Okay, so we're going to start looking at, we're going to look once again at cyclic codes a little bit differently from a, maybe slightly different perspective. Okay, so let me first, so first thing I'll do is I'll just take measure to binary cyclic codes. Then maybe let one will go non-binary outside. Let's give some part of the presentation simple. I'll do this. So what's the cyclic code? If you look at the definition, a linear code, a cyclic code. The following condition is true. If you have C1, C2, Cn belonging to the code, that implies. So you can do it in so many different ways. I'll do it when I'm so glad to look at it. Okay, so this is the definition of a cyclic code. So basically this is from here to here you go by a cyclic. Okay, so that makes the code cyclic. So I have presented that there's only one cyclic shift. But even if I take this code word and do a cyclic shift several times, say some 10 times or 20 times, I'll still get another code word. Because I can do it one step at a time and then I'll get. Okay, also I can do a cyclic left shift. Okay, what happens if I do a cyclic left shift? A cyclic left shift is the same as n minus 1 right shift. So that is also covered. So everything is covered in this definition. So one very easy way to construct cyclic codes is the following idea. So let's say something of a naive construction. So this is not the way most people picture cyclic codes. But one way of doing the construction is to simply say, I'll take a generator matrix. I'll take a first row, let's say the first row is G1. Okay, second row will be, so I need notation here. So let me just write R of G1. Okay, so I'll use this notation here. So C is this, I'll call this as R of C. Okay, just to show that this is a cyclic right shift. So R of G1, R squared G1, what is R squared G1? I rotated cyclic codes twice. And then all the way down to n minus 1. Okay, so I could take my generator matrix to be at this top. No, no, no, I want to do n minus 1. I know there's some k and all in all. I'll come to it later. Okay, so I'll take an arbitrary resource for G1. I'm not set any property for it. Some arbitrary resource for G1. And then keep cyclically shifting it to the right. Still I exhaust all the possible cyclic shifts. Okay, so I'll put n here. But sometimes you may not need all n. If you don't need all n to exhaust all the possible cyclic shifts, you can stop some other point. But otherwise you'll have to do all the n. Okay, so the first claim is this will always generate the cyclic code. Okay, so that is not very hard to prove. Okay, so the code generated by this will be cyclic. So if you have one code word, it would have been generated by some message multiplying G. If you take a cyclic shift of that, right, all you have to do is cyclically shift the message also and you'll get the code word. Okay, so this is obviously a cyclic code. Okay, so think about that a little bit if you're confused. Basically you have some message, both set of messages multiplying this. Okay, so if you rotate a code word, if you click shift by 1, all I have to do is shift my message bits by 1 and I'll get the corresponding shift in every place. Okay, so this is obviously cyclic. It's also obviously linear. So what is the difficulty here? What do you think is going to be a bit more painful to compute? Sorry, that is fine. But even before that, what are the parameters of the code that you're interested in? Lock length is fine. Okay, so let's say then what is the next parameter interested in? Dimension, right? So how do you do the dimension? How to find the rank. Okay, and the rank might be something. You do something less than that. Usually these things will be, rank will be definitely less than that. Okay, I'm going to do cyclic shift. There will be some linear dependency somewhere. Okay, so what about k? It's not clear that you have a good handle on k. Okay, and it's also a very painful construction. If you want some guaranteed minimum distance or anything like that, you don't know how to do it. There are so many other things which you don't know about this code. You just know it's cyclic. But then, any advantage you hope for because of the cyclic nature is not really coming through in this construction. It's like, there's a construction, a generator matrix. You might as well do a non-cyclic one. What's the point in doing a cyclic if you're not using any major structure in it? Okay, so this is definitely a valid construction, but it doesn't tell you anything more about dimension, minimum distance, etc. Okay, so what's nicer is to view these cyclic codes in this ring of polynomials, modulates, parent plus one like we did in class, and then think of ideals and polynomials and all that, and that gives you much nicer construction. It also tells you a lot more about k. Okay, so it turns out if you fix an n, you don't have cyclic codes for all possible k. So there are only some k for which cyclic codes are possible. So these are highly non-trivial results. If you just start with this construction, how do you go about proving something like that? You may not even be able to convince anybody of that. Okay, on the other hand, the way we do it, we'll be able to quickly find out for what k will n, k, cyclic code exist. Okay, so how do you think about constructing it? What are the easy ways of doing it, etc. All that will be clear. Okay, so while this is a very intuitive and simple construction, it's not very informative. What we will do is a little bit more complicated, but it will give you all the information you want. Okay, so what is this idea of ring? So the first idea is to use this ring. So I'll denote our Rn. Okay, so basically Rn contains n-ary polynomials of degree less than or equal to n minus 1. Okay, so this is the binary polynomial of degree less than or equal to n minus 1. Addition and multiplication. I'm going to do modulo x bar n plus 1. Okay, so that's the, that's this ring Rn. Okay, so addition being modulo x bar n plus 1 is okay. I mean it doesn't really affect anything. But multiplication modulo x bar n plus 1 really will affect the multiplication. Okay, so it's not the normal polynomial multiplication. A lot of things will change. Anything that crosses x bar n will fold back. Okay, so one nice thing about this ring is, a cyclic right shift is represented by multiplication by, so that is the nicest thing about this ring. And that's because this ring also represents the binary vector space, n-dimensional vector space. There's nothing, there's no problem there. It definitely represents the n-dimensional binary vector space. How do I think of it as the n-dimensional binary vector space? Yeah, simply take the coefficients as 0 or 1 that becomes n-dimensional binary vector space. I don't have to worry about polynomial. One nice thing about moving to polynomials is that, you have a nice kind of mathematical description for what a cyclic right shift is. Okay, so it's basically a multiplication by x in Rn. Okay, so that's the first idea. The next idea that we saw, which is again very useful, is what happens to a cyclic code. A cyclic code becomes one ideal one other. So I defined the ideals for you, so let me define that once again here. An ideal of Rn of any ring, okay, so ideal of ring R, okay? So a subset, okay, so first of all it has to be a subspace. Let me say a subspace said to be an ideal. Okay, so let me be careful about subspace here. So a subset over a path is an ideal for following conditions as satisfied. First condition i is an additive subgroup. Okay, what do we mean by additive subgroup? Okay, ring with respect to its addition operation is a group. Okay, it's an abelian group and this ideal that we look at, the subset that we look at should be a subgroup of that additive subgroup. Okay, so in particular in Rn, in Rn the binary vector space itself is the additive group. Okay, the additive group is the binary vector space. So I can think of the subgroup as a subspace. Okay, in Rn I can say i has to be a subspace. So basically if you want an ideal in Rn, it should first of all be a linear code. Only linear codes are candidates for ideas. Okay, so after that there is another definition which is again very important which will bring in the multiplication. Okay, so if you have some o-effects on ab and b-effects on i, then other clouds, o-effects times b-effects along square. Okay, so if you take any polynomial in the ideal and any other polynomial anywhere in the ring, multiply the two together, it should still be a guide. Okay, so what's the definition of an ideal? And it doesn't seem very obvious why an ideal should be a cyclic code. And in Rn it's quite easy. Okay, so in Rn, so if you go to Rn, one will imply i is a linear code, two is what? Okay, so if you take any polynomial in Rn, okay, so I have to a-effects here, so I think I should not put a-effects here, you know. I don't know what these rings are. Okay, it's just ab. Just ab, right? So in two, everything is a polynomial. And in Rn everything is a polynomial. Okay, so what does two mean? Okay, so you have a polynomial in the ideal. Okay, and you can take any polynomial in R. Okay, so in particular I'll choose a basis set of polynomials. Once I choose a basis set of polynomials, since I know already i is a linear block code, I can add up all of them, right? It's closed under addition also. So I can add up all of them and replace this arbitrary element of R with only the basis elements. Okay, so if I have the basis polynomials, satisfying this property, that's enough. Okay, so two basically will reduce you. dfx and i implies x power i times dfx is an i for, well, i equals 1 to n minus 1. Okay, i equals 0 is kind of trivial, dfx is again written, right? So it's enough if you have this property, h power i times dfx should also be in the ideal. That's enough. So once you have that, the entire prove is satisfied. Yes, because any arbitrary polynomial in R, Rn can be written as summation ai x power i. So that's good enough. Okay, so it works out. So that's the main idea. So you notice that this is basically nothing but cyclic code. Okay, so this makes i equal to 64. Okay, so it's quite easy that way. You want to spell too hard to show these properties. So an ideal in Rn is basically a cyclic code that will help you. Okay, so let's take a couple of examples. Okay, so I'm going to start this. I'm sorry. Yeah, actually xbx is also enough. Okay, so, yeah, I mean, I think it's also easy to see why xbx is enough. So if xbx is an i, then you use the same property over and over again. You get x power i dfx. Right? I don't have to. I put this out explicitly just to say xx times dfx is also enough. Because when you use the same property over and over again, you'll get x power i dfx. Okay, so let's look at some, let's go up 4. Okay, so up 4 is going to be 0, 1, x and plus x. Okay, so we have to start writing out all these parameters. It's a little bit, not a very boring answer. Okay, and then you'll have x power 3, x power 3 plus 1. I'm sorry, x power plus x. x power 3 plus x power 3 plus 1. Okay, so we'll have 16 polynomials in R4 and there they are. Okay, so suppose I ask you to identify ideals. Yes, I've said, okay, fine, ideals in Rn are cyclic codes. And I was promising you a simple way of coming up with cyclic codes. But then, where is R4? Can you identify the ideals? Can you stop any ideal very quickly?  Okay, fine. Okay, so R4 itself is an ideal. Anything else? For instance, 0 is an ideal. Okay, these two things are trivial. Okay, so we will not consider these two as non-trivial ideals. It's just trivial ideals. Anything else? Yeah, so you have, you need some method to generate the idea. I guess you already know the answer, so maybe you're giving the answer to me. But you would ideally need somewhere coming up with the ideas. Okay, so one, one very easy method to come up with ideals is to say, set up all multiples of one element of R4. Okay, so that's called the ideal generated by an element. Okay, so that's one way of generating an ideal. Ideal generated by some AFX4. Okay, so let me just mark the R4 in Rn. Okay, so this is basically, we will denote this with this notation. Okay, that is set up all AFX, BFX, such that BFX is one. Yeah, yeah, we'll come to that. I think there are some of these results last time, but anyway, I want to emphasize some of them here. Okay, so this is one way to generate an ideal. And you can show quickly that this will be an ideal. Okay, you have to prove that this is an ideal. How do you prove it? You have to verify the two properties. First thing you have to verify that it's an additive subgroup. You add any two multiples of AFX like this. You'll again get AFX times B1 of X to B2 of X. And that again belongs to R, so it's closed under addition. That's not a problem. What about multiplication? So multiply any element of this AFX times this guy with another element of R, what do I get? Once again, get another multiple of AFX, right? So it becomes AFX into BFX. You see that? Okay, so you can do other multiplication first. B1 of X times B2 of X, reduce it, model X bar n plus 1, you'll get some other polynomial. So both of them are the same. Okay, so eventually you get the element of this idea. Okay, so this is one very nice way of generating ideals in a ring. So in general, these ideals are called principal ideals. Okay, so these are called principal ideals. Okay, and let's see some examples of principal ideals in R4. So take the idea generated by one, what will I get? Yeah, again R4, right? Is it clear? Does it seem okay? Ideal generated by one multiple of R4. What about any other element? Can I get ideal generated by any other element to be R4? Can any other element? I think for all, let me use the polynomial. The polynomial that will? Ideal has to be like, what do you want the ideal? I don't want the ideal. Okay, think about this for a little while more. What else can give me R4? So what do I need? So what does it mean if I say AFX is R4? In particular, there's one element in R4 that needs to be a multiple of AFX. What is that element? One, right? So if I have one being a multiple of AFX, that's enough. If I show that one belongs to this ideal, then entire R4 will be in the ideal. Okay, that's the crucial element. That's an argument that's often used when you want to show an ideal is the entire ring. All you have to show is that some unit or one in this case belongs to the ideal. So when can I have AFX times BFX equals one multiple of AFX plus one? Is it possible? Sorry? Sorry? AFX is x bar i. It's definitely not the way it is. It's not the way it is. It's not the way it is what they need or what. Think about it. It seems like a lot of interesting discussions are happening, so maybe I should wait for a while and go solve this. There's no GCC. No GCC. The GCC is one. Okay. So that seems like an interesting place to start. So he's suggesting GCD of AFX comma x bar n plus one should be equal to one. So that's a nice property that the GCD satisfies. GCD can also be written as a linear combination of the two polynomials. So it means if I have something like this, it implies that exist polynomials BFX and CFX, what's that? What's that? AFX BFX plus CFX. X bar n plus one equals one. So if you do mod X bar n plus one, it seems like you're going to one. So it seems like a very valid idea. Can you give me an example of an AFX in R4? X. So what multiplied by X will give me one? X bar n minus one.  You will go to X bar four which is actually one modulo X bar four plus one. So those things will come. So there are also other cases. So if you have something like... See, you cannot have any other GCD. So if it is X bar plus X, for instance. So for example, X equals four. What about this way? X times X plus one. This will not be... So it will be something contained in R4. So these are... You need a strict GCD. That's the thing. There are also other ways. I think you can take any other power here. You will get that. And there are also other things. But you will be careful if you say irreducible. If it's irreducible, it should not be a divisor of R4. X bar n four plus one. So that might work out. I mean, you might have something like X bar three plus X plus one which might work out. So there are several cases which will give you the entire R4. That's some of this general interest. It's not a great interest to us. Just ask me. Yes? Can you say that any element will be equal to the element that is written by the GCD of that element? Yeah, we will come to that. Okay. So the next question kind of based on what he's asking is to say... When can I say... This is the question that you're asking, right? When can I say that ASX and BFX is the same idea? So for that... To answer all these questions, you need something to understand something more about the idea. So all these questions can be very easily answered if you understand something very basic about the idea. So that's what we'll see next. Next result we'll see, we'll talk about something known as the generator polynomial of an ideal or some kind of a unique generator for an ideal which will fix all these things. So the problem here is the same idea can be generated by multiple elements. And you have to find that element which is in some way unique to that ideal which will not generate something else that you don't want to generate. So then you can answer all these questions very, very easily. So let's try and answer that question that comes from the generator polynomial idea. So this motion of the generator polynomial is important. If you have an ideal in Rn, we'll denote this by GFX. GFX is there. So I'm dealing with binary cases here so it's quite easy. GFX is the least degree polynomial in I. So that's called... That is something special, it's called the generator. So the least degree polynomial in I is the generator polynomial GFX. So it turns out this will be in some ways unique. Just so you can show, they'll do only one least degree. So let's go and prove some properties for it. There are a lot of properties that the generator polynomial has and you'll see some very interesting results will come out of it. So this will basically fix... Once you understand the role of a generator polynomial it will fix all possible ideals in R4 or any Rn. You can quickly generate all the ideals in Rn. You'll know exactly what they are, what they can be. So all these repetitive stuff with different things generating the same ideal and all all the confusion will go away if you always fix to the generator polynomial. So there are many properties. The first one is it's unique. GFX is unique. So all these proofs will go back to this least degree assumption and the fact that you have an idea. So how do you prove uniqueness? Uniqueness is typically proved by contradiction. Suppose you have GFX and G prime effects. G prime effects and G double prime effects being both two least degree polynomials in the same ideal I. So when you say two least degree both of them should have equal degree. One is lesser than the other then both of them cannot be least. So both of them have equal degree and then both of them are in the ideal. So when you say equal degree in time, what does it mean? The last term, the last term is some expiry. Same thing for last term is expiry. Now what happens if I do G prime plus G double prime? Sorry? So if you do G prime effects plus G double prime effects I know it's still in the ideal and it has degrees strictly less than ideal. So the only way it will work out is there has to be zero in which case G prime and G double prime are equal. So you can quickly show that. So this implies this equals zero and that implies G prime. So that's the way to show that GFX is unique. So if somebody gives you a generator polynomial and says somebody gives you an arbitrary element of Rn and says find out something about the ideal generated by that. The first thing you should try to find out is the generator polynomial. Once you find the generator polynomial it will be okay. So suppose somebody asks you a question, ideal generated by AFX same as the ideal generated by BFX, how do you answer that? Find the generator polynomial for ideal generated by AFX and the generator polynomial for ideal generated by BFX. If both of them are the same then they are the same idea. Both of them are different and they cannot be the same idea. So it's unique to the ideal. The second property is that BFX will divide by Rn. So it means that it will come before this. So I think this is fine. So this really nails down every single ideal out there. Every ideal has a generator polynomial. And I know that it has to divide X bar n plus 1. So how do I nail down all the ideals? I start with X bar n plus 1. Find all its devices. Take one, each one will generate an ideal. To be a generator polynomial for an ideal. And those are the only ideals there. Nothing else is there. Any other ideal I come up with should be equal to one of these. So I will show this GFX divides at bar n plus 1. So basically the proof is to divide X bar n plus 1 by GFX. So what will happen when we do that? We will get a quotient and we will get a remainder. The important argument here is this RFX has degree of 50 less than degree of GFX. And RFX belongs to the ideal. Why does RFX belong to the ideal? If you reduce this equation modulo X bar n plus 1, you get RFX is QFX times GFX. So if you do QFX times GFX in Rn, what will you get? Small RFX. So it belongs to I. It has degrees strictly less than the least degree polynomial in I. So it should only be 0. It cannot be anything else. So that implies and RFX is in I. These things will imply RFX equals 0 and QFX divides at bar n plus 1. So there is an even more slightly more interesting property. Which is quite useful. If you have AFX in I implies that it exists. MFX such that AFX equals MFX times GFX in S2X. So if I don't add this thing, this statement is obvious. I know my ideal is generated right. Let me step back a little bit. So let me do this more carefully. This basically implies what I wanted to say. So once this result is shown, this implies that I equals GFX. Every element of the ideal is actually a multiple of GFX. Not only in Rn, it's a multiple of S2X itself. Without the model X bar n plus 1, it's a multiple. You see that? So that's a little bit more special property. All our multiplications have been modulo X plus 1, X bar n plus 1. So what I'm saying here is every polynomial in the ideal, which is in Rn, remember it's a polynomial of degree less than or equal to n minus 1. You can write it as some polynomial times GFX and you won't have to be any modular. Itself, it will be a degree less than or equal to n minus 1 polynomial so that no modular will be necessary. This is not very hard to show. Once again, the proof is... So this is what is important. Generator polynomial actually generates the entire ideal. Once I prove that, this is also a linear. So how do I prove this result? Once again, you have to divide AFX by GFX. You know AFX has a degree less than or equal to n minus 1 and GFX also has a degree less than or equal to n minus 1. And when you divide, if AFX has a degree greater, the remainder will have a certain degree and you can upper bound the degree and you'll see really, RFX again has to belong to I. There's nothing more you can do to me. So you can again show a very similar argument that this has to be true. Is it okay? Think about this for a while. Maybe there is some subtlety there. Only the least degree polynomial in the ideal divides X bar n plus 1. AFX may not be the least degree polynomial. So it may not necessarily divide. So only GFX divides. So first of all, I started with an abstract ideal. I didn't say anything about what generates it. So it's some ideal in Rn. So what I've shown after these things is any ideal in Rn is generated by one polynomial which belongs to it. So previously, we were starting with the definition of principle ideal as in all the multiples of a particular polynomial. Here I've shown that an arbitrary ideal is also principle and it is in fact generated by its generator polynomial which is a unique element in it. And that's the least degree polynomial. Okay, do you have a question? We know that C of X is the degree polynomial in the ideal. And we also know that every element in the ideal generates the ideal. No, no, we don't know. No such statement has been told. Every element in the ideal does not generate the entire ideal. If you want to count an example for it, you take the entire R4. You take the entire R4. It's an ideal, right? Not all elements generate the entire R4. You take X plus 1, it generates only a subset of R4. Every element in the ideal will not generate the entire ideal. All that is not true. The effects will generate the entire ideal, yes. I'm sorry? Okay. Step back a little bit. Wait, wait, wait. A effects is an arbitrary element of the ideal. It might divide X bar n plus 1. It may not divide X bar n plus 1. I don't know. If it divides, then? A effects will generate the entire ideal. Okay. Why should that be true? See, okay, so I think people are mixing up so many different things. See, in an ideal, if I have an element in an ideal, it does not mean that GCD of A effects, comma, X bar n plus 1 should be equal to 1. There's no such condition, okay? So maybe I should step back a little bit and give you some explicit examples of ideals in R4, and then maybe I should come to this. I just went a little bit faster, I think. Maybe I should step back a little bit. See. So this is not a property satisfied by every element. Okay? Only some elements satisfy. It's because satisfied when the ideal generated by it becomes equal to R3. All that is at the completely different propositions. Okay? Now, I'm starting with an arbitrary ideal in RF. If I did all that, the reason I did all that was just to point out that answering questions about ideas requires you to think a little bit if you don't know anything about the generated problem. You just say, okay, what is going on? Do they affect BFX? If you ask questions about the ideal generated by A effects, the same as ideal generated by BFX, you have to do a lot of work. I mean, you have to think about what is happening and all that. Once you understand the generated problem and stuff, everything will become easy, is what I said. Okay? So then I started with an arbitrary ideal in RF. Okay? Now, in this ideal, there are polynomials. It is not necessary that all of these polynomials should either divide X bar n plus 1 or not divide X bar n plus 1. I don't know. They might divide. They may not divide. I have no problems with that. Okay? It is not necessary that GCD of the elements in this ideal with X bar n plus 1 should be one. That's also not needed. It's just an arbitrary ideal. The only thing that's needed here is any two polynomials in the ideal, if I add them, I should get another guy in the ideal. Any polynomial in this ideal, if I multiply it with any polynomial from RL, I should get another elemental ideal itself. Those are the only two conditions that this ideal satisfies. Forget about all those example problems that I asked you with all these. Those things don't apply anymore. So just to get the thinking going. Okay? So let's see maybe some explicit examples. Sorry? I know my g of X divides every A of X to nothing point. Ask for this proof. Yes. No, you don't want to ask. Because if you divide otherwise, I divide by A. Otherwise, I'll have a dependent on the RL of X. Yeah. In RL, it's okay. Yeah. How do you divide the A of X? L. I'm saying it's true in F2 itself. Yeah. How did you get to RL of X? I mean, RL of X, if it's not true, then I always have, I can find the remainder of what's in RL of X. Why will you divide? Why will you divide? Dividing in F2 of X or RL of X? What is division in RL of X? See, in RL, this ideal has really no description. I haven't told you how to, how the ideal is generated or anything like that. No. What are these g of X being the least easy problem in time to automatically make division in RL of X equal to the remainder of 2 of X for whenever I divide the g of X? That's what I'm saying in the last statement. That's exactly what I said here. That's what I said in the proof books. So, since g of X is least big deal, A of X will have a degree greater. A of X also has a degree less than or equal to n minus 1. So, when you do a division, you're not going to do a star n plus 1 anywhere. So, it will not enter the picture at all. So, everything will be in F2 of X itself. That's why I said this multiplication property is true in G of X. So, usually when I write I equals generated by G of X, I don't need that this is true. I don't need that the multiplication should be in F2 of X. See, that's if the multiplication is in RL. But I'm saying here the generated polynomial is so special that not only does it have this property in RL, it also has the property in X2 of X. So, I think we really need to look at a few examples. So, let's look at a few examples. So, I'm going to take in R4, let's say the ideal is generated done. Let's try X2 plus 1. So, this will be what? So, how do you come up with these elements? So, every single multiple X, X2 plus 1. There are 16% guys there. So, let's see how many different. 0 of course will be what? Then you have 1. Then you have X. So, X multiplied by this is X power 3 plus X. Then you have X plus 1. So, what happens to multiply the X plus 1? X power 3 plus X plus 1. And then what else will you have? What about a squared? Multiply the squared what will happen? You won't get anything more. So, now you can check everything else. I think you won't get anything else. You'll get only these four guys. I'm going to come to it. So, here we got four guys. Let me ask a few more questions. Just so that it's working out. So, what's the GCD? So, if you have AXX and I, what's GCD of AX comma X power 4 plus 1? No, my X power 4 plus 1. So, for the same AX, if AX is 0, then GCD is not a proper definition. Forget about it. The AX is X power plus 1. What is the GCD? X power plus 1. X power 3 plus X. X power plus 1.  What about GCD of this? It should be this guy, right?  Because X power plus 1 also factors as X plus 1 times this guy. So, you'll get this GCD. So, let's just answer the question that GCD of AX with X power plus 1 can be anything. It's really very hard to control. So, that's one. Then, is there any other element of ideal which will generate the same idea? So, first thing I want to point out is, if you take AX in this ideal, what will be this ideal generated by AX? It will be some other ideal, but can I say anything more precise about it? It will be contained in I itself. It cannot go outside of I. So, if I take X power plus 1, I get the entire ideal. If I take X power 3 plus X, what will I get? You'll get the entire one. What will happen if I take X power 3 plus X power plus X plus 1? You'll get only two guys. You'll get zero and that. You won't get anything else. So, you can check these things. All these things you have to check. So, you can check if the ideal generated by X power 3 plus X is also equal to the same I. I'm using this I. I think the confusion is sometimes I use I inside the example. Sometimes I use a general ideal I outside. So, you should remember that when I say an ideal outside of the example, it's a general ideal. It's not something generated by X power plus 1 is R4. So, remember that. This is I.  This will just have two guys.  You can test it out. It's a bit more of work, but you can figure out what. Okay? All right? Okay. So, let me make one more statement now. So, maybe I should see, should we look at one more example? Maybe you should look at one more example. Okay? So, once again in R4, let's try and look at the ideal generated by, let me take a simple example. Let me do X power 3 plus X power 3. Okay? So, let's try that. Okay? We'll do this. Let's try. Okay? So, let's see what happens. So, you'll have zero X power 3 plus 1. What happens when I multiply this with X? X plus 1, right? We've got X plus 1. Okay? And what happens if I multiply by X plus 1? Okay? Okay? Okay? And then X squared. X squared plus X. X squared plus 1. Okay? And then you'll have X squared plus X plus 1. What will you get for that? Yeah, it will be X squared plus X. No, no, you'll get X squared plus X. Okay? So, you won't get anything more. Okay? You'll get X squared plus X again. Okay? And then if you multiply by X power 3, here what will you get? X power 3 plus X squared. Okay? And then multiply by X power 3 plus 1. What will you get? I'm sorry? X power plus 1, right? Okay? And that's it. I think that is it. You won't have anything more. Okay? Inject that you'll have 8 elements. You won't have anything else. Right? See, the first thing is an ideal in Rn is a linear block code, which already means that the number of elements in it will be a power of 2. Okay? So, if you have 16 elements in the ring, it doesn't matter. So, multiply by 16. The ideal generated by anything will either have size 1 or 2 or 4 or 8 or 16. It cannot be anything else in between. Okay? So, once you cross 4, you know it has to stop at 8. It cannot be the entire thing, because X power 3 plus 1 has a common divisor with X power 4. So, it has to stop at 8. And once it comes to 8, you can stop. You don't have to do any more. Okay? Anything else you'd like to repeat? Okay? So, now you can try to ask the same question. Which of these other elements generate the same ideal? Which generates a sub-ideal? You can try to list out everything. So, you'll get all kinds of different ideals depending on which polynomial you get. Okay? So, next question I'm going to... So, now we're done with the examples. We're going to go through one more general idea. Suppose I look at an ideal generated by some A-sites. And say A-sites divides X power into some. So, let's look at this situation. Different situation. Initially, I said I have an ideal. I don't know anything about it. We know that we can define a generator polynomial generated by the minimal degree polynomial there and all that is fine. So, now suppose I say I start with a divisor of X power n plus 1 and look at the ideal generated by it. Okay? What can I say? Can I say anything about the generator? What will be the generator polynomial? Can I say it's equal to A-sites? Or maybe it's some divisor of A-sites. How do you prove a result like this? I don't know anything about the A-sites. What do you do by the way? X power n plus 1, no? And if not, come back and become a... F2, that's not what I mean when I say this. When I say this, I don't mean multiple and F2 alone. Of course, if I multiply only in F2, then there's no case to be made. The procedure is under the polynomial. But I'm not saying anything about F2. I'm going to multiply this with arbitrary degree polynomials. As you have to determine something else. Okay. Then that will be J of X. And it will have a larger length of confidence. Which means J of X would not generate the entire ideal. Could be fine. Here, I suppose to be generated by A-sites. Yes. So if I have a generated J of X, which divides A-sites, then the ideal J of X will be larger than the ideal J of X. That's okay. As long as J of X is a multiple of A-sites and R-n. That's okay now. See what I have to show is... No. Please, please, please. Not easily. A-sites is not least degree. J of X is least degree. J of X is not least degree. J of X is not least degree. That's okay now. I never said A-fix is the least degree polynomial. Why should both of them be right? M-fix is not the right one. Yes, generating I. Because J of X divides J, then J of X generates a bigger I. No, no, no. Be careful. Be careful. How can you say you'll be generating a bigger I? Don't just assume that very quickly. You have to prove that. You're not wrong. That means proof. That's my only point. What you have to show is... You cannot get a divisive of X power n plus 1, which is lesser degree than A-fix, as a multiple of A-fix and R-n. You have to show that. Think about it. You want to show that or not. Anyway, I think we're running out of time. We'll stop here. But that's the property that we have to show. This is true. We generally are equal to A-fix. You have to show that. That requires proving. So it's not an immediate fact. That would be very obvious to you. So used to the integers. We have a great deal of disarmament on X. Here it shows, you have to express... You have to explain things about it. n plus 1. Okay. You have to express the degree less than A-fix. You have to explain things about it. Yeah. You have to explain. You have to say I'm the second. I'm the second. Yeah. So you have to show like that. This is a way to show that it cannot happen. Okay? So you have to... I mean, so you have to go from R-n to F-2. And there you have to use arguments like what he said just now. Okay? So you have to argue that it has to be that way. And then from there you can argue. All right? So we'll come to it. Will A-fix derive the experiment percent of the problem? I'll prove this in next class. Okay? So explicitly. And then we'll pick up something.


Welcome back to my channel guys. It's your girl Coco styles and on this channel. We talk about beauty fashion lifestyle and styling and today I got an unboxing and a review of Child I've been looking well not looking Trying to order from these folks for forever And I finally was able to do it. So let me get situated here make sure my camera is on for you guys So I got three hats from Ruslan begins key if you guys did not know I have been trying to order these hats for years The issue was my bank would not allow me to order So when I started seeing all the girls getting I'm like, how are they getting it? And I know there's an issue with ordering from the Ukraine So I noticed that they were all getting it from like other websites that were US US based So I was like, you know what? I'm not even gonna do it because I wanted to get an extra large and the hats that they had here I just it wasn't the ones that I actually wanted Well, anyway, I tried it recently what I had to do and no one talks about this I had to go through a third-party app if that makes sense So I had to use Clarna and then I used a card from Clarna and I used it on PayPal if that makes sense My bank was like, I know you cannot use your card period Into Ukraine. So that was my way of getting around it just so I can get the hats and of course, you know Clona just takes the money out. So I just paid like right off as soon as everything kind of registered in the app So that's the first part about it. The other thing is when you get your package You guys already know how I feel about packaging It just comes in a box and then the bag was in the box and the hats were just wrapped in Like tissue paper And I was like these hats are not cheap So I think you guys could have did a better job, but the hats were not damaged at all I've said this before ordering from lack of color is Really really good because lack of color put their hats in a hat box. So they're never smashed Brixton on the other hand. It's a no-go But let's just get into the hats and then I'll get into the sizing when I put the hats on So I got three different hats and I'll show you all three at one time and then go into them Three different hats guys if you know my wardrobe if you've been following me I have a lot of neutral colors, but red is my pop of color. It is my favorite color I don't have a ton of it, but I do have pops of red. So two of the hats I did get in a large And then the other one I got in the extra large I have two more coming that are extra larges and this is why I wanted to order from them It's solely because they have extra large. I do not see the extra larges here in the US So I got them from there the last part that I wanted to talk about Before I try the hats on is the taxes This might bite me in the tushy later I'm trying to figure out the taxes part because when you get the actual invoice It just tells you that the receivable pay the duty and taxes. I don't know what to do at this point I've been doing some research. I don't see anything This is just saying sub total value 135 You know, I don't like I don't know What they're trying to say here. So once I figure that part out I am gonna edit the description and put all of that in there for you guys But as of right now, this does say that I have to pay for it So let's get into the hats and then the sizing when you are measuring your head for these hats They recommend you using a flexible tape, which I've said this is a closet essential in one of my other videos You need this girls like even if you're ordering wigs and stuff You need this so you would pretty much take this and start at the center of your forehead and then go around to Hold on you would go around To the center here and Then you you know cross it and then you get your measurements Mine is saying 24 now if you have gone on their website My hair sticking up if you have gone on their website and Check their sizing It is going to tell you that an extra large is a 23.2 a large is a 22.8 So I just said my head was 24, right? So The reason why I wanted to try both sizes was because I wanted to see how the hats would fit with a unit on And for those of you girls and guys who don't know what a unit is.  So I'm gonna try on this one first this one was a little expensive because I think it's because it's straw now what I noticed about the straw hat is that it's straw It's not sprayed. It wasn't like a light color and they sprayed it. It's just a straw hat Really really nice. You got the RB right there the little X, which is the other signature part and This is a large. So let me show you guys how this fit. I'm gonna just crop it up All right I'm gonna show you guys my ear for example, here's my ear and This is the head right here And I'm showing you that for a reason because the large and the extra large pretty much sit at the exact same place So that's not where the the size is going when it comes to like it being larger. It's more so with this Um, I'm gonna turn just a little bit so you guys can see the back It's sitting where it's supposed to sit on my head, which is at the Largest part of my head if that makes sense um But absolutely love it this girls Is gonna be my spring and summer hat I just can't I have a ton of stuff in my closet That is neutral colors. I prefer the darker color A straw hat because of my wardrobe wardrobe and my skin tone So this is the first hat. Let me know what you guys think. I'm just gonna come a little closer So you guys can see everything. I hope you guys can see everything because I'm actually sitting Okay, let's go top Really nice now the price went up on Their hats and I'll tell you guys how I noticed because I had them in my bucket my other bucket And um The price changed honey So here's the other hat in the large This you really can't see And this is the cotton one The cotton one kind of feels like I don't know like a denim shirt kind of But this is more of like a taupe color You can't really see the color on camera But I got this one. I didn't want a black tip on this one for a reason um when you when you're shopping for things girls like think about your wardrobe and I like to have multiple of things because you never know what you're gonna wear How you're gonna wear it? So I wanted this one and I also want one that has a dark brim and a lighter top And that's just because some days I may just want to be strictly monochrome With no other color and then some days I may want to have a black brim So this is also a large and this is the cotton one Okay And again, they it fits the same way you're gonna see the large on my head The extra large on my head and how it sits And you guys I'm looking here because my mirror is here. You guys tell me what you think And I'll show you the difference when I put the other one on to me.  That's the um the paper But this is nice as well. All right, so this hat was too It's 270 Okay This hat was 270 The price on these I mean not the price the size on this is a large Like I said, it's like more of a taupe color, but you can't really see it on camera here But it's a really really really really nice hat Okay Right now on their website Everything looks like it's available meaning all sizes. So it looks like there's a nine-day shipping I actually got these pretty fast. I was surprised. I thought they were gonna be I was supposed to get them on Tuesday when this video goes up, but I actually got them on thursday So I don't know if it's this dhl. It's just good shipping, but You know, I got them pretty pretty fast. So that's the second hat And these hats don't really have like the name on like no real naming on them Um, and I was double checking to make sure I gave you guys everything Like this just says monogram embellished baker So there's the straw one and then this last one I'll show you guys Which is the extra large And it just says red On it Okay Now this is the extra large. I do have a black or leather one coming because that one is Super dope And I got that in the extra large because she told me that it was they emailed me and said it was available So I was like, oh yay like it's been months so the difference between these hats is The straw one and the other one I have to like push it down. This one kind of slides right on and it's because of this unit guys If I didn't have a this fluffy hair on it would slide right on with my natural hair With even maybe even a straighter way possibly but this has a lot of volume in it But the red one is cool and again, it sits exactly at the same place As the other hats. So that's not where the exercising is going. It's going to to The width of the hat that makes sense Because it slides down just a little bit easier But the position of the hat it still sits exactly the exact same place I hope that is making sense for you guys So it may feel a little weird that your ear is down, but that's just the way the hat is Now I could return these two and get an extra large I don't know. I don't know how I could I don't know how long that's going to take So if I do I will let you guys know if not They just gonna stay in my closet. Okay, but this is it. This is just a quick unboxing of The ruslin baguinsky hats. They are back in stock guys surprised to go up. So any hat that was 220 is now to 50 or 260 So they did increase in price and it's probably because everything is like back in stock on the website So that is it. That's all I wanted to show you guys today a quick unboxing Packaging was mediocre Um shipment was great. It was quicker than I expected. None of the hats were damaged price is Increased. Um, so that's not a good thing. Um, but they're really nice. So this is the wool version This is a cotton version And here is the straw version Um, so that's it guys. I want to thank you guys for tuning into this video If you have any questions about the hats, let me know if you have anything about Pack shipping pricing all that good stuff. Please let me know in the comments and I will definitely get to your comment Um, but thank you guys for tuning in if you stay to the end of this video Please comment. Coco. We see you and guys don't forget to follow me on instagram at coco styles and yc and coco styles And why am I personal style blog coco styles and yc and my style services coco styles and y.com And i'm on tickety tickety talk for the moment because you guys know that they trying to shut them down Coco styles and yc and I will catch you guys in the next one


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Hey there and welcome to my channel. This is Christine with Art of Awakening and I'm going to do today just kind of a channel painting video um lately. I usually when I work I've got usually I have YouTube on um you know just whatever whatever wants to come on because it's on auto and um the day this beautiful music came on it was Tchaikovsky's him of the cherubim and it really struck me. I mean it really moved me almost to tears. I mean it was just very moving and so I started listening to it over and over and over again um it comes from a longer piece of his it's made up of a bunch of little pieces called the the liturgy of saint john somebody or other so uh so I've got that playing in the background right now um but what it did was I think you know sometimes the music the music and art they all carry the energy energy within it and in this case Tchaikovsky was definitely channeling the cherubim. I didn't know a lot a lot about cherubim usually you hear about you know you see the little cherubs and they're very cute and quaint little Victorian cherubs little babies with wings well so you know after I've been listening to this for a while and it was really I mean I could feel a lot of shifts happening in my field from this music uh so I got curious and I started looking up what are the cherubim and I found so some very interesting kind of mythology around that um they appeared in various ways depending on different traditions and sometimes they are like the babies but they're in order of angels they're supposed to be just below the seraphim and you can look them up but one of the things that I found was that they're not always showing up as babies um and in one account they show up as these four faced angels that guard the Garden of Eden and and they're they're so bright you can't even look at them they're they're very very powerful and so after I'd read that I was going around the roof and then they just said draw me so that this is a painting that's kind of coming out of that sketch that I did I did a sketch in my journal around that um after they asked me to draw them so this is a cherub probably not like any cherub you've ever seen before but with the four faces the idea is that they can go in any direction without turning so they're they're really focused right and no matter what direction they're going and kind of I'm getting a few things out of this number four well first of all that the idea of guarding the Garden of Eden um and I'm assuming I didn't I'd have to go back and read it but this would have been probably after the fall um and by the way you know I I think all all traditions all spiritual traditions are beautiful and they all have truth in them so I'm not like you know you don't have to be Christian to appreciate this I really love myth of all kinds wherever it comes from so um but you know I love the symbolism of the Garden of Eden because I think that's coming up a lot lately for a lot of people a dog here a dog here over there because really I think the shift that's happening right now on the planet it's really I think the idea is you know we're gonna be returning to that state that state and and maybe that's part of what the cherubs are about when they appear is young children or babies because that's a state of innocence and a state of um you know the Garden of Eden represents really that primal um state before corruption happens and we see that like in nature nature left alone that would be the Garden of Eden before it got messed with and it's still there it is still there like every time if you go out in the woods or go in nature there it is it's right there it's it's it's we've never left it we just have left it in our minds right I mean there's been some systematic destroying of it but I think you know if we start focusing on that destruction then we only feed into it so yes it's we need to be aware of it we may need to take steps away from that destruction you know whatever we can do but it's it's super important because I know a lot of people who are concerned about the earth they really focus on that destruction and and and they get all emotional I mean like that I like I understand because I feel that way you know I I love the earth and I love nature and I don't want to see it destroyed and it's very upsetting but when you start feeling all this despair around it that just honestly the emotions are what we create with and when we start creating a lot of despair around the destruction of nature I mean that that honestly feeds into it so I think part of what this is about what they're saying here is you know first of all if you're not you know that's what the angels always say they say fear not right and so you know there is this like yes we're here we've never left it but on the other hand there's also the sense of guarding it right of not being able to enter and what is that it's like it's it's transcending the ego because we're never gonna actually really like we're always here it's always there it would never left it but but we think we have right the ego has left it our soul hasn't left it right our our higher self or it's always been available to us we could step into it at any time we want our ego has left it and so the idea is it's like I really get this like the wings of this thing they're like swords they're like very sharp swords they're like razor sharp and I'm not this isn't blood this is more like flames right so it's flames I'm getting and sword like you know in the wings themselves and what it's really saying it's like you know you need to be you're gonna have to be pure enough like you're gonna have to pass this right to pass through it and that is what is happening right now with us and we're getting there I think a lot of us are getting there and we're getting there you know the earth is getting there and collectively as human beings we're getting there and it's not as impossible as I'm making it sound because they have this heart right the cherub has this beautiful heart and loves us and is very compassionate and wants us to get through really so it's not like this guardian that's that's that's you know against us it's this guardian that's for us and the reason they're guarding it is that they want us to be happy and we've already you know approved that you can't go in the garden of Eden you've got to be aligned right in order to see it for what it is and to enter um and so it's it's really just a visual reminder that we can we can't go back we are going back we're already there we've never left it and that all this purification that we've been doing every lightworker that I know every star seat that I know has gone through a lot and even if it's not in this lifetime if you're aware enough to be watching this and to follow me this this far you've had significant trauma in whether it's past lives or this life um so they're saying you know you're doing good work that you're almost there if you've gotten to this point of seeing this apparition that you're almost in like you're you're on the threshold or maybe you're even in and you don't like like I said we don't even know it so this is the last this is the last little bit that these represent okay and actually even if you feel like you're completely not at the threshold and spending time with like this image will help to get you there because this is the threshold right um I hope I'm making sense because I'm just sort of seeing what's coming through and I'm using a couple different parts of my brain right now so it's uh it tends to get a little more scrambled than it should but but I think I've just been guided that you know doing this along with the painting they're speaking through it and even though I'm an imperfect translator they're still asking me to do it and maybe that's part of the lesson like you we don't have to be perfect in order to be pure enough to get there I think that's a big part of the lesson like I'm certainly not being a wonderful channel right now I'm just not you know it's it's all convoluted and it's all but they're saying it's okay do it anyway so part of what the message is is if you're feeling called to share your experiences or share your insights or share you know if you have insights or wisdom that you're being called or you know your attention is being drawn to it will we will help other people um so they're saying you know don't don't feel like you're not good enough even if you haven't arrived yet because you are good enough more than good enough you are perfect and the only reason we feel we're not perfect is that we're on this creative journey altogether collectively as a human race this is a creative journey that we're on and creation the act of creation if you've ever given birth you know it gets messy and it gets painful but that's what they're saying though is like you know you you are perfect and whatever you're going through it's not a reflection of your imperfection it's a reflection of being reflections here this is like water a water pool of fire or something um it's a reflection of your divine journey that's what it is so part of the lesson is gratitude deep gratitude this is the heart here heart of it it showed me this beautiful flaming heart in the center and if you see in the center there's a heart and there's flames coming out of the heart too and flames all around but if you're in the heart and I see almost this little childlike image coming up from the heart and right there in the heart you're safe it's a safety zone okay so being in the middle like this number four four faces you can't see the other face because I have to I'm not doing this in two dimensions but it's it's a four faced angel and four is a number that is associated with the angels or the archangels it's a very powerful number it stands for it represents like solidity and stability and you know security safety amongst other things um so so this is touching on the idea of faith and trust it's trusting like if you've been drawn to watch this something drew you there you were listening to your higher self you were listening to the guidance to bring you here and that guidance is going to bring you anywhere you need to go it's always there for you it's always supporting you and right now like I've had so many things popping in just in the last couple weeks things lining up just support coming from just unbelievable support from the universe and from people and you know it's it's there and the gratitude is key because when you start really feeling into that gratitude and the encouragement is really feel the gratitude okay because sometimes we use our minds and we think things I'm grateful and that's good it's a good start it's a very good start but when we start getting the emotions involved so if you have a like a gratitude affirmation you can make it so much more powerful by really feeling that gratitude within you taking the time to really you know start with like feeling what does gratitude feel like and it's like a little exercise you can you can be still one night one day or something and lie down and and think of something you're extremely grateful for and really pay attention to what that emotion feels like and that is the frequency of gratitude and you can learn you can train your body to respond to that frequency of gratitude if you practice that so I'm just kind of recording this on an every day every day every day and we're getting every day noises behind us and that's just fine because that's again coming back to perfection like if I waited until I had oh if you listen to the background right now that's the absolutely beautiful so if we waited until conditions were absolutely perfect for everything it would never happen listen to that I don't know if this phone is going to pick it up I'm really hoping that that music comes through because it's look look it up I'll leave a link below to Tchaikovsky's Hymn of the Cherubim because it's out of this world but yeah it's like you know start where you start where you are maybe you're called to start a business or something you know just follow the guidance it'll show you what to do it may show you what to do or it may show you the you know bring you to a person that will help you especially if it's something that's really been frightening you if you're not sure exactly how it'll work sometimes asking just to be connected to the right people the right time in the right ways to assist sometimes that's well again um the safety of this child like innocence because that's that's where I'm thinking of bison buffalo when there's danger they put the children the babies in the middle the young and musk oxen do this too and they protect them and they just stand in a ring all around so I'm feeling like that's what the Cherubim are doing with this garden the garden of even they're protecting they're protecting the inner child because I'm getting a lot of inner child coming forward lately in a lot of ways and how important that is the ability to like children have a natural ability to play maybe a play is so important it's one of the keys to getting back to this beautiful existence that's what art is let's play people get up tight when they you know every time when I tell people an artist the first thing most people say is oh I can't draw like of course you can draw it doesn't have to look perfect every child can draw right show me a two-year-old or well maybe all two-year-olds yeah but yeah four-year-olds right they draw the band and they don't worry about it and there's you know they look up like what look look what I drew look look at this look at this and they're so excited and they'll tell you about it and tell you the stories about what it is and sometimes those are like the angels speaking through them but then they go to school and they get trained that what they're doing isn't good enough and they stop hearing the angels they stop seeing them they stop drawing them or whatever is they're drawing what's happened to most of us old journeys about and that's why like this whole idea of purification purging all that it brings us back it peels away the layers into the inner child and that's you know um just you know letting go of all that stuff allows you in here this almost looks like a seed opening and that's the other thing that they've been asking me to relay is something to do with seeds and rooting so i'm gonna put a root in here because the connection with earth is so important i've been getting a lot of just um messages to ground lately that grounding is very very important for lightworkers right now especially always but right now sending down the root sending down the root to mother earth connecting with mother earth it reconnects us to our pure essence nature purifies all the four elements so the four faces the four elements they each purify in their own way we have the element of error and if you've ever been in a stuffy room and opened the window you know how purifying the element of air can be and then the element of water of course washing element of fire very purifying too we don't always think of the element of earth is purifying but it is earth you know think of um a beasting and putting clay on it and how that just leaches away the toxins so every one of the elements has this purification element to it and connecting with nature you're connecting with all four of those um you know we're going back going back to our roots reconnecting and another part of this is coming to center right and and the stillness because there's so much action around here there's that old saying you can't fire fight fire with fire you can't advance into this and without just being still you've got to be still stillness is the only way you can get through here that's what they're telling me is that when you become still it's like you enter the heart it's a portal to enter the heart without having to bypass all this stuff in the stillness you can hear they call it the still small voice for a reason it's necessary to come to stillness our society does not value stillness very much our ancestors did so you can connect with the ancestors sometimes they can help i mean they still remember how to sit still well i don't know if this is done or not i may go in after it dries and work on it more but it feels like this video is done maybe not maybe i'll just get a little more violet in there maybe violet is asking for violet before i leave and reminding that the violet flame is available and if you're not familiar the violet flame it's very easy you just kind of visualize yourself in violet fire and and what that is is it just transmutes it's a it's like this pure grace kind of thing there's a lot of ways you can use it but um it's grace because we aren't you know spirit knows that we're not perfect right but but that's why we have this we can ask for grace because we are perfect but we're not does that make sense we're perfect in our imperfection in our humanity we're perfect in our humanity and we're given this grace in order to transcend what we're dealing with are you my dog it's been silent for a while and some people will probably have dropped off getting bored with the silence there's something better they could have been doing right now i think really the here is just being still enough to listen and being still enough to hear what we're being called for called to do that it's probably closer than we think in fact it's right you know you're available as as a as a like as humanity itself we've got everything that we need to create a beautiful great beautiful lives for everybody on the planet it wouldn't take a huge amount it would just take you know take some slowing down and start considering you know what is it that we really want and to take our power back as as human beings we can each do that and what they're saying is like we each need to do it as individuals take our power back and part of it is getting off that treadmill of gotta go gotta go gotta go because you're gonna find as you slow down you're gonna get more and more and more attraction so that you will find things happen so easily without having to you know run ragged and when a critical mass of individuals moves into that space it opens up the doors for everybody like the last thing i'm gonna do is i'm gonna put some gold in here because i'm seeing this little child in the middle get some gold in once i gotta go get a tomb here this is designer's gouache it's uh most watercolor is transparent but this is now this may not show up too well on the video but in person it's gonna be he actually go in and put some iridescence in the rest of it as well right thank you for watching if you've held out this long i commend you and if you are interested in working with me at all i've got links below i do give intuitive sessions there a little more i don't usually speak this slowly and it's just like it's you know i had to paint at the same time but definitely i offer healing sessions readings as well as artwork and working on getting my art side up and oracle cards so i have a link to my spirit animal deck below and i do want to send a big big big thank you to everybody who has signed up for sessions and purchased decks and so forth things that i do definitely appreciate that and just wish you a beautiful day we'll catch you again later


Evers sworn in as governor, Sheboygan police investigate car break-ins, Steakhouse bandit in court for sentencing Thursday. These stories and more coming up on Community News Review. This is Community News Review, a service WSDS-TV news content provided by WHBL. Hello, I'm Maddie Fister and this is Community News Review for January 7th, 2019. A plea and sentencing hearing is planned for this afternoon for the Plymouth woman who reportedly sexually assaulted a teenage boy at a Fourth of July party last year.  Mason told investigators that she was just playing around and that she had been drinking earlier that evening before the incident. She is facing up to 40 years in prison if she is found guilty. Sheboygan police say they are investigating a number of car break-ins in Sheboygan Southside. It happened Wednesday night in the Farnsworth Roosevelt area and police do not have anyone in custody yet, but they are asking people to remember to take their valuables out of their vehicles and lock their car doors when they are parked. If you have any information, you are asked to please call the Sheboygan police department. The grafted man arrested for breaking into Jake's roundabout steakhouse in Sheboygan last July will be in court for a sentencing hearing Thursday morning. Abby Lee was charged with felony burglary after breaking into the restaurant, destroying glass, cash registers, a jukebox, and other items. The criminal complaint says that Lee propped open the taps at the bar so that they would stay running while he made himself a steak and ate at the bar. Lee is facing more than 12 years in prison and he will be sentenced by Judge Kent Huffman. A new report says one in five high schoolers in Wisconsin have tried an electronic cigarette. The Wisconsin Department of Health Services says back in 2014 just 8% of students had vaped and now it is at 20%. Health managers are worried about the health effects of vaping and they say no one knows for sure what those clouds of vapor do to young people. Vaping products cannot be used by those under 18 years old in Wisconsin and there is a growing concern that some of the flavors and vaping products are designed to appeal to underage users. If you are looking for a new job in the new year, this state has some excellent resources on hand to help you out. Amy Hansman with the State Bureau of Job Services says there are more jobs available than people to fill them and that is an excellent starting point for anyone looking to get a better career. Literally thousands of opportunities available as well as resources that can help you be a well informed job seeker. Hansman says the Job Center of Wisconsin offers up access to up to date labor market information that can help you direct to improve your position in your current job. Maybe you want to negotiate a little bit higher wage. If I complete this training, what can I expect for a starting wage for a new type of position? She says the Job Center of Wisconsin website can help you find a position you might not have known you were qualified for just by using the skills you already have. You can do a skills match and it will run a quick job search to see what sort of position is out there. The state's Job Center offers up assistance in writing resumes, finding a new position and finding out if the skills you have might get you a job you weren't expecting. You can find out more online at JobCenterofWisconsin.com. A local organization is stopping its efforts to resettle refugees in the Madison area. Lutheran Social Services will no longer be providing resettlement services to people coming into the country. Ken Braun, who runs the LSS Open Doors for Refugees program, tells WKOWTV that the high cost of housing and difficulties finding accommodations for refugees makes it impossible to continue. Federal resettlement policies have also had an effect on these services. The LSS will continue working with those who are already living in Madison for up to five years. Jewish Social Services tells Wisconsin State Journal that their resettlement program will continue. And finally, the Tony Evers era has begun. The Democrat took the oath of office alongside four elected officials Monday at the Capitol in Madison. Evers warn in where Lieutenant Governor Mandela Barnes, Attorney General Josh Call, Secretary of State Doug Lafellette and State Treasurer Sarah Godluschi. Barnes used his inaugural address to speak of bringing the state together and creating opportunities for those who might not otherwise have them, saying a child's path in life should not be determined by their zip code. Evers spoke of working for Wisconsin values and prioritizing education, saying what's good for Wisconsin's children is good for the state. He also made funding for roads and other infrastructure a priority, which he spoke about at length during the campaign process. The 46th Governor of Wisconsin also called for kindness and respect in the political process. Attorney General Call promised to make the state safer by working with lawmakers to craft legislation that would prevent another backlog of untested sexual assault kits. He also outlined a plan for a red flag law, which he says would allow family members or police to go directly to a judge to ask for someone to be immediately disarmed if they're a legitimate threat to the public or to themselves. Call also called for more funding for mental health professionals in schools. And the other constitutional position, Secretary of Department of Public Infrastructure, was not sworn in. Evers named Carolyn Stanford-Taylor to that post last week, and she will finish out the term Evers was elected to and will be up for election in April of 2021. And that is all we have for today. Join me again next time for another recap of our local news on Community News Review.


How to get famous? 14 unique business ideas for first time entrepreneurs. Hey, you wanna be famous, right? Well, at least you want your business to be famous. I mean, maybe you want yourself to be famous, but the point is there's certain things you can do. In this video, I'm gonna share with you 14 things that you can do to make yourself famous and make your business recognizable. Now, if you're new to the channel, please do me a favor and subscribe, hit that subscribe, but now there, turn it from red to gray. Don't forget to ring the bell. Turn all bell notifications so you're notified each and every single time I go live. You're gonna love this video.   Now, every niche is different. There's always competitors in a niche and if there aren't competitors, and maybe you shouldn't be in that niche, but there's always competitors in a niche and what you wanna do is be able to set yourself apart from everybody else. And that's what I wanna show you how to do in this video. You see, your competitors out there are jockeying for this audience and you're jockeying for this audience and whoever is better at these things is going to get the customer. So let's jump right into it. Let's go over some of these things. At the end, I'll show you where you can get a free course, a $97 course, absolutely free. All right, let's jump right into this. Number one is be the head chef of the restaurant. What do I mean by that?   This is super important if you really wanna build a connection with your audience. We've seen this happen time and time again for many, many years. Everything from men's warehouse to Purdue farms, the dad and then the son and you may or may not know some of these businesses, but they always had the owner of the business on camera, on video, talking to the audience out there so that they could make a connection with the audience.  The next thing I would tell you is be involved.  It just doesn't work.  You're gonna have to be involved in the business. You're gonna have to talk to customers. You're gonna have to talk to your employees. This is required and heck, how could you be famous? How could you be well known if you're never there? Number three is learn from your audience.   Make sure that you listen to them. They have great input. These are the people that are buying from you and they're offering their free opinions. You gotta listen to them. I know when I do YouTube videos, I always ask people for comments and I'm doing a YouTube video now, so by the way, comment below if you would on this video. But I always ask people for comments because I wanna know what they're doing. I wanna know what they're working on. I wanna know how I can help them. What questions do they have? What videos should I do? If they have a particular subject they want me to do a video on, I'll actually do that video because that's what they want. This is my audience. You've got to serve your audience. Number four, of course, you knew you were gonna hear this, is use social media. There is a lot of social media out there now and it can be overwhelming.  How often are you gonna post and all that? And I can't tell you what the exact mix is, but I can tell you this, you've gotta be there. If you're not on the big ones, like say YouTube, which I'm on right now, or Facebook, or Twitter, or Instagram, if you're not on these platforms, you're gonna get left behind. Now, I use a software that actually, when I do a video, it'll post it in three or four different places. You can get a software like that. That's called Hoot Suite, so you can check that out if you want. I'm not an affiliate for them, but that's a good software so that when you create a piece of content it'll go out in several different places. But you gotta use social media. It's just part of marketing nowadays. Number five, email and email often. Email, I've heard people say email is dead, and I've heard them say it this year, last year, the year before, the year before. They always say it's dead, and yet we still make millions of dollars off emails. I would say email is not dead. I would say it's gonna be around for a long, long time. It's the way we communicate. And most customers say the communication that they get from businesses they would prefer was in the form of an email. So I think that's super important too. You gotta email often. And when I say email often, it doesn't mean always hammering them with offers. Send them some good information. If you're like we sell canned meats so we may send out recipes. We may send out things on how to protect your home since we know that we're dealing with people who want to be prepared. So there's all kinds of things we can send out that would be helpful to them. Email and email often. Number six, I mean, if you wanna be famous, get some press, right? Get articles written about you. And you say, well, that must really be hard to do. It's really not hard to do. I know when I moved my business a couple of times, I contacted the newspaper and I said, hey, we're a survival food company. Would you like to write an article about us? And they jumped all over it. It was appropriate. It was timely. It was something that they were interested in. So I would tell you, if you do those articles, and you will, you will get some articles written about you.  And that press can be very important, especially if you wanna be famous. Number seven, be unique. Uniqueness matters so much. I mean, how many years did we all just shave every day? And a razor was a razor until Dollar Shave Club came out. And they said they were gonna sell a razor for $1. And it really doesn't turn out to be a dollar, by the way. I tried it out and I didn't like the razor. But that's beside the point. He had unique commercials. They were funny. They were entertaining. Those spots went viral with what he did because he did something that nobody else was doing. And I can tell you, a lot of people picked up on that style now and are doing ads just like him, but he was unique. Try to be unique in your niche. Number eight, you gotta be consistent, okay? You can't decide, well, I'm gonna work on being famous and I'm gonna work on getting out there and building my brand. And I'm gonna do that for six months and then I'm gonna lay back for two or three months. You can't do that. It has to be a consistent thing. And I know it's hard to do. It's hard for me. I have to do videos. I have to work on my business. You know, there's a lot of stuff to do, social media, all these different things. But it's gotta be done, okay? If it was easy, everybody would be doing this, guys. If somebody's telling you it's easy to build a business, it's easy to build a brand, it's not so true, okay? It's not true at all. It takes some effort. It is not hard as far as the things you have to do, but you gotta be consistent. Number nine, and I love this one, have awesome customer service. How is that gonna make you famous? Well, it works really well. I'll tell you why. I know that if I go to, let's say Costco and I buy an item and I don't like it and I wait three weeks or I lose the receipts or whatever, it's not gonna be a problem. I'm gonna get my money back. I know I am. Same thing with Amazon. You know, they have a window. I don't know what if it's 30 or 60 days, whatever that window is. But I know I could return just about anything as a prime member. So that makes me be a prime member, right? I pay that extra fee. But I know, I don't worry about buying stuff from Amazon. Matter of fact, I have bought, you know, a shirt or a pair of shoes in two or three different sizes because I wasn't sure which one would fit and just send the others back. You can do that, okay? That is great customer service. Love them or hate them. They've got great customer service.  You look at Chick-fil-A or you look at In-N-Out burgers. They got it, figure it out, okay? They have great customer service. Number 10, stand for a charity, okay? This is a great way to get famous once you attach yourself to a charity. And it's funny with charities though, you gotta talk to them in advance. You gotta tell them what you wanna do. Some of them won't want you. I found this out the hard way when I wanted to give some money to charities. Some of them are really restrictive about what you could say and what you could do.  Number 11, you're not gonna wanna do this but be controversial, okay? If you've got some strong opinions about something, go ahead and say what those strong opinions are. And the reason being is the people who agree with you will really attach themselves to you. The people that disagree with you will really attack you, okay? Now you may not wanna be attacked but even being attacked sometimes can be helpful because people are talking about you. You think of a lot of the celebrities that are always in the news for some of the crazy things they're doing. And you know, they say no press is bad press. So getting out in front of the public, being controversial can be very helpful to you. Number 12, if you wanna set yourself apart, you gotta be out of the ordinary, okay? You've gotta be out of the ordinary. To set yourself apart, you've gotta set yourself apart, okay? Maybe it's your colors, your logos. Maybe it's what you stand for. Maybe it's what you're working towards. Maybe it's your goal. But try to stand out, be out of the ordinary. This is a great way to build a brand and actually get you famous. Number 13 is not really a direct thing. It's something that I've just always done and I've noticed it's helped me a lot is always learn new things. Always learn new things. There's always new ways, like you're watching this video right now, there's always new ways to promote your business. There's always new ways to maybe do email marketing or to do sales pages or do sales funnels or ads or whatever it is. There's always new ways and you really gotta stay on top of these things, guys.  It doesn't work that way. It constantly changes, platforms change. What works changes, all these things change. You've gotta constantly be learning. And now we're down to number 14, the very last one. And number 14 is be proactive, not reactive. I see so many businesses, so many owners of businesses react to what's happening every single day. Instead of sitting down every day and looking at their business and look at what happened and to say, what are the things that we can change? What are the things that we can do? How can we change our emails? How can we change our ads? How can we change our website? What can we do to reach a bigger audience? Sit down and actually brainstorm. This is the thing that a lot of people don't do. They're constantly reacting to their business but they're never being proactive. They're never deciding in advance what they wanna do to improve things. If they did that, they would become more famous and the brand would blow up more. Now, I told you in the beginning of this video, I've got a $97 course. It's below in the description. All you gotta do is click and learn. No credit card is required. That's a pretty good deal but it won't be there forever so you need to get down there and get it right away. I really appreciate you watching this video. If you're not a subscriber, please become one. Click that button down there. Turn it from red to gray, that subscribe button. Don't forget to ring the bell. Turn all bell notifications so you're notified each and every time I go live. I really appreciate you watching this video. If you like what you hear, give me a thumbs up. I really do wanna hear your comments. I respond to as many as I possibly can so if you've got questions, if you've got an opinion, if you just wanna say hi, you could do that in the comments section down there. Thank you so much for watching this video and I'll see you in the next one. Hey, thanks for watching my video. Don't forget to subscribe to my channel and click that little bell right there so you can be notified every time I do a new video. Also, click on one of those videos there. Keep watching on my channel.


Hey everybody, it's Hari Swaminathan from optiontiger.com. We are looking at our earnings trades. Now yesterday we closed Qualcomm for a nice $1300 profit that was 130% ROI and Today we are going to close Baidu. Baidu as you can see Baidu reported yesterday and We had the debit spread a $5 debit spread which is almost close to max profit now Just about this is about an 80% ROI. So we're going to close this This is also one of our good trades that we're closing in the earnings cycle. This is for October and November of 2019. Thanks


Want to gain more stamina, but don't want to lose any muscle gains? Watch this our first caller is Daniel from Georgia What's going on Daniel? How can we help you? Hey guys first off? Thanks for having me on so High-level overview sort of headline here is how do I increase my VO to max while having a Nicular metabolism and not sacrificing gains So the context here like I grew up Sal probably very similar to you like long-distance runner very skinny like strong wind there goes Dan and found strength training and was able to pack on a bunch of muscle and Now I've been going through MAPs anabolic, which is awesome But I've worn a like wrist worn heart rate monitor for the last seven years Whether it's Fitbit or Garmin or loop or whatever And I've noticed with weight gain and strength gains via to max Going down and resting heart rate going up and trending in a way that I don't like so I'm curious if you guys have any tips or tricks on how to do that while Not sacrificing You know mass and strength gains in the gym Dan any any particular reason? Why you want other than just you know, you know having a better VO to max Do you have anything specific? Why why you want to improve that? No, it's more of just like a longevity thing like a big fan of like Dr. Peter Tia and and some of his research on via to max being a sort of all-cause mortality reduction and closely followed by strength So I want to make sure that I can you know be Capable in any scenario that I encounter got it But I'm not like a professional athlete or anything Didn't you poke holes the other day in the Peter?  So you kind of want to be fit overall You're noticing that you're losing some stamina because you're not focusing on it That's totally normal and you don't want to lose your gains or do too much to the point where you start to lose muscle and strength Your best bet is you know sprints or sprint type activities So you could do like high-intensity interval training style Workouts a couple days a week will do a good job of maintaining a decent VO to max But it's more like resistance training than long-distance steady-state type cardio type training it so Sprints are really good on a on a bike or on yeah on at on hills or running Intervals with a jump rope where you're doing lots of speed or intervals on a rower These are short bouts the total session would be 20 minutes Maybe or 30 minutes at the most with these bouts of you know maximal exertion for 30 seconds and Then you slow down and wait for your heart rate to drop back down and then repeat it And that's this is a great way to build Stamina in a way that is more muscle friendly Then you know doing you know an hour or more of kind of steady-state type of training I want to point out to that increasing your VO to max is Something you could do pretty quick too though. So like it responds real quick. Yeah, like in a week So you you could literally improve it that fast So there I see lots of different things that we can do here We can take the advice that you're saying Sal or you can interrupt your strength training every four weeks or so with a Week of you know really pushing on the cardio and trying to get the VO to back up I mean there's nothing that says that he's not gonna lose all his muscle in one week You know so you're not gonna it's not gonna and it'll probably benefit you somewhat and your strength training So if you just want to so I don't do anything. I don't I'm not like this Hard up on my VO to max But this is I'll run the mile and I am probably not as frequent as every month But I like to get on there every couple months to make sure I can do it And that's and all I all I'm really trying to do is to say hey I want to be able to if I need to turn it on for a mile I can get after it and keep my mile time under say, you know eight minutes in case the coppers get you Well, no, I mean actually for the exact reason that Daniel saying like I I agree. I want You know, I want to be able to still get up there and move and do those things I think there's tremendous benefit as we age to still be able to do that stuff And I just don't want to lose it But I also care more about being stronger and building muscle and I tend to lean more that way But then what I'll do is I'll just you know every month or two Get on there and make sure that I can get that mile time under under eight minutes If I can't then you know for a week or so I'll train it and get after it and get it back down Yeah, I always consider this too just for athletic purposes and wanting to move Quickly and to be able to maintain that ability. So that's something I'll cycle this back in and To Sal's point doing a little bit more anaerobic type cardio Where I'm doing it in bursts because it's to me. It's always translated more to things that I'm more interested in doing and You know having those abilities to be more explosive. I want to train that so And it will help, you know in terms of like increasing that VO to max You'll get the benefit of that as well. You know, it's a great program Daniel I know you're doing maps anabolic, but have you tried maps strong? I haven't yet. No work sessions are amazing. Map strong is incredible.   So if you don't if you if you don't have that program will send it to you if you do Try it out. I think you'll really like it What I love about it's just people don't really consider that it's this is such usable endurance and usable Cardiovascular effort like when you're carrying things like you you have that kind of stamina you need, you know for everyday activities And it definitely has this massive carryover for everything else just functionally throughout the day Yeah, that would be I've got anabolic and performance and aesthetics. I'm strongest like the only one I don't have but You got it now. We'll send it over to you. Thank you guys so much. No problem. Thanks for calling Daniel Yeah, that I like You know, I mean, okay, so when it comes to training, of course There's specific skills you can train for and then there's specific I guess aesthetic goals and longevity goals and when it comes to like you like Justin was saying like usable stamina the Explosive kind of higher intensity stamina tends to be more usable in everyday life when you really need it Aside from being able to walk, right? So if you can walk good distances, that's probably pretty usable The sled heavy carries Sprints have always given me a tremendous amount of return and they and they've really never I mean you can overdo anything, right? But they've never taken away from my strength or muscle gains In fact oftentimes they actually contribute to the strength of muscle gains simultaneously while I'm building stamina In those things. Yeah, and just in terms of like doing cardio and like going long distances Like it's just there's a lot less opportunity for that now I know a lot of people like their days are just crammed with a million different Activities and work-related items and so to be able to kind of be efficient with that But also have a lot of like cardiovascular carryover doing work sessions to me makes the most sense I'd be really curious to see Some studies or research around, you know if you actually just ran a mile every day before your workout and you Less than eight minutes right and you just keep that mile time low So, you know, you can do it and you know, maybe and it obviously improve on it If you're doing it every single day, that's not gonna hinder you building much muscle I mean eight minutes are less especially when you get to the point where you're really your condition Right condition and you can you can run that mile. It's just more health focused Yeah, and I think that's his main that's what I got from this right So he listened to Peter Tia and him talk about all the research around, you know In having improved vo2 max hey run a mile every day run a mile every day before your training your training and keep that mile time down and I'm sure that you're gonna your vo2 max will be in a very good place for overall longevity It'll probably carry over into your training and support that it only make you better at your workouts you know other things that are overlooked they're like complexes three exercises put together they used to call these Giant sets and bodybuilding but or or even one look you do barbell squats at a relatively high intensity for 20 reps With a 45 second rest in between sets. It hurts gonna be screened.  You're gonna get a decent amount of stamina Now you're not gonna get that long endurance type of stamina, which you know if that's your goal Then you want to train that way but in terms of the health effects You don't need you can do those kind of shorter bursts of anaerobic activity and get quite a bit Hey, if you enjoyed that clip you can find the full episode here or you can find other clips over here And be sure to subscribe


If you're a Husky, not originally from St. Cloud, you have two really cool options. You can either vote at home, through in-person or an absentee ballot, or you can use your St. Cloud State address as well and vote right here on campus.