Thank you very much for the introduction. So I will talk about, so one of my favorite topics which is character sums and especially real character sums, so sums of plus minus one defined by characters. So I will talk about two results, which are like in two papers with Olexic Lermann from Wisterl and Jonas Lamzuri from Nancy. Okay, so let's start. So the object we will look at is we take a character, modulo q, we can take q prime, and we look at partial sums of characters. So the classical example, you take the Legendre symbol, so you take a prime and you have the application which takes, which is plus one if n is a square modulo p and minus one if n is not a square modulo p and you can kind of study other edges of these quantities up to some large n and see if you have some cancellation in this sum. So obviously this sum is bounded by capital N and we expect some randomness for the values chi of n. So we expect that they distribute randomly. So for instance, if you take n large enough, you expect half of a time plus one, half of a time minus one. So if you want to show this, you want to show that this sum is a liter of n for sufficiently large n and the important thing is, so you go up to p and how large n should be to ensure that you have some randomness in this character values. So the first classical result in number theory on this is that these sums are bounded by square root of p log p and it's a bound of poly-avionic order. It's a simple FOIA argument, which shows that essentially if your sum, you take more than square root p terms, you have like half of the time plus one, half of the time minus one. So this was the first result in this field and then the people tried to show that this sums cancels for shorter intervals. So indeed they could push it a bit and Burgess showed that you don't need intervals like larger than square root p, but you can take something a bit bigger than p to the quarter plus epsilon. And up to now it's kind of the best results apart from like some small improvements. We cannot show better consolation in shorter intervals without any hypothesis so unconditionally. So this is related to what I say, the program of the least quadratic non-residue. So you take primes p and you want to take a look at the first time you hit minus one. So maybe you will have like all your first numbers such that the characters, like the legendary symbol is equal to one. And then you look at the first time you hit minus one. So if you look, you use the barn, I told you that you have cancellation for intervals like larger than p to the quarter. You can use some trick by Vinogradov and kind of push it down a bit by some factor square root of e. And so that you are sure that the least quadratic non-residue verifies this bound. So actually the trick is just that if you're like plus one on the first, say k numbers, k primes, but you will also be plus one on smooth numbers which have this prime factors. So you can push the argument a bit and show that this bound on the least quadratic non-residue. So then there was this famous conjecture of Vinogradov which says that in fact, we expect this least quadratic non-residue and we expect cancellation on much shorter intervals. So we expect that if we take the modulus to any small power, there should be cancellation in the quadratic sum and we should be sure to find quadratic non-residues very small compared to the modulus. So this is like really open up to now. But we can, if we assume the Riemann hypothesis for the directly held function, Ankeny proved that actually we can even have a very strong bound on the least quadratic non-residue which is logarithmic bound and it's used like in some algorithm that we have this very strong bound. But unconditionally we have like some like kind of really large power one quarter and unconditionally we have a logarithmic bound. So we will look at some related problems on this but not directly on, we will not study the least quadratic non-residue but we will look at sign changes of character subs. So of course if you look at the character sum, so you take quadratic character not you don't take only the legend or symbol, you can take any discriminant D and you look at the real product quadratic character associated to D and you form the partial sum. So then you look at the number so you look when this sum changes sign. So of course if you change signs on the average means that you hit many minus one values because like if you can draw this like you can draw this as a random work.   And you wanna know if you take all the first say X values or this sum, so you sum up to one, up to two, up to three, et cetera.  So this goes back to actually this goes back to very old question of Shahola and Fakete which asked like in general for discriminants can we show that we have like an infinite number of sign changes when the discriminant goes to infinity for instance. Can we show that for some these? Can we show that for all these? Can we show that for almost all these? Can we find like some discriminant such that it never changes sign? So for instance that the random work stays up the X axis. So it never has a value of n such that this sum is equal to zero. So many questions which are which are kind of open on this problem. And if we show that they are like for instance infinitely many sign changes can we quantify the number of these sign changes? So can we say if I have a discriminant D and I look at the first D sums that maybe I will have a D to the quarter sign changes. And can I say something maybe stronger which is related to what I said? Can I localize where these sign changes happen? So can I have like sign changes at the beginning of the interval? So for instance in the Vino grad of range if I take D to the epsilon I look at the first partial sums am I sure that here I will change sign or some discriminant Ds? So it's really related to the distribution of quadratic residues and non residues modulo sum prime or modulo sum, some discriminant D. So before like stating any results on this problem like that we obtained I will like talk about some related problem that we can think about the probabilistic model for this question. So a probabilistic model for this question comes from this beginning of 20th century. Vintner introduced a famous model to model the Mobius function. So he wanted to have like a good random model for the Mobius function. So he introduced what is called the Rademacher random variables. So how does this work? So you take a sequence of random variable which are like indexed by primes and you just ask that the probability to be one is the same as the probability to be minus one. So it's like one half on the primes. But then we are interested like as a character as a multiplicative function the Mobius function is a multiplicative function. So it's not a completely independent situation it's just independent on the primes. And then we extend our function on all the integers and we define what is called the Rademacher random multiplicative function which is just extending the values as any n by multiplicativity. So for all square free integers n we define fn as a product of fp and we put zero on non-square free integers. So clearly f is giving me a random multiplicative function because f of mn is clearly f of m times f of n because I defined it as a product of fp on the prime. So this is a way to define what is called a random multiplicative function and we can ask when we have a questions on characters or Mobius function we can translate it in this random size and a random side and see if we can obtain some results for this model which should be simpler than in the deterministic side. So what is the questions we can ask?  We look at averages and we count the number of sign changes of this averages of random multiplicative function from n up to x. So these problems like get so recently there were a lot of works on random multiplicative functions especially Harper had many good results on random multiplicative function but on the specific topic of sign changes so there was this result of m on a hip and Zao which showed that almost surely your average of random multiplicative function will have infinite number of sign changes. So we did it by studying the Dirichlet series associated to this random multiplicative function and then this year like few months ago guys and Yari made this argument quantitative and showed that the number of sign changes is almost surely a triple log. So log index three is like log X to some power one over C for C strictly larger than two. So the best we can do in the random size is like triple log number of sign changes almost surely. So then we can ask can we do that for characters in the deterministic size. So let me mention all the results that are related to questions I said just before. So there were a result by Angelouxu, Kalminin, Sanda Rajan which announced it in his ICM talk but I haven't seen the paper which study the probability that there is no sign changes at all. So in both cases, you can look at this question some and ask, can it stay always positive? Can it stay above the axis? So and you can see that this you can, they have some words where you can bound the probability that this happened. So you can see that if it happened, it's a really rare event. So it should happen for very few characters for very few legendary symbols for very few random multiplicative function, et cetera. So I don't state like precisely what they prove with a bunch of results on this questions. There was a question, no, it was just, I heard something. Okay, so back to our character sum. So what we proved is a following thing that for almost all fundamental discriminant. So for almost all D, so if you pick randomly your discriminant D and you look at the partial sum you will have more than log log D divided by a fourth iterated logarithm. So something close to log log D, sign changes in a very kind of specified intervals, which is something like E to sum power of log D which is very small. So we can locate if you, so some remark here. So this is like in the literature, like Baker and Montgomery were able to produce a bounded number of sign changes in the full range. So when you look at all the sum up to D. So here we have a quantitative result of the number of sign changes, but maybe like the most important thing is that these sign changes are located for very small capital N for very small intervals. For instance, like this, these sign changes occurred in intervals which are shorter than the one predicted by Vinogradov's conjecture. So these are, these like, these E to the log D to the alpha are smaller than D to the epsilon. And our method like allows us, so we can like of course like I made it but you can forget like we made an explicit estimate on the number of exceptional discriminants which does not satisfy this. And it's rather flexible and we can like do the same thing for random multiplicative function or we could look at partial sums of for the coefficient of modular forms, et cetera and use the same kind of strategy that I will explain later. So for instance, if we go back to random multiplicative function, we can prove the following. So it's exactly the same analog of what we prove in this deterministic side. So basically you will in the proof it works effectively the same instead of averaging over these you take expectations, et cetera. And you can prove that for large x the number of sign changes of your average of random multiplicative function is larger than essentially log log x with high probability. You see like with priority essentially one. So for almost all, almost surely a random multiplicative function will have at least this number of sign changes. So the thing you can ask now is is it the right order or can we do what can we expect? So here like the thing if you look at some situation which is much more simple where everything is independent. So you have a range over, you have a range independent Bernoulli variables. So plus minus one, not only on prime but everywhere. So here's the well-studied objects. So it's a classic random work and you can see that for this classic random work we expected number of sign changes when n goes from one to x is of order square root of x. So it's much bigger than log log x. So an open problem if you have any idea what we should formulate as a plausible conjecture. What is should be something like log x should be something like square root or in both case character sums and random multiplicative function. For instance, I haven't written down but you can prove if you have a range like Mobius function that you will have like something if you go all the sums up to x you will have like more than log x sign changes. So you have like different behavior and so this is like, I haven't seen stated anywhere like any conjecture on the expected number of sign changes in this multiplicative kind of random work. So let's talk about some related objects where we can connect this to the problem I've just talked about and I will need it later in the second part of the talk. So our method allows us to treat zeros of fakety polynomials. So what are these fakety polynomials? But you still take a fundamental discriminant and you form the polynomials where the coefficients are the values of a Dirichlet character. So these polynomials were introduced by fakety to study the associated Dirichlet L function. So in particular, you observe that if this polynomial has no zeros, then the Dirichlet L function has no real zeros. So in particular has no zero and this would be fantastic results. But if formulated the hypothesis that for large D this polynomial should have no zeros between zero and one. This was a quickly disproved, quickly seven years. Relatively quickly bipolar, which showed that in fact for positive proportion of discriminant, there is at least one zero of this polynomial. And so then let me tell you about another upon problem which is related to the one I told you about the random walk which never crossed the X axis. There is a problem which was formulated by Chawla and fakety and then by Sarnac again and one in some later. Can we construct infinitely many primes such that this polynomial Fp or infinitely many Ds has no zeros on the interval? Is it possible that this polynomial never crossed the X axis? So with all methods, we can actually lower bound the number of real zeros of this polynomial and we can localize it. So it's the analog of what we do for character sums works for real zeros of fakety polynomial. So in particular, Baker and Montgomery conjectured that the number of real zeros should be of the order log log D for almost all discriminants. So there should be log log D real zeros. And we can prove that for almost all discriminant that we have at least log log D divided by quadruple log of D. And the interesting thing is that we know where these zeros are. So we know how close they are on the, how close to one these zeros are. So we know how to locate in specified intervals of these real zeros. So like you see, if you recognize this quantity like E to the log D to the alpha, it's really you see that the connection with the character sums with where we were, we had like the similar expression to locate the sign changes of character sums. So indeed, like the two proofs goes more or less in the same spirit and I will present the ideas in the character sum case and then just tell you that we can do the same for these zeros. So how do you detect sign changes? So you start from this identity on the top here that if you take your Dirichlet L function you can apply partial summation and write that your Dirichlet L function is just an integral of your partial sums divided by some function, some power function. So now, well, so it's related to study of course then this S, K, D and study the L function. So what do you do if you do, you take derivative both sides with respect to S, you obtain some identity which will involve the first derivative of L and on the right hand side, you will have another function of U and the partial sums of Dirichlet characters. So then if you apply some change of variables, I can do it, believe me, and you arrive on the blue identity at the bottom which shows you more or less that on the right hand side, you have some Laplace transform of your partial sums of Dirichlet characters. And on the left hand side, you have something like L prime over L which is like a nice object of study in I think number three and you have a factor L, S, K, D plus something one over S which will not bother us much. So then the idea is that if the left hand side changes signs a lot, it will force the right hand side, the integrand on the right hand side to have some sign changes. So this follows from a general results in analysis which tells you that, which is the following. If you have a function defined on R which has a Laplace transform, so you can define this Laplace transform like integral of G S minus Sx which converge for S strictly larger than zero. Then the number of sign changes of the integrands of G is bigger than the number of sign changes of the Laplace transform. So in our case, what we have inside the integrands are our partial sums of characters. What we have on the left hand side is something related to the L function. So if you show that the left hand side which is this L prime over L has a lot of sign changes, then you will have the same thing for the partial sums of characters. So that's actually how it goes but then we have something a bit more involved but because we localized the sign changes so we need something stronger, we need to control these integrals on restricted intervals. So how do we show that the left hand side has many variations of sign? So we need to study what is on the left which is more or less, if you forgot about the other terms is more or less L prime over L for some S applied for some Ds. And we want to understand this on average of other discriminants. So the idea will be to take points like very close to one half and see that when we take some sequence of points there will be many sign changes on the sequence of points L prime over L, S. So the way to do that is to model this L prime over L by some random object so by like, no I skipped a bit of things but L prime over L, you can write it as a polynomial. If you expand it, you can write it as a polynomial over primes and if you do use some like zero density theorems you can show that for almost all D it would be some short Dirichlet polynomial supported on primes which are around e to the gx if S is one half plus one over gx. So it turns out that this polynomial you can show that they are well approximated by Gaussian random variables with mean zero and some relatively large variance, one over two S minus one. So when S is very close to one half the things to have in mind is that this one over two S minus one is very large. So when you approach the central point the variance like blows. So this is good because we have a Gaussian random variables with large variance. So if you have a lot of Gaussian random variables which are all independent with large variance you will be sure that they will take, they will have a lot of sign changes with high probability. So what we will do, we will take a sequence of points S one, S R with some like function of all points like a tending to one half. In the way that the supports of the polynomial which approximate this L prime over L as disjoint. So different primes are involved in all the approximation. So because different primes are involved in the approximation it means on the random size that we have like independent random variables. And then if you have independent random variables which are Gaussian with large variance you can deduce that you have many sign changes. So what do you do indeed? You take many points and you form the vector of approximation and you on the deterministic side and you look at on the random side and on the associated random vector. And with high probability by some probability results you are sure that these random vectors will have a positive proportion of sign changes which is very important. They change signs, they change signs so we take positive and negative values but positive large values and positive negative large values. So when you go back to the deterministic side it tells you that the approximation will have the same behavior. We have a lot of sign changes. So to do like to compare like to the random model we need some results on the discrepancy and use the method of Lamzuri, Leicester and Ratzivil. And from the approximation you go back easily to L prime over L and you see that you have quantitative sign changes for L prime over L. So then when you go back and use the results I told you about the sign changes over Laplace transform you have a lot of sign changes for the character sums. So let's summarize. I had this identity. We selected points such that L prime over L is very large and takes plus and minus values. So on the right hand side it should also has a lot of sign changes. So we escape because every this t factor is just positive so it's not a problem. And we just need to have this L SKD which does not kill the sign changes on the left. So if L SKD would be too small it will prevent us to deduce this result. So we need to prove a large deviation results for log of L SKD. In some sense we should prove that for almost all D log L SKD is not too small. So for good discriminants L SKD is not too small L prime over L goes plus minus at different points. So the same thing occurs for the partial sum on the right hand side. So let me briefly tell you how we locate the zeros which is like actually the most technical part is to locate one of the difficulties to locate the zeros on the character sum side. So to do that you need to know that when you integrate from zero to infinity the main bulk, so the main part of the size of this integral is from two parameters very localized. So you can show that you can forget what happens close to infinity, forget what happens close to zero. And the main integral is dominated by what happens between Y and Z. So if you have that it will tell you that indeed like this SKD which changes size between Y and Z. So here like to deal with that we use like some like classical techniques on large even equalities and character sums. So I will not deal with that because I wanna talk, I see the time is running and I wanna plan to talk about something but second problem which is related to directly characters but exponential sums on the unit circuit. So let me move to something completely different which is like the measure of polynomial. So here like we seems like we don't take anymore about character but we will go back to character in few minutes. So if you can forget about the first part if you were sleeping and like wake up and like look at that. So if we take a polynomial P we can define like an object which is called the measure which is just the integral of the log on the unit circle of your polynomial. So essentially this is the limit when Q tends to zero of the Q of moments of P when you look at the moments on the unit circle. So if you want this is related to this is in some simple way related to the zeros of your polynomial. So if your polynomial factorize in this way so yeah there's roots alpha i here but then the manner measure is just the product of the roots which are outside the unit circle. So it's kind of captures the information about the modulus of the roots of any polynomial. So it's when you have a polynomial where your first thing you're interested in is you to locate the roots. So this in some way in some condensed way like captures this information. So you can say I can we compute this manner measure for some interesting polynomials. What can we say about that? So in the last century little would study the specific important class of polynomials which are the one where all the coefficients are either one or minus one. So you can form all the, it gives you a bunch of polynomials with your degree n you have two to the n plus one such polynomials and they have been studied like in all the aspects like the norms, the manner measure if you want the maximal size of the unit circle you want to study the number of real roots it's also related to some combinatorics problems et cetera. So one thing which is easy is the second moment the second moment you can just use Parseval's formula and show that it's about square root of n if you have a polynomial of degree n. Then there are like, so many open questions on this kind of polynomials. For instance recently famous conjecture was solved few years ago about the possibility of constructing a polynomial which plus minus one which is flat meaning that it stays more or less of size constants times square root of n when you look on the unit circle. So it's something which like does not very much when you look at the values of the unit circle. So this was called the little root flatness conjecture and like it took some years to solve this conjecture. So it was just like as a historical note and we will focus on not on this problem but on the norms and the malar measure for specific families of polynomial. So in general, the malar measure is bounded easily by the second norm. So we know it's smaller than square root of n. And one question which was asked is does there exist a positive constant such that for every polynomial p of degree n in fact the malar measure is smaller than one minus epsilon square root of n. So can we like approach the upper bound constructing polynomial with plus minus one coefficient? So the largest known value is given by this polynomial of degree 12 which is gives you malar measure which is relatively close to one times square root of n which is just 0.98. So nothing tells you that it's the best you can do there's no proof of that. So probably this is still not solved but people like look at this external problem and started to study families of interesting polynomials. And for instance, they looked at all the family of all plus minus one polynomial and looked at what can we say on average about this malar measure. So on average about all plus minus one polynomials they showed that more or less you can compute this average and you have this ninth constant e minus gamma over two which tells you that on average a malar measure of little root polynomials is square root of n times e minus gamma over two which is a constant which is not so close to one but it's strictly larger than half. So cause the average is like larger than half then they ask can we construct actually explicit sequence of polynomials which have a large malar measure. So Shoy and Erdely did that and constructed some sequence of polynomials with malar measure larger than square root of n over two. So let's now see what can we do with polynomials which are given by directly character coefficients. So and through this again, the fakety polynomial so if you take the coefficient given by a legend of the symbol can I understand the moments can I understand the malar measure? Can I relate it to some more general problems et cetera? So but this what is what was proved on this fakety polynomial is the following results. It was proved by Montgomery that the maximum on the unit circle of this polynomial is more or less understood in sense that is of size smaller than square root p log p and bigger than square root p log log p. And it's still unknown if the right order of magnitude of this sub norm of the fakety polynomial. So at the beginning like people looked at this fakety polynomial or some related fakety like polynomials because it was like on many points close to square root of p. So it was an idea to have like some kind of flat polynomials for with coefficient plus minus one. So what was proved about the malar measures of this polynomial? So little would prove the thing that for this specific family it could prove like this malar question, it could answer it and show that there is a constant epsilon such that this malar measure are smaller than one minus epsilon square root of p. On the other side, like Erdely and Lubinsky showed that it's like almost larger than square root of p over two for large primes. And then a series of improvements led to the following thing that we can have the malar measure is strictly larger than square root of p over two. So it's somewhere between one minus epsilon square root of p and one half plus something small square root of p. So this is like what's the state of the arts is like we have like, we know that the constant is not too small and not too large. And in what about the asymptotic behavior? Can I compute it and like show for instance that it gives me very large malar measure? So it's giving me useful polynomials for the question of malar and et cetera. So this, like if I read like I quote the paper survey of Erdely, which says that the problem of finding an asymptotic for the malar measure seems to be well beyond at this moment. So there were no conjecture formulated for this malar measure. And we solved this conjecture, I mean this conjecture which was not existing. We proved that actually the malar measure is asymptotic. So for the important thing is like for fixed p it's not an average over p, it's for fixed p the malar measure is given by some explicit constant time square root of p and this explicit constant is not the one when you average all the plus minus one polynomial. It looks very close to it, but it's not the same. It's not a minus gamma over two. So it's like a new phenomenon which like occurs for this family of polynomial. So how much time I have? I think I would skip the proof but just say something. So what is specific about this? What is important? Is that this? Think of the polynomial when you apply and when you evaluate at the roots of unity you have what is called a Gauss sum. So you know it's of modulus square root of p. So it's pretty regular on the roots of unity that's why you could think like a while you could expect for instance that the maximum modulus on the unit circle is not too large. And the thing is like the value of the roots of unity is something like square root of p times some legend or symbol. So you have some kind of randomness on the values at well distributed points. And from this information you can actually prove a very fairly general results so which will goes like that. So you wanna compute the manner measure. So you wanna integrate this log of fp. So you kind of look at what happens between two of roots of unity. You change variables and more or less what you're like lead to understood is like this thing in blue which is the integral between zero and one of log g p k t where this g p k t is just normalized values of the fake heat polynomials at the point e to the k plus t over p. So our goal is to show that this actually converges when p is large to some constant. So the thing is like I said this function like using the fact that you know what is the values at the roots of unity you can rewrite this g p k t as some sum of nice function. So makes this function of t. With some legend symbol on the left. So it's something which has some randomness inside. And using this randomness you can show actually that this g p k t with will converge to some random process. And you will we will compute this manner measure as values the same as the same integral for some random process. So as I say the shifts behave like independent random variables. So you have legend or symbols that we've seen in the first part these are some these are some consideration is some kind of random plus minus one with probability one half. The other part is some nice function that you can express by Taylor expansion. So you could say, okay this function I can approximate it by some random process that I will define. I will keep the same function and just I will replace the directly values like the legends of symbols by Bernoulli variables. So it will give me a function which is random. So function on the space of continuous function. So it will give me like a random process on the space of continuous function. So it will give me a function between zero and one given by this series. And the idea is that we can prove some this limiting distribution result meaning that this GPKT will tend in some sense to this random process. And from this results, we will compute everything on the random side and deduce that we have the same results on the deterministic side. So the main thing is the main result is the following that the sequence of random process given by this GPKT converges weekly to this process, GX of T. So what does it mean? Like if you don't want to talk about probability it just means the second thing that if you take any continuous functional even you average one of a P sum of K in between zero and P minus one of P evaluated on these speculative values So for instance, you can take fee to be to be the integral of a Q of power. So you can compute moments, for instance. This is given by the expectation of this random process. So indeed, like you can recover any information on these speculative polynomials by computing the next expectation on the right hand side. So now, okay, you can recover. So I will just keep proof because I have no time and go to application. So for instance, one nice functional you have is like the, if you have a polynomial you can compute moments. So given a function and computing these moments is like a continuous functional. So you can recover all the moments from these results. So in particular, you can show and have an asymptotic for all the Q of moments which is given as an expectation of a random process to the Q, et cetera. So it means that like a this like was solving a conjecture because about the moments of fake it a polynomial before it was only done for integer moments and it was done by a rather complicated combinatoric proof. And here you have explicit asymptotic for the moments given by an integral of an expectation of a random process. So you can tell me yes, but this does not tell me anything about the malar measure because if I have a function and I associate the integral of log this is not a continuous functional. So I cannot directly apply our results. So just if I have like some time I will move to it. So on a more, let's say on a more probabilistic if I rephrase our results it would be like that. If I take any rectangle on a complex plane and look and I look at the measure of T. So I look at the values on the unit circle of the polynomials I wanna know when it falls on a specific region. Then I can compute this by computing a probability and my process falls into this region. So this is a result on the distribution of exponential sums because the fact that the polynomial on the unit circle is just this thing. It's just sum of KDM e to the two IPN theta. So it tells you like when this, how this thing distributes in the plane. An important thing is this is like as I said is not for an average of a prime but it's for fixed primes. So finally like I have a picture this is like the picture of a realization of the process on the complex plane. So it tells you how this like GXT distributes when T varies on the unit circle. And this should be the same. I mean, that's what we prove. It's the same thing distributes the same when you look at the values of our polynomial. I see something on the chat but let me finish with this. So as I said, we wanna do the same for the malar measure like to solve like to have an asymptotic on the malar measure. We will need to prove a right I wrote here which does not follow from a weak convergence theorem. So if you wanna know like on the left hand side you have a KT polynomial and the right hand side you have expectation of log of our process which would be some kind of malar measure of our process. The major problem is this like function is not continuous on the space of continuous function. So we cannot apply our result because this functional is not continuous. So it's not continuous but the only problem is like we need to control the logarithmic singularities on the random side and on the deterministic side. So we need to show that there's some time like this integrals will not blow on left and on the right hand side. So I will not enter in the details but what we will do is like, we will like consider first truncate look at when our function are not too small. So when there is no problem like the functional nulls are continuous. For instance, if GP is larger than epsilon and GX is larger than epsilon but then the functional when I truncate and I look at only this part is continuous and I have my limiting distribution result. But then the problem and the main technical difficulties to understand have a measure of T on the unit scale such that I have very small values. So such that I have some kind of singularities. So what we prove is that this does not happen too often like the measure like of this is like some kind of control so we can control the logarithmic singularities of this two integrals on random side and on deterministic side. So this is like the main work like say here like the main difficulties to control this specific singularities. So I think it's like already 50 but essentially to control the singularities the thing is to just look at the formula and the bottom HPKT and we need to understand singularities of some specific function which are given like some of rational function with plus minus one and eventually some zeros too. And this kind of function are not too hard to, I mean with some work we can show, we can analyze the first derivative and the second derivative and see that it does not like it will not stay too much small than epsilon because it has some like control variation. So when we analyze like this thing for the deterministic side and the same thing. So for one thing the epsilon will be directly characters and the other side will be very variable but to be some kind of similar analysis we can then deduce a result on the milder measure of polynomers. So then we can, for the moments we don't need this because like we have continuous functional and we can directly compute all moments using our distributional results. So maybe I would just finish here if I have time or not. Sorry, Mark. Do I have some minutes? Yeah, you can take one or two more minutes, of course please. So in practice, so you say maybe because this looks a bit theoretical, you have this Jx, we say, okay, fine. How do I compute this Jx in practice? But you can compute it by, you can truncate the series. So truncate up to J and then computing expectation of this but it's just like taking all the possibilities of plus minus one. So here on this, like on this XM. So when you compute expectation, you just compute another edge over all the plus minus one coefficient of an integral of log of, here there's like some sign missing M smaller than J of this function. Of course it's quite something like it takes a lot of time because two to the J is like very, I mean, it grows very quickly. So if you want to, you need to compute a lot of integrals and then sum it and then if you do it, you will like kind of approximate the values that told you about the, for the malar measure of Fakete polynomial. Of course you could do the same if you want to compute the constant which appears in the moments, you will do the same, you will just have a range of our coefficients and integrate what is here between the bracket to the Q. So you can compute everything on the random size and then they use this for any asymptotic for any fixed large Fakete polynomial. And actually our method like does not need specifically could be adapted for different polynomials. The only thing you need is that the value on the unit circle, I mean values on the roots of unity or on a well set of well chosen set of points have some randomness. So for instance, as was like when we released the paper, Michelle Mosinghoff wrote to us and said, can we do the same for kind of shifted Fakete polynomials which are polynomials which appears in some extramar problems. And because like the, we still have some randomness for like this kind of polynomials, you could compute in prime and you could compute the malar measure for this kind of polynomials too. So it does not apply only of this polynomial but it just applies on something where you have randomness of on a well set, well chosen set of points on the unit circle. So I will think I will stop here.


So another thing that pressure-temperature phase diagrams are useful in illustrating is the concept of thermodynamic degrees of freedom. So we've talked about degrees of freedom before. These are a different type of degrees of freedom. These are not to be confused with the mechanical or molecular degrees of freedom that we talked about when we discussed the equipartition theorem. A thermodynamic degree of freedom is the number of thermodynamic variables that I can independently change of one another. So that's number of thermodynamic variables that can be changed independently. And we'll do a few examples to make sure that idea is clear. For example, let's suppose we have liquid water. So all I've told you is that we have some liquid water. Imagine we have a container, a jar containing water. It's liquid, so it must be in this portion of the phase diagram somewhere. Might be here, here, here. It might be somewhere in the liquid portion of the phase diagram. But if we're at 298 Kelvin, one atmosphere pressure, for example, we're solidly in the liquid part of the phase diagram. If I think about how many variables, so these thermodynamic variables are things like the temperature, the pressure, the volume, the free energy, any of the thermodynamic variables we've discussed so far. How many of those can I modify independently, freely varying some, while still being able to choose any value I want? So for example, am I able to modify the temperature of my water? Can I heat water up? Sure I can. I can heat it up, I can cool it down, and it will remain liquid. So I can modify the temperature of a liquid within certain bounds. I can't change it beyond the boiling point or it will become a gas. But within certain bounds, I can certainly modify the temperature of the liquid. I can also modify the pressure of the liquid, as is made clear from this phase diagram. I can do both at the same time. I can take the liquid, I can heat it up while increasing the pressure, and it will still remain a liquid. But those are the only two degrees of freedom that I have. So we've seen, for example, that if I know the temperature and pressure, I can calculate the free energy from them. We've seen equations of state that tell me I can calculate the volume. We've seen them more often for gases than for liquids, but there are equations of state that tell me how to calculate the volume, the molar volume, as a function of the pressure and the temperature. So once I've calculated two of these thermodynamic variables or set to these thermodynamic variables, the temperature and the pressure, all the others are determined for me, and I can't vary them independently. So I can choose two of them, for example, temperature and pressure. Likewise, let's talk about a gas. I could talk about water gas. I could stick on the same phase diagram and talk about steam being over here in the gaseous portion of the phase diagram. Or if you want a gas that you can picture a little more easily at room temperature and pressure, let's say the nitrogen gas that makes up most of the air in this room right now. So if this phase diagram were not for water, but for nitrogen, on the nitrogen phase diagram, nitrogen is solidly over here in the gaseous part of the phase diagram. So at 298 Kelvin and 1 atmosphere pressure on nitrogen's phase diagram, we're in the gas portion, or technically the supercritical fluid region of the phase diagram, but the term won't make sense for a few more lectures. So we'll say that we're in the gaseous portion of the phase diagram, but I can still independently modify the temperature and the pressure, whether it's steam or whether it's nitrogen gas, I can change the temperature, I can change the pressure without having the substance change into a different phase. So there's also two degrees of freedom for a gas. Things get a little more interesting when our system is composed of a mixture of two phases, however. Let's say I have a system that has both liquid and gaseous water in equilibrium with each other. So let's say, for example, I have an open container of water, a glass of water sitting on the counter. It's in equilibrium with the atmosphere, so there's water vapor in the atmosphere in equilibrium with the water liquid in the container, so I have both liquid and gas in equilibrium with each other. I know if I have liquid and gas in coexistence, I must be somewhere on this liquid gas coexistence curve. In fact, for water, if I'm at 298 Kelvin in this room, there's only one point on this curve, on the coexistence curve that occurs at a temperature of 298 Kelvin, so once I've told you that I have water liquid and water vapor in equilibrium with each other at 298 Kelvin, there is exactly one pressure that the water vapor must have in this room, so the vapor pressure of water is determined, is a constant, and the pressure of the water in this system is going to be the vapor pressure. So in this case, I only have one degree of freedom. I can tell you that the temperature is 298 Kelvin, and then I can't independently say the pressure is two atmospheres or half an atmosphere. The pressure is exactly roughly 24 Tor, 25 Tor, the vapor pressure of water at room temperature. I can modify the temperature. I can take my liquid vapor equilibrium. I can heat the system up to, let's say, 310 Kelvin or some other temperature. When I do that, I can't keep the pressure where it was initially in order to keep liquid and vapor in coexistence with each other. The vapor pressure increases, so the pressure of the water vapor in the room will increase also. So by constraining myself to be on this liquid vapor coexistence line, if I insist on having two phases in equilibrium with each other, I've reduced to only one degree of freedom that's possible. Once I give you the temperature, you can tell me the pressure, or vice versa.  Let's consider one more example, even more constrained. Let's say I have solid and liquid and vapor all in coexistence with each other, in equilibrium with each other as well. So how can I prepare a system that has solid and liquid and gas? Well, that would, solid and liquid is easy. I can make a glass of ice water, ice cubes floating in a glass of water. If I have that in an open container, allow water to evaporate out of the liquid. So I've got water vapor above the liquid, liquid water, and ice cubes in the water. If I allow that system to come to equilibrium, and all the ice doesn't melt, and all the water doesn't evaporate, if I have an equilibrium system with all three of those in coexistence with each other at the same time, there's exactly one point on the phase diagram where I can have solid and liquid and gas all having the same free energy as each other. That's at the triple point. So for water, that's a hair above 273 Kelvin, and the pressure is down around 100th of an atmosphere. So if I tell you we have three phases in coexistence with each other, I'm not allowed to also tell you it's at 300 Kelvin or 200 Kelvin. It must be at the temperature of the triple point. It must be at the pressure of the triple point. There are no degrees of freedom. I don't have the freedom to specify any of the thermodynamic variables independently. Just specifying that I have three phases in equilibrium is enough to guarantee that we're sitting at the triple point. So what we can see and generalize from these examples is if we are somewhere in the middle of a region of the phase diagram that's a single phase, we're always going to have two degrees of freedom. We can modify the temperature and pressure independently. If I'm sitting on any one of these phase coexistence curves, it doesn't have to be the liquid gas coexistence curve. If I'm sitting on one of these curves, if I give you one degree of freedom, you can tell me the other one. So when I have two phases in coexistence, I'm always going to have one degree of freedom. If I have three phases in coexistence, that must be a point where three of these coexistence lines come together, and then I have no degrees of freedom. That pins it down to a specific point in temperature, pressure, space. So in general, we can say the number of degrees of freedom is going to be equal to 3 minus the number of phases that are in coexistence with each other. If I have only a single phase, 3 minus 1 gives me two degrees of freedom. A pair of phases, 3 minus 2 gives me a single degree of freedom. Three phases in coexistence, 3 minus 3 gives me zero thermodynamic degrees of freedom. So there's a rule that we can use to remind ourselves how many thermodynamic variables we're able to choose independently of one another if we have phases in equilibrium with each other as well. This is really just sort of a warm-up for a more complicated rule describing the number of thermodynamic degrees of freedom that will come in much more handy once we start talking about systems with more than just one component in them, when we start talking about multi-component solutions and liquid solutions, for example. So there's something called the Gibbs phase rule. This is not the Gibbs phase rule. This is sort of a precursor to the Gibbs phase rule that we'll see later on. So next thing we'll explore then is try to get a little more precise now that we've seen how interesting these phase coexistence curves are. Try to go a little more precise about understanding how exactly the pressure varies with temperature so we can predict the locations of these phase coexistence curves.


I'm Bill Hammack. And I'm Don DeCost. And this is a commentary track to lecture four of Michael Faraday's The Chemical History of a Candle. We created this with Alex Black. And this commentary track is to help you understand the lecture better. So at this point in the first three lectures, Faraday has really talked a lot about the candle, how it works, the products of the candle. And at this point, he's going to start broadening out a little bit more. We're going to get into a discussion a lot of the atmosphere. And that's going to lead us into the next lecture about comparing the combustion of a candle to human respiration. Now, in the last lecture, he pointed out what you see here is that a flame will burn more brightly in pure oxygen. In fact, he left us with just a little bit of a mystery that if things burn in oxygen, why doesn't our world just burn up and disappear? And he's going to tackle that here. He's going to explain that air has something called nitrogen in it and explain, even though it's inert, why it's so important. And what's nice about the way that he does this, he doesn't just come out and say, oh, by the way, there's nitrogen in the air as well. He actually does some experimentation to show there's something else besides oxygen. And I think it's a good reminder to us that we need to let our students do inquiry, that when we, you know, give them experiments to do, that we don't tell them the outcome. Instead, you know, let them do some discovery. And, you know, this is one of the places where we changed what Faraday did. Faraday had mixed air with nitrous oxide, which was somewhat hard to come by for us, also very dangerous. And so we decided to do a different kind of classic demonstration. This is the so-called blue bottle demonstration. A lot of teachers may be familiar with this. We often use this as kind of an inquiry-based lab. I use this for kinetics as well. In this case, we have, as you can see, potassium hydroxide, glucose and methylene blue indicator. And when it mixes with oxygen, it turns this blue color, which fades over time. And what students can do as far as inquiry goes is, as they shake it, turns blue, fades. They shake it again. They notice that it doesn't turn blue as quickly. It doesn't stay blue as long. Eventually learning that we're trying to discover that perhaps it is actually reacting with the oxygen that's in the air. And I think it's useful to point out here that this is essentially another kind of analytical chemistry test. What he has developed or what we've developed here showing you is a concentration meter. A business of the time you wait until it fades gives you a feel for the concentration of the oxygen that's in there. So we didn't give enough time for the flask on the left, the one with pure oxygen above it, to fade to colorless, which it will over time. The other thing is we want to listen to the way Faraday introduces nitrogen without saying the word nitrogen. Because there is in air, besides oxygen, something else present. The solution left something untouched. There is in fact a gas in air which the solution cannot touch and this gas is not oxygen and yet is part of the atmosphere. I love that phrase, it left something untouched and that just shows why we kept the original language. And I also think it's just really instructive as you hinted at that he didn't mention nitrogen right away. He showed us an observation first and now he's going to show us the implications of the observation. Faraday is going to point out that nitrogen is inert but that it really has an important function in our world. One of the things that Bill and I both like about Faraday is his positiveness about everything. In the first lecture he mentioned the beauty of a candle, looking at the cup. Something very simple that we've seen before but it's beautiful. We then look at beautiful candles that aren't very useful and he finds beauty in that too. The beauty is that we learn something about how candles work. And it would be easy enough to be negative about nitrogen which Faraday is going to disabuse us of. Now let's listen to Faraday introducing the pessimistic view of nitrogen. And you might say it is nothing, it is not worth chemical attention. What does it do in the air? Ah, then come our beautiful and fine results shown us by an observant philosophy. Faraday the great showman constructs this whole alternate world of what would happen. I mean it kind of reminds me a little bit of like Ice Nine which was in Kurt Vonnegut's novel, Catch Cradle. And he's going to show us all the terrible things that can happen that greats will disappear as you see here. What else is going to happen? The trains. Yeah, that's coming up right that the trains would just be destroyed so this is an engine here. And if you think about it, his audience would have had intimate familiarity with trains. This would be something that would be astonishing to them to disappear. And then he links it to lightning if I remember coming up. Is that right? He talks about that the lightning is the only thing that can affect the nitrogen, that it's that inert. And you mentioned that he's a showman. And so he normally would show us these demonstrations, but it also shows that he's a really good storyteller as well. Because obviously we can show these graphics of the great burning and we can show the train. He wasn't able to do that, but he was able to kind of paint a very vivid picture. Yeah, here's our lightning. And if we didn't have nitrogen, what would happen? The lightning would just ignite the atmosphere. That's right. And so there's this apocalyptic scenario that he is spelling out here which I think is just fascinating. And all of these too, the fireplace, the nitrogen or the lightning, the audience was familiar with all of this. And so he was able to bring that to them as well. So he concludes this discussion on nitrogen with emphasizing whether pessimistic view is not the right one, but why you should have an optimistic view of the effects of nitrogen in the atmosphere. And it's very nicely done. It requires all that quantity of nitrogen to reduce the oxygen down. And so it has to be able to supply the candle properly with fuel so as to supply us with an atmosphere which our lungs can healthily and safely breathe for it is just as important to make the oxygen right for us to breathe as it is to make the atmosphere right for the burning of the fire and the candle. And notice what he did there right at the end. He said it's just as important to make oxygen right for us to breathe as it is to make the atmosphere right for the burning of the candle. And this is where he's going in lecture five to couple human respiration with the combustion of the candle. And now he's showing us that gases have weight, that air and other gases consist of matter. And he's going to spend the next four or five minutes reinforcing this point. He's told us the point and then he's going to show us again and again. And this is the most time he spends on a single point and the most number of demonstrations for a single idea. And it's an interesting question why he does these six or seven demonstrations and spends so much time. I mean, is it because he's a showman and he just has lots of things that are really interesting to show? Or is it because he's a teacher that knows that this idea that gases having weight, having matter, is counterintuitive notion for students rather that are new to science? You know, and we have evidence of this today. There's a classic question that's given, especially to middle schoolers, for example, in which they're shown three thin hollow metallic spheres, one with a vacuum, one filled with helium and one with air, floating in water. And they're asked which one would float the highest. And of course, if we go by weight, the one with the vacuum should float the highest. But almost universally the students pick the one with helium because they know that helium floats. So it took me a while to find the balance that's in this image here. We could have used a modern one, you know, with a digital readout, but this is very visual. And I found one at our campus surpluses where they junk old equipment. And it was pretty beat up and I actually had to paint the bottom, the gray part, so it looked good. So now he's going to show us the consequence of air having weight. And we think of a finger as sticking, you know, by suction, but really it's the weight of the air. And notice this is one of several times that he takes something that we describe in common place language but in an inaccurate way and he reinterprets it with a scientific feel. And what was really nice is that he started off by talking about the fact that air does have weight. And he showed us this. He had the two containers, both of them filled with air and he put more air in one and he showed how air actually does have weight. Well, the atmosphere of course is made up of air and if air has weight then the atmosphere has weight. And so instead of talking about suction, which kind of implies that a vacuum has this force that's driving inward, he talks about the weight of the air actually pushing down. That was done entirely by the weight of the air pressing on it and you can easily understand how that is. So what's kind of interesting to me about this demonstration just looking at Faraday's lecture technique is this is the only time that he's used a schematic demonstration. So he didn't have, you know, something actual to show us. And so he's using this just kind of illustrative way with a set of blocks to show us what's going on. And this was something similar to what we did, you know, but we had the advantage of having something like PowerPoint and so we were able to show a lot of graphics and he wasn't able to do that. So this is a very famous experiment. It was done in the 17th century by a German scientist named Otto van Gerica who lived in Magdeburg and he used very large spheres to do this. You can actually see it in the Deutsches Museum in Munich and you can buy these spheres just to do this. Like we bought them from a chemistry supply house. So it's important to point out here that it looks as though the pressure is increasing by the gauge, but that's actually the amount below atmosphere pressure that it's decreasing. You can see here already how hard these are to pull apart. In the case of the giant spheres of van Gerica in Magdeburg, the hemispheres were two feet in diameter. It's about 15 pounds per square inch pressure and that means you need 60 or 70,000 pounds to pull them apart. And so they use 16 horses and we're unable to pull them apart at all. So Faraday here uses a suction cup and as we discussed already suction isn't the right word. What's really holding us on is the pressure that is pressing on the suction cup. I was a little surprised to see him mention suction cups. I didn't think they'd have them at the time, but I learned since then that suction cups have been around since at least the third century BC. They were made out of gourds and used to suction bad blood from internal organs to raise it to the surface. Faraday actually called them suckers. And it's interesting too, it's another example of him using something that people would be familiar with and kind of relate this to the lectures. This is another thing I was surprised to see in this lecture as a pop gun, but I've gone back to the original language and what he says is that you know you can make a pop gun out of a quill or a tube or anything of that kind where we take a slice of potato for instance or an apple and take that tube and cut out a pellet as I have done and push it to one end. So kids at that time would make their own pellet guns. And you know we've mentioned this a lot, but it bears repeating Faraday was a really good showman. And in this case he actually lowers his expectations here. He tells everyone he's going to blow an egg from one cup to the other. And then he says well you know I may not be successful because I've been talking too much and of course he is successful. And he mentions that it's a fact that the air has weight so it's made of matter and if it's made of matter it can actually when it comes in contact with something move that object. Now in this particular case we have to admit we cheated just a little bit. Faraday mentions that if you want to do this you should hard boil the egg. We actually used a hollow plastic egg to do this because it was lighter and because we didn't want to go through a lot of eggs. So now he turns his attention from the weight of the air to the elasticity of the air and that's what he's showing with a pop gun and how it works. And he's going to show that elasticity in the next demonstration also and reminds us how we can get a lot of air into those bottles. First of all it's interesting that this is the first time he mentions a mechanical property I believe and I don't think he mentions one again. And then it's a little odd about why he spends his time on elasticity of air because it doesn't really come back to the candle which we're going to get back to here in a little bit by the end of this lecture and more so in lecture five for a bit. So I don't know he wants to do it maybe for completeness? Yeah I think going back if you were wondering how did he fill up those bottles or how did that pop gun work he's got to use the elasticity or the compressibility of air to explain it. Now this demonstration though is a really nice one. We talked about how Faraday is kind of the godfather of chemistry demonstrations. We still use this one today. We actually not only use this with balloons we'll use it with marshmallows or with shaving cream anything of course that's filled essentially with air and it's always a crowd pleaser. I remember filming this that motor is actually extremely loud and in the soundtrack I actually lowered the sound quite a bit and Alex Black who created this series with us kept asking me to make it lower and lower and quieter and quieter until it just is that kind of hum that you hear now. And now at long last Faraday returns to the candle. So he has a really nice setup here. This open system on the top and the bottom that allows the candle to burn takes a little bit of time to remind us one of the products of the candle is water and he shows us that we can see the water vapor collecting there at the top. But what he's really interested in is finding other products of the candle something we as he said we have not yet examined and this is also a really good demonstration like he's done before where he makes the invisible visible. Something that you know he'll show us the effect of something. Now when he holds this match up he shows it flickers a little bit. It's not mechanically blowing this out. He's going to mention this in lecture five when he uses his breath to do this. And we can see here that the flame goes out. He makes an assumption here that we might think it's because of nitrogen. He's already shown us that nitrogen doesn't support combustion but he's going to anticipate as he says a little bit. He knows this isn't nitrogen but something else that doesn't support combustion and so he's going to take the next few minutes to show a series of tests about what this gas actually is. So what he's going to do here is he's going to take the gas that's coming from the candle and he's going to mix it with lime water and lime water is just calcium oxide dissolved in water and we know today that when CO2 mixes with calcium oxide that calcium carbonate precipitates and that's how come you get the cloudy color. And you know you just called that CO2. I mean it's almost impossible for us nowadays to not use the phrase carbon dioxide or CO2 as we're talking about this because we know what this is. But true to form Faraday doesn't give this gas a name until he's actually tested for it and shown us. So again show first and then tell. And of course Faraday never really called this carbon dioxide. He used the phrase of the time carbonic acid. It's kind of obvious to us now that carbon dioxide would be made of carbon and oxygen. Now notice though what Faraday does. He does a really good job scientific method approach to this. He has a test, the test gas coming off from the candle and then he has his control using air. And remember he kind of made this assumption that we might think that nitrogen was reacting with this. Well of course he's already shown us that air is mostly nitrogen so if the air doesn't react with the lime water then obviously nitrogen doesn't react with the lime water. So this gas it's not water, it's not oxygen, it's not nitrogen. So it must be something new. Now in this next segment we did have to change the language a bit based on kind of Faraday's interpretation of what was going on. So the part that's coming up we had to change the lecture just a little bit. You may recall earlier that we mentioned that well Faraday would know the existence of atoms. He wouldn't necessarily understand their arrangement in molecules. And he's talking here about how you can get carbon dioxide CO2 from chalks and corals and other things. And his language kind of implies that the gas is trapped in there and then physically released and really it's a chemical change. You're actually producing the CO2 which he's going to do here in just the next section actually. But it's nice here though to see this kind of again this universality coming along where Faraday mentions that the carbon dioxide is coming from the candle but it also comes from corals, from shells. He's going to tell us that it comes from our breath. Also comes here from marble. So let's listen to how Faraday describes this piece of marble. Here is a substance marble, a very beautiful and superior marble. And if I put that piece into the jar a great boiling apparently goes on. So I don't know if ours is a superior piece of marble but I will tell you it comes from Lowe's, the hardware store. And what it is it's the threshold I believe for a door and you can buy these long pieces of marble and break it off. So you can do this experiment at home if you want or in the classroom. So Faraday had just mentioned that when he added the marble to the hydrochloric acid that a great boiling apparently goes on. And it looked as though it was boiling, we saw the bubbles. Just like with the electrolysis when he took the water apart to make hydrogen and oxygen he mentioned you might think that this is boiling because again it looks as though it is. But he tells us that it's not steam that's coming up but it's another gas and it's a chemical change, not a physical change. So now your Faraday uses chalk as a source of calcium carbonate and if you're going to try this at home or in the classroom you might be aware that not all chalk today contains calcium carbonate. Some of it's calcium sulfate so you want to be careful what you get. So here's another example of unity that Faraday always likes that the water is water idea. A candle can produce carbon dioxide, the marble can produce carbon dioxide and the chalk can produce carbon dioxide. And this carbon dioxide is all the same it doesn't matter from which source it came. So over the next few minutes he's going to do a couple of tests to detect carbon dioxide and the reason he puts these at the end of this lecture and reminds us of these is that in the next lecture he's going to show that when we exhale it contains carbon dioxide. Now Faraday is following a familiar pattern here as he has in the other lectures. He isolates the substance, the carbon dioxide, collects a large quantity of it and then he begins to figure out its nature. He shows us that it doesn't support combustion. He lets us know that we collect it over water so it doesn't dissolve in water. He lets us know as well that it will react with lime water to turn cloudy. And then one of the first properties he talks about is that it's a weighty gas that it's heavier than nitrogen and oxygen that are in the air. And notice here in this table that he's comparing it to the gases he already isolated in this lecture so that you should be familiar with them. And while he tells us this, he has a very visual way to show this density later in the lecture. Now here we produce CO2 from a dry ice. Of course Faraday didn't do that. He probably produced it from the marble.  Yes. And this is another great one in which Faraday makes the invisible visible and he pours the carbon dioxide, which of course is counterintuitive for a few reasons. One, we don't think anything is in that flask and two, as we've mentioned before, a lot of people don't think gases have weight at all. But you can use this actually to grab the attention of kids. They think of it as magic at first and then we can explain the science behind it afterwards. This is a beautiful demonstration by Michael Faraday. Maybe Faraday did his best visually. The alert viewer will note that that enclosure is what we use to cover the ice bomb when it blew up. And this is well worth doing with kids. And even if you don't have access to dry ice, you can always use an old aquarium and just use baking soda and vinegar to make the carbon dioxide. So that completes our commentary for lecture four. And when we meet again and discuss lecture five, he's going to discuss respiration and show how what happens at a candle is similar to what happens in respiration. So I'm Bill Hammack. I'm Dandy Kost. And thanks for listening.


In most of our churches, we've preached the gospel in such a way where it's easy. The country club experience. You know, Hayes, just say this prayer and people are like, yeah, I feel better. I feel better. But see, in order to function within the spiritual realm, the Bible says for without faith, it is impossible to please God. And when you set up the gospel in most churches to look like alcoholics anonymous and everybody's just saying, yeah, I struggle too. I struggle with gossip. Oh, yeah, me too. But here's the thing. The gospel is this. I come to the end of myself.  I'm not talking about motivational. I'm talking about miraculous. I'm talking about this is only going to be Jesus. It's not a gospel that makes somebody good, better. It's the gospel that makes somebody dead. Come back to life.


Hi, it's Bridget. Welcome to Above Life Channel. The purpose here is to inspire your spirit and to fill you with hope. Today, I am going to have a conversation with Prince from the afterlife. Hmm. You guys know that. If you've been around Above Life Channel for a while, you know that this channel is in part because of Prince because I started channeling Prince way back in 2016 when he first made his transition into the afterlife. If you are a Prince fan, check out the Prince Playlist here at Above Life Channel. Lots of videos for you and I got to honor where I come from in that regard. But I've been a psychic and a medium for over 15 years now. And so the Prince stuff, that was just my entree into channeling celebrities. Now, why celebrities? Because they have powerful connections with you. You have powerful connections with them. They are influential. They can provide insight for all of us. Unique perspective about human life from their afterlife, few points. So it gives us a lot of great information. Okay, so today is a beautiful day as you'll notice the sun popping in and out. And I will keep my sunglasses on most likely because I got really sensitive eyes here out in Minnesota. And yes, I'm in Minnesota. And this week is the week that I will be sharing this particular channeling session is the week of Prince's birthday.  He didn't celebrate birthdays, but everybody who is a Minnesotan and it's a Prince fan from Minnesota knows that they still celebrate. People still celebrate here. Okay, so in fact, I believe it's his birthday. That's actually like Prince Day or something here in Minnesota. All right, so right now the times are very turbulent to say the least in Minnesota, Minneapolis, and St. Paul and the surrounding suburbs. There's a lot of chaos and I think it's a really great time for us to get some insight, some good positive purple vibes from Prince the Purple One. After the recent death, I use the word death instead of the M word because YouTube sometimes doesn't like the M word used in things and because then you have to like flag it and have all this stuff on it. So after the recent death of a black man at the hands of a Minneapolis police officer who is now in custody and more charges to come for others that were involved or should I say who didn't get involved and lots of outrage from our community here in Minnesota in the Twin Cities metro area and around the country and lots of anger, lots of upset, lots of grief and sadness and cries for change and really a lot of intensity and today is so George died on Monday night Memorial Day and I'm recording this video on a Saturday and for the first time ever we had a curfew last night like a military state kind of curfew because there are lots of people that are pouring into Minneapolis and St. Paul. They're Twin Cities you guys like there's a river in between but they're basically two huge cities that kind of form one metropolitan area of the Twin Cities and then there's rings of suburbs around there. I'm in one of the probably the secondary second to third ring of suburbs probably is what I would be classified at so I'm about 35 minutes south of St. Paul and we'll kind of southwest a little bit yeah I'm a little bit west too but there's a lot of concern for all the people that are pouring in that aren't protesters they're like militia groups and white supremacists and anti-government groups and anarchy groups and things like that and mixing that in with the local elements of people who are not wanting to protest or not wanting to be part of the community who are just angry and stuff you know who want to cause problems problems basically are really messing it up for people who want to peacefully protest in the community they're then wants and needs to come together and it's just there's a lot there's a lot so we're going to talk to Prince because he grew up in north Minneapolis and understands the pain that's happening right now so Prince and I know you've been such an advocate for like pretty police brutality and to me I'm like abuse of power that's what it is for me and and that's how I define it and also specifically racism equality all of it and I know that you've been active in that before so do you want to talk to us can you have a conversation with us and I wasn't sure if I wanted to talk to you about this because I you know I don't like controversy you guys I don't like drama I don't want to stir the pot I don't want to be part of some bandwagon of oh you know let's make change let's do this everybody you know take an advantage of the situation that is deeply affecting Minnesota which by the way I am I am a minnesota and born and raised I am here this is my state so people who don't belong here when driving here from other places across the united states which they have been because they were arrested in st paul last night all the rest that occurred on friday night in st paul according to the mayor's press conference the mayor of st paul who's awesome by the way incredible speaker very just inspiring as far as leadership goes he said today at the press conference that everybody that was arrested in st paul on friday night was from out of state so get out of our state go take care of your own business you want to cause trouble go to your own neighborhood jeez sorry okay all right all right yeah I'm gonna get a lot of comments on this one but I'm too princess you about done yes I know I know yes I'm about done what what can't give us some words of wisdom prince like first of all what would you do if you were here what would you be doing he says I wouldn't be mixing it up in Minneapolis I can tell you that I wouldn't be in Minneapolis right now he says I wouldn't do that I'm gonna be doing that there's lots of ways you can make your voice be heard he says there's lots of ways you can make change you know and for me I would offer support through music and money but you all wouldn't have known that part when I was alive you wouldn't have known that part so it was um I in one of the um the friend groups I'm in they're very they're major prince fans major purple friends of mine and then there was a mention I think from paisley park one of their posts on facebook about you after the trevon martin shooting that you had a rally a dance rally um for peace or for change at paisley park he said yeah yeah he says you gotta raise the love that's what's gotta happen you know you gotta raise the love vibration hate hate caused this problem you know hate caused this it's not something that's going to be just you know instantly healed overnight there's nothing you can really do it's just superficial you know it's just surface the wound is deep you know many of us are born with it you don't know what it's like the color of your skin doesn't cause you to immediately have a instinct to be okay so I can't match the right word when I'm calling me to translate it you guys from what he's saying it's alert that causes like a color of my skin because I'm white and and he um is saying I didn't understand I don't I wouldn't understand necessarily how it would feel to be born with a wound with a higher level of alertness in order to keep yourself survival he says you are a victim waiting you are a target and he says back in the day when I he's like when I was you know when I was younger I was in st paul and Minneapolis and we used to go to clubs and things and you know we'd play and you know that there was discrimination I mean you know it but you just kind of you try you try to turn your head you know not real not get real uh can't let it take you off your course you know you got to focus on on what you're there for and what you want to do but I experience racism all the time and that's just like it's like it's it's it's just it's a part of of this landscape the world that we live in and that doesn't mean it's right and it's not it's not something that he says you know in the civil rights movements you would think that something would shade that it would change right you would think that things have gotten better oh it's gotten better it's gotten better but he says I would argue that it hasn't gotten better the people that have the money and the power and the people that are white and mostly the white men that are in control of everything whether it be a record company or government that's a problem okay but prince you didn't even vote like you you're very publicly you had conversations where you didn't vote you said I didn't I don't vote I don't get involved in that so it seems like you didn't really do everything you could do yourself to create change you have to do with you have to do it do with what you believe in he says you have to do with what you believe in okay so like he's saying like so he literally shows me you guys remember the version of prince that I connect to sometimes you might not necessarily instantaneously recognize because he has this higher level viewpoint and ascended viewpoint where like a teacher like almost not a teacher but like a philosophical like viewpoint where he's showing me an image like a circle and it's a light light kind of a lavender color inside the circle and then he's showing me all of these little dots these little yellow golden dots inside the circle lots of little yellow like points of light inside the circle and he shows me the circle and he says within your what you believe your belief system the circle of what you believe this this is a cycle he said it's it's it's the community that you were born into it's the culture that you live in it's the society that puts the the boundaries upon you you don't even have a choice to set your own boundaries you've got to work within the structure of the belief system that you have but for an african-american person or a person with brown skin there is an extra layer of expectation boundaries confinement restriction that is set upon that and so you don't have that is what he says to me you don't have that that extra layer okay so prince i have a woman layer i'm a woman and because of that people in like the white house for example think that women are inferior and don't really see the value that i offer so i connect with my own value and create my own belief system based around what i want the desires that i have to make a difference in the world what the kind of relationships i want to have with my family and my community in my work that kind of thing so i see that piece but i also know that that's very different than a skin color piece i know that the skin color is you can't transcend that like how do you transcend that you don't change that he says i don't know what to say to you about i don't know anything about being a woman he says i know about being with women but i don't know anything about being a woman yeah i know i know and i don't know anything about being an african-american i don't know that's a thing to print so for us white people can you give us people like me who are so vanilla have no concept of how to help support change racially how i mean because the the talk now the rhetoric now the the the messages that are shared now from people about civil rights about racism are all about how i'm a racist because that's just part of my belief system like me i was born a racist basically like that's kind of the little belief system that's being placed upon me now and while i don't intellectually i don't feel like and personally i don't feel like that's true there has to be some level of truth to that if other people who are in minority classes minority is class the right word you guys minority groups or not even minority let's that even sounds totally disrespectful to say but people who are african-american or brown skin are see i don't i don't have any kind of i i don't know how to have that conversation about well how do i say this i don't know white people like me like a middle class white woman who's had privilege and still has privilege and lives in my little suburbia area who is pretty removed from the protest unless i choose to go in and support and show up but i'm leary to do that why would i do that when people like on facebook are telling me well you don't know you don't know what's going on so you're just you're like belittling our you're belittling it because you are a racist you're just a racist and even though you don't mean to be a racist you just are so all the sudden i'm like part of the problem which i understand that the systems and the structures and things like that that i have a privilege to be part of don't see me specifically as a problem and we're created by white people to keep out people who have different colored skin i understand that that part the institutionalization of it the heritage the lineage that i have because i'm a white person yes in that way i guess i inherited that racism that i'm i inheriting that but i don't believe like i don't believe that about myself i don't feel that about myself just because i don't know or i'm ignorant even if i try to educate myself i don't know i feel like i'm damned if i do and i'm damned if i don't and this isn't about me you guys but i understand that a lot of my audience is white women just like me so prince educating me having a conversation with me helping me to hear your stories and understand your experiences so that i can hear your truth helps to enlighten me helps me to to to respect where you're coming from and it doesn't make me an expert doesn't make me know how to to make things better necessarily without your help your assistance so what like what can i do i mean can i do anything is there anything i can do really is there anything i can do besides vote vote vote which you didn't do and that doesn't really rub me the wrong way he's like yeah we you know there's one thing you shouldn't talk about it's religion and politics i know i know he's like politics project you and i will not we will not ever be on the same page with politics i know and it's not he said it's not about a party you know parties and anything like that just divides you he says so so tell me help me understand so that everybody else can also understand what do we do what do we do well the simple fact that you're asking the question means to care and you have to go beyond that you have to do more than that you have to listen you have to really listen you have to just shut your mouths and let the people who have been underrepresented let them speak let us speak listen and you got it in and and it's true you know people do learn from the stories of others so you got to share those stories you got to help to spread the words of african americans and and of and i'm going to say underrepresented classes because i also want to include like latinos and um mung and samali and all the different cultures and things like that that are here in minnesota as well as across the country and all of their their voices come together you know they form a chorus but the individual stories are what we can learn from is what i'm hearing you say right it says it's not going to happen overnight you got it you can't just do it once you got to do it all the time you got to be willing to show up shut up and listen i know i know that people mean well when they say that to me as a white woman i feel like that feels like i don't want you here then when you say shut up when you're telling me not to speak it makes me feel like i need to that i don't i shouldn't be there in the first place he says and that's exactly how african americans have felt in their entire lives that's how we were born so now you feel a little bit just a little bit and you can choose not to feel that way that's put on that's put on you by your choice by the guilt or the shame or the tolerance that you've had your whole life i'm not he says i i'm not saying it's your fault i'm not blaming you but you're gonna be what keeps it the same or what makes the change happens not just let the change happen but you amplify the voices of the african americans you share the poetry you share the lyrics you share the music you buy from businesses that are other colors than you you support organizations and charities that are other colors than you that's what you do to build the communities of color to give them strength to give them voice to give them power that's what you do you consume their products you purchase their services you donate to their charities that's what you do with your white money that's what you do and you spread the love you got to really focus on the fact that this is here this is present this is now and you got to focus don't lose sight of the purpose of all while this is happening while this is happening because it is way beyond time for change it is it is way beyond it is everything is just completely unearthed so that you can replant seeds together side by side so would you do a dance party i would i would i would do a concert i could see you on top of first avenue in minneapolis right on top of the roof there just just sing and he says um no he says no no no paisley park we have a big party at paisley park the fund's going to organizations to help clean up the city because that's what's going to be needed that's if you really want to show up and help that's what you do you come out after in the aftermath and you work side by side and get dirty in that community as you help them clean up the streets the businesses you help you got to show up for that stuff you got to show up after not just right now when you're mad and angry and when it's hot you got to do after for for a long long time you got to consistently show up and be there and be present and be willing to consistently be there and be present and be willing there's a lot of pain but don't make it no don't make them the mistake of thinking that this is a new problem but this is a new wound circumstance or situation yeah the name's different but the situation is the same situation a person in power took a life a white person took a life of a black person and the white person had the power and the black person did not it is important for you to acknowledge that the way the man died was without his voice and so now that voice is going to be so important moving ahead for all of those involved as the community comes back together again there's going to be music about this isn't there why yeah i hope so this prince said i hope so music is healing it is you know i can't i can't really offer you anything that's going to make you feel any better you gotta feel you gotta feel uncomfortable you gotta feel bad you gotta feel unsettled that's how many black people feel every day of their lives just going to school walking down the street being in a public place being in a place with white people there's always this underlying what if so if you're unsettled then good that's progress it's not a bad thing when you suffer together you can heal together i see a band sly in the family stone i think or earth wind and fire both those two bands and that music you guys whatever that music is there's something that's connected or related to that are there other spiritual teachers or mentors for us that we can look to during this time to help educate us as white people to understand how we can be better members of society together how we can bridge the gaps how we can be supportive of our brothers and sisters how of of different skin tones how can i mean i probably sound so ignorant i apologize if i do you guys but this is true this is who i am who i am i don't know you know can you give us some other mentors or or or folks that we can look up to the like mlk jr i see that right away obviously he says aretha aretha franklin i'm like wow edda jane jones or if it's jerald i can't tell edda he says edda oh thank you nelson mendela oh my gosh beautiful really beautiful energy oh wow that was really nice i just felt nelson mendela energy wow prince yeah he says yeah you want to feel peace yeah talk about a man who's been through it yeah he said yeah yeah maya angelo nat king coal natalie coal his daughter what about melcom x there's adversity you know mix things up yeah yeah i feel more um why do i say f scott f scott Fitzgerald i don't know um he said he's like he's like showing me education learning like great speakers great writers great authors great musicians like looking to them to inspire us to connect with their gifts and amplify the energy of what they have to offer instead of their skin tone basically like their gifts that they have to offer their contributions to society not limitations because of their skin tone but recognizing that and the time that they lived yeah they were limited because of their skin tone can you imagine if they weren't like ask yourself what if what if that person was white then what how successful how rich how famous how influential could they have been there's another african-american woman but i can't i i don't know exactly who it is like it's not opera you guys it's it's influential as she is and as awesome as i think opra is it's not her it's somebody else though i don't know who she is though she's alive now it's not um like michelle obama that's not who it is older an older woman that's african-american i don't know who it is though i i don't and to answer some of your questions because some of you like yell at me about her like bridget why don't you just ask them what the name is what doesn't always work like that you guys because i get information in multiple ways the clairvoyantly i see that's the easiest way for me i sense and feel that's also an easy way for me i don't always hear exactly it's it's difficult for me to hear i hear bits and pieces so it's it's tough that part i don't always get everything because that's not what above life channel is about it's not about being always being right i'm always right so by getting the information and having that the real conversations that are going to provide insight to you they're going to inspire your spirit and give you some hope prince thank you thanks for having a conversation with me this is a hard conversation to have says you're going to share it right yeah i'm going to share it do you have any insight can you share any any words of wisdom or anything inspiring to us in minnesota here as we're dealing with this there's so many different pieces you guys that can feel him like he's trying to be calm but he's saying you got to stay angry but not be violent you got to be calm but not complacent you got to be ready it's going to be real obvious what needs to happen it's easy to get lost in ambiguity and you just step back because it's too hard or you're like hey it's you know there's excuses like i'm too i'm white so i don't know nobody's gonna want my help because i'm white um um it's not oh poor me it's just it's i don't know it's true right i see he said no no no no see see see no no no no no no no no that's not an excuse your skin color can't be an excuse now he said you can't make that an excuse now that's not gonna work he says that's not gonna work you know take back the power and you got to do it through love and you've got to you got to be willing to stay stick with it he's like you got to be willing to stay with it he says i would recommend that you all take care of yourselves and each other maybe have some you know watch your favorite concert listen to some music together you know have someone be your dj and hang out with your friends online online you guys online like facebook or something and have somebody spin records and you know that kind of thing you know be creative he's like be creative you know he's like you know social distancing so he's like be creative like yeah we can be awesome in the so and work we're creative he says it will pass it will pass it's gonna take time though he says it's gonna take time it looks like at least three more days you guys oh my gosh three or five more days oh my goodness it's saturday today it's gonna take time all right you guys i wish i had more like woohoo positive inspirational stuff but this is real life stuff and i think it's kind of deep for us to be connected this way and this is real and authentic and very raw conversation so for all of the viewers here at above life channel for our community thank you for being here thank you for being part of this community and for spreading messages of love and power through love it's really important that's what's needed right now at least in minnesota oh my gosh we will accept your prayers and your positive healing energies that love prevails instead of all those crazy terrorist like groups that are here trying to take over you know the protesters that are peaceful need space and room and the community needs room to grieve so we can if you guys can help send positive prayers and energy to help create that healing part and allow for their voices to be heard and continued support for the progress of change that'd be real helpful here thanks you guys thanks for being here and thanks to prince for the conversation


Everybody forced here Ripping the sports card on the neck will be sport personal box edition volume four one box here for Lee V Thank You Lee box 39 Let's go Lee PSA slab 2021 prism to a tongue of I loa sensational signatures navy camo PSA 10 20 of 25 second-year to a prism navy camo auto Nice card Lee PSA 10 fantastic grade. We do appreciate it. We will get this out to you