I'm going to go baseball bat grip, everybody grab a baseball bat, let it butt swing, right? We all on the same page here, two hands to it. He goes low single, all right?  I'm going to catch this toe on the way out, okay? Catch this toe on the way out, pull the leg, all right? Catch the toe on the way out, pull the leg, I'm going to get two hands to it, passing it away with your butt. I should be sliding this finish every time, okay? I should be sliding that finish every single time, all right? I can go far ankle, then change to near ankle.  So he goes single, say his butt's off his heel a little bit here, and I jam and slide, this is easy money for me to score, right? Straight to it, all right? I don't like it far away from my body, I like it close, okay? So this guy with low singles, I'm dying to pass quick, right here with low singles, I'm dying to pass quick, getting my finish, get real quick. So we've already done a dive, we've already done a pass, easy money, two hands to it, straight, close. And we don't need to slide slide, as much as I need to here, clamp down on it. Questions? Let's get it going!


Piezoelectric materials are widely referred to as smart materials because they can transduse mechanical pressure acting on them to electrical signals and vice versa. They are extensively utilized in harvesting mechanical energy from vibrations, human motion, mechanical loads, etc. and converting them into electrical energy for low-power devices. This review aims to give a holistic overview of recent developments in piezoelectric nanostructured materials, polymers, polymer nanocomposites, and piezoelectric films for implementation in energy harvesting. This article was authored by Susmairi Das Mahapatra, Pritam Chandan Mohapatra, Adryanis Indred Arya, and others.


Welcome to Toffee TV. It is the starting eleven show Everton versus Brighton Hall of Albion. Goodison Park seems like ages since we played the game. It seems like ages since we were at Goodison Park watching the victory over Arsenal. Um but let's just hope that the players are refreshed, fit over over COVID and I know some of them are and I know some of them are still in the team basically based on the pitches that we've seen of them training because there's no real no real knowledge out there of who is fit and who isn't fit for this game. So let's get into it. In goal, Jordan Pickford for me was the player of 2021. Um it's been remarkably steady and hopefully that continues. Uh right back, Seamus Coleman, I think he's fit to come back into the side. Uh Seamusatoire in the scene of training. uh John Joe Kenney did a good job against Chelsea playing in that. It basically was playing right back wasn't he but um we're gonna need a little bit more. Uh lots of rumors going around that there could be possibly someone coming in to hope hopefully help with the right back role. Um but uh Seamus Coleman, good that he's had a few days off. You know he's been, he's had over, he's had over well he's had 3 weeks So, that's that's positive. Uh at Lefbach, Luka Dean is available and really should be playing when you look at our injury situation. Uh of course, um Tali, Michael Enker has come in and uh he's not eligible for this game of course and he's going to give him competition over the next couple of weeks. Dean obviously could leave the club in that time as well Michael Enker is bedding into the club. Luka Dean is available to play and um hopefully he starts this game because I think we really need them. Uh the centre backs Michael Keane has been pretty consistent recently. You know, he plays every week. Um listen, he's not the greatest centre back in the world that we're all aware of that but he he does play. He plays nearly every single week. He's consistent. We can count on him and I thought he was very good against Chelsea and alongside him Ben Goffy who um he was okay against Chelsea although it did feel like he was at fault for the goal in terms of his position and that's something he really needs to to work on. Into midfield uh Alan should return to the starting line up. He was on the bench against Chelsea but um never likely well he didn't come on because he because he had an injury. He was simply there to fill the space but again he's had a he's had a couple of weeks recovery so that's good and hopefully he'll start next to him. Uh Gomez, Andre Gomez for me uh going into that three in midfield with Ducore ahead of them and hopefully Ducore does play ahead of those two players. I think that's what was an issue um against Crystal Palace was the way we played. Ducore for me should always be the one bombing forward.  Uh I think Ducore is somebody who can get goals can get assists and we need that. When you're playing too deep I just don't think you get enough out of him so I'm going to go for those three in midfield. On the right hand side uh Anthony Gordon and he plays best game against Chelsea. I thought he was brilliant. I thought he really took the responsibility that night of being the uh the the player for the team he was going to drive us forward and he did winning as far as the assist for the goal so uh I think he'll probably start on the right hand side with the Mardi Gray on the left hand side. This obviously is favorite position. Um I think he had a bout of COVID so hopefully he's over that and um you know obviously back back to it's not going to get straight back into it obviously but you know we've all got the memory of the last time we played at Goddison Park and his last minute winner so uh he has been the real spark for us this season and we you know hope that continues uh and up front Domino Carvaluin he's he should have retained on boxing there on the bench um is it ideal starting them now but I don't think we've got anything else obviously no with Charleston um yes Ellison started against Chelsea but if there's an opportunity to play Domino Carvaluin for an hour and he's fit and we've seen him in these pictures of him in training so he must be he must be ready to go then we're going to have to start him and listen the whole team I imagine is going to be rusty for this game they've they've had over two weeks off and we've seen from other teams it's not easy to jump back into it right and only played three days ago and uh you know they they they'll be straight at us from the start but Domino Carvaluin it'll be a real lift for the team a real lift for the fans going into the second half of the season that we that he's back fit and he really is the player that we've been missing and hopefully you know he can get the goals and guide us to three points against uh very good Brighton side who uh for only lost four times this season so it's going to be a difficult game and again the squad isn't at its fullest there's no meaner of course um to play centre back um you know squad players there's doubts about some of the squad players Tom Davis should be available on the bench we get some players back we get some new silence hopefully it'll be a better second half of the season so that's the team I'm going for let me know you thought in the comments you agree do you disagree with that team let me know make sure to give this video a like subscribe if you haven't already if you want more great videos join us over on page and the link is in the description and it's on the screen see you later


On behalf of the president of the Albert Einstein Society, Professor Otto unfortunately cannot be with us today. I would like to welcome all of you to the Einstein lectures 2023, especially I would like to welcome our young audience. So at various levels from students, but even maybe school kids, fans of math, Olympians, and whatsoever, we welcome those who are attending today, but also those who will watch the podcast later, because they cannot come today. This is the 14th edition of the Einstein lectures, which are organized jointly by the Albert Einstein Society and the University of Bern, and which are in turn dedicated to physics, mathematics, and philosophy. My name is Christiane Treta. I'm professor of mathematics here at the University of Bern, and it is my great pleasure today to introduce the fifth Einstein lecturer in mathematics, Professor Marina Wiersowska from the Ecole Polytechnique Fédérale de Lausanne. With just 39 years, since two days, if I found out correctly, Marina is the youngest Einstein lecturer we've ever had. And also in other respects, her life and career stand out. She was born in Kiev, and there her high talent in mathematics was discovered and supported early in school. She competed in several domestic and international math olympiats, wrote her first research paper when she was just 21. Then she obtained a master's degree in Germany from Kaiserslautern in 2007. And then did a PhD back in Kiev at the National Academy of Sciences of Ukraine in 2010. And she got another one from the University of Bern in 2013. Only three years later, well-prepared, of course, from the PhD, in March 2016, she had completed, after some longer research work before I assume, the two research papers that got her to the Olympus of Math six years later. Within a week in March, she posted her solutions of the sphere packing problem in dimensions 8 and 24 on Archive, which is the free mathematical preprint server available to everyone. And with these solutions, she stunned the entire mathematical community because they were so ingenious and elegant and, of course, because also so many other much more experienced researchers had tried to solve these problems for her. Then a whole other launch of prizes followed, among them the Salem Prize in 2016, the Clay Research Award in 2017, a New Horizon in Math Prize in 2018, the Fermat Prize in 2019, a Prize of the European Math Society EMS in 2020, and then in 2020 to the Fields Medal, the highest award in math for scientists under the age of 14. This Fields, the Fields Medal, there are four each four years, they are awarded by the International Mathematical Union, usually at the International Congress of Mathematicians. As a treasurer of the Swiss Math Society, I had the great pleasure to be a member of the Swiss delegation to the IMU, and Marina was another delegate by chance. But of course, nobody knew what would then happen. And so because I was there, this took place in Helsinki, but not within an ICM because the ICM was scheduled for July 2022 for St. Petersburg. But then Finland was so generous to jump in and hold the ceremony and at least the IMU assembly. So this was how I came to be part of this award ceremony. And it is a memorable moment in my career to be there when Marina was awarded the prize. And of course, I took the chance after the award ceremony to invite her. I'm not sure if at the time she was aware it's three lectures and not just one. But anyway, we are very, very happy that you are here, that you accepted the invitation, and we are very much looking forward to your talk on the sphere packing problem. The stage is yours. Christina, thank you very much for inviting me here. Also, thank you for the introduction. And so today I will speak about the sphere packing problem. And so let me start with explaining in very general words what this means. So imagine that you have a very, very big box, maybe, and an infinite supply of balls. And then the question would be, how many balls can you fit into the box? And then if the box is much, much bigger than the size of each of these equal balls, then this becomes a question really of not of the exact number, but rather the density. So then we would know that the number of balls inside of the box, it will be roughly speaking the volume of this box divided or multiplied by some number, the density constant. And so what we are interested in in determining, because there are many different shapes and sizes of boxes that we can consider, but we would like really to focus only on this one number, the density. And so again, to clarify what the rules of our games are, that all balls, they're hard balls, they all have the same size, and they're allowed to touch each other and not allowed to intersect. And so here in this picture, you see this intuitive three-dimensional picture that we have, and now an interesting component that I would like to introduce is that also the notion of dimension. So what we can do, we can consider this question not only in our favorite familiar three-dimensional space, but in different dimensions. And so before we get to the dimension D, let me start it easy with dimensions 1, 2, and 3. As those are the dimensions we all know and understand. And so what is the dimension in a very simple word? This would be the number of degrees of freedom to move in the space. And so if we imagine that we live on a line, then we have only one degree of freedom. We can move only forwards or backwards. And so on a line, what is a ball for? What is a one-dimensional ball? One-dimensional ball is an interval. And the line can be covered almost perfectly with intervals. So we can think of these intervals that do not overlap but only touch each other, and they will just cover all our line. And so the density of this one-dimensional packing would be one, which is somehow the maximal possible density we could even hope for. And in dimension 2, we have two degrees of freedom. We can move, so to say here, northeast and southwest. And what is a two-dimensional ball? A two-dimensional ball, it's a disk. So here imagine that our two-dimensional space, it would be a surface of this table. And as a disk here you can imagine, for example, one-frank coins. And that we are piling this one-frank coin so that we would like to put as many coins as we can on the table and to cover as much of the surface of this table as we can. And so after a little bit of experimenting, you can come to at least a hypothesis that probably this is the best way to pack disks on the surface, so into this regular triangular grid. Sometimes it's also called honeycomb lattice.  And we will have this exact, if you like mathematics and like mathematical constants, we will have this exact formula for this approximate number. And so in dimension three, what can we do in dimension three? Probably you've all been to a grocery store and you have seen there are oranges piled in pyramids. And now if you think of these pyramids, if they will be prolonged infinitely to the whole store, then what we will get would be this perfect sphere packing. And its density is about 74%, or this exact constant, pi divided by square root of 18. And so historically, the sphere packing in dimension three, it was known as Kepler's conjecture. So it was a long time ago when people became interested in this geometric question. And the question of packing sphere seems very natural. And I guess there are some ancient Indian manuscripts with text and pictures that show us that even in ancient India, people thought about this question. I'm sure that this question came over again and again through the course of history, even though it's not my expertise demand, but in our modern European history, this question is known as Kepler's conjecture. And so there is an interesting long story behind it. So from what we know that at the 16th century, British aristocrat had a great idea of invading Spain, or at least somehow fighting seriously with Spain. And so for this, if he wants to fight with another country, he would need a lot of cannons and he wanted to pack them into a ship as efficiently as possible. So he asked his scientist, Thomas Ariot, to tell him what is the most efficient way to pack cannonballs inside of a ship. And Thomas Ariot, he did not take this task slightly. He actually thought about it. And so we know about all this because the letters were preserved and he actually developed this theory of a closed sphere packing. He gave descriptions of what the best configurations would be and he found not one, but actually two different ones. So he described all this in his letter. But also this made him think about so packing cannonballs inside of the ships. This made him think about condensed matter and this ancient idea that maybe matter around us consists of atoms. And maybe atoms are packed inside of matter in something like cannons inside of a cannonball. So he was thinking about all this and he wrote a letter to his colleague, Johann Kepler. And Kepler also thought that this is such a beautiful idea. And the Christmas was coming and Kepler was supposed to give a present to his friends. And so to one of his friends, he gave the following present. So it was an essay of six corners snowflake. And so here's the text he has written to his friend. I think it's a nice coincidence that we are also meeting around Christmas time. So maybe if you are short in money and cannot buy a great present for your friends, maybe you can just write an essay that would be quoted 400 years later. So here you see somehow this is the dedication and also how the letter starts. So Kepler says, I'm not a historian. I'm not a specialist in the Renaissance time or habits people had that time. But it seems like an apology. So he could not somehow give him a present which would be of monetary value. However, he wrote an essay. And I think this is what he refers as nothing. So he tells like, I just give you nothing. And nothing is 14 pages of text. So I will just give you some quotes. So it was the main topic. So like now joking apart, let's get back to business. And the business is the following. So Kepler was walking on this outside and they have seen snow and snowflakes. And he made an observation that all snowflakes, they all are look like starlets and they have six corners. So he could not find a snowflake that has five or seven, but almost all of them had six corners. And he was thinking, why is it like this? And again, as a person from Renaissance age, he would think that probably this has nothing to do with the properties of water or properties of vapor because vapor does not have anything. There is nothing hexagonal about vapor. It's shapeless. Even though we know this, he knew that time that snow comes from vapor. And so he realized that probably there should be an agent, some agent which really likes the beauty of hexagon that forms snowflakes like this. And so I leave it for you to decide how close does this bring us to atomistic theory or to the somehow formational principles of matter. And I think many people who study the work of Kepler, they interpret this in the following way, but maybe I will just leave it to you. And so let's better come to mathematics. So here the question is what is so special about hexagon? And so one thing that Kepler thinks is special about hexagon are these geometric optimization problems, the pecking problems. So here he starts thinking about pecking problems. And so this is how one of the additions of his essay looked like. And so what would be important for us are this, oh, sorry, is this picture, this diagram.  And so now out of these layers, he will produce the dense pecking of balls. And so this is what he writes. Again, it's interesting that in the 17th century of course people did not have all the mathematical machinery we have now. There was no algebraic notation, there was no established language to speak about geometry, so it's all really more rather poetry than mathematics. And so at first what he suggests to do is so let's look at this diagram A and let's build our sphere pecking from these flat layers. And the first thing to do is just to put the one square layer on the top of another so that each ball touches exactly one ball which is beneath and widen ball which is below. And so this is what Kepler describes. Here, so if we proceed like this, of course he sees if he understands that we will not get the densest pecking because we can apply somehow more pressure to all the structure and then the all layers could slide and become denser together. And so what he does, he suggests here another arrangement so that to put each ball of the higher configuration not at the top of another ball but rather in a deep hole. And so here he writes that then somehow then the pecking will be the tightest possible so that if no other arrangement could be pellets could be stuffed into the same container. So he just claims that this is then, if we do it like this, this would be the densest pecking. And this is actually the end. So Kepler is just very confident that this is the densest pecking. And again by the standards of science at his time, this was an absolutely reasonable thing to say. So at that time people did not think so much about mathematical proofs and the rigor of mathematical proofs.  And so what, so here are some pictures. So here's the pecking which Kepler suggested to build is to take here are these balls in layers and each layer is arranged as this square grid. And now for, as you see for each ball of the next layer we put it in this biggest hole between smallest layers. Between sort of previous smallest hole, we put it into these holes that are created in the layer below. And so here, what we see here is actually another construction. And this construction looks in the following way. So now you see that if you want to arrange our disks on a plane, then the strangler grid, it will be better than the square one. It will be denser. And so maybe if we start building our pecking not from rectangular layers, but from triangular layers, maybe we can achieve something better. And so here one problem comes that in the picture here, if we put these balls of a next layer to these big holes of a previous layer, then somehow miraculously there will be just enough space for if we just put this ball as low as we can, there will be just enough space for the next ball in the hole here. And for the strangler grid, it is not the case. Here we cannot put one more ball into each hole so that then there will be not enough place for them to touch the previous layer. So actually what we have to do, we have to choose every second hole. And so this way we see that we have these two constructions. So this construction number one, which I already described for you. And so here each layer is less dense but the layers are very, very close together. And the construction here when each layer is very dense but the layers stay far away from each other. And what I do sometimes at this part of the lecture, I ask audience to vote, which you think is denser. Is this denser or is this denser? So who thinks that this one is more dense? Okay. And who would vote for this one? And who thinks that they actually equally dense? Okay, so people who think that they are equally dense, they are right. And there is actually more to this story because these two packings, they're not only equally dense, they are geometrically identical. And it seems somehow difficult to see it from these pictures even if we try imagining the packings for ourselves. So let's convince you that they are did the same and these both constructions, they will lead to the same sphere packing. So called this configuration is called the face, the central cubic, sorry. Cubics centered face letters, sorry. Faced central cubic letters, FCC letters, yes. Faces central cubic, yes. So what the name comes from? So we have this, we take a usual cubic letters and how this one works. So we think of our spaces filled with cubes in a natural way. We just put cubes one onto another and then put them in first like in a perfect square grid and then extend also in all the dimensions. And so in the vertices of our cubes, we will locate our balls and we will make the radio of our balls, okay, so the radio of our balls, we'll think about them a bit later, we'll fix them later, but this configuration of the of the vertices of cubes, this is what will give us the cubic letters. And so we would like to put a red ball and of some radios, which we will choose later and the center in here. And, but also we would like to make our configuration a bit more interesting. So we'll look at all the faces of all our cubes and we will, for each face, we will choose its center and we will put a blue ball with a center which coincides with the center of the face. And so this way we will, now we will also think of this configuration as being continued in all somehow six directions. So here at up, down to the east to the south, to the north to the south to the east and to the west. And so now we would have, how can we see that this lattice packing is actually coincides with the two packings we have constructed before. So I suggest doing this by coloring the balls in a different way. And then we will see that it will coincide with two configurations that I proposed before. So one way, so this, let's take our cubes and draw the main diagonal in our cube. So take two points which are at the biggest possible distance. So we choose one of these main diagonals in the cube and we will draw the planes which are orthogonal to this main diagonal and we will color our balls, again, in green, in red and blue as I've done it in my previous picture. And so then you can see that this picture here actually is the same as this picture here only rotated in the space. And now it's the same, but configuration is the same, it's only a different coloring. And now what we can do, we can do yet another coloring. So now instead of looking at this main diagonal, we again look at our cube and now we will look at the planes which are parallel to one of the faces of our cube. And so this way you will see that the squares will emerge here. Here it is this rotated square letters. And so we'll see that this picture, it actually coincides with this construction here. And so these two configurations, they're actually the same only rotated. So maybe you can try to imagine how you rotate it. And then you can also see that it's not that easy to do. So we, first of all, we should give credit to Kepler and to Thomas Ariot for describing these configurations to finding them so long time ago. And maybe it also should not be that angry at mathematicians that it took them 400 years to show that this configuration is optimal. And so, okay, so here you can see it's another beautiful sentence from the essay by Kepler. So here he writes this somehow that triangular pattern is impossible without the, the triangle is impossible without the square and vice versa. So of course, at that time, probably what the Kepler would imply by this sentence is something like what I showed you before. So we start with triangles and then we inevitably build squares in other direction. And if we start with squares, we will, by doing this kind of packing, we will still, we will build the triangles, but in yet another direction. So, and so now a few more demonstrations to show you that the geometric equation, optimization geometric equations can be very difficult and counterintuitive and not that easy for mathematicians to solve. And so another problem, which is actually an easier question, easier than this fair packing is so called the question about the kissing number. So here again, the name of the subject comes from another epoch. So you see that people used to romanticize geometry a lot. And so here the question is the following. So let's consider a ball. For example, this is a two dimensional example.  So we take a red ball and once around our red ball of radius one with blue balls of radius one. And so the rule is that each blue ball is supposed to touch the red one. So this is the condition that they always have to touch. Each of the blue balls has to touch the red one.  And so then the kissing number would be the maximum number of blue balls that can surround a red ball like this. And so in dimension six, it's not a very interesting problem. So here in dimension two, it's not a very interesting problem. And here we obviously have six blue balls touching blue discs touching the red one. And already in dimension three, the question becomes much more interesting. So this was a famous discussion between Issa Akhenyuton and David Gregory. And so Issa Akhenyuton thought that in dimension three, the kissing number is 12. And the 12, it's easy to construct. So what we do, maybe you know a platonic body that has exactly 12 vertices. So this is the icosidron. And so what we can do, we can take the icosidron and put a red ball into its center. So we choose a red ball so that its center coincides with the center of icosidron.   And so this way it's not a kind of an easy computation to see that if all blue balls touch the red one, then they will not touch each other. They will not also intersect. So they will be just comfortably sitting apart. And this is a configuration for 12. However, in this we will see that there is still, it seems like there is still plenty of space left. So maybe if we choose configuration which is not so symmetrical, maybe we can shift the balls, blue balls around and still find a place for the 13s. And so David Gregory, he thought that maybe if somebody tries really hard, they could squeeze the 14s ball in. However, he could not produce this configuration himself, but he thought that maybe if somebody spends enough of time of it, it still would be possible. And so now we know that Isaac Newton was right. After all, we know that Newton did have very good intuition for mathematics. At the same time, the rigorous proof for this geometric problem, it took quite a long time to find a proof like this. And the first in somehow maybe still incomplete proofs were found in 19th century. And then the rigorous proof was written down in the 1950s, so already the last century. And so here's another explanation of what is so hard about the sphere packing problem. And it's also maybe something which is a bit counterintuitive. It's that I've showed you these two different packings and then I demonstrated that they actually coincide and they are geometrically identical. However, in dimension three, we have uncountably many sphere packings which attain this the same maximal density. And how it works, here again, our method of building packings from layers would work. And so you remember at some point when we built our packing from this triangular layers, I have told you that we cannot put balls of a new layer into each hole between the balls of a previous layer. We always have to make a choice. So we always have to make this shift. We could put them either like this or like this. And these choices, we were making them at each level. And by making these different choices, we can produce many, many packings which would be geometrically unequivalent to each other. And so it also explains why the sphere packing problem is mathematically a hard problem. That's because we don't have only unique, great solution. We have many equally, seemingly equally great solutions. And when optimization problem has many solutions, this is a difficult problem. And so now let me tell you a few words about mathematical proof. So as I already told you, if Kepler and Thomas Heriot, they did not need any proofs. They could just somehow, it was obvious to them that this is the best they can do. They tried, they are so smart and probably they would not have missed a better configuration. And of course, as mathematicians, we want somehow more rigor in our argument. And so for many years, what was an example of mathematical rigor was the elements written by Euclid. And so here is one of the oldest texts that we have that contain some elements of a mathematical proof. And so of course, and so as mathematicians, when we want to prove something, we don't want just to write somehow a poem or a text that would look so convincing that everybody would be convinced. We have very strict rules on what constitutes mathematical proof. And maybe it's also an experience you have at some point in high school, when you do study plenometry or geometry, that inside of geometry often have this argument on how to prove something, how to prove a theorem. So as we proceed in mathematics, what we usually have to do, we have to establish some axioms, so to say ground rules, what constitutes a mathematical argument and what does not, and also if we work with certain interesting objects, like in geometry, this would be points, lines, and planes, and the balls. We have to define, we have to first somehow to give the properties which are axioms, and then for things which are not the initial objects, we have to give the proper definitions, for example, for the sphere packing problem. As a mathematician, what I would have to do, I would have to really define what is the density of a sphere packing.  So you would just have to believe me that we work with nice configurations where the density is well-defined. And when we have our axioms and definitions, we are good to prove theorems. And so of course what mathematicians wanted to do for many centuries is to turn this one sentence in Kepler's essay into a theorem. And there was a lot of effort and ideas and failed attempts. And so finally the Kepler's conjecture was resolved in 1998 by American mathematician Thomas Hales. And this also in itself, this was an interesting story. As for long time people thought that this theorem is still out of reach. And then the strategy in 1960s, more or less the strategy was developed on how to tackle the problem. However it still required a lot of computational power. And what Thomas Hales did, he somehow pushed this, really pushed on this strategy. And so he reduced the sphere packing problem in dimension three to a number of problems that can be resolved computationally, either combinatorial problems or optimization problems. And the issue of these problems could be used by computer. And so his proof, it was one of the first examples of a computer assisted proof. And then it took mathematical community quite a long time and a big discussion on whether such proof should be accepted. Is it still a proof? How much computer assistance do we allow? Another interesting thing is that referees were not somehow, scientists who were supposed to read the paper and to say that it is correct. After many years of work, they could write, okay, we think that this paper is 99% correct, but we cannot really follow all the details, especially the things which are done by programs. It's too hard for us, but 99% is correct, which is not so nice for mathematics. And then somehow what Thomas Hales did is he think it's another great achievement in history of mathematics, he would write formal proof of his theorem. And so this is another interesting thing that exists now is that, so as I told you that mathematicians, we do want to be like Euclid, we want to be very rigorous. We want to start with axioms and definitions and then very rigorously prove all our theorems. And now we are at the point where we can actually write our proof as a computer program. And then the compiler could check that each step is indeed correct because for ever is they just may get old before they read the paper to the end. At the same time, and so until now, I think Thomas Hales was one of the first people who would write a major modern mathematical result in this fashion. And now it seems that with all the technical advances, this movement is getting somehow more momentum, more attention, which is also exciting and interesting. And so what Tom Hales did, he actually he wrote a formal proof of the sphere packing problem and the whole project took him like almost 10 years. So it was a really major, major thing because one of the important steps also to prove to write the formal proof of his theorem, he had to somehow formalize all the prerequisites because as you know, mathematicians were usually reliant work of other people. And if we want to write this in a mode to have proof to be checked by a computer, we also have to explain to the computer all the previous work that has been done. And so, okay, so now we are, as I told you, the sphere packing problem in dimension one is trivial. In dimension two, it's not trivial but still somehow simple and it was resolved long time ago. What about in dimension three, it was a really big adventure for mathematicians for some very hundreds of years. But okay, one, two, three, seems like we ran out of dimensions, right? We have no more degrees of freedom as human beings. And so here what comes to rescue is the mathematical obstruction. And so here maybe let me a little bit justify to you why should we look in this higher dimensions and also explain what are these higher dimensions to me as a mathematician. And so here maybe as a metaphor, I would like to use this illustration to a book about Alice through the looking, who went through the looking glass. Actually, the book which was also written by a mathematician, maybe that's why. It feels to me so much. So we have somehow our experience from the everyday world that we are intelligent creatures and of course we use our intellect to think about it to make our lives easier. And then as we think about our everyday experience, we can also come up with obstructions. I think people are also very good to coming up with obstructions and some of them are the obstructions I am interested in are the mathematical ones. And so we can bring our concepts of freedom of real life into an abstract form and I would describe this process as going through a looking glass. So we started with something which was real then we think of its possible reflection in the world of ideas and now another step we do. So now we try as mathematicians we start living on this side and this world of mathematical obstructions it is in some sense reflection of course of all our human experience. At the same time as in the book about Alice it is slightly different. The laws that govern there are different and so if as a human I have only three degrees of freedom to move myself on this side I might have more. I just have to find a correct mathematical obstruction for that. And so with dimensions, with mathematical dimensions it's actually very easy. So as I already told you the dimension it's a number of degree of freedom and what it means it means that if I want to describe a point in space I need three real numbers to do it three coordinates. So you can think of it as a GPS coordinates for example. So here this is the point with coordinates so this is the origin and this is the point with coordinates minus one, one, one. And now if I think of a point as this somehow this row of its the list of its coordinates now nothing stops me from adding simply adding more numbers to this list. And this is how dimension of Euclidean space is defined in mathematics. And so here another thing which somehow often I think confuses people is that for example if we are physicists then our whatever we are working with is supposed to have physical meaning. So very often in physics it's useful to introduce extra dimensions like Einstein. He introduced this fourth dimension which would be time. And as mathematician I don't somehow I have more freedom I have luxury to say I don't care about what force dimension is. The force dimension it could be a time or I could add two more dimensions or I could say that maybe my dimensions there are just some numbers on excel sheet with salaries of people working in the same organization and it would be totally normal for mathematician. I don't need to think about an interpretation I just think about the logical structure of things. And it turns out that the sphere packing problem can be easily generalized to higher dimensions. And so here what we should do we have to define the D dimensional Euclidean space. And so our points there are these lists of coordinates. So if you have dimension D then we have D different coordinates. Each coordinates it's a real number. And so now I want to think about sphere packing problem we need a bit more information. So what if you want to define what a ball is so in dimension in each dimension the ball it's a set of points which are at a certain distance at the most certain distance from the center of the ball. And so we already know what the point is what we now we need to know what's a distance. And so here we introduce a distance between point in this way. So this would be the usual Euclidean distance and probably if you remember the Pythagoras theorem from school then you would recognize that this is exactly the Pythagoras. If we dimension to this would be exactly the Pythagoras theorem which also works in dimension three. But now we can actually introduce as many coordinates as we want as we have decided here. And so this would be the distance between these two points. And so maybe this is something of course Kepler would not have because he didn't at his time so the algebraic notation was not developed yet and it was not used systematically by scientists at his time but we do have this luxury now. And so now after we have defined what is a distance we can define what is a ball. So it would be a set of all points. So if you want to have a ball with centered point X and radius R it would be just the set of all points which are a distance at most R from point X. And also somehow another thing which we need for our to define the sphere packing problem is the notion of volume. It's a quite useful notion but I will not somehow hear things get maybe way way too complicated. I will not write it down but I will just tell you that there is a meaningful mathematical way how to define volume in higher dimensions. And so after we have defined this higher dimensions what we can do, we can also think of configurations and it turns out that mathematicians have this ability and in dimensions for example up to 10 they have already listed the number of configurations that they believe are optimal. So it's somehow for the rows which are in green these are the rows where we actually do know the answer in the sense that we have a mathematical proof that this configuration is optimal and those which are in black these dimensions are still open so we have candidates but we don't know whether they are optimal or not so maybe if there are young mathematicians in this room maybe they can choose one of their favorite dimensions and try to start working on that. And we also see that somehow what also this table shows that the sphere packing in higher dimensions has this very strange flavor is that each dimension it's always, it feels like a completely new problem. So in some dimensions like in dimension one in dimension two, like four, eight we have only one configuration, unique configuration which does the job which is the best sphere packing and there are many dimensions which are more like our favorite dimension three where we have this many many geometrically distinct configurations. So we can do something similar to for example choosing stacking layers in different ways and we will actually get not one answer but many typically uncountably many answers. And of course when dimension becomes more than 10 then we lose somehow our confidence sometimes we still have great configurations but we don't have this confidence to say that they are really the best. And so this is now the list of the dimensions where we exactly know what the optimal density is and so okay so as I told you the dimension one is trivial and dimension two it was not a very difficult question even though it was I think it was difficult for mathematicians is to decide which proofs are rigorous and which are not because some of the proofs they were quite old and maybe there was still some hand waving there. And probably I think in one of my lectures actually I plan to demonstrate a simple because these proofs they have been somehow discussed and simplified a lot and maybe now we are at the moment when it's actually possible to explain to somebody who knows only school geometry to explain the proof. So what I found in the literature for example there is a two page proof of this result. So my goal for this series of lectures actually to demonstrate one of these short proofs. Then there is dimension three which I already told you that dimension three is very complicated. It's a dimension where we have this uncountably many optimal solutions and there is a computer assisted proof by Thomas Hales. And then there are two more dimensions where we know the answer. So this is my own work in dimension eight. We have this unique configuration that was believed to be the best and actually was able to prove that it is the best and proof in dimension eight turns out it was much shorter. So for example the proof in dimension three it is at least a hundred pages text and also many, many lines of code while this proof it took only 20 pages and a small computer program to verify certain positivity condition. And then working together with Henry Cohen, Abinav Kumar, Stephen Miller and Daniel Rachenko we were able to solve this fair packing problem in dimension 24 because in dimension 24 we also have an exceptional object, so called rich lattice and it is the unique optimal configuration in dimension 24. And so here are my co-authors. So I decided to use this somehow. Opportunity to show them to you since we already seen Kepler and Heriot we can also see my collaborators. And so let me, I still have eight minutes left. So let me tell you a little bit about this higher dimensions and also a little, advertise the new lecture. So some of these questions I will stop more in the future lectures. So this is, so as I told you in dimension eight we have this very special object, the so called E8 lattice. And unfortunately it's very difficult to visualize things in dimension eight. So here I'm using the following trick. So for example you can see my hand it's a three dimensional object and if I rise it here in front of the projector it will create a shadow and the shadow is two dimensional. And also if I rotate my hand the shape of the shadow will be changing. And so what is here it's really, it's also, it's a three dimensional shadow of the shortest vectors of the E8 lattice. So what it means, so remember I told you about the kissing problem. And so if we have to hear this is a kissing configuration which was constructed from the E8 lattice so we choose one ball in the E8 ball backing and we also look at all the balls which touch this one. So we do not try to visualize all the E8 lattice but only one ball and all other balls that touch it. And it turns out that in dimension, so for example in dimension three we could have one disk could be touched by six others. In dimension three one ball can be touched by 12 others and in dimension eight one ball can be touched by 240 other balls. So this is somehow this strange behavior of high dimension. So this is what high dimensions are about is in high dimensions we just have much more space. And so now what is done here it's a projection from the eight dimensional space into some three dimensional subspace. So it's like a three dimensional shadow. And of course here we also have this choice of subspace where we are projecting but this one was chosen so that the result is somehow symmetrical and this model is created from a constructor which is called Zoom. Maybe if you have ever been to a math museum maybe you've seen it. So it's this kind of medium to create various three dimensional models. Okay, so here is another object, the one which lives in dimension 24 and here I could not find this three dimensional model of this. So this is a three dimensional, two dimensional shadow. And by the way if you go to, so as I told you like as with my hand if I rotate my hand the shape of the shadow will change. So for example if you go to YouTube there are many, many beautifully done videos of different projections for example of shortest vector of E8 lattice or shortest vectors of Lich lattice that are projected on for example on a plane and then the plane is changed. So it becomes a cartoon and some people are very skilled in drawing it beautifully and rendering it so it's a bit like a screen saver. So of course like on one hand this videos and pictures like this they can look very beautiful. At the same time I can tell you as a mathematician unfortunately these realizations they're not extremely useful for understanding the geometry. So you'll think the best way to understand these structures is the method of mathematical obstruction and visualization it's mostly for aesthetic purposes and for entertainment. And in dimension, in the latest actually one ball is touching almost 200,000 other balls so a bit less than that. So somehow you see that somehow the dimensions they really things in high dimension behave very strangely. And so maybe the last thing I would like to tell you also one thing I would like to advertise for the future is the history of for example discovery of Lich lattice. It's a very beautiful story and I think it is beautiful because this mathematical object which has a lot of great properties and is loved by a mathematician was actually first discovered by an engineer. So this is Marcel, oh not by it was discovered, sorry it was discovered actually it's Lich lattice it was discovered by John Lich who's a Canadian mathematician but the discovery of John Lich was only possible because of a work of Marcel Gallet and Marcel Gallet he was not really a mathematician he was an engineer and his motivation for studying such objects was very practical. He was studying the error correcting codes and so he created a code which was called the Gallet code and then his code was used by John Lich to construct the Lich lattice which we are now using for sphere packing and then somehow same object led to lot of discoveries in mathematics which I hope I will speak a little bit in one of my next lectures. And so another topic I also would like to cover is this connection between error correcting and sphere packings. So this would be the main heroes of my talk of my story and yes so this is somehow the error correcting means and how it is connected to the sphere packing so this all will come in the future lectures and so this is how the slides to tell you that error correcting is very important and many of the technical advances would not be possible without it. And then again maybe one thing I should say that I am a pure mathematician so my law for sphere packing does not come from this great real life applications it actually comes from the intellectual beauty of it and also my belief that whenever something is fundamental in science it will inevitably become useful in everyday life as well. So now I think I am ready to answer the questions. Thank you very much.


The comedy is tonight's theme as your Sears Radio Theatre host, Andy Griffith, presents Jim Jordan in The Troublemaker. Your esteemed father is not about to do an El Foldo. I called again, introducing myself as Adnan Mubarak-Sophine, Finance Minister of Kuwait.  Made me very nervous about my investment. Our program will begin after this message from your local station. I'm lost in Rome. Remember when gas stations were closed on Sundays? When long lines drove you nuts? Well, listen to this, a big gas price hike may be on the way. Gas could go to a dollar a gallon. International Motors brings you gas-saving new 78 Dotsons at low 78 close-out prices. Send them to B210 and hurry in to register for the fantastic ski giveaway this weekend. A pair of bowling skis and bindings, courtesy of Bobo's Rina Ski Shop. No purchase necessary, no obligation, but you must be present to win. At International Motors, 2620 Kitsky Lane opens seven days a week. Hey, are you one of those pretty McKenzie dancers from the Bodacious Show? Yes, I am. You enjoyed the show? Sure. It's seven times. That Ken Newton on that fiddle? And Peter Vinn on Banjo? Something else. And them jokes are gal-baker. I never laughed so hard.  Woo-wee! Thank you very much, sir. See, how about coming out to the truck and I'll show you how to double-clutch? Ha-ha, your dumbers are all alike. The Bodacious at Big Hudson's Historic Riverside Hotel. At 6.45 this morning, Charles Kirby, a man in his late 60s, skips down the outside stairs that leads to his little apartment over a garage, takes a deep breath, savoring the crisp morning air, listens to the birds singing, walks down the drive-thru. Walks down the driveway, the man with a smile on his face, at peace with the world. He takes out the sports section, clips back to the financial pages, runs down the New York Stock Exchange quotations, and then, Judas Priest, no longer at peace with the world, his face a mask of fury, Charles Kirby, stormed back into his house. When Charles Kirby retired, he counted on an upward-headed computer-related company called Xtronics to support him comfortably, if not luxuriously. His broker referred to it as a tiger. Well, the tiger had just bit it. Kaufman, George, and Moss, this is Slurry Bindle. This is Charles Kirby, Mr. Bindle. What in the world has happened to Xtronics? Well, Charlie, here's the way I see it. Hold it, hold it. What right have you to call me, Charlie? Well, I know your father. He calls me, sir. You should be.  Well, and nobody else does, either. You've got a gold star, Bindle. Your honesty is refreshing, if not nourishing. I'm going to let you in on a little secret. Yes, sir. I am a lone man. I'm retired and no longer young, who is about to take on a corporate giant and bring it to its damn knees. And may the fur fly where it will. And that's just the beginning of our story. A new adventure in radio listening. Brought to you five nights a week by Sears Robuck & Company. Sears, where America shops. Your hosts, Lorne Green. I'll bring you stories of the Old West and the New. Andy Griffith with a look at the funny side of life. Vincent Price with tales of mystery and suspense. Cicely Tyson with stories about love, hate, and related things. Richard Whitmark, I'll bring you stories of pure adventure. Of exceptional entertainment every week. Brought to you in Elliott Lewis production of The Sears Radio Theater. Our star, Jim Jordan. When I need advice, I go to my mom. Why not? It's free. Now that I'm married and moving into a new house, I want all the advice I can get. So when mom says shop Sears, I listen. You should. Sears is a great help on those big items you need for your new home. Major appliances like washes, dryers, and refrigerators. They'll deliver, install, and service. I always depend on Sears. You should too. Can't believe you owe the IRS that much. Well, when things just don't add up, you can count on a Sears desk calculator to help you add up what you don't owe. Add, subtract, multiply, and divide, then read the figures two different ways. 12-digit display or tape printout. There's a two-memory system that helps ease multi-step problems. Plus, its many extras make it a great time saver. Sears two-memory desk calculator now cut $25. Just $99.99 through March 7th at most Sears retail stores. Prices and dates may vary in Alaska and Hawaii. Light it, clean it, and love it during Sears Home Center sales. With lighting by, it's shine. Save $10 on four sandaliers with colonial, transitional, or country mood. Your choice, $29.99 each. And save over 20% on a 15-pound box of Sears Laundry detergent. It removes more soil than the nation's leading detergent. So light it, clean it, and love it during Sears Home Center sales for February 24th at most Sears retail stores. Prices and dates may vary in Alaska and Hawaii. From Sears, fashion is then both a storm. For weeks to sunshine, step out military flares. These double-breasted trench coats get down to details. Choose olive green or khaki tan day-crown polyester and cotton sizes 8 to 18. Another fashion winner is a new quilt trim-sheared shoulder coat with self-belt. In Chino-Beige polyester and cotton sizes 6 to 16. Both coats come with an aisle lining. Fashion is then both a storm for weeks to sunshine in the coat department at most larger Sears retail stores. Dear question, if you were a man facing this big financial crisis, if you were in urgent need of economic advice, the sort you just don't run into any old place, where would you go? What's that? The barber shop, you say. Exactly right. Mario, I come to you because you're my barber and you claim to know almost everything. What's going on with ex-tronics? Yeah, I dumped that because I didn't know what was going on either. Well, their plan is in your area, isn't it? Don't you ever hear anything? I'm counting on ex-tronics to support me in style. And that includes this expensive haircut. There's a bar down the block where some of the guys from ex-tronics show up.  Mario, don't explain jokes to me.  Go on. Yeah, I just heard grumbling there, so I sold. Thank you, Mario Saville. And now back to you in the studio, Charles Kirby. I returned home to change my costume, and just as I was about to leave for the West Valley to see what I could find out about ex-tronics, my darling daughter arrived in her usual fashion. Good afternoon, Daddy. Damn it, Carolyn. I told you not to knock and bust in here immediately like an English maid with the paper in the morning tea. I might be entertaining. Oh, you're always entertaining, Daddy. Well, you know what I mean. At your age, Daddy? Carolyn, I'm still strong enough to flip you over my knee and let my right hand explain about respect for your father.  I just wanted to tell you I'm pregnant. Isn't that wonderful? Oh, I suppose. But that's one of the easiest things in the world for a girl to be. Daddy. Congratulations to you, and congratulations to Paul. Thank you. Remind me that I could congratulate you again in nine months. Now lock up, daughter. I'm heading for a sink of iniquity in the West Valley. Oh, Daddy, do you have to wear that noisy jacket? Thank you for your prompt reply.  Good drinks, fish and ships, darks, unfortunately for me, a medium-high-level disgruntled extronic employee. Sure. I know what's wrong with the extronics. How'd you know I was with the company? The extronics ID, pin-deer shirt, the extronics windbreaker, and the extronics cap. Pretty observant, aren't you? No, just a shot in the dark. How about extronics? All right, but I couldn't talk about the company to a stranger. I see. Miss, bring another for my friend here. No, no, no, a double bourbon and ginger. You know, somehow you don't look so strange anymore, except for that weird jacket. I stole it off a horse at Santa Anita. What is wrong with the extronics? Actually, nothing's wrong. Except we've got three vice president, or whatever they are, who blame each other for everything that goes wrong. And who are these geniuses? Listen, Mr. Horace Blankett. Hinkle, Evans, and France are absolutely indispensable. We need them. No, this is Hinkle. A sweetheart, a wonderful guy, and what a sense of humor. You should see the insulting memos he sends off from Hinkle to Evans and France. Oh, he does have a ring to it. He does them right after he comes back from lunch.  Hold it? Why does he wait a couple of hours? So when Evans and France come screaming in, he's on his way home. Funny. How hilarious. He aren't laughing. I've got stock in extronics, so it hurts me when I laugh. Yeah, now the guys in France's department have no responsibility. They have nothing to do, because old Fredo France decides on everything when he decides. It took him three days to decide what kind of paper clips we should use, but they are the greatest paper clips. Oh, excuse me. Oh, what's the matter? I think I just swallowed a cannonball. What about Evans? He's really bright. And does he ever know it?  That cuts out a lot of people. And does the chairman of the board approve of these kiddie games? How would he know? He's out peddling the product. Oh. Why my life savings were being fiddled away? I had to pass the information on where to reduce some good. So I called extronics. Martha Wilkerson speaking. Is this the office of the chairman of the board? Yes, of course. Well, my name is Charles Kirby. I'd like to speak to Mr. Fisk. Though what a great many people, in regard to what, Mr. Kirby? Well, I prefer to explain that to Mr. Fisk. It's a matter of some urgency. Mr. Kirby, what may seem a matter of some urgency to you may be of total insignificance to Mr. Fisk. I suggest you explain this urgency in a letter, and I'll see that it comes to his attention. Thank you for calling. I'd love to kill that lady. And I just might do that. E.B. White would be proud of her. Heriodite and witty, but not too witty. And after a few days, who's there? A relative. Oh, come in, daughter. Well, are you sure the coast is clear? Yes, yes. Are you in a broth, negligee, hiding in the broom closet? Oh, for heaven's sake. The mail is there, and you've got a letter from Xtronics. Let me see it. Oh, don't, man. Well, I don't believe it. It's a good one, letter. Thank you for your letter. Xtronics is sorry it cannot answer all communications in a personal way. But rising wage costs, company mismanagement, and rude employees preclude that. Oh, you're making that up. Oh, so I am. In a personal way, but your letter has been filed appropriately. Hmm, they shredded me. Meanwhile, we thank you for your interest in Xtronics. Most sincerely, scribbled initials. Well, you can't win them all. Why not? It's just an expression. Hmm, I never use it. Your esteemed father is not about to do an El Foldo. I called again introducing myself as that man, Mr. Barrick Sothine, finance minister of Kuwait.  Made me very nervous about my investment, and I decided to take more direct action. What are you doing? You first. Paul has a friend whose uncle goes to the same barber as Mr. Fisk, and maybe when the uncle gets back from trap fishing in New Zealand... Stop. Time is of the essence. Well, what are you doing? I'm going to crape myself and get delivered to Harrison Fisk's estate. I don't believe it. Not even standing here looking at the crape? Harry, don't shoot. But who knows what's coming out of the crape? Could be the ultimate blob. The goo from outer space, the toady potent cell. Why can't you be in the sports page instead of science fiction? Oh, hold it. Okay, mister. Don't make any funny moves. Where am I? And who are you? You're in the outer drive of the Fisk home, and we're the estate patrol bombs squad. I was supposed to be taken inside the gates. Where's Harrison Fisk? When this crape showed up, he and his wife went out for the Blackwell Hotel.  Daddy, they haven't arrested you, have they? No, and they're not going to. Too many papers to fill out. I think we'd better take this man in. Okay, but you fill out the papers. Sir, consider yourself a free man. Come on, Daddy, I'll take you home. By the way, of the Blackwell Hotel. Oh, base one, this is Mobile 16. Alert the Blackwell Hotel that an unwelcome man will be trying to contact Mr. Fisk. He is a male Caucasian, about 75. 67. Works every time. 67, white hair, wearing a sport jacket you wouldn't believe. Sort of a two-button rainbow. Take me home, Carolyn. I've got to institute operations squeaking wheels. And what is that going to be? Harrison Fisk is going to think God has declared Practical Joke Week exclusively on him. All week long, officer. What? It's baby week at Sears Murray. A great time for play suits, snap-side shirts, play pens, jennylin baby furniture, blankets and a whole lot more. Mothers and mothers-to-be shop Sears all during baby week.  Because my wife and Sarge Junior are probably at Sears right now. Because Sears has baby buys bundled up. Sears Budget Shop has a vested interest in value. Vested dresses, invested skirt and pants at sizes 8 to 18. Style just right for strength. Their romantic flounce dress is topped by vest. Turnip pants suit coupled with vest. Also the tunic and skirt smartly finished with a vest. The vest, the season fashion basic. Lots of exciting print and solid color combinations. Some of you can be cheesy. Invest in fashion. Invest in value. Vest in dresses, invested skirt and pants sets in a budget shop at most larger Sears retail stores. Darling, I'm a madress who knows what to wear. Solid color prequels sheets in Sears' medley collection, of course. This gorgeous sheen I'm wearing speaks for itself. The color is called Indian sand. Isn't that stunning? I wear sheets of royal blue, lemon yellow. Sears has a dazzling collection of up to 24 colors. And the fit will just look. I can't understand why madresses wear anything but these moves from a press sheet. Honestly, darling, I wouldn't wear anything else. Five is from Twin to King in most Sears retail stores and in the catalog. I love to eat. But it takes so long to cook. That's why we both love our new Kenmore microwave oven from Sears. I can cook a five pound roast medium rare in just 30 minutes. Or three strips of bacon in three and a half minutes on a paper plate. Bake two potatoes in eight minutes and cook vegetables faster than boiling them in water. That means less time in the kitchen. And more with you. Fast, clean, cool cooking with Sears Kenmore microwave ovens all with automatic defrost. Choose the right model for your kitchen from the many styles available at most Sears retail stores. For the scheme that will make somebody listen to what you have to say. Charles Kirby is such a man, which is a good thing since our story's about him. Harrison Fisk wouldn't listen. And so Charles Kirby raised his voice a little bit. So to speak. Mr. and Madam Fisk, pardon the interruption. Everything is perfection as usual, Amir. The salmon is marvelous. Oh, thank you, merci. But I have just been asked to hand this note to you. Mrs. Fisk gets all the notes in this family. This seems to be for you. Oh. What is it there? Your car is not here. It has been repart to the Acapulco restaurant in Long Beach. You'll like them, Margaritas. Do you know anything about this, Amir? Oh, they do have very good magazine. No, no, no. About my car, for heaven's sake. Well, no, sir. Well, check on it, Amir. Check on it. I'm afraid your car's not here, sir. Don't bleep. Harrison. You turn it off, Mr. Fisk. Of course there might have been a burglar. But there wasn't. Well, just this note on the front door. Oh, thank you. Your alarm system has been tested by slickie boys of Hussain Incorporated and has passed. If any of this distresses you, please call this number. Mr. Fisk, I... I asked not to be disturbed. Maybe you'd like to write the quarterly report. Very well. I'll tell Mrs. Fisk you can't see her. Well, wait a minute. What's she doing here? Trembling since you asked. She's almost hysteric. Well, bring her in. Yes, sir. Oh, my God, now what? Oh, Harrison. Everyone, Margaret. Margaret, sit down. I can't sit down. My knees are clean. So what happened? Everything. Everything. That's what's happened. Party rents wants to deliver a hundred folding chairs, an electric hospital bed, and a portable toilet. Oh, there are tree surgeons who want to give us estimates for taking down all the eucalyptus trees. And there's a man with a bulldozer who thinks we want him to knock down the front gate. There's a man who wants to clear the hillside and brush. He looks like Pancho Villa, and he's got a whole army of mercenaries with the shetties. There are four, four plumber's trot.  I've had it, Harrison. I've had it up to here, or maybe even up to here. Oh, I love you, but you seem to have a lousy karma lately. And I'm thinking of leaving you before somebody has a charity block party in my house. I'm all right. I'm all right, Margaret. I think I know what I have to do. It's all right now. It's all right, darling. There's a tavern not far from your plant called the Bloody Boar. I'd like to meet you there alone in an hour. I assure you it will be good for you, for me, and for electronics. Sit down, Mr. Fisk. Thank you. I consider the way you forced me to come here completely reprehensible. I'm sorry you forced me to be reprehensible. I realize you're a busy man, Mr. Fisk. You're inaccessible. I phoned, and I wrote this very nice letter on my best stationery. This is a carbon. Hmm. Yeah, erudite. Whitty, but not too witty. Exactly. In answer, I received this form letter. No. Oh, I began to sense the source of your animosity.  Now, the reason for all this is that I put most of my retirement money into electronics. And now I see my semi-comfortable life threatened. So I found out why extronic stock is going down. How could an outsider possibly... I talk to insiders. You have three factions here that don't get along. Groups headed by Hinkle, Evans and Trance. So your management is divided. Or as today's kids put it, polarized. As chairman of the board, you should have done something about this three years ago. You're quite right. But goodbye, Mr. Kirby.  I'll hear you out, but I know the problem. I can't fire any of them. They're too good, too important. And they're also primitoners, and if I speak frankly to them, they'll quit, and one of my competitors will fire them. Of course, if I don't fire them... Write down the drain. There's no solution. There's always a solution. Let me give you an example of a difficult problem.  Half to the oldest, the third to the middle son, and a ninth to the youngest son. 17's a prime number. It's impossible. Yes, but then a wise man rode up on his camel, and they told him their problem. The wise man thought a moment and said, suppose I lend you my camel, temporarily. Now, you have 18 camels, and half of 18 is nine. So the older son took nine camels. Next, the third of 18 is six, and the middle son took his six camels. And the ninth of 18 is two for the youngest son. And now nine, six, and two are 17. And so this remaining camel, of course, that's mine. And with that, the wise man got on his camel and rode off. Shall I run the camels by you again? No, no, I was just thinking. It's an unusual solution. Well, I have a solution to your problem, and it's a little unusual too. Well, I'm through. Hinkle's Evans, France, and you will be one big happy family again. You'd have to be a genius. I do have my bright moments, and I expect to be well paid for them. Yes, right. But about the camel. Well, think about that for a while. First, I suggest you hire me, and then here's what you do, and what I'll do. Gentlemen, I want you to meet Kirby Charleston. Well, Mr. Evans and Mr. France. Howdy, sir. I'll be brief, gentlemen. Benton Rogers and Gravett, as well as Matrix Development, are concerned about the way things are going at electronics, and they should be. Our dividend is going to be half of what it was last year. Now, needless to say, they believe we're doing something very wrong, and they suggested Kirby Charleston here. He's been a very successful troubleshooter with other companies, and I'm confident he'll do as well here. I want you to give him your cooperation and extend all courtesies to him, because he'll be working directly under me as the chairman of the board. Oh, you won't find any problems in my department. I don't have any trouble. Nor do I. How can you be so damn sure? What? Your three departments make up the company, and the company is in trouble. At least one of you must be grossly misinformed about his own department. And I'm going to find out who it is. One dextronics for three days. Asking questions, taking secretaries out to lunch. Very pleasant of us, too. Asking questions and filing the answers away. The fourth day I visited Henkel, just after his three martini lunch. Good afternoon, Mr. Henkel. Oh, hello. How was your lunch? Oh, delicious. Very tasty. What'd you have? What? To eat. Oh, what did I have to eat? Let's see, I had, um... You had veal kidney. Now, do you answer your mail now? Oh, sure, girl, do it now. Never pull off till today what you don't have time to do tomorrow. That's one of Fredo Francis' problems. Can't make decisions. Come to think about it, I ought to write him a memo about that. Yeah. I had veal kidney. How do you know? The shadow knows. Oh, he do.  So, one down. And the next day I dropped in on the Evans. A really nice, bright man. Miss Henderson! Where are you, Miss Henderson? Hello, Mr. Evans. Oh, oh, uh... Where did you come from, Mr. Charleston? I came from Ohio. Miss Henderson stepped away from her desk for a moment. What for? Guess. Well, that's another one.  And what was it that Shakespeare said about women? No, no, that's what Sherman said about war. You know, you're right. Hey, have you found out who's creating the problems around here? Indeed I have. Okay, glad to hear it. Anybody I know? You better believe it. I wasn't sure that I was right about France, but I wanted to check. I dropped in as he was on the intercom to his secretary. And tell National Oxides I'll get back to him in two days. Then tell Henderson at Formula X I'll get to him with an answer in a day or so. And tell Harvey Levick I won't be able to talk a deal this week. I've got to look over all that material he sent me. Now, you've got to put off that meet. I'll get back to you later, you. Hello, Mr. Charleston. Mr. France. What can I do for you? Answer a few questions. Gladly. Have you okayed the order for the Sandrex Corporation? No, not yet. Haven't had time to look them over. Well, the space and tolerances are what you asked for. Why is it all right? I've had other things to do. I'm taking care of it, Mr. Charleston. In a very sloppy way. What did you say? Surely you heard me. How long did Sandrex Corporation guarantee their prices? Now you look here, Mr. Charleston. Today is the last day. I suggest you call him personally and say, yes, before you get a 15% hike. I'm a very busy man, sir. Would you mind getting out of my office? Out! I can only say to you what you said to me when I came in. Oh-ho-ho-ho! Gladly. I have to solve them without losing the company and my retirement checks in the process. Radio Theater will return after this message from your local station. Who will be the skier you most admired in your 2004 years? Well, probably be the kind of one. He's an expert, a crispy little guy. A little beard. He's a skier. A fearless skier, as a matter of fact. Oh. If anybody ever calls him a chicken, you know. I get the picture. As a matter of fact, in the off-season, he used to ski down a big mountain of mashed potatoes. But, you know, Bob Rossian from Ringo's ski shop is a good friend of his. As a matter of fact, he convinced his wife Leslie to marry him. Oh. Yes, the cleaner was hobbling around the bubble store with a broken wing. Very good. Very good. It might have been his fish farm. Well, the cleaner heard this muffled voice in this rack of skier and Levi clothing. Leslie. Bobo, he had got so much great merchandise that he was about to suffocate himself. And she saved him. Took a pair of orange skis and popped him out like a shoe on an apprentice that he'd marry her. Heartwarming. When did that happen? That was last Thursday. Kids were thrilled about it. Yeah, Bobo's Ringo's ski shop, 1200 South Wales Avenue. I gained 20 pounds in two months. Chocolate and bean and butter. Yeah, I never lost that weight either. With me, it was different. I was climbing the walls, yelling at the kids. I just couldn't live with myself. Neither could Dan, could he? No, not really. He said having a wife to smoke was better than being terrorized day and night.  Right. So anyway, I'm back to a pack and a half a day, addicted just like I was. Let's hold it right there. The American Heart Association wants you to know that smoking cigarettes becomes a habit, not an addiction. Habits can be broken. Smoking is a matter of choice, not destiny. We can help you quit. You don't have to gain weight or climb the walls. Contact your American Heart Association for a free booklet that explains how to break your cigarette habit step by step. The American Heart Association wants you to know we're fighting for your life. Drip it again with the concluding act of the troublemaker. For a particular moment, none of them had to be used because only a worried Mr. Fisk was involved. He stopped me in the corridor between... Well, I can tell you more about the ex-tronics and I think it's about time to set up a meeting. Oh, good, good. Hinkle's coming this way. I'm ready for him. Hello, Harrison. And Mr. Charleston. Well, I'm glad you're here, Mr. Hinkle, because I don't like to talk about people behind their backs. Mind if I eavesdrop and drip away? I was just about to recommend to Mr. Fisk that you not be allowed to make any company decisions after one o'clock. And what does that mean? It means that those three martinis at the Simpering Bull turn your brain into a bowl of minestrone. Oh, wait a minute. Your judgment goes to pot. People sell you things you wouldn't even look at in the morning.  This is a company, Mr. Hinkle. Not a daycare playground. Hold on, Mr. Troll. Are you trying to call me an alcoholic? I'm trying to avoid calling you an alcoholic. I'll leave it to Mr. Fisk to act on my recommendation. I have more to do, but I'll tell you something, Mr. Hinkle. You and your three martini lunches are one of the things wrong with electronics. Did you hear that? Oh, every word. Clear as a bell. Well, I resent this guy. Oh, I don't blame you. Of course, I have seen some of your memos. Well, I sometimes get in the mood to write a memo. Is it usually after lunch? No, I don't think so at all, not really. Well, if there's nothing to it, don't let it bother you. I won't. I won't. But, well, he practically called me an alcoholic. I hope that was a psychic slip. Yes, it must have been. I have had lunch here. I mean, I haven't. Why is that, Mr. Chalk? Wouldn't you like to know? You would, and I'll tell you. I was planning to play a tape of a phone call in which your attitude toward the president of System 3 Semiconductors can only be described as outrageous. My God, is that who that was? Naturally, Mr. Matsumoto didn't get through to Mr. Fiss. Matsumoto! And he won't call again unless I persuade him to reconsider. Oh, I wish you would. Oh, yes. Very well. I'll just tell him that I'll tell him Martha's father and mother died in an automobile crash the day before, and she was out of sorts. I'll say it was her car. I'll go, but please don't play that tape. Very well. The point is, try not to filter out everybody. I got the point, Mr. Charleston, and I'll be more careful. I have already recommended to Mr. Fisk that Mr. Hinkel not be allowed to make business decisions or write memos after his three martini lunch. The whole thing is ridiculous. I'll miss those memos. It seems to me that's pretty strong, but... Mr. Fisk has not taken the recommendation yet. It's an easy solution, which is more than I can say for the problem with Mr. France here. What problem? Mr. France, you've got to do something about that enormous ego of yours. Ego? I? Mr. Charleston, I am the humblest of men. Yes, I know. You're the tops in humility. And considering yourself humble is just another form of arrogance. I consider that pretty insulting. Well, ladies, since we're paying for it, I think we ought to hear him out. Well, what's that got to do with my department in electronics? I'll get immediately to the point. You and your ego think you're the only ones who can pass on everything. You may be the best to decide, but anybody can order special paperclips. And meanwhile, your desk becomes a hiatal hernia obstructing the passage of vital paperwork, information, and decisions. Now, maybe you don't realize that he often works all night, Braxtronics. I know. I know. Doing things he should delegate to other people. And then he's tired all the next day. But it's interesting you should come to his defense, Mr. Evans. Why not? I don't like the way you're handling this whole thing.  I'm surprised you back up, Mr. France, when you're the most inconsiderate of the people who work for you. You ride roughshod over your assistants like a Hungarian nobleman. Now, hold it. No. What about Annabelle Potkin?  Have you found a replacement yet? How would you like to step outside? No, no. Take it easy, Eddie. Suppose you think that would prove that Braxtronics didn't need her. Won't you have the slightest feeling for your fellow human beings? Can't you be satisfied being a minor God? You'd get cooperation you wouldn't believe. I'm not going to take this. Wait for me, Eddie. And you wouldn't have to break into new secretary every week. I'm coming, too. Think about what I said, sir. I'm just going to... Eddie, come on. Let's go. I just hope I haven't lost my three top men. So do I. Just remember, I have stock in electronics, too. What would you have done if Evans really tangled with you? I would have beaten the hell out of him. I'm focused in the executive restroom. Let me tell you one thing, boys. We may have had our differences, and we may have needled each other more than was absolutely necessary, but we have got to get together and get rid of this character. Eddie, would you go in and shoot him now if we all promised to swear you were in Chicago at the time? I'm tempted. And, Harry, you may have to wait until five for your martinis for a while. I may take up jogging for lunch. I didn't know we lost Annabelle Potkin. I shouldn't have yelled at her. She's too sensitive. It seems to me somebody found your sensitive spots recently. You could say that. All right. I agree with Spritto. We've got to sign a truce back each other up and tell Harrison to wrap up Kirby Charleston and send her back to Benton Rogers and Gravett. Yeah, but we're going to have to stop taking electronics for granted. Yeah, we've all been doing it. Yeah, first things first. Are we prepared to tell Harrison or else? I am. I am, too. No, not me. Not me. I didn't go out for lunch today. I thought you seemed friendlier. Oh, wait a minute, Eddie. Now, hold it. Hold it. We've got to go back to what we used to be, the three musketeers, the can-do kids, all and all for one. Are we ready? Almost. Thank you. Now, you know the old era. I've been wondering how... Oh, here they come, Charles. You were right. Go ahead. Say anything. And, furthermore, these are the men. The men who made electronics what it is. Oh, gentlemen, I'm glad you're here. I was just about to tell Mr. Charleston that we were the team that made electronics and we aren't breaking it up of a fast psychological fix from some corporation medicine man. Now, wait a minute, Mr. Fisk. You would believe me, Harrison. He's not a troubleshooter. He's a troublemaker. Oh, you can say that again. Everything I said was right. People were screwing up with his martini lunch. France wouldn't let anything get done in his department without his initials. And Evans was alienating everybody in his radiation fallout zone. Douglas MacArthur was a pussycat into comparison. That's enough, Charleston. But wait. It's all true. These men don't realize how they've been kicking electronics down the tube by fighting among themselves. I don't want to hear it, Mr. Charleston. But you've got to hear it. No. No, I don't. Mr. Charleston, you may be very perceptive. You may be very bright. And you may even be very right. I'll have to check with my free right-hand men about that because they've always been honest with me. But as far as I'm concerned... Oh, that may be so, but all I can say is... Out! Hmm. I see. Well... I can't imagine how you're going to survive. Goodbye. Hey, hey, hey, hey, hey... I'm sorry I hired him, but everything was going down, down, down. I didn't know what to do. But I had to do something. Maybe there's something to be learned here. I just wouldn't have the slightest idea. Charles Kirby, aka Kirby Charleston, it was worth it. Retreating my car from Long Beach and suffering with the burglar alarms and all the rest of it. I think you've got my management team straight now. Well, the idea was get them all to hate me instead of each other and then get them united in their hate.  Well, they're happy, you're happy, and I'm happy. Oh, what can I say? You said the things to them that nobody could say about setting the relationships. Oh, boy, were you frank. I had to be. But they're going to remember, you burned a lot of your criticism right on their lines. Well, yes. And that was quite a number you did on Martha Wilkerson. She had it coming. I guess so. I presume you were Mr. Matsumoto? Oh, yes, calling long distance from Honolulu. It's not about those camels. Well, a half, a third, and a ninth don't equal one. So the Arab didn't give them all away. Oh. I like to think that I'm the floating camel that solved your problem. I have a check here for you, Charles. Thank you. I don't mind saying that while the squabbling was going on, I sold X-tronics short and picked up a small bundle. But now, I am happy to say I'm bullish on X-tronics. X-tronics. Some really impossible guy was hired by this company. I think it was the one you used to be interested in. X-tronics? No, no, no. X-tronics. Or maybe it was X-tronics, Mario. That's what I said. X-tronics. Well, he went over there and blistered the whole management team. Rumors, rumors. There was an older guy in his 50s. So they said that, eh? You know? Well, I don't know how to put this, but let me tell you something, Charlie. Mr. Kirby, damn it. Hey, sorry, sorry. You're going to tell me what? Mr. Kirby, I was just going to say that you ought to do something with your time. I'll think about it, Mario. It would cost to replace your car's muffler, including installation. Oh, I'd say about $50. No, wait, $45. It would be around $30. Just about $40. The aluminized Sears Muzzler is only $19.99. That's half of what I guessed. It's hard to believe on a Cadillac. That's a trippy party with installation included. I should have known it. Sears. The suit to own, if you could own only one, is on sale at Sears. The $119 four-piece vested is now $89, a $30 savings. The suit, contrasting slacks and reversible vest makes six different outfits. The four-piece vested suit, it should be the suit you own. On sale for $89 until February 24, then margin Sears Men's Stores. Style, sense, satisfaction. Sears Men's Store. Generations ago, families dined by the warmth of the old spirit with its open hearth dining room furniture. Faithfully rendered early American designs and careful workmanship give it an heirloom quality. The satin glow and warm highlighting of Sears open hearth take 26 steps to achieve. There's no shorter method to bring out the beauty of the wood. And like all good furniture, open hearth is made to last for a long time with sturdy tongue and groove and mortise and tenon construction. Choose from 16 different pieces of open hearth and Sears retail stores. I sell draperies at Sears. Yesterday, a lady came in and said that she'd been in and out of about every store in town looking for draperies and at this point didn't know what she wanted anymore. I asked questions about her tastes and decor and then made suggestions. She was thrilled. She found what she wanted and learned a little too. It made me feel good to know that I helped her out. Sears people are friendly people who help you find what you want. Sears, Sears, Sears where America's shop. This thing to Sears Radio Theater brought to you five nights a week by Sears Robot Inc. Sears where America's shops. The Troublemaker was written by John L. Green, produced and directed by Elliot Lewis. Your host was Andy Griffith. Our star was Jim Jordan. Also heard were Michael Galman, Shirley Mitchell, Mary Jane Croft, Ralph Sadan, Elvia Allman, Frank Nelson, Sidney Millard, Kerr Broadley, Kerr Beigren, and Jerry Hauston. The music for Sears Radio Theater was composed and conducted by Nelson Riddle. Argil Moore speaking. The Elliot Lewis production of Sears Radio Theater is a presentation of CDI.


Biosensors have acquired paramount importance in the field of drug discovery, biomedicine, food safety standards, defense, security and environmental monitoring. Use of aptomers or nucleotides, affobodies, peptide arrays and molecule-imprinted polymers provide tools to develop innovative biosensors over classical methods. This article was authored by Vinesh Bariyas, Sudakumari Isisi, Bala Subramani on ESN-THL, KU Amaran, and others.


etiquette in Latin America. If you want to teach English as a foreign language, it is important to make sure you are as well prepared as you can possibly be before entering a classroom. It is equally important to have a good understanding of the country and culture you are hoping to make your home while you are teaching. Below, we look at some important aspects of etiquette in different countries in Latin America. The countries of Latin America are an increasingly popular destination for newly qualified English teachers. While they are generally more relaxed when it comes to formal rules of etiquette than other regions of the world, there are, as with many countries, particular social norms that any visitor should make the effort to understand. Colombia, manners and behavior. Although Colombians are generally relaxed about punctuality, as a foreigner you will be expected to be on time. Communication, shake hands when meeting people and remember to smile.  Be prepared to take some time over greetings as it is considered a sign of respect. Costa Rica, manners and behavior. Punctuality is important to Costa Ricans and you are expected to be on time for appointments. When invited to a person's house, it is appropriate to bring a gift of flowers, chocolate or alcohol. You should avoid putting your feet up on any furniture that is not expressly designed for that purpose. Communication, handshakes are the common way of greeting each other. It is fine to talk about politics in Costa Rica but you should avoid discussing religion. Chile, manners and behavior. Bargaining is not expected in shops or markets. Punching your left palm with your right fist is an obscene gesture. Gifts should be opened when they are received. Arriving late for social events is usual and you should try to arrive 15 to 30 minutes late. Communication, maintaining eye contact in conversation is important. Chileans stand quite close when talking with you. Avoid discussing politics or human rights issues. Brazil, manners and behavior. Using the OK hand signal is considered very rude and should be avoided. Punctuality is not important in Brazil and you should be prepared to wait for people to arrive late. Personal subjects such as salary, job and so on should be avoided in conversation. Communication, shake hands with people you meet and maintain eye contact. Friends will often hug each other on meeting. Be prepared to take time greeting and bidding farewell to people. Using a person's title is not important and first names are often used.