Okay, good. So then I will introduce you now. So, okay, so welcome. Welcome everybody. Today, I'm very happy to introduce the speaker. He's doing research in harmonic and complex analysis, operative theory and interactions between analysis and probability. And today you will talk about sampling, tiling and quasi crystals. So, okay, who is yours? Okay, thank you very much, Zacharias. Well, thank you very much for your invitation for the possibility. I'm very, very happy. As Zacharias said, I want to talk about the some relationships between this classical sampling theorem, tiling and quasi crystals. So, let me begin with some prediminaries or conventions. Suppose that we have a Vorel set omega in Rd, then I will identify the space of 2 of omega with the sub space of R2 of Rd, of those elements, those representatives by Nishalmos everywhere in the complement of omega. I will use this normalization for the Fourier transform to find the exponent of the exponential. So, the inverse Fourier transform will be the same form without the minus here. Since it will appear many, many times throughout the talk, I will use this notation for the exponentials. I will note this lambda will be the frequency that appear here. So, when I consider the exponential as an element of R2 of omega, I implicitly assume that it is the exponential restricted to the set omega. So, it is multiplied by the characteristic function of omega. Well, in order to state the sampling theorem, let me recall the definition of the Bayesian spaces. So, now let me take a bounded Vorel set omega. Then the prediminary space associated to this set of omega is the sub space of those elements in R2 of Rd, whose Fourier transform, Rnishalmos, everywhere in the complement of omega. In other words, the Fourier transform of F lives in what I call L2 of omega. So, the Fourier transform, because that Fourier transform L2 is a symmetric isomorphism. So, the Fourier transform here is an isometric isomorphism between the pelibino associated to the set omega and the space L2 of omega. So, this space is the natural framework, if you want to work with signals in the nature, for instance, which do not have a really high frequencies. For instance, if you want to work with the human voice, or if you want to work with signals that the humans can listen, you don't need to work with the whole spectrum of frequencies. You can restrict your attention to the frequencies. And so, this is the natural framework, mathematical framework to work with these kind of signals. Although it is presented as a sub space of L2, inside each equivalent class there is a distinguisher representative that is very regular. Since, if you have an element in L2, the Fourier transform is causing L2, and if omega is bounded, then the Fourier transform is also in L1. And so, if you consider this integral by the dominated conversion theorem, this function here is continuous. Moreover, you can derivate this expression, and what you have at the end is that this function is not only continuous, but also it's infinite function. So, the elements of the pelibino space can be thought as very regular functions. So, you can evaluate the element of the pelibino space in different points. And it turns out that the evaluations are bounded functionals. Because if you can rewrite this integral here as in this way, what you have is that this is an linear power between this function, this function, and this other function. And so, if you use the properties of the Fourier transform created with the planchial theorem, you can transform this linear product in this other linear product. And what you get is that the value of f at x can be written as the linear product of f against this function kx here. The function kx is the inverse Fourier transform of the exponential with frequency x restricted to omega. So, it is an element in the pelibino space. The norm of kx is equal to the square root of the measure of omega. So, in this way you see that the evaluation is a continuous function, but also you see that all the evaluations are uniformly bounded. Moreover, all of them have the same, the same norm. Well, this is more or less what I wanted to say about the pelibino space. And now we're ready to recall the sampling theorem, also known as Shannon-Wittaker-Cotternikov theorem. Now, that's our trick that was mentioned to the real line, and the real line that is considered the set omega as an interval, the interval minus one-half, one-half. So, given an element in the pelibino sub i, then this theorem says that the element, the f of x can be obtained if you do the sum of the values of f of x. f at the different integer numbers combined with this function here, that is the cardinal sign. And this series converges not only in L2, but also uniformly in L3. This is a very important theorem in signal processing because it's the first step toward discretization of your function. Because observe that it's telling you that if you know the function at the integer points, you know it everywhere because you can combine these values at the integers with these functions here and you obtain the function everywhere. And for this reason, it's very, very important in applications. Let me recall very briefly the main ideas of the proof of this theorem. The first ingredient probably the most important is that something that probably you remember from the functionalized course is that the exponentials with integer frequencies constitute an orthonormal basis of L2 of i. Then if you take the Fourier of the inverse Fourier transform of this exponential, you get, you can do the computations, they are not difficult, and you get exactly these functions that appear here, this cardinal sign. Since the Fourier transform is isometric, this orthonormal basis goes to this other orthonormal basis. And so these functions constitute an orthonormal basis of your space. But also, as I said before, if you take the inverse Fourier transform of an exponential, what you get is the reproducing kernel. So this fn that you see here is exactly the inner product between f and this function. And so this sum here is just the expansion of f in terms of this specific orthonormal basis. And so it converges into, okay, so this is the first part. And the second part, the uniform conversion of this series comes from the fact that the reproducing kernels are uniformly bounded. And all of them, in this case, have no one. So using, for instance, Cauchy-Schwarz inequality, you can see that the L infinity norm is controlled by the L2 norm. So the L2 conversions of this series also implies the L infinity conversion. So you have the uniform conversion. So if you pay attention to this scheme of the proof, you see that if you want to extend this theorem to more general settings, to more general sets or even to RB, the crucial step is this one, is the existence of an orthonormal basis of exponentials in the L2 of this set. Now, the orthonormal basis in general are very, very rigid systems. So this rigidity leads us to consider some alternatives in order to be able to reach much more general sets, because otherwise we can extend this theorem, but only for some specific sets. So let me recall the definition of re-spaces and frames. Well, if you have a family of vectors in a Hilbert space, we say that this family is a re-spaces of the Hilbert space. If this operator, finding the literal tool of the set lambda to the Hilbert space, is bounded and inverted. And the operator is defined in this way. What does it mean? Okay, it means that since the operator is invertible, it starts shaping. So it tells you that any element of your space can be written in this way as a combination of the element of your basis. On the other hand, since the operator is injective, this expression is unique. The coefficients to represent the given element of your space is unique. And then this other requirement here, the boundedness, is because you have here a series. It's not like in linear algebra that you have a finite sum. Now you want that if you perturb a little bit the coefficients, you want that the result doesn't differ too much from the original result. So this is the reason why you required here the boundedness. So with the boundedness, if you perturb a little bit, a little bit in the sense of a little l2, then the sum won't change too much. This is with respect to real spaces. And what about frames? Well, look at this definition and now here is frame. These bases are frames. You see, there is only a small change and it's here. Instead of asking invertibility, the operator is injective. The idea behind frames is that you are trying to generalize a notion of system of generators that appear in linear algebra. Here, since you are asking such activity, what you have is that any element of your space can be written in this way. So in this sense, you are generalizing this idea of system of generators that appear in linear algebra. One nice fact about risk basis frames, not that any risk basis is in particular frame. The difference is that in the risk basis, this expression for given an F, this expansion of F was unique in the case of risk basis. In the case of frames, there are many different ways to write a given element of your space. But that's a unique difference. So one nice feature of frames for risk basis is that if you have a frame, then any element of your space can be written in this way. Here you have a sum and you combine these other elements in g lambda. g lambda constitute what is called a dual frame. And these elements are combined with the inner product of F against the element of your frame.  So if you have in some L2 of omega a frame of exponentials now, then you can see them, these special functions that appear here, dual frame. If you consider this expansion of the element F of L2 of omega, but now using the frame of exponential. Now you start to do the same as in the proof of the sampling theorem. So you start to use the Fourier Stanford. And so, since these are exponentials, designer products, they're not to be the evaluations of the functions in the pelivinar space at the points of lambda. The frequencies that appear here. In the sampling theorem what you have here where there's cardinal signs, these signs divided by x. Now what you will have is the email for your transfer of the dual frame. But still, you can see that you can recover your function everywhere just knowing the function in these points. In the point of the set lambda, capital lambda. So it's not to have a frame in order to have this kind of reconstruction scheme for functions in the pelivinar. Okay, so now let us try to study when you have in space L2 of omega, a risk basis of exponential, or a frame of exponential. For these, let me recall the two definitions. The first one is the definition of time. We say that z omega, where z omega in rd, size rd with translations in a set lambda, at level k, if this sum is equal to k, almost everywhere. What is this sum? I consider the characteristic function of omega, and I translate the characteristic function of omega using the element of lambda. I sum all this translation, and this sum have to be equal to k, almost everywhere. In this case, we say that omega dials the space, rd in this case, at level k, the level is given here. When the level is one, we simply say that the set dials the space, we are specifying the level. On the other hand, let me recall that in rd, four lattice, simply a lattice in this talk, is the image of set b through an inverted matrix. So in particular, a lattice is subgroup of the additive group rd. Even a lattice, there is notion of dual lattice, the dual lattice consists of those elements in rd, such that the exponential with frequency mu in this case are lambda periodic. Another way to say the same is that the dual lattice consists of those elements mu in rd, such that if you take a product of mu and lambda, this belongs to the integers for any lambda in capital lambda, in the original lattice. You can also prove that the dual lattice of this expression is, as in the case of the lattice, so it's a lattice and the inverse in the invertible matrix associated to the dual lattice is this one. This is the inverse of the transpose of the matrix associated to the original lattice lambda. By the way, I never put lambda here, this lattice here is lambda, or this is the dual of this one. With this definition, let me mention some some results about the existence of this basis or our basis related with times. So consider a lattice in rd and a Borel set in rd. Our first result is so-called Fugleder theorem, and it tells you that you have the exponentials with frequencies in the lattice lambda constituted and also an orthogonal basis for L2 of omega if and only if the set omega times the space with the dual lattice. So you have a very nice relationship between the existence of orthogonal basis of exponentials and time of this late existence of orthogonal basis of exponential with some properties, geometric properties of the set omega. More recently, there was appeared this beautiful theorem of Fugleder theorem that relates also existence now of free spaces with k times. More precisely, what they proved was that if you have a bounded set, then if this set k times the space rd with a lattice lambda, then there is a risk basis of exponentials for the space L2 of omega. This is very nice. The first result appears this relation explicitly between tiling of k tiling and the existence of free spaces of exponential. Before this theorem, Schroding-Marsow has proved a special case of the theorem where the set omega was the finite union of this finite union of qs of equal side events. After this theorem appeared, Mihailis Konosakis also give an alternative proof of this result and with his method he was able to say something more about the structure. If you see in both results, we are dealing with lattices. This is taking you that you have a group acting in both situations. When you have a group action on your problem, you know that you can have a reduction in some sense of your problem. So with Elona Vor and Carlos Cabrilli we studied these problems from this point of view, trying to use this reduction that comes from the fact that you have a lattice acting on your problem. So we made this version of Graysstatt and Leibniz there. There exist elements a1, ak in rd such that for any bounded or set omega in rd such that in k-tiles rd with Elona, then the exponentials with frequencies in this shift of the dual lattice. The dual lattice here is shifted by a1 and here a2 and a3 and until ak. These system exponentials constitute a risk basis for L2 of omega. So here what you have is that the same k-tiles elements work for any omega that k-tiles rd. And you have here that you can say something about the structure of the risk basis that you get in these spaces. This was also the structure that was found in Michalis in his paper. On the other hand, this k-tapel here are Scheneck in the sense that if you pick them at random in the unique yield, for instance, with probability one, the k-tapel that you have chosen will work for this yarn. Using again this structure provided by the lattices, you can prove also a kind of converse of this result. Which is this one that tells you that if you have a Borel set, a bounded Borel set in a v such that L2 of omega meets a risk basis of this particular form, then the set omega had to k-tile the space rd with lambda. Here you use the dual. And so in this way you close the circle as in the case of Fugler's yarn. In the case of Fugler's yarn you should remember you had an if and only if. You had the existence of a normal basis, if and only if you have the tiling with the dual lattice. Here you again have something similar, you have that you have risk basis with this structure if and only if your set k-tiles the space with the dual lattice. In particular, it's telling you that for instance in the space L2 of a triangle, you cannot find risk basis with this kind of structure, because the triangle cannot k-tile the plane. There is one important difference between the Fugler's yarn and this yarn here, that is this hypothesis of boundedness. Here we are requiring the boundedness of the Borel set and this condition cannot be dropped. You can find counterexamples. Although, but then there are some results from boundedness sets formed by Yana Karvaha and her PhD thesis. You had to ask some extra medical requirements to the set in order to have a risk basis of exponentials like this. Well, this is about risk basis. In all these yarns you have a lot of structure. What happens if you want just a simple risk basis and you don't care about the structure. Well, we don't know too much about risk basis of exponentials in spaces L2. And here is a list of what we don't know and we know shortly. Just to have an idea, look at, we don't know for instance if the L2 of the Borel meter is a basis of exponential or not. We don't know if the L2 of the triangle is a basis of exponential. I said that we know that it does not admit a basis of exponentials with that specific structure but without structure we don't know if we have a risk basis of exponential. Actually, we don't know if there exists a bounded set without risk basis of exponentials. So there are many open problems here. What we know is, for instance, if you take a finite union of the shown cubes and then the corresponding L2 of this set, admit a risk basis of exponential. And this is very deep result due to Gadi-Kosman-Shaharnitsa. And another theorem about places where we know the existence of risk basis is this one about centralized symmetric polytopes with centralized symmetric phases. This is a quite new result here last year, I think, due to... But you see, we don't know too much about the existence of risk basis of exponential. So let us move to Frank's and see what can say about that. In the finite dimensional setting, we know that if you have a system generator, it has more elements or at least the same amount of elements as basis in your space. In this setting, the intuition say that if you have a frame, you have to have more elements than in the case of a basis. But how to formalize this idea? Because both of them have to be accountable. One way to formalize this idea is through this very intensive. Let me recall briefly the definition. If you have a set lambda in RB, let me write in this way qr of x. In this way, I denote the q of center at x and with side lengths r. And then the upper density is defined in this way. You count how many elements of lambda you have in this qr of x and then you divide by the measure of the q. And you repeat this with all the cubes in the space and all the points in RB. In order to define the upper density, you take the supremum of these quantities and then you take the limit when the side lengths of the cube tend to infinity. And to define the lower density, you take the infimum of these quantities over all the x in RB and then you take the limit. You can prove that the limit exists in both cases. And these are called the lower density and the upper value density. And using these densities, you can get the following result. This is due to lambda. It tells you that if you have a re-spaces of exponentials for your space delta of omega, then it is a re-spaces then both the upper and lower density coincides. It has uniform density and we denoted only by b without the plus or minus. And these two densities have to be equal to the measure of your set omega. In the case of Fourier frames, in the case of frames of exponentials, the density of the frequencies, the lower density had to be greater or equal than the measure of omega. In this way, you can formalize this envision that in the frames, you have to have more elements in the basis. But a natural question that appeared here is, okay, in a frame, I need more elements. You can sense this lower bound here is the kind of critical value for your density. But can I find frames of exponentials, of frequencies, have density as close as I want to this critical value measure of omega? This is very important in applications because remember that I told you that when you go to the sampling zone, when you go to the Bayesian space, the frequencies that appear here are then formed in the evaluations that you have seen. And so the more points that you have, the more frequencies that you have in your frame, the more points that you have to sample from your scene. So the more complicated that will be your device to sample the signal. So you want frames with the lower density, with very small density, the smallest that you can have. So this is a natural question here. It was proved by Shorby, Shorby Marzo, that this is actually true, that you can find, given a bounded word set of omega, given epsilon, rate of zero, then you can find a set of frequencies whose density is the measure of omega. This is lower than, lower or equal than the measure of omega plus epsilon. The exponential with these frequencies constitutes a frame for L2 of omega. So you can find arbitrary frames with frequencies arbitrary close to this critical value measure of omega. The proof of the result is very nice. Let me say a few words about the proof. If you have your original set omega, this yellow one, you can approximate the set omega in measure by a set omega epsilon that consists on a union of equal silence cubes. And so this set omega multitized RD with some lattice. And therefore, by the previous terms of three spaces, you know that for the L2 of this omega epsilon, you have a risk base of exponential. And the frequencies, by the Landau theorem, have to have density equal to the measure of omega epsilon. Now, if you take this risk basis of exponential from omega epsilon to omega, what you get is a frame for L2 of omega. And the density of the frequencies is equal to the measure of omega epsilon can be taken as close as you want to the measure of omega. With Elona, Agora and Carlos, and really we could try to mimic this idea and motion settings that are locally compact. And using more of this idea of shorty, we could prove that awesome that setting we can find in four-year frames close to the to the critical density. In other locally compact gradient groups, you don't have cubes, you don't have dilation so you have some technical problems to exactly the same, but you can try to follow this scheme and using this idea you can get a similar result in locally compact gradient groups too. Well, now, if you see these schemes, if you have a set, if you want to construct a frame for that set, you can use this scheme of shorty, but what happened if you set change with the time? Suppose that your set evolves with the time. Well, it presents the measure, but it becomes a longer and longer, so the diameter of your set starts to increase. So this scheme here will give you for the different set, different frames of exponentials. So a question that could appear in this setting is, okay, if I have this domain that evolves with the time in such a way it presents a measure about the diameter increases, increases. Can I find a frame exponential for all these domains in the final time? Can I find a kind of universal frame? Maybe a little bit more precise. It's possible to find a set of frequencies such that the exponential with these frequencies constitute the frame for L2 of omega, for every omega such that the measure of omega is lower than the lower density of lambda. This is a requirement that appeared in Landau's theorem, perhaps with an extra equality in the case of Landau, but it's possible to obtain this. Well, if you don't add any structure to your set or any other condition, the answer is no. It was provided by Oleski and Lianoski. But if you add some topological condition to omega, like a compartments, it turns out that the answer is yes. This is very simple reason. Oleski and Lianoski proved that there exists a set of lambda in Rd density is uniform and equal to one such that the exponential with frequencies in lambda constitute a normal basis for L2 of omega, for every compact set of omega in Rd such that the measure of omega is smaller than one. One here is the density of one. Moreover, they prove that this lambda can be taken as a small perturbation of this regular lattice set D. This is a very impressive result because you have many, many compact sets with very different structures and it's telling you that the same frame can be used to all of them. Another surprise that appeared in this universal problem is that the intuition says that this set lambda cannot have a structure. You have to per tool set D in some random way, cannot have some structure because it has to work for many different compact sets. So one, we expect to find a set without a very rigid structure, but it turns out that you can find universal sets of frequencies. So the frequencies such as this and the corresponding potentials are frames for every compact set corresponding to a measure with a lot of structure. And this is, here is where appeared the connection with quasi-critics. Let me recall the definition of model sets. I want to construct a model set in Rd. Okay, so I expand Rd in n other dimensions. I add n extra dimensions to Rd. And so in the Cartesian product of Rd times Rn, I consider a lattice gamma. And I say that gamma is in generic position if, when I consider the two canonical projections, Rd, sorry, this should be Rn. So both projections restricted to the lattice are injected. And both projections, the image of the lattice to both projections have to be dense in Rd and Rn. A simple way to construct a lattice in generic position is to make this post that you have R and you add one extra dimension with the Cartesian product with R. And then in R2, you consider the lattice step two, and then you rotate it in a rational portion of two parts. Then the lattice that you get is a lattice in generic position. And satisfy these four conditions, two injectivities and two density conditions. Well, when you have a lattice in generic position, you can define what's called the model set. So take a lattice in generic position in Rd times Rn. In Rn, take a set omega that's usually called a window, and the corresponding model set to this lattice and in this window is this set here. And this is the projections onto Rd of all the elements in the lattice whose second projection is in the window. Here is an example. Here, the window is this interval here. And so in this light blue part, you have all the elements of the lattice that belongs, whose second projection belongs to this window. So these are the elements of the lattice that you project to the first space Rd. And the projection is what's called this model set lambda omega. If you have worked with the almost periodic functions, you will see that this construction has this flavor of almost periodic functions. It's more precise in this, but I don't want to do these details. Let me give you an example. Suppose that you take a new rational number, alpha, and you take this matrix, one plus alpha minus one, alpha minus one. And so you consider this lattice. You can prove that this lattice is in genetic position, too.  And then you consider the window as a window, the interval zero. So when the construction say, okay, you have to consider those elements in the lattice, the second projection lies in the window. So the second projection has to be in the zero one. So these part here have to be in zero one. Since n and m are integers, it's obliged that m had to be the integer part of n alpha. And so if I denote in this way the fractional part of an alpha, what you have is that this model set is n plus the integer, the fractional part of n alpha. And so this is the corresponding model set to this, this window and this lattice. And here is a picture of the dots, the elements also of lambda omega. And I wanted to show you this picture because in some sense, you have here another type of stylus. And instead of using only one type, you have two different types of styles, the light blue one and the orange one. You combine them in an aperiodic way. And this model sets are also related with aperiodic diamonds. It's related with one known pair of aperiodic diamonds. Here the model sets are the points, the vertices of these polymers that we are here. And they are also used to describe the diffraction patterns of some new material in the 80s. And that's the reason why this model set sometimes are also called classic crystals. What is the connection with our problem? Well, the connection comes from this result of Mateo Meier. First of all, consider a lattice generic position in rd times r. Consider as a window interval. And then you can prove that the density of the corresponding model set is uniform and is given by the measure of the window divided by the measure of the lattice. The measure of the lattice is the measure of any fundamental domain of the lattice. For instance, this one, if the lattice has its own. This calls in general not only for this particular model set, always the density is given by the measure of the window divided by the measure of the momentum. If you take this particular model sets where you have increases on one dimension and you take this window interval, then what Mateo Meier proved is very impressive result. For every compact set, such that the measure of the compact set is lower than the density of this model set, then the freak, the exponential with frequencies in this model set is a frame for L12K. So this set has this universal property that I mentioned in the theorem of Holley's theorem, and this set is a very structured set. It's not so grand. And so this is very impressive for me. Now to finish, one more, and we say that with Mateo and Lola Vora and Carlos Caballi, we try to extend this kind of result to local compact Aurelian groups. The theory of something can be done in these groups and some of these groups in the condition that the dual group to be compactly generated. These special groups are basically due to the structure of theorems isomorphic to this type of group. This group of isomorphic to D copies of R times L copies of the torus, and here a discrete and countable Aurelian group. And so many, many results of something can be extended to this setting, in particular land also can be extended to this setting. But the difficulties in these groups comes from this group, from B. Because this part is very nice, it's a big group, a lot of analysis here, but this group here is really the source of all the difficulties. This group is just a countable Aurelian group. So it can be, for instance, the rational numbers with a discrete of polish. Or it can be this rougher group, the B and roots of the unity with a prime number with a discrete of polish. So it can be very weird. So I wanted to study this quasi-crystals and the model sets and the applications to sampling theory. The first problem that appears is, okay, is there any lattice in generic position in this group? At that time, we asked, in a year, if he knew I was asking about this problem, he said that it would be nice to know if every group G has this property, if you extend it with R and then inside this Cartesian product you have a lattice in generic position. It turns out that it's not always possible. With Lona, Carlos and Bazarab, we proved that in order to have here a lattice in generic position, then what you need is an if and only if condition that the part here, this discrete part in your group, cannot contain a copy of set P to the power B plus M plus 1 or any prime P. Okay, let me very briefly say a few words about why this very strange condition here. The reason why this condition appears is that if you want a lattice in a generic position, the possession has to satisfy the four conditions, the two conditions of dance range. So these conditions implies that there is some more system between part of your group D and part of some R to the power R to D plus M. And the problem comes from the elements of D with finite order because in R, you don't have an element of finite order. So the problems that arise from different prime numbers can be split in different places using the fact that the dimension of R as Q vector space is infinite. So using this fact, you can split the problems of different piece. But when you work with the same P, then you have to pay a higher price to absorb each element of finite order of order P. You have to pay the price of one dimension of one copy of R. And so here you have exactly D plus M copies of R. So until D plus M, you don't have any problem, but you cannot absorb any more, any other copy of any other element of order P. So that's the reason why this condition here. Well, I think that's finished here. Thank you. Thank you very much for your attention. Thank you very much. If there are any questions, you can either raise your hand in the participants list or let me know in the chat that you have a question and I will take you in order. Let's see. Any questions. I want to ask you the first question then. In this, you mentioned some some sorts of results, especially the one that you had the omega epsilon covering the omega. And then you get, you get a frame from a race basis from the mega epsilon. My question is a philosophical one in these sorts of results. Is there a way that you extract when you send epsilon to zero some sort of pre compactness of your objects, you know to to to have the hope at the end to get the race basis at the end or maybe get the better information on the on the frame that you get for omega or these are generally very hard to quantify. Sorry, sorry. Could be repeat the question you know that you have a race basis for a little bit bigger. And from that you want to extract some information about the most more philosophical if there are any if people if there is any sort of result or in the line that you want to when you when you send epsilon to zero you can extract some you have some compactness argument for me. It is usually very hard because when when you approximate your you will say, when epsilon tends to zero, the problem is that if you have a race basis, this operator that appeared in the definition of a race basis was invertible. And the problem when when you let the science epsilon tends to zero is that the enormous the inverse of this of this operator tends to infinity.  I mean, for example, if you know a priority that omega has a race basis, and if you apply this process with omega epsilon, do you recover the risk basis of your original omega from this omega epsilon risk basis approximations, would you recover. Well, in that case, I don't know, I don't know. I was thinking, for instance, in the case of the of the ball that the natural we don't know if there is any risk basis. And so you can, you can try to approximate the ball by this, this sense of some union of the squares. But in that case, you have this this problem. But if you already know that you have a race basis in your space, I don't know if you can recover. I don't know. I don't know what happened in that case. For example, you mentioned a result where you have a risk basis for a polytope which is centered centered symmetrically with each face centered symmetric. Right. So this one would be an example that you already know you have a risk basis and you could approximate by cubes in this color on type the composition. And then probably, I don't know, maybe interesting to see if some subsequence along when epsilon goes to zero would converge back to the basis that you had. I mean, this is a philosophical question. Yes. I like very much. Okay, thank you. Thank you. So are there any other questions? Can I ask something? Yes, please. Yeah, absolutely. So I would just like to ask you about this, whether all these problems about sets, which are more singular know somehow like this kind of sets. These things work there somehow. What's the situation now. Weird results now I guess. There are some results in the relations. Well, it's still an open program to know if, for instance, in the term I can talk to the space to associated to the measure. If in that set in that space, you have a frame of exponentials. As far as I know, for some kind of counter set, you know that there exists even also normal basis of exponentials, but they are very special counter sets where there is a medical relationship between the number of cuts that you do, how many intervals that you keep in each step. For instance, in the counter set, you divide the interval in three parts and take two parts. And so two and three are co-primes. And so this made the situation very difficult. Another example is you divide the interval zero one in four pieces and you take the first on the third. You repeat this construction. This is the and in this counter set of the L2 of the measure associated to this counter procedure, you can find also normal basis of exponentials. But in the program here, one of the problems, many problems, but one of the problems is that if you have a basis of if you have an optimal basis there is no good notion of density to deal with them. You don't have a base like you don't have a land outside. You have a kind of density that's associated to a half of measure, but it doesn't work very, very well. So, I don't know. At least as far as I know, there is no many things in this direction. You can replace, for instance, frames by other types of four-year expansions like effective sequences. Yes, that effective sequences exist in any L2 of a singular measure. They have good properties. But about frames, I don't know. The limit procedures are difficult because you are changing in some sense the dimensions. If you want, for instance, to go from in the ternary context, you want to, okay, I constructed this basis for the L2 of the you divide the 0, 1 in three intervals and you construct the risk basis for the L2 of the first and third interval. Then you construct a risk basis. You go one step in the construction of the count of set and you construct the risk basis for the first, second, third and fourth interval. And you go in this way. In each step, what you have are risk basis for the first half of the dimension one. But in the limit, you go to a set that has another half of the dimension. And so this creates a lot of problems. So it's quite difficult to do this. Okay, thank you. Thank you very much. Thank you very much. Any last question? Okay, so if not, I would like to thank the speaker again. Thank you very much. And next week, we will have Ivaldo Nunes as our speaker. So looking forward to that. And yes, if you would like to answer this poll before you leave, we would be grateful for that. Thank you very much everyone for coming.


Please consider this think tech commentary. A year ago, they told us we were coming out of COVID, so it was time to plan a reopening. Vaccines or not, the reopening has been limited at best, and now the variants are spreading like wildfire, with more variants to come, as a consequence of more infections. Not only in the United States, but everywhere in the world. Just look at the numbers of those sick and dying. What's worse is that there are literally millions of Americans who have a deranged hesitancy about taking the vaccine. There's no good reason for that, none, and no rational person can understand. But these anti-vax people still refuse to be vaccinated. By not taking the vaccine, they put their lives and the lives of their family and friends in jeopardy. By not taking the vaccine they put their own community and country into another surge, another COVID crisis. Just look at the numbers. The correlation, the causation, is clear. People in states, usually red states, that don't take the vaccine are dying and infecting others at an extraordinary rate.  This phenomenon, this irrationality, is actually hard to believe. When will they ever learn? Perhaps at the moment when they are dying. It's absurd. Out of Kafka, as for the rest of us, we are living in a state of amazement, and increasingly a state of disappointment. We had hoped to get back to a semblance of normalcy, to go back to restaurants, shops, entertainment, work, career, school, travel and long-delayed tourism, for ourselves and our families. But in a world where this crisis is coming back with such a vengeance, that's not so easy, and in fact optimism is increasingly dangerous. Where we stayed indoors so much over the past 18 months, now we have to continue to stay indoors again, perhaps even longer. Where we could see a light at the end of the tunnel, now the tunnel is longer and darker and we can't see, or trust, that light anymore. We begin to imagine a long-term struggle and decline, a seventh circle of community loss and personal pain. When will this be over? We really don't know, so we lose confidence in our government, our leaders, our healthcare system, our neighbors and friends and the country in general. A loss of confidence by millions of people has a way of undermining our daily lives individually and collectively, and perpetuating the crisis and economic downturn that we suffered last year. Now, instead, we have to look forward to more of that downturn, and very likely an emerging inflation at the same time. The country, and the economy, is grossly out of sorts and likely to get worse. No one knows where this goes, and festers, when you have millions of people all being disappointed at the same time. Depression, anger, further fragmentation and divisiveness and all the unpleasantness that must surely follow. Disappointment is disappointing enough on an individual level, but national or global disappointment is far worse and has unpredictable consequences. Clearly we are at a national if not an international tipping point. The bottom line is that it's time to think of others, to think of community, of humanity, and the future. It's time to think clearly, recognize fact from fiction, distinguish truth from disinformation, disregard those who would lie to us, and think about helping people again. It's time to take the vaccine. The consequences of not doing so are likely to be dire, and tragic for us and for everyone on the planet. Thank you for listening to this Think Tech commentary.


The mining industry has a significant cultural and economic impact on the United States. Many of their modern symbols, signages and logos pay homage to the California Gold Rush of 1848-1855 or just more colloquially known as Gold Rush. There are nearly 13,000 mining operations in the United States according to the Mine Safety and Health Administration. In 2019, U.S. mining directly and indirectly generated more than 1.3 million full-time and part-time jobs. Let's take a look at five mines that you can work at in the United States. So the first mine is the Lucky Friday mine, which is an underground silver, lead and sink mine located in Molen, Idaho. This is one of the oldest mines in America as it was discovered in 1880, but production did not start until 1942. Currently, it is owned and operated by Hecklaw Mining and is expected to operate for another 20-30 years thanks to the completion of the number four shaft project at the mine which will provide access to the highest grade ore in the mine's 75-year history. Next up is Bingham Canyon. Bingham Canyon is an open pit copper porphyry mine southwest of Salt Lake City, Utah. You may recognize it as the largest open pit mine in the world or for the famous landslide in 2013, which the effects can be seen by NASA satellite imagery. The locals commonly refer to it as the Kennecott copper mine as it is the Kennecott Utah copper cooperation which operates the mine. The mine was discovered in 1846, opened in 1906 and since then it is the leading copper-producing mine in the world, having produced over 19 million tons. The third mine is the Red Dog mine. The Red Dog mine is an open pit sink and lead mine located in northwestern Alaska and is owned and operated by a Canadian mining company called Tech Resources. It is the world's largest producer of sink and accounts for 10% of the world's sink production and 55% of the mineral value produced in Alaska in 2008. The concentrate from Red Dog is shipped to Tech's metallurgical facilities in Trail, British Columbia, Canada, before being exported to customers in Asia and Europe. Next is the Morenzi mine. The Morenzi mine is owned by Freeport McMorrin and is located 16 kilometers south of Silver City, Arizona. Morenzi pioneered the large-scale hydrometallurgical treatment of mine-copper ore by dump leaching, solvent extraction, and electrical winning during 1985 in parallel with conventional treatment. Waste is mainly used to backfill the Morenzi pit while the ore which averaged a .29 copper percent grade in 2004 is hauled to in-pit primary crushers. Last but not least is the Cripple Creek and Victor Gold mine. The Cripple Creek and Victor Gold mine is the largest gold mine in Colorado history and is the only remaining significant producer of gold in the state. Owned by Neumann Corporation, the mining method used here is an open pit operation and the gold is recovered from the ore by heap leaching. The mine also has a visitor center where you can learn more about the mine and even take a mine tour. So that's it for this video on the five mines you can work at in the United States. The United States is a resource rich country with good paying wages and plenty of job opportunities making it an attractive destination for those who want to work in the mining industry in a new country. If you want more of these videos let me know in the comments and we'll see you next time.


A portion of today's video is sponsored by The Motley Fool. As a small business owner, I've always been very cautious with my money just to ensure that I'm not spending it in the wrong places. My brother is somebody who has pushed me to invest in stocks, but I never really knew what that entailed until I found out about The Motley Fool. The Motley Fool Stock Advisor is a subscription stock picking service where members gain access to a library of expert stock recommendations which are carefully aimed at multiplying a member's returns. If you're curious about how it works, once a month their legendary stock advisors recommend two new stocks to members sent directly to their inbox. In the average return of all stock picks since inception of service is over 500% as of January 2021, which is beating the market by about five times. Simply visit fool.com slash loanfox to receive a special offer and I want to thank The Motley Fool for sponsoring this portion of the video, but let's get into these DIY projects. Hi everyone! Welcome back to Lone Fox! I hope you guys are having an amazing day. I have yet another DIY video for you. I have just been cranking out the DIYs lately. The first couple weeks of January actually was very in a creative rut. I don't know if you guys have ever gone through that, but I was just not thinking straight. Like I swear to you, I also have anxiety, so my anxiety was kind of bad at the beginning of the month, which sucked, but it's kind of getting a little bit better. I've been noticing lately, which has just been helping me overall creatively, and I have four aesthetic DIY projects for you guys and I am very, very excited to share them with you. And when I talk about aesthetic, I more so mean trendy things I've seen kind of everywhere, projects that have been recreated a lot across popular social media platforms, just things that I've kind of seen a lot that I want to try to put my own take on and recreate here for you guys on the channel. And the great thing about these projects is I feel like they are perfect gift ideas. These are really great to give somebody for their birthday. I know Christmas is far out of the way now, but maybe even a Valentine's Day gift, I just thought that when I was creating each of these projects, I was like, this would be such a cute thing to gift somebody. So keep that in mind as well. But I guess we could just go ahead and jump on into our aesthetic DIY projects. I kept every project here very, very minimal, not too many steps, not too many supplies, but let's go ahead and jump on it. For this first project, I actually wanted to take a current trend, which is tiled furniture and turn it into something a little bit smaller and achievable by really anybody. So we're going to be creating some really cute tiled coasters. So as you can see here, I got this 12 by 12 sheet of tile from the tile shop. It is two by two tiles. This only cost me $1.90 for the entire sheet. And I'm able to create nine coasters from this. And basically what you could see me doing now is cutting the mesh backing to create two by two square sections. And each of these little two by two squares or I guess four total squares is going to make up one coaster. And then here I'm simply going in to remove any excess mesh on those edges there. So just cutting as close as I can to the actual tile to remove any of that extra mesh. And then I kind of realized that I needed something to put this tile on top of because I didn't want it to crack and the actual mesh it's on is not sturdy at all. So I ended up hot gluing this down to some cork material. And I wish I had a thinner cork, honestly, but this cork was a bit thicker. I got this at the craft store a long time ago and I've had it in my stash. So I just went ahead. I glued the tiles down and then I used a pair of scissors to cut around the edges. But I highly do suggest actually using an exacto knife. It just creates a little bit of a cleaner edge because cork can be a little bit messy. But I did go ahead and back all of my coasters with this cork material just to kind of create a sturdier base and also will allow us to then grout them to where they have like a base for the grout to sit on. Now, if you want to go in and color your tiles, all you have to do is simply do about two to three coats of your favorite acrylic paint over the top of all of your tiles. The first coat is going to be super, super streaky, but do not worry. The second coat and third coat will totally clean that up and create a nice clean, consistent color across the tile. Now, as you can see here, I also went along the edges as well with the paint. But I ended up actually using some cocking on the entire edges just to create a more clean, kind of cohesive look. So totally disregard the edges if you don't want to do those because we're going to fix those a little bit later. Now, grabbing a hard coat sealant, I'm going to lock in that color by applying two generous coats of the hard coat sealant. This is just really honestly going to create like a nice, hard, scratch resistant coating on top of your paint. So you don't have to worry about that. And then once it is dry, we're going to go in with our grout here. So I got this premixed grout at a flooring store close by. I've had this for a while and I actually used it on a larger scale project that's coming out in a video in probably February. But all you have to do is simply take the grout on your finger, especially for a small project like this and just rub the grout in the crack. And then the actual grout that sits on top is extremely wet and almost pliable. So you can just simply wipe it off with a paper towel once you're done grouting your section. And that is really all you have to do for these colored coasters. And of course, the white ones were a little bit more simple because all I had to do for these ones was just go in with our grout mixture, kind of press it into the cracks, smooth away any extra and then wipe off any excess with a damp paper towel to ensure we have a nice clean finish. And to finish off these coasters, all I did was go in with a little bit of caulking just around the edges to kind of bond the cork to the tile and make it look a little bit more cohesive, and that finishes off this project. This next project, we are going to be using some of these large scale shells. And this is just so cute. I'm going to share with you guys a couple different styles of how to create your very own kind of shell trinket dishes that have a hand painted pattern on them. I just love this idea. And I saw a lot of people actually using these shells as candles and they were pouring wax on the inside of them. And I thought it was just so cute. But I was like, what if we go ahead and kind of turn these into miniature little trinket dishes and I thought these would be such a nice gift idea. And I also do feel like shells in general as a shape. It's kind of a trend right now. So I really wanted to go ahead and use these shells. And then on this first one here, I used some black acrylic paint and almost created this like paint stroke, polka dotted pattern.  I think it's really cute, though. So I went ahead and I applied all the little black speckles onto this one. For style number two, we're going to go ahead and with a little bit more color here. So I'm starting off with some yellow paint, some pink paint, some green, some orange, and I'm adding just a little bit of white to each of them to almost create more of a pastel tone. And I'm going off of a simple pattern that I actually have on a wrapping paper in my craft room, and it just has some organic shapes on it. And I think that creating organic shapes with these like really pretty color ways is a no brainer way to create a very, very cute pattern. So I just went in and created the most random shapes. There is no rhyme or reason to this at all, but definitely go in and apply a second coat. Once that first coat is fully dry, that way you have a double coat coverage. In style number three, the first part of this was filmed completely out of focus, but I feel like you guys totally understand what I did here. I just went in the crevices of the entire shell and added this peach toned paint, which I think kind of created a more pastel, a little bit lighter and airier feel, which I love as well. And of course, the last step is just going to go in with a hard coat finish, something that has like a little bit of a gloss to it or a sheen, just to protect your surface, makes it wipeable.  I think it's such a cute idea and it's super simple and affordable to recreate as well. Something that I've been seeing everywhere on social media right now are these really, really cool molded kind of structured candles. So I ended up picking up my very own mold online. And what I'm doing here is actually inserting a needle into the bottom of the mold, which is actually going to be the top of the candle. And the reason that I'm doing this is so that my wick can go from top to bottom and I'm able to trim it on the bottom and still have some of it exposed on the top. I do feel like these candles are more so just little decorative elements. You wouldn't actually burn something like this. But now that my wick is inserted, we can go ahead and melt down our wax. So I am using some soy wax here and I heated up some water. I put the wax inside of a little measuring cup and I'm melting this down double boiler method. I believe that's what it's called, where the boiling water is essentially going to be melting down at the candle wax. Just make sure to stir occasionally and mix up that wax so that it gets fully melted down and dissolved. I also picked up these concentrated colors for candles, and these are essentially like dyes that you can pour into your wax to color them. So I opted to go ahead and use this brown one because I thought a brown candle would look quite nice. So I mixed in a couple droplets of that brown color and I ended up adding just a little bit more so I got a darker shade. So that's what I'm doing here, just mixing that in. It melts down and it colors the wax, which I love. And here I thought I had a genius idea of opening the bottom of a paper cup. That way, my entire mold can sit on top and the wick can go underneath the cup. And the other part of the wick can stick out the top. So I did exactly that and I poured in my wax. But something I didn't realize was that this mold is very flimsy and the cup had a ridge on it, so it ended up actually indenting it. Now, when I'm holding the wick in place, I'm using one of these little wooden clips, but you're going to want to go ahead and let that wax fully cool. And you're going to see exactly what I meant by those kind of impressions on the top. So when you unmold this, it is the most satisfying thing ever. It just peels right out, which I love. And then there is that wick. That's why I was telling you we needed the top and bottom. But as you can see on all of the corner balls, they're broken. And then I dropped it, which is just lovely. So I wanted to go ahead and do another one. So I remolded up my mold with the little wick in there. And I just left this sit on a flat surface as opposed to on top of the cup. And it actually worked so much better. So after about an hour, I was able to unmold it. And this one is a little bit lighter than the last one, which I love because now I kind of have two different brown toned candles. And honestly, they just turned out so, so cute. I think these look so expensive and they only cost a couple of dollars to create. For our last and final project, I am doing a DIY version of a neon sign. So this is really fun. I'm starting off with a canvas. It is 12 by 16, I believe.  We're going to be building the sign right on top of this canvas. So I opted for this peachy pink color. And I suggest giving around two to three coats for a full coverage. I then went in with a pencil and I just traced out what I wanted my neon sign to say. And I ended up just doing like a very loopy cursive. Hello, because I thought that would be kind of cute with the pink background. And I'm using this red LED strip, which is actually going to be the light up portion of our neon sign. So this is a very kind of thin cable that has a light running through it, essentially, and it's very pliable and bendable. And you can also adhere it to any surface.  And this adheres so quickly and simply, you guys, like right as the actual LED strip touches that superglue, it's bonded, which is really nice. You do have a couple of seconds to kind of peel it up or move it around. However, it just works really, really great with the superglue. So I went all the way around, went around my cursive word, and I did this. And if you don't have a cursive word, you can actually kind of sew the LED strip in and out of your canvas, if that makes sense, by poking holes. And that's how you can get some more block like lettering. And next, what I'm going to be doing is threading back down my strip underneath the canvas. That way it goes in the backside. This is how you're going to hide any of that extra LED strip. And you could still run it back up through the front in case you want to add more of it, which is exactly what I'm going to be doing. But first, just kind of fixing that letter O and making sure everything is in place and, guys, any excess glue. I'll show you how to clean that up in just a little bit. But I'm popping my LED strip back out for the back through the front once again, because I wanted to add a line underneath the word hello, just to kind of add emphasis to it and almost make it pop a little bit more. So I thought the line just, you know, grounded the entire neon sign and just the word in general.  But if you want to clean up any like overflow of superglue, I just went back in with my same exact paint color and painted right over the top. It hides any of that kind of shiny finish that the superglue gives. And then on the backside, all you're going to want to do is just tape off any extra LED strip. You're going to want to plug it into the battery pack, tape the battery back down on the backside and you're good to go. You could turn it on and off and hang it on your wall. And that guys finishes off today's video. I hope that you enjoyed this one. I definitely do feel like these projects are more of a younger demographic. If you look at them, however, just keep an open mind because they are great gift ideas. They're perfect for Valentine's Day and they're also perfect. Just DIY projects in general that you can totally change up, change the colors of things, change the placement, the finish to kind of match your own interior design style. And maybe you found something in here that you would like to create for your own space, which is my goal in every single video. You are not already make sure to subscribe to my channel. I post brand new home decor and DIY content every single week here on Lone Fox and click that bell icon next to the subscribe button. That way you get notified when I upload brand new videos. And I'm going to go ahead and let you go for the day. I hope you have a great creative rest of your week and share with me your projects and DMs over on Instagram. I would love to see that. I'll catch you in the next one. Bye guys.


Mahatma Gandhi on his 75th death anniversary. Mahatma Gandhi on his 75th death anniversary. Mahatma Gandhi on his 75th death anniversary. Mahatma Gandhi on his 75th death anniversary. Mahatma Gandhi on his 75th death anniversary. Mahatma Gandhi on his 75th death anniversary. Mahatma Gandhi on his 75th death anniversary. Mahatma Gandhi on his 75th death anniversary. Mahatma Gandhi on his 75th death anniversary. Mahatma Gandhi on his 75th death anniversary.


Kathy and Elliot Lewis on stage. Kathy Lewis, Elliot Lewis. Two of the most distinguished names in radio. Appearing each week in their own theater. Starring in a repertory of transcribed stories of their own and your choosing. Radio's foremost players in Radio's foremost plays. Ladies and gentlemen, Elliot Lewis. Good evening. May I present my wife, Kathy. Good evening. Tonight, a nostalgic play. Come back with us to the days before World War I. The days before we girls used to bob our hair. The days of the Model T Ford. When we fellas wore a starched collars. And all it cost for two ice cream sodas was ten cents. Come back with us to Greenwich Village in New York City. And meet all the young hopefuls. The artists and writers and musicians who live there. Our scene, a converted boarding house deep in the village. Our people, two boys and two girls. The boys live upstairs, the girls downstairs. Our story, take my hand, my love. By Samuel B. Harrison. Where you been? I got a haircut. I was growing down over my collar. You got home early. Yeah. Hang if I know how you can work on this old rattle trap. I manage. Any mail for me? No, just that. For me. From Harper's. Hmm, rejection? Yep. But I don't get nearly rejection slips. I get the nicest letters with my returned manuscripts. Read that. Hey, Fred. Hmm? I'll have to owe you my share of the rent till I get paid for the last batch of pot boilers. No, that's all right, but golly. Well, I'll have to check in a few days. Hmm. Walt, you know you can't earn a living writing blurb for a greeting card. Yeah, I know, but it keeps me alive until I get... Till what? Did you get a book of poems published? How many of the greatest poets of all time died popular? I never thought to be rich. All right. But you can't have a decent suit of clothes, a couple of dollars in your pocket, be able a Saturday night to take a girl to Lushaw's for dinner, maybe a Broadway show. You. When's the last time you had better than a 30 cent dinner at Marlborough Lotties? What's wrong with being a newspaper man? You don't hear a word I'm saying, do you? No, sure, but I don't see anything wrong taking a girl to Marlborough Lotties, even if it isn't very soignier and dinner costs only 30 cents. Oh, what's the use? Just look at Norma. Oh, it's a fetching hat, Beth, but isn't it a little old for you? I really think it is. Well, I have to make an impression, don't I? Did you impress anyone? I certainly did, and you'll never guess who. Whom, darling? David Bellasco. No. Yes. We talked for half hour. Oh, he's the grandest, kindliest man. He asked my age. I told him 19 going on 20. How did he believe me? Did he ask you to sing for him? No, he doesn't do musicals. Well, then why? I wouldn't mind doing a dramatic role, like Maude Adams and Peter Pan. Where'd you get that old-fashioned tin bathtub? Borrowed it from the janitor's. Whatever for? I'm going to paint you in it. You mean like September more? Yes, but I'm going to call it Saturday night. Why? Because farms and small towns, a Saturday night bath and a tin tub on the kitchen floor and in the stove But gracious. Will people see me like that in the museum or art gallery? I hope. The folks back home ever see you. I'll disguise you, honey. I'll give you a snuff nose and auburn hair. When do we start? As soon as I finish this, I'm working on. There. There's as good a bow as I can tie. Ah, thanks. Fred. What? Have you got a speaking acquaintance with the girl, artist I think lives below us? Uh-huh. That cute little trick staying with her cat's your eye? That 17-year-old infant, certainly not. Uh-huh. No, no, no. Seriously. No, I mean Norma Saunders, according to the name of the letter box. Don't you know each other? No. I said something once, you know, just joshing, and you were really interested? I'd like to know her. She's a prude vinegar in her veins. Because she isn't easy to know. Ah, she's an old maid. She's under 30. She has dignity. A lovely, sensitive face. Skinny as a rail. Looks consumptive. No. She has an ethereal quality. Wow. Love comes to Walt Kidridge. Invite her out to dine in continental atmosphere of ma birdlotties, or write her a sonnet.  Hot here. Uh-huh. You miss the breezes of Vermont? Oh, no. I love Greenwich Village. Everything about it. That basement place you took me to. Guido's Den. Mmm, just candlelight. Of course it's a little smelly, but so romantic. Artists and writers sitting around, even on the floor talking, arguing. I think it's wonderful. Yes. It is wonderful. There's also a heartbreak here. I want to paint. You hear that? She wants to play. That man upstairs pecking away at his typewriter. Thousands who come few succeed. Most ultimately give up the struggle and vanish into limbo. New, hopefuls come to take their place. If I don't get anything, can I come stay with you again next summer and try?  I'll be 18, man. Norma, don't push away your plate. Oh, I've had all I want. My goodness. I don't know how you keep up your strength. Now, you finish that. I have no appetite. I can't. You must force yourself to eat. Go on. The room's cooled off. Close the door. Snooper. It's like the tendrils of a grapevine. Your lips are rose petals kissed by the morning dew. Your form isn't from a woodland. There's your line. Oh. Other nights, look at the mud you've tromped over my kitchen floor. Is that the place for your gauntlets? Chop some wood, delch, you'll know yesterday's hand bone for dinner. Go on. Oh, good I did. That means good luck, a surprise. Yes. Aren't you ever going to marry? Worried and being left on the shelf? No. What I mean is, well, as long as I've been here, a little man caller or gone out with one and you don't seem to... What in heaven's name? That, hanging eggs, something in it. Wow. It's not come down on a string from upstairs. I'll find out. Hello? Let's finish the dishes. Ignore it. Maybe it's something like a drowned kitten. Nonsense. Kids, I remember one. There are no children upstairs. Let's take it in and see what it is. Encourage the idiotic... Certainly not. Close the window. No, I'm not going to stifle here just because of that. Give me those scissors. Now we'll see. It's a bottle. Maybe there's a note in it. Distress, like island cast away. Silly. Wow. Italian port wine. There's a note on it. You snatcher. Read it. Dear Miss Saunders, please accept this. A glass of port before a meal stimulates the appetite. Humbly, Walt Kittridge. The eavesdropper heard when we were having dinner. He couldn't help it. The door was open. Which one of the two upstairs is this? The taller one. I know the other one's name Slade. He's nice. He smiled at me. Of course, of course. It's undoubtedly to be introduced to you. Oh, no. Oh, yes. Just tie that bottle back on the string. Wait, I'll write a note to go with it. Put him in his place. No, Aunt Norma. You don't hold yourself in very high esteem if you jump to the conclusion that the man is interested in me. We couldn't possibly be you. You put yourself on the shelf and I know you're only 29. Going on 28. He finds himself humbly. He does a sweet thing like this. You want to put him in his place. If you're the person I think you are, you'll keep that wine. Write a nice note of thanks and tie it on that string. Yes. Sit down, Joe. I'll be shaved in a jiffy. Where did your border walk over the weekend? White planes. A cousin lives there. You know you've got a fine talent. But head in the clouds. Now, last week he sent in 30 pieces of dog earl rhymes for greeting cards. They took nine. A dollar and a half a piece. Hey. Hey, Fred. What? This by the typewriter titled Take My Hand, My Love. Read it. A sonnet. Love ballad. That's lyric. It's much too impassioned to be written without some particular female in mind. Is there someone? Oh, yes, yes, my jingo. She lives right below us. Come here. I'll show you something. Yeah, window sill. See that nail? He ties a string on there and sends down notes, hauls up the answers. Can you beat that? A grown man. Oh, but why if she lives downstairs? Oh, the lady is unapproachable. Look, obviously he wrote that Take My Hand, My Love for her. But he'll never have the gumption to send it down. But if he finds it missing... Type of copy he'll never know till the walls of Jericho come tumbling down. That's enough of this for now. Finished already? Oh, no. I thought you had to deliver those drawings by Monday morning. They will be. Now I'm going to set you in that tub and paint. Tonight? It's Saturday night, isn't it? Night of the bath. This canvas isn't a masterpiece. It'll at least be authentic. Little of the left. Now a kettle on the stove. I'll do that. You start getting undressed. When you stop giggling, I'll believe you're really 17 going on Twitter. Yeah, I must get a kerosene lamp. Gives a wonderful yellowish glow. Actually, water in the tub? It's not necessary this first sitting. Oh, pull down the shade. And Norma, there's a note. I see it. Let's see what I'm doing. There. There's your Mr. Kittridge. I don't know what he said in that note, but I think you're wrong. Please, best, stop playing Cupid. I don't want to hear anything more about him. I'm sorry. You are listening to Kathy and Elliott Lewis on stage. Tonight's play, Take My Hand, My Love. If you knew of some neighbors in financial trouble or perhaps in emotional straits, you'd tell them if you knew which social agencies in your community could see them through their difficulties, of course you would. In the same spirit, Americans contribute annually to community chests and the United Fund to see to it those agencies continue to be there when needed. Give the United Way to make your community all it needs to be for those in need. A wog. I say wog. Yeah. Seen the lady downstairs since you're back? No, once or twice. Why? Just curious. Any change in her attitude? Yeah. Strange, I used to get a smile and a word of greeting. Cold shoulder? Excuse me, a feeling I'm not there. Well, if you want my advice... I don't. Oh, Miss Saunders, I'd like to talk to you. Yes? I hoped we were going to be friends. Is there any reason why you're again looking the other way and passing me by as if I were a pariah? I'm obliged to make explanations. Well, of course, but... let's be adult. Yes, let's be adult and stop the childish game. Notes on strings. All right, but you still haven't told me why. Because you're impudent. I impudent? Well, how? This past Saturday night... Go on, please. I don't want to discuss it. I made a mistake when I accepted that bottle of wine. I was out of town Saturday night. I must really ask you not... Did you say you were out of town last Saturday? That's right. I spent the weekend in White Plains. Oh... Oh... Will you forgive me? Sure, sure.  I'd be delighted. Good, let's go. Say, uh... Won't you tell me what it was Saturday? You thought I'd... Let's talk about something else. Okay. Is that, um... that chap you live with a very close friend? Well, not really close. He has his interests. And I have mine. You write. Well, I try mostly poetry. Published? Or occasional short pieces in Scribner's Argosy, The Post. So little I've got to write blurbs for greeting cards. Perhaps I should have stayed put teaching school. Where? Dexter, Ohio. Where are you from? Bennington, Vermont. Long here in the village? Seven years. You? Eight. Oh, here we are. Is this all right? Yes, thank you. Just coffee, please. Are you an illustrator? I like to think of myself as a painter. My cupboard would be there if I didn't have this commercial work to do. Is that it? Yes. May I see? Yeah, animals. I try to do amusingly for a toy manufacturer. Yeah. Mm-hmm. These are amusing. Fine. Truly fine. Must be fun doing them. Yes. It pays fairly well. Say, look. How would you like to do a children's book with me? Thing publishers like Best I'm Told is to get the illustrations with the story complete. Oh.  Got any written? Yes. Oh, thank you. But yes, but I've got a corking new idea about a little chick who didn't want to be born afraid to face life lying snug and safe in his shell. Uh-huh. He envisions danger of the fox and the hawk a bleak winter fight for a possession of a worm he finds. Do you see it? Yes, yes. Of course, he finally emerges and finds there's beauty in the world. Yes, very much. When? I'll do it tomorrow. And I'll put it in your letterbox. No, bring it in or send it down the usual way.  Yeah, all right. Shame to let these tickets go to waste. Oh, give him the wall. He goes for high-brown music. He can take her downstairs. Any repercussions on another thing? Say, Walt. Yeah? Our music critic gave me a pair of tickets for the opera for tomorrow night. Traviata, Borsmi, Stiff. You want the tickets? Well, I've got a business dinner date tomorrow night, but, well, yeah, maybe I can make it. Good, good. Well, here you are. Thanks. Thanks, Fred. Well, so long, fellas. So long. Hey, look, Joe, I think that love ballad we sent down backfired, but she wouldn't refuse the opera. I'm going to ask her. Let's do it for him. Oh, what if she accepts? Oh, I'll tell him and he'll kiss me for it. If she refuses, nothing lost. Yeah, but he said... I know what he said. Sit down there and type as I dictate. Darling, you finished reading the little chick who didn't want to be born? Oh, it's cute. The cutest story. Are you going to do it? I've already started. Really? Yes. There's a sketch of page one. Here. Do you like the worried look on motherhand's face? Yes. Both your names will be on the book. If it's bought and published. And something else. I'm going to be opera tomorrow night with Mr. Kittridge. When did he ask you? While you were reading in the bedroom. There's an invitation swinging out there. I simply wrote yes with my initials and tied it back on. He hasn't pulled it up yet. He will. Of course I'll have to get some things. All right, Blue Taffeta. I saw just the thing. Organdy, collard and pout. Fred, how much longer? I mean the note. It's 6.30. Take another look. See if she took it in yet. No. Not still there. Same sheet of yellow paper. Well, all it up and skip the whole thing. Hey, Fred. The wind blew the note. String caught in the next house fire escape. Boy, I gave it a yank. The note tore off. There it goes over the roof. It's gone. Well, that's that. Nice perfume. He isn't coming to take you. Because, darling, he has a business dinner appointment up town. It's blowing to rain. It's been blowing to rain for two days now. There. How do I look? Lovely. Just lovely. Turn around. Perfectly divine. Taking your umbrella? This is blue and I... I'm off. Enjoy yourself. I simply wanted to ask... What are you doing here? I came in out of the rain. Why aren't you at the Manhattan Opera House where you asked Aunt Norma to meet you? I? You. I didn't ask. You sent down a note on the string yesterday asking her to go to the opera with you tonight.  I'll go there now. What's the least you can do? Why didn't you take a cab home? Could you get a cab? Oh, poor dear. You're chilled. Off with that. There. Steaming hot bath and getting to bed and hot tea. And I... Don't. Don't cry. He isn't worth it. I don't care about the dress or anything. It's just a humiliation. I know. Corned drug store to get some fine iron. I'll be right back because she's back upstairs in bed. She's never been very strong. And if she gets pneumonia, you're gonna be... Let's go. Let's go of my arm. Listen to me. I didn't send any note. Do you hear? You tell her that. I didn't send it. If you didn't, who did? I thought you went to the opera. Did you send down a note in my name asking her to meet me there? You did, didn't you? Well, yes. Why? Why? I wanted to help you.   She never got the note. The wind blew it off.  I'm not carrying you anyway. This is my place and you can get out. I will now. I'll be back tomorrow to give you the $12.  If my wire's that permission, I'm in the show. Rehearsals start next Monday. Five weeks, then we open in Philadelphia. Oh, that's wonderful, honey. You think you can take care of yourself? Of course. Norma, you're not going to start working the second day out of bed. Oh, this is no exertion. You're not going on with that book after what he did? Yes. Mr. Kittridge didn't do it. Then who did? The fellow he lived with up there. Stupid, practical joker. I know. Don't ask me how. I know. I just know. No, Mr. Marlowe, you're the first publisher to see it. Fine. A happy union of talents, your illustrations and Mr. Kittridge's story. Why didn't he come with you? He moved. You quarreled? No, we didn't quarrel. But I'll find him and we'll make it up. Good. When you do, here's the contract. Here's us the option on the next book you do together. Three copies. You both sign. Bring them in and we'll give you the checks. Advance on royalties. Goodness, I thought I'd have to go without saying goodbye. Oh, it's a great day, darling. I found out where to reach Mr. Kittridge. Really? Maverick Lottie, a woman who runs that restaurant on McDougal Street, told me. He's working for the New York Herald. Now the Grand Central Station. I'll send him a telegram from there. Aunt Norma, you're in love with him. There. You keep one, one for me and one for the publisher. Not so fast, please. First, some of Ma Bertolotti's good port wine to sharpen the appetite and celebrate the occasion. There. Will you toast it? To the little chick who didn't want to be born. We're afraid to face life. But learned it could be joyous. And many more books we'll do together. Since we're collaborators.  Please do. Walt? You were laid up 10 days. August 14th was your first day up. How did you know? I'm a newspaper man. I... I hope you won't think I'm taking advantage of a situation, but... Well, some time ago this summer, as I sat at my typewriter, dreaming as a poor scrivener does, I wrote a little sonnet to you. That's very nice. What did you call it? Take my hand, my love. Will you, Norma? Yes, Walt.


It all started with my grandma. I was just a kid when the doctor sent my grandma home in a wheelchair to die. Diagnosed with in-stage heart disease, she'd already had so many bypass operations that the surgeons essentially ran out of plumbing. Confined in a wheelchair, crushing chest pain, her doctors told her there was nothing more they could do. Her life was over at age 65, but then she heard about Nathan Pritikin, one of our early lifestyle medicine pioneers, and what happened next was documented in Pritikin's biography. It talks about Francis Greger, my grandma. It was a live-in program where everyone was placed on a plant-based diet and started on a graded exercise regimen. They wheeled her in, and she walked out. In a few weeks, she was walking 10 miles a day, went on to live another 31 years until age 96 to continue to enjoy her six grandchildren, including me. Her miraculous recovery not only inspired one of those grandkids to pursue career in medicine, but granted her enough healthy years to see him graduate from medical school. So it's really all thanks to her. During medical training, I was shocked to find out that this whole body of evidence on reversing chronic disease with lifestyle changes, opening up arteries without drugs, without surgery, was being largely ignored by mainstream medicine. Wait a second, if effectively the cure to our number one killer could get lost down some rabbit hole and ignored, what else might there be buried in the medical literature that could help my patients? I made it my life's mission to find out. That's what led me to start nutritionfacts.org, and that's what led me to write the book How Not To Die. Surfaces show people wildly overestimate the power of pills and procedures to keep them healthy. Patients believe cholesterol lowering statin drugs are about 20 times more effective than they actually are in preventing heart attacks. No wonder most people continue to rely on drugs to save them. But our leading killers aren't caused by drug deficiencies. The dirty little secret is that most people surveyed said they would not be willing to take many of these drugs if they knew how little benefit these products actually offered. Whereas cleaning up our diets is not only safer and cheaper, but can be more effective in preventing arresting and reversing some of our leading causes of death because you're treating the actual cause of the disease. Each year the CDC compiles the 15 leading causes of death and so I have a chapter on each, How Not To Die From Heart Disease, How Not To Die From Lung Disease, How Not To Die From Brain Diseases, Digestive Cancers, Infections Diabetes, High Blood Pressure Liver Diseases, Blood Cancers, Kidney Disease, Breast Cancer, Suicidal Depression, Prostate Cancer, Parkinson's disease, and How Not To Die From so-called iatrogenic causes, which is essentially death by doctor. That's the first half of the book and the good news is that we have tremendous power over our health, destiny, and longevity. The vast majority of premature death and disability is preventable with a plant-based diet and other healthy lifestyle behaviors. I didn't want to just write a reference book, though. Yes, there's thousands of citations to peer-reviewed scientific papers, but I also wanted to be a practical guide on translating this mountain of evidence into day-to-day decisions. And so that's what became the second half of the book. First I started out with a traffic light system to classify everything into red light, yellow light, and green light foods, though there are exceptions. I talk about the best available balance of evidence suggests the healthiest diet is one that minimizes the intake of meat, eggs, dairy, and processed junk and maximizes the intake of fruits, vegetables, beans, which are split peas, chickpeas, lentils, legumes in general, whole grains, nuts and seeds, mushrooms, herbs, and spices, basically real food that grows out of the ground. These are our healthiest choices. Some foods, though, have particular medicinal qualities, and so I then center my recommendations around a daily-dozen checklist of all the things that try to fit into my daily routine. So for example, I recommend a quarter teaspoon of the spice turmeric a day, a tablespoon of ground flaxseeds, berries every day, greens every day. I talk about the healthiest beverages, the healthiest sweeteners, how much exercise to get. The whole daily-dozen list with recommended serving sizes is available as free apps for both Android and iPhone. Just search for Dr. Gregor's daily dozen. There's only one diet that's ever been proven to reverse heart disease in the majority of patients, this plant-based diet. If that's all a plant-based diet could do, reverse the number one killer of men and women, shouldn't that be the default diet until proven otherwise? And the fact that can also be effective in treating, arresting, and reversing other leading killers like high blood pressure and type 2 diabetes would seem to make the case for plant-based eating simply overwhelming most deaths. The United States are preventable and related to nutrition. According to the most rigorous analysis of risk factors ever published, the global burden of disease study funded by the Bill Melinda Gates Foundation, the number one cause of death in the United States is our diet. The number one cause of disability is our diet, which has now bumped tobacco smoking to number two. Smoking now only kills about a half million Americans every year, whereas our diet kills hundreds of thousands more. So obviously nutrition is the number one thing taught in medical school, right? The number one thing your doctor talks to you about at every visit, right? How could there be such a disconnect between the science and the practice of medicine? Doctors have a severe nutrition deficiency in education. Most doctors just never taught about the impact health and nutrition can have in the course of illness, and so they graduate without this powerful tool in their medical toolbox. There are also institutional barriers such as time constraints and lack of reimbursement. In general, doctors aren't paid for counseling people on how to take better care of themselves. Of course, the drug companies also play a role in influencing medical education and practice. Ask your doctor when's the last time they were taken out to dinner by Big Broccoli. Probably been a while. It's like smoking in the 50s. We already had decades of science linking smoking with lung cancer, but it was largely ignored because smoking was normal. Most doctors smoked. The average per capita cigarette consumption was 4,000 cigarettes a year, meaning the average American smoked a half pack a day. The American Medical Association was reassuring everyone that smoking in moderation was just fine. There was the same disconnect between the science and public policy. It took more than 25 years and 7,000 studies before the first Surgeon General report against smoking came out in the 60s. You think maybe after the first 6,000 studies, it could have given people a little heads up or something? Powerful industry. And today's meat, sugar, dairy, salt, egg and processed food industries are using the same tobacco industry tactics to try to twist the science and confuse the public until the system changes. We have to take personal responsibility for our own health, for our family's health. We can't wait until society catches up to the science again because it's a matter of life and death.


We're gonna give you a couple of players to keep your eyes on because we're keeping tabs on them. That's right, we're doing all the work. I know you're thinking, oh, who gets to go on vacation for four weeks? We do, but it's not because we didn't finish our homework and turn it in before we left. Now, these are guys who are kind of flying under the radar or rookies who I do think might be, I'm not gonna say might be. These are guys that are gonna be a factor this year. I'm just gonna go ahead and say it. And we're going to start with the guy that sort of spurred this idea. A friend of the show, Buccaneers running back Rashad White, okay? Leonard Fournette out of the way. The lane is cleared for him to be the guy in Tampa and we hope he is. Leonard Fournette was supposed to be on our show and he no showed. He overslapped Leonard Fournette. So we're all for Rashad White, okay? Last year, no, we love Leonard.  He and I talked about that versatility. We talked about him being a stud in that bat field and you can imagine how everyone down there is excited to see him be the lead dog. In early April, Jason Light said that Rashad's going to be a stud and it wasn't just lip service. And you know how I know that? Because the Bucs didn't draft a single running back in the draft a few weeks later, despite plenty of opportunities to pull that trigger. And while there are some doubters out there, fine. I'm sure there are. Rashad's drawing inspo from that Bucs fan base and seems more than ready for the opportunity. Listen to what he said to me a couple weeks ago.   It makes, you know, it's heart, it's heartwarming for sure. He's one of my favorite guys we've talked to, really. I'm really rooting for him. And I think when we're looking at fantasy a couple of weeks ago, he's really under the radar still. Yeah, he's outside the top 24 running back. So he's like running back three in fantasy. I also love, by the way. In a PPR league, what are we doing? Yeah. Like you gotta grab. And we think Baker, Trask aren't gonna be trying to get him the ball as soon as possible. How's their O-line? Yeah, the O-line, they've made some adjustments there. Didn't have a great year last year, but yeah, they've kind of overhauled it. I like the bucks colored roses, by the way. You're really going all in on Rashad. I would call them up in Adams colors and I think they stole it from us, but that's okay. Okay, let's move on here. Lions running back.  Talk to me. Jamir Gibbs. Tell me. Love him. Why? Versatility. All right, so kind of same thing as Rashad. Right, we love that. Okay, so let's do it here. Here's the rookie, everybody's so excited about. Explosive as a runner, ability to line up at receiver, but I mean, let's not act like Gibbs doesn't possess some of the very same traits here. I mean, look at this, Bijan Robinson sort of comparison vibes, right? This isn't the only guy with highlights out of OTAs. Detroit signed David Montgomery. Sure, we love David Montgomery. I wish him the best. So there will be a little competition there for touches in the backfield, but we're gonna see Gibbs lining up all over the field just like he did at Bama. He's jumping into the Detroit offense and already, you know, they're set. They were top five last year as far as the offensive side of things go. So we should see big things out of him this year. And I'd like to understand the feel as long as possible and give that defense a little rest. Like let's not roll the defense out there every three seconds, Dan. That's a good point. Yeah, definitely help that defense out, keeping him off the field. But if it wasn't for Bijan, we'd be talking about Gibbs so much more because I think he's that special of a talent. Yeah. So I love that you're giving him some love here because I think I'm with you. I think it's gonna be a big year. They used him at Receiver a ton at Alabama. We talked about Bijan so much, all that footage out of OTAs of them lining him up in the slot. We're gonna see the same thing out of Gibbs and I think it's gonna be- Too many guys in the backfield though for me to get excited about him? I don't think so, because I think he could be out there with Montgomery. You could see him line up in the slot with Montgomery at tailback. We loved Montgomery. He was coming out, remember? Yeah. And I was at a division with a division rival. Okay, now let's get to a couple of receivers and this is the first time we've mentioned it. We're just putting our stamp on it as we head out the door for some vacation. It's Calvin Ridley. Out of sight, out of mind and meaningfully so, but I just wanna remind everyone of what he's capable of when he is on the field. I'd forgot about this a bit until Hamilton put it on my radar during one of his breakdowns. Look at the numbers he put up last year, a full year. That's back in 2020 was the last time we saw that. It was top five in yards, top 10 in touchdowns and that was in less than a full slate of action. So now you put Ridley in a dove Peterson, advisor offense. You know, he's the number one guy. Trevor Lawrence will take another step up and I think we're going to see a career year out of him. Another receiver I like. We like the young guys who told you we're gonna bring this here with Vikings rookie Jordan Addison Vikings fans. You're insane. First of all, we'll get to that another time, another day. And I do love Jackson Smith and Jigba. We are Seattle Seahawks show record but we gotta give Jordan some love. And yeah, you got Justin Jefferson there but situation matters so much when you look at which rookie receivers pop up. And he is because of Justin Jefferson and a whole other bunch of reasons in the best possible spot to kind of have what kind of year though like monster year? What does monster year mean for a rookie? I can see like 1100, 1200. Wow. Yeah, cause they throw it, they throw it around a lot. Will he have a better, will you put your name on him having a better year than Jackson Smith and Jigba because it's a more balanced offense where he's not, you know. Yeah, I think. In Seattle. Because I think statistically when you look at it, the Vikings were the third heaviest passing offense last year and now no Dalvin Cook, you think they're probably gonna air it out a little bit more, right? I don't know. A lot of opportunities and with all the attention that Justin Jefferson is gonna be getting, you gotta think he's gonna get some favorable match ups. I mean, they're gonna pass the ball even more I feel like with the Vikings. You get rid of Dalvin Cook, I know he caught the ball a little bit but I mean having the best receiver in the game and Justin Jefferson helps him. So hopefully it frees up Addison to make big plays out the gates and you're right about that. Okay, the last one, oh my gosh, Irv Smith. When he got injured a couple of years ago and he was on the Vikings, more heartbreak for the Vikings. He's, we were robbed, I think, of a couple years of greatness. That doesn't matter, throw that out the window. He's still only 24 years old. He missed the entire 2021 season. He had to complete with Lopes with Kyle Rudolph and TJ Hawkinson last year. But now it's a bit of a fresh start. And where else to go but Cincinnati, love it for him, okay? And he's talented, previously underutilized. We can make the argument of freshness even though he's coming off of an injury plagued four-year stretch with the Vikings. Look at what Joe Burrow has helped his two predecessors achieve, okay? He loves a tight end. CJ Uzama put up career highs in catches, yards, touchdowns in one full season with Joey B. Hayden Hurst had a career high in reception just by missing four games. Chase and Hayden's demand so much spotlight. Not only on yachts with Instagram models, but also on the field between the hash marks. You know, they get lots of shine and when that happens, the tight end position can grab some balls, can do what they want to do. And Earthsmith Jr. has shown the potential. If he can stay healthy, I think we're gonna see him become a major factor, hopefully, in this offense. So these are just some guys that I like for your fantasy, for your, I don't know, for your odds, for your whatever, for you to sound cool at the bar, picking up chicks. Hey, Janice, I think Jordan Addison's really gonna pop off this year. Wanna go on a date? Swipe left. This is, there you go. Keep tabs on these guys. And who did I miss? I'm sure you'll let me know. Okay, coming up next on the show, the US Open is in LA. PGA Tourist star, Will Zellatoris. Oh, that's a nice smile. All right, let's talk about it with Will. Rising star. Janice loves a Vikings number two this year. Janice is like, oh my gosh, hey.


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