My co-presenter Charlie Downoff is I think on an airplane on his way to Singapore he had travel changes. So he was going to be there in person giving the talk and I was going to be joining remotely as it turns out I'm remote and he's also even more remote. So but that's okay we've worked on this together and we're going to talk today about a workshop that we've run in several different formats over the over the last year. But rather than just give an account of the of the what we've done in the workshop in the past I'm going to walk through the methods that we've used. So we call it the open future design workshop. There's a pre-print here if you write down this number 2306.08426 that will describe some of the case studies that we've that we've done so far. So but I'm going to talk with you today about the methods. So and they're written down on these little cards so I'll kind of zoom around on the cards here to explain what we're talking about. So they're methods for envisioning and exploring the future together in in your minds and in groups. So it's fun to start with this one here called Derive Comics. So you want to develop some kind of future together to explore as a group. You have your group but everyone has their own experiences and their own lives. So what are you going to do to make a shared experience? Gather some data. So go for a walk or look out the window or just close your eyes and explore something in your mind around the theme of the workshop. Assume you know what that is. And then document okay document what what you see in your mind in a way that makes it possible to share with others. So this could be photos, a screenshot, post-it notes or whatever you might like. So if you're there in the audience you might take a second to close your eyes and imagine you know something that you encountered on your way to the conference that you might like to share with other people. We won't have time to report back but you could imagine these things and have that already do that exercise. And then if you're going to do the workshop we try to build a meaning map of the of the things that people have collected in their interactions of the previous activity and distill some kind of shared meaning. So again everyone has their own experience. So let's talk together about the problems and opportunities that everyone sees in those explorations they did. So maybe some of these will cluster together. So maybe you saw a case of disparate on the way to the conference you might have seen something that made you feel upset and you want to share those things. Or maybe you saw something that makes you happy and you want to share those things. You might find common themes around how you're feeling about that. Of course everyone's going to have their own perspectives on that. Some things someone might find something upsetting that someone else finds gives them pleasure. That's possible and that's okay too. The map can contain all of that information. Just make sure you document those different viewpoints to get everyone on the same page. And if we were doing this in the workshop we might do this around tables in smaller groups rather than all on one big map. And so that's this card here called structure conversations. Having convened the workshop if unstructured conversations are likely to take lots of time without yielding some concrete benefits then structure the conversations around your shared interest to move things forward more effectively. So maybe it's necessary to find the themes and create breakout rooms or breakout groups accordingly. When I ran this once at university at work I just did group people by faculty. I've had some of those conversations per faculty. And now once you've got all these different groups broken out you may want to or need to increase the participant control because if you only have one facilitator they're not going to be able to be in every group all at once. So you need to have people doing things inside their groups. For example, taking notes. So we have a nice system for taking notes across layers of meaning. So in these little groups you can explain to them how to take these structured notes. They can be responsible for that and responsible for reporting back to other groups as well. So those are two of the things you can do to increase participant control that there are other things to do as well. Something else to mention here is this concept of re-infusing expertise. So what I like to do in these workshops is you're trying to build this meaning map but you want to get everyone on the same page but you actually want to kind of incorporate their thoughts and feelings as human beings before you incorporate their expertise. So this is why I'm talking about the things you've seen on the way to the conference or the things you might see in your neighborhood. If it turns out that you're also a sociologist or a climate expert or a building an architect something like that then those pieces of expertise can be used to flesh out the map. That might be a second phase of activity and I'll talk a little bit more about that later. So this is something we can move towards. And I think the last one on this on this set of ideas here is to do your research. So if you're running a workshop you can already start doing the research before the workshop starts by interviewing participants and starting to build your own meaning map which you might share with them beforehand. So these are good instructions for basically getting people on the same page. So if you want to do the research to start with then that will help. And you could do that. You could help as well with getting people on the same page with some context setting. So the theme might come from a workshop sponsor convener or from this previous research and you can share that. So it's all about how we kind of get started starting to build a perhaps a visual picture one visual picture or multiple visual pictures getting everyone towards the same page. This is the next set of instructions for exploring the future. So now we've thought about these ideas that people brought to the table like things that they find exciting or things they find concerning related to the topic that that they've been tasked with or that they've brought a variety of topics. So how do you explore that? We like the idea of handing out some functional roles. So having tried this a few different ways one of the ways we tried with some kind of role playing like stepping into the persona of you know a type of imagined character that didn't really work as well for what we were trying to do as taking these functional roles which are giving people things that they can do inside of a conversation while they remain themselves and they continue to have their own expertise. So the first one here is the wrinkler and their their role in the conversation is to think about what could go wrong. So as we brought everyone onto the same picture the same map before we're envisioning all these different themes that people had brought to the table hooked up and aligned with each other in some way. That already is starting to tell a sort of story about what how these things fit together do they fit together you know again someone people might have conflicting views so their perspectives on on what could go wrong or what could go right may be very different. But for example if you're envisioning a future in which a certain set of local problems are being solved that's nice but what if what if the local government or the local authority doesn't doesn't get on board with the views of the citizens and the citizens have to stage uprising or something like that or they at least have to stage a big campaign and that's the kind of thing that the wrinkler might might propose in terms of what could go wrong and this is helping to envision different futures so possible futures in which or maybe they're quite unlikely but things aren't going the way we thought they would inside of this shared scenario. The next rule is the time traveler who asks the question what has happened in the past and what could happen in the future and they're providing some historical context and anticipating alternate futures so again in the in the context of kind of local project maybe you could imagine some local movements that have worked in different ways in the past achieving different kinds of goals you know some might be more urban some might be more rural they may have different ways of working together you might realize that in the future we'll be working across context more effectively through technology so you could imagine a future in which different local groups are networking together around themes that are shared and so forth so again this is just a kind of conversational role and you could have this physically have this card out to someone and they could trade the cards during the conversation so they don't have to be stuck as the time traveler forever. The next role is this analyst role so what are the what are the moving parts inside of your scenario that you've envisioned so maybe we focused in on a specific part of the future like you know for looking at local issues maybe we're focusing in on on housing so you could focus in on all the current challenges and components of the potential solutions that are being devised in terms of local housing so who are the actors what are the resources what are the institutions and you could identify and look at how those things work together so an example in in houston to address homelessness they formed a a coalition of all the organizations that had anything to do with homelessness and they got them talking to each other and as a result of that the outcomes for people using their services significantly improved and that's all due to this kind of internal orchestration but the analyst also has another role which is to consider the challenges beyond the current focus and to identify and orchestrate the integration of those components outside so if you if you were looking at things related to homelessness you may also want to be looking at things like jobs and mental health and families and migration or whatever the other things are that touch on the concepts related to having a home but aren't necessarily the you know they're not necessarily the the services and roles and actors specifically focusing on this topic um and then to make this all hang together we have these facilitator roles so linkers and reflectors these are looking at the proposed scenarios that people are coming up with like resolving homelessness or changing the way we use other local resources these are things that are coming up in the conversation um how do these start to uh fit together so you know maybe we um want to change how we use certain local resources and that's going to create some jobs which for people for example so these linkers have to be able to stay on their feet making these connections between the ideas that other participants are coming up with so I call them facilitator roles because they're people who are facilitating the conversation but like I said in the spirit of that increasing participant control there's no reason not to hand out this card to someone who's just showed up on the day and ask them to start making some links um and similarly this other facilitating role is a reflector who can look at how that scenario is evolving so maybe you know we've exhausted the topics uh or at least our knowledge about those topics and we realized you know to make some progress you'd have to ask an expert so um here we talked about this thing called a project action review where you say well all right we've done our initial exploration that's coming to an end it's now time to um take up another topic or look at another um look at another proposed problem that might come up etc so these reflectors can um help kind of steer the conversation along so that's it in terms of how the exploring works according to this um this system so again we promise way to get everyone on the same page and start to look at what what works and what doesn't work and then explore what could work better what has worked in the past um so here's some examples that we we looked at in these are three examples all from the same workshop so these were some of the scenarios and problems that people came up with so contested space so-called public space doesn't always feel welcoming to all members of the public it can be overrun with antisocial behavior it can feel exclusionary or uninviting it can be the site of a conflict although the uses of public space are complex each space does not need to support every use equally so um I think this is looking at the example of a public space which is used for public drinking and maybe that's uh okay at least for the people who are drinking there they want to place the to gather but other people may not feel like that so the point is when you have a public space it can be used in different ways and this becomes a feature of the space that the uses are are contested um and that's interesting right so that it doesn't force any particular resolution of that problem or challenge um but uh it gives things to talk about um another one here is funding of public space even though public space is known to increase wellness in the population well-being priorities that would lead to increased funding for public space aren't universally adopted um in order to make the benefits of such investment clear uh we should increase transparency around investments in public welfare for example creating a register of impacts of local social enterprises in the case where the government isn't investing in that so this is again proposing a proposing a solution rather than um looking at uh you know these these kind of strategies so this is saying here's a more developed uh possible solution but it's also outlining some of the problems so this is this is something that was coming out of that kind of exploration that I was talking about before um and here's another one rebalancing social services uh welfare-related services should be supplied and balanced with local needs um they're often or not uh could varied expertise be integrated in a similar way to the domain specific skills practiced by medicine sans frontier to address uh complex local challenges um so those are some of the things that came out of the conversation I'm going to skip these for the moment and come back to them uh here are a number of other miscellaneous patterns that are are useful and I'm realizing uh this one I really wanted to look at um now so these are called pattern language components um and all these cards that you're seeing here when they have a context then if then on them they don't all have those but some have this kind of structure uh for talking about where these strategies can be used you're collaborating in this case where people are new to design patterns if these new people are being invited to create their own patterns like these are starting to become you know problem solution pairs um but they don't they're not terribly acquainted with design patterns you can introduce these dynamic keywords however uh because therefore and specifically to describe a gap or conflict you allow you know help them talk about those gaps or conflicts in their situation um to talk about some clauses and that are operating in the situation um and to describe some sort of rationale for what to do about it based on what's worked in the past or what maybe works in another place um and then specifically to describe next steps they could take in their in their um in their scenario so these ones although they've been re-abstracted down into text form short paragraphs were outlined by people using those keywords uh using cards on on a table um so I thought that was uh a useful one to share right away is is these cards can also be printed out and shared around um so um other things to say about it so like uh how does this work in terms of its broader broader process and one of the more recent ones I I ran I ran it at the university I work with and they're working on creating an open research action plan at the university so um what I did was I I interviewed a bunch of people uh to gather some initial themes and make that an initial meaning map um and then at the workshop I ran through the activities uh like we've talked about in the context of our life at the university I'm talking about what works and what doesn't work for people there how they'd like to see things evolving differently and this is structured per faculty and then I took a bunch of pictures so they had laid all these themes out using pieces of uh no papers and some of those cards I mentioned over here so I took pictures and then I was able to analyze that into an outline using the supplied uh template so um having gathered themes for the participatory project uh they may have some um uh explicit uh I think they should say structure they may have some explicit structure in in them uh because the way the information was gathered so by faculty or by using these cards but additional structure can be created linking the intermediate artifacts into a relevant template so we had these this linker role kind of making a map of every everything um but now we are have leading as output from that process uh a draft open research strategy using someone else's template so that's a that's a useful um way to present the results um not not conclusive and saying this is the final thing but it could also be structured as a as a project plan or a set of working groups um you might create uh online a new online community who will work on the emergent themes which have been created um this one talks about adapt layers as needed so when I mentioned that I have a template for structured um analysis of a given uh uh theme this is called causal layered analysis and the layers uh this is uh uh created devised by something called so how in yatula who's used it in in various uh consulting contexts including with the un and so forth um the layers are litany system worldview and myth and that allows you to kind of drill down from the problems that people encounter every day to what where those problems are coming from how they fit together and the kind of deeper stories that people are telling so in the in the case of wikipedia we want to create a educational resource that everyone can access uh and the worldview is that well that that can be done with a wiki you know or maybe that that's the system that can be done with a wiki the worldview is it can be done openly online giving everyone the the right to edit things but you know we're all familiar with the litany of problems that can come up like edit wars or vandalism or whatever so these are the problems that everyone's familiar with um but you know by and large those who are highly involved share the kind of same story and myth is not meant to be um a put down it's meant to be inspiring so like what's the what's the driving metaphor here you know maybe it's an encyclopedia or an educational resource or maybe it's a more profound thing like access to uh knowledge and knowledge creation process so but that said this card is saying you can adapt the layers as needed so if you're if you're talking about a challenge that has a more technical flavor your layers might be something like the real-time updates to the system uh information or the dynamic user interactions and you're you're instead of talking about worldview you might talk about what languages uh like concrete programming languages you're going to use and what actions you're going to take so this is a nice uh starter pack of of things to analyze and so people can take notes and you can easily teach them you know talk about the problems that um come up in your situation and try to drill towards the shared meanings and then in terms of this meaning map if you have notes from different groups in that format you can try to harmonize them and you know when I did this across faculties the different faculties came up with very different pictures uh and that's that's inherently kind of interesting looking at those diverse perspectives but like I said you can adapt the analysis layers as needed um some of the other things is is in a in a in certain groups if they're already oriented around a type of action or a type of um agenda um we've developed some possible next steps but if we leave without any concrete commitments then then maybe nothing will happen it was just a good conversation so um is it possible to introduce some early actions that people already do together in the workshop format turning it into a working context that begin to get people doing things um so um yeah this I think I've covered almost everything on this page except going meta going meta says that in the course of working on our project together um can we go meta to apply the project's methods to itself so here what I might do is take these cards and this conversation if we had a bit more than half an hour we could work together and kind of having acquainted you in this in this briefing with the methods we might try to fill in some new patterns and we might try to critique these methods and we might say well how are we going to use these methods in the future can we use these uh at at a future wikimania for example to uh to you know accelerate our way of working together um or can we use these as we take them but can we use these methods if we go back home to our other professional or um volunteer context social context could we use some of these ideas and other contexts and really get into using some of these concepts like what could go wrong etc to uh to workshops and methods themselves and so that's this process of going meta and we might say well for example we might realize um these roles these uh roles are great but maybe we need some other roles beyond these ones to get the most impact out of the methods or we might say well you know in half an hour maybe it's been okay to have joe talk at us a lot but actually we could have made a micro workshop that just did one or two of these activities and and that was useful so okay now um because i'm running out of things to cover um i'll leave a couple minutes i think there may be a chance for questions as well if they're facilitated if uh if anyone wants to put questions in the chat if that's possible or otherwise facilitate them um we'll just share a little bit so we ran we ran another context we ran the um workshop in was a course and we brought our paper along and shared it as a as a pre-reading so that was this kind of context setting pattern everyone had read our paper and then we came along and talked to the students who were developing um projects at a computing course so one of the student students enthusiastically said uh these authors actively participated in our class shared expertise and created a collaborative learning environment their presence allowed us to gain deeper insights into the concepts and methodologies of paper leading to innovative project approaches um by closely studying the patterns of patterns they identified in their research i gained a fresh perspective on project organization and established the logical and coherent structure to my uh work well that that's great as a as a narrative and it sounds it sounds very lovely but really what it's saying is is how are we uh providing engagement and guidance what are some better ways that we already have tried to provide engagement and guidance to people who are developing projects so um although there's a way to envision futures together again if people are working on projects they've already envisioned a future so what can we do to support that work as it goes along um and the same uh student said the authors insights helped me to navigate common project development pitfalls um through their emphasis on effective documentary um documentation regular testing and thorough project planning i was able to avoid costly errors their guidance ensured a consistent progress trajectory and maintained the professionalism of my final project so again what kinds of scaffolding and structuring um do we can we can we build or can we help people build that help them avoid mistakes and the same thing with the skill ability and adaptability so that's it that's the tool kit that we have here that's primarily like i said uh set up as a as a workshop but some of these um concepts go a bit beyond the the workshop format um here's the pre-print again if you want to see the developing paper this was uh been submitted to um pattern sorry pattern languages of programs which will take place in uh illinois um very soon um okay so i'm gonna stop sharing and just see if i see anything coming in on the chat um i've got according to my my talk i've been talking uh 24 minutes but we started a little late so i want to make sure um we're respecting the clock but if there are questions i'll read them in the chat yep please wrap up jump to latest okay um looks like we're on time i don't see any questions coming along here um but i hope that was useful for you no question from the room nida um if there are questions i will uh drop my um email in the chat so you can email me it looks like the next person's coming out maybe so okay thank you very much i'll say goodbye thank you joe


I'm going to start. So I'm going to be talking in this lecture about polynomial approximation. So I'm going to move off track of the quantum algorithm stuff that we were talking about before. And the reason why I'm talking about this is sort of we established before, or hopefully I established to you, that you could get from using your quantum algorithms, you could get from blocking coatings of some matrix A to a blocking coating of a matrix that's this polynomial applied to A. And roughly speaking, if this was like a Q blocking coating, meaning that the gate complexity is Q, then this turned into like a DQ, sorry, NQ, NQ blocking coating of P, where N is the degree of P.  And the issue is that normally, for a lot of the applications that we want, we're actually wanting to apply non-pollinomial functions. So in this Hamiltonian simulation example, we wanted to apply this function from H to E to the minus IHT. This was not a polynomial. Similarly, there's another example, which is this linear regression algorithm that's sort of central to quantum machine learning. And this performs the map A to A inverse when A is some sparse matrix. And so if you think about what this is applying, this is just applying the function that's 1 over x. And so in these instances, these are not polynomials. And so we're going to have to work a little bit harder to find polynomials that approximate them. So in this lecture, I'm just going to be discussing how can you take a function and then get a polynomial out of it. Yeah, so our tool for doing this will be Chebyshev polynomials. And basically, you can do this, use these Chebyshev polynomials to get any sort of approximation that you might want, up to some details of actually just the computation of working things out. OK. Yes. Yes, so there are more simple ways to see this in more complicated ways that are more efficient. The simple thing is to say that your Hamiltonian is a linear combination of unitaries. So here, these Pauli's are unitaries. There's another approach that basically goes any S sparse matrix you can implement in S with an H over the sum of lambda A. There's another approach that's if you have an S that's H that's S sparse, you can get a block encoding of H over S. All the entries are bounded by 1. And this is, I think the argument is basically that just that you somehow decompose it into Pauli's in an insufficiently nice way. Are there any other questions about why I'm doing polynomial approximation or anything else?  And what they are, how I'm going to define them, is that the degree n Chebyshev polynomial, which is called Tn, is the function that satisfies for a complex Z, this equation. And OK, so this is how it's defined from this definition. You might ask why is this a polynomial? Well, you can see that this is a polynomial by observing that it satisfies this recurrence relation. And yeah, so what you can do to see this is you can plug in your value of x and then you can see that this recurrence basically becomes the following, which is these are twos. I don't know if I can continue doing this. Yeah, hopefully, I'll keep this in mind that these two are similar. So this recurrence I wrote is equivalent to this equation or you can just check that this is true. And furthermore, you can see that T0 is equal to 1 and T1 is equal to x, like T1 of x. Yeah, OK. And so from this, you can conclude that all of the Chebyshev polynomials are indeed polynomials from this definition. OK, if we plug in, if we take z to be equal to 1 half, sorry, if we take z to be something on the complex unit circle, then this 1 half z plus z inverse is equal to cosine of the angle. It's like a real part. And so you get a definition that might be more familiar to you, which is that Chebyshev polynomial applied to cosine theta is cosine n theta. And from this, you can conclude that, OK, first of all, OK, is this possible to see? Yeah, from this, you can conclude that Tn of x is bounded by 1 for all x in minus 1, 1. I'm going to start abbreviating this by saying Tn minus 1, 1 is bounded by 1. I'm just going to use this notation to refer to the above. And the reason for this is like for any x in minus 1, 1, there's a corresponding cosine theta, or corresponding real theta, such that cosine theta is equal to x. And therefore, cosine n theta will be between minus 1 and 1. Another thing you can observe is that Tn has parity n. So you can see this from the recursion that your T1 is odd, T2 is even, T3 is odd, and so on. So the property of Chebyshev polynomials that will be important for us is that it's basically that any sufficiently nice function from minus 1, 1, to reels can be written in terms of Chebyshev polynomials. In this way is a series of polynomials. So k from 0 to infinity of ak, Tk of x, where these ak's are like absolutely, are like they're absolutely converging. And sufficiently nice here is a very mild restriction. I think what you need is some Lipschitz continuity. So basically, you just need that the derivative of f is always finite, or as finite as it has an upper bound between minus 1 and 1. And the reason for this is for the same reason that you can have like Fourier series of functions. And in fact, there is essentially parallel theories. Because if you have this function f from minus 1, 1 to r, you can define another function g, which is, I guess, from the unit circle to r by saying that g of z is equal to f of the real part of z. And so if f of x has a decomposition into Chebyshev polynomials, then g will have a decomposition of the form I'm plugging in the value of the series here. So it's ak, Tk of 1 half z plus c inverse, which is the sum from, it's a sum over z to the k and z to the minus k. And the thing to note is that this is like a Laurent series. These are like Laurent series. And furthermore, you can define some h from minus pi pi to r by saying that it's equal to, by taking the corresponding angle of g, the corresponding, applying g to the value on the unit circle with the corresponding angle. And then you'll get that this is equal to k from 0 to infinity of, what is it here? It's e to the ik theta plus e to the minus ik theta. And these are Fourier series for h, which is a 2 pi periodic function. And so what I'm going to do later is I'm going to show properties that these ak's are decaying. And so you can truncate them to get polynomials. But just keep in mind that there are these three parallel theories. So if you have intuition from Fourier analysis or any other of these, there's sort of parallel statements for all three of them. So that's what I wanted to say on that. So from these theories, you might suspect such a thing. But there is an orthogonality property that these temperature polynomials satisfy, which is that these are orthogonal under some inner product. And I'll just write down what the equation I want to say is. So here, this is the inner product. So if I have two functions, tk and tl, then I'm taking their inner product to be this integral. And I'm saying that this is orthogonal under this inner product, which means that this is going to be, I guess, I guess, orthonormal. But it's going to be 1 if k is equal to l and 0 otherwise. And I think you might need it. Actually, I think it evaluates to 2 if k equals l equals 0. So there's some slight issue. So there's always some slight constant things that happen at 0. But basically, they're all orthogonal. OK, how you can see this is you can say, I'm just going to take this and I'm going to substitute. I'm going to do a u substitution. x equals cosine theta. And then dx equals minus sine theta. And then what happens is that my integral becomes 2 over pi. I think this goes now from pi to 0, tk of cosine theta, tl of cosine theta. And square root of 1 minus cosine squared is sine. So this is sine theta. Am I off by some thing here? There's, sorry? That's true. I'm worried about this absolute value. Is there some absolute value here that I'm, I don't think so. I think we're fine. Yeah, OK. All right, so this is integral from 0 to pi of, this is tk of cosine theta. By what I said before, this is cosine of k theta. And this is cosine of l theta. And OK, I'm going to leave it as an exercise to show that this is equal to 1 when k is equal to l, and then 0 otherwise. You can get this with a cosine addition formula. Yeah, any questions on this? The form of the inner product is like, yeah, the way that you should think about these polynomials is that they belong on the unit complex circle. And so that's why you get these terms of square root of 1 minus x squared, which is like, if you were working with these on the complex unit circle, then these terms would disappear, and then these would become like polynomials in z, or I guess Laurent polynomials in z. Another way you could think about it is that like, another way you can think about it is that this inner product is like the normal inner product, except when your function is near minus 1 or 1, in which case things start to get strange, like things are more heavily weighted. And then this is a behavior of Chebyshev polynomials that's common that you'll see in other places as well. I hope that's helpful. Any other questions? OK, so the nice thing about this orthogonality property is now you can actually take a function and then figure out what Chebyshev coefficients are. Because if I have some f of x that's written as this sum, this Chebyshev series, then I can imagine I integrate this against some, I use this integration inner product thing. And I take the inner product with tk, and this will give me the value at the coefficient at tk. So if I take this inner product, tk of x, then what is this? This is like some, I'm interchanging the sum in the integral, so don't get mad. Hello, hello. OK, great. So as I was saying before, if we have some function with a Chebyshev series, we can figure out which coefficient, we can figure out coefficients via this integral, because we can take the inner product with tk, and then we get this sum. And then this expression is the integral we evaluated before, which is 1 whenever L is equal to k. I guess out here. And so you get that this is equal to ak. OK, so if you have this integral here, if you compute this integral here, you can get the value of ak. Now, I don't recommend doing this computationally, because I think it's not the best way, but in any case. So one thing, but from this, we can conclude a lot of useful properties about these Chebyshev coefficients. For example, something that we can conclude is that if my function f is bounded, then it turns out my Chebyshev coefficients are bounded. So here I'm just writing what my value of ak is. OK, what can I do? I can then bound this by, I take the norm of f in here. I take the norm of tk in here. And then this is still 1 over 1 minus x squared. And this is equal to 2 over pi times the integral 1 over root 1 minus x squared. And if you believe me, this is just pi, so the value is 2. So I've just shown that for all of my Chebyshev series, if my function is bounded, then the coefficients are bounded. And this is one reason why you see it all the time in these polynomials all the time in applied math, because this is different from what happens in the monomial basis. So for example, you can have polynomials like the Chebyshev polynomial, where your Chebyshev polynomial is bounded. But if you write it in the monomial basis, the leading term here is 2 to the n minus 1 x to the n. All of the coefficients are exponentially large in the degree. And so you get some instability, which is why it's better to work in the Chebyshev basis where things are more stable. OK, so this suggests the reasons why we're looking at these. OK, so the main polynomial approximation we're going to be looking at is that of Chebyshev truncation. So if we have some function minus 1 1 to r, and it has a Chebyshev series, then we can find this truncation to be just we take all of the coefficients up to n. And this will be our degree n approximation to f. And well, you could ask, how well does this do? And we can measure it. We can see how well it does in the uniform. And it's like a norm that point-wise. So how poorly this approximates, well, since it comes from the Chebyshev series, we just need to bound the tail. As we said before, all of these tk's were bounded by 1. And so you can bound this by k equals n plus 1. You can bound this by the sum of the tail of the Chebyshev coefficient tail. And this is sort of what our goal is to bound, because we know that for our series, this does converge. And then maybe if we choose n sufficiently large, then this bound will be like, because we know that this, because we know that the whole series of ak's converge, we can choose n large enough so that this is epsilon. And this will give us a Chebyshev polynomial approximation that's good up to epsilon. So basically the only thing is that we want this to be epsilon. And our broad vision here is that if we have some a, and then we want to apply some f of a, but we can actually only apply fn of a, like this polynomial, then using this argument, we can show that how much we're off by in operator norm is bounded by epsilon. Yeah, so that's the high level viewpoint from the quantum algorithms perspective. OK. So we want to show that these ak's are decaying. There's a bunch of theorems that basically state something of the following form, which is that the smoother that your function f is, the faster that these Chebyshev coefficients decay. So for example, there's these Jackson theorems which say that if I am singly differentiable, or if my first derivative is bounded, then my tail, then these ak's decay is like 1 over k squared. If they're doubly differentiable, then they decay as k cubed. And the nice thing is that you don't really need to try to get a handle on what explicitly computing these coefficients, which can be difficult. So I'm going to prove a statement of this form, which is that this theorem comes from a book of Triphethan. And what this is is if f is a function and it's analytic and it's also analytically continuable to the interior of this ellipse. So I'm defining this ellipse as going to be a little confusing at first. And here it satisfies on this ellipse that f of z, I guess, is bounded by m on this ellipse. Then what we can conclude is that our ak's are exponentially decaying for all k. OK, so what am I saying here? So OK, we look at some analytic function. So for example, e to the x, e to the tx for Hamiltonian simulation. Or I can look at other applications. You want to look at some smooth version of the inverse, or you can look at some version of arc sign if you rescale it. These are things that appear in applications of quantum algorithms. And we want it to be continuable to this ellipse here, and I'll draw what it looks like. So here is my complex plane. Here's minus 1, and here's 1. And what my ellipse sort of looks like is it looks like this. And here if rho is like 1 plus delta, then this width here is like delta squared, and this width here is like delta. So it's not drawn to size. Basically, the idea is that I want my function to be bounded a little bit outside of minus 1, 1. And how far I can go outside of minus 1, 1 and still be bounded dictates the rate of convergence. So if rho is 1 plus delta, that means that I need my size of my approximation to be like 1 over delta. Right? So what's going on here is that then we can choose. Then we can notice that by this theorem, my error is bounded by the sum of these AKs, which is bounded by 2M. Or you have this exponential tail. I'll just write down what the answer is here. I believe this is right. Maybe it should be an N plus 1. Anyways, all this to say that you can choose N to be equal to 1 over log rho log of M over rho minus 1 epsilon. If you want this to be bounded by epsilon. So if I can extend to this rho as 1 plus delta, then I get a 1 over delta degree approximation. And here, when I say exponential convergence or exponential decay, this will give you a log 1 over epsilon term here. OK, so to see where this is coming from, I'm going to prove it. As I wrote before, we have this integral for our value of the coefficient. And what I want to do is I want to lift it to something on the unit circle. So the way that I do that is I say x is equal to 1 half z plus z inverse, where my z is on the unit circle. And if I do this change, what I'll get is that this is equal to, let me see. I'm just going to tell you what it is. So I'm integrating around the complex unit circle once in the counterclockwise direction. And here, this is something that you might recognize as the Cauchy integral formula. And what we can say is that we know that f, as we said before, is analytic in some interior of this ellipse, this Bernstein ellipse. And so what I'm going to assert is that we can actually just take this and we can expand our contour to be the contour of the circle of radius the row. And you'll see that the space that you pass through is the space in the interior of the Bernstein ellipse. And so we can still make this so this is an inequality. And then what we can use is that this function here is bounded by m on the ellipse. So it's bounded here as well. And we have that this guy is bounded by row to the minus k plus 1. Is that right? Yeah. And then the length of the contour that we're integrating over is it's a circle of radius row, so it's 2 pi row. And so this will give us a total 1 over in magnitude. This is bounded by 1 over pi times 2 pi row times row to the minus k plus 1. And this is precisely what we wanted. This 2 row to the minus k times 1. OK. OK. Any questions about this? Right, so this you might be wondering at this point what the difference is between this Chebyshev approximation and a Taylor series approximation. The Taylor series are probably more familiar. And then you see them a lot as the tools that you use for doing polynomial approximation. And there is a corresponding statement that you can make here about Taylor series. Roughly speaking, what you need is that you need your Taylor series coefficients to also be decaying. So if you have your, for example, if your Taylor series coefficients are bk, then what you need is bk. I'm going to center it at 1, at 0. You need that this guy is bounded by some m. And you can sort of think about this as saying, OK, I have my Taylor series approximation, or my Taylor series, and then somehow I need to bound it on a circle, or I need to be bounded. And this is slightly outside of minus 1, 1, something like this. And so I think you can get similar results. They might not be optimal, but you can get a log 1 over epsilon convergence. And the main annoyance here is that sometimes you don't want to necessarily go to figuring out what all these coefficients are. And in this case, you can just figure out what this bound is of your function in some region. And this might be easier. So this is just trying to explain how this works. So sometimes this isn't enough for us, though. As I mentioned before, sometimes we want to approximate functions that are like 1 over x. And 1 over x is a function that's not bounded in the Bernstein ellipse at all. And you might wonder what you can do here. Well, the thing is that our criteria of what we want to approximate has changed. So here, if we are trying to apply this to a linear system, what we are allowed to say is that our matrix that we are performing this on is well conditioned, meaning that none of the singular values go beyond this value, say, like delta. And so what we really want to do is we want to approximate it on this region. And then we don't really care what happens in this region. We maybe need that it's bounded between minus 1 and 1 because then we can apply our QSVT. But apart from that, we don't really care what happens. And so in these settings, when you have a different criteria, then there's different tools that you need to massage it into some other form that you can then use. In approximating this particular function, people have been pretty resourceful. So I think there's sort of an ad hoc way to approximate this, which is saying f of x is approximately 1 minus this function. And the idea is that this 1 minus x squared to the b thing is going to be sending this to 0, but it's otherwise like, sorry, how should I say this? This is some function that's peaked at 0 and then decays very fast. And so if you combine these two in such a way, then you're going to get some function that sort of bridges this gap in the middle between minus delta and delta. So there are ad hoc ways that you can do this. There are also theorems that basically say that you can take some f of x, which is like 1 over x. And you can approximate them on smooth areas and then sort of gloss over areas of discontinuity. So imagine I can do the following, which is I can take my function. I want to approximate it in this region, outside this region. And imagine I could do the following, which is I have two different functions. And one function performs this approximation. And then it immediately goes to 0. And then it's just 0 for the rest of the approximation. And then I can imagine I have another function that approximates this negative piece and then it goes to 0 and then 0 for the rest of the approximation. Then what I can do is I can sum these two and then I get my approximation for 1 over x. That's good on both of these regions. Basically, now all you need is some way to get a good polynomial approximation, but one where it goes to 0 outside of the area of approximation. And I'm going to state a theorem and not prove a theorem, this theorem, that says that you can do this. So if you have f that satisfies, OK, yeah, sorry. So what I'm going to say is it has the same properties and continuable to this Bernstein ellipse, e-row, and then bounded by m on this ellipse. Then you can find some polynomial approximation, and it's going to be of degree as here, such that my function is, maybe I'm going to say m is equal to 1, so it's just going to be a function that's bounded by 1. And so I'm going to get a good approximation on f for f. For the region that's slightly outside of f, it's bounded. And then finally, on the rest of it, so I can extend it even further, this is bounded by epsilon. So here I'm imagining I have my inversion polynomial. And what I've done is in order to use this, I rescale so that my function is like, I sort of shift everything so that the region I want to approximate is in minus 1, 1. And then I can indeed get this approximation that's good on minus 1, 1, and then it's bounded in some region that's like delta size. And then it goes to 0, or it goes to near 0 very quickly after.  And this is interesting because if you took a naive approximation, like a Chebyshev truncation of this polynomial, the Chebyshev approximation would approximate it well on minus 1, 1. But then it would shoot off very quickly. And this is, Chebyshev truncations are generally very poorly controlled outside of minus 1, 1. And these are scaling as x to the n, so it could be very large. OK. And let me see. So I think I'm going to, I think I'm not going to discuss the proof for time. But basically, the idea is that you take your Chebyshev, you take this x to this bad scaling guy, and then you just take this and then multiply by some rectangle function that's 1 in the region minus 1, 1, and then quickly drops down to 0 here. And the idea is that if this is like, you can choose particular sine functions, such that this guy is actually scaling like e to the minus x squared. Or that's like the way it's scaling. And then so it beats out the x to the n. And so you multiply this by this error function. And then when you do that, you then do polynomial approximation again. And using this, as I mentioned, how you do this before, you can get these approximations to 1 over x. Notice that compared to what I was able to do before with the standard theorem, the standard theorem was giving you something like this expression, which I said it before, but it's 1 over, when rho is 1 plus delta, it's like 1 over delta log, when m is also a constant, it's like 1 over delta epsilon. 1 over delta log, 1 over delta epsilon. And so if I'm requiring this boundedness, I'm only losing this factor of b, which is the amount, how long I want to stay 0 for. And so using this, I can then conclude as a corollary, but it's like a computationally heavy corollary, that you can get some approximation of 1 over x. Here I'm scaling by delta so that my function is bounded between, so that my function is rescaled to be size like norm 1. And so I can say that there exists some odd polynomial p such that both I have that p is bounded in minus 1, 1, and also that p is close to f on this region of delta 1. And because it's odd, it'll be nearly the same on minus 1 to minus delta. So this is how you might get these polynomial approximations. The final thing I wanted to discuss is lower bounds. So it's a question of how you know that your polynomial is as good degree as you can get. And there's a couple of main things that I use as intuition for these polynomial approximation problems. And there's inequalities called, OK, there's one called the Markov brothers inequality, which says that if p is a polynomial that's degree n and p is bounded between minus 1 and 1, then you have that the derivative of p is bounded by n squared and also in minus 1, 1. And there's another theorem called, I think it's like the Bernstein inequality, which says that in the same setting you get that the derivative is bounded by n over root 1 minus x squared. And so what this means is that if you have some polynomial approximation between minus 1 and 1, in the region of a constant from, say, 0.9 to minus 0.9, in this region, your derivative has to be bounded by n. And then in this region, it can be larger. It can be up to n squared. And then so something that you can note is that, for example, in this setting before where I wanted to approximate delta over x on this bounded region, then what I wanted is I wanted some function. So if this was my approximation of delta over x, then I wanted something that here was minus 1 at minus delta. And here was plus 1 at delta. So that's what I want for my approximation. I can be epsilon off of this. This is basically what I want for epsilon sufficiently small. And so by the mean value theorem, this means that there must be some value, some point in this interval with derivative that's like a derivative like omega of 1 over delta. OK? So from this, you can conclude that any polynomial that approximates this delta over x sufficiently well needs to have a derivative of 1 over delta. And therefore, using this Bernstein inequality, you can conclude that it must have degree 1 over delta. Now there's one more thing that I wanted to mention, sorry, which is that if suppose that you were able to, this sort of suggests that if I have some ill-conditioned part of my function, then it would be better if it was on the edge. And this turns out to be true, which is that if you have your function and it looks like this, where it's like this happens near minus 1, for example, it could be like minus 1 minus delta. Then the degree of approximation you can actually get is you can get quadratically better of 1 over root delta. And this is the principle behind why sometimes you can get a quadratically better dependence on your condition number for positive definite matrices. And in the lecture notes, I've linked to a reference that says that if you're given a particular kind of blocking coding for a positive definite matrix A, then you can get an algorithm that scales as root k, root kappa instead of kappa. OK, that's it. Thanks.


Locke's presents Hollywood Makers of Luxe toilet soap bring you the Luxe radio theater starring Maureen O'Hara Richard Conte and Veronica Lake in Slattery's hurricane Ladies and gentlemen your producer mr. William From Hollywood ladies and gentlemen One of the most dangerous and adventurous assignments in the United States Navy is the hurricane service In which flyers chart the course of threatening hurricanes to warn those in the path of the oncoming peril Our play tonight Slattery's hurricane is the story of one flyers battle Against the elemental storm in the sky and the emotional storm within himself And as the stars of this exciting 20th century Fox drama, we bring you three accomplished players One of your loveliest favorites Miss Maureen O'Hara the dynamic Richard Conte and in her original role provocative Veronica Lake Besides being a favorite topic of conversation the weather also has its effect on delicate skin Smart women protect their complexions with Luxe toilet soap Because they know that in any weather Luxe soap is a safe complexion care Now here's act one of Slattery's hurricane starring Richard Conte as Slattery Maureen O'Hara as Aggie and Veronica Lake as Dolores To describe a hurricane It's one of nature's nightmares a windstorm of great violence Often several hundred miles in diameter with a dead calm at the center called the eye Every year from July to December the Caribbean crawls with these evil offspring of the elements Usually the hurricanes waste themselves over the ocean But several times in a generation the protecting hand of God is withheld and the hurricane heads towards shore and the homes of men This is the story of such a hurricane and of a man named Will Slattery On the coast of Florida is the United States Naval Station known as hurricane weather central Manda Kramer, sir. Yes, all planes have been evacuated to carry point sir as you ordered all except lieutenant Hobson's plane Sir, see that the crew stands by Hobson will be here in a few minutes I beg your pardon sir, but the radio tower has just reported contact with an unidentified plane using lieutenant Hobson's call Huh pilot claims he's reporting for hurricane reconnaissance to replace lieutenant Hobson He says Hobson is too ill to fly sir and request instructions Tell that fool whoever he is that we're in an emergency condition. Tell him to get up the second once I've already done that. What does he mean Hobson's too sick? I called his home not 20 minutes ago I spoke with him. What's this all about mr. Harding? I don't know sir I suggest you talk to the pilot yourself. He still has contact with us tell the tower. I'll be right there Yes, sir. He's coming through now Sambo three from Ruffneck Sambo three from Ruffneck Do I get instructions or don't I stand by commander Kramer will talk to you. Here you are sir Navy tower to aircraft. This is an order get off circuit and report it nearest airport There's a hurricane heading for the whole coast with a hundred and twenty not winds burying approximately a hundred and ten with a Center two hundred miles offshore out. Thanks for the bearing. That's just what I wanted to know Grumman mallard 2975 60 miles south east Miami 600 feet Changing course for 110. Sorry. I can't obey orders.  What do you know about it? May I take the microphone? Yes, go ahead. Slattery. This is Hobson. I know what you're trying to do. You can't get away with it Slattery your friend who had the metal. Yes, sir. You can't let him do it This is Kramer I'm warning you for the last time you're obstructing a Navy weather flight and you're jeopardizing the lives of Everyone in this area. You've got a reserve commission slattery and I'll have you caught marshaled. I'll put you away for 20 years Yes, sir. I'll contact you later, sir over. Are you coming in slattery? Answer me. Are you coming in? plane is capable of mission Who are you kidding slattery? You got a nice airplane, but I wasn't built for this kind of stuff No and neither were you but you're in it What are you gonna do about it? What can you do about it? This is how it's all gonna end slattery, but how did it begin? Don't you remember? Oh, you were really lucky that day 150 million people in the country Just can't believe it. What are you doing in Miami? I live here. I've got a nice soft job Why don't you keep in touch with a guy? Don't fly what I want to keep in touch with you Come on over to my car. I want you to meet somebody. Hey Dolores. Remember the guy I was telling you about Here he is honey Hobby Hopson Dolores Greaves. Hello, Mr. Hopson. I know all about you. How do you do? I'm still too surprised. I know what to say. It's sure good to see you hobby Look, this is where Dolores lives and we were just going in she's going to cook dinner So how about joining us? I wish you would if Wilkins standard I'm sure you can too. I sure like to miss Greaves, but I'm just going out to the naval station I'm on duty in half an hour. You got a car. Yeah, just down the street. Look honey I haven't talked to this guy in a long time. I'm just gonna walk him to his car be right back Glad to have met you miss Greaves. Give him a rain check for dinner Wilkins. Are you kidding about duty? You're not still in the Navy. Sure. I am. Holy smoke jets. Uh-huh private tears weather squadron private tears those flying freight cars What are you gonna do trying to catch up on your reading? I managed to keep interested I got a little flight to make right now as a matter of fact You want to come along if I can fix it why not? What about your girlfriend? Oh, I'll see you a little later. What kind of a flight hop patrol? Something like that reports of a big wind about 300 miles out at sea. Maybe a hurricane. It's part of our job to find out Okay, weatherman wait here to light square things with the law Sattery some fun quite a flight Harvey that's as close to a hurricane as I ever want to get close You were right in it son. How maybe you see why we need big planes a small plane in a wind like that be torn apart I still don't know what you guys are trying to prove you can't stop a hurricane by flying through it No, but if our dope shows that it's heading for land then we can warn all cities within 150 miles of the track to board up And you cut off the lives of that many people even for a day it costs millions and millions of dollars What do you do now write up a report? Ah, that's all been taken care of radio. Oh, yeah Sandal three from roughneck. Well, what goes now roughneck? Well, let's get cleaned up first I better phone my wife. Yeah, I guess I'd better give the law to ring to this way I Could you just said goodbye and walked away from that guy?  Well, what's the occasion? Who cares? Haven't you two got an anniversary or something? Say Dolores, I thought you and Harvey were gonna dance. That's right lieutenant not backing out Are you well, we'll just have to let these two get acquainted by themselves and save some champagne for us Well, Mrs. Hobson Small world isn't it seems to be doesn't it? I've got to congratulate you You didn't bat an eye when you just saw me and you're quite an actor yourself I lose you in San Diego find you here my Annie married to a guy I met in China very happily married to yeah, so I see What happened to you agi you might have telephoned me to say goodbye At least I've left a note in the milk bottle you have the nerve to sit there and ask me what happened to me Yes, you ran out on me. I chased all over town trying to find you and then I had a ship out. I Never did find out what happened to you. Now isn't that a pity? I ran out on you Maybe you've forgotten all the night you stood me up at the time you went out to get cigarette And didn't show up to two weeks later. Well, that was different. You just disappeared off the face of the earth You're darn tootin that was different The wedding bells had started to ring the last time will or had that slipped your mind too, agi I told you I'd be gone for three or four days. I had some things to attend. You told me that's so many times I Let's just forget it. Shall we it was long ago and far away And now you found somebody else So have I look you can't brush me off like that.   I'm not laughing, but you've got to listen. I'm married now. It's all over It wasn't easy for me either, you know it hurt for a long time. Well, but I got over it I had to so you're married hobby I could figure you married to a lot of guys agi, but somehow not to him What's wrong with him? Nothing he just swell guy just doesn't seem like your type Did it ever occur to you that I might have changed? I'm off that roller coaster. Mr. Slattery I'm off it for good. Take it easy How'd you two make out? Hobby's a wonderful this girl make anybody look good. She sure would say hobby. I just got a terrific idea You want Judy tomorrow? No, why? So how about me picking you and agi up in the morning? Take you for a ride in the boss's plane. I'll show you the best private ship in the world today I forgot to tell you agi will still a pilot. All right only works for some millionaire. So does the Laura as a matter of fact That's how we happened to meet the boss's secretary and the boss's pilot wait you see that plane. Mrs. Hobson sounds good, honey. Hey, what time will you pick us up? Will sorry, but you'd better count me out I have to go to the hair dresser in the morning. Oh, come on honey. You can go there anytime If you'll forgive my saying so mrs. Hobson, I can't imagine your hair looking any more beautiful than it is right now Well, Dolores, how about making me look good on a dance? I'm glad you called me tonight will they're both so nice, aren't they? Yeah Nice couple. Well, I have have you ever met her before? Mrs. Hobson. Oh, oh no not until tonight. She looks like quite a girl, doesn't she? Honey, I had a feeling you two knew each other in the service. I mean all three years I guess I thought I told you all about slatter. He wasn't quite a thing down in the Marianas He broke flight formation one day went off by himself and sunk a Jap cruiser. Oh big hero, huh? Yeah, the only trouble was his wing cameras were shot up and he couldn't show any proof Few of us saw what happened, but it didn't do any good So all he got for sinking his ship was 10 days with disobeying orders and breaking flight formation Oh look, honey, can't you put off that appointment tomorrow? I like that guy. It'll be fun flying with him again I'll be I Oh, all right, darling, whatever you say, she hadn't changed that much I drove them out to the beach. The boss is asleep. Dolores was waiting for us She wanted to show them around a good look, Aggie. Oh, we may never be so close to paradise again. Boy. Oh boy. What a layoff It's really something isn't it, but don't let the surroundings impress you too much You see will and I only work here. Of course will lives here too. The boss does a lot of flying He likes me around Those are my headquarters over there that college this must be very hard for you to take mr. Flattery I told you I was riding a gravy train Let's get down to the hangar aren't you coming to laura? I haven't made the gravy train I'm still just a secretary. Maybe I'll see you when you get there. Well, thanks for the tour What's your boss do? Well, not a great deal.  Oh, baby. Sure. He's the guy who makes them RJ millen Just think harvey the nation's sweet tooth built all this just like our dream house. You're thinking of building We've got lots of plans. I want to settle down somewhere have a nice steady life Three or four babies on navy pay. Oh, we can drink can't we? Yeah Would you please? Yes, mr. Millen Where's mr. Slattery going on the plane? Yes. What do you know about this? Why did he take the plane? I'm sure he won't be gone very long He took some friends along with lieutenant and mrs. Hobbson Lieutenant the navy mr. Gregory. They were in the war together. Well, he had no right to do this Well had to check the plane anyway, but get to loris Sure breakfast mr. Gregory. I'll take care of this. Don't go in miss caves I'll have some letters to dictate in a few minutes. Yes, sir Plane is 60 degree bank on one engine. I still can't get over it. Well, not again I dropped 10 years of my life up in that wild blue yonder.  I want to hook a home and get some rest I've got a date too. Remember. Oh, yes the hairdresser Look, harby. Why don't you take my car? I can pick it up later and we can The options have a nice flight. Oh, yes, mr. Millen. We had a very good flight, sir I'm glad loris has been telling us about your friends Good morning. Good morning. Very nice of you to let us impose like this mr. Millen I have a very special reason for being grateful to the navy lieutenant during the war most of my family was in shanghai Oh, but that's selfish. I should say the whole world is grateful Oh, uh, wouldn't you like to say goodbye to laura? She and mr. Gregory are in the patio Like miscreaves that last paragraph We regret to advise you that the last shipment of chocolate was in every way inferior This happens again that it will result in immediate cancellation of all our orders and type it up And there's the envelope to her banner headquarters and a copy to singapore Oh, mr. Millen's coming. Oh Lieutenant and mrs. Hobson. It's my business associate mr. Gregory. How do you do good morning, sir? That's a wonderful plane you people have. We certainly appreciate the ride. I'm delighted So you and mr. Slattery our old friends we met in the navy mr. Gregory Well, we won't be needing the plane today, mr. Slattery. So if you wish to be with your friend, thank you very much, sir See you later. The loris. Goodbye. Goodbye. Goodbye. Your friends are all right the loris And the next time you bring guests first you ask me And I will say no Gregory tell it to slattery No point in upsetting to loris. You know, she isn't any too well. I'm all right, mr. Millen I've been meaning to ask you no more headaches my dear no more dizzy spells I'm all right. There's nothing wrong with me. And in that case you won't mind typing those letters right away, mr.  Even with my heart see how calm I am. That girl if you ask me I'd throw her out. She's a rope around our mix. Yes Yes, she is Gregory That's exactly why we can't throw her out In a few moments, we'll bring you act two of slattery's hurricane Now here's our hollywood reporter Libby Collins. Greetings Libby. I have a nomination to make mr. Keeley How about making Clark Gable our permanent mayor? Oh, you're talking about the metro golden mayor comedy key to the city. Mm-hmm Well, I'll go along with you on that Libby, but let's make Loretta young honorary mayor at the same time Well, why not? She's delightful as a lady mayor in the picture very dignified and hardworking Till cart comes along and then romance begins and some hilarious comedy with frank Morgan as the fire chief And Marilyn Maxwell is a very beautiful but not very bright bubble dancer I love the scene where Loretta goes to the costume ball as a little girl in pigtail And Clark complicates things with his surprise masquerade costume so much so that they both land in jail But no matter the lady mayor wins her man that we take for granted. Let me with Loretta young is the heroine of key to the city Yes, john She's so thoroughly feminine always has the fresh beautifully groomed look that's so appealing You know, she's a luxe girl. All right. Indeed. She is Loretta has used luxe soap for years for complexion care And now that luxe toilet soap comes in the big bath size. She's more devoted to it than ever It's a really luxurious bath soap isn't it certainly is where that rich creamy lather and a perfume screen stars love The luxe soap perfume is exclusive It's a blend of many flower fragrances And the luxe soap beauty bath makes you sure of all over luxe loveliness That's important these days when the new strapless dresses are really high style And arms and shoulders must be at their prettiest You can't buy a finer soap than the generous new bath size luxe toilet soap Your store is featuring luxe toilet soap now Be sure to put both the bath and the complexion size on your shopping list Let the whole family enjoy this big longer lasting bath cake Remember nine out of ten screen stars use luxe toilet soap Here's our producer mr. William Keely act two of slattery's hurricane starring marino hara as aggy richard conty as slattery And baronica lake as the lorries Most of florida a raging hurricane mounts and fury And headed straight for it buffeted by the lashing winds is a small airplane at the controls is will slattery Do you read me navy tower? Did you get my message? Do you read me over? We're at your slattery. Are you coming in or not? You want the exact location of the center of the hurricane, don't you?   Kill myself. It's an idea, isn't it? What would you do commander kramer, sir? Only you'd never get yourself in a spot like this to begin with, would you? But then you don't know aggy And you'd never pull the stunts that I'd call would you like the time I invited at the dinner that intimate little roadhouse Just you and I We're forming aggy I can remember when you were never on time for a day. Oh, no will Wow This is a nice place. I was hoping you'd like it Hopping it off. All right. Oh sure these three-day patrols are just routine Well Where's the law?  Well at the last minute she got a headache You mean she's not coming at all. You didn't really expect to find her here.  What's the harm in having dinner together? You've got to eat someplace. What's the matter with you?  Look will I don't intend to get myself in any spot where I have to trust you or myself I'm going home. All right. I'll drive you home. I'll get a cab. Why we're not a couple of kids. You want to go home? So I'll take you Too bad they serve the best food Sorry if I spoiled your evening will forget it Maybe I am being a little silly but with hobby out of town It was my fault. I tried to pull a fast one. I didn't get by with it I'll try again some other time. I wouldn't put it past you. Thanks for the warning. Is that the house?  You've got quite a memory I thought about that night a lot of times That blue dress you wore Sequins and a little black ball right right about there You looked awfully pretty that night agi There's not much you forget. Is there there's nothing about you. I can't forget Well, aren't you going to scream slap my face or doesn't mean anything now when I kiss you? Well, please Please I Good night. Now wait a minute There's one thing you've got to get clear on your head what I told you the other night I meant every word agi I've got you in my sister and I can't get you out ever Oh The boss keep you late tonight Your door wasn't locked and I thought I'd wait for you. I was worried about me. I always worry about you Uh, did you make your friend for dinner? Who did you say it was? Well, you're getting at the floris You wouldn't lie to me. Would you will it wasn't agi?  Look do me a favor. Will you drop it? All right? We'll drop it But there is something else I wanted to talk to you about I want you to quit this job quit the best job a pilot ever had You got it for me, didn't you? What's the matter with you anyway? You know, it's a bad job. It's doing the same thing to you that it's already done to me.  That's all I know. That's all I want to know I don't want to hear your ideas on it or anybody else. Is that clear? You have such a nice way of expressing yourself Yes, it's very clear. I won't bother you with it anymore. Good. I won't bother you with anything anymore. I'm quitting tomorrow Yeah, I know you were going to quit last week and the week before and the week before that Why don't you relax? So I had dinner with another girl. So what and you were with her? Yes, I was with her You want to know something else? You were right I knew her a long time before harpson ever said eyes on her Would you mind getting out of here? I'd like to get some sleep I seem to remember you're telling me harvey was your friend He is And the fact that he's married to aggy doesn't mean anything to you It means that he can take care of himself and while you're asking questions. I'll tell you something else Aggy's the only girl that ever meant two cents to me. She always has and she always will And if harpson is married to her, that's just too bad I used to think you were a man will I don't care now what you do to yourself and I don't care what you've done to me. I asked for it But what you're doing to aggy and harby is rotten and I hate you for it.  Well, yes, mr. Mill. I guess we are and I know you must be worried about it Too worried sir because she isn't here. She's gone. Oh come in Gregory. No word nothing Maybe she's gone into town. Is anything happened between you two? No, nothing important It's very important to us that we will find the sleeves as you know, mr. Slattery She handles all my personal matters and the private files It's not like her to go off without any word if you should hear from of course, mr. Mill I'll let you know at once. Thank you. You see having no relatives no place to go I just can't imagine. Well, let me see what I can do mr. Mill. Thank you, ma'am for her apartment house She's gone any forwarding address none Thanks One fast phone call really didn't think she'd walk out on you. Did you not unwilled slattery? Anyway, maybe it was a good idea if the laws did get herself lost for a while There'd be nobody now to check up on you nobody to tell you the kind of a heal you were Nobody to say after all she is hobby's wife You ran true to form didn't you willy boy? Why didn't you just stick a knife in the hobby's back? Certainly. Well, that's all right, mr. Mill. Any word from Dolores? No, no nothing I uh, I need playing this morning in about an hour. I'll be ready, sir You may have to consult your maps. It's a very small island Curito Santos No, I'm not far from Cuba. I have a dear friend who's a planter there. If it's on the map, we'll find it mr. Mill Excuse me, sir, but do you think you should go you don't look so well? I'm not well, mr. Slattery Perhaps this little trip will make me feel better Uh, we'll be going alone just you and I mr. Gregory is in town today. Yes This is nc 2975 calling miami radio nc 2975 calling miami radio over nc 2975 this is miami radio go ahead miami radio. This is nc 2975 I'm 90 miles southeast miami coming in 1000 feet Will Slattery pilot R. J. Milne m i l n e owner of plane died a few minutes ago during flight Heart attack, please make arrangements for ambulance at r. J. Milne state Notify car on his office confirm, please. This is miami radio. Roger will call out Something he had it under his shirt just before he died. He begged me to throw it out. Didn't I? Now where are you? Are you all right? I know all about it. I know what i'm doing I just want to know where you are baby. Don't say that just listen to me for a minute. Will you? Hello Hello Operator Operator the possible to trace the call for me. I was just speaking Milne or has carried a tape to his chest business papers. I want it.   We've been paying you no questions asked no trouble very comfortable Let's keep it that way. This used to be an easy job before I knew anything about it But now I know that makes it tough so I want more dough 500 bucks a week and I want you to talk real nice and friendly to me like mr.  Slattery I like that when you give me the envelope you have news for it now sure i'll give it to you But then what happens?  Don't try it mr. Gregory.  It's the rest of the cops. It tells them all about you So if you want to keep healthy, I've got to keep healthy too. Yeah, and there's nothing to worry about is there We have a deal mr. Slattery. You make a lot of money. You'll be very happy Oh, I almost forgot this letter came for you today I'll give me for opening it And my congratulations Navy department. Yeah, it looks very official Your hero They're going to give you a medal Navy cross You just try to keep sober for the ceremonies mr. Slattery. You talk some more business after you get to admit That was the end of the free ride Gregory punched your ticket and he punched it good You were ended up to your ears now and on top of that you had the guts to accept the navy cross The navy thought a lot of you that day. Didn't they slattery? Full dress parade people in the grandstand Even an admiral making a speech United States Naval Reserve For extraordinary heroism in an air sea engagement off the marianas islands on august 6 1944 For single-handedly sinking the japanese-like cruiser zagaho Thereby contributing materially to the success of his squad His skilled airmanship courage and devotion to duty Appell the highest traditions of the united states naval service For the president of the united states John L. Sullivan secretary of the navy Congratulations lieutenant. Thank you, sir. You don't think i'd miss this do you?  What was that for? For a lot of things I'm proud of you will hobby still at pentacolor. Yes, he'll be back tonight Come on. Let's get out of here You decided that slattery the big hero with the navy cross Somebody else reviewing this morning at master field mr.  Miss graves was found at the field unconscious I made the diagnosis. It was reasonably easy He's in one of our psychiatric wards. I I just can't believe it. No nobody ever does I had quite a talk with miss graves You're aware she was very much in love with you That as a matter of fact it was more than love She depended on you I don't know whether she did or not anyway What's that got to do with it the fact that she might not be here if you did now look doctor I had nothing to do with it. I didn't say you did I only said that it appears that miss graves was very much in love with you But apparently she felt she was someone she could depend on now wait a minute. I suppose you wait a minute I know i'm exceeding my authority in talking to you like this, but If you treated her like a human being using a little sympathy a little understanding Things might have been quite different for her Now if you care to tell me your side of the story there is no other side May I see her please? I don't want you to see me like this Don't turn away honey. It's all right Why didn't you tell me I tried to but you wouldn't listen. I didn't know you didn't want to know You didn't want to know about anything not about me or milner. Gregory not even about yourself I'm not blaming him. It's my own fault He didn't ask me to follow you wrong like a baby You even warned me it was just a home for laughs I made a mistake I want to help honey. I'd do anything You can't help me will Nobody can't don't talk like that You've got to get well Dr. Rose told me you were at the field that must have handed you a laugh watching me get a medal But don't kid yourself that I think medals mean anything It's what people do that counts and I I didn't do so good did I Look baby this medal I don't deserve it Keep it for me. Will you? We'll make a deal You get well, and I'll try to make up for the things I did Then I'll come back for it. Okay Just leave me alone I'll leave it here. Anyway I'll be back tomorrow honey in case you want to see Oh, well We'll I'm so glad you're here hobby's back back Yes, and he saw us in the car this morning. He started asking questions. What did you tell him? I told him we'd known each other before I had to well, what did he say? That's just it he won't say anything He's been drinking and now he won't even listen to me. He'll listen to me Well, we've made a terrible mistake. You know that now as well as I do We'll before it's too late. We Hobby I want to talk to you you want to talk to aggy Go on talk to her. Oh my me. You never did all right, but you're gonna listen to me You've got the wrong angle on this hobby. This has all been my doing not aggies Sure, we knew each other before but that was a long time ago I'm the one that started it up again. I'm the one who tried to take her away from you. Nice going. Yeah, but it didn't work out You want to know why? I'll give you a very good reason Aggy's not in love with me. She's in love with you.  You don't have to believe anything. Just look at him I don't want to look at her You're the one I want to look at Come on fight Fight what's the matter with you fight Hello, this is commander crema. This is commander crema. Oh, oh, no, no, sir. He's not here. Give me the fall hobby. No, no Hello commander. This is hobbson sir I just got in that storm has intensified itself into a major hurricane hobby. It's an emergency. I wouldn't call you hobby there, sir They really prove to evacuate all the other planes to cherry point. You're the only one left. How soon can you get I I'll say 30 minutes, sir. You better make it 40 in this rain. Good. I'll have you cruise standing by.  You've got to believe me get away from will you've got to stop him. Yeah That'll hold it for a while The rest is up to you waggy. I've got to get out of here hobby You're trying to contact slattery commander. We can't get a signal. Yeah, there's no use trying He refuses to answer besides if he's still alive. He's beyond transmission arrangement Why would he do a thing like this hopson? Why? Help me out. Helped you out. I was in no shape for duty, sir. When you're calling or not I've been drinking slattery well, but you got here as you said you would yes, sir. I got here too late You got any chance come in a light plane in a hurricane. You know the answer as well as I do All right, hardy. Keep trying to contact him, sir. I'll let your crew hopson We'll give slattery 10 minutes after that be ready to take off. Yes, sir Now for station identification. This is cbs the columbia broadcasting system. Well, the slattery is hurricane Our guest this evening is blonde blue-eyed marion marshal Who used to be a cover girl before she came to the attention of 20th century fox? It's more exciting to be a starlet isn't it marion? Definitely mr. kealy I've enjoyed every part I've played and for inspiration. There's always a set of a new picture to visit Take comedy technique for instance I had a wonderful time seeing them do mother didn't tell me what a gifted comedian dartha maguire turns out to be She's perfect as the young wife who discovers that being married to a doctor makes home life a little uncertain William londigan is the absent minded physician who unintentionally neglects his wife Yes, and I love june havoc is the other doctor's wife. She has a dry wit that delights you I understand june's been so busy with stage engagements. She hasn't had much time for pictures It's a pleasure to see her in mother didn't tell me a really accomplished actress and Also very easy on the eyes. She certainly is mr. Kennedy a real blonde beauty with fair delicate skin And I happen to know she's a luxe girl has used luxe for years June is like so many famous stars who wouldn't trust their complexions to any other soap That's because luxe toilet soap gives delicate skin the gentle protecting care it needs Indeed it does I find those luxe soap facials really work. Mr. Kennedy. They leave skin so fresh Really make it softer and smoother Recent tests by skin specialists prove how effective those facials are in actually three out of four cases Complexions improved in a short time Thank you, miss mary and marshal for being here tonight Now here's a shopping hint for women everywhere your store is featuring luxe toilet soap now Be sure to put both the complexion and the big bath size on your shopping list See what quick new loveliness this fragrant white soap will give your skin Remember nine out of ten screen stars use luxe toilet soap Now mr. William keely our producer the curtain rises on act three of slattery's hurricane Starring richard conty as slattery Marina hara as aggy and veronica lake as the lorries Will slattery is taking flying out alone to meet the hurricane to report its position in progress To slattery the hurricane holds the answer to all his problems the chance to square himself For a million to one chance of coming back alive Friends of the hurricane gathers itself with a thrust shoreline and the light plane is little more than a leaf in the wind But suddenly slattery sees a break in the storm A patch of light and a quiet sea beneath a great pocket surrounded on every side for the roaring elements This is it the center the heart the eye of the hurricane But what do you know slattery you made it? Now you've got something to tell them back there Master tower from slattery master tower from slattery. Do you read me over? Master tower from slattery. Do you read me? Tower from slattery have positioned. Do you read me? Come in please This is operations. Go ahead, sir There has been no further contact with grumman mallard 2975 pilot slattery Order direction finder networks to trace transmission, please grumman mallard 2975 Advise coast guard and all ships in area. That is all for repeat the order No further contact with grumman mallard 2975 pilot slattery All message operations stand by Master tower from slattery. Do you read me? Five five four three two one half position Over slattery. This is kramer. I read you over.  The storm center at 25.8 north 76.9 west over slattery repeat position Repeat position over Latitude 25.8 north Longitude 76 point roger hold on. I'll check the charts He's right that puts the center of the hurricane 240 miles north or east to where it was last night Slattery, can you see the ocean? Can you see the surface? Sure? I can see it. Can you see anything else? Just a lot of sky and a lot of water I'm in the middle of america round. What else can I give you stand by slattery stand by Well, he's got the wrong dope razor back islands only a mile away from his position He's where he says he is. He couldn't miss it Slattery your position does not check your instrument must be wrong. Your position does not check I've double checked my instruments. That must be the correct reading over. They must be lost Slattery, I cannot confirm your position Return to base. Do you read me? Return to base. Wait a minute. Hold everything Slattery, what is it? I almost hit something looks like an island Shape like a big sleeping heart razor back. He's got it slattery your position verified. Do you hear me position verified? Yeah, thanks return to base for what caught marshal 20 years Is that what you said? We'll talk about that later. Are you all right slattery the plane holds together I'll be in slattery give tower 10 minute reports on wind velocity and direction roger Hello, hello, give me operations Operations, this is commander crema teletype urgent to combine forecast center from navy hurricane weather central Hurricane definitely will strike the coastal area order evacuations and boarding up procedures start immediately Present position hurricane I follow I'm lieutenant hopkins wife. Has he taken off yet? I don't think so, ma'am Then he must be up in the tower. Is it all right? I'm sorry mrs. Hobson. You can't go up emergency condition If you care to wait here, I'll see if I can contact Mallet navy tower from grumman mallet mayday mayday. Do you read over? How do you think they're gonna help you run up a ladder? The starboard engine's dying wise guy. You're cooked. That engine isn't going to catch Just fly her down You're the guy that wanted it this way Hobby, I was so frightened about flattery. Don't worry. He made the eye of the storm. He's on his way back. No hobby It wasn't will I was afraid you might be flying. I know how you feel. You have every right in the world I don't know what to say except that But I love you hobby I always have and I always will I I have to stay here for a while Can I can I wait for you? Wait for me I just can't think eggy Come in flattery Come in Good not read last message. Are you all right repeat? Please good not read you over Tower from flattery Tower from flattery Do you read me Mayday lost one engine Is lieutenant hobson available lieutenant hobson, please over stand by flattery lieutenant hobson, please Slattery's got guts. All right, not if I see rescue. No private plane is built for weather duty. All I can do now is I'll flattery wants to talk to you hobby. Yes, sir Come in flattery. This is hobson come in tower from flattery Mayday won't be able to make it Take this message hobby Have information on narcotics smuggling ring information on narcotics smuggling ring Information on their activities is in my safety deposit box In bank corner of collins and 15th In bank corner of collins and 15th Do you read me? Do you read me? I read your flattery over You've heard it, sir. I don't know whether he's crazy or not to pass it on to the miami police right away. Yes, sir This is kramer flattery. Do you read? Give me your position Come in flattery Come in Aggy you You better get downstairs again. I'll ask commander Come in Anderson calling commander kramer commander kramer, please over go ahead anison ground control approach unit reports aircraft Sir has no radio approximately three miles from master's field. Holy pete. He made it. He's coming in hobby I don't know how he'll ever bring it down in this anybody can he can plane approaching sir plane circling the field All right emergency crew on station Emergency crew on station. I walked off the field. Didn't I that tells me you're all right. Did you get my message? Yes, your position was right. We lost contact with you right afterwards No, no not that message. I sent a message about a statement. That's been taken care of too Incidentally flattery uh, you know charges filed against you. Thank you Hobby Hobby, I'm sorry. I'm sorry for everything.  Okay, lieutenant I'll help him out of here. Yes Turn up the radio mr. Hardy. I'd like to hear what the weatherman has to say sir Within the next two hours the storm has risen to hurricane intensity Thanks, however to the advanced warning provided by the efforts of the air force the navy and the weather bureau Residents of the coastal areas have ample time in which we evacuate to higher ground The authorities urgently warn against an update according to recent Well, we've forgotten something sir. It's hurricane boiled up so fast. We never gave it a codename What was it on the last one messy tassie sir. Yeah the one before that Whirling joe Well, there's nothing funny about this one um That's just name this one Slattery's hurricane Oh slattery. Well, hello commander. It's all right. I've cleared you with a dispatcher. I uh, just came down to wish you luck Thank you very much, sir. I uh, also want to tell you that I'm glad that you have Back in the regular service again. It's good to be back, sir.  What's that head of yours made out of anyway bones solid bones, sir You can't hurt that Well, if somebody else would like to wish you luck. He's all yours.  I'm all well now I'm going to stay that way. You don't have to worry about me anymore You look pretty wonderful yourself. Well, that's the uniform I brought this will your navy cross. I don't need it anymore I thought you were coming back for it. Keep it for me. Will you aren't you going to kiss me?  What's the old saying if march comes in like a lion? Does something like going? Well, it came in like a lion in the luxe radio theater in oklahoma. It came in like a Leopard a very spotty month The marine every time you visit us. I'm reminded how gorgeous your luxe complexion looks in technicolor Thank you bill. I've always been a luxe girl veronica. There's something I've always wanted to ask you What became of that hairdo you used to have you know where you wore it over one eye Well, I'd say I just combed it back. I wanted to see how the other half lives Well, it's very becoming veronica and we enjoy looking at the other half of a lovely luxe complexion You know luxe has always been my favorite complexion care Can you tell us what play you have for next week bill? One of the best love stories ever written marine Because next week's drama is little women Our play is adapted from metro golden mayor's superb motion picture And we'll have three of the original stars of the picture June allison peter lawford and janet lee Three great stars in the story enjoyed by generations of americans Perfect entertainment for the whole family So don't miss june allison as joe peter lawford as lorry and janet lee as meg in little women We'll be looking forward to it bill. Good night. Good night and many thanks to you all Here's a passion flash from hollywood an all-occasion glove with a novel drawstring Adjust to three different lengths short pursuits Brace it length for afternoon and full length for evening Marina hara has ordered several pairs and pastel fabrics to complement her spring and summer wardrobe Of course her gloves like her washable dresses and blouses get gentle luxe flakes care She knows how easily wrong washing methods fade delicate colors actual washing tests prove Miles safe luxe flakes care keeps colors excitingly lovely up to three times as long Be sure to get luxe flakes tomorrow. Give all your knife washables that lovely luck luck fever brothers company the makers of luxe toilet soap Join me in inviting you to be with us again next Monday evening when the luxe radio theater present june allison peter lawford and janet lee in little women This is william kealy saying good night to you from hollywood Through the courtesy of 20th century fox producers of under my skin starring john garfield Richard conty also appeared through the courtesy of 20th century fox Whose next release will be the technicolor production wabash avenue starring betty gravel and victor mature Third in our cast tonight were steven dunn as hobson bill conrad as kramer alan reed as milne And frederick tozer larry dobkin bill john sone george niece william green eddy mar and bob briffon Our play was adapted by sh. Barnett and our music was directed by louis silvers This is your announcer john milton kennedy reminding you to join us again next Monday night to hear little women Starring june allison peter lawford and janet lee Stay tuned for my friend ermah which follows over these same stations The columbia broadcasting system