Thank you, Philip. And the other two organizers, of course, for inviting me to speak. So just a couple of words before I start. I mean, so the title of this talk, hopefully you can see my slide here, is a transcendental birational dynamical degree. If you have been around NTWeb for a while, then you probably saw a talk with a very similar title from my co-author Jason Bell a couple of years ago, except it did not include this word. And so what I'm going to tell you about today is, well, first of all, first of all, joint work with Jason and his co-authors from that previous work, so Jason Bell, Jeff Diller, and Matthias Jonsson, where we construct a transcendental number related to algebraic dynamical systems, in fact, a family of them. And so the talk is really, I mean, from a number-theoretic perspective, what you should think of this talk as being about is this relatively rare object in number theory, which is a transcendental number that arises naturally, somehow mathematically, that we can also prove is transcendental. So with that said, the first portion of the talk will essentially be me trying to convince you that this number arises naturally. So to understand a question about some integer sequences and their growths, which are related to dynamical systems, and then the latter part of the talk, I will discuss how we actually approach the transcendence of these numbers. All right, so the integer sequences that are the fundamental objects that I'm going to talk about today are these so-called dynamical degree sequences, which is to say that they will arise from algebraic dynamical systems, that is, dynamical systems, which are discrete and arising from polynomial maps. So here is a basic example of a dynamical system in two variables, formed by iteration of the map, which sends xy to yxy, the product. So iterating this map, we obtain a collection, so accountable family, of polynomial maps of the following form, so xy maps to yxy, which then maps to xy, xy squared, and then to xy squared, x squared, y cubed, and so on. And the feature that we're going to, sorry, the feature that we're going to be interested in for these is the degree of the iterates of these systems as polynomial maps, which is to say the maximum of the polynomial degrees that occur. So in this particular example, of course, before we iterate, we have degree one, and then here we have degree two, degree three, I don't know why that keeps happening, I think maybe my phone is too close, I'll throw it away, sorry. Degree five and degree eight and so on, and you can see exactly or at least predict what you hope will happen, which is that the sequence of degrees that we obtain here is the Fibonacci sequence. Okay. And that's not hard to see, by the way, by definition of the map, when we apply an iterate, we take the degree of this component and add it to the degree of the component before it. And since we started with one one, we obtain the Fibonacci sequence for our degrees. Okay. So that's my claim that they're interesting integer sequences that arise this way. What is actually happening geometrically, what is this this degree actually encoding? Well, it's actually telling us what the dynamical system does to curves or more accurately how it pulls back curves. So here, for example, if we look at the iteration of the line say x equals one, then we can start looking at the iterates of x equals one under this map and we see similar degree. I'm really sorry, I don't know what to do about that or why it's happening. I'm just going to keep getting rid of it when it pops up. The degrees that occur precisely the degrees of those iterated maps, right? And so geometrically, what is the interpretation of the degree of one of these maps, let's say a map of C2 or more generally affine two space, it's the degree of the pullback of a generally chosen line, right? So we take some line and we pull back under our map in this example. We have the same one from the previous slide and this preimage, of course, then if we look at the collection of points whose image satisfies some linear relation will be something of degree two. Okay. All right. So if this is the geometric interpretation, the degree of some curve in this surface, then in fact, we had better work with compact geometry. And if anyone knows what is happening with my iPad and wants to tell me how to make it stop, then please feel free to put that into the chat. Okay. All right. So let's continue on with this example, but now viewing it as a rational map of projective space so that we actually have some nice compact space to live in and compute degrees. So if we homogenize the map from the previous slide, here's what we get. And computing the degree sequence from this map and projective coordinates, well, of course, composing two degree two polynomials will give us a degree four polynomial. But once we look at the second iterate, for example, we see that these polynomials have a common factor, right? And so as a map of projective space, we need to remove this common factor. And what we're left with is that the second iterate has degree three. Okay. And so these degrees then will precisely match the degree sequence from our first slide. But now somehow we're working in this nice compact space. And geometrically, what's happening here when we fail to have full degree, which is to say we have a cancellation here in the second iterate, for example, is that some positive dimensional thing like a line is mapping to eventually to some point in the indeterminacy of the map. Okay. So this map, for example, is not defined at the point zero one zero. Since all of my polynomials take the value zero there. And this line z equals zero maps to that indeterminate point and causes the degree of that iterate to be lower than say expected. Okay. And so the punchline here is that the degrees, the degree sequence. So the degree of these iterates is telling me something about the geometry of the iterated pre images of the indeterminacy set of the map. Okay. All right. Now the good news in projective space is that of course when we compose to morphism or to rational maps more generally, we're composing polynomials and the degree cannot increase by anything more than, sorry, rather the degree of the composition can't be anything more than the product of the degrees. Okay. And that means that the following quantity is going to be well defined for any dominant rational map, namely the first dynamical degree of a dominant rational map defined by say n plus one polynomials f not through f n will be the limit as n goes to infinity of the degree of the nth iterate taken to the one over nth power. Okay. So we have this degree sequence associated to our rational map. And all I'm doing is pulling out a properly normalized growth rate for that degree sequence. Okay. So just a comment in the literature, often the logarithm of this number is referred to as the algebraic entropy of this system. And we'll see in a moment why that might be an appropriate name. For those who are comfortable with the algebraic geometry involved here, of course, there's a higher tech perspective here, which is not about degrees of polynomials per se. But we don't have to just look at the geometry say in the surface case of what happens to curves or more generally, in a projective space of higher dimension, what happens to co-dimension one sub varieties, right? We could certainly try to understand how the map pulls back linear sub varieties of any co-dimension. Indeed, we do do that. One can construct exactly the same definition and obtain these things which are known as higher or intermediate dynamical degrees. Okay. We're not going to use these in the talk. But of course, it's important to know they exist. And I will mention them just in one brief comment about entropy a little bit later. Okay. All right. So let me make a few comments first. Okay. So because of course, we only have a failure of multiplicativity for the degrees of the iterates when we have indeterminacy, if we have a map which is actually a morphism, so has no indeterminacy on projective space, then this sequence, the degree sequence will just be of the form, say a degree of F to the N. So this D is supposed to be degree of F. Okay. And so taking the 1 over Nth power, of course, we obtained just the constant sequence in that limiting definition for dynamical degree. And so we have dynamical degree equal to D. Okay. So we can obtain integer values of dynamical degrees in this way. The example we were carrying over in the first few slides with the Fibonacci degree sequence, of course, has dynamical degree equal to the golden ratio. Okay. Okay. More generally, if instead of just taking these particular choices of polynomials, we consider a general polynomial map of the plane, then it's known due to work of Favre Mjönsson that in fact, so long as these are polynomials, so we have this sort of plain polynomial transformation, then this dynamical degree will be not only algebraic, but a quadratic integer. Okay. So this golden ratio is of course one example, but we have control over these dynamical degrees over these growth rates in this more general setting. Even though we have an upper bound of exponential growth on the degree sequence, other growth rates can occur. Okay. So here is an example to illustrate that for you. So this is a very straightforward map. It's a map which sends xyz to xz, xyz squared. And if you go ahead and write it down, which is quite easy to do, you can compute the degree of the nth iterative of this map, and it's in fact linear in n. So this is n plus one. And so of course, because this dynamical degree is measuring exponential growth, we do not have that. And so this dynamical degree is just equal to one. Okay. So this loses some information, of course, when you don't have exponential growth. All right. And then here's one of particular interest I think is quite fun, actually, and is only known in the bi-rational surface case. So if you take the projective two space and you take a bi-rational map, so a rational map with rational inverse on the projective plane, so long as it's defined by integer polynomials, say, then we can consider the reduction of the map at any prime. Right. Now, at all but finitely many primes, we will again obtain a bi-rational map. If we view this as, say, a bi-rational map of P2 for Fp bar, and we can again consider what is the dynamical degree of that reduction at any prime P. And what Shunya proved is that the limit as P goes to infinity of this dynamical degree of the reduction actually agrees with the dynamical degree. In fact, he proved a more general lower semi-continuity statement for bi-rational surface maps where the surface is over an integral scheme. Okay. And the reason I find this example particularly interesting, or this theorem particularly interesting, is that this is not a sequence which is eventually stabilizing. Okay. These these lambda of sps are actually limiting and can be strictly limiting to the, say, complex or characteristic zero dynamical degree. And so here is an interesting example that he provides of when that can happen. So this is an example with dynamical degree two as a map of, say, complex projective space. But for every prime P, the reduction has dynamical degree strictly less than two. Okay. So I found that quite interesting. All right. And I should say it's also interesting to me because I'm unaware of any generalizations, right, higher dimensional or removing the assumption of bi-rationality. Okay. All right. So why are we interested in this quantity beyond it being very sort of straightforward and natural once you encounter a polynomial dynamical system? A lot of the initial interest in this actually came from integrable systems. And without getting into what integrability is, I mean, a sort of algebraic integrability of a dynamical system would be something like preserving a vibration or, well, you can see these examples in the bi-rational surface case. So here in this chart, which I have stolen from the paper of Chef Diller and Chauffin, is a characterization of possible growth for that degree sequence in the case when we have a bi-rational map of a surface. Okay. So two dimensional invertible maps. And there's three situations where we have dynamical degree one, which is to say sub exponential growth of that sequence. And two situations where we have dynamical degree, which is greater than one, so exponential growth. And what I wanted to point out, the reason I've included this table for you, is that in a dynamical degree one case, there is in fact some preserved vibration for the map, unless okay, maybe it's dynamically trivial. But in the sense that we have an iterate, which is essentially the identity. But dynamical degree one, in the case of bi-rational surface maps, characterizes this type of algebraic integrability, which is to say that the map actually preserves some kind of vibration. Okay. And all of these things can happen. It's worth saying, it's not so easy actually, to construct this example. I already constructed for you an example with linear growth, right? This one is a bit tougher, but they can all happen. Okay. Without commenting much on what topological entropy is, it's simply a dynamical way to measure how this system separates orbits of nearby points. Okay. I just want to say that it's controlled, in a sense, by the dynamical degree, or rather, this generalization to all co-dimensions of the dynamical degree. Okay. So the dynamical degree, at least in some cases, being greater than one or not tells us something about integrability. It also controls, so gives an upper bound for the topological complexity of the dynamical system. To more perhaps of greater interest to this crowd, so the dynamical degree also controls the arithmetic complexity of orbits. Okay. So in the situation, I guess maybe I've written this so that I'm still over projective space, so of course you could just take some projective variety and all the machinery goes through. So in the situation when your map is actually defined over some number field, okay, then if you take an algebraic point inside of projective space, the set of iterates will form a collection of algebraic points, so long as the orbit is well defined, right? You don't run into the indeterminacy. Okay. And if you consider the height of those points, I've got a little plus here because I'm taking the nth root and I don't want to do that to zero, so the plus just means the maximum of the height and one. Okay. Maybe that's worth writing down instead of just saying quickly. Okay. That if you look at this height growth, which will have, which is properly normalized by taking nth roots here, that in fact this height growth is controlled, so bounded above again by this dynamical degree. Okay. So first of all, I should say it's not known in full generality that the soup is unnecessary, that in fact, so the expectation would be that this sequence, if say the orbit is the risky dense, should form a limit, should approach a limit. Okay. But we don't know that in full generality and that's why the limb soup is here. In fact, it's also conjectured in this work of Kawaguchi and Silverman that not only does this sequence, or rather, sorry, this sequence have a well-defined limit when say we have a risky dense orbit, but in fact that that well-defined limit is equal on the nose to the dynamical degree, that there's a connection between the arithmetic of orbits and this notion of algebraic complexity for the map given by the dynamical degree. Okay. So this is known in a number of cases, but not in full generality. Okay. The other connection that I particularly like, not because I know so many applications of it, but I just find it very nice, is a theorem due to Blanc and Conta, which is that again, restricting my attention to the birational surface setting, I'll say a word about why all these theorems are easier to prove there in a moment. This dynamical degree controls say a minimal degree representative in the birational conjugacy class. Okay. Which is to say that if you look at the degree of any conjugate of the map that you're working with, and you allow your your conjugacies to vary over all birational maps of P2, then this cannot be any larger than a relatively explicit polynomial in this dynamical degree. Okay. So again, there's some notion of complexity here. I suppose a more geometric notion of complexity is controlled by this algebraic dynamical degree. Okay. And maybe I'll make the comment. So, so why is this, why these assumptions? The reason, of course, is that this this group, the group of birational transformations of P2 is something that we have reasonable control over in the sense that we understand a normal set, a good set of generators, which is to say the Cremona involution, which we'll see on the next page, and all the linear transformations. That statement is wildly false, as soon as we take something of dimension greater than two. Okay. And so the control over the structure of this group is what allows all of almost all of these of these cases where it's known for birational maps of P2, almost all of those to go through. Okay. All right. So why else might we care about this dynamical degree? So it's interacting with the geometry in an important way. So first of all, the dynamical degree is a birational invariant. Okay, if you take your map and you conjugate it by something birational, it will not change the dynamical degree. It does, of course, they could, of course, change the actual degree of the map, but the dynamical degree is a birational invariant. Okay. And if you think back to what I said about that Fibonacci example, the reason we had that cancellation in the second iterate is that we had a line, say Z equals zero, which mapped to a point of indeterminacy. Okay. In the surface case, this is essentially what always goes wrong when your degree feels to be multiplicative. And so here's an example. So this Cremona involution, which if you wanted to, you could write as say one by X, one by Y, one by Z. And then you really believe me that it's an involution. This Cremona involution has indeterminacy at the intersection points of the coordinate axes. Okay. And in fact, it maps each coordinate axis to one of those indeterminate points. And so its dynamical degree does not behave well with respect to iteration, right? However, we can blow these three points up to obtain a new surface on which we no longer have just a rational map, but in fact, a regular map of morphism, right? So we can resolve the indeterminacy of the Cremona involution by blowing up these indeterminate points and obtain something which does not have any indeterminacy. As soon as we do that, we know what the dynamical degree has to do, because as soon as we fail to have any indeterminacy, then, well, maybe let me wait till I get to the next slide. But as soon as we don't have indeterminacy, this is no longer projective space, right? And so I'm not necessarily saying that the dynamical degree is an integer, although in this case it is. But as soon as we can resolve indeterminacy, we can compute the dynamical degree, okay? All right. And so as I'm sort of heavily laying on here, the general strategy then for trying to understand the growth rate of this sequence of integers is to try to bi-rationally conjugate the map to one that doesn't have any indeterminacy or whose indeterminacy you can control. All right. So the basic question then, at least from our perspective, is, well, okay, which integer sequences arise as degree sequences for rational maps of projective space? I have here a note that this is a countable set of sequences, okay? Because the set of rational maps of a given degree of projective space of a given dimension is a finite dimensional thing, right? So this is some finite dimensional quasi-projective variety. If I want to say, hey, the degree is not multiplicative at the fourth iterate, right? The degree drops at the fourth iterate. That's imposing a Zersky closed condition, okay? And you can only do that finitely many times. All right. So there's only countably many integer sequences which arise this way. We could ask a weaker question of what are their possible growth rates, right? Which real numbers arise as dynamical degrees for these kinds of maps? In fact, historically, sort of the original question and what tends to be of a bit more interest is imposing the restriction of invertibility. So if we require our map be invertible, we saw this example of the Cremona involution where we were able to resolve the indeterminacy. If we make this assumption that we have by rationality, then how do these sets change, right? So which integer sequences arise for by rational maps and which real numbers arise as dynamical degrees for them? All right. So let me give you some more interesting examples. The first one will generalize the Fibonacci example. So just a definition. So to any n by n integer matrix, if it's not singular, I can associate a rational map, which I'll call the monomial map associated to the matrix, which sends coordinates t1 up to tn to the product of those coordinates where the powers here are coming from the matrix entries. Okay, so this Toric map. This, of course, can be homogenized to a rational map. So it defines a rational map of pn to itself. I'll call it h sub a, where a is my matrix. And it has the useful property that the nth iterate of this monomial map is easily expressed in terms of the matrix A, right? So it's just the monomial map associated to the nth power of the matrix that I started with. And the dynamical degrees of these things because of this property turn out to be relatively straightforward to understand. So first, just noting here that that Fibonacci example, of course, is one of these monomial maps. Okay. This does not happen to have any negative entries, but of course, I'm allowing that just so you know. So if I have one of these monomial maps, then in fact, the dynamical degree of the map is equal to the spectral radius of the matrix that we started with. Okay, we'll see a little bit on the details of why that's the case, but I'll just say a word. Given one of these matrices, the degree of the monomial map can be written as an expression in the entries of the matrix. Okay. And of course, the entries of the matrix have say the nth power of the matrix have a growth rate, which is a big O of the spectral radius to the nth power, right? And so you can utilize that argument to show that the growth rate of the degree sequence will also be exponential with growth rate equal to the spectral radius. This is the approach that Hasselblad and Prop took. Okay. But as a corollary, in particular, the dynamical degree for any of these monomial maps will always be an algebraic integer. Okay. And so in a partial answer to this question, the sort of basic class of maps that we have to work with produce only algebraic integers for dynamical degrees. So the actual statement that Diller and Favre proved is that in fact, we still get algebraic integers if we consider any birational map in two dimensions. Okay. So if we take a projective surface and a birational map of that projective surface, then the dynamical degree is either one or the degree sequence satisfies an integer linear recurrence. And consequently, well, first of all, the dynamical degree is then certainly algebraic. But in fact, by the Hodge-Index theorem, it follows that it's either a solemn number or a piezo number. Okay. So recall my sort of shoddy picture here of solemn numbers and piezo numbers that we have a unique real conjugate amongst all the Galois conjugates of this algebraic number, which is on the real axis and outside the unit disk. In the case of solemn numbers, the requirement is then that the reciprocal is the only other conjugate lying off the unit circle and the rest lie on the unit circle. And for a piezo number, we require that all the other conjugates lie strictly inside of the unit disk. This is supposed to be even though my graphic doesn't quite show it. Okay. So the special class of algebraic numbers are precisely the ones which arise from birational maps of projective surfaces. We'll see in a moment this is in stark contrast with the rational case, which is why we tend to separate out the two possibilities. Okay. All right. So the strategy that Diller and Favve successfully implemented was to try to find a better model for the dynamical system to conjugate the map to a new birational map on a new surface, which had the property that the behavior on co-dimension one sub radius, so curves in this case, was functorial. So which is to say that the pullback on the Picard group commutes with iteration. Okay. And if you can prove this, then you can show that the degree sequence actually satisfies a linear recurrence arising from the linear operation on cohomology and that the dynamical degree is actually the spectral radius of that linear operator. Okay. And so this was viewed, I think, as evidence toward a conjecture that had been made a few years earlier in the context of studying these things from the point of view of integrable systems, which is that in fact, if you take a birational map of projective space, so they didn't quite say that way, then the degree sequence always has to be a linear recurrence sequence, so finite linear recurrence sequence. And consequently, the dynamical degree is always an algebraic number. Okay. But the thing is that this Diller-Favve strategy is valid for birational maps in the surface case, but in fact is not always possible if you remove either of those two assumptions. Okay. So if you move the assumption, remove the assumption of birationality and you're just dealing with a rational surface map, or if you increase the dimension and you allow yourself to look at birational maps in higher dimension, then this conjecture is false or this part of the conjecture is false. Okay. And this was shown in the same papers that dealt with those monomial maps, because in fact, it's false even for the most basic, most basic maps you can think of, which are these monomial maps. Okay. So there exists monomial rational maps of projective space in two dimensions, or if you want birational maps, you need to go up to at least three dimensions whose integer sequences of degrees of iterates do not satisfy a finite linear recurrence. So the work of Jason's that he spoke about a couple of years ago, so the work of Bell, Diller, and Jonson back in 2020 showed that in fact, removing the assumption of birationality and dimension two, so allowing yourselves rational maps. So if you're birational, of course, then you have to be one of these piezo or solemn numbers for your dynamical degree. But as soon as you allow yourself noninvertible maps, they show that you can obtain maps with transcendental dynamical degree. Okay. And their construction is very similar to what we ended up doing for this generalization, but just to give you a sort of brief sort of hook to hook on to their construction was related to monomial maps. So recall that it's possible to construct monomial maps whose degree sequences do not satisfy a finite linear recurrence. What Bell Diller and Jonson realized is that it was possible to twist those monomial maps to obtain dynamical degrees, which satisfy a functional equation like this or an equation of this form. Okay. That we have this degree sequence, which is not a finite linear recurrence. And the dynamical degree of a related map of a twisted map satisfies this kind of equality. Okay. And what they were able to prove is not only that they could do this construction, but then in fact, the resulting number has to be transcendental. And I'll talk a little bit more about that in just a moment. Okay. So I saw an early version of this paper and I thought it was really something spectacular. And I was interested in in the birational case because well, first of all, because of the initial motivation was really about invertible systems from the from the integral integrable systems perspective, but also because the transcendence question involved turned out to be much more intricate and general. All right. So a short time later, well, it was during COVID, it felt like a very long time later. We proved that there in fact exists a birational map of projective space, in fact, in any dimension higher than three, sorry, higher than two, with transcendental dynamical degree. So there's really this breaking point in dimension two, birational maps of projective surfaces, must be piezoarous alum numbers, birational maps of higher dimensional projective space can include ones with transcendental dynamical degree. Okay. All right. So concretely, the construction mimicked their previous work, which was to construct a map related to a monomial map, which had this property that the degree sequence of this monomial map, if you look at the monomial map associated to this particular matrix, I'll tell you a little bit about how we came up with this example, that that degree sequence doesn't satisfy any finite linear recurrence. And the map that we work with has some relationship to that degree sequence. And so we can deduce transcendence. But the relationship is more complicated than what I have up here. All right. So what does the dynamical degree of our construction satisfy? So let's take a brief, sort of very brief historical diversion here, on transcendental values of power series. Okay. So I've sort of combined three pieces of work here into a single theorem. So I recall that Mahler was interested in the case when theta was a quadratic irrational. I think Bamer was able to deal with a case where theta was an irrational, which was well-approximable. And both of them were assuming that we were just taking an identity polynomial here. And then the full generality here is due to Loxton-Randford. Okay. And the statement is this, that if you take an irrational number and an algebraic say B, base B is what you should think of here, and P a polynomial with algebraic coefficients, assuming that the series converges, then the series which evaluates the polynomial at the fractional part of n times theta, so fractional part of n times theta times B to the n here. So if you like, you can think of this as my base, although of course it's strictly less than one. So I'm in, you know, sort of the small parts here. And taking coefficients, which are the values of this polynomial at these fractional parts, then this series will evaluate to something transcendental. Okay. And I think that, yes, B not equal to zero. Thank you. Very good point. And I think that the somehow most believable description of this and one that's also related to our own approach is probably the setting of theta being well-approximable. Okay. If theta is well-approximable, so we have rational numbers, which are very good approximations of theta, then those rational numbers here, if we approximate theta by those rational numbers, then what we will get is a series here, which is very, very close to the series that we're interested in for a large number of n. So the partial sums will be very close for a long time. And being a little vague here, what you want to take is, you know, if say theta is approximately equal to say P over Q, what you'd want to look at is, overloading P a little bit here, my apologies, n mod Q theta. So this series will provide you with an algebraic approximation to your power series, which is good enough that, well, in Bramer's case, a Leaville type argument was enough to produce transcendence. Okay. In general, of course, you need some stronger statements about the failure of well-approximability for algebraic numbers by algebraic numbers, which we'll get into the details of in a moment. All right. So what Bell Diller and Jonson showed is that this theorem also holds not just for polynomial values of P, but they were interested in a particular choice of continuous piecewise linear function in, you know, this value modulo one. Okay, sorry, I wrote it on the circle, but equivalently, this value modulo one. And this piecewise function really arises from this degree computation I mentioned earlier that the degree of a monomial map can be written in terms of the entries of the matrix. Here it is. Okay. And so dealing with this function, they were in the two-dimensional case. You can really write down very explicitly the particular, going back a moment here, the particular power series here, which the dynamical degree that they constructed satisfies. And so the transcendence theorem they approved then argues that this value, if lambda f is algebraic, is necessarily transcendental. Since one is not transcendental, you obtain a contradiction to the algebraic city of lambda. Okay. All right. And again, sort of similarly to Bamer here, they were using the convergence of theta to provide these high-quality algebraic approximations. Okay. So what is the transcendence statement in a similar vein that we approved? So let me just sort of digest it for a moment here out loud. So again, we take some irrational number, and we are interested in a function of the fractional part of multiples of that number. Okay. So those are the coefficients of our power series. But now, rather than working in one dimension, we are going to take a vector of algebraic numbers. Okay. And so this is replacing our B in the previous page. And replacing the P of the previous page will be some piecewise constant function with finite discontinuity set. Okay. So it takes finitely many constant values, say, modulo one. Okay. And there's a technical assumption here that can probably be removed. Let me say this is about the interaction of theta with the discontinuity set of gamma. Okay. As I say, I believe this can probably be removed. Maybe let me take a moment before I deal with the question in the chat. And if you know that your function has the property and your algebraic numbers have the property that this map is real valued. So I'm allowing general algebraic numbers here, so it wouldn't necessarily be, but I'm going to force it to be real valued, non-constant and maximized by, say, J theta. Okay. This is for sufficiently large values of J, of course. Then the sum here of this is just the vector product of this piecewise function evaluated at the fractional part against this vector of powers of the rho i is actually going to produce a transcendental number. Okay. I see there's a question in the chat. Philip, do you mind if I take a quick look? Yes. Yeah, go ahead. Yeah. So it's maybe honest, maybe just don't mute and ask the question. Yeah. I was wondering if it's possible to estimate the gap between the transcendental number and the closest algebraic number. I would be curious how that... So the answer is sort of yes. It's yes up to the ability to estimate how well approximable theta is, right? So you can get a bound, but it's going to depend on, so if you have small... Sorry, what am I trying to say here? It's going to depend on the continued fraction expansion for theta, I think, but now that I'm saying it out loud, probably you could get rid of that dependence because having unbounded continued fraction expansion is actually the good case, is the easy case somehow, and having bounded continued fraction expansion is the tough case. So actually probably the answer is just generally yes. Yeah. Is that enough of an answer? Is it possible to construct that or from what I see, this cannot be... It's not a constructable estimate. So the sense in which it's a constructive estimate is that the mechanism of the proof is to take convergence of theta and to produce analogs of this object that I mentioned on the previous page. So looking at instead these periodic things, which of course evaluate to algebraic numbers and controlling the quality of the estimate between those two things. Now that comes at the end of the day, down to finding how far out these sequences actually agree with each other, because our gamma, because we're not dealing with a polynomial, we're dealing with a piecewise constant function. And so if I take something which approximates theta very well, then the fractional part of n theta and the fractional part of that approximation times n will evaluate to the same thing under gamma. So in fact, it's not just in this case, because we're dealing with piecewise constant functions, it's not just that the values are close to the approximating series, which takes an algebraic value, but in fact, it's on the nose in agreement up to some n for the partial sums of the two series. And finding that n where they start to diverge, yes, I think that we do in the course of the proof control the size of the tail of that difference. Yeah. Okay, thank you. Yeah. All right. So I think I have said enough about this, I mean, right, so why is this so much more difficult than the previous slide? Well, the very obvious issue that could arise is cancellation, right? I mean, I'm taking a sum of three things here. Okay. And of course, just because you know that one of those sums evaluates or each of those sums evaluates to a transcendental number doesn't mean they don't cancel each other out into something algebraic. Okay. And so some kind of assumption here is certainly necessary. Really, the key one is this maximization property. Okay, well, and then I guess by default also the real value so that maximization is meaningful. Okay. And just another comment that in fact, this argument is simpler when we have unbounded continued fraction expansion when theta is actually well approximable, just as it is sort of historically the case with these transcendence of power series statements. Okay. All right. So the key tool of course, some deep theorem in defunding approximation, the one that is particularly well suited for us is ever to form this bound on sum of S, well S integers, but it's particularly powerful in the case of S units. Okay. So this is a consequence as he showed of the so-called Piatic subspace theorem. And the statement is here, what I want to point out, you can read it if you happen to not be familiar. So the statement is about approximability of these sort of non-degenerate sums of S integers in terms of the various input data. Okay. The important thing of course is that all that matters is the length of the sum once you've fixed your base field in your set of primes that you're integral outside of. The useful thing in our setting, because we were dealing again with a piecewise constant function taking only finitely many values, those coefficients in our power series are then S integers for some choice of primes S. In fact, they're S units for some choice of primes S. And in the case when you're dealing with S units, this statement will lose this term. So by the product formula, this term will evaluate to one. And so the lower bound that you get on sums of S units, and you should think here that we're applying this to partial sums in the power series from the previous slide, will give you a lower bound assuming you can prove non-degeneracy, which is the tough part, will give you a lower bound, which is what we use to prove the transcendence, which conflicts with the good approximation from the previous slide. Okay. All right. So where does this fit into the dynamical degree story? So it fits into the dynamical degree story with that construction that I mentioned of twisting a monomial map. So in fact, what we show, well, in fact, I've only written down the three dimensional case, but there's higher, it works in any dimension with proper assumptions, that if you take an invertible three by three integral matrix, and it satisfies some say geometric assumptions, which is to say that it has strictly leading complex conjugate eigenvalues, no power of which land inside the reels. Okay. This is precisely how you construct a monomial map, which doesn't have degree sequence, which is a finite linear recurrence. Okay. Then we can cook up a birational involution of P three, so that, okay, maybe not just twisting the original matrix monomial map, but some power of that original matrix, matrix or some iterate of that original monomial map, will satisfy that the twist here has dynamical degree, which is the unique positive solution to a power series equation of this form. And my function here is in fact precisely the type of function that we dealt with on the previous page. Okay. And so the transcendence theorem of the previous page tells us that the value on the left hand side here is necessarily transcendental whenever x is algebraic. But of course, then my dynamical degree cannot be algebraic because the value it takes, the series takes as algebraic. Okay. All right. So just to comment here on where this all comes from. So what is the map doing geometrically? Well, as I said, we have this monomial map with this sort of irregular degree sequence, and we're composing it with an involution. Okay. The point is that this involution is very carefully chosen so that the indeterminacy of this map actually provides a recursive relation between the quantities that we're interested in, this function which I understand, for lack of a better way to say that, okay, which takes the form of one of those functions in my power series from the previous slide. And the degrees of the monomial maps, which I also understand. Okay. Here is the relation. By the way, this is a picture. Sorry, I should have said this is a picture of the indeterminacy of this map is that you would expect it to be some kind of general line. But in fact, it's very carefully chosen so that the lines that lie in the indeterminacy locus here are intersections of particular planes inside of p3. Okay. And that's enough to get you a relation of the following form, which notice is almost is basically an infinite linear recurrence sequence in the degree of the nth iterate of the map. Okay, almost not quite. But this is sufficient to then yield the power series formula that we use to prove the transcendence of the dynamical degree. All right, so just some final questions. We're dealing with these birational maps in p3, of course. Is it possible to construct a map which has transcendental dynamical degree whose inverse also has transcendental dynamical degree? If you recall, the monomial map constraint, the monomial map we sort of built these things out of, had to have complex conjugate eigenvalues, leading complex conjugate eigenvalues with no power that landed in the reals. Okay, the very silly reason why you cannot do this for birational maps of p2 is that you cannot put two of those on the unit circle in degree 2 and obtain, what am I trying to say here, that a matrix with complex conjugate eigenvalues that satisfy this will never be invertible. That's what I'm trying to say. Okay, if it's two-dimensional, you have to go into three dimensions at least. And so I suspect you can do this easily with our construction in even dimensions, or maybe even just dimension four or higher. But it's not obvious to me whether or not this can happen in dimension three. Okay, certainly our construction will not suffice to do this in dimension three. The second question is, in the case when you do have transcendental dynamical degree, do we still hope that that conjecture of Kawaguchi and Silverman is true? Is height growth along orbits still controlled by that dynamical degree, even though it's transcendental? And somewhat surprisingly, so Matsuzawa recently has shown that not in our birational p3 case, but in the rational p2 case, the original case of Beldiller and Jonson, that yes, the maps with transcendental dynamical degree, which are bi-rational maps of, sorry, rational maps of p2, do admit orbits whose arithmetic degrees have transcendental growth, okay, which agrees then with that transcendental dynamical degree, which is a bit surprising. In fact, Kawaguchi and Silverman had initially made two conjectures, sort of as sub-conjectures of the same conjecture. One was that the arithmetic degree is always an algebraic integer, and the other is that the arithmetic degree always agrees with the dynamical degree. And so now we know that we knew from the transcendental dynamical degree construction that one of those parts was false. Matsuzawa told us that, in fact, it's the algebraicity. Okay. The third point here is to look for finer information about the collection of dynamical degrees. This is understood somewhat for bi-rational maps of p2. Okay, so first of all, that there's a lower bound with the property that all bi-rational maps of p2, if they're not equal to one, are greater than or equal to this lower bound. L is lemur, by the way. So this is related to some interesting number theoretic questions. Nothing is known outside of the p2 setting there. Okay, essentially the same question here. And then finally, from a dynamical perspective, we use this degree to construct invariant dynamical tools like invariant currents and variant measures, things that help us study these objects. And just recently, Diller and Reuter have made some progress on those constructions in the case when the dynamical degree is transcendental, which is somewhat surprising. Okay. And I think with that, maybe I will thank you for your attention.


Good afternoon, folks. Steve Rhodes coming to you live from the shores of sunny Delray Beach, Florida. This is your 1 p.m. update and a bit of a mixed bag out here. You've got the Dow, the S&P, the Nasdaq, and the Russell, which just turned green out there. The semis are off 19 points. That was a 53. S&P 14, Nasdaq 100 to 69. Russell, again, just turned slightly green. Semis are off 20. Trainies are down 85. We have gold trading out at $18.40. That's up $2. Silver's off $0.31 cents. At 21.45, lights we crewed up $3.26 cents. 106.26 is the print there. Natural gas up $0.07 cents. And a 30-year treasury up two points. Boy, two and a half points out here trading out at $1.35. That is a big move there. Let's go take our nine-panel market update chart. We begin by taking a look at the ES mini in the upper left-hand side. You can see that there's that new profile that formed. We talked about that yesterday. Bullish and structure closed above the center of that profile. We're trading above the center of that profile now. That suggests we're headed up to $38.41. That's the top of that box. If we take a look at the spot follow totics, it is falling. It's moving back towards its 50-day exponential moving average at $2806. If we ever see that get below that 50-day exponential moving average, we should see the markets blast off to the upside. If we take a look at the NQ, they're just consolidating between support and resistance strong support at $11.396.27. And resistance to the top of that profile at $11.716. U.S. dollar index just consolidated with inside its daily profile. That's between the range of $103.62 and $105.56. Gold is consolidated with inside its daily profile as well. That's between $18.13 and $18.79. Silver, really kind of consolidating between the bottom of its weekly profile, $21.03. And the top of its daily profile, which is $22.33. Light Tweed Crude is trading below the bottom of a daily profile, but I believe it is holding the TD9 breakout level. We'll certainly take a look at that during the Trader Zed Show. Natural gas, a brand-new profile that is formed, this profile forming above the prior profile. That's a slightly bullish signal out there. We can see the 30-year Treasury is back inside its daily profile. That suggests the next run to about the $136.26 or $138.09 level. So, folks, stay tuned for the Trader Zed Show. But if you're off to start your Wednesday, we want you to have a wonderful one. Thanks so much for joining us. We'll look forward to seeing you again soon. But stay tuned for the Trader Zed Show.


3D-concrete printing can produce lightweight and strong functionally-graded material for construction, and an approach was developed to create tailored material by examining filament properties with printing parameters, resulting in a 50% improvement in strength-to-weight ratio. This article was authored by Yi Wei, Daniel Te, Jian Huilin, Mi Yang Li, and others.


Hey everybody, welcome to the 24th Qt tutorial with C++ and GUI programming. Today we're going to be covering the slider. So let's just drag one of those on our dialogue and the progress bar. And let's just set the layout here. Now, these may look familiar. Slider, you just grabbed a little knob here and move it back and forth and the progress bar shows a progress. Let's actually just make a signal and slot here. So we can see the horizontal slider or Q slider. We can say action triggered, range change, slider mode, etc. Let's just do a value changed and on the progress bar we'll do set value. And you've seen this before, this is nothing new, but I wanted to compile it and show you real quickly what we can expect from these. That's the basic functionality here and you notice this text. On Windows, that's kind of annoying because it's off to the side and on Linux it's in the center. You can actually turn that off to show text. Display text. We'll find it eventually. I'm not going to waste 10 minutes at your time trying to find it, but it's in here somewhere, I can guarantee you that. Text visible, there we go, sorry about that. That's how you hide the text on that. Now these controls, although they serve different functions, are actually very, very similar in design. Let's actually go in here to our dialog, CPP. You can see when you go UI, horizontal slider, there is a value and that gets the value and the same thing is true with the progress bar. Now to set the value you simply say set value and it's the same on both of those controls. Like I said, they're very similar. They have a range. One allows you to manipulate the range. One allows you to display the range. We're going to actually just get rid of this signal slot here and delete that and we're going to set that up manually just so you can see how this works. We'll say connect. Remember the first argument is the sender. We say UI, horizontal slider and we want to send the signal and the signal is going to be value changed and then the third parameter is going to be the receiving widget which is the UI progress bar and we're going to set the slot which is going to be set value. You can see how that works. The horizontal slider is emitting the value changed and the progress bar is receiving it in the slot set value. Notice the parameter and these have to match or this will not function at all. You won't get a compiler error. It just won't work. That's one of the annoying things about Qt that I really don't like. If it's mixed matched or if something weird happens it just won't work. Let's go ahead and run this and make sure it works as expected and there it is. There's our signals and slots hard at work. Now you notice when we started this the values were off. What we're going to do here is we're going to say UI progress bar set value and we're going to say UI was a horizontal slider value. Let's just run that again. That way when it starts they'll be the same value. Voila. Two very simple yet powerful controls. I'm sure you can think of a million and one uses of each of these controls. Progress bars I'm sure you've seen them all over the place here. Not as heavily used but they're pretty useful. This is Brian. I hope you found this tutorial educational and entertaining and thank you for watching.


What can we do to start journaling from a leadership standpoint to raise that self-awareness? Most people don't understand the value. And so they don't do it. They're like, yeah, journaling, I heard it's good, but I don't really, whatever I've heard these sayings, journal, journal, journal, what about it? So the utility value in journaling and what journaling does is it helps you practice extracting your own inner wisdom. So as a leadership coach, when I'm working with individuals, AJ, I just noticed this, you swallowed a few times. The topic is emotional. What did you notice? Did you notice that you swallowed? Was there something going on with that? So I may ask a body question. I may ask, hey, as we've been talking this conversation, there's this theme of this, it's come up a few times. What's your thoughts around that? Like, oh, I didn't notice that theme or I didn't notice I did that. So journaling gets you in this practice of asking questions. So moving from statements to questions. So so many times in life, I'll go back to my 17-year-old driving around the corner with the music blasting. I can make a statement, I can't believe he's doing this. Again, I've told him several times and now because of that, I'm gonna go down there and yell at him. Or I can say like, ooh, I wonder what's going on, how often does he do it, where did he just leave the gym, is somebody else or is he trying to, I don't know, like what's going on, let's go explore. So we're actually moving from judgment to curiosity. And so what helps with moving from judgment to curiosity is moving from statements, making statements, whether it's an individual showing up late, whether it's somebody not meeting an expectation. And that's where conflict essentially comes in on expectations, making statements to being curious. And so curiosity shows up in asking questions. And so in journaling, you start practicing the art of asking questions and you start with yourself. Once again, everything starts with you. So the art of asking questions, know why questions.  How I look at that is it is a great start. So if you're like, hey, I'm really, I'm thinking about doing jiu-jitsu, awesome.  That's where why questions fall in. Like I want to start asking questions. Hey, why questions? Sure, that's a great like white belt place to start. But now if you want to like continue and develop this art, we actually move away from why questions. There's research on actually diving into and constantly asking why question creates a negative feedback loop. So we move from why questions to what questions? Once again, which takes time and practice. So even for me, it's simple in language where I'm like, oh, why just in a conversation? And now I've done it so many times why I have to reframe it to a what? And it becomes difficult because now you're moving sentences around to form it. So it takes a lot of practice and then you notice because what is a knowledge question? What has motivated both of you to continue this podcast for this long? Vice, why have you continued this podcast for so long? And so the other thing that why questions do is it tends to put people on the defensive. A.J., why'd you ask me to come on your podcast? A.J., what about me led you to asking me about asking me on your podcast? So it's a different, so just that first word reframes how the individual receives it. And so same thing for yourself. So if I'm sitting there journaling, you know, why did I do this?  Well, why did you do this? So you can really create a self-spin up even in your own journaling. So know why questions. Stream of consciousness is the easiest way to start. So just sit down, pick a place, just like any other goal. I wanna work out, you know, probably you're probably not gonna start out working out for an hour. With journaling, I'll ask clients, you know, what can you commit to? Oh, I can commit to 20 minutes, you know, every morning or every evening. I'll go, awesome. How about five minutes? And I'll just, I'll slash it because with goals, you have to calibrate new goals. There's a calibration in there. And so I'll slash it because all I want them to do is start doing it and see the value in it and get the repetitions in the practice. And then I'll create a window. How about three to five minutes? And they're like, yeah. I'm like, is that doable? They're like, oh yeah, it's totally doable. And even with that, that still takes, there's a week they'll come back, nod and journal, and then I hold them accountable. And then they start journaling a couple of times and then it starts growing and expanding. And then they start seeing the utility value and they start going through and like, oh wow. And now it prompts our conversations as well. So we'll start talking about the things that comes up in their journaling. And they're like, hey, I noticed I keep going back to this one event every time I sit down to journal or every time I go to journal, I avoid it. I'll even ask questions about that because they're now thinking about certain things when they go to journal, which is once again, we're just extracting information and helping them dive into themselves. So the key to journaling is you start the art of asking questions and you learn how to really excavate your own inner wisdom.


First up, MVP right now. Okay. I want to stress right now. Eddie, Giannis or Luca? I'm gonna go Luca for two reasons. One, he's killing.  He's leading the league in scoring. He's an absolute monster. I've watched him up close once this year and then I watched him break my heart again last night. The other reason is because, you know, it's the MVP is a funny award. It's kind of just his turn, you know, like Giannis has two, it's gonna be a lot to get him a third one. Luca needs to be anointed in this sense. And it's not that he's undeserving either.  So it's just working out. I'm gonna go Luca. All right. All right. Chandler, most improved player, Maxi or Markinin? To me, it's got to be Laurie Markinin. And here's why. First of all, they're nine and three, their first place in the Western Conference. Do they sustain that? Probably not. But yeah, also, they're now getting to the point where the bottom four or five teams the last couple of years have lingered around 20 wins. They're already at nine. So at this point, I think they got to be committed. This kid here went from a high draft prospect with a lot of potential to a guy that just kind of got thrown into a trade that now is proving himself to be a valuable, valuable player who's doing it all. He's knocking down threes. He's rebounding. And he is the main reason to me why they've got off to such a hot start. People expected Maxi, people expected Maxi to be doing this. I didn't see Laurie Markinin coming out of nowhere averaging 24, nine and four and the best team in the Western Conference. So that's, I got to go with him. I would love to be a flyer on the front office. He might not, although he's such a good moment, such a good moment. Six man of the year. This, yeah, Eddie Westbrook or pool between those two, I'm gonna go Russ. Russ has been killing it in that role and he's really stabilized their offense when he's on the floor. Jordan pool has been, has, has a little bit of a down season thus far. I think he'll snap out of it and Russ might be in another jersey when he wins this award later this season. But he's, he's been great. Also, you know, Benedict Mather and Andy and Christian Wood and Dallas, he's been amazing off that bench. But yeah, out of these two, I'm gonna go Russ.


Here we go in this video. What we're going to talk about is putting it together, eliminating excessive movement. So how do we eliminate movement to pick up better technical patterns in the shortest period of time? We're going to talk about it in this video. So let's check it out. Hey, everybody. It's Eric Johnson. In today's video, what we're going to talk about is what we call in the throwing chain reaction system the wind up one start. So what we're going to be doing is teaching our athletes how to just be more on balance and how to come from our start position into a really nice, what we refer to as pillar two. So we have that really nice long wide sweep, great entry and we're setting up a great sprint. So one of the core things that we're going to do is we're going to lock up the hips and we're going to clean up a lot of extra motion. So even with Carson, one of the things we're working on again, if you watch the videos we're working on is orbit and you're kind of seeing how we apply our system and how there's issues in his throw that we can see as we work through this basic progression. So that progression pillar connection because what we're doing is we're always connecting each piece and we're looking at our triggers and our start and piecing it all together and putting together our full throw. One of our huge things about the throwing chain reaction is the setup is the trigger and that sets up the entire throw. So we've been working on a couple of different things and one of the things is when he winds up, we want to see his chest being a little bit more on top. So we're going to be able to create a longer entry path and he has a tendency to start almost two upright. And so he takes a path into the throw more like this. Okay. So one of the things we're going to do is like we just said, he has a tendency to put his shoulders a little too far back. We don't want them too far forward. We don't want them too far back. We want to find that sweet spot. And again, this is one of the things we refer to in our system as windows. We have windows of execution. We have these positions, different torso length, different leg length, femur length, arm length. These are the little individual things. So this is why it will always vary and there's a range per thrower. Okay. So what we're going to do is have Karsten focusing on that. He's going to keep his shoulders so he can get his shoulders moving and he's going to be more balanced. There we go. So not bad. You can see, again, he's generating some good stretch on that two kilo ball and you're going to see how it wants to come out. And if he can connect and drive his delivery side through, he's going to pull that ball a little straighter instead of over to the side. So again, these would be the things you would want to be focusing on as you're working on your wall throws to get in our reps, our technical focus, always that rhythm, not worrying about how far it's going to go. So again, what we're going to do is one of the things I'm going to point out. So as he comes in, again, we're trying to stay a little straighter. So I'm going to have him when he winds up and we keep the hips straight. We're just going to have him shift it as he winds the ball. So he's going to be able to get his hips in the ball a little bit more. So he likes Gerd Cantor. So we're going to think of his Gerd Cantor warm up. Okay. So let's do it this way. Gerd is going to be here and you're going to see him taking the taking the wine more like this, right? So he's kind of more like this and he's going to be here. So we're going to set up more like to your best Gerd Cantor imitation. Okay. There you go. So not bad. So you're going to see how he's working, working this way. And so what we got to have him do is cut back to the middle. He closed off the feet. So his hips are facing more this way. So again, this is a perfect example. We're going to be able to get in our reps and focus on our technical position. So what we're going to do is we're going to make sure that we're moving around the corner a little bit better. And what we're going to be doing is pushing his left knee around and into the throw more, and we're going to be activating the sweep a little bit more. Okay. So we're going to do that. Let's go. That was much better. So if you notice the rhythm of better rhythm, the top top, the sound of his feet and he pulled that discus from over there or the discus ball and it pulled it more into the sector. So that felt better. Right. Yes. Okay. Again, we're going to be concentrating on that nice long position. So we're active and we're a little bit more active in our pillar too. So as we shift this way, we're going to be getting that sweep leg moving and watch the position of the left arm. That was much better. So now we pulled it around so you could see how we progressively moved in. So when we're using the wall, again, we can look at that feedback. First throw, he was locked out. Second throw is a little better alignment. Third throw, he pulled it right down the pike. He's making adjustments. That's what you want to do with your wall throws. Remember, this is how your training is supposed to be. You want to have a plan. You want to practice with specific focuses. And again, one of the things we like about our system is that we're trying to teach other throwers and coaches is that so many of them aren't sure of they're still trying to learn. And then what the system allows you to do is to actually learn, apply. And you can see how we're making results right on the fly. I think for that, we'll conclude our wall throws. Remember just a great training tip and get your wall throws in, get your lots of reps, reps, reps, good reps, quality reps, make the right adjustments. Again, for our online members, we'll go through and we'll break down this stand throw, this pillar five, six, and we'll talk about some of the specific drills, pillar drills and pillar connection movements and kind of break this down in more detail. And if you'd like that, go ahead, hit and log in to the site. See that. If you're interested in becoming a member, be sure to click the link. Look at all the info and details that comes with it. It's a step-by-step system for teaching you how to throw and learn how to throw farther, faster. Hopefully this video was helpful. Be sure to hit that subscribe button, thumbs up, and we will see you on the next video. How to counter in rotational shot. One of the things we talk about is winding the left side. So one big mistake people make is they try to wind.


Happy July 4th, everyone! And welcome! This is Melissa Arma with the Stock Swoosh. It is Independence Day weekend and pretty much Independence Day week. A lot of people taking off because July 4th is on a Saturday. Rarely happens, actually. So I decided to do some July 4th specials for everybody for the next week. And this is a great time because it's ramping up right before earnings season starts, which is actually the second week of July. If you have any questions, you can email me at melissa at thestockswoosh.com or you can call me at 929-3200-GAAP. And you can follow me on Twitter, Facebook, YouTube, or Skype. So happy 4th of July, everyone! It's a great time to be an American and it is a beautiful, beautiful weekend here in New York City. So we're six months into the year. It's hard to believe. 2020 has been like a flash, flash in the pants between everything that's happened with COVID and just everything that's going on in the world today. So I'm really looking forward to the second half of 2020. Why? I think the second half of 2020 has a huge potential to be way better than the first half of 2020, if you know what I mean. Now if you have been trading or if you have not been having a good year 2020 financially, maybe it's time to get your year on track. You know, right now with everything that's happened with the pandemic, working from home has been not just a luxury. It's been a blessing, to be honest with you, because so many people have been unemployed out of work and if you are able to still work from home, the nice thing is that you can train from home too and work your regular job from home and transition into becoming a trader if that is really what you want to do. And I think in some ways that's really a benefit to some people who can't normally trade at their office, but now are working from home and can trade from home. So if you want to become a career trader, now's the time to start. You still have six more months left of the year. And again, like I said, earning season begins in two weeks. And what's the difference in the significance between earning season and times like this coming week? You just have more volatility, more momentum in the market, more trades, more gaps. And for those of you that don't know what I do, I focus on gaps. So I look for the gaps every day. So whether you want to be a full-time trader or a part-time trader, now is a good time to join. You can do swing trades, you can do options, you can do day trades, I say make money doing it all. It's one system that's all that I do, depending on how you apply it, whether for an option trade or a day trade, really is up to you. I offered both. So I teach my method in a class, it's called the Golden Gap course. I teach this class once a month. It's a 26-point rating system that teaches you how to pick the best gap each morning. And that is what I do. So I decided to run a couple, July 4 specials for everybody out there who is interested in joining and jumping in. They will run through today, July 4, 2020 through one week, July 12, 2020, okay, if you want to sign up. So here's everything I'm doing. This is a huge package for a great price and a really big savings. This is an Independence Day package. I'm offering the Golden Gap course, the Gap Options newsletter through 1231, and the stocks will show live trading room through 1231. The Trends class, the Wealth class and the Gap Options course for one package price, $79.99. This is a huge savings. The class alone is $69.99. So you're getting all of this help, all of this education, all together in one package to help you get right into the end of the year and to jump into trading and really get all my trade calls, options and day trades until the end of 2020. The Trends is about long-term trends. The Wealth class is a great class for your mind. The Gap Options course helps you with the options, and then the Golden Gap course teaches you the strategy. This is a huge savings. I'm also doing, if you want to just do the Golden Gap course alone and have all the day trade calls, the room is free until the end of the year. This package price is $69.99. Again you're getting the room free to the end of the year. Great deal. Again this is through July 12. If you just want to do options, I'm doing the Gap Options newsletter. Options Independence Day package. You sign up for the Gap Options letter for one year and you receive the Gap Options class free and the Trends class free. This is a great deal too. $69.99. That's a normal cost of the newsletter for one year and then you get the two classes free, which really will help you train better. Again education is so important. Then I'm doing 50% off the other products and services I offer. Market report, normally 500 a year, 50% off for 250. That's once a week. It's a weekly market review that comes out in a Monday. The Trends class, which is normally $99.99, almost $1,000, 50% off $500. The Wealth Manifestation course, $599, again $300 only. That includes one month free in the live trading room. The Entries course, $14.99, again 50% off, $750. The Gap Options course includes one month free of the options newsletter, which is huge. You've been wanting to get on the newsletter. I only offer an annual subscription for this. This is a great deal, $12.50 for the class and one month of the newsletter. If you would like mentoring sessions with me, no prerequisites. You could be a student or a non-student. Ten sessions an hour a piece, normally $4,000, ten sessions for two grand. This is a great deal. Five sessions for $1,000 and three sessions for $600. This is huge. Again, if you're new and you need the extra time with me, it's one hour for one session via phone in the trading room or Skype, whichever you prefer. There was, oh gosh, there's been so many trades in CCL, but this was the one, most recent one here we did the day trade. This wasn't even that big of a mover when you think about it, really about $0.60, but you could have made $3,100. This is a day trade in, out, in, out on the same day. A beautiful, beautiful move, drop down, fell, we got it, okay? So I'm just going to briefly go over here what I look for in charts. I look for gaps. This was the gap in CCL, closed here, gap down, fell, okay? So this was a short, okay? This is a short when you're looking to trade. And each day I'm looking for gap ups and gap downs. Here was a gap up, here was a gap up, here was a gap down, here was a gap down, here was a gap down, okay? So this is what the system teaches me, how to find the best pick each day, which on this particular day was CCL. Nice trade, love getting these fast trades. Spending to make $3,000 in one day in less than an hour is fabulous. Now this was a much bigger move in a bigger play in Apple. Beautiful, beautiful, beautiful, closed here, gap up, rallied. This one here you would have had to haul into lunch. But still to work for two and a half hours and to be able to make almost $9,000 is spectacular. I mean just so great. But the risk here was $2,400. But still look at how you almost quadrupled your risk, almost, okay? This is a fantastic call. Entry was $364.20, stop was $362.20 and this was one where I mean basically when I'm looking back at this I'm thinking gosh, you know, we could have done an ad in it. We definitely, definitely could have taken more of this. But you know, again it allowed me to just continue, hold with it, stick with it and it worked. It pretty much went to the dream target. So $8,880 on Apple and again this was a gap up. Then this is, if you get the options newsletter package or even any of the specials that includes the options newsletter, the options newsletter is trades that are emailed to you. So this one was sent in the morning, 10.26 in the morning, June 23rd it was Boeing. It was to buy the Boeing putts. The strike was 185, okay? So anyways, this was a put, the stock gap down. Again I called it on the 23rd and when you look at this you can see the drop off here, okay? When you're doing an option, typically a column within one week to two weeks. Now this was called pretty tight because I expected the move to grow fast, okay? So more time doesn't necessarily help you in options because ideally you would get the move between 24 to 48 hours, okay? But it was a really, really nice quick short that happened. The advanced trader risk in this was $86.40, so this would have been 18 contracts sold at $13.20, profit was $15,120. This happened from the Tuesday just for that one week, okay?  That's what a put is. A put is a short. But if you just wanted to do beginner, one contract, $480 risk sold at $13.20, profit $840. And again, this is a nice way to make extra money besides the day trades because some of these options trades can be huge, okay? Just can be so big. Again, almost 200% return on investment just within 48 hours. So you can do the options newsletter, you can do the day trades, you can learn the classes. I think education is extremely important if you want to do the best that you possibly can. But everything I do is based on gaps. Everything I do is based on momentum and I'm very, very focused on making money. That's one of the reasons why I have such great calls. So 80% of my calls, I have about an 80% win ratio for the strategy itself. So you figure out of when you're looking at your trades, so you take 10 trades. You figure two were going to be losers and eight are going to be winners. And that is how I would size yourself when you're taking the trades. Right now during this period, for everyone 2020, it is about financial freedom and just freedom period. So give yourself the gift of education, the path to freedom and it is about investing in yourself. Education is a gift to yourself and trading or doing something where you can be financially independent even if it's in the future, okay? Till you learn it and you do it and you build your account up to the point where you can start to make $15,000 in an option trade. If you don't start doing it right away, you're continually putting off where you want to go.  And again, July 4th week is a great time to do it. So again, the class dates are as following for July, July 18th and 19th, 9 to 5, normal class tuition is $6,999, class is online and the options newsletter normally is $6,999. Again, trades are emailed to you. If you're interested, email me and Melissa at thestockschwisch.com. Have a great holiday weekend everyone and please everyone stay safe.