So, let me recall where I ended last time. I began discussing the problem for moments of L functions on the critical line. So, for zeta, there are these conjectures of the 2kth moment behaves like some constant ck times t log t to the k squared. And the ck was factors as the product of two relatively well understood things. a k is very easy to understand. It can be written explicitly in terms of an Euler product. So it can be given as an Euler product, which converges absolutely. And then this gk was this mysterious factor which I wrote down a formula for a conjectural formula for last time. And then I also mentioned that you could look at analogs of these in other families of L functions. So for example, you could look at chi mod q. So these are all conjectures. Let's say if you average over primitive characters and this should be asymptotic to some other constant ck times q log q to the k squared. Let's say q is prime so that I don't have to worry about how many primitive characters there are. Or if you look at fundamental discriminants and you look at quadratic characters, then this should be asymptotic to some constant times. If you make a q, I'll actually know the state. If you make a q, I will mention that, sorry, this should be k squared, of course. If you make a q average, you'll know it for one more value of k. Actually, maybe, so I'll mention results in that case. And you could look at twists of a modular form, where you expect something like x log x to the k times k minus 1 over 2. So first, let me explain where these conjectures come from and then I'll explain what we know towards them. So these are representative of many other kind of conjectures that you could write down. So why do we have these shapes of these asymptotic formulas? Well, so there is also a connection between these moment conjectures and the distribution of zeros of L functions and random matrix theory. So let me just say, let's say for the zeta function, for zeta, if you look at the zeros and suppose we assume the Riemann hypothesis and write the zeros as half plus i gamma and arrange these gammas in ascending order, then there's the conjecture which goes back to Montgomery and then with numerics by Othlisco and then extensions by Rudnik and Sonak. That if you look at the, so these zeros get closer and closer to each other as you proceed up the critical line. So the nth zero is roughly of size 2 pi n over log n. Because there are about t log t zeros up to high t. So you can normalize this by looking at gamma n times log gamma n and divide by 2 pi. So the nth normalized zero is about size n. And you can ask for the spacings between gamma n's. So Terry mentioned that for primes if you take the normalized spacings between primes they should behave like spacings between random numbers thrown down with mean spacing 1. This one is expected to be different. The spacing between these ordinates should correspond, is expected to correspond to spacings between eigenvalues of large random matrices. So maybe to give one kind of ensemble that you could take here, you could look at the unitary group and choose a matrix uniformly from our measure on this group. So, and then it'll have eigenvalues which look like e to the i theta 1 to e to the i theta n. And then these numbers theta 1 to theta n lie between, let's say we arrange them so that they lie between zero and 2 pi. And then what you're interested in would be the normalized spacings between theta j times n over 2 pi. So then the difference between two consecutive eigenvalues would be, if you normalize these eigen angles, the normal, the spacing would be 1. And that, whatever spacing that you get should correspond to the spacing. So in other words, for each matrix you look at the, the spacing distribution here. And then you average that over all matrices in this group, u n. So when you average it, there's some measure on how these theta 1 to theta n are chosen. The measure is not simply d theta 1 to d theta n. It's something that keeps track of this, e to the i theta j, let's say j and k go from 1 to n and then times d theta 1 to d theta n. So, so the difference here from just taking angles independent of each other is that the angles here repel each other and you don't want two angles to get close to each other. The space of, the space of the group where any two angles are very close to each other is very small, comparatively. Okay, so that's a conjecture on, on the zeros of, of L functions. And the conjecture says that you should look at the spacing between zeros of random matrices and you want large random matrices. So at the end you let this parameter capital N go to infinity. So Keating and Snaith had the, had the idea that you should make this a bit, bit more precise. And you can make it as an actual model where you say that if you look at statistical properties of zeros around height t, then that should correspond to, so, so you don't just want to say that as t goes to infinity it corresponds to something on the random matrix side. As n goes to infinity, you want some relation between t and n. And the relation that they, that they, that they postulated was that, well the spacings between two consecutive zeros here is roughly 2 pi over log t. And the spacings between two consecutive eigenvalues here is roughly 2 pi over n. So that kind of gives you this, this equivalence. If you take, except these two things equal to each other, you should take n to be roughly speaking on the scale of log t. And you look at random matrices of size log t. Okay, so if you, if you believe this, then that gives you a way of thinking about other problems on, on L functions. So, for example, if you want to think about properties of zeta of half plus i t, around height, around height capital T. Then you might expect that this is modeled by, by some other object, by the object here whose zeros are the eigenvalues of your random matrix. And that would be the characteristic polynomial of a random matrix of size n, which is about log t. And then you can imagine that maybe the conjecture for the 2kth moment of zeta of half plus i t should correspond to whatever we can find for the average over the unitary group of the characteristic polynomial of some matrix g to the 2k average over the group g. So, so now this side, one can calculate exactly. And in fact, what's done by, what's something which was computed by Selberg in a famous paper that he wrote in a Norwegian high school journal. And, and so, so he computed that this has a poll, has, is a polynomial, well, or from his work it would follow that it's a polynomial in n of degree k squared, okay? And so this should correspond to log t to the k squared. So you can see this matching and the leading order asymptotics for this involves this constant gk times n to the k squared. And this is the motivation for the Keating-State conjecture. So notice that in this, in this analogy, you don't see this arithmetic factor ak, which I said was easy to understand. And you could write in terms of some Euler product involving primes. Of course, there are no primes on this side of the, of the random matrix side calculation. You can form an analog of this in all the other cases that I wrote down. The group that you average out might change depending on the family that you're, that you're interested in. And you can form analogs of these, of these conjectures. So, so the other conjectures. So, what are you evaluating in k squared on y? Yeah, okay, you can evaluate it anywhere, let's say, just at zero, let's say. It doesn't matter. It would make a, so in the t aspect, it makes no difference where you evaluated. In, in one of these other families, you might have to evaluate it up. If the, if, because there's no special point in this, in this situation. So the other conjectures are formulated similarly, and you would, so, so, so there, there were refinements of this conjecture of Montgomery and Rudik, Rudik Sarnak by work of Katz and Sarnak on, on family. So given some family of L functions, Katz and Sarnak prescribe a, a symmetry group, which kind of tells you what family of random matrices you should look at in order to understand this pacing, zero spacings of L values in that family. And you use that to model the moments as well. And it, and it, and it gives you these different predictions that it seems to work. Okay, so these are, you should, these are of three cases. Unitary, which is where examples are like the zeta function in t aspect, or all Dirichlet L functions. Or you could look at orthogonal. And here, the typical examples would be like the twists of a, of a modular form. And you see this feature of k times k minus 1 over 2 coming in the, in this example. And the last would be, would be some plectic where the prototypical example is something like just quadratic Dirichlet L functions, okay. So, so this is one way of approaching these, these moment conjectures. It, it's very, it's very pretty in, in, in, you know, kind of telling you conceptually where this constant gk should come from. But it also, maybe one part of it which is unsatisfying is that you have to put in these aks by hand. That there's no way in which you see that arising from this heuristic. So maybe one, one heuristic along these lines should be that maybe one should think of the zeta function as having two parts. One is that it has a, it looks like an Euler product up to some point. And then there's another part which comes from the zeros. The Euler product one does separately and must match something from the ak times some, some well understood thing. And then the average of the zero part should give you something which just gives to use this random matrix calculation. So there is work along those lines by, by Hughes, Gonic and Keating. I'm not sure that I'm remembering the names correctly. So maybe, maybe Farmer, Max would know. No? Okay, I think it might be Farmer rather than Keating. Okay, we can give John Keating some credit as well. So, so these, so they formulate some hybrid models of trying to write L functions in terms of both primes up to some point plus some contribution from zeros and then you can study each one of these, each one of these separately. And then that, that produces the, the contribution from A, the primes will produce the ak and the contribution from the zeros will produce the, the gk. Okay, so, so now I want to tell you a different way of approaching these, of approaching these conductors, which is due to Conry, Farmer, Keating, Rubenstein and Sneith, which for, just for integer moments. So, so this argument actually will work for real moments as well. And for integer moments, there's a conjecture which will identify, say, let's say one advantage of this is that it identifies all the lower order terms in the moment conjecture. So, so for example, in the case of the kth, 2kth moment of the zeta function, it'll produce that there's a polynomial pk in log of t. So, of degree k squared, and then it'll identify for you all the, all the terms in the expansion form of all the coefficients of pk. But it's a very messy polynomial, it, it, it won't just be this polynomial, because as you can see it'll also have to include arithmetic coefficients corresponding to generalizations of this ak. And this conjecture is also very, very simple in a, in a certain sense. So, I want to explain how, how this conjecture works. So, let's, let's, I'm gonna do this in a special case. And we're looking, going to look at all Dirichlet characters chi mod q. Let's say these are primitive. Assume that q is prime if you like, and something large. And I want to make a, a conjecture for the sum of l half chi to the 2k. And the idea here goes back. So, I mentioned last time that Ingham computed the fourth moment, but he computed the fourth moment of the zeta function with various shifts with four, with four parameters. So, the idea here is to do something similar here, not just to look at this as the, as a 2kth moment of one object, but to introduce 2k parameters. So, it's convenient if I look at the completed l function instead. So, let's say we multiply this by, oh, there's an epsilon chi here.  So, this is either 0 or 1, depending on whether the conductor is, the character is even or, even or odd. And this is a Gauss sum whose sign is 1, absolute value is 1. Okay, well, I won't worry too much about this, this a. So, suppose instead of looking at l half chi to the 2k, we look at, we look at the following object instead. So, I introduce 2k new variables, alpha 1 to alpha k and beta 1 to beta k. And of course, if I set all the alphas and the betas to be equal to 0, then I recover the 2kth moment that I am interested in. There are some factors of these, of these guys, but of course I can just remove them at the end. Now, okay, so this object, whatever it is, you can see that it has, that I'm allowed to permute some things and the answer should not change. So, I can permute all the alphas and clearly nothing changes. I can also permute all the betas and nothing changes. But also I claim that I can take, I can take this. So, let's say I put alpha k plus 1 to be beta k, beta 1, alpha k plus 2 to be beta 2, and so on. And I claim that if I actually, I can think of permuting these 2k variables, and then still nothing should change, okay? And this is the step that you have to check. And the reason for that is that if I, if I make a permutation of, of all of these 2k variables, then some of the pluses will become minuses. Let's say some alpha 1 becomes beta 1, and beta 1 was back to alpha 1. If I do that, then I'm using the functional equation twice. Once to replace this by a half minus alpha 1 chi bar, and then using the functional equation once to replace this by a half plus beta 1 chi. And if I use it twice, the sign of the functional equations will cancel out as well. Epsilon chi times epsilon chi bar will be 1, okay? So, so nothing changes can, so what we see is that there's a slightly non-trivial relation that I can permute alpha 1 to alpha k, alpha k plus 1 to alpha 2k in this notation. And then again, nothing should change. So the, whatever answer that I have, whatever answer that I want to make here, this answer should be symmetric under the action of S2k, of permuting by these 2k variables. Okay, so now let's make a guess as to what the answer should be. And we'll make a very naive guess. We'll expand all of these into this Dirichlet series, even though nothing might actually converge. And then just identify diagonal terms. So we have various factors coming from, so you have these q over pi to the half plus alpha j. So again, I'll forget this a that appears. So let's, so this is just some nuisance factor. Let's not worry too much about this. And then what I have is a sum over chi mod q. And then I'm going to expand out all of these, these L functions as Dirichlet series. I'm not worried very much about convergence. So I have a sum over k variables m1 to mk and another k variables m1 to nk. And then I have something like chi, something like that. Okay, so now we do one more piece which is not justified. We take the sum over the characters chi mod q and we use the orthogonality relation. The orthogonality relation will give you that the product of m1 to mk should be congruent to the product of n1 to nk, mod q. For this sum to contribute. And we're going to think of that orthogonality relation as just picking out the diagonal term m1 to mk equals n1 to nk. Okay, so the guess from summing this might be, so the number of characters times whatever you have here, times the diagonal terms. Well, these should also be co-prime to q if you like. So variables co-prime to q. And then you have now this object is something which is nice and multiplicative, if I look at a prime p, it must appear the same number of times on the left-hand side as on the right-hand side. And on the left-hand side, let's say if it divides one factor, it might divide some, it might appear as p to the half plus alpha seven, and then it might appear as some p to the half minus beta 11 or something like that, right? So you can write this as a product. So this one can formally think of as a product over all primes p of various expressions which would look like, let's say, maybe it can appear in this product more than once. But let's say the terms which appear once would look like one over p to the one plus alpha j minus beta k, beta l. And the sum over all j and l. Plus higher order terms. And you can see that, okay, the higher order terms will involve larger powers of p here. So think of alpha j and the betas being relatively close to zero so that if I have a one over p squared or so, that gives me convergence series. So the main contribution is this p to the one plus alpha j minus beta l, which I can think of as a zeta of one plus alpha j minus beta l. So we would compare this with the product over all j and l going from one to k, zeta of one plus alpha j minus beta l. And then what is left over will be something analytic in a wider region. So, okay, so that's going to be my guess for what happens if I just blindly expand the Dirichlet series here and then just restrict myself to the diagonal terms, okay? So I would make a conjecture that the answer is something given by this product of zeta functions times some other function which is nice and tame, can be understood easily. And this is roughly of the right shape because if you imagine alpha j and the beta l being small, then this is like having a pole of order k squared for the zeta function. Now, okay, but now if you look at this answer, this answer is not really symmetric in the variable. So you're allowed to permute the alpha j's, that's okay. You can permute the beta l's, that's okay. But you cannot now make some of the alphas become betas and some of the betas become alphas. So you miss some symmetry that the original answer must satisfy, so much better. So now the idea is, now let's, so we are, so this answer, the guess is symmetric under sk cross sk, but not. So the conjecture of Conry-Farmer Keating Group in Sinonsite is very simple. You just symmetrized this answer. So symmetrized the guess. So you take a sum over all permutations in this quotient and just take the guess, let's call that z of alpha 1 to alpha 2k. And then you do that, you apply the permutation to these parameters. And so each term in the guess, so if you just look at this object, this has a whole bunch of singularities. It has singularities whenever one of the alpha j's equals a beta l. Or if one of the alpha j's happens to be 0. So there's a lot of singularities of that expression. But if you symmetrize it, then the object is actually regular in all the alphas and betas. So this object, so this has many singularities, but the symmetrized answer is entire in alphas and betas. So all the symmetries cancel out. And where do you get powers of log? Well, you can see that the powers of q will vary in this answer. You might have a q to the alpha 1 plus, the sum of the alphas minus the sum of the betas in one situation. And maybe that exponent will get flipped around in other cases. So the powers of log will come from the power of q that you have there. And then finally, if you take all the variables equal to 0 in this answer, it matches exactly the Keating-Sneith convention. So in the alphas and betas, don't they use the functional equation? So this uses the functional equation in a certain sense, right? Or you mean do they use the- Yeah, of course, they use the functional equation in the alphas and betas. Right, but that only takes, that gives you, so that accounts for one permutation where you replace all the alphas by betas and all the betas by alphas, right? Because it's clean on this one. Yeah, I think it's clean on this one. I can't disagree with that, right? But it's the same idea. So in a certain sense, this moment conjecture is the simplest conjecture that you can write down, which is likely to be true, right? You must have this S2K symmetry and this is the simplest object which has that symmetry. So it's a very intuitive conjecture in some ways. But what we are missing in all the ways that we know to attack these problems, we start with, as Emmanuel was saying, with something like the approximate functional equation, where you start with an expression for your L functions. Which if you put in these variables, you start with two terms, one of which is, so one term will be symmetric under pi being the identity permutation, that's what you have. And the other one being the permutation where all the alphas are replaced by all the betas, just an involution there. And you miss all the other symmetries. And so whenever you have to make a long computation, which eventually recovers all the symmetries that you had, but you don't know how to do that. We don't know how to do that in any straightforward way. We don't know how to keep the symmetry from the start. So what do we know about this conjecture? Well, so we have theorems for small values of k. So for zeta, we only know for k equals 1 and k equals 2. The second and the fourth moments. If you look at the family of quadratic twist, quadratic Dirichlet L functions, well, maybe let me do L have chi to the 2k for chi mod q. So this is kind of exactly analogous to the zeta function. But already there are some features here which are harder than for zeta.  But for k equals 2, for the fourth moment of the zeta function, for a long time we know that you can get an asymptotic formula with a power saving in the error terms. So in this case, to get precise information on error terms, seems quite complicated and it's only known fairly recently by work of Matt Young. And in the case where q is prime. Then one can look at moments of quadratic Dirichlet L functions. So here we know the first three moments. So here you can take just integers, they don't have to be even. So we know the first three moments. These are maybe the first two are due to Utila, and the third is due to me. And then the fourth one is very much on the edge. It looks like one should be able to prove the fourth moment. There's a nice result of Heath Brown, his large surf for real characters. Which implies that if you look at the fourth moment, you can get a bound which is almost sharp. You can get the right power of x, x to the 1 plus epsilon. So in situations where you can get a sharp upper bound for the moments, you might hope that you'll be able to refine it to get an asymptotic. So this is one such example, but we don't know how to refine it to get an asymptotic. And another result in this direction is that on GRH, Matt Young and I can get an asymptotic. Actually, this is not strictly speaking correct, so maybe let me put this in quotes. We have a method which we worked out in a different case, which should extend to give asymptotics in this case. But nobody has written it down. You're using quadratic reciprocity. You do use quadratic reciprocity, right? So another example where we know very little is to take the quadratic twist of a modular form. So f is a fixed eigenform. And in this case, we know only the first moment, k equals 1, which was done, so it's not strictly speaking correct, but maybe, but in principle, this was done by the Muthys and Bump Friedberg-Hofstein. They dealt with derivatives and obtaining the first moment in this case, but the same method would work for just these values. And knowing this for k equals 2 would also be an interesting extreme case. We are almost there. The same result of Heath Brown would give you the right upper bound for this moment to give you an x to the 1 plus epsilon bound here. And again, we don't quite know how to do this, but on GRH. So this actually is what Matt and I proved, given asymptotic. So this asymptotic is, it doesn't verify the actual polynomial that you're supposed to see, it only provides the leading order term, the first term. And you can't say anything more. That's a lost guess, okay. Now, another kind of results which have been done recently is work of Conry, Evaniates and myself. And this is a question on what happens if you average over q as well. So you might, you would like to prove something over where one can see this kind of 42 that appears in the moment conjectures. And we would like to see that this is 42 times something. So, so there are q characters of an average over q. There will be q squared characters. And then the power of log should be log to the nine. And so this is to account for the a k's or a 3's average out in some sense. And we would like to be able to prove exactly this result as stated. We don't quite do that. But we do something pretty close to this. So, let me replace this by, this is the completed L function again. That's half plus i y. And then I need to introduce a little integral here with respect to y. And then one can prove, sorry. Without any weight, if you like. Because it's a completed L function. So the completed L function, the gamma factor will decrease exponentially as y gets large. So you should think of this as a small average of the L function in an interval of length one. But, and then this we can verify an asymptotic formula for this. In fact, what we prove is something more precise. We actually get the full asymptotic expansion that's conjectured by Keating, by Conry Farmer, Keating Rubenstein, and Snaith with an error term, which is something like q to the 1.9. So you get a power saving in the remainder term here. And then there is very nice work of Chandi and Lee from maybe a couple of years back, where they prove the analog of this with the eighth moment. And the main thing is that they see the 24,024 that was conjectured. So but in this result, you don't get a power saving in the remainder term. This is really just checking the asymptotic. You don't know the full asymptotic expansion in this situation. And also, this is on GRH. But maybe this is the largest moment for which we can check these moment conjectures so far, okay. Now, so those are the special results that are known. And what I'll do in the next couple of lectures is to discuss failing asymptotics. We might want to get good upper and lower bounds for these moments. So I'll discuss now lower bounds in the next lectures. I'll discuss upper bounds for these moments. So the rough maybe the say one line summary of this is that is that we can get the right upper bounds, right lower bounds for essentially any family in which we can do something. So whenever we can compute some moment. So usually you would like to compute this maybe the smallest moment that you can. The first moment, like if you like, let's say the first moment plus epsilon. Which I'll explain what this means. That you should be able to put a little bit of a Dirichlet polynomial as well. And also be able to compute the moment. Then this will imply the correct low bounds apart from constant for all larger moments, okay. So let me try to explain how this works. And this is something which was developed by Rudnik and me and with the recent extension by Maxine. For GV and me. So results like this for the zeta function are classical. They go back to maybe to titch march. That you can get a lower bound for zeta half plus IT to the 2K. Which is of the right order of magnitude. And you can think of this as just being an application of Bessel's inequality. So you can look at zeta of S to the K. It becomes a complicated object when K is large. But at least you can understand how it's in a product with functions like N to the IT. You can understand this for small values of K, for small n, up to t to the 1 minus epsilon. And then you can use Bessel's inequality to say that if you understand these inner products then the mean square of zeta of S to the K is bounded below by basically the averages of these mean squares. But this proof doesn't by itself extend nicely to families of L functions. And you can also ask what happens when K is not a natural number. And here there was work by Ramachandra and Heath Brown who got the same lower bound whenever K is a positive rational number. So you can ask if you have the bound for all rational numbers, why don't you have it for all real numbers? But the reason is that the bound that they obtain will depend upon the height of the rational number K. So the constant that goes in here will depend upon K. But it doesn't depend just on K, it depends on the height of K. So you have a slightly weaker result for all real numbers K. So now let me first describe this work with Rudnik. And let me illustrate it by first giving you a proof that if I want to look at the sum of L half Chi D to the K, D up to X. So let me just say first for all K which are natural numbers. But then I'll extend it to rationals and the rationals in a little bit. That I can get here a lower bound of the form X log X to the K times K plus 1 over 2. So the main thing that we're going to use in the proof is that, okay. So I said that if I can compute the first moment plus epsilon, then I'd be able to produce results of this type. So let me illustrate what I mean by that. I would like to be able to compute the sum of the first moment L half Chi D, say fundamental discriminant, and twisted by some number Chi D L, let's say. And L is some number which I'd like to be some small power of X. Maybe L goes up to X to the 1 over 100 or something. You can do much better, but let's start with this. So how should I estimate these? Well, first you can start with some kind of with a relation for the L function L half Chi D, which is called the approximate functional equation. But one can, there are many exact versions of this. So there's nothing approximate about it, except the name. Which says that you can essentially understand this L function by two sums, which go up to roughly square root of the conductor. So the sum goes up to basically square root of X Chi D L N over root N. And there are basically two sums of this. There would be the sign of the functional equation involved. But for quadratic characters, the sign is always one, so I can write it as just two copies of this kind of object. So, and this should not really be root X, it should be like root D over pi or something like that. But let's not worry about that. And so if you put that in here and exchange sums, so you get that this moment is the sum over D up to X. And then a sum over N up to root X. So let me write this as D over N, and then there's an L over N. So now I would like to take the sum over N outside and bring the sum over over D inside, okay? Sorry, what are they right here? D over L, okay? So I take the sum over N outside and bring the sum over D inside. So I would like to understand something like twice the sum over N up to root X. And then a sum over D up to X, D over L. So now I can view, so instead of just viewing this as a sum over character's chi D, by reciprocity I can view this as a character sum to the modulus N times L. And N is fairly small, it's only of size root X. L is fairly small of size only some small power of X. So it's a character sum where D is pretty large and this NL is pretty small. So I can really understand character sums of that type quite easily. By even something as simple as Poliovin-Ugradoff. So of course, the larger the moment, the more complicated your argument to handle this should be, would be. But in this case, even a simple Poliovin-Ugradoff inequality argument would say that, would imply that I only have to worry about the terms, where this is the principal character that gives you a main term. Everything else gives you an error term. So you only have to look at the terms where N times L is a square, is the main term, and everything else gives you remain the terms. Okay, so what would that be? If I write L as, let's say, L1 times L2 squared, where L1 is square free, then N has to be L1 times some other number M squared. And so you can now understand this fully. The answer should be, so the answer should be, apart from some constants, 1 over square root of L1. And then a sum of 1 over M in some appropriate range gives me a log of root x over L1. So that's the answer for this moment. And you can see that we can evaluate this quite easily with a very good main term and a good remainder term, at least for small values of L. Okay. So now we can use this to get bound for all the moments, as follows. So I will look at the sum of L half chi D. And then try to think of something which looks like an approximation to this, which I'll call a of chi D to the k minus 1. Let's say k is an integer. And a of chi D is going to be a truncation to something that I can think of as being an approximation to the L function, but not quite. So this little x, if it were like root x, then it would be really a good approximation to the L function. But I would like to choose it to be some small power of x, maybe x to the 1 over 100 k. So if I choose that, then if I expand A of chi D to some large power, that's still going to be a short Dirichlet polynomial that I have here. And so I should be able to evaluate all of this, all of this object. So I should be able to evaluate this. So I'll finish this calculation after lunch. So we can evaluate this. And then on the other hand, I also get just by using Holder's inequality, that this is bounded by L half chi D to the k, to the 1 over k. And then the sum of A of chi D to the k, to the 1 over k minus 1 over k. And I can evaluate this too, because this is just a short Dirichlet polynomial that I have to evaluate. So this can also be evaluated, okay. And then of course, if you compare those two, then you get a lower bound for the kth moment of L half chi D. And this moment, you don't lose anything in this argument, as I'll show. And you get the right lower bound. So let me stop maybe here and then I'll continue, I'll finish off this proof. Okay, are there any questions? Do you have a heuristic for fractional moments? A heuristic for fractional, what? The heuristic is CKR at best. Yeah, I think they don't have a good heuristic for fractional moments, I think, based on this extrapolation. Right. But so for example, I think it will not be a polynomial, right? So we don't, yeah, I think it's not a good, yeah. I don't think anyone has a good idea of what the conjecture should be there. Something that obtains a power saving of, so they conjecture that CFKRS for these integer moments also conducted that there should be an error term of size square root of T or square root of X or whatever. And this conjecture may or may not be true. But there's no even plausible conjecture for fractional moments of that. Thank you again. All right.


This paper proposes a dynamic right-of-way priority control approach, PDRE versus, for emergency vehicles to improve their efficiency on basic road sections and cities. PDRE versus uses a movable physical function area in front of the emergency vehicles to prevent connected vehicles from entering a lane or to request them to change lanes to avoid a collision. Additionally, a dynamic monitoring area is set up in front of the emergency vehicles to monitor the connected vehicles headway distribution in the inner lane. When connected vehicles enter the buffer area, a lane change request is sent and the traversal search method predicts the optimal time and rate of speed to change lanes and guide the connected vehicles ahead of the emergency vehicles to merge into the target gap. Simulations with the Sumo platform showed that the priority of the PDRE versus reduced the average delay of the emergency vehicles by over 70% under certain conditions without significantly increasing the delay of the connected vehicles. Furthermore, the simulations demonstrated that the long buffer area was suitable for low volume conditions whereas the short buffer area was suitable for high volume conditions. This article was offered by Jirui Kuang, Xiyan Zhao, and Lei Feng. We are article.tv, links in the description below.


The following is a presentation of TFNN, The Trader's Edge with Steve Rhodes, all now toll-free at 1-877-927-6648 or internationally at<PHONE_NUMBER>. The Trader's Edge, now Steve Rhodes. Good afternoon folks, welcome to the August 1st, the magnificent Monday edition of today's Trader's Edge show. I'm your host, Steve, Perseverance Rhodes, who absolutely knows and each of us should always be pioneers of our future versus prisoners of our past. Hope everyone out there is having a great day. Let's make sure we have an extraordinary one and the easiest way to do that, well it's to always remember that life is happening for us, not to us. That's right, when you and I make that one little two by four shift it means we can find a gift in every set of circumstance that life is going to toss at us. Now today you and I are going to check on the circumstance of these markets. We'll go figure out what those bulls and bears, what those buyers and sellers are communicating to you and I just past one o'clock in the afternoon. Do want you to know I'm actually grateful for your presence here. You're more important than that and that's this. During this next 60 minutes I'm here to serve you. So feel free to pick up that phone. Would love to hear from you at<PHONE_NUMBER>. Now if you can't call him we've got you covered there too. You can always send me an email, send that to Steve at TFNN.com inside the subject heading police put radio show question. Of course inside our Tiger's Denimal, any and every ping we'll do. So let's go ahead and get this show started on Magnificent Monday of course this is Tiger, financial news network. I'm Steve Rhodes welcome to last show and I got a slightly mixed bag out here that mixed comes from the S&P off four points in New York Stock Exchange zone 33. The Wilshire off 23. Otherwise you got the other markets the upside. That was up 32, 110th percent, NASDAQ 123. About two tenths percent, three tenths for the Russell. Six points out there, three tenths for the Summai's. That's nine points the upside. Tremors are basically flat out there. Gold's up six bucks, that's three tenths of a percent. Silver is up eight tenths of a percent, that is 16 pennies. The place we grew back nearly five percent off four dollars and forty four cents. Natural gas is back nine pennies and the 30 Treasury trade not at one forty four seventeen at seventeen ticks to the upside. Lead the charge, dollar wise the upside. You've got Liberty Trip advisor holdings of 55 bucks, ran at forty three percent, AMTD digital of 32 bucks, eight percent, Aspary automotive group of six percent, ten bucks, E-Pam systems up ten bucks to three percent, Boeing up about 11 bucks, six and a half percent to the upside. The downside is booking holdings of 42 bucks or two percent. Chipotle's down 14 or nine tenths percent by a pioneer natural resources up ten. Four percent, Aeon PLC is up eight seventy. That's three percent to the downside. So we've got some movers and we have some shakers out there. Let's do this here. Let's go take a look at our NQ chart. See if we can figure out what might be going on in a moment. We'll trade, we'll switch over to those screens. White background charts, multi time frame set of charts. We'll look for the NQ out here. So as we take a look at those sets of charts, daily time frame, although I can't draw it in, there is an A to B equal CD to the upside. On the next price projection level for the NQ out here would be 12382. So it's not shown on the screen out there, but that would be the one to 1.618 A to B equal CD. So that's the price target. The only thing it could get in the way of that would be some type of bearish reversal candle. If we got that at day's end, because we have an A to B equal CD, that's above 1.272 expansion level, we would have a confirmed sell the D point. Five-hour chart is saying, yeah, I don't see anything just yet. Four-hour chart says the same thing. The two-hour chart is potentially by 2 p.m. Could potentially confirm erosement to indicator top. In the case of the 60-minute time frame chart, it has already done that. And price is taking on its first level of support. Now it would be the top of its profile. The price closed below this area, which really where it's trading right now around the 12,986 range. The price closed below that, then we're looking at a move likely to 12,831 to 12,876. 30-minute time frame chart out here has got what? Just make sure it's populated properly. It's got nothing. Road's been to indicator top on the 15-minute chart. Says the price closed below 12,960. Look for 12,851 to be a target level out there. So we've got some short-term topping signals. No levels of support that have failed just yet. But we'll want to certainly watch those throughout the day. We've got turning signals on the short-term time frame charts. That takes us up to the two-hour. 240, not so much. But we might get that. Now the 240-minute chart, the bar that I'm currently in, completes at, also completes at 2 p.m. So you've got a little bearish shooting star, but it may be way too early. It is way too early to call. That's only 111 in the afternoon. We've got to wait for the next 49 minutes to play out before we take a look back there. But I wouldn't look at that because if we're going to see a turn in the marketplace, then we will see it begin on the short-term time frame charts. So the short-term time frame charts, most of them, not all of them, have a topping pattern that is in place. Just now we're watching the support levels out there. So that's what's going on inside of the NQ. Can't imagine it's much different elsewhere. Let's take a quick peek and we'll go take a look at the ES mini. See if there's any other signals out here. Let's take just a few moments. Now, it also has a daily A to B equal CD pattern that's underway. Price is really targeting, it's trading near, it's trading near, it's 1 to 1.272 A to B equal CD, which is a $41.19. Price is trading at $41.25. Next upside price target is $42.27. Again, that is short of some type of bearish reversal candle forming for the daily time frame. Intraday charts, nothing on the 5-hour. 240-minute has a TD9 count top. 2-hour chart by 2 o'clock may confirm a roadsman to indicator top. 60-minute has already done just that. 30-minute, not so much. And so we have really similar patterns out here. I would say that the key level to be watching the ES mini should be 4081. 4081 is the TD9 count breakout level for the 120-minute time frame chart. Now, we're trading at 41.25 out here right now. So that's another, what is that, about 40 points or so give or take out there to the downside. But that would be the level of support. And if that level fails, then we're likely headed lower out there. In fact, that could also generate a bearish reversal candle. So I certainly want to watch that going throughout the day. Now, each of the equity future contracts, I'll just change screens here momentarily so you can see the patterns. Each of them have A to B equal CD patterns that are underway. And they're at various different levels. Again, here's the ES mini upper left, 1.2, 1 to 1.272 area. Same pretty much for the NQ. Well, same pretty much for the Dow. And the Russell 2000 is above it. It being 1876. Likely its next target or destination is 1930. That's a 1 to 1.618. A to B equal CD. Now, this is something I just noticed, wasn't here this morning. But it is now, and that is that a new daily profile is attempting to form an S below price for the Russell 2000. As long as price stays above 1851, 66. And we won't have a confirmation on this profile until this evening. But right now, let's assume that it does hold. That's a bullish indication. And that suggests that the Russell 2000 wants to head to higher ground 1930. Now, the interesting thing here is if we take a look at, we took a look at the NQ and then the ES mini. The reason why I was focused on the NQ is because truly it's the only one that is sitting at weekly resistance. Weekly resistance being the top of its weekly profile, 13001. Price close above that on Friday. Well, that would be suggesting that we have a change in trend underway. So we're not there. It's just Monday.  Friday morning. So we'll need to record Friday show between eight and nine in the morning. Steve Rhodes with TFNN. We'll be right back. Now, let's take a look at Tesla for one of our dinners. That's Rossi. With booming inflation, we have purchasing powers eroded. There's no better place to protect your hard earned money than ain't gold. This is the gold flagship asset is the Monk Todd Gold Project in the Northern Territory of Australia. This is Australia's largest undeveloped gold project. We are talking a world class gold project in a tail one mining district. This is a large scale, low cost project with significant existing infrastructure in a politically safe and friendly mining jurisdiction. This is the gold just completed the Mount Todd feasibility study, which resulted in a 7 million ounce gold reserve and a 16 year mine life. All of this combined with the approvals of all major operational as well as environmental permits. This distinguishes Mount Todd as an attractive, devious party, ready development stage gold project. This is the gold trades on the New York Stock Exchange under the symbol VGZ. Steve Rhodes started his trading career as a student almost 20 years ago and the student has now become the master. Steve won the prestigious Timer of the Year award in 2018 and barely missed that mark again in 2019, finishing at number two for the year, an amazing accomplishment. Steve Rhodes is committed to sharing his techniques and knowledge with anyone who wants to learn and he shares his vast amount of trading knowledge every day in his mastering probability newsletter. Steve's award winning newsletter, Mastering Probability, is delivered every trading day with updates throughout the afternoon. Sign up for Steve's market newsletter, Mastering Probability, and you'll receive access to seven of Steve's educational webinars absolutely free. At TFNN, all our newsletters come with a 30 day money back guarantee so you have absolutely nothing to worry about. Visit TFNN.com and try Mastering Probability for 30 days risk-free today, TFNN Educating Investors. Are you looking for a way to consistently add winning trades to your portfolio? Tom O'Brien is here to help. Tom O'Brien has been successfully trading markets for over 30 years. A frequent contributor to TD Ameritrade Network and CNBC, Tom O'Brien founded TFNN over 20 years ago to help educate investors just like you. Tom's Daily Market Newsletter, Market Insights, is published every morning when the market's open to give you the competitive informational edge you need to succeed. These newsletters are packed full of Tom's advanced technical analysis and are geared to deliver comprehensive strategies for a successful portfolio. Get Tom O'Brien's newsletter, Market Insights today and try all of our products and newsletters 30 days risk-free with our money back guarantee at TFNN.com, TFNN Educating Investors. Toll free at 1-877-927-6648, internationally at<PHONE_NUMBER>. Good back up, folks. We've got the charts for Tesla up on our screen right now. That is for Rossi inside the Tiger's Den. Rossi went short at about 930 bucks out here. So he's looking for some information as to what the charts are telling us. So the very first thing that we should notice out here, Rossi, is resistance. Where is resistance? So the weekly timeframe, we can see that price is trading right into the top of that profile. That profile reading, at top of that profile is 90356. We're trading right now at 897 out there. So you'd certainly love to see that hold as resistance out there. If you see a close above that, it's a weekly close. So it really doesn't matter what it does on Monday or Tuesday or Wednesday or Thursday. Even though it might matter to you, it's not going to matter really until Friday. But right now you're off to a good start and you'd like to see that resistance level hold. We don't have a resistance area that's up in this range here for either the monthly or the daily. So you want to really kind of keep yourself focused there. If price does close above that, what that suggests is, and I'm looking at the monthly timeframe chart and then the weekly is a run to either 969 or the 1092 level. So you don't want to watch that area of that nine, what did I say it was nine? Did it hear momentary, 90356 out there. If we look at the daily timeframe chart, again, no pattern out here to identify a top. We'd have what looks like a bearish shooting star candle that might form today, but too early to make that call. If I look at the intraday charts, you've got a TD nine count pattern on 130. That says you're looking for, the first area of support to fail would be around 873. Now it's going to be less, that's the oscillator and change line, but as price moves up and down, that number changes. So use that as a guideline. If price were to close below that, then you're looking for run to 831, 808 or 786. You have a TD nine count on the 65 minute timeframe. Price is sitting at support as we speak. If price closes below this level, we're trading at 894. Price closes below 894. That suggests you could see Tesla pull back to 835.30. That's where it would likely pull back to. That's the 65 minute TD nine count breakout level. The 30 minute chart has a TD nine count top. That says with price right now, below the top of its profile, a run for the 858.99 to 871 to 877 area is the range where price will likely target. That's the 30 minute chart out there. That's all that I see inside of Tesla. So I do hope that helps you out. The daily support level would be around 809.07. That is its green oscillator and change line. If price pulls back, Tesla rejects that green oscillator and change line, then it should be off to the races to the upside. So thanks so much for the request, Rossi. Do again, hope that helps you out. Next question coming in from David H in Tom Ball, Texas out there. And he wants to take a look at a Niko Eagle. So let's get those charts out here. I think I might have those up on the screen. Oh, Michael, yeah, we do. Okay. Let's read David's question. It goes like this. Is a Niko Eagle in an A to B equal city down on the monthly timeframe chart? If so, what would price need to do to negate the A to B equal city down? And two, what are the support areas on the way down that could stop price from achieving the one to one A to B equal city price target? If you have time, can we take a look at LNG, Chinear Energy? Yeah, absolutely. So the question is, we're going to go change over because I can draw the A to B equal city patterns on the other charts. So we're going to go over to the black background chart. What we'll do here is we'll get the Niko Eagle going. AEM is the ticker symbol. And the very first thing we're going to do is focus on David's question about the monthly timeframe chart. So it's simply expand this out. In essence, what David is asking me to do is to draw an A to B equal city pattern. So the first one that it would be drawn in there would be with the high out here from the trading week or month of September, 2020. Then what I would use for a B point would be the low from March of 2021. The C point would be the retracement up to May of 2021. It was a 58% retracement that typically leads to more than a one to one A to B equal city. But look how price kind of fanned out on the right side of the C to D leg out there. So it said it was really losing energy or steam. What price actually has done, and it did this last month, is it that tag the one to one area. Now the one to one area is 39.93, price got below that. Last month price got down to a low of 38.02. What this does not have on a monthly basis is a bullish reversal candle to confirm the buy the D point. But you were asking, so what needs to be done to negate the A to B equal city pattern? Actually, nothing needs to be done to negate it. It needs to be confirmed. So on a monthly basis, you would be looking for a bullish reversal candle. Are we gonna get that? Well, I don't know, but it is one possibility. If we go take a look at the weekly timeframe chart out there, now it's possible that what David was looking for was an A to B equal city that looks like this. And I can't rule it out. The A point would be the same, but the B point would be all the way down at the lows from January of 2022. And then the C point would be the retracement up into April the first. Well, that didn't work out real well. We gotta try to draw that in there once again. Do what I'll do is, yeah, I just deleted both of them. Let's start here again. So this might be the A to B equal city pattern that David is talking about. And that would get us down into the 2333 level out there. Now, from a monthly standpoint, the month of January at 49 million shares, and last month we did 82 million shares. So that is certainly in play. Both the A to B equal city patterns are in play. But I like the first one that we looked at in the monthly basis. And the reason I like that is because that actually confirms the weekly timeframe chart. If we look at the weekly timeframe chart, it did confirm A by the D point pattern. So the A point again, being the high from September, 2020, the low being the March, 2021. And then the C point being the run-up into May of 2021. There's your one-to-one level. What we can see here is you've got a bullish piercing candle. And so you do have a confirmed by the D point on the weekly timeframe. That might help out the monthly chart that we were looking at. The weekly chart, because we can see the A to B equal city pattern here, what we can assume is that in fact, the daily timeframe generated that same signal. And what we had here was a gap to the upside. That gap to the upside took place on July 28th. That would have confirmed an A to B equal city pattern to the upside out there. So to summarize, Nico Ego, where do we want to go from here? Well, price is trading above the top of its daily profile. This could be two consecutive sessions in a row for that. If that's the case, that is suggesting price once will move higher. We confirmed that the weekly already has a by the D point pattern. So what that then says is, oh my goodness, well, we're on the white background chart. Is that whole time? Were we on the white background charts that whole time? Give me a moment here. Sorry folks, I thought it had changed over. So here's the black background charts. You've got the, so here's the monthly. You'll see, I got the larger A to B equal city pattern out there. So let's put the smaller one, the original one. The original one, again, they're both in play out here. You got A point up there on September of 2020. The B point is March of 2021. The C point is the week of May. This is the monthly chart, May of 2020. So you can see the one to one level at 39.93 was hit. And again, on the weekly timeframe, if we open this up, you'll see that same one to one A to B equal city. You'll see the bullish reversal candle. That suggests that price should move higher. Now, higher to where? Now we want to go change over to the white background charts. And that's going to be that weekly oscillator and change line. That should become the price target area. Why is that the price target area? Above the top of the daily profile confirmed by the D point pattern for it. The weekly timeframe was confirmed by the D point pattern. The monthly price pulled back to support, basically, which was near its breakout level at 35.12. So you should see Ignico Eagle. David, trade up into around the 46.37 level. We'll come back to this break. We'll go take a look at Sheneer. Energy, ticker symbol there, LNG. If you want to take advantage of this sector, now is the time to subscribe to my Gold Report. The Gold Report is a comprehensive look at the metal sector, as well as the markets that move gold, which is the currency and bond markets. New subscribers get a 30-day money back guarantee so you have nothing to lose. Every Monday morning, I publish the Gold Report with coverage of gold, silver, bonds, the XAU, HUI, GDX, as well as more than 30 different mining equities. To see for yourself the types of profitable trades that are recommended within the Gold Report, sign up now by visiting tfnn.com. Don't miss out on the next great gold trade. Sign up today. Tfnn is excited about our new software charting program, the Art of Timing the Trade Chart. In collaboration with Tom O'Brien and using his best-selling book, The Art of Timing the Trade, your ultimate trading mastery system, David White has programmed an outstanding piece of software that will complement any trader's methodology. Using this first-of-its-kind program, the Art of Timing the Trade Chart allows you to scan thousands of stocks for Fibonacci formation setups, including guardleafs, ABCs, butterflies, and much more. The Art of Timing the Trade Chart is designed to help you when scouring the markets for stocks just beginning to form the trading patterns that many investors spend days, weeks, or even months searching to find. And right now, we're offering licenses available at only $79 a month. We are so confident that you're gonna love this new charting software that will even give you a 30-day unconditional money-back guarantee. Don't miss out on this incredible new piece of software. Get your copy of the Art of Timing the Trade Chart today by visiting TFNN.com. Sharpening your skills as an investor is like getting better at playing a musical instrument. You have to practice, sure, but you also need excellent instruction from experts. At TFNN, you'll get advice and guidance from the authority in technical market analysis, and it's not just dry, tedious text either. TFNN airs live financial content streamed live on TFNN.com and TFNN's YouTube channel with Tiger TV, live every market day from 8.30 a.m. to 4.00 p.m. Eastern for free. Each host is an experienced trader and gives their take on the market while taking calls and questions live from around the world. From the moment the market opens until the closing bell sounds, Tiger TV has eight different shows with expert hosts to help you make the right moves with your money. Watch online at TFNN.com or on TFNN's YouTube channel and become the investor you were born to be. TFNN Educating Investors. This segment is brought to you by Think or Swim. For more information, just click the Think or Swim banner on the front page of TFNN.com. Welcome back, folks. U.S. equities turn to the downside. The only one trading in the positive is the Russell. That is up 1.00058, S&P is down 15 as we speak. So let's go to our next question, where I really follow up from David, which was a take we get to Cheneer Energy. LNG is the ticker symbol. A specific question is, does this stock have enough strength to take out the highs? Well, if we take a look at the weekly timeframe chart, we're gonna see here is a nice consolidation pattern. The first time up on the weekly chart, March of 2022, nice wide-ranging bar, volume 13.4 million. Next time price was up here, the week that began May 2nd, 13.6 million shares, same volume. Last week, price tested in the top and rejecting the top of its consolidation, a test with 13.5 million. So got the consolidation in place. The tops are being tested with similar type volumes out there. I guess the answer to that question is, David, doesn't appear that it has that strength to do that, but price is above the top of its weekly profile, above the top of its monthly profile. So that is a positive out there. You'll see a new daily profile has formed. So your resistance level there is 148.61 in support being down at 138.61. So it doesn't look like, from a volume perspective, that it has the energy to push through the top of this. But this is gonna be tied to natural gas as well, and that is likely headed higher. But it's dealing with resistance as well out there. So the charts are somewhat similar. So I do hope that helps you out. Thanks for taking the time to write in. Again, there was David in Tombow, Texas. Our next question, the only one that I see that is in is coming from Kota, inside our Tigers Inn. Kota wants to take a look at the GDX. So let's punch up the GDX. Let's do that on the white background screens. Let's go ahead and change over to them. Give me a moment, we'll get those up here. And so with regard to the GDX, says, please look at the GDX next to life, you get a chance. We're gonna do just that. So in the case of the GDX, here's what we know. We know that it's a tough call. You don't have a bottoming pattern for the daily, the weekly, or the monthly timeframe. Price right now is consolidated with inside it's daily profile. It's a bullish structure daily profile. A close above 26 bucks is gonna suggest that price should make a run for 27.34 to 27.61. You don't have a completed buy the D point pattern. That's the A to B equal seeding to the downside inside the GDX. So even though you got a bullish reversal candle for the month of July, it didn't complete any kind of a pattern out there. So that's kind of a difficult one. And on the monthly timeframe, no bottom signal as well either there. So right now, now look, these patterns that you and I take a look at a road's momentum indicator bottom, the CD9 count, wave number seven. There is a wave seven pattern on the daily timeframe for the GDX. That's the only pattern that would be out there. But these patterns that you and I take a look at are not present at every bottom. It's just when they are or top. It's just when they are present, then we pay attention to them because they are signaling to us some important information out here. But nonetheless, price may be targeting at 27.34 to 27.61 and the close above 27.61 and the GDX suggests a further move higher. So hope that helps you out, Kota. Your next request was to take a look at the industrial sector, XLI. So let's get that fired up on the screen out here. Excuse me. So in the case of the industrial sector, today is going to complete a TD9 count top. Code, I don't know what today's high will end up being. The current high is a 96.04. Let's assume for our discussion that price does not take out that high. You should see a short-term top form. You can also see that oscillator and change line change colors two bars ago. Price and oscillator and change line have a date with each other. I don't know exactly where that's gonna take place. It should be somewhere south of 95.44 and north of 90.95 out there. Well, we'll take things one step at a time. On a weekly timeframe for the industrial sector, you've got a rogment to indicator bottom that's been triggered, but no bullish reversal candles. Therefore I can't say it's a confirmed bottom out there, but with price above its oscillator and change line, if the TD9 count top fails on the daily timeframe, Kota, the next level of resistance you'll see is 96.75, the price can clear that. Then you're looking at around a 97.09 or 98.59. The monthly timeframe actually does have a completed by the D point pattern, so that's interesting out here. That's gonna look like this. It's just drawing the A to B level out here and then we can just move that line to the C to D level, so it's about there. We move that over. You'll see that was more than a one to one A to B equal CD. You got the bullish engulfing candle last month. So you do have a confirmed bottom on the monthly chart. So, what you've got that daily TD9 count top. Hey, sometimes these patterns fail, right? So watch today's high, whatever that is. If there's a close above that, still know that you've got resistance that needs to be taken out at 96.75 out there. So I hope that helps you out Kota with regard to the industrial sector, as well as the GDX. Let's go to our first caller and that is Jim in Palm Harbor. Jim, thanks for calling. Thanks for holding, how are you today? I'm great, Steve, how are you doing? Excellent, nice to hear your voice. How's your summer going?  Yeah, I'm sorry to hear about that. That is the second K9, 17 years old I've heard in the last two days out there. So I don't know, 17 is a magic number out there but my condolences to you for that. Oh, well, that's okay, it's part of life. Yes, yes it is. So you want to take a look at Acoma? Acoma? Yeah. Go right ahead. I was looking at it, it looks like it might have bottomed on a weekly chart and it's got a, I've got a TD2 up on a weekly chart and if you go to a days, it looks like it might have bottomed. I've got a TD3 up there and the Fibonacci, it's on the weeklies, it's above the 23.6 and got a ways to run for the 382 and 50% that's 1.8. I just want to get your thoughts on a swing trade, really. Okay, so let's do this here. So I've got the white background charts, we might just change over to the black ones in a moment but if we take a look at the white background charts out there, we'll see a nice confirmed approximate indicator bottom that formed on July the 15th out there. So that was your bottom signal. Price was able to take out the old profile, that was, took that out the tie, the resistance level on July 21st. A new profile formed a few days later, that was on July 25th. Price right now is trading above the top of that profile. The top of that profile is measured out at 95.47. You close above it on Friday, you'd love to see a second close the day above that and that would suggest a further move higher. Now further move higher to where? That really takes us to the weekly timeframe charts. So you had indicated that you saw some type of bottom pattern out there. I don't know that I've got one of those but that doesn't matter, you've got one on the daily, that was very clear. Yeah, don't really have a signal on the monthly time, on the weekly timeframe chart that I'm looking at but here's what we do know, price above it's red, acid, and change line. Here's what we also know, for a series of weeks, price had been below its weekly profile. So your first level resistance, we already talked about the daily timeframe which we're trading right now at the top of its profile, 95.47, turns out that the bottom of the weekly profile is also resistance. We can see that two weeks ago, Price tested and rejected it. Last week did the same. By the way, that level out there is at 96.69, 96.69. So that's the area where the battle is taken and price can get above that. Then you're looking at moving to 99.64. 99.09 is the bottom of its monthly profile. So let me step back for a moment. First thing you wanna see is a close above the top of the daily profile. That's at the 95.47 level. If we get that, Jim, price should make a run to 96.69. If price get above that, we're looking at a run to 99.09. If price get above that, then we're looking at a run to 99.64. Now, we're about to go to break here and I don't wanna just leave you like that in case you have further questions. So stay on the line, if you would. And we'll come back to Jim and Palm Harbor and discuss Akamai Technologies, aka AM. TFNN has been your trusted source of analysis for bonds, metals, stocks, commodities, and options for years. And we are happy to announce that we're bringing that same caliber of analysis for the Forex Market. Teddy Keckstad has 30 plus years of experience in Forex trading, commodity risk management, Forex hedging, volatility, and so much more. Teddy releases his weekly Tiger Forex report every Monday morning with elite coverage of all major currency pairs, including the DXY, Eurodollar, Pounddollar, Aussie Dollar, Dollar Yen, Dollar Swizz Frank, and so much more. Teddy will recommend specific trades when the market presents them and provide updates throughout the week when warranted. For the month of July, inaugural members to the Tiger Forex report will receive 25% off the monthly subscription for as long as they're subscribed. Just use promo code Teddy25 to lock in the added savings. This offer is good only for the month of July, so do not miss your opportunity to save on the Tiger Forex report. TFNN, educating investors. The technology around us is changing every day. With so much happening, it can seem impossible to keep up with all the information. David White's investment newsletter, The Technology Insider, is designed to give you all the information you need to understand the technology that shapes today's markets and tomorrow's future. David White has made his living staying on the cutting edge of technology. His weekly newsletter will give you specific recommendations for valued tech stocks, as well as entry prices, target prices, and stops to set for each trade. Dave delivers his weekly newsletters every Friday with updates throughout the week. You can get The Technology Insider at tfnn.com for only $37.50. Sign up for David's newsletter, The Technology Insider, and get an inside look at everything the technology sector has to offer. Try it risk-free today with our 30-day money-back guarantee. TFNN, educating investors. Biotech is booming, but for how long? Whether you think Biotech Bull has room to run, or has run its course, trade LABU or LABD, directions daily S&P Biotech three times bull and bear ETFs. Visit directioninvestments.com slash biotech today. An investor should consider the investment objectives, risks, charges, and expenses of the direction shares carefully before investing. The prospectus and summary prospectus contain this and other information about direction shares. To obtain a prospectus or summary prospectus, please contact direction shares at<PHONE_NUMBER>. The prospectus or summary prospectus should be read carefully before investing. An investment in the funds is subject to risk including the possible loss of principal. The funds are designed to be utilized only by sophisticated investors such as traders and active investors. Distributor, Four Side Fund Services, LLC. This program is brought to you by Vista Gold, traded on the NYSE American and TSX under the symbol VGZ. Welcome back folks. You got the down off 80 points, the S&Ps down 19 NASDAQ 100 off 33. Let's go back to Jim in Palm Harbor. We're taking a look at Akamai Technologies, aka M is a ticker symbol. So Jim, the information I shared with you, do you have some further questions or anything else that I can assist with out there? Well, I just saw, you know, when you mentioned monthly, on the monthly I showed a hammer candle for the month of July and it looks like it wants to go up, but it's only the first day of the month for August, so. But and then if I drew a FIBO line down to the time to mark four down with that hammer candle and if it looks like to me that that was the indication of a bottom on the monthly and I don't know if it's a true hammer candle, but it looks like it. So if your candle looks like mine, which it should, but maybe you've got some other type of data feed, it is not a hammer candle, but that's it. You know, if it looks like mine, then the answer is no. And there's really two reasons for that, Jim. The first reason is that the wick of a hammer candle needs to be twice the size of the body of the candle itself. And so there's clearly that is not the case here. And then the second is there can only really be a very short upper shadow or upper wick to a hammer candle out there. So this just is that it is not a bullish reversal candle on a monthly timeframe out there. So again, if yours looks like mine, it acts technically speaking, it is not a hammer candle out there. What doesn't change our view with regard to where prices trade in relation to resistance, the top of that daily profile. And then we're resistances on the weekly profile. And then if price moves higher, where resistance is that, which is the bottom of the profile for the monthly timeframe. Is that what I'm sharing with you, as far as a hammer candle versus not a hammer candle, is that making sense to you? Yeah, that's why I said I wasn't sure if it's a true hammer candle. Now, my top wick is very short, but at the bottom one is not twice the length of the body. So. Yeah, yeah. It just doesn't fit the classic definition of that. And here at Tf&M, here at Tf&M, we don't change our definitions. It's a hammer candle, thank you very much. No problem. Jim, always good to speak to you again. My apologies and sorry about your pet out there. And hopefully it's blue skies from this point forward for you. So we'll hopefully speak to you again soon. You bet. That was Jim. You bet. That was Jim in Palm Harbor. So I don't have any other questions at this moment here. So let's go take a look at gold. You mentioned that during the, okay, we've got a couple out here for Intel. Let's go take a look at gold first anyways out here. See what Goldilocks is doing. So we take a look at that monthly, or the daily timeframe out there. Today is bar number eight of a TD9 count. That says we shouldn't expect and anticipate a short-term top for Goldilocks to form between today and Wednesday out there. Remember, the high of that pattern needs to form on bars eight, nine, the bar following nine. So it looks like that is going to come to fruition. We just don't know which day is going to be the high of that pattern. Sometimes we can look at the intraday charts and make that determination. Well, on a five-hour timeframe chart, we don't have any kind of topping pattern. On a four-hour timeframe chart, we do not have a topping signal. On a two-hour chart, you do have a confirmed erosement to indicator top. You've got the same in essence on the 60-minute, nothing on the 30. I'm not going to worry too much about the 15 and the 10 out there. So what Gold has a couple of short-term tops it needs to break through support if that is going to mean anything. Support would be between 1775-10 and 1779. So those would be the levels to be watching to the downside out there. If there was another downside level, it would probably be 1762. So those would be areas of support, potential support for Goldilocks. Back to the next question that came in, which was a take-lead of Intel. So let's go back to our three timeframe charts out there. INTC is the ticker symbol. Let's go see what's out with earnings on Thursday or Friday. I can't recall which day. I do recall that it had gapped down or it traded lower. But let's go, yeah. So it did gap down out here and what it was doing was it was testing that prior swing point. That prior swing point was from July 5th. That swing point has volume of 33 million shares. And even though we got a test and rejection of that swing point on Friday, it was with 125 million shares. So not exactly the test and rejection that you're looking for. That suggests that we should see that candle tested again. Well, that's taking place today. Now today's volume is 33 million shares. But again, that's going against 33 million shares on the trading day of July 5th. So it's still too much volume to the downside for this to be a rejection of that swing point. It doesn't mean it can't reject it and move higher. You've got resistance at 37.85 and 38.51 on any move higher. If you look at the weekly timeframe chart here for Intel, I don't have a bottom signal. And we can see that today's move higher ran right into resistance. Co into that was at its red oscillator and change line. So that's a level you'd certainly want to see price close above to suggest that there's some rally or counter trend movement, which could or should take price to the 4024 level. Nothing on the monthly timeframe to show us a bottom, although price did pull back to its breakout level. That breakout level was established at 34.93. So another price got all the way down there was pretty close at 35.24. So prices pulled back and we've got no bottom signal. And in fact, we have a negated TD9 count bottom. So Intel is showing some trouble out there. Absolutely. Where's a price target to the downside? If that were to be the question. So for that, what we would do is switch back to my other screen out here, the black background screen. So if you give us a moment, we'll do that. We'll take a look at the longer term picture. That's a monthly timeframe. The A to B equals CDs. What would we draw on there? What fits my eye is really, what I'm looking at is the largest move higher as price was moving down. And that for me, that sticks out as being the day or the week of January of 2022. The week that began January 1st, 2022. So that becomes my C point. Now for me to use that as a C point, I have to use that low as the B point as well. Which I don't really like doing, but it's the only thing that visually fits my eye out there. It fits my eye out there. So here's the A to B equals CD pattern for Intel, which gets us down to about the 3409 area. So that's what I see when I take a look at Intel. Is it a bottom? Do I see some kind of bottom? You got wave number seven Coda on the daily timeframe. What you want to really see there is a light volume test of that swing point. Let's go on to John in New York. John, thanks for calling, thanks for holding. How are you today? Okay, how are you? Excellent, thanks for asking. DSL is what you're calling about. Tell the folks what you're doing, how I can best help you. It's Diana's, Diana's shipping. Double-eyed, you've got solutions, okay. Go ahead. I had a good move this week. Yes? Should I sell some or hold on? Okay, so great question. Here's what I can share with you. What's the chart you were looking at? The black one. So price above the top of the daily profile was bearish in structure. It's been above it for three days. That's a bullet signal. Price above the top of the weekly profile. Did that last week? That was at 12.50. That's a bullet signal out there. Don't have anything that great to report on for the monthly timeframe. So your question is, should you sell it? Well, the only reason to consider lighting up on it or adjusting your stop would be the white timeframe chart that shows that today is bar number eight of a TD9 come. That suggests that DSL should form a short-term top between today and Wednesday. Don't know which day that might be. It doesn't guarantee a short-term top. But do me a favor, John, hold on through this break. We're gonna go to it in about 10 seconds here. We'll come back and try to answer your question. I'll look at some short-term timeframe charts for you as well. So we'll be back with John in New York to finish up a double-line income solutions. VistaGold owns and operates the largest undeveloped gold project in Australia, the Mount Todd Gold Project. VistaGold just completed their feasibility study resulting in a seven million-ounce gold reserve. VistaGold has all major permits approved and has retained CIBC capital market assistance in evaluating alternatives and in completing an accreted transaction. VistaGold trades on the NYSE American and TSX under the ticker symbol VGC. VistaGold executing a strategy to create shareholder value. You might think that if you want to be successful at trading in the stock market, you're going to need a crystal ball. After all, it's impossible to predict the future, right? Like any endeavor in life, before you decide it's impossible, get some advice from the experts. You might find that it's not so impossible after all. For daily market overviews that give you direction on the key indices, selective stocks and commodities, subscribe to the opening call newsletter at tfnn.com. The opening call newsletter is written by Basil Chapman, creator of the trading methodology known as the Chapman Wave. The Chapman Wave up-down sequence gives you an edge in identifying price turns, finding the peaks and valleys in stock prices.  First-time subscribers also get a 30-day money-back guarantee. If you're not satisfied, let us know and you'll get a full refund within 30 days of signing up. tfnn.com, educating investors. Everything in the universe is governed by the Fibonacci sequence. This mathematical principle is responsible for everything from the most aesthetically pleasing artwork to patterns in the stock market. To stay on top of stock patterns you can take advantage of, sign up for the Fibonacci 24-7 newsletter at tfnn.com. When you subscribe, you'll get a weekly report from Veteran Day Trader Larry Pezzavento on stocks you need to pay attention to and you can trust Larry's analysis. After all, he's got 45 years experience as a day trader. Larry will also provide daily charts, videos and data on the key markets that he's tracking. Expect notifications from Larry on market movement you need to act on at any time. First-time subscribers also get a 30-day money-back guarantee. If you're not satisfied, let us know and you'll get a full refund within 30 days of signing up. Subscribe to the Fibonacci 24-7 newsletter today. tfnn.com, educating investors. tfnn has launched the Tiger's Den. Hosted at Discord, tfnn has been educating traders for more than 20 years with live programming hosted by a variety of professional traders during market hours, the Tiger's Den. Available to all tigers and tygruses for just $1 for the year. There's no catch or any costs when you join our community of traders. Sign up today and become a part of this educational community of traders. Just visit the front page of tfnn.com. Don't forget, you can listen to tfnn live on your mobile device 24 hours per day. Go to tfnn.com and hit Watch Tiger TV. That's tfnn.com and hit Watch Tiger TV. Welcome back, folks. So, John, when you entered this trade, can you give me a feel for what your trading plan was? I'm just trying to understand. Go ahead. It's just for the income. It pays a nice dividend, like 10%. Yeah, okay. But was this too, when you entered, is this for a long term? I'm just trying to get a feel for it. I'm retired, so I'm a long-term income. Well, here's what I can share with you. Here's what I would be concerned with if I were you. First of all, understanding what's inside this. And so we take a look at this. The top 10 holdings out here, you say you've got debt from Brazil, debt from Argentina, more Brazilian debt out there. I just get comfortable with regard to the makeup of this portfolio out here. The US dollar index is gonna continue to move higher. I know that it's pulling back today, but longer term, the US dollar index is the flight to quality. And that should put pressure on these other, should create a bunch of sovereign debt defaults that are out there. And so that may roll into this specific holding. So you're dealing with a daily TD9 count top that should form, should complete between today and Wednesday. So you wanna consider that. I would say right now, and I look at short-term timeframe charts, the only one with real signals are the 15 minute, the 30 minute and the 130 minute chart. This suggests to me that price should pull back to test 12.68. If price gets below 12.68, then what we're likely looking at is a further move lower and that could take it to 12.18. So, but first if 12.68 holds, then the TD9 count likely hasn't topped just yet out there. Does that help you out? Yes, it does. Perfect. Do you have time for another question or are you finished? Well, the show is over, but I'll certainly be here tomorrow. If you wanna send me an email, Steve at tfn.com or give a call on the show, happy to look at anything that you'd like. Okay, John. Yes, thank you. You bet. That was John in New York. Have a great day. And that goes for everybody else out there. Stay tuned, you got two more great hours left. Your favorite polar bear, David Whites up next. Tom O'Brien will take us on home and I'll be back with you on terrific Tuesday. You have a marvelous Monday, folks.


Hey, this is Brett with Summit Auto in Fond du Lac, Wisconsin. You're new and used Jeep and Jeep Wrangler headquarters and I am stoked to show you this awesome looking 2021 Jeep Wrangler Rubicon in the limited run color of Gekko Green Clear Coat. This 2021 Jeep Wrangler in Gekko Green has the 3.6 liter Pentastar V6 engine with start-stop, the 8-speed automatic transmission, pumps out 285 horsepower, it's going to get you 24 miles per gallon on the highway, 20 city, and an average of 21 miles per gallon on this Jeep. Once again, Gekko Clear Coat is the color. This is a limited run color for 2021. We were lucky enough to get one of them and I'm super excited to do the video. Any code on it is PFM if you were curious and just a really good looking color and it looks just as good in the picture studios as it does outside and it's really an amazing looking color. They used to have it on the JKs a couple of years I think around 2012 or something around there they had Gekko Green but they brought it back for the JLs and it looks fantastic. We're going to walk around the vehicle. I apologize for the noise that's going on in the background. We're actually building on our standalone Jeep building and that's going to be coming in early 2022 but they're starting construction today. Same day we get this Wrangler so it kind of makes sense. You get the little sandals on that plastic piece there, a little Easter egg for you. You get the Rubicon stickers on the hood. These are a $995 rim option for this Wrangler with the polished aluminum lips. It has BF Goodrich All Terrain TA KO2 tires. These are LT28570R17s and they got a pretty nice all terrain tread pattern to them. This one also has if you notice the color match fender flares as well as the color match hard top which are options on the Wrangler Rubicon. It also has the 3M tape on the back fenders which I really like. The Jeep is thinking about your investment a lot of times on the used end those come in pretty chewed up on the JKs but on the JLs if they're color matched they put the 3M tape. You get the rock rails on here as well. We'll take a look at the original window sticker. Feel free to pause this at any point. I'm not going to go over everything but just some of the main points 2021 Jeep Wrangler Rubicon 3.6L 8 speed automatic transmission. You get the Jeep Wave membership backup camera, the 4 to 1 rock track heavy duty part time 4 wheel drive system, 410 gears, off road plus mode, performance suspension, all that good stuff there. And then the optional equipment the limited edition Gekko Green clear coat is $395 leather trim bucket seats are $1625. The customer preferred package of 25R which gives you the premium sound in 8.4 with nav is a $1995 option. You get the Alpine premium audio, the Uconnect 4C with nav, GPS nav, the Sirius Guardian for a year, Sirius Travelink and Travis Plus for five years and emergency assistance call, 4G LTE Wi-Fi hotspot, safety group is a $995 option and that gives you the backup parking assist system and LED tail lamps. Somebody just asked me about LED tail lamps on these vehicles and the only way to get them or one of the ways to get them is with that safety group. Advanced safety group which gives you automatic high beam headlamp control, full speed forward collision warning plus adaptive cruise control stop and advanced brake assist is $795. You get the 8 speed automatic transmission which is a $2500 option. The 3.6L is a $750 option. Remote proximity keyless entry or the enter and go system is $695 so you just have to have the key fob in your pocket and doors will automatically unlock and it's a push button start. Body color three piece hard top is a $2500 option. You get the storage bag and the rear window defroster and wiper and washer does not have a soft top. The satin black grille by Mopar is an option, it's $325. Body color fender flares are $695 and those special rims are $995. This one does have remote start as well which is a $495 option. Total MSRP is $55, $820 suggested retail price. With the limited availability on vehicles right now the prices are kind of crazy. You might know that we do have this one listed for over MSRP just because of the high demand for brand new vehicles right now and the limited availability. But you can get your summit price what we are asking for this vehicle in the upper right hand part of the screen is a link to our website to this vehicle so click that and check us out there. This one does have the three year 36,000 basic limited bumper to bumper warranty five year 60 powertrain. There your miles per gallon 24 highway 20 city and 21 average. You do have the front crash test rating of four stars and the rollover of three stars. We're going to continue with our outside walk around here and that color match hard top is color matched all the way on top there. So it looks really good. The LED tail lines will turn the vehicle on at the end of the video or at least the lights on and then we'll do a final walk around with all the lights on. You do get those backup parking sensors the red tail hooks the matching spare tire and rim there's your HD backup camera and then you can see that tread pattern a little bit better here on the spare. Back here you have the factory subwoofer there is a little bit of storage but mainly for your roof bolts door hinge bolts and windshield bolts these seats fold down and then they fold up for extra storage or you can just grab this bar and then take that whole seat out completely if you wanted a lot of extra storage and then you can see that all the bars are color matched in here so when you have the hard top off it all matches really nicely and then you can see that you can take these hard top pieces off so there's three bolts on the side two on the top and then there's two quick disconnects to get this off. We actually did a demo on a JL on how to remove your hard top and you can check out that video in the upper right hand part of the screen really in-depth and shows you step-by-step how to remove your hard top without damaging it and it's a really good video I think people have really liked it on YouTube anyways so this closes like so and these doors they've made lighter than the old JKs. You have a wheel base for the two-door four-door overall length water 40 and all that information on that badge right there thing I like about these doors is they close nice and smoothly now they used to kind of have a hinge on them and they didn't always close the best the metal fuel door right here somebody said they were plastic that's not plastic that's metal anyways coming down the side here you have the enter and go system if it's got the little dot here notice you do get blind spot monitoring on here heated mirrors now it doesn't say anything about blind spot monitoring on the window sticker but it does have it and I think that's part of that rear cross path detection system inside the leather trim seats give you the red stitching the red Rubicon stitching on the backrest and that is actual stitching on there if you have your HD capabilities on turn those on driver seat height adjuster lumbar these are still manual seats you do get side airbags right there the back seats are trimmed out the same way you get the latch child safety system for any child car seats there is your bag for that hard the front hardtop pieces you can see when the headrests are down you have a nice view out that window power mirrors power locks will hop inside auto headlamps tilt telescopic steering wheel turn the ignition on so you can check out the instrument cluster and the radio inside here notice you get the cool Rubicon animation at the beginning there and the seven inch LCD display digital speedometer compass outside temperature display and you can change those to read whatever you want you have the leather wrapped steering wheel cruise controls on the right adaptive cruise controls right there so it'll maintain a certain distance between the car in front of you and yourself so if they speed up it'll maintain that distance if you have your cruise set to faster than that or if they pull off in front of you so that you're open it'll go to your normal cruise but it will always maintain the distance that you select on there so that's a really cool safety feature information center controls and Bluetooth controls on the left and you do get audio controls on the back of the steering wheel and I do like that red stitching on the steering wheel as well this is the 8.4 4C radio with the navigation gives you AM FM and serious XM radio capabilities on here you saw the factory navigation remember it gives you serious guardian for a year and traffic and traveling plus for five years that comes with this radio you can also do your backup camera here I like to point out just how nice and crystal clear that backup camera is those grid lines are dynamic meaning that they're going to turn as you turn that steering wheel to kind of show you where you're headed to in the future if you're backing up and then you have your different apps here including your off-road pages which are really cool this one also has they always have some good animation there as well you get your pitch and roll you also have your latitude longitude your altitude you have your different accessory gauges on there and you have your drivetrain right here so it tells you your steering angle so as I turn that you can see that changes and also you know if you're off-road and you're not really sure where you're at as far as your sway bars and axles are being locked and stuff you can go to this page and it will tell you right there so really cool stuff and then you also have your different apps you have obviously the Wi-Fi hotspot projection manager where you can project your cell phone to the screen via Android Auto or Apple CarPlay we actually did a demo on Android Auto on an 8.4 if you want to check that out that video is in the upper right hand part of the screen climate controls down here if you don't like doing them on the radio you also have volume and tune controls start stop capabilities you can turn those off stability control parking sensors you can turn those off speed crawl control and you can turn the screen off all together down here your window switches Augs USB USB-C jack and then down here this is what makes a Rubicon a Rubicon is the sway bar disconnect the rear lockers or the front lockers you can do both or just the rear and then you get this off-road plus mode which is pretty cool system too there is your 8-speed automatic transmission 4x4 shifter on the floor you get the keyless entry with the remote start on there you get the hideaway key that flips out nice bulky key fob I've always liked these key fobs since this body style came out and it you can actually store your key right in that little holder there so you always know where it is because this is a proximity key you just have the push button start you get the red dash on this particular one notice the little wrangler climbing up your windshield there has all LED lights on the interior now which are pretty cool and bright and then of course your quick disconnects for your front hardtop pieces once again you can check out that demo on how we remove the hardtop that's on how to get these off that's in the upper right hand part of screen again so click then check that out you get your home link buttons for your garage door security systems and line systems and you get assistant SOS buttons in the rear view mirror or above the rear view mirror and then you get your auto dimming rear view mirror and once again notice seven slots right above there so they always keep that sort of theme going factory floor mats come with this Jeep the other thing you have is your doors they come off with the t50 wrench we actually did a demo on how to remove your hard or your doors on your JL if you want to check out that video that's in the upper right hand part of the screen and as promised I will do a final walk around here with all the lights on notice you do get the LED license plate lights and those LED tail lamps look really good especially on the gecko green and I would personally like to thank you for checking out the video today and I hope from this HD video you will have been able to get a really good idea of the overall looks styling options and color of course of this particular Wrangler if this was a four-door I'd be extremely tempted to buy it I got two twin boys but two door probably won't cut it for me anyways thanks again for checking out the video remember to like subscribe subscribe and share click the bell notifications we do videos on our new and used inventory each and every day here at summit auto.com and in fact if you want to subscribe to our YouTube channel in the upper left hand part of the screen is a subscribe button it links more Wrangler videos like this one in the upper right a link to this vehicle on our website in the lower left and a link to one of our latest YouTube videos in the lower right click those check us out and we're super excited to be offering the brand-new 2021 Jeep Wrangler two-door Rubicon in gecko green clear coat thanks again for checking out the video and remember to like subscribe and share


Hello, I'm Marisa from Philippines. I took a hundred and twenty hour online course without tutor support and recently I was able to complete all the units and got certified. I applied this course even if I'm not fully confident if I can make it. While I expected this course would be mind-blowing that it would be difficult for a native speaker like me to understand the lesson. But when I'd received my lesson, I found that it was interestingly developed. Each unit lesson were outlined articulately, simultaneously, and the task in each unit tests my understanding and serves as my guidance to keep me going. The reasons why I took this online course is that I wanted to study but I want to go to school and because I don't have much time. Well ITTT is an option for me because I can study wherever I am and whatever time I prefer and I can even download the lesson to study offline. And I wanted also to have a career which is to teach English as foreign language abroad. But my problem is that I don't even have a university degree. Well ITTT gives me hope because they provide a teaching certificate once course completed and it is internationally accredited to use for applying a teaching job abroad. I highly recommend this to those who want to get an online course, not only to those who want to teach English as foreign language abroad, but also to those who aspire to become a teacher or for those who have a teaching job right now. Take one course from ITTT International to acquire an additional knowledge, additional skills, and learn effective method to use in the classroom to learn and execute effective ideas for an effective teaching. Thank you for ITTT for being part of it and for sharing all for sharing all those learnings. I highly appreciate it and I gain self confidence and I'm happy with a really outcome. All for the best. God bless.