is Melissa at thestockswish.com. Have a great evening everyone and I'll see you tomorrow.