Great, thanks very much. And thanks for the invitation to speak here again. I have given variants of this talk a couple of times before. So if you are one of the people that have suffered through something like this before, I hereby swear this is the last time that I speak about this material. The paper associated to it is now on the archive. So if there's anything in here that sort of piques your interest, you can go and take a look. I want to begin, so this is going to be joint work with Julian Leitschek, a former postdoc here who's now in Bath, Roman Sarapin, who was an intern at IST for a while and Arne Schmetz in Lebanon. And I want to go back and talk about where this problem comes from before I pause and speak a little bit about some basic SIV problems. And then I'm going to return to the problem and motivate it a bit more and ultimately end up explaining some of the new conjectures that we've been developing. Sorry. Yeah, so this problem really goes back to this very influential AIM meeting in 2002. I think it's the only AIM meeting that took place over two weeks. It was about rational points. It had lots of motivating open problems. And in particular, there was this survey written by the late great Peter Sunuddin Dyer. He had a Hilbertian list of 23 open problems that he presented and this was problem 21 and sort of motivated by the Manin's conjecture, which is about counting rational points of bounded height. He phrased this question. So you suppose that you have a variety which has a map down to P1 whose fibers are all conics and you just try and count the number of points in P1 of bounded height, number of rational points in P1 of bounded height such that the fiber has rational points. And he asks, what can you say about this counting function? Can you actually prove anything about this counting function? And in fact, can you do something similar if instead of asking about fibers that are conics, you have fibers that are curves of genus one? So we're gonna address this question in this talk, but we're gonna tackle a slightly easier question, more much easier question, where we sort of switch out the phrase containing rational points in the fiber or containing periodic points in the fiber. And I think both questions are interesting, but Peter's question is a lot harder. So before getting onto that, but that's where we're going, but let me just go over a couple of SIV problems which are quite classical. So if we're given a polynomial with integer coefficients and two variables, one of the basic proto-SIV questions one could ask is, can you show that this polynomial takes infinitely many prime values? And can you give an estimate how often it takes a prime value? A variant of this question, which will be more relevant in this talk is to try and ask instead that your polynomial is only divisible by primes that lie in some predetermined set of rational primes. So the set of rational primes that crops up here are these kind of Frobenian sets of primes. So these are just primes such that the Frobenius-Condruxity class lies in some determined union of contrast to the classes of the Galois group under some extension. Anyway, some set of primes coming from algebraic number theory. And so can you, so yeah, what the, I'd say that traditionally SIV methods of focus very much on question one there's a lot of very substantial important work on that. But one of the goals of this talk is to give a bit of agency to this second question because that's what happens to come up in this application. So let me just state it again. If, can you get an asymptotic formula or estimates for the number of times, sorry, the number of pairs of integers M and N whose absolute value is bounded by T and for which this polynomial is only divisible by primes lying in this, this Frobenian set of primes. And actually the particular set of primes that we'll care about are just the set of un-ramified primes that split in some fixed extension number field. So let me give you a couple of examples. So here's a theorem that will permeate this talk actually. It's a result of Landau from 1908. I'm sure you must have, many of you have heard of this before and it pertains to the univariate polynomial f of x, y is equal to y and we're going to take K to be the Gaussians. So the primes, this is a Galois extension and the primes that split in this quadratic extension are just the primes congruent to one mod four. And so the second question on the previous slide is just asking about the number of integers which are only divisible by primes congruent to one mod four. And his sort of complex analytic argument tells you that, yeah, 0% of them are of this shape. And in fact, you get an asymptotic formula of the shape t divided by square root log t. Also, I should maybe also reflect on question one. So here question one is just the prime number theorem. Here's a second example, found in the Operator-Creebro book by Friedlander and the Vagnettes. So we're taking a more complicated polynomial this time. f of x, y is equal to x to the six plus y squared and we're going to take K to be a Galois cubic extension. And then we're asking about, how often you get these integers m and n such that m to the six plus n squared is less than or equal to t and such that this polynomial is only divisible by primes that split in this cyclic cubic extension. I think getting an asymptotic formula is still out of reach for this problem but we do have upper and lower bounds and that's what's carried out using the beta sieve in this book. And you get an asymptotic formula of the shape t to the two thirds divided by log t to the two thirds. So yeah, question two has some sort of resolution for this particular scenario. However, I should point out that the corresponding question one is completely wide open. So I guess from the point of view of sieves, I'm trying to promote the idea of question two and here I'm showing you that it should be easier. In general. I mean, it soon becomes difficult though. So I think this would be interesting and useful but I don't know how to do it. So can you handle this same problem for cubic extensions which aren't cyclic? That's one question. A second question. So if you picked any binary quadratic form q for which k and sorry, for which little k and for which number fields big k and you handle question two for these kind of special polynomials which are q of x to the k comma y. Okay, so those are my sieve theory questions and I'd like to go on to say a bit about this question of local solubility and yeah, the goal is basically if you're given a family of diaphanetone equations whatever that means, can we say something about the probability that an element of that family is soluble of every piadic field? Okay, so this is not an entirely new subject. I want to motivate it by this, this I think very attractive example. So you take the family of diagonal conics and so they're parametrized by these three coefficients a x squared plus b y squared, c z squared equal to zero and we just take all of these diagonal conics we order them by height so a b and c run over integers. So we have a very nice criterion here for when this conic has a point everywhere locally and in fact, actually here because of the Hassell principle that's equivalent to having a rational point and that condition for piadic solubility at least at an odd prime is just that any of the primes, for any of the primes dividing A, the corresponding genre symbol minus b c over p should be one and similarly for the other two coefficients. So yeah, it's a very simple criterion and you can use that to get an upper bound for this density of conics which have points everywhere locally and this was done by Seher in 1990 and so he counted the number of primitive triple vectors, primitive triples of integers, a b c which are bounded in modulus by b and have this property that the corresponding conic has piadic points everywhere and so he was able to show that you, that actually 0% of these are everywhere locally soluble and he was able to make that 0% very precise in the sense that it grows at worst like b cube divided by log b to the three over two. Okay, so he proved this using the large sieve 1990 and in fact, you get a similar result if instead of just fixing sorry, just studying diagonal conics if you looked at the full family of conics with five coefficients, you get the same kind of result but with a different power of log b but still a negative power of log b. However, it becomes different if you look at instead of the family of plain conics you look at the family of plain cubics, you get different behavior. So this is now you have 10 coefficients defining your ternary cubic form and you order them all by height and then recently, Bargava, Cremona and Fisher were able to show that a positive proportion of these plain cubics defined over the rational numbers have local points everywhere and in fact, they were able to pin down that positive proportion. So I should say actually that the fact that a positive proportion of these plain cubics are everywhere locally soluble goes back to earlier work of Poonan and Voloch in 2002 as part of a more general discussion of hypersurfaces but the main thrust of this paper is to really work out what that constant is and it happens to have this oiler product so it's kind of nice that you get the same oiler factor for every prime peak and essentially what this is saying is that 97% of plain cubics when ordered by height are everywhere locally soluble. So what's, it's kind of interesting what's going on here, why are we getting a positive proportion here and yet for the family of all conics we're getting 0%. So this talk is really going to be driven by families of equations over the projective line. So it's very interesting to ask about high dimensional bases and I will touch on that but most of the motivation is around families of diaphanetone equations over P1. So we saw, we've not seen any examples of that before so in SES result everything was parameterized by P2 and in the bark of a Cromona Fisher paper it was parameterized by P9. So this is an example of I guess a threefold which naturally has a vibration over P1 and I'm just calling these Landau threefolds for the purpose of this talk and you can sort of see that they have the same sort of structure as the sort of basic problem that was considered by Landau. So I'll return to this example. Let me introduce a tiny bit of notation. So in this talk rather than writing all of these things out every time by using the word standard vibration I'm just going to mean a dominant morphism from X to P1 and I'm going to assume that X is smooth, it's projective, it's geometrically irreducible and it's a variety defined over the rational numbers and I'm always going to assume that the fiber above a generic point is absolutely irreducible that's going to be my setup throughout the talk and for these the counting function that I'm going to be interested in is this. So we are counting points in the projective line rational points in the projective line which have height at most B and so the height here is just the naive standard height on P1 where you just take the maximum of the components after you've taken primitive pairs of integers as represented in coordinates and the key thing is that we want only those rational points which lie in the image of the idyllic points of X. So that's completely equivalent to just asking about the rational points in P1 above for which the fiber above them has points everywhere locally. So it's exactly that kind of counting function that Swinnipe and Dyer was discussing at the start of the talk except that we are asking only about the local solubility everywhere rather than global solubility. Good, so we could think about this counting function again in the context of these landale three folds. So we know that S for example so S and T are co-prime integers in that example at the top of the screen and we require S to be a sum of two squares. So any odd primes dividing S have to be congruent to one mod four and similarly for T. So we essentially have like an application of landales result will give us that for this particular family, this counting function should grow like B squared divided by log B. That's the example that I want to just mention here. And sort of motivated by like a Ceres example and these kind of landale type constructions. There was a lovely paper in 2016 by Dan Loughran and Arnais Schmetz where they made this conjecture about what should happen in general. So we're going to be working with standard vibrations again. It kind of makes sense to assume that the total space has points everywhere locally otherwise this question is not going to be very interesting. And there's this kind of technical assumption which asks that if you pick any of these points in P one and look at the fiber above it among the components that exist in that fiber there should be at least one which has multiplicity one at least one irreducible component of multiplicity one. So that's some sort of technical assumption. And if you have these two assumptions then this counting function should grow like B squared. So B squared is like the total number of rational points in the projected line of height B but it should die off by a log B to this power of this power of capital Delta. So let me define capital Delta quickly because it's going to be kind of important. Oh yeah, before doing that I should say that they give evidence for this so they prove an upper bound which matches this prediction using the large sieve. And there are some lower bounds in the literature but I have to say I think that's a much harder problem and it's a little bit more patchy. So what's this Delta? So it turns out it's a rational number and this is how you construct it. So you pick a point in P one, a closed point in P one and you look at the set so you look at the fiber above this closed point and you look at all of the sort of irreducible component absolutely irreducible components of this fiber which have multiplicity one. Okay, we know that there's at least one of them by this assumption in the conjecture. And then you pick a finite group gamma through which the action of the absolute Galois group on these irreducible components factors and you define this rational number. So it's just the proportion of elements of this subgroup with the Galois group which act with a fixed point on this set of irreducible components. Okay, and with that to hand, you can now define Delta. So Delta is just the sum of like one minus Delta and one minus little Delta as you go over all of the closed points of P one. So maybe a couple of comments here. So firstly, if you have one of these standard vibrations so one of the assumptions was that the generic fiber is irreducible. So that means that there's only going to be finitely many D for which the fiber is not irreducible. And as soon as the fiber is irreducible that means that any element of the absolute Galois group is going to act with a fixed point on this set of fibers. There's only one irreducible component in the fiber. So for all but finitely many closed points D this little Delta is equal to one which means that this sum over Ds over closed points D in the definition of capital Delta is really just a finite sum. Maybe I could give an example of its calculation. So let's go back to this Landau example. So if you, so that there are, the fiber is a, is a quadric surface, is a quadratic surface. It's, you know, for most, so it's only going to be reducible when S or T vanish. So when S and T are non-zero then you, the fiber is irreducible and so this little Delta is equal to one. If S is zero, so that corresponds to the closed point zero. So if S is zero, then, you know, the, this fiber is, is, is, you get these two components which are conjugated under the Gaussians and quadratic extension. And so therefore you really only have to look at, yeah, there's really just this C, Z mod two Z which acts on this, on these fibers and only one of them is going to fix the components. So this little Delta is equal to a half. So it's just the identity which is going to fix the, fix the component. And so it's the same for the closed point at infinity. And so that's telling you that Delta is equal to a half plus a half, which is one. This is, this is, this is consistent. If we just go back to this, this conjecture. So the conjecture is telling us that we should expect for this example B squared divided by log B and that's indeed what we, what we saw at the top of the screen. Okay, so I think this is a really nice conjecture. It's sort of generated a fair bit of research, you know, and anytime you have something like this you can immediately think about how you might, how you might extend it. And that's what I want to do today and I'll just mention a couple of ways that you can extend it. So firstly, you could try and make some predictions for the leading constant in this asymptotic formula. And that's now been done. This was done last year to do some beautiful work of Dan Locke from Nick Rome and Epithemius Sophos. So firstly, they make a conjecture for the leading constant and they back it up by upgrading and says up a bound to an actual asymptotic formula. So they calculate the leading constant here and it's kind of interesting as it has this structure as a product of local densities but there's these other two factors like the number two and the number pi over three, pi to the three over two, which take a bit of explaining and then that's what they end up doing in their paper. So I recommend taking a look. You could also ask about moving away from P1 as a base and maybe looking at some higher dimensional things. And I should say that in their paper, Locke and Schmetz actually do work with five vibrations over PN and you get the same sort of picture arising. But this was something that I spent a bit of time with when Roman Serpent was in my group and we looked at an example. So the basic question is, what can you say about other vibrations from X down to Y? But now you just assume that Y is some fan of variety. So it's not P1 anymore, but just some variety maybe for which you know how to prove Manning's conjecture. So you know how to count the number of rational points of bounded height on the variety. So can you say something about, the number of rational points on your variety, the bounded anti-economical height such that the fiber above the point is everywhere locally soluble in this vibration. It's a natural enough question. So if you kind of naively conjoined the Manning conjecture and this Locke and Schmetz conjecture, it would suggest that this counting function should grow like B times some powers of log B. And so you should get log B to the row minus one. So this is what you would see in the Manning conjecture where row is the rank of the Pickard group of Y. And Locke and Schmetz would be suggesting that it decays by this log B to the delta. So delta is this invariant that's associated to the structure of the vibration and the behavior of the fibers. So we were, sorry, yes, we explored this in a very particular case. So if we take a split quadric in P3, so this is a smooth Fanno variety. It has higher Pickard rank. So it has Pickard rank two. And we took a particular X. So this was the X that we took lies inside P3 cross P3 and it gives you this morphism from X to Y, the fibers of which all these diagonal quadric surfaces. And it turned out that you get something slightly weird happening. So as part of the summer project, we calculated that delta for this problem is two. But the counting function is bounded above what basically has order of magnitude B. So I, yeah, so just to point out that the expected exponent of log B according to that conjunction of conjectures on the previous slide is row minus one, minus delta. As we saw, delta is two. So we were expecting to see like B divided by log B for this counting function. So that's kind of slightly surprising to us. We've looked at it a little bit and it seems like there might be some thin sets inside Y, thin sets of rational points inside the rational points on Y, which if you remove them, you would get an asymptotic formula, which is a little bit more in keeping with the expectation we had on the previous slide. So I think there's some forthcoming work in the pipeline here of Cameron Wilson, a PhD student at Zimios Sophos, who's been studying this. And I think that's the sort of co-thin set that you need to take to make these work. Okay, well, that was two ways in which you could think about generalizing this conjecture or extending the conjecture. The one that I want to focus on, actually, and the one that this previous on the archive is about is this technical condition that they had to think injecture. So they had this condition that above every closed point, there should exist at least one irreducible component, which has multiplicity one. So what happens if you drop that condition? Yeah, so that's what I've written there. Well, it turns out that it seems to have a very strong effect. A very strong effect indeed. And in some sense, our paper is trying to make more quantitative, an observation of Koryo Telen, Skorobogatov, and Swinness and Dyer in the late 90s. So they played around with these situations where you have vibrations with multiple fibers and they wrote down a number of examples. This is one of them. So you take basically a standard vibration that is built from taking a proper model of this affine variety. So T is the variable for the affine line. And so C and D and F, they're polynomials with coefficients in the rational numbers. And you can see that for fixed T, this is an intersection of two quadrics. And if you sort of think of it projectively, it's an intersection of two quadrics in P3. So it's a genus one curve. So this is a vibration in whose fibers are all genus one curves. So it's a bit more complicated than the examples that we were looking at before where we had conic vibrations or quadric surface vibrations. We have higher genus curve vibrations. What they actually proved was, well, two things. So firstly, they proved geometrically that you get double fibers occurring above the roots of F inside X. So this is an explicit example where double fibers actually occur. And moreover, so what they actually proved was that there are only finitely many integer points inside P1 in this situation that, sorry, in the situation that the degree of this polynomial F is at least six. Then they showed that basically the rational points upstairs, the image of the rational points upstairs lie in, you know, finitely many points downstairs. So that is, well, sorry, yes. And I should point out that in fact, something stronger follows from their argument. This was something observed by Loughran and Livian Madison, was in fact, even this local solubility counting function is finite. So whatever the value would be, there's only finitely many elements of P1 which lie in the image of idyllic points on the total space. Provided that the degree of F is at least six. I think there's some conditions on C, D and F as well. I think F needs to be even degree and square free and maybe co-prime to C minus D, something like that. So yeah, so that technical condition that they had in their conjecture, might have seemed somewhat strange, but if you remove it, you get drastically different behavior. And that's what we're seeing here. And yeah, that's what we wanted to do was to try and investigate this a bit further and put it on a more quantitative footing. So maybe I should, I mean, one of the things that we struggled with in this, I mean, it's always nice to have sufficiently explicit examples so that you can kind of see what's going on. Like in the Landau example, we could very easily see the kind of multiplicative constraints that were placed on S and T in order that the fiber is everywhere locally soluble. And it would be nice to have some examples in this setting too, but it turns out that there aren't the same kind of conic bundle type examples in this world. So there's a famous theorem of Grave Harrison Starr, which implies here that as soon as you have double fibers or multiple fibers, the generic fiber cannot be rationally connected. So in particular, it cannot be rational. So in particular, it cannot be a conic or a quadric. So they are necessarily more complicated like this. So yeah, so having these fibers that are higher genus curves is the norm in this world. And it can be a little harder to see what's going on explicitly. So here we closely follow some ideas of some very influential ideas of rhetoric campana. It was a geometry, but explored this kind of questionnaire, ethnically as well. So if you have a standard vibration, you can associate an overfold to it. And the way you do that, so an overfold is just a pair of a curve and a Q divisor. But here we're having a curve is just P one. And we build this Q divisor in a specific way. So for each closed point in P one, we just write down M, M sub D. So it's a positive integer. And it's just gonna be the minimum multiplicity that happens in the fibers. So above D, you've got a bunch of components, irreducible components. Some of them are gonna be, sometimes there's not gonna be any multiple fibers in which case you take MD to be one. Sometimes there are gonna be multiple fibers. And you just take MD to be the minimum multiplicity that occurs in these fibers. And then you just define this, it's like strange looking Q divisor, D sub pi by taking a sum over these closed points of one minus one over M as a coefficient of that divisor, of that closed point. So again, generically these things that the fibers are absolutely irreducible. So generically that means MD is gonna be one. So again, this looks like it's an infinite sum but really it's just a finite sum. Good. And then the basically key idea, which again goes back to Campana really is this idea that we can interpret this local counting function, sorry, local solubility counting function as a problem about counting Campana points on this orbit fold. Okay, so that the topic, so independently of this story, the topic of counting Campana points on orbit folds has sort of received increased interest and attention over the last few years. It's sort of a world where it kind of lies in between the world of rational points and the world of integral points, which I think everyone agrees are too difficult, but Campana points are still expected to exhibit the same sorts of things, properties and behaviors that you would see in the world of rational points like a man in type conjecture, some kind of like a brown man in obstructions and all of this stuff. And in fact, there's a nice paper by Piro Pan, Smith's Tenemoto and Arie Albrado where they phrase or they articulate what a version of the man in conjecture would look like for counting these Campana points of bounded height for much more general things than P1 and this divisor really just for, in a much more general setting. So anyway, there is that story, which I think is important, but what's gonna be key here is that, yes, we certainly need to be counting Campana points, but in order to really detect this local solubility business, we are also having to throw in some extra conditions, some extra multiplicative constraints, which are gonna come from some kind of application of the Chebotarov theorem. So with that in mind, so these are associated to any of these vibrations, we can associate this all before base. And then we have this, this is the first conjecture. So if you assume that the, that this divisor written there is ample, then this counting function should grow at most like B to the two minus the degree of D plus epsilon. So actually the condition that minus KP1 plus D is ample is equivalent to two minus the degree of D being positive. So it's equivalent to that exponent being positive. So this kind of general topic actually goes back like this sort of bounding points on, campana points on varieties goes back to a beautiful talk by Bjorn Poonan in 2006, which I remember seeing at the, there was an MSRI program. And this pertains to a particular base. So you take the protective line and you just take three, three divisors above which you have some higher multiplicity. So you take a half times the close point zero, half times close point one and half times the point of infinity. And when you unravel the definition of, campana points, what is a campana point? What's a rational point in this setting? It just turns into this very charming and concrete problem, which is asking about counting positive integers, A, B, C in some range. They have to be square full. So if, you know, the prime divides A and P squared should also divide A. And they satisfy this linear equation, A plus B is equal to C. So Bjorn was popularizing this back in 2006. And I think we still don't know how to solve it, sadly. In this case, the sort of prediction for the exponent is two minus the degree of this divisor, which is three over two. So the expected exponent is a half. So we're really kind of expecting that this counting function grows like B to the half. You could think of this different, I mean, square full numbers have the same approximate density as squares. And we know how to handle this problem for squares. It's just counting Pythagorean triples. So certainly B to the half is a reasonable thing to believe in this setting. I think the current record is still B to the three fifths plus epsilon. So you're not even able to handle this, this very simple looking problem. However, what we'll see is that it is much easier to handle problems when you have just two divisors, sorry, yeah, two close points above which there's some higher multiplicity. This problem of three, three multiple points is much more challenging. Yeah, so anyway, I want to try and, you know, the motivation in this conjecture, well, the motivation next is to try and make this conjecture a bit more explicit, you know, is this epsilon acuding some explicit power of a logarithm and what is it? So the first ingredient that we had to sort of develop in this setting was a sort of more general criterion for testing local solubility. So if we pick a point in P1, a rational point in P1, we take a sufficiently large prime P where sufficiently large is just relative to the data of this vibration. And we suppose that, you know, the fiber above that point has peatic points. Okay, so what can we say then about X is there's some condition on X that we can write down? Well, this is the condition that comes out of the work. So if you pick any close point above P1 and, you know, a close point in P1 is basically the same thing as a binary form, H. So this close point we're thinking of as a zero set of a binary form. Then there are two possibilities. So either the peatic valuation of H, this binary form evaluated at this point, X is bigger than MD or it's equal to MD, but then you get this extra information that the Frobenius element at P has to fix one of these components in the fiber. So this is a slightly different kind of time. This is a slightly different set to the one I defined before. It's very similar, but we are, you know, above this close point D, we're looking at all of the geometrically irreducible components, but we're only gonna look at those which actually have multiplicity equal to this minimal multiplicity in M sub D. Tim, there's a question in the chat. Sean, maybe you wanna unmute and just ask away directly. Yeah, on the previous slide, could you show the previous slide?  Yeah, well, that's literally the definition I gave before, which is counting points in P1, which lie in the image of adelic points. So there are no questions, there's no question of container points at that point. Not, there isn't a question of container points at that point, but it is gonna follow from the sparsity criterion that I was just describing that you are basically reduced to counting container points. On the top, on the total space? On P1. On P1, on P1, okay, okay, thank you. Yeah, thank you. Thank you for the question. Yeah, so that's, well, that's kind of what you can see here. So I guess, you know, to illustrate this, if we were looking at the close point above, the close point at zero, so the point zero comma one, then the binary form is just X naught and you can sort of see that it's saying that, well, basically that, at least with respect to sufficiently large primes, X naught has to be M naught full. And, you know, if the prime divides X naught exactly to the Mth power, then you get this extra information that this Provenius element fixes one of the irreducible components in the fiber. Yeah, so it's basically telling you that you are reduced to a container point on P1, but you have this extra kind of Provenium condition. Good, so let me say a little bit about what happens for one fiber, one multiple fiber, because that's a little easier to see what's going on and we prove an upper bound. So we sort of refine that epsilon, you get B to the two minus this expected exponent. And then you get the same log B that occurs in this Locke and Schmades conjecture. So here, I'm really just taking the, assuming that there's one multiple fiber at the origin, sorry, at zero. So yeah, just rapidly, I guess I already kind of discussed this, so the sparsity criterion tells us that we're just counting now pairs of integers, which are co-prime, which lie in a box. And I get that X naught, so the first integer has to be M naught full. And for any of the other divisors that occur in this vibration, this binary form H evaluated this integer, X naught, X1 should lie in this multiplicative span of the primes that split in some appropriate extension associated to the divisor. So that's just what that notation means. I just mean that any prime dividing this binary form evaluated at X naught and X1 should be a Frobenian prime for some predetermined Frobenian set of primes. And yeah, it turns out that this is something that you can also use the large sieve to estimate. Good. So here's another example. So these are the health and surfaces, sort of nice family of surfaces. I think they were introduced in the late 19th century as I think they're classes of surfaces which admit vibrations to P1, the fibers of which are genus one curves. And I think they, so they're elliptic vibrations, but they're actually all rational. So they're kind of a very special class. And so I'm gonna assume that there's a triple fiber now above the point of a closed point at zero. And I'm also gonna assume that above the point one I, there's a reducible fiber, but I'm gonna assume that it's split by a cyclic cubic extension. So I just mean that yeah, it's not geometrically irreducible. And if you work over this cyclic cubic extension then all the components are defined over it. So again, if you go through that sparsity criterion, you get a similar kind of conclusion. So firstly, X naught, yeah, so now I, yeah, sorry, I think that I mentioned a sparsity criterion. I just mentioned a necessary condition, but you can also, so part of the work was to get a necessary and sufficient condition under suitable assumptions. And that takes a bit of work. It's a bit above my pay grade. It uses sort of log geometry and things like that. But you can use it here to actually determine that X naught is not just cubed full, but actually a cube and the binary form associated to this closed point is just the sum of two squares. So we're asking that X naught is a cube and that X naught squared plus X one squared lies in the multiplicative span of primes that split in this cyclic cubic extension. But that's a problem we saw before right at the start. So that was one of these sieve problems that I was trying to advertise. And this was sold for us by Feedlander and Appanards. And we get upper and lower bounds, which are matching. But yeah, it'd be nice to have more success stories with the sieve, which could then be used to investigate this particular problem. So I'm on my last slide. So I just mentioned it's very natural to try and think about this epsilon when you have more than one multiple fiber. So yeah, let me just go back. So when we had one multiple fiber, we were able to prove this upper bound and it was a kind of a predictable upper bound. It was kind of B to this exponent, which can be less than two. And we've got the familiar exponent of log B in the denominator, just as we saw in the Locker and Schmetz work. But yeah, but it turns out that if you have more than one fiber, that all goes out the window and we sort of get this kind of slightly surprising and crazy looking conjecture. So we are gonna assume again, that we've got the standard vibration, that the relevant divisor example, obviously it makes sense to assume that the total space is everywhere looking soluble. I haven't defined all of these terms. They are, if this peaks your interest, you can look in the paper. But we basically construct a morphism or polymorphism and the conjecture is that this exists and it has the structure of a G-torser, the finite tail group scheme G. And actually this counting function should grow like it's the same thing on the top. So B to this exponent, two minus the degree of the divisor. But on the bottom, you get something slightly wild. You get like an exponent of log B which is a minimum of all of these different rational numbers, capital Delta, associated to different elements, different sort of twists, V in this parameterized by this H1. So yeah, these, if you, so the V's parameterized these twists of this morphism and the pi V that occurs in that exponent is a standard morphism you get by normalizing the pullback of this morphism pi along this twist.  I just wanted to make a couple of comments verbally. So if it turns out that if you are in the situation that there are no multiple fibers occurring, this immediately recovers the original Locke-Rensch-Maitz conjecture. If there's one multiple fiber occurring, then it turns out that this orbit fold is simply connected and so this H1 is trivial. And you actually just collapsed down to the exponent that we saw, sorry, on the previous slides, so just the Delta of pi. And we, I guess in the paper, we try and prove some evidence for this slightly weird looking thing and we prove an upper bound which matches this when you have two multiple fibers over zero and over infinity. And we also provided an example which shows that you definitely need to take something smaller than what the Locke-Rensch-Maitz analog would suggest. So you definitely need to take this, it seems like you need to take this minimum. So we proved and provided an example which shows that this is correct and different from the kind of naive analog of the Locke-Rensch-Maitz conjecture would suggest. Good, thanks very much for your attention.


Delighted to have you here with this deep act. I will hand over to you. We look forward to your presentation. I welcomed you all. Good morning. Good afternoon and good evening to all the participants. So let me begin. So let's begin by looking at what we refer as this digital first or rather we can also term it as digital revolution. But when we talk about the disruption that this digital revolution has brought or you can call it the changes right so is that the is this the only time the world is witnessing disruptions or changes at this scale when the answer is no right from very early on since the days of agricultural revolution and then followed by industrial revolution and of course then by internet or computer revolution. So we have as a world seen you know huge amount of changes in the way businesses operate and with each of such revolutions there are newer possibilities there are possibilities to redefine every aspect of the businesses while there are some negatives right while we see that you know we feel a lot of businesses getting bundled out and are not able to withstand the changes that come at such a fast pace at the same time on a positive note. Let us look at a perspective of how we can embrace these changes how we can accept the changes and use the opportunity in hand to define new way of doing business that brings me to the next slide which is the business 4.0 from a perspective defined by us at TCS and I am happy to share that some of the details that I am going to talk about from how digital first is impacting commercial aviation would be based on this perspective that I am going to explain now if you look at the key for business behaviors from business 4.0 perspective so on the left you see embrace risk right so we all have seen how it has impacted so many organizations which I just mentioned you know had to get bundled out right so moving beyond rigid planning and operational barriers with an agile strategic approach is what we term as embracing this it is very very important in today's perspective of doing business in business 4.0 at all leveraging ecosystem which means collaborating with partners inside and outside the supply chain to create new products and services on the right I have customized which is another business behavior that you can relate to which is nothing but personalizing the products and services to a market of each and every customer and even to the level of transactions that customer do and not just that but at a scale at a very large scale and finally but an important behavior which is to create exponential value which is nothing but adopting business models that leverage value from transactions at multiple levels and address the new markets and the new possibilities and to enable these four new behaviors or the behaviors this new era we have also coined four technology enablers first one of course being you all have been a part of it that it's it's in our day-to-day lives now so use cloud in IP and processing services I don't have to talk more about it intelligent is nothing but to develop new business model and explore that how we can give provide more and more customer value agile this is in particular very important as of today so initially we started implementing or utilizing the agile concept only from software or IP perspective but as of today it is all the aspects of operating businesses that need to be agile and then automated which is free of workers or our people so that they can focus on more valuable time well you can go on to tcs.com and get more details on this perspective and I use these terms at the start so that we can utilize these terms as we go forward in the presentation now let us look at how commercial aviation pairs against other industry verticals in the adoption of these four business behaviors that I just mentioned and you will see that there are of course leaders early adopters and followers in the category that we have put this entire section of industry and if I have to compare commercial aviation so we can see 10% of the leaders are from the travel and hospitality which is commercial aviation and the related supporting industry and if you look at the early adopters so yes there is again a sizable amount 12% of the early adopters comprises of commercial aviation organization and in the last group which is the followers there are 8% of it so as you can see there are you know these statistics are part of a very you know a detailed and a global research study that was done by ECS for you know senior executives across the industries and my thought was to bring in the flavor of how this adoption is happening for the commercial aviation industry but now let us look at the adoption of digital technologies and capabilities here I have added specific technology capability so starting from the cloud automation AI IoT and blockchain special mention to AI is that this is one idea where commercial aviation has been adopting at a faster pace than the other vertical right so with this it just brings things to perspective that where the focus for each of the industry verticals are and it also tells that you know nobody is far behind so everyone is trying to write this wave of digital transformation with a very little margin right and the current time is going to be very very important how we shape things for the businesses of future now while we are talking about businesses it is very important to also understand about the customers or the users or consumers right and I thought this perspective would really help understand a key element of the business 4.2 perspective that I shared at the center of the picture I had mentioned abundance and this is a perfect example of abundance since 2007 and you know the expected smart phone user in 2021 is going to increase to almost 50% of the entire population of this world it's a huge huge shift from the consumer behavior or the way consumer has changed from how they used to operate earlier versus now right so this brings the very very important perspective for the businesses as well to know and so now when the customer or the consumer is using smart phones what are their needs what are the kind of challenges the businesses may have right so that brings us to the perspective of the customer side of things which is you know they are looking at you know the products and services to be priced very competitively they want very good experience they want their services to be near their time best value flexibility personalization and and you know new demands and challenges keep on adding right so with that perspective the organizations also are the businesses also have to be prepared and provision for such demands and challenges and hence digital that brings me to the how part of digital while we talk about you know various aspects of digital first or digital transformations let us look at a view of how any particular organization or enterprise needs to talk to this new business 4.0 technology enablers while there is a need for digital platforms to be developed by any enterprise this is the perspective of adding the the various elements into the digital first of it in the form of digital building blocks of any platform let me start with enterprise data the data that the enterprise is receiving or is generating is you know ever increasing and it's increasing in leaps and bounds this data needs to be transformed so that it can be consumed as an information so the first thing that is required as the part of this platform is to have a set of digital transformation service which can enable this transformation of data into information once we have information but all of it can be consumed or can be utilized by everyone so we need to come to a point where specific visualization of data is important so that that gives me my KPIs or that gives me you know dashboards of for various you know operations or planning or you know any of the aspects of my business right and while I get these key indicators right the KPIs are my key numbers these numbers in silos are not enough for me to take any decision I need to have a view where I can correlate my information from various sources which requires or rather essentially means that we need integration services to correlate things and services I'm using terms that not only mean integrating a few systems amongst each other but it can be events it can be you know any data it can be a real-time alert or notification it can be an event or data in any form that we may have to integrate and correlate and once we have that correlation configured we will be able to make much more informed decisions and take actions related to my business now all these actions and decisions needs to drive my core business services and through that those core business services then I can see realize my business outcomes and the cycle goes on I need to then look at the outcomes get the feedback see where I need to increase where I need to maybe fine-tune a little bit so that I can go full circle and redefine my processes in such a way that I know what kind of outcome I am expecting or I need to achieve so that calibration needs to happen you know in a ongoing manner with that let me quickly take you through two case studies which are related to digital transformation that I was part of the first one is about identifying digital building blocks and a roadmap for an airport which had faced a similar problem or a similar challenge that any enterprise faces without having an overarching digital transformation strategy on how they are going to transform their business it's not just about IT systems but it's about how you drive your businesses through your IT and as you can see in this particular slide at the core we had the key in the information systems and the landscape from the airport's enterprise landscape and then surrounding that we had given them a roadmap to develop 5 key enabling or you can say digital building blocks the first one being the enterprise chain management and the data application services the next one was integration the integration integrated service management and a common security model and all these services were enabled through a composite platform services so the point I was trying to drive here is that whenever we face any challenge at enterprise level it needs to be addressed with not only from a particular part of the organization but it has to be looked at how it will impact at the enterprise level and that was precisely we were trying to do in this particular exercise or assignment that we worked on and on the right hand side we have shown how we have given a roadmap of parallel work streams for the organization to start executing on the path of digital transformation so those parallel work streams where first one to have the preparatory work and organizational readiness of this is a very very important aspect is it's not just that a particular team within the organization or particular department in the organization is ready it needs to be the readiness has to be at organizational level and many times it can just be a cultural thing which needs to be handled very very in a sensitive manner and that is very crucial to start once you have this readiness you need to have a core technology team who will take care of the various technology transformation streams right so the ones that were identified for this particular scenario was the technology transformation for data, BI, integration and security the next level was to come up with the operational and the governance of the data and BI and along with that there were a few strategic areas which were identified from the list of the whole lot of initiatives which were in pipeline now if we go step by step completing the first four steps it was much easier for us to identify which initiative or projects should be prioritized right so that kind of you know comes along with the planning that you do in the initial phase of the first four streams that we did so therefore you know one of the consulting assignment that we recently did now I will come to another engagement where we were helping with our journey for an airline again a very similar scenario where the organization wanted to utilize and grow in the robotics process automation capability whereas we figured that this was coming from a very specific group within the organization and we tried this not only a solution that fits to the needs of that particular department in the organization we felt that there is a need on taking the solution and scale it up to the enterprise level in the phase one it was about the evaluation and selection process and we also helped the enterprise and the team in the organization to look at you know the various strategies on getting the right product and we completed a first POC in the phase one to ensure that we are able to demonstrate how this is going to bring change to the business of the customer in the second phase we then set up four processes and then did this completed this project on time as per the needs and requirements and I mean to everyone's delight it had the results we were looking for in the phase three we took the learning from the first the second phase and then created a factory model to be able to govern and execute large-scale automation of similar requirements at enterprise level and then finally in the phase four which is the current phase and currently ongoing we were able to institutionalize the RPA across enterprise and take this journey towards an advanced automation beyond this time level so I guess as per the time limit given to me I also had to leave some time for the question answers so over to Steve. Thank you Deepak this is Steve that was that was great thank you and yes you're right right on time thank you for staying on time it's very important so first question that came in for you Deepak is is the airlines have been obviously greatly impacted by the by the pandemic do you see the need for their digital transformation to be even greater now or are they just simply going to have to put everything on hold because the money is not there well I mean I would say that you know the airlines are the worst hit from all the industries if you look at the impact of the current pandemic and I say I would see that airlines are taking a measured approach as of now there are two aspects to airlines right so one is the operations and the other is you know all the other business to generate you know revenue and other aspects of airlines now as you can see that you know the revenue stream has cut and the operations are at their minimum it is very important to take measured steps because this situation is not going to stay forever and the moment things improve I think airline needs to be prepared to take that you know increase as well as there will be changes in the behavior from from customers right so at the very initial phase I would see the importance of having prepared for getting the business back to normal right and then in the second phase to deal with the changes in the customer customer behavior and that both of us this would need the organization airlines to be prepared in their digital status absolutely coming out of this is going to be so important next question that came in can you talk to any AI use cases in the aviation industry you've come across all right and in fact there are many so we have been talking about bots right so the first the reference that was there in my introduction was about the chatbot that we developed and it was a very basic use case to start with which was just to you know do a wave with the lengthy FAQs that typically any airline has can I do this can I take this into my along with me in my journey so that's so on right so for travelers there are so many questions and go through those you know pages of FAQs was very cumbersome to be kind of digitize them all although they take you into a box where it was a conversation and where it was also available across channel so be Facebook messenger or any other digital channel that was a simple example okay okay thank you so in the in the second example that you get case study example that you gave Deepak the RPA one what sort of processes were automated was it frontline business processes or internal IT operations and can you give any specific examples right so it was a core business status so I think I gave an example when I was talking to the adoption slide so when we talk of automation right we are in the looking at how we can free up our workforce from doing you know mundane things right that can be done otherwise so for example if you look at airline industry there are so many you know standards and operating procedures and you know certain steps that needs to be completed before you know we can give a go ahead to the next task right some of these involve manual intervention but there are many that can be done without and as the initial step what we did was to eliminate those you know processes from the you know people where there was no manual intervention just to give an example then an air a new a new airplane is inducted there are a host of processes involved in this activity a lot of that can be automated can be done by applications then waiting for someone to just do a tick mark on that diary so that was an example that I wanted to cite okay all right great well we are right on time and deep back thank you for keeping us that way and I very much appreciate your your presentation and a virtual round of applause thank you very much be back


If you're new to the channel, I'm Gulf Pony, I do new car truck SUV reviews on YouTube. And today, we are in the brand new 2023 Mitsubishi Eclipse Cross courtesy of younger Mitsubishi in Hagerstown. Marilyn, for more information on their inventory, please feel free to check out the link in the description box below. And so we are in this one today because there are actually several changes for the 2023 Eclipse Cross. Of course, we'll be going over those in this video, but not only that, you get America's best warranty as well being five years, 60,000 mile bumper to bumper, 10 years, 100,000 miles on the powertrain, but not only that, if you were to get this at younger Mitsubishi in Hagerstown, Marilyn, you also get a double powertrain warranty being 20 years, 200,000 miles that you can apply to anywhere in the US and have it completely valid. So I wouldn't mention that as well because it doesn't get any better than that. Who really keeps their cars for over 20 years anyways? But ultimately in this video, we will be testing out and going over everything about this one from acceleration to braking, steering, feel, ride quality, sound system, exhaust clip, all that fun stuff. So having said all that way, you guys say let's just go ahead and jump right into it and as always, let's start with pricing. And so as you can imagine, there are several different trim levels for the 2023 Eclipse Cross. First one being the ES starting at $25,795. Then you have the LE for $27,045, SE for $28,045, SE package starting at $29,045, which is the one we are in today starting at $29,295 and the SEL Touring starting at $31,395. And so for the 2023 model year, that super all wheel control system is now standard across the board. Unlike for 2022 where front wheel drive came standard, that super all wheel control is now standard across the board. So that is definitely a big win in my personal opinion. But anyways, powering the beast is a 1.5 liter turbocharged inline four cylinder putting out 152 horsepower at 5,500 RPM, 184 pound feet of torque coming in at 2,000 RPM.  It was originally created for rally racing. So racing on the snow and the dirt. So that pretty much tells you what was meant to be an off road system to start with. So that is pretty cool. Sent to the ground through a CVT zero to 60 time coming in at approximately 8.6 seconds with MPG numbers coming in slightly different headed upon the trim level that you go with. So for the ES you get 25 of the city, 28 on the highway, but for all other trims, 25 of the city, 26 then on the highway taking regular unleaded fuel. So that before we do any kind of fun acceleration or paddle shifter test here in the Eclipse Cross. Wanted to mention there's a couple of drive modes.  But ultimately there's drive modes will adjust things like the shift points of throttle response and actually the traction control settings then as well. So like I alluded to there, we do have paddle shifters and that's just going to be for the SDL trim level. So having said that, let's go ahead and find a straightaway. Let's put the paddle shifters and acceleration here to the test at the same time. Keep in mind this is going to be simulated shifting because we are in a CVT after all, but still let's see how quickly they react and let's see how quickly we can get our new 2023 Eclipse Cross here up to speed. All right, so before we do this to put it in full manual shift mode just slide the shifter all the way to the back and to the left. It is then going to display what simulated gear you were in up on the gauges there. I'm also going to go ahead and slide it into first gear and these paddle shifters are absolutely massive and they feel kind of high quality as well which is pretty cool. So right now it is holding the gear. It's not shifting, which I'm kind of surprised. That's awesome, but here we go. Okay, paddle shifters are quick. Surprisingly, not the quickest vehicle in the world but the paddle shifters are dang quick. It's kind of cool. I kind of like them because like I said they do feel high quality. They're absolutely mammoth like Maserati style paddle shifters these things are gigantic which is really cool. My personal opinion, but again, not the quickest thing in the world. It'll be enough to get you merged onto the highway but I'll pretty much just leave it at that but then to give back full control to the Eclipse Cross you just simply slide the shifter back to the right in case anybody was curious. But anyways, to go along with that acceleration as always braking is equally important. So upfront you will find 11.6 inch ventilated front discs in the back, 11.9 inch solid rear disc. As far as that 60D zero stopping distance goes it's going to come in at 125 feet which is pretty much right on par for the segment there. A lot of SUVs will come in in the 130s and sports sedans come in in the one teams. So 125 feet is probably right where you want to be at for the Eclipse Cross and having said that braking feels not a soft braking feel definitely leads to the firmer side of things which I personally appreciate. It feels just like I personally would want the Eclipse Cross to feel like. So 100% braking is on point without a doubt.  As far as ride quality goes in my short little test drive here today it's actually good. I know that Mitsubishi reworked the suspension to kind of soften up the ride for the 2022 model year. So that carries on of course for the 2023 model year. So it is a pretty darn smooth ride for not having an adaptive suspension or an air suspension or anything fancy like luxury vehicles give you. So no issues for me as far as Cabanoys goes we are going 55 miles per hour, 55 stay alive and it's okay. It's not bad. I get a little bit of wind noise coming from the driver's side window here but honestly it's nothing that would annoy me. It's perfectly fine. It's pretty much as expected. I'll put it that way now. I've actually heard warriors so definitely no issues there. As far as steering feel goes again it's right where I would think it would be. It's not a loose steering feel. Believe it or not a lot of SUVs do give you loose steering feels. It's not a heavy steering feel either. It's just where it should be basically. Then as far as visibility goes 100% on point I think it was the 2021 Eclipse Cross that kind of went across the middle pack there where you kind of impeded visibility a little bit but then last year they changed that. So now visibility at least out the back is 100% perfect. Did what I also mentioned then when it comes to forward visibility rain sensing windshield wipers come in the LE trim level and up. Having said that there has been no rain today whatsoever and they have already come on for me so I don't know what's up with that but I do like rain sensing windshield wipers though. I will say that they are incredibly convenient so you don't even have to touch the windshield wipers when it starts to rain kind of like automatic headlights I guess you could say. And believe it or not you can actually get a head up display for the Eclipse Cross as well if you were to go with the SEL or SEL Touring it is optional for those two trim levels I'll put it that way it doesn't come standard but there's a package option that can allow you to get that if you wanted it. But anyways that about rounds out the performance segment of this review guys let's now go ahead and take a look at the exterior of our brand new 2023 Misubishi Eclipse Cross. All right so here she is you guys the new 2023 Misubishi Eclipse Cross finished in Labrador Black Pearl. In case anybody was curious of our exterior color. Another little fun fact though before we get started on the exterior this one is made in 96% Japan. Typically when I look at the window stickers of these vehicles a lot of the times they are spread across the globe as far as where the parts come from where the assembly is done all that stuff this almost is a 100% JDM vehicle so basically everything on this thing is coming from Japan so a little fun fact for you but anyways let's go ahead and start up front of the Eclipse Cross here. LED headlights are now standard on all trim levels of the Eclipse Cross it did not used to be that way in the 2022 model year because last year the ES and the LE trims came with halogens but now even the ES the bottom trim level comes with LED headlights so that is a huge win for the Eclipse Cross and I absolutely love that. LED daytime running lights also coming standard LED fog lights also standard for all trim levels across the board as well. Automatic headlights coming with the SE trim level and up automatic high beams coming with the LE trim level and up and then silver accenting can be found on the front bumper for particular trim levels. I do like the black accenting that we have on this particular Eclipse Cross though because we have the black exterior and I think that looks dang good. I also like the chrome accents surrounding the headlights because it goes very well with the chrome misubishi badging up front and the chrome Eclipse Cross lettering found on the front portion of the hood so definitely a very nice looking front end to the Eclipse Cross without a doubt but that pretty much rounds out the front end. Let's now go ahead and make our way to the side. Let's open up since we are around to the side of this one roof rails are going to be optional for all trim levels with an SE and it meaning the SE and the SEL trim level so they're gonna be optional we don't obviously have them today but chrome windows surrounds do come standard rear privacy glass does come standard for all trim levels as well. Taking a look at the side mirrors they are body colored power adjustable side mirrors that will get integrated turn signals they will be heated as well. All of that comes standard for all trim levels usually don't find that on other manufacturers so big fan of that. If you were to go with the SE trim level and up the side mirrors are also gonna be power folding then as well that's pretty cool. Matte black side skirts will come standard can't really tell the difference though with the black exterior if you were worried about that. Taking a look at the wheel setup 16 inch two toned alloys coming with the ES 18 inch black painted alloys coming with the LE and 18 inch two toned alloys then coming with the SE and SEL trims therefore that is obviously what you guys are looking at right now but again a very good looking side profile but now let's go ahead and make our way to the back because it is gonna differ slightly dependent upon the trim level that you go with let's go ahead and make our way to the back. So all the way to the top of course you will find a body colored shark fin antenna but just below that that rear spoiler is going to substantially differ amongst the trim levels and so you will get a kind of more tamed rear spoiler for example with this SEL trim level that we have today but for the SE trim levels and then it's also optional on like to say the LE trim level as well but you get a much more aggressive rear spoiler with the SE and again optional on the LE as well so LE detail lights do come standard you do have some trim level badging on that rear tailgate as well so if you end up wandering onto a Mitsubishi lot that is a surefire way to distinguish what trim level you are looking at so obviously the easiest way. Down below you will find a single exhaust outlet tucked away underneath on the passenger side there nothing fancy there but having said that I do believe you guys know what we have to do next as always here is that exhaust clip. So now since we are around to the back of the Eclipse Cross when it comes to opening that rear liftgate it is a manual liftgate so simply just lift up on the liftgate itself and it's gonna obviously open up for you but once opened up cargo capacity comes in at 23.4 cubic feet which is a decent amount of space if that was not enough space though there is a 60-40 split meaning the rear seats do fold down bumping that up to 50.1 cubic feet obviously with the rear seats down there there is a cargo cover that comes standard back there I like seeing that there's some cargo lighting there's several tie down anchors there is a grocery bag hook and then if you were to lift up underneath of that cargo floor you will actually find a spare tire which I personally love as opposed to the fixer flat so that is a big win for me personally then taking a look at the rear legroom that comes in at 37.1 inches so for reference I'm an even six feet tall this is how much space I had in the back there there is a passenger side seat back map pocket as well rear center armrest with cup holders there is a 12-volt power outlet heated rear seats are actually available on the SEL trims we don't have it today but another one of those package options available for the SEL trims if you wanted to spoil your rear passengers a little bit but what surprised me was that there is no rear ventilation for the rear passengers so Miss Abishi if you were watching this wouldn't mind if you added rear ventilation I think that would definitely help with the sales of the Eclipse Cross overall not a deal breaker but definitely something I would personally add to this one but then make your way to the front seats cloth seating coming with the ES and LE trim levels leather suede combination I love that look I reviewed that one last year for the SE trim levels leather seating coming with the SEL that is what we are looking at right now heated front seats for the LE trim level and up eight-way power driver seat for the SE trim level and up eight-way power adjustable passenger seat them for the SEL trims overall for our SEL that we have today seating was plenty fine certainly comfortable so no issues with taking this thing on a short test drive that I did today so then take a look at the steering world is tilt and telescoping it is leather wrapped for the SE trim level and up it is heated then for the SEL trim level and up so Western Maryland here we definitely get some cold weather so heated steering wheel is definitely where it is at so love the steering wheel on this thing then make your way to the startup let me start by showing you guys the key here it's actually a pretty basic key you got your Mitsubishi logo on the one side when you flip it over lock it on lock and that's about it it does all keyless entry with a push button start though if you go with the SE trim level and up otherwise you're gonna get that traditional turn key start so having said that we do have that of course all I'm going to do is simply put my front of the brake and press that engine start button which is located directly to the right of the gauges and so speaking of once started up tachometer is all the way to your left speedometer is on your right there is a small digital display front and center giving you your average miles per gallon at any given time of course what gear you're in if you were selecting your own gear through the simulated gears through the paddle shifters there outside temperature trip A, trip B pretty much your basics that you would need on the digital portion of the gauges there then make our way to overall interior quality if you wanted a power panoramic roof that is available for the SE and SE L trims meaning it's an option with a package option for those two particular trim levels it doesn't come standard with either of them though homelink controls is gonna be optional again for the SE L trim level dual zoom climate control comes standard on the SE and SE L trim levels gotta love that so both driver and passenger can each set their own temperatures up front there but if you were not to go with one of those trim levels the ES and LE still give you automatic climate control which is a big win because a lot of times that'll be manual on other manufacturers so you can still set your own temperature and it will get to that temperature automatically for you but you only got one particular temperature that you can set as opposed to the dual zoom climate control I guess. Another cool little fun fact about the eclipse cross is if you open the glove box you actually have two different tiers so I guess you could put your misubishi motors eclipse cross manual up top and then you could put whatever else you want down below or really you could put whatever you want in there but I do like that it's two tiers you usually don't find that within the glove box there just in front of the shifter you got some rubberized storage you got dual USB charging ports, 12 volt power outlet you have just to the right of the shifter your heated seat buttons your super all wheel control button to lock it in that four wheel drive mode let's say if it's snowing out here in Hagerstown, Maryland so that's definitely an option as well electromechanical parking brake just behind that also dual cup holders within the center armor it's definitely a decent amount of space in there as well and I like the kind of carbon fiber-ish look surrounding the power window buttons on the doors there and I love the gloss black look surrounding the cup holder so many manufacturers out there leave this as a matte gray plastic so I love that misubishi added at least a gloss black design because it really is pretty darn easy to clean I have that in my own personal vehicle so it's very easy to clean so I do like that but now let's go ahead and make our way to detect the infotainment screen I should say seven inch color touch screen display coming with the ES eight inch color touch screen display coming with the LE trim level and up but I will say either display screen that you go with you still get Bluetooth and audio streaming you still get Android Auto, Apple CarPlay if you go with the SE trim level and up you're gonna get factory navigation system and of course you can check out your radio information up there as well and so when it comes to the sound systems there are two of them the ES and LE trims are going to give you four speakers then the SE trim level and up is going to give you a six speaker sound system so therefore that is the one we have today so what do you guys say? Let's go ahead and turn on the radio see what we got playing today and let's test out the clarity of this one gotta be honest, not that bad for a six speaker sound system I've heard worse six speaker sound system so quite honestly it's not gonna blow you away but that is not bad honestly for the size of this vehicle that's right on point for what a sound system should sound like for the eclipse cross but anyways, last thing I wanted to mention to you guys on the infotainment screen at least is when you do put this vehicle in reverse you will find a rear view camera coming standard across the board but you will also get that bird's eye view you guys are looking at on the right there if you were to go with the SE L trim level so that is pretty darn cold and that gives you that panoramic view basically but as always that is going to lead us into safety and so to start front side side current airbags do come standard driver's knee airbag up front as well in the back you're gonna have latch aka lower anchors and tethers to children for the rear car seats rear child door locks, tire pressure monitoring system but there is some new standard safety for the 2023 eclipse cross another big change here lane departure warning is now standard for all trims across the board along with automatic emergency braking with pedestrian detection whereas in the past you had to go with upper trim levels now it's standard across the board so that's pretty cool SE trim level and up is going to add to that a blind spot monitoring system with rear cross traffic alert and then the SE L that we have today is going to add to that adaptive cruise control as well so overall when it comes to my final thoughts of the eclipse cross this is a very good looking SUV in my personal opinion it's got very nice design to it at least on the exterior America's best warranty as well you can't argue with that you get five years, 60,000 mile bumper to bumper 10 years, 100,000 miles on the powertrain and younger Mitsubishi is 20 years, 200,000 miles on the powertrain so that is wonderful LED headlights now coming standard I think I remember saying exactly that in my 2022 eclipse cross video recommending that Mitsubishi should add that standard for all trim levels and what do you know, they listened to me and now it is standard for all trim levels though that is pretty darn cold but now hopefully they listened to me again on this next part when it comes to room for improvement rear ventilation and my personal opinion is and maybe a deal breaker cause I got kids I want them to be comfortable in the back seat and this isn't a super small vehicle like a tiny little compact car really think rear ventilation should be something that should come on the eclipse cross and the only other thing I can think of is the availability at least of a wireless phone charger so at this price point I definitely think that should be an option as well but anyways, let me know what you guys think of the eclipse cross in the comment section below and that is about it for this one you guys, thank you so much for watching if you're afraid to follow me on social media at the bottom of the screen if you wanna see what's coming next on the channel before it actually gets to YouTube be sure to hit the subscribe and the bell notification button if you're in a new car reviews that is what we do here on this channel after all do appreciate you guys watching more than you know and I will see you guys all in the next video stay gold


The other way of understanding this is through the notion of eigenvalues and eigenvectors. You see what are we saying in effect, we are saying here that when you put the sequence into the system what comes out is the same sequence multiplied by a constant. Now this is not going to happen for all kinds of sequences. If you had you know a sequence which is square like in nature, square wave like in nature, it is not going to happen for LSI system in general. But for complex exponential this happens. When you put in a rotating phasor into the LSI system with impulse response h n out comes the same rotating phasor but multiplied by a constant. Now such a sequence which goes into a system and emerges as the same sequence but multiplied by a constant is called an eigen sequence of the system. Sequence that goes in that is unchanged in form as it emerges saying that the action of the LSI system is decoupled for each of these sequences. You understand this better if we can do one more thing by whatever means now. You could express any sequence, you see remember you did this earlier. You expressed any sequence x n in terms of these impulse sequences delta n minus k. So you wrote any sequence x n as summation k x k delta n minus k. So you were able to express any sequence in terms of the unit impulses. Now suppose by whatever mechanism you are able to express any sequence as a linear combination of such rotating phasors, rotating with different angular velocities. Then what we are saying here is if I take any one of those phasors that comprise the input, the response of the system to that component of the input is only the same phasor multiplied by constant. So if I decompose the input along each of these phasors so to speak for different amegas, then what I am saying is the response to the phasor at a particular angular frequency omega has nothing to do with the angular at the phasor at some other omega. They are all decoupled, they can be treated separately. It is slightly deeper issue but we will understand this better as we go along. So in other words what we are saying is I mean let me try and bring this a little more, you know it takes a little time to absorb this. What we are saying is take the same system S, LSI, HM. Suppose instead of giving e raised to power j omega n you give a1 e raised to power j omega 1n plus a2 e raised to power j omega 2n plus a3 e raised to power j omega 3n. Out would come a1 h omega 1 e raised to power j omega 1n plus a2 h omega 2 e raised to power j omega 2n plus a3 h omega 3 e raised to power j omega 3n. This is not at all difficult to show from the property of linearity or additivity and the property that when you put in e raised to power j omega n out comes e raised to power j omega n multiplied by h omega. So what we are saying is the action of the LSI system on each of these components is decoupled. I do not have to worry about what a2 and a3 are when I am calculating the output to a1 e raised to power j omega 1n. When I am calculating the output to a2 e raised to power j omega 2n I would need to worry what a1 and a3 are. They are decoupling. Now carry this argument to its limit. So if you could here I am taking a very specific input which had only 3 such e raised to power j omegas only 3 such frequencies. Suppose I use a continuum I use all the frequencies omega from 0 to 2 pi or all the frequencies from minus pi to pi. Incidentally now we make a remark about what range of values omega can reasonably take. Omega is the normalized angular frequency in the frequency axis. There is elastic or there is repetition omega between minus pi and plus pi. You see the reason for that is very simple. What does omega equal to 2 pi correspond to omega equal to 2 pi corresponds to sampling frequency. And we have agreed that every sine wave is comprised of 2 oppositely rotating phases. 1 rotate if a sine wave has frequency omega it comprises of 2 phases. 1 rotating clockwise with frequency omega and 1 rotating anticlockwise with frequency omega. In other words if you take both to be in the clockwise or anticlockwise direction 1 with frequency omega and 1 with frequency minus omega. 2 phases come together to form a sine wave. So on the phase of angular frequency axis each pair of frequencies omega and minus omega come together to form a sine wave. And obviously there are magnitudes the 2 phases have to have the same magnitude and opposite starting angle or opposite phase. Now we see in that when you sample you take the original set of frequencies or original frequency axis whatever it is shifted by every multiple of the sampling frequency and these shifts are added. On this normal scale the sampling frequency is 2 pi. So you are going to take the original spectrum shift by every multiple of 2 pi and add up these shifted versions. Let us show that graphically. Whatever there is you are going to shift this by every multiple of 2 pi and you are going to add them. Now obviously if you do not want these shifts to overlap this must remain between minus pi and pi. And of course that is also obvious from what we have seen in the sampling theorem. The maximum component that you have in the original signal should not be more than half the sampling frequency. So that means the so called unique omegas that we can deal with are only between minus pi and pi. So when we take any input x n and ask whether it can be expressed as a linear combination of e raised to power j omega n we need to worry about the omegas going from minus pi to pi. The frequency axis is continuous. I am not going to be able to use a summation now. I need to integrate the limit of a summation as the variable of summation becomes continuous even integral. So what we are saying is suppose I mean I am trying to motivate the whole idea. Suppose I take the same LSI system once again, give to it a combination x omega e raised to power j omega n. You know what I mean by x omega x omega is the component of x n along e raised to power j omega n. So what I am saying is suppose you are able to decompose just like you decompose the impulse response. Suppose you are also able to decompose the input along different e raised to power j omega n. And you did this for all the omegas going from minus pi to plus pi. Then it means that you know since minus pi to plus pi is exhaustive putting those components back should give you back x n. So what we are saying is this is what we are trying to ask is what comes out here, isn't it? And we are saying in particular when you take x omega e raised to power j omega n, what will come out? So x omega e raised to power j omega n is going to give us x omega h omega e raised to power j omega. Is that right? And therefore if you integrate this, if you integrate over omega here, you can also integrate over omega here. How can we say that? That is because of the property of linearity. If for each omega I can do this, I can do it for the combination of the omegas. It will lead us to a very interesting property of this inner product. This inner product of a sequence with the rotating phasor gives us a new domain called the discrete time Fourier domain or discrete time Fourier transform domain. We shall see more of this in the lecture to come and build up in greater depth the whole idea of the discrete time Fourier transform in the coming lecture. Thank you.