Hi everyone, Sean with jaspyscasebreaks.com here doing 2020 Bowman Sterling mini box break number 10 random team edition where We're giving seven spots away in the flawless random team break of 2017 flawless Coming up after this 30 spots everyone gets a random team at MLB They will be a mini box pack break of 2020 Bowman Sterling at the end of the break will randomize all 30 customers names Randomize them top seven names each win a spot in the 2017 flawless baseball random team break Thank you So here is the list of customers from mark new bower down to Jeremy port and last spot mojo and the teams From the Diamondbacks down to the Washington Nationals Dice world randomized both lists and pair them up six to six twelve times separate dice roll separate randomizer at the end of the break to give away the seven spots in the break in the flawless 12th and final time from Eric jock down to Stephen Kendrick you see 12 times there 12 times on the dice now 12 times on the teams 12th and final time from the Giants down to the twins 12 times and No trade window not going to read through the whole list, but let's get it all on screen Kyle Lewis Derek Valenciic Boba Shet Mike Tower Adley Rushman Kevin Crouch Louise Robert Stevie Kendrick Bobby with junior Nick Stover Jason Ming is Franklin Mackenzie Niko Horner Roy Shappek Gavin Lux Nick Stover And all the rest print Rip give away spots. There's our box Let's find something nice in here and nail bite right here at the end of this game There's our final list Good luck everybody and Cubbies Brennan Davis nice prospect auto there for Roy Shappek and the Cubs Niko Roy Very nice. Now. Let's give away those seven spots Here is the original list of customers names from Mark Newbauer back down to Jeremy Port top seven As it says here at the end will win Randomizer four to one five times fifth and final time five times five times Top seven so from spots eight down to 30. Thank you guys very much for getting in on this and hopefully I can do better for you next time Thank you for getting in and special congratulations to Stephen Kendrick Jeremy Port Derek Valencik Justin Humphrey Roy Shappek Jarell Taylor and Derek Valencik You seven have one spots in the upcoming 2017 flawless baseball break that was 2020 Bowman Sterling mini box break random team number 10 from jaspercasebricks.com. I'm Sean jaspercasebricks.com Thanks for hanging out. We'll see you next time


OK, so the proof that we still have to complete is this. It's the fact that small l2 is complete. So yesterday we remained here. We have a Cauchy sequence xn. Hence, this is the definition of being a Cauchy sequence. And we proved that there is convergence of all components. So for any k, we proved that there is convergence of the components. So what remains? So we have something in the limit, xk. For any k, we have a number. Then we define the sequence xk. This is a definition. So what remains to show is that x is in l2 and that the convergence actually is not of all components, but is the convergence in l2. So this was what remains. Now, for any l in n, I have that this, for any l, for any epsilon positive that exists, such that xk n minus xk square m less than epsilon square for any n and m. So in particular, since this infinite sum is less than epsilon square, a forciori, this is less than epsilon square for any fixed l. So letting m going to plus infinity here and using star, it follows that for any n bigger than n bar. Now, this is true for any l. And epsilon is independent of l. Therefore, for any n bigger than n bar. So we have at least something similar to the convergence. You see, we are very close to say that xn converges to x, actually. We are almost there. The only thing is to observe that we have this, but then let me write this. So I want to show that x is in l2. So I have to control the sum of square. I have to control this, right? Once I control this, then this is the definition of convergence in l2. So let me write it xk square. I add and subtract x, OK? I add and subtract this. And so this is less than or equal than xk by the triangular probability of the absolute value xkn square, OK? And this is less than or equal than. So there is also another inequality which is, say, a plus b a plus b square less than or equal than 2 plus square plus b square, OK? This is clear. So apply this inequality with this choice of a and this choice of b. So this is less than or equal than 2 times xk minus xkn square plus 2 xkn square. And so we see that when I sum this over k, when I sum this over k, the sum over k of xk square is less than or equal than a number, even a small number, 2 epsilon square, plus 2 times the infinite sum of xkn square, which is finite, because each xn for any fixed n, each xn is in L2, because this shows that x is in L2. So we have proven 0.1. Hence this finite, which is, like to say, x in L2, which implies 0.1. So to conclude the proof, we have to show that this convergence takes place actually in L2. But this is exactly what is written here. Now let me, for any epsilon positive, there exists n bar such that this is less than this. So this sentence here in the box is exactly as to say that this converges to x in L2. So this shows the theorem. So our small L2 is an interesting, reasonable, infinite dimensional Hilbert space. I left also another exercise yesterday on the semi-norm, P. Did you try to do that? Is it OK? Everybody solved the problem on the semi-norm? Yes? So P of x1 minus x2 is larger than or equal than P of x1 minus P of x2. This is because P is sub-additive. So remember, P of x plus y is less than or equal than P of x plus P of y. And therefore, this is simply obtained by writing this like this. So from this, we have this. But then we can exchange the role of x1 and x2. So P of x1 minus x2 is, remember also that P of lambda x is equal to P of x. And so P of x1 of x2 is also equal to P of x2 minus x1. OK? So therefore, this is larger or equal than P of x2 minus P of x1. We obtained that this, say, P of x2, P of x1 minus P of x2 is less than or equal than P of x1 minus x2, but also larger than or equal than minus P of x1 minus P. So this means that P of x1 minus x2 is larger than or equal than the absolute value. So this gives us a little bit more surprising that this happens. OK?  And the answer was yes, that it was no, we cannot. So this cannot contain a ball centered at the origin. Do you have the proof of this? Which is the proof? Yes? So let's assume that I have a ball contained here. OK. So now we will obtain, yes, it's true. If there is a point here, say, a point, actually, 0, 0, 0, a point epsilon over 2, because this is open, actually. So if there is a point with one component, like one component equal to epsilon over 2, say, then I take k, so that 1 over k is less than epsilon over 2. And then I take the point 0, 0, 0, 0, and I take k component, I put epsilon over 2. This is a point here, but it's not a point here. So there is a sort of isotropic object. And this isotropic object cannot be contained in this object here. However, if you have not followed this proof, we will obtain, once more, this fact as a consequence of the next theorem that I will say. However, the proof is direct, as he said. OK. Let me show you this theorem. So let x be a point in L2, and let r be a positive number. OK. And so Br of x, as usual, is the set of all point y in L2 such that the distance from x to y is less than r. This is a usual notation, open bold. And this is the distance in L2. So whenever I d, d means distance in L2. OK, so this object here is not, sequentially, relatively, it's not relatively compact. OK. OK, this is maybe, so you see, what it is saying is that neighborhoods in the topology of the distance are not local. This essentially says that if you have a neighborhood in the, if you take, say, the closure of this neighborhood in the topology of the distance, then this cannot be compact. Or if you want, if you have a compact set, then it has empty interior. And so this says that this topology has, the number of compact sets for this topology is not enough, in some sense. There are many open sets, but not so many compact sets. And this is maybe not good. So one would like to have something, some new topology that has more compact sets. And this is the origin of the so-called weak topology. So at the end, one needs to put on this space another notion of topology, weaker than the topology of the norm, but with more compactness properties. So behind, after this result, there is the problem of choosing another topology in A2. You don't see this distinction in finite dimension. In finite dimension, weak topology and strong topology are the same. You don't see anything. But in infinite dimension, of course, in finite dimension, this is relatively compact. If you close it, then take a sequence here, then there is a limit, some sequence converging to, OK? So I mean, this is, I put in parentheses sequentially because for metric spaces, we can always use sequences, OK? So if you want, you can put sequentially, or you don't put sequentially, it's the same because we are in metric space, OK? So proof. So you have to keep in mind this result because this is maybe one of the starting points of functional analysis, meaning that compact sets have empty interior. Compact sets have an empty interior. And this is a sort of origin, one of the starting points. So proof idea, find a sequence. So let me call it sequence Yn, find the sequence of Yn in Br of X having no converging subsequence. Try to find the sequence. So if you want to make the proof more transparent, so X is a point in L2. So X is a sequence, is a point in L2, OK? Now, if you want to make the proof more transparent, maybe you can take X equal to 0. If you take X equal to 0, then you show that this holds at the origin, but then essentially you have the result. So if you don't like now the proof that I will do, take X equal to 0, which makes it more transparent, OK? So I have X at the center of the ball. And I want to find Yn sequence of sequences. So for any n, we define Yn as follows. Yn as follows. So the component K of Yn is equal to the component. So if K is equal to n, or if K is different from n, I split this, and then this is equal to the n component, plus say r over 2 here, and then Xk. So if X is equal to 0, just to understand a little bit, if X is equal to 0, then Yn K is equal to 0, if K is different from n, and r over 2, if K is equal to n. This is just to understand a little bit better. Notice that any element then Yn is in L2, clearly. And also, the distance of Yn from X is less than r, as you can see. It is immediate, because it is actually r over 2. Therefore, Yn belongs to be for any n, for any n, OK? Now we can consider the distance between two points of the sequence. So take now n different from m, and consider the distance between Yn and Yn. It is clear what is the distance between these two points. I mean, because the distance, so the distance between Yn and Ym squared, if you want, it is by definition this infinite sum of Yn K minus Ym K squared by definition. Now, I remember now the definition of 1. So this is equal. So this reduces just two contributions, only when K is equal to m or n. Otherwise, this is 0. So this is equal to Yn n minus Ym n squared plus Yn m minus Ym m squared. I mean, the infinite sum reduces only for those index K equal to either n or m. And this is r over 2 squared. This is r over 2 squared, right? So this is 2 r over 4 r squared over 4, OK? And so this is equal to r squared over 2. Therefore, the distance is always equal to r over square root of 2. And therefore, Yn cannot have any converging subsequence. If you prefer, you can just think about this case. x equal to 0, it is more transparent. So Yn cannot subsequence. So the fact that you have at your disposal an infinite number of dimensions allows you to make some kind of tricks. In any direction, a constant distance from the origin, in any direction. And then this sequence cannot have converging subsequences. So exercise, let C be a subset over 2. If the interior of C is non-empty, then C is not compact. Or in other words, K, K compact, K sequentially compact, OK? So let us try to do the exercise. Assume by contradiction, C is compact. And assume that its interior is non-empty. And assume its interior is non-empty. So take a point x. So if the interior is non-empty, there is an interior point. So there is a point in the set, and there is a radius. The topology is the one induced by the norm. And so the ball is contained in C, OK? Now, a compact set in a house of topological space is closed. And this is a metric space. Therefore, it is a house of space, right? So we also have this inclusion at C is closed. So this shows that this contradicts exactly the previous, because this shows that then this is compact. This is the closure. This is contained in a compact set. So this is compact. Yeah. This is a closed subset of a compact set. Therefore, this is compact itself. So I am using just this. So then we are x is compact. Which is, in contradiction, what we had just proven before. Is it OK? So this is something that should create some difficulties to our intuition on what is smaller 2. Because now smaller 2 is complete, is separable, but compact subsets have empty interior. So this is something that is not so easy to imagine, I believe, when one study topology. So this is a very good interesting example of topological space with this strange somehow property. And this is also the starting form, as I said, of introducing a new topology, having more compact sets. However, as I said, from this you can start to understand how difficult the function analysis is, because infinite dimensions probably something outside our intuition, in some sense. For instance, let us do this exercise. Let f be a function continuous with compact support. f is identically 0. So there are no continuous functions on L2, no non-trivial continuous functions on L2 with compact support. And so when you see measure theory, for instance, or the space of continuous functions with compact support has, in general, a lot of importance. Well, in this case, you have to replace continuous functions with compact support with something else, because there are no non-trivial continuous functions on L2 with compact support. So let us try to see why. Do you help me? Can you help me? Do you have some? Yes, per image. And where do you use compactness of support in this argument? Yeah, but then you have to use compactness of support. Yes, take a point exactly. Take x in L2, such that, say, f r. So assume by contradiction that f is not identically 0, OK? So assume by contradiction not identically 0. Hence, there is at least one point in L2 where f is not 0, OK? OK, without loss of generality, we can assume that f of x, say, is positive. By continuity, since f is continuous, there exists a neighborhood called y, neighborhood of x, such that f of y is positive for any y in u. Now consider the closure of u-bar. Now the closure of u-bar is necessarily compact, because it's contained in the support of f. Because u is contained in support of f. So that, as before, u-bar is contained in the closure of the support of f, which is compact, which is equal to the support of f. And therefore, this shows that there is a compact set with non-empt interior. The support of f has non-empt interior contradiction. So all of this is in order to say, pay attention to infinite dimensional spaces. Something that you think to be extremely natural, like this, does not exist. And so if you want to take some test function over L2, you have to change the class. Theorem L2 is not sigma-sequentially compact. It's not sigma-compact. This means that it's not the union of a countable number of compact sets. Proof, assumed by contradiction, that you can write L2 as the union over n of kn with kn-compact. Then we know. So I assume that you can cover L2 with a countable union of compact sets. So not only L2 is not locally compact, meaning you cannot be covered by the union of a countable number of compact sets. Because this we know, then for any n, since it is compact, it has empty interior. Do you see the contradiction? Now, the contradiction is the contradiction with bare theorem. You know, bare theorem. That says that under this condition, at least one of the compact sets has no empty interior. Because this is complete. The completeness over 2. If you have a complete metric space, which can be covered as a union of compact sets, at least one has no empty interior. At least one. So we know that there are not so many compact sets in L2. So after this discussion, this discussion, we deduce that there are not too many subset sets in L2. And this will be a problem for us. But we also know, because we know, that they have empty interior. However, we started the lecture by observing that there is a set, strange set here, contained in L2. So let me call it C contained in L2. That could be compact. This could be. Why it could be? Could be compact. Well, we don't know exactly if it is. The previous was minus over k, 1 over k, because it was symmetric. Now, this is exactly the same. So the fact that there is zero, it is because this is more standard. Maybe this is really called the Hilbert Q. So we have already observed that it is in L2, because the extremum is made by a sequence convergence in L2, the right extremum, the right extrema. We know this. We know that this is closed. This is closed, convex, closed. We don't know it is compact, actually. But we know that it has empty interior, at least, because this was observed at the beginning. So that is why, I mean, we know that interior C is empty. And this is why we hope to have at least one example of non-trivial compact set. So maybe this is a candidate. This is a candidate of non-trivial subset. So now, this is called the Hilbert Cube. And let us try to prove that it is compact. So we have a theorem here. Are there other questions, problems for the moment? Is it OK? Do you see some proof of this? It's not easy. You already know. So is there somebody else that see the proof? What about the following proof? This is a product of compact sets. Yes, what about using the Theon of theorem? Theon of theorem says that the product of compact sets is compact. This is general, big, extremely strong. That is the point. So I assume that I want to prove this using Theon of theorem. This is the product, countable, even countable. Not more than countable. Actually, just only countable product of compact sets. Theon of says it is compact. Yeah, it is compact, but in the product topology. And we know that the product topology has more or less compact set than the topology over 2. More, because it has less open sets. Is it clear? No? I mean, there is a theorem in topology, very big, which says, in general, the product of compact set is compact for the product topology, however. So one could invoke this theorem and say, well, this is compact. The problem is that you have proven it is compact in the product topology, not in the topology over 2. And the topologies are different. In particular, it is more easy to prove compactness in the product topology that compact is in the 2. So we have to change. By the way, in this countable, I mean, there is also no need of so big theorem like Theon of for proving compactness in the product topology of this, because this is a countable product. And therefore, one can do this by hence taking a sequence and the diagonal argument. So actually, maybe Theon of is not necessary, because you take a sequence of points here, countable sequence. Then you take the first component. The first component lies in 0, 1. So x, 1, n. This is a sequence, first component. Then you can extract a subsequence such that you can extract, say, a subsequence indecised by n1 such that the first component is converging. So that, say, x1 n1 converges as n1 goes to infinity to a point n1 is a sequence. So n1 is a sequence of indices, sequence of natural numbers. Maybe the notation is not perfect. Just to say that this is converging to a point at x1. And since 0, 1 is compact, actually, this belongs to 0, 1. Then I consider x2 n1. This is a sequence. It belongs to 0, 1, 1, 1, because I'm taking the second component. Therefore, it has a subsequence converges to a point in 0, 1, 1. So let me call n2 this subsequence extracted from the sequence n1. So n2 is a sequence extracted from n1. Therefore, x2 and 2 converges to x2 in 0, 1, 1, 1, 1, et cetera, et cetera. I mean, this procedure can be repeated the countable number of times to extract just one sequence such that one sequence n prime, I don't know, n prime, sequence extracted using a diagonal that sequence of indices such that xk n converges to xk as n goes to infinity for nk. And this belongs to 0, 1 over k. So this shows by hands using, however, a diagonal argument, which is not so trivial, show by hands compactness in the product topology. So you don't need, actually, for this countable product, you don't need invoking, maybe, Ticanoff theorem. So now, well, let us prove that this Hilbert cube is actually compact in a topology of L2. So there is a theorem of topology which says the following, which actually is an interesting characterization of compactness. Theorem of topology is that compact if and only if. It relates compactness with completeness. If it are if, complete and not totally bounded. This is an interesting result of topology. Now I'll tell you what does it mean. Maybe I recall you what does it mean, totally bounded. For any epsilon positive, there exists, say, an epsilon. OK, c contained in a topological space x is so definition. Let x be a topological space. Capital X, a subset of a topological space xc is called totally bounded. If for any positive epsilon, there exists an index and epsilon. And there exist points x1, xn epsilon points of x such that c is covered, the union of pi xi contains c. Ah, sorry. This is a ball of radius epsilon, totally bounded. It's called totally bounded. Remark, maybe home matrix space. So x matrix space, because we are using the balls. So since we are using the balls, we have a matrix space. Home, totally bounded, implies bounded. But bounded does not in general implies totally bounded. But the converse is not always true. Well, this implication is very easy. Because say, take epsilon equal 1. Then you can cover your set with the finite number of balls of radius 1. Therefore, there is a big ball of radius, I don't know. You can cover your set with the number capital N of balls of radius 1. So you can cover your set with the number of balls of fixed radius, and your set is here. Now by the triangular property, there will be a big ball containing all these finite number of balls.  And then there will be a big ball of radius 100 plus 1.  And therefore, this means that it is bounded. Bounded means exactly that it is contained in a big ball. So the proof of this implication is easy. And indeed, the word totally says totally stronger than without totally. But the converse is not always true. It's not always true. And therefore, you cannot just prove completeness and boundedness in order to have compactness. You have to prove completeness and total boundedness in order to prove equivalence to compactness. Now, do you see why converse is not always true? Well, but now you are using a lot of things, the existence for an orthonormal base, for instance. I mean, only one point is one. The rest is the many. They are called the many. Yes. And we take each of them some small ethylene over which is joined each other. So let us try to see if I have understood. So you take L2.  You take the ball bi. You take? 0, centered at 0, radius equal to 1. The ball bi is centered at the origin with the radius 1. Yes. Yes. Then? Yes. 1, 0, 0, 0. Oh. You take one. So points in this ball. Yes. Take the closure. 1, 0. Yes, closed. OK. This 1, 0, 0, 1, 0, 0, 0. OK. These are elements of this ball. So let me call ei. Ei at this point. Where were? So ei is this point. And then I take the neighborhood 1, third centered ei. These are disjoint. Yes. They cannot cover all b0. And b0 is bounded. Well, there is another maybe example that we can do. Let me see whether it works. Take on r, the distance between two points, to real numbers, is equal to 1. If x is different from y and 0, if x is equal to y. Take the following distance. So all points of r have distance 1. If x is different from 0. And 0 if x is equal to 0. So all points that are distance 1 from the origin. Discrete space. So this is bounded space. But if I take a ball with less than 1, I cannot cover it with a finite number of balls. More or less the same. OK. Well, he claims that this is a bounded set. This take is x1h. Maybe it is better to take the discrete space. So the ball covers the sequences so that the sequences is bounded. And if it is any number less than 1, then we only cover that if we need to choose the ball that covers, finite ball that covers the other sequences, it cannot be because the sequences is finite. And not definitively constant also. Well, I think that these kind of examples can be constructed simply taking this degenerate distance in which all points have distance 1, unless they are equal essentially from the origin. So you really don't need an infinite dimensional space. You just can work this in the real, but just take a distance very, very, very singular. Now, of course, if we prove that C is compact, then this will imply that it's interior is empty. And so we will have another proof of what we have said at the beginning of the lecture. Beginning of the lecture was have you proved that this is an empty interior? Yes, that's right. So this will give another proof of the fact that the interior of C is empty. Now, when you said epsilon net, maybe you were referring to total boundedness, I think, something like that. So let us try to do the proof. Because if you want to work, I mean, we have something. I can take a sequence here. And I know there is a subsequence such that all components converge. This is the Ticanov type argument that I made before. So what would remain is that to prove, actually, that subsequence not only converges in the sense of product topology, but it converges also in the sense of L2. What I'm saying is that we have already half of the proof without epsilon nets. So maybe you can try by yourself. Take a sequence here. We know there is a subsequence such that all components converge to a point of a 2, diagonal argument. This is not enough. That subsequence must converge also in L2. Try to prove this by yourself. Now we do another proof. OK. So, well, we have to show essentially that C is totally bounded. Because we already know C is closed. L2 is complete. Therefore, C is also complete. OK. So C L2 is complete. C is closed. C is contained in L2. This we know. And therefore, C is complete. So what remains to prove to show C is totally bound. Bound. OK, so fix epsilon positive. Take n bar, a number n bar in n such that the tail k from n bar plus 1 and infinity 1 over k squared say less epsilon over epsilon over 2. I don't know. Epsilon over 2, epsilon over 2. OK. Now I fix an n bar. I don't claim that this n bar is then n epsilon I'm looking for, for the total boundedness. I fix just an n bar. I can always do this. OK. Now let me define a sort of projection of C on a finite dimensional space. So I take the following projection of C, all point of C having components after n bar, which are 0. So xk equal to 0 for any k larger than n bar. So I am projecting the Hilbert cube on a finite dimensional space. This is contained in a, this is contained. I mean, it's a subset. So a can be considered as a subset of r and bar. And a is also, therefore, a, just the product from 1 to n bar or 0, 1 over k is totally bounded because it is compact in this finite dimensional space. OK. So a is totally bounded. So I can apply now the definition of total boundedness. So any point z in a belongs to at least one of the balls of radius. So let me just take. So there exists points a1, a now, and epsilon in a. So there exists an epsilon, comma, there exists point in a.  So maybe when I wrote total boundedness, I don't remember if I took the point x1, xn epsilon in c or in x, maybe in c. OK. So any point z belongs to one of the balls which is equivalent to say that the minimum of the distance the minimum j1 and epsilon, the distance in l2, normal over 2, z minus aj is less than epsilon over 2 for any z in a. So this is equivalent to say that there is at least one index j such that z is in the ball centered at aj of radius epsilon over 2, this minimum way of writing. This is true for any z. So actually now I have a candidate, an epsilon, corresponding to the previous choice of epsilon. So now the idea is to show that this and epsilon is OK. So OK, now take any x in the cube and write x bar as the projection of x, x1, xn bar, 0, 0, 0. So this x bar is an element of a. Now I consider the distance from one of the aj's. And the distance now is just this plus the sum. Now any aj is in this finite dimensional space. Therefore the components of aj after n bar are all 0. Therefore here there is no aj. So there is just the sum from, let me use the index k from n bar plus 1 to infinity of xk square. OK, now this is square. Now any xk is an element of 0, 1 over k. So this is less than 1 over k square. And so this is less than bar plus 1 infinity 1 over k square. OK, so by our choice of n bar, this is less than or equal than epsilon over 2. Ah, sorry, there is a mistake here. So let me take the square here. So let me take this, it's of course the same. Maybe I should take epsilon to the square, it doesn't matter. OK, so now for any j, for any j, for any j, for any j from 1 to n epsilon, this is true for any j 1 n epsilon. So I now take the minimum over j. From this inequality I find that the minimum, this is a minimum because it's a finite number of indices. So the minimum for any epsilon, the minimum over j, x minus aj square is less than or equal than the minimum from 1 over n epsilon x bar minus aj square plus the tail plus epsilon over 2. But I know from the finite dimensional argument that my projection, the finite dimensional space, is less than epsilon over 2. So this is actually less than or equal than epsilon. So that should be an epsilon square, I'm sorry, because then I want bolts of radius epsilon. So if you adjust everything by putting here epsilon square, epsilon square, epsilon square, epsilon square, and epsilon square. OK, sorry for that. So what I've shown is the following. So conclusion is, the conclusion is, min j from 1 to n epsilon x minus aj square less than or equal than epsilon square for any x in the cube, which is exactly like to say that I'm covering with a finite given epsilon, I'm covering with a finite number of bolts of radius epsilon centered at these projections, a1, an epsilon. So this shows, hence, given epsilon, I found n epsilon and points in c, actually in a, but a is contained in c, obviously, such that this, which is equivalent to say that it is totally bounded. And you see, the proof is not easy. There is always the idea, so that the difficult part of function in this kind of argument is that you always have the idea that infinite dimensions count less and less and less. As the dimension increases, they are not so important. So what we are doing here is that we are covering a big part of c, finite dimensional, using totally bounded. And then we have something which remains, something which remains that, however, is not so important. It's the tail. So the point is to make this argument concrete. It is never so easy to do this. In this case, it works. And so there is at least one proof that c is compact. Maybe you could try by yourself another proof using sequences. We already know that the sequence converges in the product topology. Then it remains to prove that that converging subsequence actually converges also in the stronger topology of L2.


Income Tax 2021 2022 Inventories. Get ready to get refunds to the max, dive into Income Tax 2021 2022. Most of this information can be found in Publication 334, Tax Guide for Small Business 2021. Looking at the income tax formula, the top line, the income line, noting that it would be supplemented with another schedule, basically an income statement, income and expenses. The expenses basically being the deductions, the net then rolling into the top line of the income tax formula, as well as the first page of the 1040, the tax return. We would generally have a schedule C rolling into the schedule one, rolling into page one of the form 1040, line number eight. Here's our schedule C profit and loss, basically an income statement. We're now looking into the inventories. Remember that inventories we mentioned a couple of times that if you're looking at the methods, cash method, accrual method, some kind of hybrid between the two methods, a combination method as they call it, then the inventories is one of those things that you might think would push you over to be using an accrual method, at least for those inventory type of transactions, because inventory itself is an accrual account normally. In other words, when you purchase inventory, even if paying cash for it, you usually put it on the books as an asset, not as an expense, which is what you would do on a cash basis, and then you expense it in the form of cost of goods sold when you use the inventory, selling the inventory, in order to generate revenue. That's an accrual thing, and that's why we typically think that if you have the inventory, then you might be choosing that accrual method unless you choose an exception or choose some reason not to. But remember, whatever method you put on the tax return, you want it to kind of mirror what you're doing, of course, on the bookkeeping side, one, because that's basically what the tax code wants, and two, because you don't want two different methods to be trying to keep track of as you're entering the data. So generally, if you produce, purchase, or sell merchandise in your business, you must keep an inventory and use an accrual method for purchase and sales of merchandise. That's the general rule, but exception here for small business taxpayers. If you are a small business taxpayer, you can choose not to keep an inventory, but you must still use a method of accounting for inventory that clearly reflects income. So you could have certain situations then where you have the inventory, but you might basically still be kind of on a cash method. So you can imagine situations where possibly the inventory is not, you're not accumulating a lot of inventory, like you don't have like a stockpile of inventory, for example, and so you're like purchasing something and then selling it pretty close in time frames, for example, maybe in a situation where you can make an argument and say, well then I'd still kind of like to use a cash method, even though technically I have inventory involved because I don't want to be tracking inventory under the normal inventory tracking method because I have kind of a just in time type of system. I'm not holding on to the inventory that much. So once again, exception, if you are a small business taxpayer, you can choose not to keep an inventory, but you must still use a method of accounting for inventory that clearly reflects income. If you choose not to keep an inventory, you won't be treated as failing to clearly reflect income. If your method of accounting for inventory treats inventory as non-incidental material or supplies. So basically you're saying it's not as big a significant factor in that instance, and therefore the IRS isn't gonna be like worried about it in that point in time to be on an accrual method, for example, or confirms your financial accounting treatment of inventory. And so you could see what the IRS would kind of be worried about here. Obviously if you had a lot of inventory, you can kind of manipulate a cash method with inventory because you could, for example, purchase a bunch of inventory that you're gonna sell by the end of the year and then record it as a cash expense at that point in time, even though you haven't used it in order to generate revenue, you're gonna use it to generate revenue when you sell it kind of in a future point in time. So by stockpiling increase in the supply of inventory, for example, on a cash method, you can kind of try to distort your income. And if that becomes significant, of course the IRS is concerned that people will manipulate their income in that instance. And so if, however, you choose to keep an inventory, you must generally use an accrual method of accounting and value the inventory each year to determine your cost of goods sold in part three of Schedule C. So you might say my inventory is small and so maybe I don't do that, but again, you wanna be careful because if your inventory gets larger and significant to the point where you're tracking it in your bookkeeping, you would like your bookkeeping to mirror what's on the tax return and you would have to then request that the IRS allow you to move to switch methods, which they would probably grant switching from a cash method to an accrual method, but it's still kind of a pain to go through that process. So pick a method that kind of mirrors with your bookkeeping, of course, keeping the exceptions in mind, but also maybe thinking out into the future what your bookkeeping might be in the future, noting that you can change the methods and that would probably be a reasonable reason to change from a cash to an accrual method, but still kind of a pain. Small Business Taxpayer. You qualify as a small business taxpayer if you A, have average annual grocery seats of 25 million or less, or the three prior tax years, and one more time, A, have average annual grocery seats of 26 million or less for the three prior tax years and B are not a tax shelter or defined in section 448D3. If your business has not been in existence for all of the three tax year period used in figuring average grocery seats, base your average on the period it has existed. So obviously they're looking at the average income basically of the 26 million, and then you can have this special situation saying, well, what if I've only been in business for two years or one and a half years? Well, then you figure the average generally on whatever the years that you have been in business. And if your business has a predecessor entity include the gross income of the predecessor entity from the three year tax period when figuring average grocery seats. So you might say, well, what if I changed my business entity from last entity to this entity? Well, then you should probably take the prior entity for the last three years would be the fairest thing to do. So if your business or predecessor entity had a short tax year for any of the three year period, annualize your business grocery seats for the short tax year that are part of the three year period. So in other words, what if I've only been in business for a year and a half? Can I take the, you know, the 50 million I made this year and add it to the 25 million I made last year or the 15 million I made last year, but it was only for six months or the other way around, you know, this last whatever. You can see half the income and then take the average. No, you can't do that because that would mean your income would be low your average would be that would be a way to manipulate the average because you didn't have a full year than that you're calculating the average. So you have to account for the fact that you only made revenue for a partial year and that instance to get an accurate number of the average for the test to be fair. Treating inventory as non-incidental materials or supplies. So remember, if it was incidental, that might be one of those kinds of exceptions that you can basically have kind of like inventory that would be an exception. So you can still use it a cash method. This is non-incidental. So treating inventory as non-incidental material or supplies, if you account for inventory as material and supplies that are not incidental, you deduct the amounts paid or incurred to acquire or produce the inventoryable items treated as non-incidental materials and supplies in the year in which they are first used or consumed in operation. So we're talking about the consumption or use of them at this point in time, they're not claiming the purchase point in time. Inventory treated as non-incidental materials and supplies is used or consumed in your business in the year you provide the inventory to your customers. So that's when you give it to the customers, that's when you actually, that's an accrual kind of component. So financial accounting treatment of inventories, your financial accounting treatment of inventories is determined with regard to the method of accounting you use in your applicable financial statement as defined in section 451B3 or if you do not have an applicable financial statement with regards to the method of accounting you use in your books and records that have been prepared in accordance with your accounting procedures. So generally you want your tax treatment to be lining up to your bookkeeping treatment which if you're required to make financial statements your financial statement treatment. So changing your method of accounting for inventory. So what if I need to change things? What if I marks it off as a cash method and I need to change it to an accrual method or something like that? Well, you can't do it without permission. That's for sure. So if you want to change your method of accounting for inventory, you must file form 3115. You can see change in accounting method later for that information. So items included in inventory, if you are required to account for inventories include the following items when accounting for your inventory. Merchandise or stock in trade, raw materials, work in process, finished goods, supplies that physically become a part of the item intended for sale.  But if you're making something then it gets a little bit more complex because you got the raw materials like if you make a guitar, you buy the wood or something like that. And then it's in process when I'm gluing stuff together, I'm gluing stuff together and I got glue and stuff that's like small materials which might be like overhead or something like that. And then you've got the finished product which is my finished guitar. And then I'm gonna sell that guitar all the stuff being included in it, the raw materials, which includes things like the labor, I mean the raw materials, the wood and so on and so forth. And then I include stuff like the incidental glue and stuff, work in process, the overhead kind of stuff and the labor and then the finished products. So that gets a bit more complicated when you're making stuff. Valuing inventory, you must value your inventory at the beginning and end of each tax year to determine your cost of goods sold. So we want the good old, this is schedule C line 42, the good old cost of goods sold calculation, beginning inventory plus purchases or if we made the inventory, the goods that are meant gives us the goods available for sale minus the ending inventory. That's gonna give us then the cost of the goods that are sold. They want that calculation. So to determine the value of your inventory, you need a method for identifying the items in your inventory and a method for valuing these items. Inventory valuation rules cannot be the same for all kinds of businesses. The method you use to value your inventory must conform to generally accepted accounting principles for similar businesses and must be clearly reflect income. So obviously valuing income gets a little bit tricky because for example, if you're buying inventory and then selling it, then you get in a situation you might have old inventory and it's like, well, what if the inventory got old? If it goes below the purchase price value, then you might have to revalue it, the lower of the cost or market or something like that for the inventory. So you gotta use whatever general accounting principles are for your particular business with regards to the valuation of inventory. Your inventory practice must be consistent from year to year. Once again, consistency. That's a key principle in accounting and in life, I would argue, but we're talking about taxes here. So taxes too, even though the tax code hasn't been very consistent with their laws and regulations lately, but we strive for that consistency.


Hello everyone this is Ahmed Dan of AhmedDan.com you are watching one of my studio travel videos on the sovereign hill playlist and today i'll take you back in time 200 years ago sovereign hill in Ballarat a small city from the 1850s gold rush in Australia is preserved at us back in those days experienced time and life in this recreated gold township of the 1850s with me in this episode you see red coat soldiers they were the soldiers on the gold fields protecting gold wagons in transit from Ballarat to Melbourne they also held the police to enforce the law sovereign hill gold mine city located in the suburb of Ballarat and our drive from Melbourne there will be more videos coming up from the sovereign hill gold mine city thank you for watching forget about subscribing this channel forget about liking this video i'll see you shortly 14th second Somerset Shire rests upon the battlefield of Waterloo where this particular regiment was personally congratulated by the Iron Duke himself the Duke of Wellington


This episode of HCC 788 brought to you in part by the Dicast Enterprise. Join us each week as we discuss the sexual proclivities of Commander William T. Riker. The bravado comedy of Lieutenant War. And the adorable monkey shines up one Wesley the Sweater Crusher. Or maybe we'll just talk about the Golden Girls. Or hairstyles. Or cartoons. That's equally likely. We also like G.I. Joe. There, we tied that in nicely. Well done everyone on that. By all our play sets and toys. Cobra Commander 788 here. Back to your so great cop here. Hello G.I. Joe. I'll SWATERIZE IT! I'm excited! Ready. Ready. Ready. Silence kill. Hello everybody, good to Cobra Commander 788 here. It's time for another Battle Force 2000 review. And we are almost done with Battle Force 2000 Month. Can you believe we've made it this far? We have two more to go including this one. And for this review we are looking at the Eliminator four wheel drive vehicle. And the driver sold separately blocker. All of the Battle Force 2000 vehicle drivers were sold separately from the vehicles that were intended to drive. But we are reviewing them together. And all of the first wave of Battle Force 2000 vehicles had a special gimmick. You could remove pieces from them and then reassemble them to form the future fortress. With each of these Battle Force 2000 reviews we are adding a piece to the future fortress. So this time we will see how the Eliminator contributes to the fortress. Also with each of these Battle Force 2000 reviews we have a special guest. Joe Fan 82 will provide a preview of the upcoming Joe Con Exclusive Battle Force 2000 figures. Hey guys, stay tuned after HCC reviews the vintage blocker. Because I'll be looking at the modern version we'll be seeing at Joe Con this year. Thank you Joe Fan 82. Don't forget to check out his YouTube channel if you haven't already. Joe Fan 82 has been so helpful with these previews. I wonder if he could educate us about other things. Joe Fan, what would you say are the geopolitical implications of Britain's interest in the Rock of Gibraltar? Hey guys, stay tuned after HCC reviews the vintage blocker. Because I'll be looking at the modern version we'll be seeing at Joe Con this year. Fascinating. Another question for you. Do you really think the existence of the Higgs Boson was confirmed by the large Hadron Collider in 2013? Hey guys, stay tuned after HCC reviews the vintage blocker. Because I'll be looking at the modern version we'll be seeing at Joe Con this year. Ha ha, Will Wonders Never Cease. Let's go ahead and look at the toy. HCC 788 presents the Eliminator and Blocker. This is the 1987 Battle Force 2000 Eliminator four-wheel drive and the driver blocker. This vehicle and the driver were sold separately and they were available in 1987 and 1988 and were discontinued for the year 1989. Blocker was never sold with the Eliminator. All Battle Force 2000 figures were sold as carded figures, not packaged with their vehicles. In 1987 Blocker was sold as a single carded figure. In 1988 he was available in a two-pack with his Battle Force 2000 teammate Maverick. As you can see we have two figures here. That means we will be looking at a variant. Hasbro was clearly hoping Battle Force 2000 would have legs. They were expecting it to last beyond this first wave. There were a couple later Battle Force 2000 items released in 1989 after the first wave had been discontinued. In 1988 Destro's Iron Grenadiers were released with the Battle Force 2000 logo on the packaging. But Battle Force 2000 never faced off against the Iron Grenadiers in G.I. Joe Media. It's probably just as well. The Iron Grenadiers were elite troops. They were not Cobra cannon fodder. I don't think Battle Force 2000 would have fared well against them. We're going to look at the Eliminator now so we're going to set Blocker aside and look at him later. The Eliminator is Battle Force 2000's general purpose vehicle. It is in the same category as the vamp, the awe striker and the desert fox. Although it is slightly larger than the vamp and the awe striker, it's closer in size to the desert fox. Let's look at the parts and the features of the Eliminator starting here in front. We have this push bar. It is in black plastic and it does extend out pretty far. Then we have these little green guns and the blueprints call these deflector sound deflector pivoting laser guns. And I think the description is kind of funny because it uses the word deflector twice. They are green and they do pivot a little bit. On the starboard side of the vehicle we have what the blueprints call forward looking infrared or FLIR radar detected laser cannon. And they got something right. Forward looking infrared or FLIR is a real thing. It is a thermographic camera that sees infrared radiation. They're often used to help pilots navigate through night and fog. It is black. It does pivot a little bit more than 180 degrees. But it's really only useful for targets straight ahead or on the starboard side of the vehicle. It doesn't pivot around far enough to hit targets approaching from the port side. Next we have this roll cage in black plastic. It looks like it is hinged here in the front so you ought to be able to pop this up and swing it forward. But apparently not. Mine is firmly fixed in the body of the vehicle. Just bend the action figures knees and wedge him through the bars and into the driver's seat.  They just kind of rest on the sticks. But he is fully seated in the driver's seat and that gets to my problems with the roll cage. The driver has no protection at all. He's totally exposed to enemy fire. There's hardly any point in having an armored vehicle at all. And the driver's head sticks out above the roll cage. So if the vehicle rolls, the cage would give him no protection at all. It misses the point of even having a roll cage. Now let's get to the wheels and the wheels are a major feature and a major problem with this vehicle. They are plastic wheels with a hubcap design that's similar to what we have seen on the dominator and the marauder. But these are in blue and both the front and the back wheels turn. They ratchet in a right, a center and a left position and that's not bad. But the problem is the connections where the axles connect to the body of the vehicle are extremely weak and prone to breakage. And that has happened on my back wheels. So the back wheels no longer ratchet. They just kind of swing freely and that is a major problem. Because now the vehicle will not roll properly. The back wheels do not want to stay straight. This is an unfortunate case of inadequate engineering. When you turn the wheels, it puts quite a bit of pressure on those posts and the posts are too weak and they break. And if you have a broken post, the wheels don't align properly which reduces the functionality of the vehicle. This fire control station has a seat. It has a computer screen with a sticker. And it has a control stick here with some other instruments. The figure sits in straight legged and you can put the figure's hand on the control stick. The computer screen has a keyboard on it but it's impossible for the figure to use unless he can type with his feet. There is a stairway to the fire control station and I like that. There's a foot peg on the bottom step. Moving toward the rear of the vehicle, we have the main armament. We have this big green gun which the blueprints call a remote operated dual 20mm repeating auto load cannons. It is big and green. It can elevate and it can pivot all the way around on the turret. There are a couple foot pegs on the turret so you can place an action figure on there and usually that's a pretty cool thing. However, the gun itself only comes up about to the action figure's thigh level so it doesn't look like the figure would be operating the gun from that position. Finally, in the back we have a universal GI Joe tow hook. This entire back section with the fire control center and the gun turret are removable but I wanted to take one last look at the vehicle with that piece on because I do think it works quite well. It follows the lines of the vehicle perfectly so I think the eliminator looks good with that piece on. To remove that piece, just lift up to reveal another set of weapons underneath. The blueprints call this a removable tactical ground station and we will look more at this later but let's finish looking at the main vehicle. Removing the ground station reveals a lot. First of all, we still have that stairway but we have another foot peg on the top step. Swinging the vehicle around to the other side we have two additional foot pegs and a removable engine cover with some engine detail. We also have the lock unlock detail that we have seen on all other Battleforce 2000 vehicles. Most importantly we have this hidden missile rack. It lies flat in this recessed trench that runs down the center of the vehicle and it swings up on a couple hinged supports. You can swing this missile rack up and it holds in this about a 45 degree angle and you can see there is some additional molded detail under the missile rack so that's nice. The missile rack has six removable missiles. They are all the same. They peg in on a very shallow peg which means it can be a little bit difficult to get on and off. Unless I am missing something, the blueprints do not have a name for these missiles. They are just small generic green missiles. I like the idea of this missile rack more than I like the execution. I like the idea of this pop-up hidden weapon. I think an enemy would be surprised to find this vehicle has six hidden missiles on it. However, the way this is set up, the blast from the front missiles will hit the ones in the back. Now let's look at the removable tactical ground station. We've already seen what's on it. It has a seat and a control panel and a pivoting turret. But in order for it to work as a ground station, you have to pop out these four little feet. There is one on each corner and once you have those popped out, it will sit flat on the ground. That really is all there is to it. There is no other transformation. It just pops feet out so it will work as a unit separate from the main vehicle. This is also the eliminator's contribution to the future fortress. Now let's look at how the eliminator fits with the future fortress. We have already assembled the pieces from the dominator, the vector, the skysweeper and the marauder. Now the blueprints suggest that you place the eliminator's piece in this position and it just rests right there. Once again, it does not connect to anything. It simply rests near the other parts of the fortress. It must have been at this point when kids really caught on to the fact that they were being conned with this future fortress gimmick. There's only one piece to go and only one position for it. If you're disappointed at this point, it's very unlikely the fortress will redeem itself. Now let's look at Blocker. Blocker may take his name from American football offensive linemen, which are also sometimes called blockers. I have the full card back for Blocker, so let's take a brief look at that. We have the Battleforce 2000 logo and the GI Joe logo. We have a price sticker for Murphy's. This figure was $3.99 and frankly, I think that's a little steep. Unfortunately, a lot of the paper is ripped off the front here. But we have this artwork for Blocker and Blocker is firing his laser. But oddly, it seems he's not looking in this direction that he's firing. Slipping the card around to the other side, we have the cross cell and a partition for Battleforce 2000 here. We have our flag point and Blocker's file card. We will look at that later. Let's look at Blocker's accessory. He came with what the card contents called an XL-13 light refraction submachine laser. It is in silver plastic and it has a big loop. Submachine laser makes no sense, but it's not a bad looking accessory. There are worse Battleforce 2000 weapons. Let's look at the articulation on Blocker. He had the articulation that was standard by 1987. So he could turn his head from left to right and look up and down. He could swing his arm up at the shoulder and swivel at the shoulder all the way around. He had a hinge at the elbow so he could move at the elbow about 90 degrees and he had a swivel at the bicep so he could swivel his arm all the way around. The figure was held together with a rubber O-ring that looped around the inside. That allowed him to move at the torso a bit. The figure could move his legs apart about so far. He could bend his leg at the hip about 90 degrees and bend at the knee about 90 degrees. Let's take a look at the sculpt design and color of Blocker starting with his head. And on his head he has this non-removable helmet or hat. It is in dark gray with an orange camouflage pattern. And this looks like it's supposed to be a hat. It looks like the top of his head is covered in cloth. And I got to be honest, I thought originally this was a helmet, but it seems a little less cool as a hat. The face sculpt is adequate. I don't mind that at all. Here's where we have the variant. Some blocker figures have a clear visor attached to his hat. Other blocker figures do not. That visor can be removed with some effort, but I don't think it was intended to be removed. There's another odd variation that went along with this visor. The figures that had the visor also had a red paint application on the left wrist that was gray on the non-visor variant. I don't know which of these was released first, but I suspect it was the version with the visor because the extra piece and the paint application would have increased the cost of the figure. So I can see those features being cut for cost. On his chest he has nicely sculpted armor. He has what looks like canisters attached to silver shoulder straps. And I do like this look for Blocker. This ridged armor pattern reminds me of the Baroness, but I'm sure that's just a coincidence. On his arms he has gray sleeves, and on his right arm he has an orange band. He has orange bracing over gray gloves. Now this orange color isn't too bad. It's kind of muted. It's not a bright orange. On his waist he has a belt that is unpainted. A spot of silver paint might have been nice there. He has more of that orange camouflage pattern. On his legs he has gray trousers and that same dark gray color. And he has more of that orange camouflage pattern. He has large pockets on his thighs. You would expect these to obstruct the figure when he is in the vehicle, but they don't really. He has gray boots with orange shin guards. And on the right side he has a silver knife. Odd placement for that knife right in the middle of his shin. This gray and orange camouflage pattern may seem strange to you, but that's because you may not know that in the future lots of things will be gray and orange. In the far distant future, all the way in the year 2000, plant life will change to this gray and orange color due to global warming. So Blocker will blend right in. Right, Joe Van? Hey guys, stay tuned after HCC reviews the vintage Blocker because I'll be looking at the modern version we'll be seeing at Joe Con this year. Truer words have never been spoken. Let's take a look at Blocker's file card. It has his faction as Battle Force 2000 and GI Joe and a portrait of Blocker here. It says his codename is Blocker and he is the Eliminator Driver. His file name is David B. McCarthy. His primary military specialty is Mechanized Recon. His secondary military specialty is Special Services. Special Services refers to the entertainment branch of the armed forces. His birthplace is Boston, Massachusetts. He is a New Englander. In the case that the file card was not written by Larry Hama, the writer of the GI Joe comic book, because Hasbro employees liked to make all of their characters New Englanders because Hasbro headquarters is located in Rhode Island. However, this file card does seem to have Larry Hama's fingerprints on it. His grade is E5 Sergeant. This top paragraph says Blocker drove a taxi in Boston for three years. After seven armed robberies, three backseat bursts and one near fatal plunge through a collapsing roadway, he decided it was safer to drive something armor-plated with the ability to shoot back. I'm not 100% certain what this collapsing roadway is a reference to. It may be a reference to an accident on the Tobin Bridge in Boston, where a gravel truck crashed into a support, collapsing the upper deck and killing the driver. But that was in 1973 and Blocker was unlikely to be a taxi driver at that time. There was a collapse on the so-called Big Dig, that was in 2009, long after this file card was written. It doesn't paint a very good picture of Boston, but this card was written by a New Yorker, so there may have been some rivalry there. He is qualified to drive every wheeled vehicle in the Army inventory and was the test driver for half a dozen prototype vehicles at the Aberdeen Proving Grounds. As noted on Knockdown's file card, Aberdeen Proving Ground is a U.S. Army Ordnance Testing Facility in Aberdeen, Maryland. This bottom paragraph has a quote. It says, I have seen him go over into a complete role at high speed, recover and keep on going. He takes on the slalom minimally at 85 miles per hour and doesn't tip nary a cone. The skid pad, don't even ask. All that and he does it with vehicles built to military specs. Amazing! So he drives really fast. We get the same idea from a lot of other GI Joe vehicle driver file cards. He doesn't have any description of his special services specialty. I would have expected that. Now I'll turn it over to Joe Fan 82 for a preview of the modern blocker figure. Thanks, HCC. Here is the new blocker and he's ready to go hunting. He's got the dark brown uniform from the original figure with a red-orange camo pattern. And he's ready to go hunting with some rabbits with his hunting hat, which is not removable. In addition to the new head sculpt, he comes with a removable vest, tactical shotgun, and his original XL-13 laser weapon. This looks like a pretty decent attempt at updating and modernizing the original design. Personally, I would have preferred to see the uniform and camo color more closely match the rest of the team. But that's how the vintage figure looked, which is what they're going with here. That's it for this preview of modern blocker. Back to you, HCC. Helpful as always, Joe Fan 82. If you've liked these previews, he does this kind of thing all the time on his own channel, so check out Joe Fan 82 on YouTube. Let's talk about blocker and the eliminator's appearances in GI Joe Media. Battle Force 2000 did not appear in the cartoon series. They were only animated for commercials. The Sunbow era of the animated series ended with the 1987 GI Joe animated movie. Battle Force 2000 came out late in 1987, so they missed out on that. I have looked through the Battle Force 2000 commercials, and they did not see blocker, so blocker may not have been animated even for the commercials. Blocker did appear in the GI Joe comic book series whenever the rest of the Battle Force 2000 team appeared. He didn't have any noteworthy solo story arcs. Of course, the character died, along with most of the Battle Force 2000 team in issue number 113. Looking at the eliminator and blocker overall, the figure and the vehicle are evenly matched in that they are both slightly above average and not quite top tier. The eliminator is a good general purpose vehicle and Battle Force 2000 needed one. I like the versatility. The hidden missile rack is not a bad idea, but it could have been executed better.  I just feel like something more could have been done there. The driver is exposed in a tiny roll cage, and since his head pops up above the roll cage, it provides no protection at all. And that's really the case with a lot of these Battle Force 2000 vehicles. You have armored vehicles that do nothing to protect the driver. My biggest problem with the eliminator is the front and rear axles. The connection points are way too weak. I have a couple eliminators, and they are both broken in the same place. A piece that was going to have that much pressure put on it should have been engineered better. I like blocker, not as much as Dodger, but I would say blocker is probably my second favorite Battle Force 2000 figure. I think in a previous video he was my favorite, but after looking at Dodger more closely, I have reconsidered that. A blocker is still okay though. I like the colors on blocker. I can even tolerate the orange. It's not a bright orange, so I can live with it. And I think I prefer the variant of blocker with the visor, because it makes his hat look a little more like a helmet. For all of Battle Force 2000's fault, I will give it one thing. It was one of the few GI Joe sub teams that was all new. And that's why it's surprising that Battle Force 2000 wasn't more popular. Tiger Force is remembered more fondly than Battle Force 2000, and all Tiger Force did was repaint and reissue old toys. But Tiger Force reissued toys that kids liked. I mean, we liked the original vehicles and figures, so I guess we didn't mind getting reissued. But the all new Battle Force 2000 did not succeed. But I think this goes to show that no matter how much marketing effort is put into something, if the kids don't like it, they're not going to buy it. That was my review of the Eliminator and Blocker. One more review for Battle Force 2000 Month. It is coming up on Sunday. Thank you again, Joefan82 for imparting to us your wisdom. Only one more to go, my friend. One more to go. Just a reminder, after Battle Force 2000 Month, I am taking a week off. There will not be a new review the following week. But the week after that, I am coming back with a new Returns Choice review, so watch for that. Please like this video on YouTube, subscribe to the YouTube channel, share this video, like me on Facebook, follow me on Twitter, support the channel on Patreon, and check out HCC788.com Thank you for sticking with me for Battle Force 2000 Month. One more to go. I will see you in just a few days. To protect America, a top secret team creates the battle vehicles of the future. We're in real trouble here, Hawk! Call it in! Battle Force 2000! Battle Force 2000! Secret Force of G.I. Joe! They're splitting in two! And reforming! Battle Force 2000! They can take on any pose of G.I. Joe! Well, Battle Force 2000 finally defeated...


For many people, their first experience of coding and data science is with the application SPSS. Now, I think of SPSS and the first thing that comes to my mind is sort of life in the ivory tower, though this looks more like, you know, Harry Potter. But if you think about it, the package name SPSS comes from statistical package for the social sciences. Although, if you ask IBM about it now, they'll act like it doesn't stand for anything. But it has its background in social science research, which is generally academic and truthfully, I'm a social psychologist and that's where I first learned how to use SPSS. But let's take a quick look at their web page, IBM.com slash SPSS. If you type that in, that'll just be a alias that'll take you to IBM's main web page. Now, IBM didn't create SPSS, but they bought it around version 16. And it was very briefly known as PASW predictive analytics software that only lasted briefly. And now it's back to SPSS, which is where it's been for a long time. SPSS is a desktop program. It's pretty big. It does a lot of things. It's very powerful. It's used in a lot of academic research. It's also used in a lot of business consulting management and even some medical research. And the thing about SPSS is it looks like a spreadsheet, but it has drop down menus to make your life a little bit easier compared to some of the programming languages that you can use.  Otherwise, SPSS costs a lot of money. But if you have it one way or the other, when you open it up, this is what it's going to look like. I'm showing SPSS version 22. Now it's currently on 24. And the thing about SPSS versioning is in any other software package, these would be point updates. So I sort of feel like we should be on 17.3, as opposed to 22 or 24, because the variations are so small that anything you learn from the earlier ones, it's going to work in the later ones. And there's a lot of backwards and forwards compatibility. So I'd almost say that this one, the version you have practically doesn't matter. You get this little welcome splash screen. And if you don't want to see it anymore, you can get rid of it. I'm just gonna hit cancel here. And this is our main interface looks a lot like a spreadsheet. The difference is you have a separate pane for looking at variable information. And then you have separate windows for output and then an optional one for something called syntax. But let me show you how this works by first opening up a data set. SPSS has a lot of sample data sets in them. But they're not easy to get to and they're really well hidden. On my Mac, for instance, let me go to where they are. In my Mac, I go to the finder, I have to go Mac to applications, to the folder IBM to SPSS to statistics to 22 the version number to samples, then I have to say I want the ones that are in English. And then it brings them up. The dot SAV files are the actual data files. There are different kinds in here. So dot SPS is a different kind of file. And then we have a different one about planning analysis. So there are versions of it. I'm going to open up a file here called market values dot SAV, the small data set in SPSS format. And if you don't have that, you can open up something else, it really doesn't matter for now. By the way, in case you haven't noticed, SPSS tends to be really, really slow when it opens. It also, despite it being a version 24 tends to be kind of buggy and crashes. And so when you work with SPSS, you want to get in the habit of saving your work constantly, and also being patient when it's time to open the program. So here's a data set that just shows addresses and house values for and square feet for some information. This I don't even know if this is real information, it looks, it looks artificial to me. But SPSS lets you do point and click analyses, which is unusual for a lot of things. So I'm going to come up here.  So I simply click values, put that right there and I'll put a normal curve on top of it and hit okay. And then it's going to open up a new window and it opened up microscopic version of it here. So I'm going to make that bigger. This is the output window. And so this is a separate window. And it has a navigation pane here on the side. It tells me where the data came from. And it saves the command here. And then, you know, there's my default histogram. And so we see most of the houses were right around $125,000. And then they went up to at least 400,000. I have a mean of 256,000, the standard deviation of about 80,000. And there's 94 houses in the data set. Fine, that's great. The other thing I can do is if I want to do some analyses, let me go back to the data just for a moment. For instance, I can come here to analyze. And I can do descriptives, I'm actually going to do, excuse me, I'm going to do one here called explore. And I'll take the purchase price, and I'll put it right here.  And it goes back to the output window. Once again, made it tiny. And so now you see beneath my chart, I now have a table and I've got a bunch of information to stem and leaf plot and I've got a box plot to great way of checking for outliers. And so this is a really convenient way to save things, you can export this information as images, you can export the entire file as an HTML, you can do it as a PDF or PowerPoint. There's a lot of options here, and you can customize everything that's on here. Now, I just want to show you one more thing that makes your life so much easier in SPSS. You see right here that it's putting down these commands is actually saying graph and then histogram normal equals value. And then down here, we've got this little command right here. Most people don't know how to save their work in SPSS and they kind of just have to do it over again every time. But there's a very simple way to do this.  I'm going to go to new syntax. And this is just a blank window. That's a programming window, it's for saving code. And let me go back to my analysis. I did a moment ago. I'll go back to analyze, I can still get it right here. And descriptives and Explorer. And my information is still there. And what happens here is even though I set it up with drop down menus and point of click, if I do this thing paste, then what it does is it takes the code that creates that command, and it saves it to the syntax window. And this is just a text file. It saves it as dot SPS. But it's a text file that can be opened in anything. And what's beautiful about this is it's really easy to copy and paste. And you can even take this into like word and do find and replace on it. And it's really easy to replicate the analyses. And so for me, SPSS is a good program. But until you use syntax, you don't know the true power of it. And it makes your life so much easier as a way of operating it. And yeah, this is my extremely brief introduction to SPSS. All I want to say is that it's a very common program, kind of looks like a spreadsheet, but it gives you a lot more power and options. And you can use both drop down menus and text based syntax commands as well to automate your work and make it easier to replicate it in the future.


Organic photodetectors, OPDs, have become increasingly popular due to their unique features such as tunability, cost-effectiveness, flexibility, and ease of fabrication. Recent developments in device architecture and operation mechanisms have focused on improving performance and applications such as spectrally selective detection, wearable devices, and integrated optoelectronics. Future prospects include addressing research challenges and exploring new opportunities for OPDs. This article was authored by Haoran, Gingdechin, Yan Qingli, and others.


Most educated people in this country, especially if they've graduated from our elite universities, like Stanford, as I did, like Yale or Harvard, will not know these things. Not only will they not know about the evils of communism, they will have been taught that it is a system that is admirable in many respects and that the only reason it's failed is that it has never been properly instituted. And so they fall again and again for this deceit, this deception that man himself can create heaven on earth. Man, with the help of God, can create a society where evils are restricted, but not totally eliminated.