A warm welcome to the 39th lecture on the subject of advanced digital signal processing, specifically multirate processing and wavelets. In the previous couple of lectures, we have looked at some very interesting expositions by students who looked at applications of wavelets and multirate digital signal processing. One of the successes and one of the joys in conveying a subject is to see how well it is reflected in the understanding of the student and more so if the student could actually challenge the teacher by giving an alternate proof or by rectifying an inaccurate statement and by providing a more accurate statement with a more accurate exposition. So among the audience of the course has been other than the students who presented some excellent application presentations in the previous couple of lectures. One student who pointed out an inaccuracy in the way in which we dealt with a very fundamental principle but you know perhaps not as accurately as it should have been. So he pointed out that although it was correct that a function cannot be compactly supported both in time and frequency, he also pointed out that the reasons for this had not been accurately explained and I was very happy to see that there was this alternate proof which he brought out which I shall now ask him to present in this lecture, the 39th. So I present to you Soham Bosu one of the students one of the audience of the course who then looked carefully at this issue of trying to be compactly supported in both domains time and frequency and came out with a much better explanation of the inability to have compactness in both the domains. I have asked Soham to present himself both the theme which he is going to talk about the concepts that underlie the proof and the proof itself. So here we have Soham Bosu to follow. Hello everyone, today we will be looking at a particular proof of a theorem based on the uncertainty principle in Fourier transform. So here is what we are going to look at proof that a non-zero function cannot be both time and band limited at the same time. Basically by this we mean that a function cannot have a compact support both in its time domain and its Fourier domain. So this is already a well known theorem some proofs of this exist, we will have overview of the existing proofs and also the new proof that I have found. So here is what we mean by time limitedness and band limited or compactness in general. So suppose we have a function f which we draw like this by the support of the function we mean the interval outside which the function is 0. For example if I take a rectangular pulse like this the support of the function is this say it is a to b and the support is just the interval a to b. On the other hand if I take the Gaussian function the support is a whole real line. So function is say to be limited or say to have a compact support when this support is finite in length for example for the rectangular pulse. So let us first look into the general background on which we will be working. So we will be considering Fourier transform which is one way of looking into a signal from its frequency domain by frequency we mean sine waves, sine waves define frequency. So over here we will be considering translates of sine waves and their linear combinations. So if we do it in the complex domain we can bring out exponentials because we can split up sine functions like this sine omega can be written as e to the power i omega minus e to the power minus i omega by 2. So if I want to write the complex exponential e to the power i theta I can write it as i sine theta plus cos theta but again cos theta is a translate of sine theta which is pi by 2 minus theta. So basically like all other transforms the general idea of Fourier transform is to project a function on to the space of e to the power i omega t or e to the power minus i omega t. The mathematical formula is given like this for function f the Fourier transform if it exists is given by f of s equal to f t into e to the power minus 2 pi i s t integrated over d t over the whole of the real line. The function f is taken over the whole of the real line the domain of the function. For the inversion of this formula we need some conditions on f s. So if the inversion holds for generally when a function f belongs to l 1 or l 2 r then the inversion formula holds and it is just given by a similar formula there is just a change in the sign in the dependent variable of the exponential part. So now we have come across l 1 and l 2 or in general we also consider the spaces l p for functions. So let us define that spaces first by a space l p if we say that a function belongs to l p r what we mean by that is the p th power of the modulus of the function when integrated over the whole real line say this is t is finite. For example the function which we took rectangular function so it goes from minus a by 2 to a by 2 this belongs to l p basically all l p's for p real numbers. On the other hand the function sinc x by sinc x we mean sin pi x by pi x. So if we draw it goes at x equal to 0 the limit goes to 1. So it goes something like this but this function does not belong to l 1 because if we take the modulus of this the areas become positive here the areas cancel out a bit and the final integral becomes pi by 2. On the other hand here this parts become positive and if we fit in triangles into this we can see that the area of the triangles diverge. So the area of modulus sinc x is also going to diverge. So in this new proof we will be extending we will be first considering functions which belong to classes l 1 and hence we will be extending the class of functions to general l p p from 1 to infinity. So there is a very general inequality which states that when a function belongs to l p p between 0 to p between 1 to infinity it also belongs to l 1. So mathematically if belonging to l p p belonging to 1 to infinity implies that if belonging to l 1. So for the time limited function this can very easily be proved with holders inequality. Holders inequality states that for two functions f and g this integral is less than equal to so here are the conditions for holders inequality to hold. The main point is 1 by p plus 1 by q has to be equal to 1 and of course p and q both positive from this we can get the range of p and q. So if we take p to be less than 1 then q becomes negative. So p can only go from p and q both can only go from 1 to infinity equal to infinity is also allowed if I allow 1 on one side. So here is the proof of this theorem this another inequality called Young's inequality which states the following fact given any two real numbers positive real numbers a and b a b is less than a to the power p by p plus b to the power q by q. This thing is very easy to prove if we use the generalized a m g m inequality of course the same conditions hold over here 1 by p plus 1 by q equal to 1 or otherwise p by q is equal to p q. So for the proof of this we will be using the generalized a m g m inequality. Here is the generalized inequality given two numbers a b and alpha 1 alpha 2 are the weights of a and b we have this result a and b are positive real numbers. So over here let us try to fit in the numbers which we have been given.  Similarly, on the other side we get p q and the p with root of a to the power p whole to the power q and b to the power q whole to the power p taking real value real values of p and q and as they are not 0 we can cancel this and what we get is a to the power p by p plus b to the power q by q on the left hand side. On the right hand side we get the p q with root of a b to the power p q this is greater than equal to. So finally, we have got the Young's inequality. Now say there are two functions f and g such that f belongs to l p and g belongs to l q with the same conditions 1 by p plus 1 by q equal to 1. So, from Young's inequality let us define redefine f as f is equal to f by the p th norm of f the p th norm is this over the whole of d t g is also redefined the same way g by q th norm of g. Now let us put in this f and g now point wise as numbers f t and g t in the Young's inequality. So, f t to the power p by p g t to the power q by q is greater than equal to f t g t f t and g t can be complex all functions also. So, we take numbers by defining the module. So, if we integrate this over the whole of the real line we basically get out whole less inequality because on one hand we are going to get 1 because is the weighted p th norm of f t already we get 1 by p on one side plus 1 by q on the left hand side which equals 1 and on the right hand side we have got mod f t into mod g t the integral where f t and g t are weighted. Now if you bring out the weights and multiply with 1 we get that the p th norm of f into the q th norm of g is greater than equal to the integral of the modulus. So, this basically proves whole less inequality how are we going to use this in our proof. So, say a function f is time limited can we show that if it belongs to l p class it belongs to l 1 class also yes we can by choosing a suitable g. So, say the function that we are interested in is non 0 on this particular interval c only compactly supported in c and 0 everywhere else. So, we can write down only on c because everywhere else the function is 0 the whole less inequality. Now we can choose g in such a manner that g is 1 only on the interval c and 0 everywhere else in that case on the LHS only mod f remains on the RHS there is the p th norm to the power 1 by p of f and g has been replaced by 1 on the compacts support c. So, it is just 1 into d t to the power 1 by q which will basically give me out the measure of c to the power 1 by q. So, on the right hand side it is a norm of f p which we already know is finite and the measure of c is also finite as f has been assumed to be time limited. On the LHS as this is less this has necessary f should necessarily belong to L 1. Next we will move on to some tools that we will be using for this whole thing. The tools are a particular Fourier transforms and in linear algebra the Vandermont matrix. So, we will be considering the Fourier transform of the rectangular poles it is defined as to have a value 1 between a by 2 to minus a by 2 and 0 everywhere else. So, this is a real symmetric function. In Fourier analysis we know that a real symmetric function gives a real symmetric functions otherwise what we can do is in the Fourier integral for a symmetric function f t we can just take the cos 2 pi omega t part because the sin omega t will give us a 0. So, let us see what comes out over here. Now the interval changes from minus a by 2 to a by 2 because all the rest is 0 and the way we define the sinc functions this is just sinc a of omega and this structure of the sinc function the factor a is only inside the sin part is very crucial for our proof. Next the Vandermont matrix this matrix is defined as c j's are the different values for different columns and their corresponding powers in different rows are i minus 1. So, when it is written down the matrix becomes like this a 1 and square up to say a n a 1 to the power n on the other side if it is a n then it goes as a n square and stops at n to the power n. So, whenever we have a linear equation a x equal to y vectors we are always looking at invertibility of a so that when we know y we can solve x like this. Now Vandermont matrix have some very nice properties for which it can be shown to be invertible if we just know the entries. So, here it is how it works. So, a 1 to the power a 1 to the power n and in between let us replace any particular column by x. So, it goes as x x square till x to the power n. Now, let us take the determinant of the matrix. So, we can take the determinant here already let us call it d x. This will be a polynomial in x with the maximum power x to the power n. If we put x equal to a 1 if we put x equal to a 1 these two columns are going to be the same and the determinant will come out to be 0. Similarly, for all other a n's except say we have put this x at the j th column except a j for all other a's this determinant is coming out to be 0. So, the polynomial d x will be of the form some constant which is minus 1 to the power some power into pi x minus a l's goes from 1 to n, but not equal to j and the rest part. This rest part can be found out. Now, let us replace x by a j. So, it is a j minus a l. Similarly, in this matrix we can replace other columns by x also and do the same operation. So, for each and every column there will be a product like this, but in the determinant the highest power of a 1 that we can have is a 1 to the power n. With that limitation what finally comes out by combining all these products is a i minus a j say i less than j. So, that we do not repeat and i is not equal to j ever in that case it is just 0. Now, matrix m is invertible whenever the determinant mod m is not equal to 0. So, over here whenever my entries in the first row a 1 up to a n are mutually distinct we get that the determinant is not equal to 0. So, it is always invertible in that case. Next we will move on to our theorem that a function cannot be compactly supported both in its time and its Fourier domain. So, the function only non-zero functions we will be considering because in our proof we will be showing that the only function which is allowed to have this property is identically 0 function. So, let us first consider some very smooth functions which have compact support. For example, the function e to the power in between minus 1 to 1 and 0 otherwise. So, it can be shown the function has a structure like this e to the power x false of when x goes near 1 this part goes near infinity and because of the minus sign c x tends towards 0. Not only that if we take the derivative of c s c x what we get is the minus parts cancelled out. Now, as x goes near 1 this thing goes near infinity as before and the exponential part falls off. On the other hand the lower part of this thing goes towards infinity, but exponential falls off faster than any polynomial that can be made. So, overall this thing tends towards 0 if we take the limit and this holds for all derivatives not only c prime x it holds for all c p x p belong to natural number. From this it can be shown that the function is very smooth. So, minus 1 and 1 this function and all its derivatives tend towards 0. So, can we have a very well behaved function like this also for which the Fourier transform is compactly supported. So, there are various proofs which use various sort of things for example, analyticity, holonomicity and all those things. So, I will first state one proof and give a brief overview of it. When a function is band limited and say the function f belongs to l 1 then as f s can be written like this we can have a bound on the modulus of f s. Now, this is 1, but this is the l 1 norm of f 1. The Fourier transform of the derivative of f f dash t if it exists is going to be i omega f into e to the power minus 2 pi i omega t g t. Now, we have assumed f to be time limited. So, instead of the whole real line the only interval we can handle now we will be handling now is the compactly supported interval c over here also because omega is in c only t sorry t is in c only when you take the modulus again we will get a bound for f dash t. We can show that all these derivatives f dash t hold at all points when f is both time limited and band limited. So, over here f was band limited. So, for only some interval omega 1 to omega 2 it would have been non 0 f omega and for the rest it would have been 0. So, if that holds we can write the function f t by the Maclaurin expansion as f 0 plus f t into f t 0 and so on. Now, due to the assumptions on f all this number exists. So, this is basically a power series expansion and this holds for all t. In that case by the identity theorem if this power series is 0 on any open interval it is going to be 0 everywhere on the real line. For compactly supported f of course, the power series is 0 outside the compact support c. In that case only the 0 function f equal to 0 is allowed because the function reduces to 0 by the identity theorem. This theorem states that a function f which is holonomic which has a power series everywhere is defined completely by any its values on any open interval. Now, the whole motivation of this proof is new proof is that all this previous proofs that are there using holonomic city or celebrated Parley theorem this require complex analysis and requires master in complex analysis. You have to know quite some bit of complex analysis to know this basics. On the other hand we have many other ways of looking into signals. By unlimited signals of course, we have come across the Shannon 9 quest reconstruction formula or the Shannon we take a cardinal series whatever it is called the reconstruction formula is a way to look into this theorem with that. So, in Shannon's revolutionary paper in signal processing he gave this formula which is called the Shannon 9 quest formula. What is so revolutionary about it? So, this formula holds when f is band limited this f h k is the samples. So, say there is a function f like this can you have an operation for measuring the function f t where to take the value of f t at each and every point. Instead of that if there is some limitation on f over here the limitation is in the Fourier domain it is compactly supported. So, can we remove the redundancy if there is some. So, indeed there is some redundancy in the Fourier domain because it is band limited we can take only the samples at regular intervals. So, h is a sampling time it goes over all case and from that we can get back our f t we will show graphically how this works. So, say a function f t is compactly supported in its Fourier domain. We will be using distribution theory for this function for this proof, but in the actual proof I have used just real analysis no distribution theory, but for the sake of brevity we will be showing here with distribution theory which equally holds. So, if you take a train of delta functions by delta functions we mean say a function delta function is defined like this the integral over the whole real line is 1, but the function is 0 at all points except t equal to 0. So, this is not actually a function in the normal sense it is a distribution. For Dirac delta train on the other hand side we also get another Dirac delta train a series of pulses like this. So, if this is t the difference between this is 1 by t. Now we know that multiplication in the Fourier domain implies convolution in our real set the time domain and vice versa. So, over here we multiply in the time domain with delta functions and this gives a sampling it just picks up the values at those particular points and everywhere else it is 0. So, you get some weight delta functions a i a i plus 2 something like that. On the other side capital f is getting convolved with delta function, but there is another property of delta function that it is convolution with any function gives the same function back. So, over here this whole capital f will be repeated along the delta train at each and every delta it will be repeated and the whole thing will be summed up final outcome is something like this. Now, can we reconstruct the signal from this say we had the repetition repeated lobes like this. If we did ideal lobes filtering with the rectangular pulse over here on the other hand side on this side we would have got our actual f s back now Fourier transform is a 1 to 1 operation. So, on the other side the corresponding operation should give us back f only. So, over here we can already see what is going to happen the threshold conditions say the repetitions of f is overlapping like this we cannot do the ideal lobes filtering. So, the condition for ideal reconstruction is that the capital f lobes have to be well separated and this is the threshold condition where they exactly touch say this is band limited by s minus s to s. In that case our repetition rate has to be good enough so that they become well separated. If we make the delta trains closer in the time domain in the frequency domain they are going further apart. So, threshold condition this is 2 s and 1 by t. So, 1 by t should be greater than 2 s greater than equal to 2 s. So, t should be get less than equal to 1 by 2 s and when this equality is there that is the shun nyquist threshold condition. Now, let us consider functions which are time limited and band limited. So, after multiplying with the delta train we get lobes like this and sampling over here. For f t which is completely supported in the time domain we get only a finite number of samples which are non-zero everywhere else it is 0. On the Fourier domain if we take t to be sufficiently small or h which will be using over here on the other side they are well separated with 0s in between them whole intervals which are 0s. Now, we have freedom on the reconstruction we can multiply this whole repetition with a rectangular function which goes from this point to this point. Because this is just 1 it is going to give us f s till this rectangular function remains within this range. So, we will have some other freedom let us call the rectangular function it goes to sigma prime and sigma is the band limit. So, we will define a parameter m which is 2 h sigma 2 h sigma prime. On the other hand in the time domain by corresponding operations we get the series to be the cardinal series or the shun nyquist series to be f h k sinc of here the m comes in and this is a factor of m in the form. Now, f is time limited. So, k will not go from minus infinity to infinity it will stop somewhere say at n. So, now this is a finite series. Now, what I told before the property of sinc function this sinc becomes sin pi m sin sin sin sin sin sin sin sin sin t minus h k by h due to this structure of the sinc function this 2 m's are going to cancel out which is very necessary for our proof. Now, what was the condition on m? We did not find out exactly, but here was sigma here was the repetition. So, sigma prime had to be between this was say 1 by t. So, if we take sigma prime lies between 1 by 2 t to sigma then this exactly holds. We can rewrite that with m whenever m lies between 1 and 2 h sigma the reconstruction formula holds. If the reconstruction formula holds the series cardinal series converges to f t. Now, k goes from some finite number n minus n to n. Now, when this condition is satisfied the m for all real m like this this holds exactly in that case we can take this cardinal series to be a function of m and t and if we fix t f t becomes a number t minus h k by h becomes become numbers. So, we have a freedom of m which we can manipulate. Let us take the first derivative of c m t where t is fixed. So, we would not write t anymore. So, c prime m t minus h k by h comes out it gets cancelled pi also gets cancelled and we are left with f h k cos m pi t minus h k by h k going from minus n to m. Now, f t was already fixed. So, when we take the derivative with respect to m it is just 0. So, c prime m is 0 and here is this finite series. If we write it graphically. So, this is a plot of c m t with respect to m. So, when m lies between 2 h sigma and 1 this is going to give us f t. We do not know what is going out going on outside, but over here we are sure that this is a constant function. So, we can take as many derivatives as we want of c m t with respect to m. So, let us take second order derivatives of c prime m second order derivatives with respect to m for this cos it will just bring out pi minus t k h by h to the power 2 i. So, say this is the ith second order derivative 2 i plus 1 and all of them are going to be identically 0 because c m is a constant function when m is like this. We can write this whole thing down as a matrix equation. So, say how many samples did we have? It is minus n 0 and n. So, 2 n plus 1 we take 2 n plus 1 equations like this doing the derivatives. So, the Vandermont structure Vandermont will come later on. So, we get a structure like this and c k. So, what we are going to get that V c is equal to 0 writing down all the 2 n plus 1 equations. Now, this V has a Vandermont sort of structure. It is going to be invertible as long as t minus h k square k i square is equal to t minus h k to k j square. This thing does not hold and this thing only holds for a finite number of values of t i not equal to j. Now, we can work on the whole of the real line other than this values of t. Let us call this set of t as b. For all other t we get that c is equal to V inverse into the 0 vector. So, it is absolutely 0. So, c is 0, c k is 0 for all k. In that case either f h k is 0 or the cos part is 0 or both of them. Now, the cos part can be 0, cos part can be only when m pi minus this thing, this whole thing equals odd multiple of pi by 2. So, only for some finite countable number of values of m this can hold, but in our proof we have a whole range of m. So, this is not going to hold for all m if we take any other m. This will give that for all the other m f h k is equal to 0. Now, it can be showed that for a time limited and a band limited function, the functions are very smooth. So, all the derivatives exist they are bounded on both the sides and all these things. So, if these things happen our cardinal series C m, our cardinal series is going to converge exactly not only point point wise, but exactly it is going to converge to f t. So, if you have f h k is equal to 0 all these things are equal to 0. So, if it converges to f t on the other side I am left with only f t equal to 0 nothing else, but we are missing the set b t belonging to b, but we already know that f is continuous. So, if only for a finite number of points it is non-zero and for the rest of it is 0 that cannot hold it has to be 0 on the whole of the real line. So, here it is we have finally, got that only the function f equal to 0 is allowed to have this property. We have had here a very interesting exposition of the proof of the inability to have simultaneous compact support in time and frequency by Soham. I must appreciate the effort by this young man to look again at the concept carefully to identify a number of basic lemmas inequalities and equalities that underwrite the proof and to present it in such a lucid manner. Not withstanding the fact that there have been small glossing over of points during the proof which we can condone the idea has come out beautifully and he has been able to put across the ideas in a very accurate and a very beautiful manner. So, even though at places we might have found small inaccuracies in writing which you know one can condone I thought I should mention that because when the audience listens to it it might be found that there are some places where symbols have been missed and so on or you know written with a little bit of overloading of the symbol and so on, but that happens I think we can condone that, but otherwise I think the proof has been brought out beautifully and I think we should appreciate this young man for the beautiful proof and the beautiful way in which he explained the idea and it is satisfying in two ways. It is satisfying because it tells us technically how worthwhile it is to pursue the subject of at least trying to be reasonably compact or you know going towards a reasonable support in one domain when the other is compactly supported. We know that there are wavelets that are compactly supported. Now, we also know from the discussion from Soham that in the time in the other domain they cannot be in the frequency domain they cannot be compactly supported, but the challenge is how close you can go to constraining them in the other domain. So, the challenge becomes evergreen once again that is the technical reason. The personal reason is that here is a young man who has looked at the concepts of the course carefully and put in his own thought his own creativity to augment the ideas of the course. I encourage all students who at some time of the other listen to these lectures to take an example from these students who have enthusiastically participated and participate as enthusiastically. Thank you.


Breakout is here, there's only one breakout. And everything else is follow-through, is continuation. So, Amazon was a different trade here than it was here. Welcome to Access a Trader, the number one community for those who are committed to taking control of their trading in order to achieve success, profitability, and longevity. Thank you for joining us. Here's Dan Shapiro to help you find your edge, master your process, and own your future. Hey guys, good evening everybody, and welcome to another edition of the Access a Trader dot com nightly wrap up show, hope everybody is doing well. So let's talk about the market. So we've been on a really, really big run, right? The last seven, eight days, if you've been watching this broadcast, pretty good levels reclaimed. And the more important part was there was tremendous opportunity in the videos of the world, the apples, the Teslas, the Amazons of the world, the quite a handful of others that really did aggressive projections in their moves to the next supply zone. And they did fantastic, right? It absolutely fantastic. And the one thing I always maintain, markets don't go straight up. And no matter how good a market is, no matter how bad a market is, right? Even during 2007, 2008, when you felt like the world was gonna collapse, the financial system was very, very close to collapsing. Even those days that had monster moves to the downside and people are capitulating, unfortunately, losing their accounts, losing their homes, their livelihoods, all that, it never went straight down. There was always violence to the upside. And the same thing happens when a market is strong, right? When they reclaim levels, and we talked about that two days ago, it was kind of a day early with my interpretation of what could actually happen and it actually happened today. So we talked about a couple of days ago that the market was going to fight kind of in this area here of this 339 level that I believe there was gonna be at least the pregnant pulse, right? We talked about those levels. We talked about make sure that nobody is getting aggressive on names that were up six, seven days in a row that kind of mirrored the NASDAQ 100. Because usually what happens is when a stock or an ETF or whatever the case may be, whatever asset class you're trading usually gets in that battleground into supply. Usually what happens is it does get rejected. Now, again, I was one day off on my interpretation, but we kind of got it today. And it's not a point of I was right, I was wrong. It's to the point of kind of knowing where areas of supply are and where kind of danger zones are. It's kind of what we always talk about. We never talk about worst case scenarios. We just want everybody to make sure they're conscious of levels, important levels, either whether it's macro confirmation or apparently levels that could get rejected at a very good reflection point. So we wanna make sure everybody's prepared. That's all it is. And that you're not doing anything which arise wide shut that you're always keeping your eye on the ball and making sure you're getting involved with names that are macro or just coming out of channels instead of going into a potential of parabolic quote unquote blow off area. And today was the first time I can honestly say in the last seven, eight days that I really didn't find a lot of value, right? We'll get to the pivots in a second. Bowling was fine. Overstock was fine. Obviously congratulations to all you guys who had Tesla overnight. We said, I thought the stock could reach around that 790 area before supply kicked in and traded into the 786, 787 level pre-market. But the more important part, not in any individual name, the most important part was that stocks got tired today. You could see it. The Amazon ran out of juice. Netflix, they got upgraded this morning. Couldn't even put up a fight, right? If you get about even confirming macro just couldn't even put up a fight. Names that square were very, very strong really started selling. Names like Apple in the video. And the one thing that you'll see when you're doing your chart work tonight and this is kind of where you have a pretty good idea of what's gonna happen tomorrow. There's certain times I'll do my chart work and I go, there's nothing to think about, right? You don't need to be creative. Apple, Facebook, Amazon, Tesla, Netflix, Facebook, you don't have to be creative. Just look at those names. They start confirming channels. It's a green light. There's nothing to talk about. Today's not that day, right?  If you are looking for a lot of really great value tomorrow to the upside, you're probably gonna be very, very disappointed. Now look, can a stock go up? Of course, we're still in a very, very good, aggressive market, but stocks are tired. You can see it tired. And just to show you how a lot of these names really mirror what's going on in the QQQs, you can see it perfectly, right? Qs broke out and Amazon broke out, right?  And I don't want to use the word top. We're not close to the top. We're not even talking about revisiting any lows. We're just talking about day to day of what potentially could happen next. So if you see what happened with the QQs and you'll notice the same thing happened to Amazon, they tested and they broke below the five day moving average. And if you are new to this channel, number one, I'm not an investor for long term. I love Amazon. I love Tesla. I love Facebook. I love Apple. I love the whole economic structure. I think everything is higher. If you like something Apple tomorrow, it'll be higher in five years. We're not talking about that. We are trying to take advantage of price points and short term technical analysis in a 24 hour period that could give you a macro move. So if you believe in the five day moving average is the purest form, the shortest term of sentiment, then you have to really pay attention on what about the same next. So if you guys notice every single time on this big move upside, right? Every single time it hit the five day moving average, it bounced. It hit them five day moving average, it bounced. Hit the five day moving average, it bounced. Hit the five day moving average bounce. Today it hit the five day moving average and closed below. So what does that mean? Again, we're not talking about the market is going to zero. Let's just put that out there. This is still a very, very good market. We are way above the 324 area and the longer we stay above the 324 area, the bullish will be. So even if we do, and again, we know what he knows, we're just preparing for what the data is telling us. So even if we do back tests for a couple of days, and you can see it, maybe get a move down to 332. And again, this is assuming tomorrow's price action continues and we confirm taste channels and the cues. And this 335, 50 level is going to be important for tomorrow. You know why? Because that's number one, that's underneath the five day moving average. And number two, it's going to confirm it for a possible move down to 332. And if we start really getting aggressive, maybe it goes down to that 27 and if we really start getting aggressive, maybe it goes back down to the 24 where this whole rally started. But again, before we put the cart in front of the horse, we just want to take it day by day, right? Just try to take advantage of channels. So when you're looking at a lot of charts tonight, you're going to see a lot of really tired names, right? Amazon had a huge move. Remember guys, the breakout was right here. And this is what we kept on talking about for day after day, right? The breakout is here. There's only one breakout and everything else is follow through, is continuation. So Amazon was a different trade here than it was here, here, here and here. And now it's here. So if you bought the stock anywhere in this level and you scream at yourself, Amazon stocks, it failed the breakout. No, it didn't fail the breakout. It broke out here. You're just now in the position of a continuation rally that's stalled from a very short term. So if you believe in the theory that if the five day moving average is the shortest term sentiment in a chart, well, what do you think is going to happen tomorrow? If Amazon starts confirming down and starts losing this channel here, then yeah, there is 60, 70 points to the downside. I know it sounds scary, but Amazon ran 450 points in seven days. So macro-wise, if you are an investor, right? All it's doing is just back-testing some of the move it put in the last week and a half, healthy, right? That's the one meaning you want to kind of drive forward if you are an investor of this market. Tech needs a rest. That's all it is, right? And if it doesn't rest and starts reclaiming channels tomorrow, well, what's the worst case scenario?  You are still maintaining a very, very good grasp in the bull case and the bull thesis still exists and it's still alive. But again, one thing that you have to pay attention to if it does start losing today's levels, then you do have a lot of room down. Look at names like NVIDIA, right? NVIDIA had a monster one, right? Huge one. We've covered NVIDIA and not the bull, the nausea, right? It broke out right here, the 541 level. It didn't break out here or here or here, right? So now we're kind of in the same mirror images on the cues. And again, it's not at the five day moving average, but if it starts losing a couple of candles, why can't you get a seven to 15 point move to the downside? It's still healthy. Think about it. The breakout happened at 541. Why can't it go to 580 and still be good, right? It's still incredibly, incredibly strong. Look at Apple, right? A name that we covered through nausea as well. Reclaimed the 50 day moving average went on a really great run. Apple put up, let's see here, Apple put up nearly a $9 move in seven days. That's like the equivalent because Apple is so thick. That's like the equivalent of Tesla going up 60, right? It really is. So the idea that, again, it closed on the five day moving average. And if the bears, well, I don't wanna say the bears, but if the buyers continue their quote unquote strike for two days in a row and needs a breather, why can't it go back down to like 129? It'll still be two, two and a half dollars above where it broke out. So resets are healthy, right? Resorts are really, really healthy. Just the way stocks don't go straight down. Stocks don't go straight up either. I don't care how aggressive the market is. Stocks need breathers, it's organic. Names need to reset. Buyers need to come back. Sellers need to kind of react. And the most important part is that you're looking at the market from a non-buy sort of way. Instead of just saying to yourself, shut up, then Apple's going to 150. Maybe it will, but it's probably not going 150 tomorrow. And that's the most important part to understand. We're not trying to predict the future where Apple is gonna be in six months a year from now. We're just trying to take advantage of what the data is telling us for the next day. So not only can you be prepared as an investor if you're trading both sides of the market. I trade both sides of the market for the last seven and a half, eight days of it is extremely bullish. And now I'm just careful. That's it. I do like some names to the upside tomorrow, but I'm very, very conscious. I'm trying to keep things very simplistic for tomorrow. So for example, if Apple loses the range to the bottom, I'm gonna sell some Apple. If Amazon starts, you know, can't rally and it puts in an opening range low, blows today's range and rallies and takes that opening range low. Yeah, there is a lot of room in the stock. And if you go through a lot of names, you'll see they're all the same, right? Square, close, blow, you know the lit. Again, who's gonna feel sorry for Square? Square just rallied in the last seven days from 236, right? From 236 to 280. Okay, you're telling me, even if the stock has one more day of backtests and it goes to the rising 10 day moving average of 250, the stock is still $15 where it was when it started this whole rally seven days ago. So again, if you're an investor, there's not a lot of red flags yet. The longer we stay above, the longer we stay above 324, the more constructive the rest is. And you know, look, if everything goes well and we do backtests maybe a couple of days, right? The market holds those levels, reclaims those levels, starts going back up. Who's better than you? So the bullish thesis is still very, very much alive. The longer we stay above the 324, we'll have a different conversation if we ever lose that 324 and close below the 50 day moving average, but that's not here nor there. And it's obviously not for a discussion for tomorrow. Tesla, right? Had this monster, monster breakout, huge breakout, yes, they reclaimed the 50 day moving average, put up a $68 candle, right? We talked about the levels yesterday in the video, 770 the first stop, 790 is the next supply. And what the 786 today, and for all you guys who made some sales, great job. I mean, that's exactly what you have. If you have a runner, the most important part to understand is again, the longer Tesla stays above that 720 area, the better it'll be if the market stabilizes and catches a bit in the next couple of days, let's go back higher just like everything else. And the one thing, especially on Tesla, it's a very volatile stock. If you are appreciating levels, if you believe in the theory that stocks go from supply to supply to demand to demand, well, that's exactly what Tesla did in the last 24 hours, took out this whole supply at 720, went to the next supply zone, right? 786, 787 pre-market and kind of went down just like with everything else. So the idea that a stock is good, a stock is bad, it's only as good as its technical level. So all in all, I do believe the market is tired. I do believe, listen, if we start confirming to the downside today's levels, yeah, we could get a pretty decent move to the downside with some decent cashflow opportunities. But again, nobody's calling for a bear market, nobody's calling for to retest the lows. All we're trying to do is put ourselves in a position to take advantage of short-term price action. So let's talk about today's pivots. Again, not a lot, there really wasn't a lot here. So again, this was the pivot to the upside yesterday, that 710, 719 area, the stock went up what? 75 points in 12 hours, right? I mean, great move.  As the stock was coming in 20, 30 points or so, you start still seeing this week's expiration for Friday, the 800s next week, had some pretty big bets. So they're buying it into weakness. So again, as long as Tesla holds that 720 area and doesn't close below it, I think the thesis on that is fine. But again, a 75 point move in 12 hours, really, really hard to complain there. Anyway, so eBay 65 never came close to that. Good job for all you guys who traded overstock. 80 needs to reclaim, 81.58 is a very, very big area. It's the previous highs off the channel, so it needs to confirm as well. So here was overstock, right? It took out that 80, took out that 81.58 and traded all the way up to the 84 and change area before everything got pulled. So good job for all you guys who traded that. Boeing, nice little move on Boeing. They were coming to some aggressive call buying on Boeing. They were coming for the 65s, the 75s, 255 needs to build. Here was Boeing, right? Here was Boeing, here's the whole area of 55 went right into supply at 258 and change. Again, not a huge move, but again, it is what it is. ZS never got to the 200 level dash quick pop, not a huge pop, but a quick pop, right? 150.5, 151 needs to build. Here was dash, right? Took out the 51.5, 50, 51.5, went to like 53 and change and then everything got pulled all together. And this is where I could see, this is kind of my whole point that the moves weren't sustained today. You could see how tired a lot of these names are. Peloton never got to the 125 area. Roku traded right to supply guys. If this thing ever reclaims that 96, 97 level, it should go. It got rejected perfectly right into supply. So nothing there on Roku, dash take on the way up, Boeing take on the way up. Like I said, it will come in for the 260 weeklies. Tiger, 21 only put up like a 35 cent move and they completely got destroyed. So nothing really on Tiger, Qs, they look. Again, it traded down to this area, had an initial bounce and then it lost it. Again, this is why, again, I say stocks are getting tired, there's not gonna be a lot of value tomorrow. So before you get superly aggressive, if you're only a long biased trader tomorrow, if you are only one side, tomorrow might be a day you kind of wanna step back and maybe let go of the gas because again, if you look at charts today and I really encourage everybody to do so, you're not gonna see that big value.  So be very, very careful. But all you guys who do trade both sides of the market, again, we are watching the downside ranges and if these things do confirm, especially the bigger names of the Amazons and the Vidya's, the Apples, the stocks had a really, really big runs. Those are the ones we could probably take advantage for cash flow moves back to the downside. Guys, have a great night everybody. God bless and I'll see you all tomorrow.


Scientific Advertising. Written by Claude C. Hopkins and narrated by Paul Jones. Scientific Advertising. Chapter 1. The time has come when advertising has in some hands reached the status of a science. It is based on fixed principles and is reasonably exact. The causes and effects have been analyzed until they are well understood. The correct method of procedures have been proved and established. We know what is most effective and we act on basic law. Advertising, once a gamble, has thus become under able direction. One of the most safe business ventures. Certainly no other enterprise with comparable possibilities need involve so little risk. Therefore, this book deals not with theories and opinions, but with well proven principles and facts. It's written as a textbook for students and a safe guide for advertisers. Every statement has been weighed. The book is confined to established fundamentals. If we enter any realms of uncertainty, we shall carefully denote them. The present status of advertising is due to many reasons. Much national advertising has long been handled by large organizations known as advertising agencies. Some of these agencies and their hundreds of campaigns have tested and compared the thousands of plans and ideas. The results have been watched and recorded so no lessons have been lost. Such agencies employ a high grade of talent. None but able and experienced men or women can meet the requirements in national advertising. Working in cooperation, learning from each other and from each new undertaking, some of these people develop into masters. Individuals may come and go, but they leave their records and ideas behind them. These become part of the organization's equipment and a guide to all who follow. Thus, in the course of decades, such agencies become storehouses of advertising experiences, proven principles, and methods. The larger agencies also come into intimate contact with experts in every department of business. Their clients are usually dominating concerns, so they see the results of countless methods and policies. They become a clearinghouse for everything pertaining to merchandising. Nearly every selling question which arises in business is accurately answered by many experiences. Under these conditions where they long exist, advertising and merchandising become exact sciences. Every course is charted. The compass of accurate knowledge directs the shortest, safest, and cheapest course to any destination. We learn the principles and prove them by repeated tests. This is done through keyed advertising by traced returns, largely by the use of coupons. We compare one way with many others backward and forward and record the results. When one method invariably proves best, that method becomes a fixed principle. Mail order advertising is traced down to the fraction of a penny. The cost per reply and cost per dollar of sale show up with utter exactness. One ad compared to another, one method with another. Headlines, settings, sizes, arguments, and pictures are compared. To reduce the cost of results, even one percent means much in some mail order advertising. So no guesswork is permitted. One must know what is best. Thus mail order advertising first established many of our basic laws. In lines where direct returns are impossible, we compare one town with another. Scores of methods may be compared in this way measured by cost of sales. But the most common way is by use of the coupon. We offer a sample, a book, a free package, or something to induce direct replies. Thus we learn the amount of action with each ad in genders, or cost per dollar of sale. These coupon plans are dealt with further in the chapter on test campaigns. Here we explain only how we employ them to discover advertising principles. In a large agency coupon returns are watched and recorded on hundreds of different lines. In a single line they're sometimes recorded on thousands of separate ads. Thus we test everything pertaining to advertising. We answer nearly every possible question by a multitude in its traced returns. Some things we learn in this way apply only to particular lines, but even those supply basic principles for analogous undertakings. Others apply to all lines. They become fundamentals for advertising in general. They are universally applied. No wise advertiser will ever depart from those unvarying laws. We propose in this book to deal with those fundamentals, those universal principles. To teach only established techniques. There is that technique in advertising as in all art, science, and mechanics. And it is, as in all lines, a basic essential. The lack of those fundamentals have been the main trouble with advertising of the past.


But you already knew that if you had to figure out how to cure it. How is it possible to turn such a shitshow into something good? This is one of my most asked questions in the comments in my DMs about mental health. Therefore today is a really special episode. So this guy Dan Mace thinks he knows how to defeat depression. Well let's find out. What is up everybody this is Chris from the Rewired Soul where we talk about the problem but focus on the solution. And if you're new to my channel my channel is all about mental health. What I like to do is pull different topics from the youtube community and try to use those to teach you how to improve your mental and emotional well-being. So for any that kind of stuff make sure you subscribe and ring that notification bell. So all of you who have been subscribed for a while you're like hey the set looks different again. Yeah I'm on the couch right now recording this until we get finished unpacking and all that I've just been moving around.  And that's actually something we're going to touch on in this video where we're talking about Dan Mace's new video.  So I love Dan. I think he's in recovery too. So those of you who don't know me like I've been sober for the last six and a half years almost but yeah I think Dan's in recovery. He's made some other videos on mental health. He had one about like falling down and like talking about or it was like falling off a building or something like that and it was about anxiety. He's an awesome film creator. I was introduced to him like a lot of people were when he helped Casey Neistat launch 368. But yeah Dan's a really cool guy and I love I love his videos. But yeah throughout this move and stuff I wasn't able to catch this video but just now I just sat down I'm like okay need to watch this and need to talk about it all right. But anyways there will be a link down to his video down below. But yeah I just wanted to kind of break down some of the ideas and things that he was talking about in there. And there's definitely some things that you guys can learn from it and it's really interesting too because there's things that people don't say and I wonder if they know what they're talking about. Like he knows what he's talking about but I wonder if they know what it is. But anyways I'm gonna quit jumping ahead of myself. Let's get started. Out of nowhere I just started feeling really sad when on the outside my life seemed great. It seemed perfect.  But none of that mattered. I couldn't feel it. Almost in a figurative sense it was like I couldn't see color anymore. And after speaking to a professional I figured out that I have depression which then turned into anxiety. So one of the first things that we can all relate to, not all of us, but some of us can relate to is that that first clip that I just showed you where he's talking about how he had stuff but he wasn't happy. And this is something that I've been trying to beat into your guys' heads forever. You know what I mean? And this is important to realize, like just for a couple reasons. Realize that accumulating stuff isn't going to necessarily make you happy, right? You know, the girlfriend, the job, the car, the money, the stuff, the things, the fancy hobbies, whatever they are. There was no guarantee on that happiness. The thing that always reminds me of that is just when you see celebrities who are struggling with depression, right? They seemingly have everything. Like yeah, it's nice to not have so many financial burdens and all that, but there's no guarantee on your happiness. The best metaphor for this is surfing. The ocean creates this pandemonium and we ride that ship. We don't try control it or tame the beast but rather flow with it. Mindfulness in a second, but I love that. So like that surf analogy where he's talking about riding it, yes, yes, yes. So some of you who have been starting to meditate or getting into mindfulness just in general, here's what your homework is. Find the mindfulness meditation called rain, okay? So something I've talked to you guys about is that we're constantly in a state of attraction or aversion, right? Give us the things that make us feel good. Get rid of the things that make us feel bad. The rain exercise for meditation teaches you how to ride your emotions no matter what it is. You're no longer trying to run away from it. You're learning to embrace it. And what you're doing is rewiring your brain. So rain stands for recognize, accept, investigate, and note, okay? And the biggest part of that is the accept part, right? And basically when you get to that stage of the meditation, you're just noticing what's happening and you're just letting it happen.  Sometimes it's even your anger. This is actually an exercise that I teach to clients at my drug and alcohol treatment center. This is something that's very good for cravings as well. If you're somebody who's trying to quit, you know, eating certain foods or whatever, like this is a great exercise to kind of ride those waves, okay? I remember somebody who was in a meditation class with me, like she was trying to have better eating habits and she timed her craving, which I thought was genius. She timed her craving during the rain exercise and she said like her food craving lasted like, I don't know, six minutes and 45 seconds. But she said it was very helpful because she was present and aware for it. So now she knows like, okay, now that I have this craving for chocolate or whatever it is, it's only going to last six minutes and 45 seconds. It's not going to last forever. It's going to last this long, you know what I mean? People train tirelessly through cognitive behavioral therapy to get this right, but it's actually really easy. You don't think about what's going to happen in a few moments time. You don't think about yesterday. All you think about is now. So right there, right there, everybody. Like this is what I'm saying. Like I don't know if Dan knows it or I don't know if people are just more subtle about trying to get you all to practice mindfulness. But what he's talking about right there is just be present. That's all mindfulness is. Mindfulness is being present, being here, being now. All of you who struggle with anxiety like I do, like anxiety is typically in two different modes, right? We're either tripping about the future, things that haven't happened yet, or worried about things that we already did in the past, that we have no power to change, right? So mindfulness is a practice where we're constantly bringing ourselves back into the moment, back into the moment.  Then you bring it back to the right now, right? It's like, oh my God, what's going to happen on that date? I go on later. Nope, you bring it back to right now. Oh no, what am I going to have for dinner? Nope, you bring it back to right now. And that's all it is. You're training your brain to come back to the moment and recognize what is happening right now. This is my secret potion. It's the ability to think slightly deeper, to be more connected on an empathetic level.  Those last part right here, like, this is something that I've been trying to do on my channel. I've made a few videos about this. And it's weird because like Dan, I think changed the name of this video. Like when I first got linked it, it said like, you know, mental health is your superpower and now it's like called defeating depression. But yes, this is something I've been trying to teach you.  I think I've talked about bipolar being a good thing. And you know, even though there's struggles, like we have to look at these and say like, how can I benefit from this? So somebody like Dan, he takes this as some kind of creative outlet. So let me share with you my experience. So I have a brain that does not stop, okay? And I hated it forever. Again, I wasn't accepting it, right?  And I was just like, why is my brain always racing?  Like I couldn't go to sleep because my brain just wouldn't stop. Well, like what Dan's saying, I've turned that into a positive. I just, I get comments all the time like, Chris, how do you have so many ideas for videos? Chris, how do you have so much energy? Chris, how do you do this? Because I've learned to take all these racing thoughts and get them out somewhere, whether it's videos or writing it or doing it, like in some way and expressing it. So I've taken this thing that was, you know, a weakness and I've turned it into a strength. Now I have a YouTube channel that's growing. I'm able to help other people and stuff. So this is what we have to do. We have to reframe this stuff and say, okay, maybe this isn't a weakness. Maybe this is actually my strength. Lastly, good for you, Dan. Good for you, brah. Or Brew, how do you say it? Anyways, good for you. Like I'm glad, you know, that he's supporting BetterHelp and BetterHelp's helping him out. You know, like BetterHelp is an organization that I've stood by this entire time. Tristan, my amazing girlfriend, she uses BetterHelp online therapy. She actually met her therapist here in Las Vegas through BetterHelp online therapy. I have other friends who use it. So anyways, like you guys, if you would like to, like Dan said, like no pressure, like I, like here's what I would say. I don't care where you get help. I literally don't care. I don't care if you get help just watching my videos, joining our free Facebook support group, going to a therapist somewhere else, using your health insurance or whatever. I don't care where you get help, just get help for the love of God. Just get help, okay? But if you want to try BetterHelp online therapy, there's a link down in the description below. It's an affiliate link and help support the channel. Like I said, I know a lot of people who have great experiences and like it can't help you, especially if you know, there's not therapists in your area. You just want to do it from the comfort of your own home, whatever it is. Anyways, that's all I got. But yeah, let me know down in the comments below, like how have you been able to turn your mental illness, if you will, or your weakness into a strength, right? Does it help you be more creative? Like do you have some kind of outlet? Is it music? Is it making things? What is it? You know what I mean? So like let's have a conversation down below. Anyways, that's all I got for you with this video. If you liked this video, please give it a thumbs up. If you're new, make sure you subscribe and ring that notification bell because I make a ton of videos. And a huge, huge, huge thank you to everybody supporting the channel over on Patreon. You're all amazing. And if you want to, boom, check out the Rewire & Soul Shop. You can click or tap right there on that icon. You can get one of these shirts. All right, anyways, thanks again for watching. Turn your weakness into a strength, and I'll see you next time.


As of recording this, we're only one week out from Guardians of the Galaxy Volume 3, a film that James Gunn promises will be an emotional ending for our characters. Whatever that means. I'm excited. And I'm equally excited by these early impressions from movie critics that are better than me and got to see this film early. Let's dive in, see what they have to say. Spoiler free. As of right now, the Tomato Meter, it's a weird thing to say, sits at 81% for Guardians of the Galaxy Volume 3, which, if you don't know, means 81% of the critics that saw it thought it was better than it was worse. I know, it's complicated over there. That doesn't mean that the movie is only 81% great. It just means that of the critics that saw it, 81% say it's a fresh film, not a rod and film. And we're gonna dive into a few from each right now. First, let's start with some negatives. Rodrigo Perez says, and I'm gonna do voices for these people, because why not? In James Gunn's Overt attempt to make Volume 3, the most heartbreaking Guardians movie, and make you feel all the feels, the writer slash director just tends to overcook a lot of the scenes with weepiness that feels forced. Rodrigo was not too impressed. You know what I'm not impressed with? The fact that he gave Thor Love and Thunder a fresh score. Yeah. We're gonna be discrediting all of these reviews. Let's move on.  The new characters in the saga are hollow and horribly ridden. Most of the returning hero characters are often smug and irritating. You know what else is irritating, Carla? You gave Ravenous Sua a negative score on Ron Tomatoes. I don't actually know what Ravenous Sua is, but I agreed with you on pretty much everything I saw you review. So, okay, maybe that one's a legit, scary, worrisome review, but we're gonna move on. Peter Martin says, a clumsy, overlong series finale that never strikes a balance between the touching story at its core and the numerous moving parts hovering in that story's vicinity. He gave the Super Mario Brothers movie a rotten score.  Get back on your high horse and ride over to your cigar lounge with your buddies and talk about how great Bo was afraid as you prick. Let's move on. El Merino says, cue the Sarah McLaughlin music because James Gunn is trying hard to make you feel things. This ends up as a total mess that relies on Disney-style emotional manipulation. Gave Blonde a fresh score. Gave that Anna Deyramas pile of crapa positive there on Ron Tomatoes. So I don't really care what you have to say because that movie was awful. Let's get some positivity in our life. See what some of the pro critics had to say because I'm definitely not biased or going in thinking that this movie's gonna be great. That's not what a good critic does. Chris Joy says, when the movie is focusing on what it's done so well in the past with the topic of family, the story beats are emotional and impactful, shining with real heart. And just to show I'm fair and honest, he also gave Ant-Man and the Wasp Quantum Shidia, good old fashioned positive on the site. So. So maybe I don't really care what Chris has to say. Kristen Akuna says, Gunn delivers an emotional send off for his reg tag group of misfits that possesses the humor and heart of the first two movies. Devoid of any real stakes. Volume three is not the best in the trilogy, but it's better than the last outing.  Gave that one a fresh. Mark Hughes. For some reason I'm sarcastic clapping even though he genuinely liked the movie. Bravo! Everyone involved. Everything you've come to know, love and expect from a Guardians of the Galaxy movie is here again.  Actually, you know what? He said, and as usually, Gunn has ramped it all up even better than before. And this is a Forbes. This is a person that works at Forbes apparently.  And he also gave Ant-Man and the Wasp, Quantimania a positive, a fresh. You're done. I don't care what you have to say. One more positive. Oh God. Grace Randolph. America's sweetheart. The Marvel magic is back. James Gunn continues to evolve as a filmmaker, growing more confident and ambitious. The cast also does stellar work to deliver a thrilling, but also moving superhero movie. Discredited? It's Grace Randolph. Those are just some of the thoughts and critiques by the experts out there. The ones that saw Guardians of the Galaxy volume three early, I'm not bitter. I gave you some pros. I gave you some cons. I think the overall takeaway we can get from this is that the movie fulfills a lot of promises. It's emotional, maybe too emotional, maybe a little over the top.  I'm okay with that. It's Gunn. He likes to have some fun. Let Gunn have some fun is what I say. I'm excited for this film. I'm not overly optimistic because the MCU has been kind of a disaster since Endgame. There's been some good ones, but for the most part, I've not really liked what they've been putting out. But in Gunn, I trust. Haven't really watched any of the trailers, except for I think the first one that released, haven't watched any clips. I know I'm gonna see it. I wanna be surprised. I wanna be wowed. And these critics so far aren't really turning me away. And that's good. We should always make up our own mind at the end of the day anyways. So I wanna hear from you in the comments. Are you concerned? Are you worried? Have you already seen the movie? What did you think of it? Are you Grace Randolph? Let me know, Grace. That's just why, no, no, no. I don't wanna hear from her. Please like the video if you enjoyed the commentary. Subscribe if you haven't already as I post tons of movie content each and every week. Would love to have you stick around. And hopefully, see you next time.


Hello, Namaste Myself, M.S. Basargaon, Assistant Professor, Department of Humanities and Sciences, Walsh and NISTOR Technology, Solapur. Now, learning outcome. At the end of this session, students will be able to solve homogeneous linear equation. Now, we will see what is Cauchy's homogeneous linear equation. An equation of the four x raise to n d raise to n y by dx raise to n plus k 1 x raise to n minus 1 into d raise to n minus y by dx raise to n minus 1 plus so on plus k suffix n minus 1 into x dy by dx plus k n y equal to x, where k 1, k 2, k 3 and so on k n are constants and x is function of x is called Cauchy homogeneous linear equation. Now, such equation can be reduced to linear differential equation with constant coefficients by using the substitution x equal to e raise to z, that is z d is equal to log x. Then we can calculate the value of dy by dx, that is dy by dx is equal to dy by dz into dz by dx. Here, dz by dx is 1 by x, which is equal to dy by dz into 1 by x. Multiplying throat by x, you get that is x dy by dx is equal to dy, where here d stands for dy dz. And differentiating both side dy by dx with respect to x, we get similarly x square d square y by dx square is equal to d into d minus 1 of y. Similarly, we get x cube d cube y by dx cube is equal to d into d minus 1 into d minus 2 of y and so on. After making the substitution in equation number 1, that is in Cauchy homogeneous equation, we get a linear differential equation with constant coefficient, which can be solved by the method of linear differential equation with constant coefficients. Now, solve the following linear differential equation with constant coefficient d square y by dx square minus phi u dy by dx plus 6 y equal to e raise to 3x. Pause the video for a while and find the solution of this equation. Now, I hope you have solved here given equation d square y by dx square minus phi u dy by dx plus 6 y equal to e raise to 3x. And in symbolic form, it can be written as d square minus phi u d plus 6 of y equal to e raise to 3x. And here, we have to calculate first CF, that is with the help of auxiliary equation. The auxiliary equation is here, d square minus phi u d plus 6 equal to 0. Now, factors of this polynomial are d minus 2 into d minus 3, that is d minus 2 into d minus 3 equal to 0, that is d equal to 2 and d equal to 3. Now, we can write the complementary function as CF is equal to c 1 e raise to 2x plus c 2 e raise to 3x. Now, we can calculate pi, equal to 1 by d square minus phi u d plus 6 of e raise to 3x, which is equal to x into 1 by 2d minus phi of e raise to 3x. Since, we know 1 by f of d of e raise to ax is equal to x into 1 by f dash of d e raise to ax if f of a equal to 0. Here, a is 3, if you put d equal to 3, here the denominator will become 0. Now, which is equal to x into 1 by, now we can replace d by 3 here, that is 2 into 3 minus phi e raise to 3x, which is equal to x into e raise to 3x. Therefore, the complete solution of the given linear differential equation is y equal to c f plus p r, that is y equal to here c f is c 1 e raise to 2x plus c 2 e raise to 3x and pi is plus x e raise to 3x. Now, we will solve the homogeneous differential equations, solve x square d square y by dx square minus 2x dy by dx minus 4y is equal to x raise to 4. Now, this is a homogeneous linear equation and to express this homogenous to linear, the standard substitutions are x equal to e raise to z, that is z is equal to log x. So, that x dy by dx is equal to d of y and x square d square y by dx square equal to d into d minus 1 of y, where here d stands for d by dz. Now, substituting all these values in the given equation, we get d into d minus 1 minus 2d minus 4 of y equal to e raise to z raise to the power 4, that is by simplifying this we get d square minus 3d minus 4 of y equal to e raise to 4z, let us call this as a equation number 1. Now, which is a linear differential equation with constant coefficients and its auxiliary equation is d square minus 3d minus 4 equal to 0, that is the factors of this polynomial are d plus 1 into d minus 4 equal to 0, that is d equal to minus 1 and 4. Here both the roots are real and different, therefore you can write cf as a equal to c1 e raise to minus z plus e raise to 4z. Now, we will calculate the second part, that is pi, pi equal to 1 by d square minus 3d minus 4 of e raise to 4z, again if you replace here d equal to 4 here, in place of d square and d we get 0, therefore here we can multiply by z, which is equal to z into 1 by derivative of d square is 2d and derivative of minus 3d is minus 3 and derivative of minus 4 is 0 of e raise to 4z, since 1 by f of d of e raise to ax is equal to x into 1 by f of d of e raise to ax, if f of e equal to 0, which is equal to no, we can replace here d by 4, that is z into 1 by 2 into 4 minus 3 of e raise to 4z, which is equal to z by 8 minus 3, that is 5 of e raise to 4z. Hence, the solution of equation number 1 is y equal to cf plus pi, that is equal to c1 e raise to minus z plus c2 e raise to 4z plus pi is z by 5 e raise to 4z. Putting z is equal to log x, that is e raise to z equal to x, we get y equal to c1 x raise to minus 1 plus c2 x raise to 4 plus log x by 5 x raise to 4, which is the solution of the given equation. Now, we will solve one more example, solve x square d square y by dx square plus 2x dy by dx minus 12y is equal to xq log x. Now, we can make use of the solution x equal to e raise to z, that is z is equal to log x, so that x dy by dx equal to d of y, x square d square y by dx square equal to d into d minus 1 of y, where d stands for d by dz. Now, substituting these values in the given equation, we get d into d minus 1 plus 2d minus 12 of y is equal to e raise to 3z into z. By simplifying this, we get d square plus d minus 12 of y equal to e raise to 3z into z, let us call this as a equation number 1 and which is the linear differential equation with constant coefficients. Now, its auxiliary equation is d square plus d minus 12 is equal to 0, that is factors of this polynomial are d plus 4 into d minus 3 equal to 0, that is the d equal to minus 4 and 3. Therefore, complementary function for these values, we get c of e is equal to c1 e raise to minus 4z plus c2 e raise to 3z. Therefore, now we can calculate p i, p i is equal to 1 by f of d, that is here f of d is d square plus d minus 12 of e raise to 3z into z, which is equal to, we can write e raise to 3z and we can replace d by d plus 3, that is d plus 3 whole square plus d plus 3 minus 12 of z, since we know 1 by f of d of e raise to ax into v is equal to e raise to ax into 1 by f of d plus a of v, which is equal to e raise to 3z 1 by d square plus 6d plus 9 plus d plus 3 minus 12 of z, that is equal to e raise to 3z into 1 by d square plus 7 d plus z. Now, by simplifying it, p i equal to e raise to 3z 1 by 7d, now shifting this to the numerator, we can write 1 plus d by 7 raise to minus 1 of z, which is equal to e raise to 3z by 7d. Now, expanding this by using binomial expansion, we get 1 minus d by 7 plus d by 7 whole square minus so on of z, since we know the binomial expansion 1 plus x raise to minus 1 is 1 minus x plus x square minus x cube and so on, which is equal to e raise to 3z by 7d. Now, operating this bracket on z, 1 into z, that is z minus 1 by 7 into d of d, which derivative of z is 1, that is 1 by 7, which is equal to e raise to 3z by 7 into bracket, what we get z minus 1 by 7 d z, which is equal to e raise to 3z by 7. Now, integration of this z minus 1 by 7 is z square by 2 minus 1 by 7 z, thus the solution of equation number 1 is y equal to c 1 e raise to minus 4z plus c 2 e raise to 3z plus p i, that is e raise to 3z by 7 into bracket z square by 2 minus 1 by 7 z. Now, putting z is equal to log x and e raise to z equal to x, we get y equal to c 1 x raise to minus 4 plus c 2 x cube plus x cube by 7 into bracket log x whole square by 2 minus 1 by 7 x, which is the solution of the d 1 equation.


Okay, so let's look at the residuals. That is to say what's left over in the data when we remove this regression line. Now we can use the regression model tool here. The regression that we've done is where our independent variable is year. Our dependent variable is temperature. And so we can run that regression here. And that gives us two things. First of all, it tells us the autocorrelation in the residuals, which as we can see is pretty small. It's minus 0.11. And if we look up the statistical significance of a negative correlation of 0.11 with on the order of 100 degrees of freedom, we'll find that it's statistically insignificant. So that means that in this particular case, it doesn't look like we have to worry about the added caveats associated with autocorrelation, when our residuals do not look like uncorrelated white noise, but instead have this low frequency structure. Now we can actually plot those residuals, and I'll make a new plot here. I go down to model residuals, and so I'm going to plot the residuals as a function of year. We no longer need a trend line here, and that's what we have. So when we remove the trend, we accounted for a statistically significant trend, and when we removed that trend, this is what was left over. These are the residuals, and they look pretty much like Gaussian random white noise, which is good. That means that the results of our regression are basically sound. We fulfilled the basic underlying assumptions that what's left over after we account for the significant trend in the data looks random.


Ladies and a gentleman coming to you live from New York City. It is your man's note. Hoot Silent Mike here again with another Coaching fix your form if you want to get involved. We need a three reps at seventy percent email to ask mi KKE at gmail.com. Here we go. Looks like we mostly got some squats. We've got a nice little walkout Not bad. We're already chilling my man squat pretty good He's hitting depth very easily what I'd like to see in the upper body is to have you kind of shrug back and down It may be the angle, but it looks like you're a little bit shrugged up. I just don't want to have those traps Too uncomfortable and we want to make a platform for that bar to lie So if you shrug similar to a bench press Scapular retracted down and back think about pulling your shoulders behind your back and downwards It'll give us a perfect little wedge, especially with a low bar squat just to put that thing right on the top of your rear Delt Overall the control the form Bar path and depth all look very very solid another piece that we always every single person I think you continue to improve upon and get better is our breathing and bracing Every set every rep I always do in the gym I'm constantly trying to remind myself and figure out new ways to get tighter in my stomach and my lats Now again the tighter that we can become between our hip and our shoulder kind of our midline As rigid as we can become there and all three lifts any lift really even clean and jerk snatch Overhead press all that the tighter we become the more force we can produce with our legs and the more that force will transfer into the Bar or the way we want Another squat another really really clean squat shout out to everybody send them videos lately because shit's looking good Overall say really really solid what I would say though is in the beginning And I'm not I don't want to get into the debate Anywhere of whether you should break at the hips or the knees or simultaneous But for you my friend I would recommend pushing into your knees just slightly more In the beginning because right here you break at the hips for a while And then your knees almost separately and I'd like both to move more at the same time This will just allow you to get a little bit deeper a little bit cleaner And it'll also allow as you're coming up Not to have that middle Hiccup that you can see right there where your quads lock out before your hips on the squat We want to lock our quads our knees excuse me and our hips at the same time pushing I constantly think about pushing into the bar with my back and pushing the floor away from me So overall I'd say really really solid a1 a form 4.7 stars out of five stars But from the beginning let's have you push into your knees a little bit more and again for everyone everyone And I repeat everyone Let's get to keep on focusing on how tight we can squeeze our back and our lats And how tight we can brace and breathe into our stomach sides and low back In between each rep on each rep for the entirety of every set As my first gentleman with a little bit of a walkout issue Probably took you about seven steps with some knee bends and some hip bends I'd like you to stand tall and strong and be a little bit more efficient I would like two to three steps to get into your squat stance at most Overall looks really solid looks like you're maybe descending a little bit fast for you Um, that one was maybe a little bit slower But what happens is on the first two you descend a little bit quick You lose some tightness and now your hips shoot up a hair early again We want our hips and and knees to move simultaneous We want them working together so that that torso angle Where our body is relative to the floor Stays as much the same as we can throughout the entire lift We don't want as you see right there the hips to fly up a little bit early or back and our chest to fold Now a torso angle for majority of people is going to be just fine Having a little bit of a lean with a low back or a low bar squat will be okay But it's when the hips shoot up too early And the chest drastically Falls closer and closer to the ground on the way up or the way down That we may come into some issues potentially Injuries but mostly just inefficiencies in the squat and obviously the lift the most amount of weight to not get injured And to be able to handle the volume that we want to build the most amount of muscle We want the most efficient lift we can Pretty decent walk out. We got a little bit higher bar position Overall fairly solid looks like you got some flat shoes on i'm not sure if you're struggling For depth with that ankle mobility weightlifting shoes may be an issue But we're looking like we're a little hot Something I would recommend for people if you don't know what depth is is turn your camera Straight at you from the side And raise the camera to your hip level. I know there's some safeties there So you might have to move those lower the weight and do some pause squats This will allow not only visually from the camera to see where depth is Pause squats and getting comfortable in the hole will allow you to progress over time and get more comfortable And more body awareness of where you should be or where depth is A lot of questions on should I go as to grass low bar as to grass high bar as to grass should I do this Should I do that should I try to cut my squat higher to lift more weight? And that's just a little bit too individual of a question but I think whether your goal is hypertrophy or Strength and powerlifting that we for sure want to go to full range of motion meaning hip crease below the knee For hypertrophy the longer range of motion you take a muscle through Obviously, you're putting more stimulus on it You're hitting different fibers and over time you can build more muscle through that and for powerlifting or strength Obviously, that is just the standard. That's what you have to do to complete one rep So for both parties whoever or whatever your goal is I recommend hitting full depth and a good way to find it is either have a friend there Watching you from the side or put your camera there on the side And lighten the load to hit some pause squats. My man here can squat. I can already tell by them.   Omar says I popularized it I would I would actually say I invented it in the world popularized it fo me Sorry, bro. I missed your whole first rep. Let me try to help you out now Uh, just complimenting them thicky thighs.  You can often see a lot of faults I think it's a really clean squat, dude My one cue or recommendation for you is don't be afraid to push into your knees or think about the cue of pushing into your knees This will allow you to stay a little bit more upright. Your hips just move a hair funky You keep nice and tight Good clean walk out and just focus on pushing those knees Not only will you be able to activate your quads a little bit But it'll allow room for your hips to go straight down and a very clean bar path I guess from this angle at full speed one more tip would be work on your timing Kind of cadence pacing whatever you want to call it in the squat is very important how fast you can control yourself Controlled chaos on the way down Hit and rebound out of the hole with good timing will allow you to lift the most amount of weight So it looks like you are a little tentative going into the hole I'd say if you're comfortable it looks like you're already handling a good amount of weight Just be a little bit more comfortable out of the hole and try to hit it hard Get it in and get it out Backs tight looks again like everybody we can always keep on working on our stomach getting tight Um, you could if you're not comfortable in the hole You could try some pause squats or even a slightly narrower stance Like I said, you do have some thick of thighs my friend So you can get into it moving on to another squad here. What do we got my friend? Squats straight into the mirror. Who's mad at it? I'm not mad at it Overall really really good. Uh, this angle it does look like you're a hair high It could be the angle of the camera that one was fairly deep But overall my friend, I would say your squat is really really solid Sometimes people in the comments will say, Mike, you didn't see the butt wink on that And what's actually happening is because of his belt and actually hitting depth Sometimes people lose a hair of tightness in the hole But their actual Butt or butt wink is not happening Butt wink is when your pelvis is actually slightly rotating forward under load in the squat And that's something we probably want to avoid just because you don't really want your sacrum your pelvis Moving around under load and that about wraps it up for this fixture form Ladies and gentlemen, be sure to subscribe and give this thing a thumbs up Comment below if you enjoy this type of footage and I'll catch you guys in the next video my rascals. Peace