Start small So money geeks mr. V here welcome to another video guys so into this video I want to answer a question that I've gotten from some of you guys on the channel on how to day trade with a small account So if you again when I mean small account I'm talking something between a thousand dollars and five thousand dollars How do you do a day trade with that amount of money on Weibo? Before we get started guys if you're new to the channel will talk about how to earn money how to save money How to invest and build wealth? So if that's something that interests you go ahead and hit that subscribe button and the notification bell so you don't miss out on new content So how do you day trade on Weibo with a small account again? Like I said a small account in this case. I'm talking between one thousand and five thousand dollars For starters once you sign up for Weibo There are two types of account that you possibly will get you get a cash account or you get a margin account So I've actually done a video here on the channel where I break down the differences between the two accounts between Cash and a margin account and hook that here so you can definitely go check it out But just for this video, I'm gonna explain a little bit so a cash account Once you sign up and deposit money you get unlimited trade with a cash account But the only thing is there's a cash to it But when you sell your stocks it takes two business days For your money to settle back in your account before you can use it to trade again Whereas with a margin account if you have over 25,000 dollars, then you get unlimited Trade and if you have less than 25,000 dollars you get three day trade in a five business day cycle So if you take you can if you on Monday if you do three day trades and United until Five business days you will not be able to do another day trade. So that's just the catch up So in your case if you have less than a 25,000 dollars So in this case again between 1,000 and 5,000 dollars I would recommend that you switch here if you have a margin account today I'll say switch it to a cash account or if you haven't already opened the account open Cash account once you open your cash account and deposit your money and the money settles Here is how I want you to structure it because this is how I started you take for instance your 5,000 dollars. I want you to split it up into two parts. So $2,500 a piece So on day one let's say on Monday you only trade with $2,500. That's what you would have for trading that day And then on Tuesday you trade with another $2,500 So if you use $2,500 to trade on Monday, you have $2,500 to trade on Tuesday And then the other $2,500 that you traded with on Monday goes through Tuesday and settle on Wednesday And it's ready for you to trade with again. With that strategy You would never ever miss a day of trading because your money hasn't settled yet The implication here is that you are gonna make less profit because you're only trading with half of your Your buying power, but the good news is that you still be able to do your trade on a daily basis As opposed to having to wait a day for your cash to settle So that is a very simple strategy that actually started with I implemented that until my account grew to the point Where I could convert my cash account to a margin account in an insight trading unlimited trade So that works for me just fine So you should definitely give it up that strategy a shot And that is another way for you to avoid what I call the PDT that pattern day Trading valuation that you would get and also you can avoid good faith valuation So good faith would be that you buy a stock today with money that hasn't settled and then sell it today Before the money settled. So that way they'll punish you for doing that. So you you're using one stone to shoot two birds So you take that money you split it up in in two parts You are no you have enough money to trade every single day at the same time You are avoiding getting good faith valuation because you're not using unsettled cash to to trade So let me know in the comment section. What do you think about this strategy again? Like I said, it works for me. What if you don't add a honest other strategies that you are applying or implementing that is helping you Grow your account. Let me know the conversation Give you if you're new to the channel and you want to learn more about kind of day trading here on Weibo That's what I talk about. That's my that's my Experience and so we use Weibo and I'm gonna put a link in the description below where you can sign up and get two free stacks Just for signing up and depositing a hundred dollars to get you started and as always guys Stay motivated


card number 25, card number 25, William Buckley, card number 25 of the Iran Contra scandal trading cards from eclipse comics, William Buckley, and these came out in as before 1988. Card number 25, CIA station chief William Buckley. William Buckley was the CIA chief of station in Beirut, Lebanon, when on March 16th 1984 he was kidnapped by Muslim Shi'at fundamentalist. His abduction sent shock waves through the intelligence community, for Buckley was the CIA's top terrorism expert. CIA director William Casey's decision to send Buckley to Beirut in 1983 had been doubly dangerous. First, Buckley was an obsessive womanizer whose flamboyant ways had led to breaches of security while he was stationed in Vietnam in the early 1970s. Second, his cover had been blown in Pakistan in 1979 when Muslim fundamentalists had ransacked the US embassy in Islamabad gaining access to high secret US intelligence. Casey's decision violated a CIA rule not to send an agent who had been burnt into the same theater of operations for at least five years. In November 1984, ex-CIA agent Theodore Shackley, C.C.R. 24, who had known Buckley in Vietnam, went to Hamburg, West Germany, to meet with Manutcher Gorbinafar, C.C.R. 26, an exiled Iranian wheeler dealer living in Paris who claimed he could arrange an arms for hostages ransom deal for the release of Buckley. In October of 1985, two months after the first US Israeli shipment of TOW missiles to Iran, the CIA received reliable information that Buckley had been taken to Iran and tortured to death the preceding June. It is still not known what secrets, if any, he revealed to his captors.


Now, let's see what effect will be on the CML and the SML if any of the assumptions related to the KPM are relaxed. We know that the KPM assumes boring and lending at the same risk-free rate, but in real world we see it is impossible to lend and borrow at the same risk-free rate. So we see two different lines going to the Markovitz efficient frontier. The first line risk-free rate F that is R, FR, F shows investment opportunities that combining the risk-free assets and the portfolio F on Markovitz efficient frontier. Now it is impossible to extend this line without further borrowing at RFR or the risk-free rate to acquire further units of F at the same level of riskiness. So in order to get more units of F, we need to have borrowed some amount at the rate of RB in order to invest in portfolio K so that we can extend our CML along the line segment KG. This means that now CML is made up of RFR, K and G that is it is a line segment of RFRF and we have now a curve segment which is RFR, which is FK and another line segment that is KG and that KG is possible to the additional borrowing at the RB. So we see that as we have the slope of the borrowing line KG lesser than the slope of RFRF. So in this case the investor is at the loss means his net return is the loss because the slope of KG is lower than the slope of RFRF. The assumption to relax the risk-free rate means we have a zero beta model. In this case if portfolio M is a mean variance efficient portfolio that is an alternative model by black which does not require a risk-free asset. In this case within the asset of feasible alternative portfolios many exist where returns can completely uncorrelated with the market portfolio. In this case the beta of these portfolios with the portfolio M is zero. So zero beta portfolio does not have any systematic risk however it may have some unsystematic risk. The availability of zero beta portfolio will not affect the CML but it will allow the construction of a linear CML as we can see this line in the diagram. A zero beta capital model can also be presented like the regular capital model. In this model we have an equation where expected return on security i is equal to the expected return on a zero bitum. This is the zero rate of return and it replaces the risk-free rate. In replacement of RFR we use the term RZ so we see that zero beta model generally does not require any risk-free asset. The KPM model also assumes no transaction cost but if we assume that there exists some transaction cost then the investor will buy as the KPM assumes no transaction cost so investor will buy or sell any missed price security until they plot them on the security market line but if there exist any transaction cost the investor will not try to correct the missed pricing because in this case the cost of buying and selling the missed price security may exceed any potential access return which will investors return into loss so the securities will plot very close to the SML as we can see in the diagram securities are plotting alongside this thick line at a interval with the transaction cost SML will be a band of securities rather than a straight line so we see that we have a band between these two lines above the SML and below the SML this means that the width of this band is basically a function of the amount of the transaction cost. Heterogeneous expectations and the planning periods investors with different expectations about risk and the expected return would definitely have a unique CML and the SML on on individual basis we know that KPM is a one period model so corresponding to the planning period for every individual investor a one year planning period the KPML the CML and the SML would also differ for someone with a one month planning period so if we have an expectation difference difference and the timing difference the CML and SML will also be differing for individual investors taxes KPM assumes no taxation but in real corporate world taxes are there means that the expected return in the KPM are pre tax return as we have seen earlier in the model whereas the actual return for most of the investors are after tax if we see this model this model provides the returns after the deduction of capital gain tax which is levied on the capital gain or the price appreciation and in the other part of the equation we see a tax on the dividend so we are taxing two items for the return the first tax is on the capital gain that is the price appreciation and the second tax is on the dividend receipt tax rates differ between individuals and and corporations or the institutions for institutions that do not pay taxes the pre-tax model is correctly specified because their pre-tax and after tax gains are the same in this way we can say that tax rates can cause major difference in the CML and the SML among the investors whether they are individual investors or the institutional investors


JCOC offers a unique opportunity for influential civilian leaders who have limited knowledges of the U.S. armed forces to share their hands-on experiences with their military. I didn't know what to expect. The things that struck me were the caliber of people I met. I probably feel the wind from the Osprey. I probably feel the the G that we pulled or too easy but I really think that the most memorable things the things that were etched into my heart etched into my mind was the interaction with the servicemen and women. Sometimes in life we we provide support by cheering on the team but I'm really inspired to go back and and let some people know that I bear influence with to say hey don't cheer on the team get on the field let's be a part of what's happening. We saw a tremendous demonstrations from all five branches and met an incredible group of people who were incredibly passionate about their jobs. I think the one thing that really struck me about all the people that we met was that you know it seemed more like they were excited about the opportunities that they were having in the military than being motivated by a sense of duty to do that they were doing. There was a real proactive passion from all the people who met. My expectation was that it was going to be more formal and not as intimate and what I found is that this is a very intimate close organization. Sometimes we get caught up when listening to the rhetoricals on television and we forget the level of professionalism and seriousness with which each of you all take your jobs and it makes me very very proud to be an American. I personally spending a week with these people and very very confident each and every one of them is bringing back clearly a new sense of understanding about what the men and women in uniform do for you. One thing that I think I have failed in my 14 years of pastoral ministry is helping to support my men and women in ways more than praying for them but also encouraging other persons to see the value of our military. People in the military are just people and they need to understand that because I don't think everyone has that perception of it and I and I hope to change that perception. When I'm 80 years old looking back over my life this will be an experience that will come to my mind.


Hello, Didier Stevens here with another video on GNU Radio Companion that you can use without any dedicated hardware like an SDR. In previous videos we used different sources like a sine wave, a noise source. Here we are going to use a source that reads a WAV file and uses that audio that you can then further process. So you need a WAV file, a source and that's something you have to find here in the modules. Now with CTRL F you can search here. So CTRL F and let me type WAV like this. Okay, so in audio I have a WAV file source. So I select this. I'm going to do this without repetition so that the WAV file only plays once. And then here in test user documents I can find here a recording like this. Now this is a WAV file that was recorded at 44.1 kW. So the typical frequency for WAV files and that's also the frequency of a compact disc audio if you're still familiar with that. So I need to change my sample rate here to 44100. And then in my audio I need an audio sync for the speaker like this. I can connect this, save it. So WAV and then I can play it. This is WAV recording. So that was a recording that I just made here and I'm playing it back using here GNU radio companion. Now if I change the sample rate here, if I put it at 48000 like I did with the other examples, then I will actually use a higher sample rate than the sample rate with which the WAV file was recorded. And let me play this back. This is WAV recording. So now you hear my voice with a higher pitch and would I have used 32000? So a sample rate lower than the actual sample rate. This is WAV recording. So now you hear my voice much lower. So let's come back to the actual sample rate 44100. Okay. And now I'm going to put in a filter. Remember we talked about filters and I use the bandpass filter here. Like this. So I'm going to insert this here. So float, float. And then again I'm going to use the frequencies of the telephony system. So 300 to 3000 hertz. Transition of 10. And then I can play back my WAV file. And now you will hear that it sounds like coming out of a telephone handset. This is WAV recording. And this here is just actually the same as in the previous video that I made, the previous examples. But instead of using a noise source here, now we are using a WAV file. And then you can process any sound that you wish as long as you can create a WAV file for it. And just one last thing that I'm going to show you here is to make some echo. So let me remove this. So an echo is actually a repetition of the signal but with the delay. So you have in a new radio companion a module to introduce delays here. Let me use that module. And then what I want to do is combine the WAV file without any delay together with the delayed WAV file. So for that I need to add. So that's something we also did here, mat operators. And we are going to do an addition like this. And everything needs to be in float like this. And the delay here, if you look in the documentation, the delay is specified in the number of samples. So since my sample rate here is 44.1, I'm going to use that as a delay. I'm going to delay this for 44100 samples. So that's one second. And instead of typing 44100, I can say sample rate like this. And you see here appearing 44.1. So my original audio is going from the WAV to the addition to the audio sync. And then I'm going to delay this same audio and put it here in the addition to the audio sync. And now we can play this and you will hear an echo of one second. This is WAV recording. As you can see here you heard the recording twice. But it's not a really natural echo. So I'm going to speed this up a bit and I'm going to do this one tenth of a second. So device the sample rate by 10. Here you can actually input Python expressions. So this is a variable divided by 10. So let's play this and you will hear that this sounds more like a natural echo. This is WAV recording. And now what also typically happens with echoes like that is that the second and sometimes a third and you can have more than one echo. So that the second signal is a bit fainter than the original one. And that's also something you can do by multiplying here with a constant. So let me add this and so float float. And I'm going to divide this by two. So I'm going to multiply this with 0.5. And so my signal here will be half like this. And then I can again listen to the echo. This is WAV recording. So here you had some simple examples to show you how you can use GNU radio companion here with WAV files. And then do some simple tricks like filtering to have a certain audio effect or all echoes like here I did with the delay.


I was recently inspired to create this look using Makeup Geek eyeshadows, which have been discontinued, the brand doesn't exist, but I was recently watching one of my absolute favorite podcasts, The Sesh. I love these girls so much, go and check them out. But they were interviewing Marlena from Makeup Geek, and it just reminded me how much I loved Makeup Geek eyeshadows, so I thought I would share with you how you can add some new life into some of your old shadows. Because these have been discontinued, they're really old, they're losing their pigmentation, they're losing their texture, but there's ways that you can still make them work. So if that interests you, then just keep on watching. And if you're new here to the Makeup Geek channel and you enjoy tutorials like this, then I might suggest hitting the subscribe button below. It's totally free. And if you're already subscribed, then thank you, and welcome back. Now let's get started. So we're going to start off with an eyeshadow primer because primer is a key. This will give the shadow something to hold onto, and this will then increase the pigmentation that older shadows tend to lose. And because this is tinted, it's also going to disguise any natural discoloration on my lids, which allows the shadows just to pop that little bit more. So I'm going to keep the lid slightly damp, and this is just going to add a little bit more grip for us.  So take your favorite blending brush, and my next tip is to make sure you're holding it a little further down on the handle. Older shadows will lose their consistency within their texture, so you might end up with some patchiness. I'm pairing that with a damp base, and you're kind of asking for trouble, so holding the brush a little further down on the handle will soften the pressure and allow for a smoother blend. I'm just working this into the crease of the eye, leaving a little gap right underneath the brows, and then just slowly bringing it down onto the lid. Next I'm going to move on to using this little baby blending brush, which I love so much, and I'm going to use the shade Mocha. So what I'm first of all going to do with this is use this to mark out my lid a little bit more, so I'm kind of squinting so I can see where the hood of my eye is, and then I'm just going to slowly build this shadow up to kind of recede that hoodedness. Think of it like contouring your eye shape. Now I know what you're thinking. You're using a blending brush to apply this, and then you're going to use your other blending brush to blend it out even more. Wouldn't be better just to use a flat brush or your fingertips, and yes, definitely to build up to the point where I want to get this to. However, because these shadows are slightly older, I find it's better just to keep layering using your blending brushes, because if you go straight in with a flat brush or your fingertips to kind of pack the shadow on there, the unevenness of the texture can cause a lot of fallout, and then you don't get that same buildup of pigmentation. So I'm going to be using the two blending brushes to slowly build up to get the shape that I want, and then you should eventually end up with something that looks like this. And this is going to last a lot longer for us as well. Next, I'm going to take a flat brush and I touch a concealer and just apply this on the lid where I want to apply shimmer. By applying a small amount of concealer, we're actually going to add some moisture back into our shimmer, which then helps it shine so much more. And I'm using two different shades of shimmer, a lighter one in the inner corner and then more of a golden one right in the middle. But that small amount of concealer underneath is just going to help them shine so much more. And then I'm going to take a fade shade. If you've been following my channel for a while, you know what a fade shade is, but basically it's something with a lot of pigment that we use to create this sort of pro finish. So older shadows can look a little bit muddy. So applying a really pigmented shadow like an orange, a peach, a pink, a red, it will create so much more depth and create that really professional finish. So we're just going around the edges of our shadows and it basically creates the illusion that we have all of these different tones on our lid, but really we just have this extra one. And my final tip, and this is my absolute favorite, is to use a setting spray. Close your eyes and spray over your shadows and watch the magic happen. It adds so much more moisture. Now not everybody can do this because some people have sensitive eyelids, but I do find that this is your finishing step. If you want to add some life back in to those older shadows. And then I finished off with mascara, the rest of my makeup. I applied a little bit of mocha just underneath the eyes. And then there you go. That is how to add some life back into your older shadows. These are maybe seven, eight years old. They're really old shadows and they're still going strong. If you want to see more looks using some of the other shades that I absolutely love from Makeup Geek, definitely let me know. As always, my friends, be kind to yourself, be kind to others and go and check out the Sesh Podcast because they are so lovely and so kind and I know that they will brighten your day because they brighten my day all the time. And I'll see you in the next one. Bye for now.


I don't think that's work. Go. I can't. Tell him to go ahead with the hell that she is. It's great to be back in Montana and at Charles M. Russell High School again. I have to be honest with you though, some of my hard-working aides recommended against leaving the Capitol and coming all the way out here. So to keep him happy, I decided I said, okay, we'll flip a coin to decide whether to visit your beautiful state or stay in Washington. And you know something? I had to flip 14 times before it came out right. So then I got them all together and I said, boys, we're going where the people think big and the sky's the limit. And you know what I meant. They didn't. They all headed for Tip O'Neill's office. But it's also wonderful again to see the incredible beauty of this state and of the wilderness. And I don't feel so bad now about where we're sending Tip and the Democrats next Tuesday. But besides getting to come to Montana, there are some other advantages to being president. On the way out here, I told Air Force One's pilot to fly low over Mount Rushmore. I just wanted to see if they were adding any new faces. Seriously though, it really is refreshing to see the Rockies again. You know, we don't have anything like that in Washington. We have one mountain. The big spenders have been working on it for 40 years. It's called the national debt. But our incredible national debt and what led to it is one major reason why I'm here today. We're approaching an election that is every bit as crucial as the election of 1980. Americans remember all too well the days of double digit inflation, skyrocketing interest rates, the days of hostages in Iran, and the national malaise. The days of leadership in Washington that blamed you, the people, for their mistakes. Now none of us wants to return to those days, but make no mistake. The decisions made by Americans next Tuesday and throughout this decade will determine whether we stay the core of the national renewal or whether we stagger off on one more economic binge, a binge that we and our children would have to pay for, and we'd pay for it with another pounding hangover. Take that mountainous debt as an example. Today it stands at more than a trillion dollars that the government will spend this year just in interest payments on the debt. That's as much as it took to run the whole federal government only 20 years ago. But our trillion dollar debt is only one small part of the chaos, confusion, and all-purpose mess left us by the failed policies of the past. Farmers, miners, housewives, and small businessmen are still paying dearly for these mistakes. Remember when inflation was in double digits for two years in a row? Remember when interest rates hit 21.5 percent? Can you remember when the growth in the gross national product had gone down for the third year in a row and the money supply had increased by 13 percent in the last half of 1980, the highest rate in post-war history? Remember when unemployment was already a serious problem? Business failures were increasing, and a recession was on its way that would hit us with hurricane force, a recession that was the legacy from all those years of boom and bust, of wild spending, erratic monetary policy of tax and tax and spend and spend and borrow and borrow. Now I'll admit these statistics aren't at the fingertips of every American, but I do know that in their own quiet way Americans fully understand the mess we inherited in 1980 and the obstacles we faced in trying to straighten it out.  In the last 10 years, federal spending had tripled. In the last five years, federal taxes had doubled. In 1980 alone, federal spending had gone up 17 percent. The budget for the Department of Health and Human Services, the one that does all the charity programs and so forth, had become the third largest budget in the world, just after the national budgets of the United States, the total budget and of the Soviet Union. Those automatic spending programs we call entitlements were completely out of control. Those are the programs that the Congress doesn't have to change them every year. They're there with automatic increases built in. The food stamp program alone had grown in 15 years from $65 million to $11.3 billion, an increase of more than 16,000 percent. On an inaugural day, just minutes after I took the oath of office, I went back into the Capitol and I signed a memorandum, without even taking my coat off before I did it, that put a freeze on federal hiring. In the next three or a year and three-quarters, I should say, we cut the rate of growth in government spending by nearly two-thirds. We got individuals and businesses their first real tax cut in nearly 20 years, and achieved the historic reform of tax indexing. That was stopping government from moving people into a higher percentage tax bracket, just because they'd gotten a cost of living increase. We cut into the thicket of federal regulations that were smothering economic growth, and we saved businesses and citizens at least $6 billion annually. Now, that's not a cut in the spending of government. That was a savings out in the private sector to the people who had to do all the paperwork connected with those regulations, and we reduced the number of man-hours of paperwork the citizens were performing by nearly 200 million man-hours. We knew it would take time, but even though our economic recovery program has been in effect for only 13 months, the dollar is stronger than it's been in any time in 10 years. Inflation wants the number one concern of every American, double digit for two years, is down to 4.8 percent so far this year. You know it's a very interesting coincidence, because 4.8 percent was exactly what inflation was when the last Republican president, Jerry Ford, left Washington. And that 21.5 percent prime interest rate is all the way down to 12 percent and even in some banks, 11.5 percent. Now, we're still working to bring them down further, but let's face it, we've already come a long way from that 21.5 percent that we inherited. And you know, the source of our economic problems is even beginning to dawn on the baffle gabbers and those fancy dudes in Washington. They're finally realizing what Montanans have known for a long time. Government is too big, it spends too much of our money. You can't drink yourself sober or spend yourself rich. You can't prime the pump without pumping the prime. But all these gains that we've made are now in jeopardy. Already the national leadership of the Democratic Party is talking about taking back your third year tax cut, scheduled to go into effect on July 1, and revoking the historic reform of tax indexing, which I mentioned. If they have their way, the average family of four would lose over $2,500 in tax relief over the next five years. So that's why the election here in Montana is so important. We have to maintain control of the Senate. We couldn't have done what we've done if we didn't have a majority in the Senate, as slim as it is. And we must have control of the key committees of the Senate, which comes with having a majority. We have to put men and women in the House of Representatives, who have a record of being part of the solution, not part of the problem. And I ask you, is there anyone who's fought harder against big government, anyone who's fought more eloquently against higher spending and more taxes than your own candidate for the United States Senate, Larry Williams? As head of your National Taxpayers Union, Larry has led the battle here in Montana for the tax indexing initiative and for an amendment to balance the federal budget. Montanans aren't going to have to worry about where he's going to stand and how he's going to vote at the crucial moments in the next Congress. Now, let me tell you, he's no rubber stamp. I know that. And it's probably going to be times when Larry and I won't always agree. But I do know this. I know his personal integrity and his conscience will always be his guide. I need Larry Williams. Montana needs him. The country needs him. We need a proven fighter in the battle to get government off our backs and back to the standards of excellence once envisioned by the founding fathers. And I especially want to address some remarks today to those voters in Anaconda and Butte and the other towns around this state, many of them minors and traditional Democrats. This is their chance to send a message to the national leadership of the Democratic Party. This is their chance to tell them, leave the tax cut alone, stop stonewalling the balanced budget amendment and the school prayer amendment, and stop bottling up those legal reforms that would crack down on professional criminals. This is your chance and their chance to tell the national leadership of the Democratic Party quit bowing to the liberals in the grandstand and quit playing deep left field. This is their chance to tell them we've had enough of warmed over McGovernism. We're voting Republican this year not because we've left the Democratic Party, but because under the present leadership the Democratic Party has left them. And I say that as someone who spent more than two-thirds of his life as a member and a worker in the Democratic Party. I just couldn't follow the leadership any longer when it took the turning it took and the path that it has taken in these recent years. Now just let me add something right now that I hope all voters from all parties in this state know. There isn't a single congressman who's done more to help not just this district and state, but the entire country in the areas of agriculture, defense, and controlling big government. Not one more than Ron Marlenay. We need Ron back in the house. And I don't have to ask you, it's apparent that I ask you for your support, but when you go out of here start button-holding friends and neighbors and make sure you get their support too. One other prediction for you. If we can get our message out to the independents and the Democrats in this state, another personable and strong-minded Montanan named Bob Davies is going to steamboat with Larry and Ron right into the next congress. Let me insert just one thing. It's a kind of warning. I can predict that in these last few days of the campaign nationwide, and this touches on something that Larry spoke of a pure moment ago, I can predict that our opponents are going to broadcast widely one of the most dishonest canards that has ever been fostered in a political campaign. They're going to tell you that we, and I really include myself because I'm kind of a target of that, that in some way we're on our way to changing or reducing or doing away with social security. Now that is sheer demagoguery and it is an outright falsehood. I pledged back in 1980 during the campaign and before that in the primary campaign that social security, we all are aware, has some fiscal problems that have come upon it that must be resolved to keep the program's integrity. But these problems must be solved and will be solved not at the expense of taking any way thing away from the people who are presently dependent on social security. And you know, I said once during the campaign, anyone who will tell a lie about that is just like a fellow that says he enjoys a cold shower in the morning. They'll lie about other things. Well, so here are the stakes. Do we go back to tax and tax, spend and spend? Do we go back to economic stagnation at home? Do we go back to dishonor and retreat abroad, as we've known it in recent years? Or do we stay the course? Do we offer hope and incentive and opportunity? Do we keep alive the beacon of faith and freedom that is the American dream? I know the answer we'll get from those of you out here in the wide country and under the big sky. I know that all of you believe in these things. The love of freedom, the belief in basic values, the sense of daring that we've returned to Washington. It's in your very being.  Believe me, this election is important to our future. Help me, help your state, and help this country. Send Larry Williams to the Senate. Ron Marlene and Bob Davies to Congress. And together we can make America great again, not just for ourselves. But I'm so happy to see all these young people here because that's what this election is all about. America great again for them. Thank you and God bless you. Thank you very much. I ask you, I ask you in the next four days to go forth with the message that you had today and help me and Ron and Bob to go back to Washington, D.C. to do what needs to be done for the greatness of this country. Please help us these next four days. Carry the message for us. And on behalf of the Chamber of Commerce, I have something to present to the President, a pair of silver spurs. Ronald Reagan has earned his spurs. Just waiting until Tip O'Neill hears about those. Use them, President. Thank you all very much for being here. I hope you've enjoyed this day as much as we have. CMR, great falls high. We thank you very much. Thank you.


Ok, primer vse, da vši malo zelo? Ovo je nekaj zelo, da je, da je, da je... ...zaprej vse? Ok, vse je zelo. Vse da tamo zelo, da smo počeli, da, ...zato, da je pravda, ...pozirušnjenja o tezno teori, ...po polymer, ...po polymer, ...po polymer, ...po polymer, ...en vse ino za, ...ko se dopreluje vse, ...en polymer. Tko vsem, da je neko... ...bej pa vse, da sem... Tako, imamo zelo svoje energije, kaj je odstavljeno z dva kontribucija, jedno z energijem, ko je odstavljeno z interakciju tubovi, minus zentropi, ok. In smo zelo, da energija kontribucija je, da je odstavljena z interakciju tubovi, kako je terga, da je entropija, je pripravila je, je bilo, je bilo, je bilo, gašan, logratem do n-20 gašan, n-20 function do gašan polimer. Kajte si to površati, minimizujte, in se si, da je, da je, in do no. We found that no is equal to 3d plus 2, and we, which means that the equal 1 is no equal 1, d equal 2, no equal 3 over 4, d equal 3, no equal 3 over, 3 over 5, and d equal 4, no equal 1 alpha, and for d larger than 4, it's expected to be no equal 1 alpha. So one can say that for d larger or equal to 4, no has to be 1 alpha, OK? And so we discussed this a little bit, and we said, we noticed that the prediction is remarkably, in spite of the simplicity, is remarkably accurate because it's exact in d equal 1, where the exponents are not trivial, let's say, is, I told you a little bit to trust me on this, this is exact in d equal 2, and is exact for the larger or equal to 4. Well, for d equal 3, it's not exact, but it's a good approximation because the exact value is about 0, 59 instead of 0, 6, which is this number here. So one now, OK, first two things. We noticed that the critical dimension here is d equal 4, namely for d larger or equal to 4, the conformation of the polymer is Gaussian again, namely the extruded volume effect do not count too much, do not count, I mean they are not relevant, and in order to see that in a very simple way, one can just look back to this expression, to the two body interactions, right? And in order to see why these are irrelevant, one has to do in the following way, one has to insert here, in this formula, the expression for the unperturbed radius, OK? So, namely, I insert this here, and that goes like kB t, d0 n square divided by n to the d over 2, OK, b to the d, which is v0 b over d, n 2 minus d over 2, or v0 bd, n if 2 is minus d minus 4 divided by 2. So what you notice is the following, that if d is smaller than 4, this contribution goes to infinity, for n goes to infinity, for n going to infinity, right? So that means that two body contributions are important and are relevant, so they cannot be neglected, which is the kind of information contained here. While for d larger than 4, this thing goes to 0, which means that two body interaction can be neglected, OK? So very simple. For d equal to 4, you can imagine there are logarithmic correction as a tiny crossover, let's say. So one can say, OK, but here, in this expression, I mean, we, so one can say, OK, two body interactions are not important in d equal to 4, but what if I consider 3, 4, 5 body interactions? Because this, I have been neglected here, I just counted two body interaction. So those kind of contribution count for d larger than 4, and one can do, one can actually can do the same exercise. So approximately three body interaction, so if I write, yeah, I write it like 3, for instance, here, KVT. So that will be like V3, there will be some coefficient, right? Question? V3, and then there could be, I mean, the contribution to the three body interaction can be calculated exactly the same manner. So it will be like nRd to the power 3, so it's like rho cube, the density to the cube, times Rd. So it can be calculated exactly the same way. And this is V3 n cube divided by R2d. So now to, yeah. So sorry? Ah, just to distinguish by this, I say it's like three body interaction, if you want this is E2. Just a notation, just to distinguish, okay. Yes. Ah, yeah, more or less, yeah. This is basically, these coefficients are the virial, they fall from the virial expansion. Okay, so this is the second virial coefficient, which, okay, should be like V2, if you want. That's the third virial coefficient so on and so forth, proportional to. So if I insert here, now again R, let's say the ideal behavior, what I get is like something like that. V3 n3 R to the d, sorry, n to the d, right? And then this is equal to V3 n to the minus d minus 3. So as you can see, actually these contributions are even less relevant because they become relevant only below d equal, sorry, they become relevant only for d smaller than 3, okay. But they are still, so for d equal 4, actually they do not contribute at all, they contribute even less than those. And so on and so forth. So adding new terms to the virial expansion does not add anything. So if you want to have, sorry. Yeah, exactly, you can neglect it. It's just for a polymer in a good solvent, you can take only the two body interaction, that's it. Okay. If you want, it's the only fixed point. If you do some kind of renormalization group, it's the only fixed point. For d less than 4, for d larger than 4, it's the Gaussian fixed point. So everything is ideal. Of course, I mean, for this particular condition, what I mean is that you have this kind of behavior for a self-abiding polymer in, I mean, we were only to body interaction count. So now you can also play a little bit. So now that we have verified that three body terms do not count too much, actually you can play a bit. So with this term here. So in general, you can a bit generalize the, the flority, or in so, because it's, it becomes very accurate in can be even for other conditions. So it does not, it's not only accurate for the particular case of, let's say, of the self-abiding polymer. So for a polymer in good solvent condition, but this becomes accurate also when you have, okay, this will be probably more clear later, but it becomes also accurate when you have, when practically there is a partial screening of two body interaction. What I mean is the following. When you have a polymer in a, so I mean, this kind of model, I mean, this kind of, let's say, theoretical framework that I told you now, it holds only for a single polymer chain in a solvent, but the situation can be more complicated. For instance, in the case of, let's say, chromosomes, which is, let's say, the next topic of these lectures, you don't have only one single chain. We have more chains, I mean, more, more chromosomes, so it's a, so it's a many chains system, which means that the situation that describes the physics of chromosomes is not the one of a single polymer chain, but it's one of multiple polymer chains that are all together in some, let's say, restricted space, and they interact together, so it's more like the situation of a so-called polymer solution. Okay. So by polymer solution, I mean the following. And suppose that you start from, if you have a container of some volume, and you start adding polymers, and you start from very dilute condition. By very dilute condition, I mean the following, that you have your polymer chains here, polymer, no, let's change here, and these polymer chains, so the typical size of your polymer chains is, that we know, this is b to n nu, right? And what it happened, that suppose that you take an enormous volume in such a way that, let's say, that the chain are, I mean, the typical size of this chain is much smaller than the volume, which is, let's say, the typical volume available for one single chain.  So the physics that describe this single chain is like the one for a single polymer chain.  Then what it happened, that these polymers becomes, I mean, they start to become to overlap, actually, okay, like this. And no, no, no, it's, oh, maybe I did, I did a smaller, I should do a smaller volume. So I suppose that you take a smaller volume, you start to compress the system. Okay, okay, let's take a smaller. Then what it happens, no, in the chains are always there, then they start to get packed, right, in like this. And what it happens, eventually, they do not longer behave like single polymer chain, because they start to interact with each other, right? And then what it happened, that because of the kind of interaction between each other, and because in the end the polymers are soft object, they start to interpenetrate, okay? And so by interpenetration, eventually, it means that if, so I move from here, so this is called a dilute solution, dilute solution. Then when they start to interpenetrate, it's called a semi, well, yeah, it depends on the language, but in general it's called a semi dilute solution. And then eventually I go to maximum compaction, if you want, even smaller, when it, let's say, in the volume span by each chain, there are other chains. So everything is interpenetrating, everything is compact. And that's called concentrated solution. It depends a bit by the language, but concentrated solution.  So in the extreme case where all the solvent has been removed by the solution, and you have only polymers, you have a melt. So namely the only molecules that you have inside your system are molecules coming from the polymers, okay? So these particular conditions, oh, actually these conditions, and these conditions, is the one that is more, I mean, this one that you can, you have to use, and you will see it later, to describe, to start modeling chromosomes inside the nuclei of the cell. This is the kind of condition, because you have many chains inside the restricted amount of volume, which is the nucleus. And in order to model it, you have to use the physics of semi-dilute solution. And what has to do with fluorine? I mean, because fluorine can tell you something about how this chain would behave. And that's the following. So suppose that, so we use a very extreme situation. We use the situation where we are in a melt, yes, semi-dilute. Yes, we will see it a bit later.  I mean, why these conditions are interesting? Because they are described, I mean, you can imagine that a bit, but they are described by different, I mean, you will find different physics than one of dilute solution. It's more complicated. You can expect that, because you will have a lot of interaction because of the presence of many chains here, right? And actually, the condition where, the concentrated solution condition is a bit simpler to describe the semi-dilute one. In spite of the fact that you have, I mean, you can expect that you have much more interactions here. And the reason is the following. And that was the first realization of this thing was done by, actually by Flori again. And the fact that in concentrated solution, so when you have only the polymers, almost only polymers, you have a partial, if not complete screening of tuboid interaction within the single chain. And so what I mean is the following. It's like tuboid interactions. Count much less than the situation when the single chain is isolated. And this looks like a bit of a surprising result. But the explanation actually is quite intuitive. And again, it's due to Flori. And the reason why it's like this is the following. So in a concentrated solution, or let's say briefly in a melt, the concentration, so suppose that you take, so let's say in general, the concentration of your monomer, because everything is packed, so you have a real compact system. The overall concentration of your monomer, c, I call it c over r, is roughly constant. See, r is the position of, it's some position in your system. The concentration of your system is roughly constant. But you remember, the interaction of, to say, the tuboid interaction, is proportional to c r square, to the concentration square. But as I told you, this is roughly constant. So this is actually a constant square. So this is for tuboid interaction. So that means that the force between two monomers, which is just, let's say proportional to the derivative of u with respect to r, just if you want the gradient, is just zero, is negligible. Okay, so it's a bit of an intuitive argument. So that means that tuboid interactions do not count in a melt. Because of, for one single reason, because the volume occupied by one single chain is actually penetrated by the presence of the other chains. And the other chains remove the effect of the tuboid interaction within one single chain. So it's like they are effectively screened. This is a genuine, many-body interaction. So because of it, so that requires a bit of a proof, but I mean, you can somehow, you have to believe me now. So the flow return here does not go, so I call it e2, just to say that it's tuboid interaction, is no longer of this kind, but is v2 over, so it has a screening, so it has a one over n term, times n squared over times rd. And so this is effectively like equal to v2 n over rd. So that's, so, okay, this is for single chains, while this is for a melt. So as you can see, contribution to tuboid interaction in a melt is much more depressed, because it has a one over n term. And that comes from the fact that you have, basically a lot of interaction are suppressed because of the presence of the other chains inside the volume of one single chain. That actually, it's not immediate to derive, it comes, well, it comes from, so basically you have to study the system as a, it's called, as a mixture of different polymers, okay. And then, so it's like, so actually you start from a system in like of a and b particles, where things can, so you study the entropy of mixing of the system, and you, but you treat the system, so in the end you treat the system as, like any part is equal to the rest. And then that gives you a second virial coefficient, which is one over n, the single virial coefficient of one single monomer, let's say. And it's not immediate to derive, it's a, we can, maybe I can show it tomorrow. So for time being, let's take it this way. I wanted just to show you, actually I wanted more to point out the fact that you can still use the fluorite or even for, for this polymer solution that somehow they apply, they are more, this is the correct description, if you want to use it for chromosome. And, but it's not, no, it's not obvious, it's not like something, something trivial. And actually that, so because of it, actually, well, because of it, actually you can see immediately one thing. So if I put, so suppose that I, again, I put the impersturbed radius here, so I just do the same exercises before. So I have that r goes like n, so it's b, n over one alpha. So what, sorry, what I have here, I substitute this thing here, and I get that e2 over kbt goes like v2, n, b to the d, again, n d over 2, and that goes like v2 over bd, n one minus d over 2. Or n minus d minus 2 over 2. And that means, so it's set consistent, which is to some extent, that means that for d equal 3 now, this term is no longer important. So that means that, and that was a consequence of the results of Flory, that in d equal 3 now, and in a melt, polymer chains are ideal. So that means d equal 3 melt, r goes like b, n one over 2. And that's actually, which is the correct result. So in a melt, linear chains are ideal. OK, not a completely ideal, because there are, but they, the sides behaves like a Gaussian chain. So you have a different exponent than the, compared to the situation where the single chain is isolated. And that's the consequence of the fact that you have the screening of a student volume effect. So it's a, this is quite important. I would suggest that you, it's an important result in polymer phase, so you should, I mean, depends what you want to do in the future, but it's an important result to take into account. This thing, so what I did is, so this is, so maybe tomorrow I can provide you, I don't know, maybe tomorrow, but let's say, for timing, just believe me that this means that this, this, sorry, the two-body interaction are screened because of the presence of the many chains, OK? You know, this thing, right? So why it's v2 over n, and why v2 to square root of n, this is not obvious, this, I mean, requires a bit of more formalism, but, OK. So then, that means that the second, I mean, let's say, the contribution to the energy is v2n to r over d. So then I can repeat, because I want to know what is the, which is the critical dimension for this, let's say, new flow theory, OK? So I substitute the unperturbed radius here, I mean, as the same way I did before, and then I get this formula, and which means that for d equal, so this, this two-body interaction count whenever d is smaller than 2, yeah, is smaller than 2, for d larger than 2, in particular for d equal 3, the two-body interaction do not count for polymer in a melt, OK, which means that the chain, a single, a linear chain in a melt, so when you have many other chains within which interact with it, is a Gaussian polymer, OK. So as two-body interactions are, because two-body interactions are screened, and so the behavior of a single chain in dilute condition or in a melt is completely different, and that is something one has to take into account when, for instance, describes, I mean, one will see for the physics of chromosome, is a relevant condition, because we have many chains packed together basically. So that's why, and I wanted to tell it this thing now, because in a sense it's a good, it's timely because you can use, you can still use the flow theory. So you can see the flow theory is very powerful, actually predicts, even in spite of its simplicity, predicts a lot of things. Any question? OK, so I want to stay a bit more on the flow theory. So we go, so now, OK, for this, we go back to the case, because I mean, I want to also stress about criticisms on the, I want to tell you about criticism for the theory. I mean, where are, I mean, which, what are is the weaknesses of the theory. So we remain on single chains, because that's a good example, actually, where things do not work, I mean, which things do not work well. And that's the following. So we said that this case nu is equal to 3 over 2, right? So first of all, why the theory, which is important to stress, why the theory seems to work so well, because in the end we have neglected a lot. First of all, we have for sure made a very bad answer about the entropy, because as you can imagine, the entropy of a self-avoiding polymer is much less than the entropy of just a Gaussian polymer. No, you have actually, for a Gaussian polymer, you have much more configuration that you can achieve compared to a self-avoiding polymer, because basically the conformation of a self-avoiding polymer are done in this way. You take all the conform, possible conformation of a Gaussian polymer and you remove conformations, which overlap. But as you can imagine, the conformations which are overlapping are much more than the ones which are not overlapping. So this one, it's a serious hypothesis. So why things work well, and the explanation you typically find in the textbook is the following, that for possible reasons, you have either overestimated the entropy, which is what I'm saying now, but you have also overestimated the energy term. So you have some sort of two kind of overestimation that compensate each other by some magic somehow, by some series of fortunate events, and then you get an exponent, which is almost right. But nonetheless, the things do not work perfectly, because one assumption of the Florida theory is the fact that the end-to-end distribution function, Pr, is basically Gaussian, because this is the basis of the theory. So it's R squared and B squared. Yes, it's basically a Gaussian. That was basically the foundation of the theory. Actually, this thing is wrong. I mean, it's really wrong. So what I mean by wrong, I mean that suppose that you are simulating a self-avojding walk, and you can measure the end-to-end distribution function, you will never find this formula. So you can spot that the Gaussian approximation is completely wrong. And actually, so yes, your question. No, it's wrong also on a short scale. Yes, and now I was, why it's wrong? I mean, what is the correct expression? The question? Yeah, just interrupting. No? Yes, for a self-avojding poem. So this expression is wrong. It's the one that I've used, of course, Floria's used. Of course, it's not me that invented this thing, but it's wrong. And the correct one is, which I have to remember now by heart, so it's something p over r. Capital thing I just know. So let's say the correct one, p over r, is something that has to go like r to the theta times the exponential of minus k, so it is a bit r to b n to the nu, 1 over 1 minus nu. So I say, OK, that's a bit, looks a bit ugly, but that's the correct expression. And that's a function, it's called red and r. That provides a very good description of the end-to-end distribution function for a self-avojding poem. And why to come to your question? Well, this function, first of all, OK, first of all, this function is a sort of generalization of the Gaussian function, of the Gaussian function, for the following reason. So this nu is precisely the same exponent nu. So in particular, if you have theta equal zero and nu equal one-half, OK, you get exactly back the expression for the Gaussian, which is, OK, which is the correct, why I tell you now. While, so this is for a Gaussian, OK, I know theta is another exponent. Then I will tell you what is that. Because it tells us also why the flow rate, where is wrong the flow rate theory. While, for a Gaussian or idea of polymer, OK, let's say idea of polymer. While for a self-avojding polymer, we have theta larger than zero, and ooh, I mean larger than one-half, let's say, for a self-avojding, so, well, actually that should be written more precisely this way, be n to the nu. All right, I mean, the dependence is the same, but just to be annoyed, why it's important that factor, I will tell you now. So, to answer your question, why, you know, just to answer your question, why this, I mean, this is the correct format, even intuitively, because actually, whenever, actually this term, actually is depressing the fact that you can make the two ends close together, because that's impossible, actually, because we have extruded volume effects, even between the two ends, right? So then this exponent theta has to be larger than zero, OK, first. And second, your, so that term here, basically come from the fact that extruded volume effect in some sense overstreč the chain compared to the, let's say, to the case where you don't have those extruded volume effect. And so, that's why this function has this some sort of funny behavior. That was proposed for the first time by these two guys. Yeah. It's not, no, it's not totally empirical. So, sorry. So, this is actually for r equals zero that has to be so, because that means that you have, basically, the probability to close the chain has to be zero. For larger r, OK, of course this, you can neglect this term because it's just for a large r. Then for larger r, that comes from the fact that, as I said, so it's because you are so, let's say, you have the extruded volume effect tend to overstretch the chain. So, if you take this to contribution and to account, this is, actually, that this formula is not completely XR, but it provides a very, very good description of the data. In particular, you mean take this to go, to use that in flow? Or, what do you mean? Ah, you mean, yeah, exactly. So, what do you mean that I use this here? Or no? Or go back? Or you mean the reverse? Yeah.   That's because you are, actually, the reason for that is because you are correcting only this term, but you are not correcting this. So, this, that's remain uncorrect. Of course, you can correct this, but then you have to put some kind of effective correction because you cannot predict that priori. And then of course, you can do that, but then it's a fit. I mean, then you don't, you need something that you need to fix and you don't have it. But my point here, I mean, more than that, is that this formula. So, this is correct. I mean, if you have, for instance, a simulation of a safe avoiding walk, you can use this formula to, for instance, to fit to the data and find theta. Okay, no, you have, and you can find theta. Okay, you don't have no, but you can find also no from the day, from your simulation, of course. But then, what I mean, is that you can find either no and theta. And actually theta is, is very, it's quite different from zero. So, there was one question. You, yeah, sorry. I will just. K is a constant and actually, to find it, you fix it by imposing that this function has norm one. So, it's a constant. So, if you integrate this, you impose that this is normal one, this k is given in some terms of, in terms of some gamma function, but it's a constant. Yes. Well, the dimension enter in no and theta, because that, those depends on the dimension. But the shape is the, let's say, no, the shape also will change. Depends on dimension, because depends on theta in no. But the dimension is hidden here. Okay. And, yes. Well, that's the kind of explanation that is given in test book. To be honest, it's not never been completely clear to me, but it has to be something like that, because for sure, this thing, for sure, this has been overestimated, right? Because we have much less conformation for a, for a, for a self avoiding chain with respect to a Gaussian chain. And so then this has to be, was underestimated as well, in order to compensate for the right, for the, for the, for the reason why we get in a very good exponent. But in the end, that's the only exponent we get because we have no way by using the flow theory to get the theta exponent, which actually is quite important as well. Because that gives you the, let's say, that is correlated to the formation of contacts between the ends of a, of a self avoiding polymers. This, this theta. And, just to conclude this, actually theta is, it's, actually is related to the entropy of of the self avoiding work, in which way. So, so it's, it's connected in, in this way, actually is, so if you, so for a, in general for a polymer, this is quite, quite general for a polymer, the number, so let's say the, the partition function of a, of a polymer, typically goes like mu to the n, by some n to the, some exponent gamma minus one. Okay. So this is, the now, the total, let's say the total number of work for some polymer model. This is quite general. That are composed by n monomers. Okay. So that, that's how they grow. So there is a, let's say an exponential part and a power law part. Okay. So, this in general is quite understood if you do it on a lattice. So, suppose we go a bit step back. So suppose that we are analyzing random works. So, super simple. On a, on a cubic lattice. So for a random work on a cubic lattice, d dimension, random work in d dimension. So this function goes like 2d, 2d n. Okay. Or if, and there's no time here, if you want gamma is equal to zero. Okay. And why this is so, because anytime I'm here, I have 2d possible direction where I can go. So that's the, that's actually, this is exact is no, is not in an approximation. This is for the random work. So as you can see that, this formula works well for a random work. For a self avoiding work, always on d dimension. This is, I mean, it is as this kind of former. Okay. And this move here, it's a constant, which depends on, which depends on the lattice. It's not universal. It depends on the lattice you are using to model the system. So if you are on a cubic lattice, it's different that you are on anicom. The depends on the still exponential, but depends on the kind of lattice you are using. While this is actually the interesting dependence, because this exponent does not depend on the lattice you are using. It's universal. So you can use the more convenient lattice if you want. In order, it depends on your code. So that's it, seen quite well in for a self avoiding work. And now, this exponent here, actually gamma is related to theta, because theta is equal to gamma minus one divided by nu. So this exponent theta is actually connected to the entropy of your polymer, which is, if you want this bit obvious, because that's related to the contacts you can form in your chains. And if you want the contacts, it's the only thing that distinguish a self avoiding polymer from a random walk. Because the contacts are forbidden in, in a, let's say, yeah, forbidden in point contact. You cannot overlap two monomers by constructing a self avoiding polymer. So it's quite natural that theta is related to the entropy of your work. Okay. Gamma, as far as I know, so in a generic expression from gamma does not exist, but I can give you the value, for instance, in 3D. Well, here it looks like it's almost exact. So for instance, in 3D, just to show you that this thing is not negligible. So gamma, gamma, of course for a self avoiding walk in 3D, it's equal to, well, here the value is 1 dot 160. Okay. It's approximated like seven over six. So it's larger than one, of course. Which means that my exponent theta, it's one over six divided by, and I have to use nu in 3D, which I can use, and I can use the flow approximation. No, it's three over five. And so this is equal to five over 18, which is about, oh, I tried this slide. It's zero to seven five. So as you can see, it's not negligible at all.  Or are going to zero, sorry, this way. And then it increases. Okay. So it's an important effect. It cannot really be, cannot really be negative. Okay, yes. This one, think, but to be honest, now don't remember. And, but I don't think it's, it's entirely, but yeah, I think you should, yeah, let me see, I think you should start from here. To be honest, I have not really reviewed how to derive this, but it's, let me see, maybe it's here. I think you're probably, okay, so this, as I said, you derive it from, let's say, it's because self avoiding polymers are overstretched with respect to standard Gaussian polymer. And, but then this, so this exponent theta is basically related to the, yeah, when you, it's when you go to, to, to small distances, and I think, if you go to small distances, you can, so you can relate, so, so you can write the, ee, yeah, I think you, you relate the, ee, yeah, you write the number of conformations that have small distances, through these, and then I think it is the path, and that's path, and then you get the relation between these two kind of response. So you can see, they are not independent, but theta and nu are independent. So I mean, they are two, so once you have the two, you can derive gamma, or if you have gamma, you can find theta, but you have, these two exponents are, are not, are completely independent, so they are, they are both necessary, and in particular, the theta exponent, that's, that's why it's profile, the, the flow theory, it's, it's where the flow theory is profoundly wrong, it's wrong that, it can, it doesn't predict, it doesn't have a prediction for the exponent theta, or if you want the exponent theta, it's zero for the, for by construction, for the flow theory, and that's it's more severe defect, but okay, in spite of that, it works, apart from that, it works quite, quite well, and I want just to conclude with, with this, maybe then we do just a small break with the following, so suppose, because this is something that we have said yesterday, so suppose, suppose that I have, sorry, actually I, you remember yesterday, we had, so for a, for a Gaussian chain, so let me, yeah ideal polynomial, ideal polynomial, so you remember that I derived the, the looping probability, you know, the fact that we have two ends, and I want to know what is the probability that for two points on the, on the, on the chain, I mean at the ends of the chain, what is the probability that this thing close, and I told you that this, so the, I don't remember which notation I've used, yes, but the, the closing probability, p that r equal zero, for a Gaussian chain that goes like one over, n, b times d over two, this is in the dimension, right, so in the, as someone of you observe yesterday, this is just related to the return probability for a random walk, that's a, that's the day, the dependence for a, for a, for a random walk, yes, but how that changes for a step avoiding walk, so also as a polymer, so if you, then that's quite easy to derive now, because we have everything, it would just use this function, and then you have that p for r, well, here we cannot use really zero, because we know that for r equal to zero, that just, I mean basically the probability is zero by construction, because two more of this cannot overlap, but you can always say that I took a, let's say a microscopic distance, let's say the size of, roughly the size of my monomers in order to derive my closure probability, and that will go like nb, it will be like nu d plus theta, okay, if I use, and you can see why, because nu theta comes from here, right, while the terms d come simply from the normalization of my or my function, and if you want, this is a perfect general, you can see immediately, this is a generalization of this, because if I use here, theta equal zero, and nu equal one half, which is the prediction of a foregatian polymer, I just recover this formula. So this is quite good that you take into your mind, because this property, I mean the contact, probably some people can actually measure now in chromosome, so now we will, I mean when I will tell you a bit more on chromosome, we can, we will actually compare these two functions to real biological data, and we will, there are some conclusions about it, if they are adequate, no, not adequate, I mean we will see. Actually, if you put numbers here, you, for instance, in d equal three, the solar we can just compare, so what you get here, so suppose that I have d equal three, so that goes like one over nB, one over five, sorry, one, yeah, three over two, which is one dot five, while if you put d equal three here, let's put numbers, so I have nB, so I have, for nu is three over five, d is equal three again, and theta, I said it's forgot, is five over 18, horrible. Five over 18, so now I have two bit, so it's one nB, so it's three over five, plus five divided by 18, so I simplify this, so this is one over nB, so it's 59 divided by 30, so it's about one, so that's about, so it's 59, it's not 60, but it's very close, so it's one over nB to the square, roughly, I mean it's a bit less, so as you can see, and as you can expect, this law here is much steeper, not much, it's steeper than this one, which is just nB one dot five, so this property will be very useful when we analyze, I mean we discuss experiments on chromosome, because we will compare to those laws, because it's exactly what people can measure in chromosome, this is a close, let's say contact probability, and so that's very useful to compare to this kind of theoretical prediction, okay, what time is it now? Do you want to do s-break? Yeah. Half past three. Can we make five minutes break? Do I take some water, too? Theta, yes. Yes, so in fact, maybe I can, sorry. Ah, it's just a constant. It depends on the, actually it's not universal, it depends on the lattice you have. So, no, that describes, we can talk later on, or tomorrow, I mean, I'm here, so. So, yeah, I think it's useful if I tell you this thing that I have in mind, but I need a bit of more time, so I will be more specific tomorrow, because otherwise I will be, I have to rush a bit, sorry. So this is something, actually I have not told you when we mentioned, so we go a little step back to the, let's say, to the ideal polymers, because this is also, somehow it's useful.  So, either if it is a Gaussian chain or a self-avoiding walk, and in general, even a polymer in solution, it's, actually these properties are for a polymer in bulk. Even when I did the picture, when I took a polymer switch, I started to compress, the system is always meant to be, so I just repeated the drawing a bit larger. Even in this case, so for a met, I mean, the typical side, so this is the container, but the typical sides of my polymer chain, I said that, I said that R goes like B and one-half for a polymer, for a linear chain in a melt. So, but still, this quantity is much smaller than, if I call it L-box, than the typical size of, let's say, of my container. So, if you want a confinement effect can be neglected. So, it's real, like, it's like in a bulk. So, why I said that? I said that because, actually, this condition is not obvious, that is relevant, for instance, for, again, for chromosome, because we know that we have a nucleus, okay, and we have our polymers here, I mean, our chromosome, which are more or less swollen. But now, this is in real, I mean, we need to wonder, if the confinement is important, namely, if the equilibrium size of the chain is much smaller, comparable, or, so, to say, so, if the chain, if the size of the chain, when it's free, is much smaller, comparable, or much larger than the nucleus, than the size of the nucleus. So, just to give you an example of a, I mean, of the biological systems we are going to discuss. And that's very important, of course, because in the first case, I mean, if the typical size of my, let's say, of my chromosome, when it's free, is much smaller than the size of the nucleus, then, I mean, this is like a real solution, if you want. So, the nucleus does not impact, at least does not impact too much on the size of the polymer. If it's comparable, starts to have some impact, and if the free size, so, the size of the chromosome, let's say, when it's free, I mean, without the confinement, much larger than the size, when it, I mean, is much larger than the size of the nucleus, then, I have to expect that the confinement effects are important, OK. So, we are going to discuss a little bit, what it happens to polymers, so, I go back again to, to generic features of polymers, when they are in confinement, OK. It's different from the melt. Well, the melt, yeah, it's a good question, thanks. But, of course, the situation is not completely similar. OK, I will, sorry, who asked that, because I was, ah, yeah. So, I will be more specific to tomorrow, because today I don't have the time. But the situation is the following. So, in a melt, rigorously, I have, as I said, I have still many chains. And as I said, ah, so, you expect, because of the Flory theorem, that the, let's say, the size of my chain at equilibrium has to grow, as I said, as a Gaussian polymer. But then, so, this, I mean, if you calculate, if you put here the typical side of a monomer, times the number of my monomers, then I get a number and then I have to compare to the size of the container. If, this can be much smaller than the size of the container. I have many chains, right? So then, in this case, the confinement is practically negligible. Maybe I have some confinement effects at the border of my container, but in the end, but inside, that doesn't matter, no? Any chain does not feel the presence of the confinement, right? While, I mean, this is not always the case. So, yeah? I guess. But confinement, so, they are both confined, but in confinement, I mean that, the, let's say, the size of the polymer when it's free, let's say, so, in free space, it's much larger, or larger, or, let's say, of the size of the box. And in that case, the things change a lot, actually. And as I said, I mean, tomorrow will be on specific, but one thing that you can immediately see, so, actually, there are some equations that are exact, and tomorrow, I will tell you more about that. But let's take the most simple system, which is, again, the Gaussian chain, right? Because we know everything, and it's, there are no extruded volume effect, and it's simple. So, for the Gaussian chain, we know, no? What is R? As to go, like, B and to the one-half. OK. And then, suppos, I want, I confine this thing to a sphere of some diameter, so I have, and I have my chain. Then, this thing has to be smaller, or, if you want, in order to get, to stay hamperturb from the condition to be free, has to be smaller or larger. Of, let's say, of the diameter of the sphere. So, then, it's a bit of a matter of definition, because I'm neglecting, let's say, I'm neglecting pre-factors. But if I call this diameter D, then this thing has to be, in order to be hamperturb, to remain hamperturb, has to be smaller, let's say, of the diameter of the sphere. Which means that polymers, remain hamperturb, if n is smaller, or comparable to d to the b square. Right? So, this is the condition of the polymer. So, polymers, so there is, let's say, crossover length for the polymers. And for this length, if I took a polymer, which is much smaller than this length, so I can call it, and see, just to make like the crossover.  But if they are larger, of course they, I mean, it's a serious ingredient, it's a serious actor of the game, right? And that's very simple, if you want to describe, I mean at least for a Gaussian chain, how things have to go. Well, one can be more rigorous, but in the end it's, I mean, what you should expect is very simple. So, if you have, so actually it's very useful, if you, so you have your, okay, let's do it, yeah, let's do it here. So we have our sphere. I just take a sphere because it's similar, but of course, I mean, you can take a cube, I mean, that doesn't change too much. So I have my chain here, and then, and then goes like this. And so the physical, I mean, the way you can describe the system is the following. So then, now it's important, so you take any, if you take pairs of loci, oh loci loci, pairs of monomer, sorry, on the chain, then you should expect the following. If you calculate the spatial distance between them, and I call it this, think n, no, this is the, sorry. This is a single polymer folded, is a Gaussian polymer, so it's an ideal polymer, folded inside the sphere. So there are no extra polymer, so I can put any, I mean, any length can stay there, because there are no, so it's a bit unrealistic situation. But okay, for time being, let's do this, because it's, that can be done, I'll show you tomorrow, it can be done exactly. I mean, it can be done everything, it can be done exactly. Or for time being, just let me, let me just be qualitative. So if I, so I define, so I took two pairs of monomers on my chain, and then I calculate the distance between these two monomers, the average distance, let's say, between these two monomers as a function of n, and of course, I mean, you can see what would be the result. I mean, that has to be like n, b, n to the one-half, okay. When, then I can use this result, when n is smaller or equal to d over b square, right? And then, of course it's simple, what can do this polymer, can only have a distance, which is the order, of the diameter of my container, right? And it should be like this. Sorry? Yeah, it has to be of this order. Actually, you can derive the precise relation, but it's the same order. It's the maximum length that it can get, no? Because it has to explore, and on average, has to stay on this order of magnitude. No, R is the average distance between two pairs of monomers on the chain, and I'm asking how this quantity grows in terms of the linear distances on the polymer, right? So for a free polymer, this is always like this, because it's a Gaussian polymer, but here is no longer, the polymer is no longer free, because it's in a container, and then this thing, because of this relationship, has to be like a Gaussian relationship up to this contour length, and then has to be order of D for larger length, because simply the polymer is confined. So any, let's say, it's like having any two random monomers, any two random points, if you want inside the sphere, and then the average distance between them has to be over the order of diameter. That's it. But so then, this is quite important, because you can see that there is a, actually a crossover length that tells you when confinement effect becomes relevant. Okay. And tomorrow, because again, I need to be, I mean, I want to be a bit more specific on this. You can derive the, and actually for Gaul, only for ideal polymers, you can derive an exact formalism. Actually, I think it's quite useful. We'll tell you about tomorrow, and we will also discuss a little bit about confined polymer, because even by using extruding volume effects, because they are quite useful, and also people can do experiment on that, apart from the fact that, of course, they are related to chromosome, it's not the, I mean, they have more general application than just chromosome. And, but we can do it tomorrow, because I need a bit more time. So if you have questions, maybe I can ask where we are. Yeah. This is a decolidation assumption, right? Yes. So overall thing doesn't matter the smaller things. Yes. Is it because of the Gaussian nature? It's because of the Gaussian nature, yes. So because when you go here, right, basically it's like having, it's reflecting, it's like having a mirror. And then, because this direction is the correlated. Actually, things become more complicated. I didn't talk about it and yeah, I'll see if I have the time, but things become more complicated. If you, so this is a, so Gaussian polymer, or, I mean, this, or, also Frigendi chain, whatever, they are example of flexible polymers, right? But suppose I have a, what is called a warm like chain, which is a semi flexible polymer, then I have another, yet another lamb scale, which is the persistence lamp, or the cool lamp, whatever. And then, this is, this also, also play a role, because basically the semi flexible polymer is characterized by a lamb scale. It's like, if you want, it's like a, yeah, it's like a rigid rod, it's like a spaghetti. So below some scale, my polymer is very stiff, above some scale becomes to be, because of thermal fluctuation becomes more and more flexible. And of course, you can imagine, if you have this such kind of polymer, and my confinement, I mean, D is comparable, or smaller, than these persistence length. And then the conformation of the object is completely different. It's no longer described by this. But of course, I mean, it becomes more complicated. But since it's, but I don't know if I have the time to do that. I will probably not, since it's not really relevant for, for the, let's say, for the chromosome, for the biological part that I have to tell you, I mean, in the last lectures, and we will not talk about that. But it's relevant if you have, for instance, DNA in very strong confinement. So that will be important. But I mean, it will not be important for what I'm going to tell you in the rest room. We will not talk about that probably. Yes. No, it's much smaller. Yeah. For what? For DNA, LP, DNA, it's equal to 150 base pair, which means it's about 50 nanometer. Yes. 50 nanometer? No, sorry, the opposite. Is, no, no, it's right. It's 50 nanometer, which is 150 base pair. You know, it's correct. Yes. No, no, it's correct. It's correct. And this is for double-stranded DNA in, let's say, in normal conditions, in normal salt. Because the persistence length change, change, I think not twice, but change can change considerably according to the to the salt conditions. Because DNA is a charge. But those, the experiments where people have measured this were in, it's a physiological circuit with, I think they use the magnesium ions, Mg2 plus. And that should be more or less, I want to say something, not correct, but it's, I mean, like saying, like saying physiological condition. And in this condition, they found these values for, this is the accepted value for DNA. And that's of course, it's not small. It's, you can see it's 150 base pair. That means that you really, I mean, it depends by what are you doing. But since you can prepare, actually, for instance, the people can study DNA on a slab, or in channels, or nanopores, whatever. And in that case, and they are very nanopores, so they are very small. They are much smaller than these, like the linear sides. They are much smaller than these. So then, this physics, I mean, this physics does not apply there. But for chromosomes, this is not relevant because, I mean, even for bacteria, bacteria have, the DNA of bacteria is one mega base pair, which is, this is, has to be, it's basically DNA. So, which means that it's about 1,000, not even more, no? Is, no, cimandamila is 5,000. Larger than these. And so it's doesn't, doesn't matter. I mean, the confinement is not tight. So, while it matters, for instance, for viruses. So DNA, viruses with DNA, obviously there are also viruses with RNA, but viruses for RNA, sorry, with the DNA, the viral capsid is actually is comparable, or smaller than the 15 nanometer. So, the DNA is basically make some kind of order inside the viral capsid. Then for viruses we have to take into account confinement and the semi flexibility is not simple, but we are not going to talking about vile. So, which is a totally fascinating subject. It's not because I don't like it. It's, it's, I mean, it's a different physics. Okay. For, I mean, for chromosome doesn't matter. I mean, it's not really relevant. I mean, so we can use this physics. Okay. So it will continue tomorrow. If you have a question, just so you can just ask.