Hey all, welcome back to my channel. Today's video, as you can see from the title, it is going to be another hair review video. This video is sponsored by Miss Lynn here. So thank you so much Miss Lynn for sponsoring this video. They sent me over their 13 by 4 lace front wig in their transparent lace. Super cute. So it does come with an adjustable elastic band that you can sew on, wig cap, of course the wig. The knots were not bleached. So I did go ahead and bleach the knots. I just use regular 30 volume developer. Oh, it comes with three combs on the top of comb in the back. And yeah, but anyways, I use 40 volume developer and then just a powder bleach that I had laying around. You want the consistency to be not runny, but not so thick either. Just kind of whatever in between is that. So I like to add the powder first and then go. Wait, I'm ready for it? Doing content right now as we speak, bleaching the frontal to do a hair review.  This is for me. Okay, so back to what I was saying before I was really interrupted, but yes, I'm using an applicator brush. I know some people like to use a knife or a spoon. Never use anything metal. I always use something plastic when you are doing this. I did let the bleach sit on the knots for an hour. I noticed that when I used to do like 20 minutes, the knots don't be bleached enough for me. So this time I did let it sit for an hour and some knots still wasn't as bleached as I would like them to be, but it definitely tinted the lace literally to mimic my skin tone. So that was a plus. And then I like to use the shimmer light shampoo and conditioner to rinse that out. I went ahead and shampooed and conditioned the hair also since I had to get everything wet. And as you can see, baby, yes, the knots are bleached, the lace is tinted. It literally looks like scallpiana now. So we're good. Then I went ahead and decided to blow dry the hair. I always recommend air-drying the hair, but I wasn't a time crunch at this point. So I did blow dry literally the cap and the hair. The hair kind of air-dried as I was plucking because plucking did take about 30 minutes in itself and it still wasn't plucked to my liking and plucked to how much it needed to be plucked. But that's not in the hair nor there. So I did go ahead and put it on my mannequin head while I'm watching Raven Elise shout out to her. And yeah, I decided to just kind of blow dry it a little bit. And then as I was plucking, it kind of just finished up air-drying. Start plucking, you guys should know to part out the front section so you don't over-pluck and bolt out your frontal end or closure. Then you want to pluck behind that line. And as you can see, those knots are bleached mighty well. Like I did a really, really good job this time. So I like to do the pluck-pluck-skip method, pluck-pluck-skip. You guys know what I like to do and literally just do that around the whole perimeter of the head until it gets to whatever hairline you're trying to mimic, either it's yours or somebody else's. And you're good. Bleached the knots, but I did want to go in with this lace tint just because some of the knots weren't bleached. And this was my first time using this so I kind of wanted to just see how good it works. This spray or tint is a little bit darker than my skin tone, but I feel like that'll work out in my favor in the longer run. Like if I don't decide to bleach the knots, like you don't make it really, really tinted. So I went ahead and cut off the back part of the lace because you don't want that in the back of your wigs. And then I do also cut out the comb that's in the front of the wig because you don't want that in there because it just makes it really, really bulky. Then after doing all of that wig maintenance, I like to put the wig on to see how the wig is fitting to see if I need to adjust anything like the adjustable bands. And the elastic band wasn't already inside this one, so that lets me know if I need to add it or not to add it to see how tight it is. So this one wasn't as tight as I would like it to be. So I did decide to go ahead and add the elastic band. And right here I am going to show you how to sew in an elastic band. The way I like to do it is obviously fit it on your head first and then see put the band right above your ear. So wherever your ears at, I like to kind of hold right there on the wig and then I'll put my finger right there and then just flip it over to the other side. You do want to sew it on into like a smile. I used to sew these on wrong all the time and it used to annoy the heck out of me.  So this wig wasn't too big.  This one was a really good constructed wig. Like some wigs companies have sent me has been really, really small but this wig literally fit me perfectly. So I really, really love that about it. And you always want to clean off your forehead with alcohol, witch hazel, something because you don't want any dirt, makeup or any residue on. So you need a clean base for the application. And that was from the lace tent. The lace tent I always tend to kind of bleed. So I always make sure to clean off my forehead before I lay down any glue or spray. I am using the even New York glue and then I went ahead and used the even New York spray. This is my very, very first time ever using the even New York spray. And I have to say it literally worked so much better than the got to be glue spray. Like baby, I'm about to throw out got to be glued. Okay. But no, I think I'm going to keep the got to be glue hair spray and use that for like if I have curls because I don't really like to use pump it up. So I'll just use the got to be glue spray and use that for when I curl my hair. But the even New York baby, that's a 10 out of 10. Why y'all ain't tell me about it before, sis? Like it literally works like a charm. I use the red one. I know there's a red and a black one. I make it the black one to see what the black one's talking about. But girl, this red one, you need to go get it. It's going to make your application so flawless. And baby, the hold is holding. Like if I didn't even use the glue, the freaking spray would have held this wig down. I love it. Then right here, I just wanted to show you guys how flowy this is. Like it's so cute, super soft, has a lot of volume and density. I love it. And I decided to go ahead and run the flat on through it just a little bit. Since I did shampoo and conditioning, it was kind of like fluffy, like a fresh blowout. So I decided to run the flat iron through very, very quick. You know, like I'm not actually like trying to get the hair bone straight. Just wanted it to be a little bit straighter for when I get ready to curl it. I did decide to do loose kind of like beachy wave curls. I did use the flat iron. And then when I get into the bathroom in the bathroom scene, you're going to see that I did use a wand curler. I was just trying to see which process was faster for me to be able to curl my hair the fastest. Okay, so I had to come to the bathroom to use the hot comb. You want your closure front or whatever you have on to lay flat. Okay, so the hot comb I'm going to do it every time. And then I was sitting in the mirror noticing that my right side was plucky better than my left side. So I did go ahead and do my left side. This image is flipped. So that's why I look like I'm doing my right side. But this is really my left side in real life. So on my head, I was able to pluck. That's how I know that even though your spray was really good, because when I try to do this, we got to be glued free spray. My lace will be lifting when I start plucking on my head.  So yeah, I just tried to do the pluck, pluck, skip and it's very much giving scalp. It's very much giving bleach knots. It's very much giving my skin tone and my skin complexion. Absolutely love bleaching your knots because it makes a freaking difference. So I really, really do recommend bleaching your knots. Then I wanted a little bit of layers in my bangs.  And then this is me curling my hair for the, no, this is me finishing this side of my hair. And then I did go ahead and go in with some concealer. I either use my fancy foundation or concealer. I can remember, I think I used my foundation so it can kind of mimic my skin tone a little bit better because the concealers usually a little bit lighter. So I did want it to be just the tad bit darker. Then I cut the lace off, sprayed a little bit more that even New York spray to hold everything down. And after that, we're done. Back to my channel. Today's video as you can see from the title, it is going to be another hair review video. This video is sponsored by Ms. Lynn Hair. Thank you so much for sponsoring this video. I did want to sit down, talk about the hair and get into all the goodies that they gave me. So of course I got the wig cap, then I got this thick elastic band that they sent. And this is a separate goodie bag separate from the actual bag that the hair came in. I got a wide tooth comb. Next, I got a edge brush. This one is really cool. I haven't had one like this before. So I'm excited to use this. I did do no baby hair just because y'all know I don't really care for baby hairs, but I never got a wig brush or edge brush that's like this. So I'm excited to try that out. They also sent some cute little press on nails. Then they sent a pair of lashes. And I haven't got a full on like makeup brush set ever from my hair company. So that's totally different. And it literally comes with like everything that you would need like beginner friendly makeup brushes that is nothing, you know, too crazy. But definitely something that you can use. These are like mini. So like these would be perfect for if you're traveling. I'll probably take some of these. They're mostly kind of like eyebrow brushes and like eyeshadow and stuff. But hey, thought that was really cute. So the wig that I'm wearing, they did send me their 13 by 4 straight lace front wig within HD transparent lace. As I was trying this wig on and putting it on my head, I think I did talk about the lace like before even cut the lace. It melted in really, really good. I did have to bleach these knots. I'm pretty sure I already talked about that and showed that, but I had to bleach the knots because the knots did not come bleached in baby. The knots were knotting. Okay. So I bleached the knots and I used some lace tinting spray and then I got it to look like this. So I think it looks really nice. I did apply this wig lower than my liking. Like I feel like not a lot of my forehead is showing and I don't really like that. So next time I apply this wig, I will apply like kind of in my hairline, but I didn't want to like pull my edges out or anything like that. So I did apply it a little bit lower, but I don't have a big forehead to begin with. So me applying a wig lower does not look right on me. But yeah, they sent me, how many inches did they send? I have it on the screen. I can't remember the length that they sent me, but super cute. The hair came bomb straight. I did do a little bit of loose beach waves with it. I think it looks so cute. I did decide to do a middle part because a side part, this is only a 13 by four. So you don't have much parting on the side to make a deep side part.  And yes, super cute.  Like I said, I did pull my hair down, my hairline down a lot lower than my liking, but as far as like the wig application and the way it lays, it lays pretty good. I actually really, really liked the lace. I didn't have any problems laying this wig down. I did use that new even New York spray also. And baby, that's a 10 out of 10. I definitely recommend so I really, really like the application. It's just, I moved the wig a lot lower than my liking. That's what I don't like. Other than that, baby, I don't have any complaints about the wig. I did cut a little bit on the bangs just to have a little bit of layers to kind of shake my face since I did put my hair line down a little bit further than my liking, but the hair holds a curl. Okay, it is raining so I did have to go outside with my dogs and like obviously it's raining but the curls did fall, but yeah. I do like a looser wave so they wasn't really tight to begin within the first place, but you know, it's still super cute. I like it. Thank you so much for watching. Don't forget to like this video, comment, and subscribe. And until my next video, I will see you lovelies then. I will be showing pictures and videos at the end, so


Okay, anyway, so let's do this problem. And we're going to use the Rydberg equation a couple of times in this problem. So I'm going to kind of manipulate it to the way I like to set it. So it says, what is the energy difference in kilojoules per mole between the first and second shells of the hydrogen atom if the lowest energy emission in spectral series with n2 equals 1 and n1 equals 2. And then it gives you a wavelength. But I want to calculate that wavelength, okay? So let's just calculate that wavelength and then we'll calculate the energy, okay? And I'll show you where that difference is that we were talking about, okay? So again, you've got to have the Rydberg constant given to you. So you're not going to have to memorize that. You're going to have the Rydberg energy constant given to you, okay? You're not going to have to memorize that. But you do have to know what the Rydberg equation is. So there's the Rydberg equation. And it tells us that it says n2 is 1, okay? So we should be able to figure out the wavelength. And this problem actually gives us the wavelength, but I'm not going to give it to you. So let's see if we get the same answer. Okay, so we'll get the Rydberg constant. But also remember, wavelengths are usually given in nanometers. So you're going to have to convert the means to the quarter, that's 0.75 to nanometers right now. So tens of a negative 9 is 1 nanometer. So that's going to cancel down. And then let's move this up here to run it out of space. So what have we got? 1.97 e7, well times the conversion to nanometers times 0.75. Okay, so I got this answer. I'll take it out to a number of digits because really there's no significant figures in this equation. So that's 8.2275 times 10 to the negative 3. But that's herd nanometer, okay? So you got to watch out about that, right? So if you want to think about it, if you want to think about it on the quick, you want to think about it that way. But if you really are having trouble doing that, you can always erase and be like, it's not, just invert this thing, right? So we say 1 nanometer divided by 8.2275 times 10 to the negative 3. And that should give us our number. They're all product because there it is, 121.5, we'll go to, man. Okay, so it said look for the, I said look for the wavelength. Then if we look at the problem, sure enough, the wavelength is 121.5 nanometers, okay? Just like you would expect, okay? Is everybody okay with that portion of it? Why do you divide it by itself? Because it was per nanometer. Nanometer and per nanometer are not the same thing, right? This is divided by nanometers. So we had to get the nanometers up at the top. So what did you think the inverse to get the 1 over that? So this is this. I've just written it out a little different, okay? It's the same thing. Yeah, but you said that I like flipped the whole answer. You're going to flip it at the end over here, okay? I promise you that's the right answer. I promise you, I wouldn't lie to you. I lied to you before. You don't ever, right? Okay, so are you cool with that? So go through it. If you're having trouble understanding why to flip it here instead of there, you're really doing the same thing, okay? You're just doing it at a different time. I just like to do it step by step by step so it's like people can be like, okay, if I get it, you know, okay? So, and of course, like I said, we've confirmed it. The wavelength is 121.5. So it really is asking us for what's the energy, okay? And like I said, you could go through the whole problem that you got in putting it into Planck's equation and getting your energy done. You could use the Rieberg equation again to get the energy. So can I erase all of this stuff? Is the Rieberg energy constant on this part? Yes. Yeah. All constants, you'll be good. It's really hard to remember all that stuff. I mean, it's good to try to though. So I just write that up because I won't forget it. So now let's change the Rieberg equation to the energy form of it. Sometimes you'll see delta E, that just means the change in energy, okay? When it's the energy, you're going to use the Rieberg energy constant. So you just plug in the same transition, same thing, okay? The Rieberg energy constant is not per joules though. So you don't have to do that flip, okay? Okay. So it's actually much easier and you should get something like negative 1.635 times 10 to the negative 18 joules, okay? It's a very small number like you would expect to get, okay? Why would you expect this to be a small number? Because it's one electron going from one energy level to another energy level, okay? Notice the negative here. That says that it's releasing energy because when you're falling down energy levels, energy is being released, okay? So this makes it more stable when you release energy. If you want to think about it that way, okay? This is the first time we'll be talking about that. But again, this is per one electron, okay? So what you guys were doing the other day was we'll multiply it by Avogadro's number and figure out what per moles of electrons. Every one mole of electrons, we've got 6.022 times 10 to the 23rd. But joules are one mole, okay? But it asks for kilojoules per mole. And you should get this number, negative 96 kilojoules per mole of electrons. So that was probably the numbers that you were getting. And you weren't realizing that you were instead solving for a mole of electrons as opposed to one electron. So I have bad money even though nobody showed me the calculation, okay? Are there any questions about this? So again, this way is really cool because we could figure out well, what's the energy of the electrons that switch it? And the Riberg equation. Alright, Riberg equation, we don't have to plug it back in the points. So if, like, the Riberg or the energy equation is actually going to change if it's negative, does that mean it's going to change? Yeah. Yeah. Because there's going to be a negative times a negative, and that's a positive, right? So that would be where it's going up in energy level. Okay, that makes sense, right? Because it takes energy to climb up a hill if you want to think about it that way. But it doesn't take any energy to fall off, you know, and splash on the ground or whatever. Okay, are we cool with that? Any questions on that one?


Hey Virgo, this is Kelly from the House of Virgo. Welcome back to my channel. If you're new welcome Please do hit like share and subscribe. I would greatly appreciate it So this is a reading for all Virgos Virgo sun moon rising or Venus if you would like to get a personal reading There's a link in the description box below this video Yeah, all right Virgo, we're doing the Celtic cross So let's take a look and see what the energies are for you I'm gonna lay out the spread and then I'm gonna pull some romance angels cards to see what the love messages are Here for you. All right, we're gonna this reading will be for the next 24 hours But and also can play out through the rest of August. Okay, so let's take a look spare. Please show Virgo past present and future outcome Or Virgo. Okay. The past is the fool Present energy is the king of wands. The challenge is the three of wands Driving the reading is the ace of pentacles crowning the reading the ace of swords the near future page of cups Page of pentacles in the position of your advice. What's external and the other person the knight of pentacles Hope some fears the king of cups the outcome is the six of wands with the knight of wands. Okay, so Let's see. We've got the king of cups Cancer Scorpio Pisces you are the king of wands Aries Leo Sagittarius also have Aquarius energy Oops All right, let's take a look if those signs resonate for you great. If not, don't worry about it All right, let's take a look and see what the love messages are here for Virgo for this time period What does Virgo need to know in regards to love? Mmm. Look separation same as yesterday So you and someone are in separation perhaps at this time or were and Healing family issues. Okay. There's some kind of change that's happening here There's some healing going on during a separation period. All right Be sure of that. Okay Forgiving parents somebody's forgiving their parents for Perhaps the way the difficulties that they have in or have had in future relationships Taking a look at a parents. Maybe there was divorce or toxicity in caregiver relationships that set a precedent for them going forward It very well could be that the person that you're connected with Has this They may not realize or may not have realized that there was maybe the way they were in a relationship with you Was unhealthy or they may have just been mimicking what they saw in their past. Okay There could be for some of you you're separated from your person because they're trying to work on Some family things whether they're going back to If you know for a fact they're going back to a spouse to try to fix the relationship For others. There's some healing stuff going on. Okay so healing in the sense where There could be a family member or someone they're connected to is in some kind of recovery or was ill and They needed to separate from you or and just remember these roles could be reversed. Okay So the past position is the fool Okay, so the energy was ripe in the past for some type of a new start a new beginning A leap of faith that was taken or risk some kind of an adventure there was like hope in the past Okay, and also a sense of freedom. Okay, so let's take a look. I want to clarify the fool here and see what was Was this fool energy about this? This is somebody who was off on their own Maybe they were hiking or this is something that they enjoyed to do in the past or they were traveling a lot But kind of like This person Irresponsible behavior in the sense like there were consequences to every step they took but this person was very I feel like in the past Just oh, they're just gonna do it anyway, you know Might have been a bit of carelessness, you know, like oh well, you know, I don't I think my parachutes. Okay I don't really need to I'm over thinking it. I don't need to tighten it like this person was not cautious with their risks that they took they just it looks like in the past they just went ahead and You know did whatever they needed to do But I kind of get a sense that I'm feeling possible absence. So maybe this person in the past left you or walk, you know, like Abandoned your connection in the past.  Please show me the fool Okay, the emperor. Hello Emperor. So this is Aries energy Okay, so you they could have went back in the past to deal with a father figure or To release themselves from a father thing or they went on some kind of a trip with a father or it was just your person who just abandoned your connection and Went about their merry way or maybe they were seeking to be a Father or a better parent. Look, there's children there Okay Or to get more stability this person might have disengaged from your relationship in the past to Come into becoming Someone who is Respectful or someone who is like well respected, you know, it feels like a like some kind of a level up Energy trying to level something up about themselves male or female. It really doesn't matter. I'm just gonna pull Another card here. Okay, the son. All right, something to do with a child possibly or children again this person was on a quest for Seeking validation or unconditional love Either from a child or reconnecting again with a child in the past or hoping to It feels a lot like looking back because you know this energy here Yeah, these two cards as I see them It's like they're both looking away from this reading away from the future I feel like for a lot of you this person turned inward to look back on their own childhood or how, you know healing some family issues taking a look at their The way they were raised or how they grew up as I was saying in the beginning of things that might have been Affecting the relationship as you both know it or as it was or maybe it wasn't progressing in some way I'm gonna get one more because he's our major Archana's Okay, it's Queen of Cups. Okay, so they could have been taking a look at something with the parents. Okay Queen of Cups energy looking at the nurturing aspect of a mother. I feel like this person maybe traveled to see parents in the past and Seeking some type of validation or approval in some way because the full card makes me feel like like childlike energy playful, you know Or going back again, like I said to heal past relationship with a spouse or a partner Or also seeking So a lot of messages are coming to me and I'm please forgive me for that And I know I'm trying to tap into everybody's energy But there's an energy. There is something here. I Definitely feel like they left. I definitely feel like this person for whatever reason You know could be a thousand different reasons or whatever the things that I'm seeing here But they definitely pulled their energy and Disengaged now currently The King of Wands is the center of this reading. Okay So this person, you know, it doesn't matter what their sign is what matters is this person is showing up here in your present Okay, as someone who is very much now in a leadership position very powerful charming and courageous Let me take a look here Let me clarify in card for the King of Wands. Well the Queen of Swords a lot of people showing up in this reading So Virgo, I feel like this is your energy. Okay This person I feel like they're they have their focus on you at this time, okay, this person is waiting for the right time or When they it could be there might be something here about travel this person mate is coming back Around and let me get the three of wands clarification here. Oh Yeah, waiting for the right time to make some kind of offer or have some type of new start with you here. Okay King of Wands and the Queen of Swords Very different type of energy the Queen of Swords is it very much like you Virgo very truthful Can cut right to the heart of things. You're very good listener and I feel like there's a coming together the the obstacle here has to do mostly what I feel like With distance. I don't know if you guys are across the water from one another.  This person it feels like Waiting for the right time waiting for the perfect opportunity to reinvest in something here with you Okay, so if they went back to try to repair something with a former relationship, I or try to seek Someone else or try to connect with someone else. I don't really feel like that Yeah, that really didn't work out. That was the situation that actually left them Trapped Okay That was it almost kind of it feels like whatever they were trying to do it just didn't work out Hence the reason why they're showing up here and you're reading again. Okay, there's somebody showing up in your reading Ace of Pentacles is here. Ace of Swords is here There's definitely this person in their mind and what's driving them They want to communicate with you and tell you the truth and tell you like clear the air and Try to have a new Start or a new initiation of something here. Let's get an Ace of Pentacles clarification. Yeah, put an end to these this some type of burden Okay, this person is like Carrying a huge weight on their shoulders right now and Coming towards you The drive what's making this person come towards you is because they it feels like They're carrying all this Baggage or this weight by themselves and they need you they need your help with it and I know Virgo.  Look Look at that door. There's no mat on that door So yeah, but they want to come into your house they want to come to your home This person might even be knocking on your door very soon They could be bringing you some firewood. It's really weird. Yeah, I just feel like they're traveling But they have like a lot of baggage with them that they're coming in here with for you to bring I For some of you it may not be they're just dropped They might be coming towards you to drop some things off to your house Maybe they owe you something or they want to bring some things to you It's interesting. It's interesting. That's what I feel like they're there I just feel like expected knock on your door Now if you're at a long if you're at a distance some of you it might be a It might be like a text message that has a knock sound like a text alert that Knock thing. I don't know if I'm just hearing that knock like a sound Text alert of some sort or a phone call or something It's something to do with like the sound of knocking on a wood door Maybe somebody here has a full straight. Maybe a wood door with an arch in it okay, they might be coming to offer to fix your house or It could even be like bringing their stuff back is what I'm hearing Now I know that's not that may not resonate for all of you So like if you are at a distance to one another this could be It just feels like specifically for some of you the knock sound on a door Okay, somebody here might live in the mountains Or it might maybe in an invite to come to where they are. It's very strange Okay, the ace of sorts Wow, you guys the king of wands is back again to clarify. So this definitely is this person same person and The truth that they're bringing to you to kind of like clear the air bring some clarity It's basically that they're ready to put effort and try to make something work here with you That's what they're that's what they've been thinking now They're also probably going to tell you they've been very busy and distracted and focused on Whatever they're building in their life or creating, you know at work or crafting or polishing up They've been polishing things up in their life Okay, they may have had to go back and sort some things out in the past with family or a spouse or just their life in general take a look at some childhood traumas or Things like that, which is why you guys have possibly been in separation But here comes the page of cups. So here comes this sweet little Communication from this person a new love beginning a new love start this person And they might even send you a selfie look to take a picture of themselves with their phone They might even send you a picture of themselves with a nice background or something at a beach Or by some water if you I feel like some of you are kind of at a distance to one another Let's take a look at this page of cups now this person So the page of cups can represent childlike like some immaturity So it it's not really a child that feels mostly like this person is coming in to offer Some peace as well. This is what some type of love messages come in They bring peace with them It could be an apology for some yeah, I'm picking up an apology. What is this about? This is the near future What is this about? The night of cups. Oh my gosh Virgo. I Just heard I'm sorry. I didn't make the offer that you needed me to make I'm sorry. I didn't come in with what you wanted and what you needed. Okay, the night of cups Romance a proposal. I'm sorry. I didn't propose then or you know I know not everybody is trying to be married, but I'm I'm sorry. I didn't like bring our relationship to this like the romance aspect of it I feel I do feel like I just keep I just keep getting drawn back to this past energy That this person For some of you they went back to help a spouse with a child and for others They had to help with some family thing. Okay So, yeah, this is this is what I feel like this person is coming in to really bring in some romance. I Want to say like come in and it could be a proposal for some of you Okay, whether you want it or not that's up to you But it feels like they're coming in in the near future. Maybe sometime like the third week of August They travel to see you That's like they want to come and knock at your door. They want to come and spend some time with you Bring gifts for the kids for some of you if you have children if not Our grandchildren are just, you know come with all their stuff is what I feel like Being in the ten of one's energy It's like finally saying yes A lightning up freeing themselves from whatever they were before okay But not wanting to be like freedom like not wanting to have that freedom with you Freedom from others or freedom from some past struggles or burdens That they were carrying around with them for such a long time. All right, that's strange You're giving me a lot of messages here Okay, the hermit. Yeah, this person has been in isolation. They've been thinking about this for quite some time reflecting on really needing to fulfill something for you Bring you that love Give you what it is that you truly wanted, you know, maybe they wanted something you wanted something else But truly coming in with this energy now could be another Virgo for some of you Let's take a look your advice page of pentacles. I feel like when I look at this card your advice is to Definitely You can look Look, but don't touch. That's what they just said look, but don't touch Virgo You know, you see how that pentacles just kind he's he can't touch it So it's kind of like more of a concept and idea Look at this what it is that they're coming in with and find out You got to find out if this is real Before you jump in or say yes or agree to anything to once yep That's that's where you're at before you make any kind of a move here before you make any decision You got to look and see if this offer is real because there's something about The fact that this pentacles floating and this guy cannot touch it. There's something about it That makes me feel like it's just a little bit out of reach And it looks good and it looks like the real deal here But it's it seems like from your perspective. It's you can't touch it. It's not fully tangible yet It's just kind of floating there So your advice is you know See what this person is bringing in and but you you need to make sure that it's real Do you know what I mean? Or that they're not just coming in for any other reason because They're struggling and they need your help Okay Because again, there's no look. I'll say it again There's no doormat the foot of your door and if there is You should remove it as a symbol that you are not a doormat A night of pentacles not that you feel like a doormat because you're willing to help whomever needs it but not at the expense of your Not having anything for yourself or without reciprocity I feel like this person is willing to come in like they've been whatever they had to do in the past They're cleaning up that energy The challenge is when and I feel like this person currently is thinking about you and thinking about themselves And they do see that you are someone there that is worth investing in But I'm not exactly sure they feel like right now. They can't come in. Look at the two roses up there Okay They can't come in just yet They got to wait, but they are coming. All right night of pentacles this person. Listen. This is them They are coming in with this solid offer. Okay, this is what's external to you But Virgo like they're holding it in their hand here and they're taking their time coming in very slowly with us This is not an energy of someone who's lazy, this is someone who is very steady takes their time thinks through things Doesn't rush into things. They're coming in with this Pentacle that's you can hold that you can touch but Virgo. You're like, I'm not sure. I don't know if I want to touch it You're just You're not sure you're you might even Be thinking well, that's all great. That's a good idea. That's great, you know, it looks good shiny beautiful pentacle, but I Don't know. Maybe some of you are fearful to To take a hold of what they offer there might be a bit of fear. I Don't know why I'm feeling that You've got the high priestess. Yeah, see there's something to do here with your intuition and the hanged man You're not doing anything and that that is your advice you're like, okay I gotta maybe see this situation from a different point of view because I don't know if you want to Take something that's coming in from someone who I feel here Just left you. Okay Whether they communicated to you that they weren't going to progress the relationship or they just completely Left this is more for some of you this the fool I mean, he is somewhat responsible But because in the in the reverse he's not so they may have communicated with you that they got to fix some things or do Take care of some stuff in their past and left the relationship or they were dealing with other things or they just left Okay, and for some of you Either way, it's still struck you the same way See what else going on with this person six of wands. Yeah, see they're coming in and they want to win They want to try to win your heart. That's what their intention is here victory with you Okay, and ten of pentacles. Here you go They want to build build this brick house with you I don't know if you guys live in a brick house or not, but you know structure balance security Financial Security safety family Could be an offer this person might be coming in and inviting you to Have dinner with a family or inviting you to come work. Look, this is marriage There's two people here with a baby and a grandfather Okay, this might be what something that they're seeking at this time Very well could be If that's not the story of this person Maybe they're a grandfather and want you to meet their new grandchild or maybe they want to meet yours Or have a child with you It's all about family structure and balance. It's interesting. I feel like this person is Yeah, they've been going within Making that ready to do this compromise here with you I just feel like there's like some kind of compromise And like I said earlier, I think this person wanted something and then you wanted something maybe two different things But this decision is being made, you know, I feel like okay, let's do it So let's take a look the king of cups hopes and fears so the king of cups is you know Very nurturing and supportive leader You might be hoping this person comes in in that energy very tolerant and pathetic Coming in healed hopefully Yeah emotional emotionally balanced Definitely you're hoping that whoever your person is that they come in like they might have been emotionally imbalanced in the past Meaning very inconsistent on an emotional level in and out of your life in a lot of ways So that's such a terrius energy there. Let's take a look. I want to get one more card here Yep, two of cups again balance Coming in together maybe starting the relationship over romance friendship starting over having a spark Coming into union a meaning of the minds agreeing Being in alignment wanting the same things Partnership both of you choosing together reflecting there's diversity in this relationship So there's differences between the two of you but coming into union anyway and finding common ground now the outcome of this reading Looks beautiful. I I see celebration victory Harmony I see self-sacrifice and loyalty here and this night of wands like all of that Coming in on that horse with a passionate drive from this person And for both of you I feel like this person is really going to try to make some kind of sacrifice Like a sacrificing self whatever it takes to come towards you And it might be travel here for some of you, you know I'm kind of picking up like this very Respirated, let me just get another card here King of pentacles. Well, I can't believe all the people showing up in this reading here So I will tell you yeah this person has Maybe emotionally and emotional instability, but they do show up here as this king of pentacles as an energy Okay, however, the energy is this person has a very passionate strong energy they also have a very Deep emotional side to them, but they're also coming in very stable here at the end and the king of pentacles is solid Solid solid and ready to invest and also this person could be coming in with a lot of financial gains as well. It feels like Very resourceful, you know able to give has from can give an up From a pool of resources for something that maybe you are hoping to have with this person It just feels like they're coming in with this these solid offers here for you with these pentacles But you're hesitant Virgo You're hesitant, you know, you're trying to See it see it from a new perspective like going within because I feel like your mind is kind of set in a particular way At this point, especially if there was separation That maybe you gave up and thought well, it's never gonna happen It's I'm not gonna invest in this anymore My person told me that they wanted to break up and they were going back to somebody else So they had other things to take care of and Virgo You might have just gotten to a point where you just said forget it And for some of you this person just left or just abandon you which is very traumatic Experience for you to have, you know, so it's kind of difficult for you to come back from that, right? Even if they told you they were leaving it's still difficult for you to come back from that And I don't feel like I feel like this person is I don't know if they needed this separation And to sort some things out in their heart and mind about what they wanted But they're coming back fully healed here Which is what you were hoping for but you know, where are you now Virgo? You're hesitant Okay, yeah So I wasn't asking anything so I just pulled the card. Let's get a couple of love messages. Oh, excuse me Let's see what the energy is here. I love messages from this person Okay, this one was flipped. I don't know how to let you go. Okay, this person They had no intentions of letting you go Virgo Not for the long term Okay, I fear that you will reject me They do they have a strong fear of that and I feel like that's why the hermit is here with a lot of Internal reflection about how to come towards you and to make this offer Here and now your true love is already part of your life Okay You came closer than anyone to this person Virgo you speak to this person through music Somehow I don't know what the attachment is to music, but they can hear you they do wish I wish I treated you better There's an apology coming here For whatever they did or choice they made because it hurt you. I know I messed up everything Detached with love to clearly see the truth Virgo You got to get out of some kind of denial stage like you're trying to make something real In your life manifest real things in your life and this person coming in with the solid offer You're not sure if you either want to touch that again You're taking some time to yourself to gain new perspective on your life about your emotions How you feel deep inside what you really want you might be in mystery mode right now Not communicating with this person holding back some secrets Because I feel like as yesterday's we're here to crossroads and you're not exactly sure which way you want to go at this time All right, I'm sorry. There's the apology coming in and and With making it up to you with some kind of solid offer Couple of charms you guys. Let's see what the charm energy is. Oh Okay, well, we've got a dragon hmm So I'm kind of feeling like third week of August that this energy is gonna start Playing out, but these are the messages I'm putting up today, but here's a dragon So I don't know how a dragon would resonate for some of you Like a dragon fire Fierce I'm feeling like very fierce energy from this person or the energy surrounding it I don't know all that much about dragons. Honestly If you guys want to leave comments what your interpretation of because all I'm picking up They're telling me fierce fire-breathing dragon like passion and fire This person's coming in like that I feel like dragons can be very protective in their defenders as well So the very defending type of an energy from this Just got chills Okay, and we have a padlock here Yeah, like they're gonna try to melt your heart It's all locked up and you threw away the key Virgo They're gonna try to come in with this dragon energy and just try to melt your heart maybe you have like it some of you might actually have like a dragon like a Bearded dragon or a pet Kind of thing or maybe have statues around your house where there's dragons or somebody might have like a dragon tattoo But this padlock kind of feels to me like there's no key here So you you might be just very casual with this person, but your heart is completely locked up Now from them not from other people Just from them Okay, so we have a camera a baby foot. Okay, so let me see the energy here from this camera hmm, I Just heard that song photograph from deaf leopard All I have is a photograph I don't want you I don't need you all I have is a photograph So I don't know How that Would resonate for you. Maybe that was a song somebody liked or a song you like or maybe if you read the lyrics I feel like it's coming from this person Well, not the part that they don't want you or need you.  Okay, and this baby baby Footprint here, and I'm just hearing footprints in the sand that old poem very significant, but this baby footprint is Like a like Like on a on a birth certificate It feels like a birth certificate, but this is like a step. There's a step being taken here baby steps Okay, this full card always reminds me of like it's this it's the zero. It's the oh, you know, it's like the It's like the egg You know, it's like the beginning it's like that the beginning I feel like your person went back to take care of some business or left to take care of some stuff and They were seeking perhaps trying to find happiness and somebody else in the past and you know, but this person For a lot of you it feels like they were I keep I'm sorry, but I keep getting drawn back to this past energy This person went back to to sort some things out with some family stuff here that was going on with them But I know it's like crazy. I don't ever once I'm done with this past reading I always move on to the next but something here keeps drawing me back to that and I'm not really sure sure But I will say there is definitely like this person was a fool Definitely They went back like the fool. He knows nothing. He doesn't think about things He's like a baby walking through the house and climbing up on the countertops without even thinking of the consequences You know, and and I feel like they were very foolish in the past whatever they were trying to sort out for themselves Definitely taking a look at that self-sabotaging energy or playing the victim. Okay? So Virgo, I'm gonna leave your reading there. Wow. This went on almost 40 minutes, which is That's what happens when you do Celtic cross readings, but I hope you enjoyed that I'm gonna leave it right here for you and I will see you guys tomorrow. Take care Don't forget to like share and subscribe. Okay. Bye


What's up, everyone? It's your girl, Kenzie here from Layton Sports Cards, and I am ripping a personal here for Christian C. We're ripping a 2022 Don Russ baseball hobby box. All right, Christian, best of luck. Thanks so much for your order. Here we go. Again, no base cards ship out of this, but does include rookies. So you'll get all your hits and numbered cards. Base does not ship. 2022 Freddy Peralta. And the blue, the blue foils do ship. Stand the man. The number there. That's a 2022 George Brett, Jake McCarthy, blue foil, right of record. Shrending to 349 Tyler O'Neill, Max Freed. Shohei Otani to 2022 blue foil. We have Red Sox signature auto Jared Durin. There you go, Christian. Nice one there. That's Ricky Otto, Tyler Stevenson. Second auto here. We've got Angel signature series auto Reed Dettmers. Reed Dettmers for the Angels. Stan Mioza. We're going to rookie to 2022 Matt Barash. Unleashed to 999, Vladigarero Jr., Joey Votto. To 2022 Ichiro, Jonathan India. That's Retro to 88, Retro, 88 Retro. Sorry, Brendan McKay for the Rays patch there. Brendan McKay, Trevor Rogers foil. 2022 Dougie. Luis Frias on the blue foil. We've got Fire Emoji to 75, Adam Wainwrights. 2022, Aaron Nola, Lance Lynn. Got Independence Day, Jose Abreu, Raider Rookie Seth Beer, blue foil. Got Dominators, Golden 99, Anthony Rendon, Angels, Bregman, blue foil. 2022, Cal Ripken, Wainwright. Got Independence Day, Byron Buxton for the Twins, Jim Tomey. Season stats to 22, 23, Aaron Nola, Phillies. Gonzalez, blue foil. One 317, Merrill Kelly. Got our glass now. Got Independence Day of Jose Berrios, Pete Alonso. We've got Liberty, Raider Rookie Vito Brujan for the Rays. Nice hit there, Christian. And the liberties are one per box. Raider Prospect to 25, Royce Lewis. Nice low number there. Those are hard pulls, for sure. 2022, Catel Marte, my Piazza. Last pack here for you, Christian. Independence Day, Diamond Kings, one solo, and both flows on the blue foil. All right, Christian, that'll do for your box. Congrats again on your hits. We'll get these right out to you.


I grew up with these two blankets in my house. My mom used to say that I grew up crawling on them. I don't know, maybe these things have meaning, and I absorbed the power of these blankets. But I also listened to the stories, which is what led me to want to pull back the layers of telling the story of slavery in this region. Those two blankets, according to the family tradition, were made by Manuelita Cisneros. She was a matriarch of my branch of the Rael family that lived in Abacue. She was taken when she was about five years old, and she learned to weave in the master's tradition, as many individuals did. They learned the trades in the master's house. She was baptized as Cisneros, that wasn't her name. We will never know her name, and that's part of the story of slavery anywhere, and certainly true here, that we lose their origins, we lose their name, we lose the story. She was captured, brought into Abacue, enslaved with a Cisneros family, and she eventually married another mestizo, someone who himself was mixed, Antonio José Rael, whose grandfather is listed as a mestizo, and whose grandmother was listed as a coyota, and so she marries him, and that family moves from Abacue into the village where I was raised in Cuesta. And so for me, that was my great, great, great grandmother. And so as I learned the story of these blankets, I just wanted to recover more and more of those individual stories. For many years hung above my desk when I was a state historian. Now it hangs above my dining room table, and it reminds me, every time I break bread, of the create, it reminds me, that blanket reminds me of the genetic weave that I come from. In this case, it's a literal weave that she made with her hands, but it's also, I'm the result of a genetic weave that she was part of. So that's the story of those two blankets that I have.


All right guys, it is getting to the end of February So I'm gonna give you a behind-the-scenes look at all the cool stuff that's been happening this month I'm currently coming to you from Wanaka, New Zealand, which is on the South Island Maybe I don't know in a few hours outside of Queen's Town This has been a crazy month We've done a lot of traveling did a lot of interviews had the whole Jason Brennan ongoing Jason Brennan Situation I wanted to thank you guys for being big-time supporters. I deeply appreciate it This upcoming month we are leaving Wanaka in I think four days then we are going to Christchurch for five days and then we are going to Australia for a month, which I have been looking forward to this leg from Or on the Pacific when me before we even came to New Zealand I had in mind Australia because there's a mathematician there That actually I don't know if it was a supporter or somebody on YouTube said hey You got to check this guy out his name is Norman Wildeburger. He has From what I've seen almost identical views on mathematics than I do but he isn't a crank outside the system He's actually got his PhD from Yale and he teaches at the University of New South Wales So I am so stoked and with him haven't shared this really with anybody But with him he is offered to use the University video system So we're gonna have like video to do an interview. We're gonna do some audio. We might make a multiple So this nest next leg is gonna be awesome. I'm gonna get some cool behind-scenes footage there So enjoy this compilation of the some cool video that has taken place at the last month And I'll see you guys next month So this is me talking with one of the academics before our interview. He's giving me a tour of his place Including goats Who knew we would be putting goats today? Yeah, right? and Then you'll see a shot of us in the midst of an interview You're out in this very beautiful picturesque place this point We're talking about the liar's paradox for a solid hour or so that didn't make it into the interview that he didn't want posted But it was fantastic conversation In the process of narrating square one for the audiobook version I can already hear my voice is horse Not a very glamorous process kind of stinky house sweaty Sticking my head in the closet talking to a blanket So I'm going to continue vagueness Another objection comes from a criticism of logic being quote to black and white So I also got a chance while I was in Auckland To give a talk a local public working space a listener of the show invited me out there And it turned out really well. So I made some friends made some good connections It's good public speaking experience for me to talk about professional podcasting This is the real exciting behind-the-seats footage stuck in Auckland traffic in the rain after a shitty experience U.S. Embassy Because of course everything is overpriced and then you got to sit down for an hour to get a little signature notarized and this is a boat ride that Julia and I got to take to Waikiki Island one night Which was just a blast. So that's the view of Auckland in the evening gorgeous All right, this is our last day in our Cute little New Zealand apartment It served us very well. We're gonna miss it parting words So unlike any place that we've been in the world so far we moved into an actual apartment we had a lease in Auckland and It was it's one of my favorite cities in the world. We moved in felt like it was home. That's just the view from the balcony It was safe. It wasn't very like hip because we're not very hip So it was a perfect place for us and now here's gonna be some footage of after we left Auckland We took a road trip on the North Island We went to what's called the Coromandel Peninsula, which is gorgeous and we went down to see some traditional Maori reenactments in a city called Rotorua So this first clip is a guy reenacting the welcoming ceremony that the Maori would have given When visitors arrived, you're also gonna hear right after one of the Maori war dances slash songs Which is super intimidating and awesome Pretty fucking crazy isn't it? So we really enjoyed that trip. That's a picture of My wife and I of course me and the derpy looking hat my noggin is too big for any of the hats They had on sale.  We all we stopped overnight in a city called Fidyanga and our Airbnb host had a bunch of Guitars laying around and he was like practicing guitar while we were staying there. I was like, hey, man You want a jam? Because he was doing kind of bluesy jazzy stuff and he was like, yeah, so this is footage Of our jam session, but she was way more advanced than I am as a musician, but it's still pretty damn cool So this is us playing like a slow jam together. I think his name was Brian, but don't hold me to it Hey, send me some feedback if you like the stuff that you'd like to see stuff that you wouldn't like to see You should get some awfully cool transit footage for this next month If you'd like to see the glamorous process of me editing videos or audio Whatever it is, shoot me an email or message me on patreon and I'll add it into next month's clip Alright and another sincere thanks for me You guys are helping make this show possible in a profound way and I deeply appreciate your support There's lots more stuff coming your way. So stay tuned


Okay folks, Larry Pesavento, mid-afternoon update for TFNN. Our guest today will be Peter Lides of Lides Stock Market Cycles. He'll be on at 1.30, all we have some great information. I posted the chart here of the Dow Jones e-mini as of yesterday when we made that 3.82 retracement about two hours after the market had opened. We talked about that and of course we've went down substantially from that level today. We'll update that. When we go into the charts for today as we go into the second part, folks, we have a situation with the Federal Reserve will be coming out today and we have two days here of talking about what they're doing. It doesn't make it too much difference what they're talking about, folks. Interest rates are going to be going higher. The bond market is telling us that and we'll look at that also but these markets are not acting as well as you might think they are. In fact, we've come down from 35,360 to 36,000 in just a matter of three trading days and that is not usually a very good sign. That's from where we were last Friday. If you remember, everything looks so good early in the morning but then it didn't look good a little bit later. Everything's getting hit a little bit. With the exception of gold, it's up a very, very tiny bit and it just made the 61 retracement up there at 19,000, try it again Larry, 19,588 was the exact number and the high on the gold today was 19,588. We were fortunate enough to get the sell off there at 35,8. We got it now. Stop it. Break even has a small lead in it but that's better but we've got the Fed out here and of course anything can happen and it usually does. So with the worst, you have to stop it, break even on that position to see what's happening. Stocks continue. We're coming in for some type of a bottom here for a short term down here around 44,60 level in the S&P but I don't think that's going to last very long because the price objectives that we're looking at look much lower actually. So let's keep those in mind. It's not how much money you make folks. It's how much money you don't lose. So stay tuned<PHONE_NUMBER>.


I think antitrust is essential, but it is not sufficient to deal with issues in this area. Remember, antitrust is a very targeted statute. The Sherman Act and the Clayton Acts have you in Congress telling the agencies to prevent a reduction in competition and to protect, prevent efforts to monopolize. It's not about promoting competition. That's what you put in the Communications Act. It's not about preventing all discrimination that is unreasonable. It's about getting at it when it is blocking entry, or when it's cartel-type behavior, or when it's actually allowing for a dominant firm to expand its dominance. So it can get at a few things, but not everything, even in the competition realm. And then beyond competition, as Dr. Turner Lee says, there are a lot of other values at stake here. This is about opportunities and equity across our society. That is something the antitrust laws only address through trying to promote consumer welfare. It doesn't address the other attributes. So I think the FCC plays a critical role. Title II is a portion of it, the principles and the rulemaking authority. 706 as a mandate was offering the ability to extend broadband and possibly even expand it, as Dr. Turner Lee said, to low-income customers. I think that could be extremely valuable as we move forward and make it affordable for everybody. So it's the combination of the tools I think that's so critical.


In the previous lecture, we developed a solution to the Fokker-Planck equation for the primary case of a free particle undergoing random walk, but random walk described by continuum mechanics along x direction. We arrived at a Gaussian solution to the probability of finding the particle at some location x at some time t. Today, we discuss other than Gaussian solutions which are in principle possible, but they will not come from the Fokker-Planck equation one has to take a step back and go back to the Chapman-Colmograph equation. So, today's topic is general non-Gaussian solution to continuous time random walk. So, we take a step back and rather than use the Fokker-Planck equation which will give you a Gaussian solution, let us see what general possibilities exist. For that, we go back to the Chapman-Colmograph equation, we go back Chapman-Colmograph equation which is written as the transition probability to a point x 2 from x 1 in a time t is given by via the Markov assumption by the Chapman-Colmograph equation which says it is the probability that it has come to x 2 from some point x in a time tau, where it had come to x from the original point x 1 in the time t minus tau integrated over all possible values of x. So, this is the Chapman-Colmograph way of writing the Markov transition process for continuous variables via integral formalism. So, picture here is particle has come to x 2 it started from x 1 and we say that it came from x 1 to x. So, x 1 let us say time 0 this is x 2 at time t. So, at some time t minus tau it was here and then in the remaining time tau it reached x 2. So, this is the meaning of this equation. Let us consider a special case consider spatial homogeneity of the system spatial homogeneity which means the transition probabilities from x 1 to x 2 do not really depend on from where it transits. In other words this basically implies the transition probability from any point x 2 from x 1 in a time t is given by is equal to identically the difference between the two positions only. So, it does not depend on the position from where the transition took place. So, if space is completely homogeneous it is completely invariant then then what matters is only the length by which the system has to undergo a transition. So, that length is x 2 minus x 1. So, it is a function only of that length of transition. So, we used this definition in our previous derivation of the Fokker-Planck equation also. So, with this assumption we can write with this the C k equation takes the form P of x 2 minus x 1 in time t equal to minus infinity to infinity P of x 2 minus x it will be true for all points in a time tau and P of x minus x 1 in time t minus tau for all points x. So, P is now a function of difference between the spatial positions only. Denote x 2 minus x 1 by a new variable say x 2 minus x 1 equal to xi. So, xi is a new variable and let and also we denote x minus x 1 x is our intermediary variable by xi prime. Then we can always write for example, x 2 minus x then x 2 minus x becomes x 2 minus x 1 plus x 1 minus x and this can be written then as xi minus xi prime. Hence we have the C k equation can be written as xi t which is integral minus infinity to infinity P xi minus xi prime tau P xi prime t minus tau d xi prime. So, this is almost like the formulation we made when we derived the Fokker-Planck equation we are expressing the transition probability by the jump length xi here. So, this simply says the probability that a jump xi has been made at a time t is given by the probability that it made a jump xi prime in a time t minus tau and in the remaining time tau it jumped by x xi minus xi prime integrated over all xi prime values. Now, let us why we did this will be apparent now let us Fourier transform this equation that is definitionally the Fourier transform is we use the notation P carrot say k t is defined as integral minus infinity to infinity e to the power i k xi P xi t d xi. So, Fourier transform with respect to the spatial variable or jump length variable xi. So, e to the power i k xi P xi t d xi we have been doing this and that is the definition. So, upon applying this to C k equation we have left hand side for example, will be LHS is simply e to the power i k xi P xi t d xi which is P tilde k t to be more precise P carrot k t and hence we get P carrot k t will be equal to now already there was one integral in the C k equation. So, this transform introduces another integral. So, there will be two integrals 1 over xi prime and 1 over xi. So, we can write e to the power i q xi. So, i q xi can be written as i q xi minus xi prime plus i q xi prime you can always do that because i q xi prime anyway is going to cancel I had multiplied only by e to the power i q i k xi. So, we decided it is k. So, i prime then the transition probabilities xi minus xi prime in time tau multiplied by xi prime in time t minus tau integrated over both d xi and d xi prime. So, this is now simply becomes minus infinity to infinity minus infinity to infinity. So, if we define xi minus xi prime as xi double prime for the xi integral it is e to the power i k xi double prime P xi double prime tau d xi double prime and then e to the power i k xi prime exists as such xi prime P xi prime t minus tau d xi prime where we use xi double prime equal to xi minus xi prime and d xi is nothing, but d xi double prime. So, now these two are two independent integrals there is no coupling between them. So, they simply become the products. So, we get that is we get k t will be 1 integral e to the power minus infinity to infinity i k double P xi double prime tau d xi double prime multiplied by another integral minus infinity to infinity e to the power i k xi prime T xi prime t minus tau d xi prime. Now going this is exactly the Fourier transforms of the respective quantities which leads us to a neat algebraic, but it is not just an algebraic equation it is an algebraic product difference type equation P tilde k t is P P carat k t is equal to P carat k tau multiplied by P carat k t minus tau of course, let us recall that tau should be lying between 0 to t with the end points included it cannot exceed t. So, this is a very neat reduction in the transformed space of the Chapman-Kolmogorov equation for the special case where there is a spatial invariance. So, this is like a this is a convolution theorem equivalent of a convolution theorem of Fourier transforms. For example, if we choose the classical Gaussian packet which we have been discussing quite often P carat k t equal to e to the power minus k square dt is satisfies the above equation. So, this is just to reassure ourselves that we have whatever we have done so far is consistent with this because we know that if it is exponential e to the power minus k square d into tau it will be e to the power k square dt minus tau the tau term cancels and you get back t which is again e to the power minus k square dt. But this is not the only solution that it is so, that this is not the only solution we can obtain by converting this into some kind of a differential equation. So, given this equation we differentiate with respect to tau only to find other possible solutions find other possible solutions we differentiate the convolution convolution solution we can say convolution with respect to P that will be d P carat by dt k t and on the right hand side the first term does not depend on t at all. So, it will remain just like this P carat k tau it will be and the second one will be d P carat by dt k t minus tau. Now, we are free to choose a tau tau is an intermediary variable only thing is it should be less than it should be more than 0 and less than or equal to t. So, we make use of this possibility that it can take any value and we decide that it can we will put tau equal to t. Now set tau equal to t since tau can take any values less than or equal to t positive values less than or equal to t. So, when we do that the left hand side becomes then we have d P carat by dt k t it is going to be on the right hand side we will we will see here. So, when we put the first second term will remain just the same, but here when we put it will be completely independent of any time. So, it will be 0. So, this will remain P carat k tau, but tau is t. So, we can replace this tau with t and the other one therefore, will be d P carat by dt k 0. Now this is an entirely an independent quantity which represents the gradient of P carat at time 0 and we have no prior information on this quantity. So, we can justifiably assume it to be just a function of k the quantity d P by dt k 0 is unknown a priori we do not have a priori information on this. Hence, we assume it to be just a function of k we assume that it is a general function of k that is d P carat by dt k 0 identically we say it is some lambda of k we will discuss what is allowed. Hence, we get the equation d P carat by dt k t will be lambda of k of P carat k t this is a first order equation in time. Hence, solution is P carat k t equal to some constant of integration and in general it it will not be a function of time, but it can be a function of k we cannot stop because k is just a parameter and e to the power lambda k into t. So, it comes as a product of lambda k. So, this is the most general solution for the Fourier transform of the probability density function in the spatial homogeneity case arising from as a solution of the C k equation. Now, let us see what how to determine a k to find a k or to determine a k we use the initial condition that is we know that P x at time 0 was some delta x it has to start from somewhere let us say started from the origin since it is spatial invariant it does not matter we can always put x equal to 0 as a starting point. Hence, if you take the Fourier transform P tilde k at t equal to 0 that should be the Fourier transform of delta x which is unity. Therefore, from the solution on top of this page we have 1 equal to a k e to the power lambda k into time was 0. So, that is a k. So, a k is unity hence a k is 1. So, we have the solution P carrot k t equal to e to the power lambda k into t as general solution to C k equation under translational invariance. Now, some properties of lambda k can be estimated for example, what should be the value of lambda 0 properties of lambda k let us say what is should be lambda 0. Now, as k tends to 0 if you go back to the from the definition we have P tilde k at t equal to 0 is integral minus infinity to infinity P k t e to the power i k x P x t dx. So, as k tends to 0 this will tend to integral P x t dx then k equal to 0 and this we know is equal to 1 normalization of the probability density. The probability should be normalized because the particles should exist somewhere in space all the time. Since you have started from some and there we have no loss mechanisms you should be normalized unity which basically means hence P tilde k t limit k tends to 0 should approach 1, but from our solution P carrot k t equal to e to the power lambda k t will approach 1 if lambda k approaches 0. Hence, one property is that lambda k lambda 0 equal to 0 we can say that lambda 0 equal to 0 is an essential property. We cannot have a constant there. There are now many other properties will emerge when we examine in detail the behavior of the solution for various situations. We will discuss in the next lecture. Thank you.


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